WIND EFFECTSSTRUCTURES

ON

Vicw ol'C--hicago with Standard Oil Company (lncliana) huiltlilrpi lr('iu ('('ntr'r (Alt lritet'ts:Pcrkins and Will, and Edward Durell Stone and Associirtt.s)

EMIL SIMIUNlsr Fellow, Building and Fire Research Laboratory, National lnstitute ofStandards and Technology, Gaithersburg, Maryland

ROBERT H. SCANLANProfessor, Department of Civil Engineering, The Johns Hopkins

University, Baltimore, Maryland

WIND EFFECTS ONSTRUCTURESFundamentals and Applicationsto Design

Third Edition

A Wiley-lnterscience publicationJOHN WILEY & SONS, INC.New York / Chichester / Brisbano /

" ';:-l:nEltA]ltr DO pOtsTOi:acuicjacle de Engenharra

BIBLI' TECA I?Hr.' :3 6o't

oata .1lL *,{- I tg

Toronto / Singapore

This text is printed on acid-free paper

Copyright O 1996 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyondthat permitted by Section 107 or 108 of the 1976 UnitedStates Copyright Act without the permission of the copyrightowner is unlawful. Requests for permission or furtherinformation should be addressed to the Permissions Department,John Wiley & Sons, Inc., 605 Third Avenue, New York, NYt0r58-0012.

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Library of Congre ss Cataloging-in-Publication Data:Simiu, Emil.

Wind effects on structures: fundamentals and applications to design/ Emil Simiri, Robert H. Scanlan. - 3rd ed.

p. ciTl.Includes index.ISBN 0471-12157-6 (cloth : alk. paper)l. Wind-pressure. 2. Buildings-Aerodynamics. 3. Wind resistant

design. I. Scanlan, Robert H. II. Title.TA654.5.S55 1996624.1'76-dc2o

Printed in the United States of America

1098765432

96-5238

Soave sia il ventotranquilla sia I'ondaed ogni elementotranquillo rispondaai nostri desir!

Cosi Fan TutteAct l, Scene V (Terzettino)

For Devra, Erica, and Michael paul.

PREFACE

'f 'lrt' tlrird edition of Wind Effects of Stuctures reflects the many developmentstlrrrt occurred during the last decade in the wind engineering field. The text haslrt'en rcvised, updated, and augmented to include new information and/or ref-crrnccs on, among other topics, windstorm damage and insurance, hurricanerrricnrrnctcorology, aerodynamics of circular cylinders for large Reynolds num-lrt'r's, c<lmputational fluid dynamics, tail-limited extreme value modeling ofrrorrtnrpical storm and hurricane winds by "peaks over threshold" methods,t'rn;rirical aeroelastic models, progress and limitations in wind tunnel modeling,tLrrrrping of flexible buildings, across-wind and torsional effects on tall struc-Iule:;, low-rise buildings, behavior of roofing, power lines, and wind loadl:rt'lors. The material on suspended-span structures has been almost completelyrt'wlillcn. A new chapter on standards has been added, which includes a usefulrt'lt'rcncc to a diskette, appended to this book, containing an interactive com-lrrrtt'r vcrsion of the ASCE 7-95 Standard provisions for wind loads.*f

Wc thank the many colleagues who used our book in their professionalprirt'ticc or as a primary text for teaching wind engineering and gave us thelx'rrclit of their comments. Special thanks go to Dr. R. D. Marshall, whoset'xpelicncc and judgment, particularly in the areas of wind tunnel-modelingrrrrtl winrl-loading codification, are reflected in several portions of this book,

'll Il. St'lnlan was rcsponsiblc lirr rcwriting Sccrs. 5.3.3, 6.1.1, 6.5, 6.6, and 13.1. E. Simiurv;t: tt'slxrttsiblc lirr all othcr rcvisions antl ltltlitions. His contributions to this book are made inlrr: privitlc c:tllrcity ancl cio not ncccssruily lrl)lcsont thc position of the U.S. Department of( onrnr'rcc ol ol tlrc Nirlional lnslilllle ol Slirntl:rnls irrrtl 'l'cchnokrgy.rllrr'tliskt:l1c's witttl loutlittg srtllw:trt', tt'lttrxlrrtcrl lirrrrr ltcl. l7-5, is in rhc public rkrrnain anclr: r()l iul AS('li llrrlllic:rliorr ol tkrr'ttttrr.ll

vll

vial l'llllA(,1 I() llll lllllll)ll)lll()l'l

ilu(l {o l)rrrli'ssor .l . Krrttrllr ol lltt' [ )trrvcrsily ol 'l'okyo, wll(r llirirrs(irkirrgly.t,c.t c,l (lrt, tcx( ol' tlrc sr:t'oiltl crliti0rr rrrrtl krrrrlly lxrintccl tlut a nutnbcr ol'

typogrirlllriclrl ornrrs. wc irlso wislt lo (h:tttk rtttt'txlittlrs, charlcs Schmicg, Ira

tir,*irt y, arrtl Nancy Lin, krr lhcir-hcllllirl cooPt.r'rrtion._. .

'.lkrwing the Russian translation irl'oLrr trixrk, a Chinese translation was

publishcd b! Tongli University Prcss, Shanghai' We welcomed it while re-

[..tting the latter';disregard of intemational copyright norms'

Etuttl- StvtuRoeBnr H. ScnNl,qN

Rockville, MaryLandBaltimore, MarylandMav 1996

PREFACE TO THE SECOND EDITION

lrr tlrc alnrost ten years that have elapsed since the writing of Wind Elfects on.\tt'ttt'torrcs a number of significant advances have occurred in the wind engi-Irr.e ring lield. These include the development of the following: improved micro-rrrt'tcrlrological models, particularly for atmospheric flows over the ocean, which:rle ol'interest in the design of offshore structures; procedures for the estimationol e:xtrcme winds from short records; new information on the modeling ofr'xlrcnlc winds in hurricane- and tornado-prone regions; improved procedureslor r:stitnating the along-wind response of structures; new procedures for esti-rrurting the across-wind and torsional response of tall buildings and the across-rvirxl rcsponse of towers and stacks; simple and probabilistically rigorous meth-,rrls lirr taking wind directionality effects into account in design; practical pro-, t'rlrrrcs fbr the risk-consistent design of cladding for wind loads, which makert lxrssible to achieve more economical designs for any given safety level, ors:rlr.r clcsigns for any given cost; methods for estimating the response of offshore:itru('lurcs to wind in the presence of current and waves; and new informationt,n wirxl cffects on various types of structure, including trussed frameworks,lrylrrbolic cooling towers, and semisubmersible platforms.

'l'hc tcxt has been expanded to reflect these and other advances. It nowrrrt'lrrtlcs fivc new chapters, as well as a new appendix that is intended to providetlrt.r'cirtlcr with a brief introduction to modern structural reliability concepts.'Ilrc original chaptcr on wincl ttttttrc:ls wlts substantially revised, and much newrrr;rtt'r'iirl was aclclcd to thc ollrrrl t'lt:tP(cr.s, plrrlicularly those on the atmosphericlrrrlrtlirry laycr, cxtrcrttc wirttl t'liltrrrlokrgy, blull'body aerodynamics, aero-t.l:rslic ltlrcrxlntir, tall builtlings, lrrrtl (orrrlrtLr ellecls. Most of the new materialr'orrsists ol'prlrclicirl tlcsigrr irtlotttr:tltotr ;tutl trtt'lltrxls. As in thc first edition, a

tx

I'lll ln(il l() llll :;l (:()Nl )ll)l ll{)l!

consistcnl cllirrt lurs lrr't'rt nt:rrlt'l() l)()lnl ottl rrttrl tlist'ttss (ltr: ttttccrtrtitttics,limitatiorrs, rrrrtl crnrls irrlrt'r't'rrl irr vrrriorrs rllrllr, rrrt'llrtxls. luttl lcclrrriqttcs.

Thc aulhors woulcl likt'l() ('xl)r'('ss lltt'rr wlrrnr:rpptccirrtiott lo I)r. I{. D.Marshall, who initiatcd antl tlcvclolletl tlrc wintl cngirrccring prograrn at thcNational Bureau of Standanls; I)r. N. lsyrrnrov ol'tlrc Univcrsity of WestcrnOntario, fbr contributions to Scct. 9.-l; rrrrtl l'rolcssor l). A. Recd of the Uni-versity of Washington, forcontribu(ions to ('hrrp(cr ll. Spccial thanks are alsodue, for valuable comments and criticisrrr, t<l l)rol'cssor E. A. Arens ol'theUniversity of California at Berkeley; Dr. R. l. Basu of H. G. Enginccring,Inc.; Professor O. Ditlevsen of the Engineering Academy of Denmark; Dr.B. R. Ellingwood of the National Bureau of Standards; Professor Y. Fu.jino olthe University of Tokyo; Dr. M. P. Gaus of the George Washington Univcrsity;Dr. P. S. Jackson of the University of Auckland; Dr. F. Mahmoodi of thc 3MCompany; and Professor B. J. Vickery of the University of Western Ontario.However, the responsibility for any errors or omissions lies solcly with thcauthors. We also wish to thank our Editor, E. W. Smethurst, Editorial Super-visor, Balwan R. Singh, Designer, Lee Davidson, and Production Supcrvisor,Linda Shapiro, all of John Wiley & Sons, and Technical Editorof the Russiantranslation (1984), Dr. B. E. Maslov.

The references to the authors' affiliations are for purposes of identificationonly. The book is not a U.S. Government publication, and the views exprcsseddo not necessarily represent those of the U.S. Govemment or any ol' its agcn-cies.

Etr,tlt- StnaruRonp.nr H. Sr',rur.a.N

Rtx'kvilla, Mur.ylund

PREFACE TO THE FIRST EDITION

l'lrc wind loading of.civil engineering structures involves, in certain cases,t'rnsiderable complexities that must be taken into account in order to achieves;rl'c and serviceable.designs. Examples of wind engineering problems that.r:quire special attention include: the dynamic .".poni" of tati structures; thepcrformance of exterior glass and curtain walls, particularly in high_rise uuito_i'gs; the serviceability of pedestrian areas in clrtain types or t'uilt environ_rrcnts, the oscillations and flutter ofsuspension bridges; it. u.tion oftornadoesrrr nuclear power plants; the estimation of the piobability of occurence of('x(reme winds at a given site.Motivated by the need to provide rational descriptions of the phenomenarrv<llved and to develop. appropriate analytical and design tools, a vast spe-t'i.lized literature-not always easily accessible-has emlrged in the last twotlcc.ades. wind Effects on structure^s is an attempt to preseni a synthesis of therrrrin trends of this literature in the form of a texi designed for use by advancedstrdcnts of engineering,and by practicing structural lngineers and architects.l'hc tcxt devleops its chosen _topics independentry anal as often as possible,llirrn fundamental principles. In addition, extensive references are provided to:r widc range of primary sources.'l'he level of preparation assumctr .r' rhc rcader corresponds approximatelytr that of thc bachelor's degrec irr st'it'rt.c .r cngineering. a "onrirtent effortlr:rs bccn rnadc to avoid unncr'cssiuily t'l;rlxrrr(c rnathematical formulations.Silnplc n<ltions o1'probahirity rlrt.'ry. sr;rrrsr it.s :rrrtl thc theory of random pro_t't'sscs ctllpl<lyccl in tlloclcrlt wittrl t'ttlirrrt't'rirr11 :uurlysis havc been prescnted inlrpltt:ntliccs, irr which iltluitivc itl)p1ry.1q lrt.:, lr;rvt. lrct.rr slr-6ngly clnphlrsizcl.'l'lrc lilsl 1xrt1 ol'llrc lcxl tlist'rrs:,t.:, trrr.tr.nrolo;r11.:tl, rrrit.lrirrc,tcgr.gl69iclrl, lrnrlt'lirlr:r(ologit';rl :rspt't'ls ol'llrc wnlrl r.trlnorrrrrt.lrl lllrl luc ol.inlt,t.t.sl irr wilrrl

xt

xll plrEl ACt r() llll t llisll FDltloN

engineering. 'l'hc scconcl part proscnls l)itsic (:l(:tttL:ltts ol'acnrclyttittttic:s, struc-tural dynamics, and aoroclasticity, lillkrwr:d by applications to thr: clcsign ofvarious types of Structures and structural tnctttbcrs. Scparate chaptcrs are de-voted to a discussion of wind-induced discotttlir( in and around buildings, andto assessments of the wind tunnel as a design tool.

Wind engineering is a new and rapidly developing field. Cunent proceduresfor estimating wind effects, and the information on which they are based, shouldtherefore not be regarded as definitive. It is the authors' strong feeling thatareas of unceftainty must be carefully defined, and that the limitations inherentin current procedures must be stated clearly. This has been done throughoutthe text.

The division of responsibility for the work has been as follows: E. Simiuhas written Chapters 1-3, 5, 7,9-ll, and the Appendices, and R. H. Scanlanhas written Chapters 4, 6, and 8. * The authors have, however, shared editorshipand extensive critical exchange on all parts of the text.

The authors' sincere thanks are extended to the following persons who readporlions of the manuscript and offered valuable criticisms: Professor H' A.Fanofsky, Pennsylvania state University; Dr. N. J. Cook, Building ResearchEstablishment, U.K.; Dr. J. F. Costello, U.S. Nuclear Regulatory Commis-sion; Dr. H. L. Crutcher, National Climatic Center, National Oceanic andAtmospheric Administration; Dr. J. J. Filliben, statistical Engineering Labo-ratory, National Bureau of standards; Dr. J. c. R. Hunt, Cambridge univer-sity, U.K.; Dr. G. E. Mattingly, Institute forBasic Standards, National Bureauof Standards; Dr. J. M. Mitchell, Environmental Data Service, National Oceanicand Atmospheric Administration; Dr. R. N. Wright, center for Building Tech-nology, National Bureau of Standards; and Professor J. T. P. Yao, PurdueUniversity. All of them should share the recognition for the many improve-mcnts their comments brought about. The responsibility for all errors or im-perfections rests, however, wholly with the authors. Many thanks are also dueto Devra Simiu and Robert N. Scanlan for careful reading and editing of thetext, and to Mrs. Sue Murray, Mrs. Rebecca Hocker, and Mrs. Nora Scanlanfor their capable typing effort. The authors also wish to express their indebt-edness to the late R. S. Woolson, Editor, J. Frances Tindall, Editorial Super-visor, Joel L. Bromberg, Editorial Assistant, and Debbie Oppenheimer andSandra Winkler, Production Supervisors, all of John Wiley & Sons.

Evn SrvItuRoeEnr H. ScaNI-aN

Washington, D.C.Princeton, New JerseyJune,1977

*Chaptcrs 4, (1, ilnrl ll ol llrc lirst ctlilioil trrrrr:spond in lltt'sctotttl irrtrl llrinl oditions to Clhilptcrs4, 6, antl 13. l,9r'llrc st't.orrtl t.tlition, l{. ll. Scunltn lrirs tevist'rl lltc r'ltirplcr ott witttl lttttncls. antlIi. Sirrriu lrls bcrlr n.s;xrrrrihh'lor tlrt'rrllrr'r'r'cvisirttts ittxl ttrltliliorts l() tltc t('xl.

CONTENTS

INTRODUCTION

PART A THE ATMOSPHERE

1 ATMOSPHERIC CIRCULATIONS2 THE ATMOSPHERIC BOUNDARY LAYER3 EXTREME WIND CLIMATOLOGY

PART B WIND LOADS AND THEIR EFFECTS ONSTRUCTURES

I Fundamentals

4 BLUFF-BODY AERODYNAMICS5 STRUCTURAL DYNAMICS6 AEROELASTIC PHENOMENA7 WIND TUNNELSB WIND DIRECTIONALITY EFFFC IS

5

33

9'1

135

195

216273

308

xlll

xiv (,( 'N

il N tl

ll Applications to Design

9 BUILDINGS: WIND LOADS, STRUCILJHAL RESPONSE'AND DESIGN OF CLADDING ANt] TIOOFING

1O SLENDER TOWERS AND STACKS WII'H CIRCULARCROSS SECTION

11 HYPERBOLIC COOLING TOWERS

12 TRUSSED FRAMEWORKS AND PLATE GIRDERS

13 SUSPENDED-SPAN BRIDGES, TENSION STRUCTURES,AND POWER LINES

14 OFFSHORE STRUCTURES

15 WIND.INDUCED DISCOMFORT IN AND AROUNDBUILDINGS

16 TORNADO EFFECTS

17 STANDARD PROVISIONS FOR WIND LOADING

APPENDIX A1 ELEMENTS OF PROBABILITY THEORY ANDAPPLICATIONS

APPENDIX 42 RANDOM PROCESSES

APPENDIX 43 ELEMENTS OF STRUCTURAL RELIABILITY

APPENDIX A4 PRESSURE COEFFICIENTS FOR BUILDINGSAND STRUCTURES

INDEX

ABOUT THE DISK

327

383

404

420

446

487

511

551

576

591

629

643

665

676

684

INTRODUCTION

l'lrc rlcvclopment of modem materials and construction techniques has resultedrrr rhc cmergence of a new generation of structures that are often, to a degreerrrrkrr'wn in the past, remarkably flexible, low in damping, and light in weight.srrt'' structures, as welr as uu.iou, nou.r typ", of rigid structures, exhibit anrrrt're'scd susceptibility ro rhe action of wina. acirdt;;,r,1;has becomerr('(('ssilry to develop tools enabling the designer to ".ti*ai"'*ind effects with;r lrig'cr degree of refinement than-was previ,ously ,"quiJ. wini "ngrn"".rngrs rlr. discipline that has evolved, p.imarityau.ing ilr"iurt-e."d; fiom efforts;rrrrctl a[ developing such tools.

rsu! uvvsuwD'

It is the task of the engineer to ensure that the performance of structures:'rr'jt:crcd to the action of wind will be adequate during their anticipated rifelr.rrr rhc standpoint oftroth^structural safety and serviceibilitv. io'u"tieve thist'r(,, rhc designer needs information regarding (r) tt" winJ'environment, 12;tlrt' re l.tirn between that environment arid the iorces it induces on,h" r,*",u.",;rrrrl ('1) thc behavior of the structure under the action of these forces.

rHE WIND ENVIRONMENT

lrrlr.rrr(i<ln <ln the wincr cnvinrnrrcnt needed in design includes elements de_r r,u'cd lhrrn .mctconrlogy, nticr()nlcte()tl)l()!,y, and climatology.Merc.nrl'gy pnlvirrc.s lr trcsr.r'iPr i.r,,,,,,i'"^prunation of the basic f.eatures of'rlrttosltltcrr-ic lklws. Such ll.rrlrrt.s rrrirv lrt.ol.c.gnsirlcrabl" ;j;;ifi.;nce fiom a.'lttttlttt':tl tlcsigrr vic:wP.irrl. li,r't'rrrrrrIlt'. irr llrc t.usc.l,rr",nrnu,ir, the prcs_t'ttt t' ol'ir rcgiorr ol' low irlrnosplrt,r,, 1,,,.r,n,,,.,. ;tl lllc cctllcl- <ll. lhcl slrlrrrr is lrl:rt'tr''l'rrrrj.r irrrp.rrirrrr'e irr tir,, ,r,.,,,j.,, ,r rrrrr.rt.lrr.rx)w(rr prirrrrs.

lN il i( )t )l,o l l( )N

MiCrornCtcgnrftlgy lrltr.lrrlrls (p rlt'sr'rilrc llrr' rlt'lltrlt'rl sltlt('ltllc ol ltltttospllt't ic

flows near the gnruncl. 'l'oltics ol rlirt't'l ('()n('('ltt {o lltc sttrtcltrrltl tlcsigrrt't'iltclude the variation of nrcan spcorls willr heip,lrt irltovt: gtttttlttl, tlrc tle:sct-illtitlllof atmospheric turbulence, and thc tlcl'rctttltttcc ol' (ltc tttcitrt spectls ancl tll'turbulence upon roughness of terrain.

Climatology, as applied to the wind cnvirotrtttcttt, is ctlnccrncd with thcprediction of wind conditions at given geographical locations. Probability state-ments on future wind speeds may be conveniently summarizcd in wind maps,such as are currently included in various building codes.

WIND-INDUCED FORCES ON STRUCTURES

A structure immersed in a given flow field is subjected to aerodynamic forcesthat, in general, may be estimated using available results of aerodynamic theoryand experiments. However, if the environmental conditions or the propertiesof the structure are unusual, it may be necessary to conduct special wind tunneltests.

Aerodynamic forces include drag (along-wind) forces, which act in the di-rection of the mean flow, and lift (across-wind) forces, which act perpendic-ularly to that direction. If the distance between the elastic center of the structureand the aerodynamic center (i.e., the point of application of the aerodynamicforce) is large, the structure is also subjected to torsional moments that maysignificantly affect the structural design.

STRUCTURAL RESPONSE TO WIND LOADS

Because the aerodynamic forces are dependent on time, the methods of struc-tural dynamics may have to be employed to determine the response. Further-more, the random character of this dependence requires that elements of thetheory of random vibrations be applied to the analysis. In certain cases. it maybe necessary to perform an aeroelastic analysis, that is, a study ofthe interactionbetween the aerodynamic and the inertial, damping, and elastic forces, withthe purpose of investigating the aerodynamic stability of the structure.

From the foregoing it is seen that the design of modern structures subiectedto wind loads requires the use of information and methods derived from a broadspectrum of disciplines. It will not be suggested here that complete answers tothe questions involved exist at the present time. However, considerable prog-reSS has been made toward an understanding of some of these questions. As a

result, procedures and techniqucs have been devclopccl lhat have significantlyimproved the designcr's irhility to ostinratc thc cll'ccts ol wirttl l-rorn thc stand-point of both strcrrgtlr ltrrtl scrvicr:lrbility. lt is lht: itittt ol tlris lcxt to prescntthesc proccdurcs llul tt't'lrrrit;rrt's, to llrovirlc lltr-: lrirt'kgnrttrrrl trtlttcrial rccluircdfirr rrntlcrsllrrrtlirrg, llrcir r':rliorr:rlc, rrrul lo cxrttttittt' t'rilit':rlly llrt'ir clrplrhilitics irs

wcll rrs tlrt'ir lirtritlrli()ttri it,'i th'si1'.rr (txrls.

PART A

THE ATMOSPHERE

CHAPTER 1

ATMOSPHERIC CIRCULATIONS

Wirrrl, or the motion of air with respect to the surface of the earth, is funda-rrrt'ntally caused by variable solar heating of the earth's atmosphere. It is ini-trirlt:tl, in a more immediate sense, by differences of pressure between pointsol t't;ual elevation. Such differences may be brought about by thermodynamic;rrrrl rncchanical phenomena that occur in the atmosphere nonuniformly both intilil(' irnd space.

'l'hc energy required for the occurrence of these phenomena is provided bytlrt' sun in the form of radiated heat. While the sun is the original source, the.,()rr'('c of energy most directly influential upon the atmosphere is the surface,l thc carth. Indeed, the atmosphere is to a large extent transparent to the solarrrtli:rtion incident over the earth, much in the same way as the glass roof of aI'rccnltouse. That portion of the solar radiation that is not reflected or scatteredlr:rt'k into space may therefore be assumed to be absorbed almost entirely bytlrc crrfth. The earth, upon being heated, will emit energy in the form of ter-rcslrirrl radiation, the characteristic wave lengths of which are long (of the orderol l0p) compared to those of heat radiated by the sun. The atmosphere, whichr' lrrlgcly transparent to solar but not to terrestrial radiation, absorbs the heatr:rtlilrtccl by the earth and re-emits some of it toward the ground.

I.1 ATMOSPHERIC THERMODYNAMICS

1.1.1 Temperature of the Atmosphere'lir illtrstlutc thc nrlc ol'thc lctttgx'r'lrltul'tlistribution in the atmosphere in thepttxlttt'liott ol'wincls, a sirtrplilit'rl tttrxlt'l tll'atrnosphcric circulation will be

n lMolit't il illo ollrot,l All()Nti

prescntcd. In this rnoclcl thc cll'ccls tll'tltc vctlicitl vrtriittion tll'itir tcttlpcraturc,of the humiclity ol'thc air, ol'lho nrlirliorr ol lltc eirrllt, artd ol'l'riction will beignored, ancl the surfhcc of thc carlh will bc rrssrttttctl to ho unilorm and smooth.

It will be recalled that thc axis ol'nrllliou ol tlrc carlh is inclined at approx-imately 66'30' to the plane of its orbit an)rrrKl tltc sun (planc of the ecliptic).Therefore, the average annual intcnsity ol'stllitr ladiation and, consequently,the intensity of terrestrial radiation and thc lcnlpcralurc of the atmosphere willbe higher in the equatorial than in the polar rcgions. To explain the circulationpattem that arises as a result of this tempcraturc difl'crence, Humphreys Il-l]proposed the following ideal experiment (Fig. l.l.l).

Assume that the tanks A and B are filled with fluid of uniform temperatureup to level a and that tubes 1 and 2 are closed. If the temperature of the fluidin A is raised while the temperature in B is maintained constant, the fluid inA will expand and reach the level b. The expansion entails no change in thetotal weight of the fluid contained in A. The pressure at c remains thereforeunchanged, and if tube 2were opened, there would be no flow between A andB. If tube I is opened, however, fluid will flow from A to B, on account ofthe difference of head (b - a).Consequently, at level c the pressure in A willdecrease while the pressure in B will increase. Upon opening tube 2, fluid willnow flow through it from B to A. The circulation thus developed will continueas long as the temperature difference between A and B is maintained.

If tanks A and B are replaced conceptually by the column of air abovc theequator and above the pole, it can be seen that, in the absence of other efl'ects,

FIGURIT l.l.l, ('in'ulittiotr l)irllcnr tluc to tcmpcrattrtc tlillercrtcc bctween two col-r.rrrrns trl'llrritl. lirrrrrr /'/ry,rlr',r tl tlrt Ait hy W. .1. lltrtrrplttcys. ('opyright 1929, l94Ohy W..l . llrrntlrlrreys. llst,rl wrllr lrcrrrrissiort ol Mc(irrrw llill lixrk ('otrtpany.

W0r rn

I I AIM( )r;t,ilt ilt(; il il ttM()llyNAMt(il;

FIGURE 1.1.2. Simplified model of atmospheric circulation.

rrn atmospheric circulation would be developed that could be represented as inf iig. 1.1.2. In reality, the circulation of the atmosphere is vastly complicatedby the factors neglected in the above model. The effect of these factors willbc discussed later in this chapter.

The temperature of the atmosphere is determined by the following processeslt-2, I-3, l-4, l-5, l-61:

o Solar and terrestrial radiation, as discussed previously in this chapter.o Radiation in the atmosphere.o Compression or expansion of the air.r Molecular and eddy conduction.o Evaporation and condensation of water vapor.

1.1.2 Radiation in the AtmosphereAs a conceptual aid, consider the action of the following model. The heatrrrtliated by the surface of the earth is absorbed by the layerof airimmediately:rlrovc the ground (or the surface of the ocean) and reradiated by this layer intwo parts, one going downward and one going upward. The latter is absorbedIry the next higher layer of air and again reradiated downward and upward.'l'lrc transport of heat through radiation in the atmosphere, according to thist'onccptual model, is represented in Fig. 1.1.3.

1.1.3 Compression and ExpansionAlrrrosphcric pressurc is pnrtlucctl by tlrr-: wcight of the overlying air. A smallrrr:rss (or particlc) of clry irir rrurvirrg vr.rtit'lrlly thus experiences a change ofptl'srittro to which thcro ctlrtcsltottrls lt cltirnp,e ol'lcrnpcrature. To determine thel;rltcr. lho oc;uation ol'stirle lirr'1x'r'lt'r'l llits('s iul(l thc lirst law <lf thcrrnodynamicslrlc ttsctl:

n lM():;l ,lll lrl{i (:llr(:t,t Ail{}t..ti:

Ilcirl rir.lrillrll Irt(,lriltlr'r !lr.r r.--f --

l,'l(;tlRl,l 1.1.3. Transport of heat through radiation in the atmosphere.

pu-RTdq:c,flT*pdu

(l. r. r)(1.1.2)

In these expressions p is the pressure, er the specific volume, R the spccificconstant for dry air, Z the absolute temperature, dq the amount of heat trans-ferred to the particle, and c, the specific heat at constant volume.

Differentiating the first relation and substituting the quantity p der thus ob-tained in the second relation, there results

dq:(c,,+R)dT*udpComparing this relation with

(r.r.3)

dq : crdT (l .t.41

which cxpresses the first law of thermodynamics in the particular case of anisobaric (constant pressure) process (co is the specific heat at constant pressure),it is easy to see that c,, * R : cr. It is therefore possible to write, if theequation of state is used once more,

dq : cpdT - (l.l.s)

Processes for which dq : O are referred to as adiabatic. For such processes,the previous relation becomes

dnRT -:_p

qI_4oo _oT cpP

which, alicr irr(cgltrtion, yicltls tlrc cquation

,1,

't',, (';,)""

(1. 1.6)

I I n l M( 'r,t

,ilt ilt{ . I ilt nM( )t )yNn Mt(.1

kttowtt lrs Poissorr'r, or llrt'rlry rrtli:rbalic c(pr:rrr()n. lior tlly ;lt, ll/t.,, o.llitJ.A lirrrriliar e,x:ttttplr'ol tlrt'e'llecl ol'prcssrrrc r'llrnp,c orr llrt' lcrrrPcrirltrrt: is llrt'lrt':rtirrg ol'cornprt'ssr'tl :ur irr lr tirc pultrp.

ll', in thc itltttospltr:t't'. tlrc verlical rnotiorr ol'irrr rrir'pirrccl is sullicicntlyrrrpid, thc hcat cxcltiurgc ol'that parcel with its cnvir'onnrcnt tnay bc consiclcrcdt. bc negligiblc and thc assumption dq : 0 is appnrximatcly correct. lt thenIolkrws from Poisson's equation that since ascending air experiences a pressurerltrcrcase, its temperature will also decrease. The temperature drop of adiabat-it'rrlly ascending dry air is known as the dry adiabatic lapse rate and is ap-proximately 1"C/100 meters in the earth's atmosphere.

consider a small mass of dry air at position 1 (Fig. 1.1.4). Its elevation andl('nrperature are h1 and z', respectively. If the particle moves vertically upwardirt some reasonable speed, its temperature change will effectively be adiabatic,regardless of the lapse rate (temperature variation with height above ground)Plcvailing in the atmosphere. At position 2, while the temperature of the am-lricnt air is 22, the temperature of the element of air mass is T) : T, - (hz -/r,)'y,,, where 7o is the adiabatic lapse rate. Since the pressure of the elementrrrrtl of the ambient air will be the same, it follows from the equation of statetlrirt to the temperature difference T5 - T, there corresponds a difference oftlt:rrsity between the element of air and the ambient air. This generates a buoy-rrrrcy force that, if rz I Ti, acts upwards and thus moves the element farther;rway from its initial position (superadiabatic lapse rate, as in Fig. 1.1.4), or,tl 'l'2 ) Tl, acts downwards, thus tending to retum the particle to its initiallxrsition. The stratification of the atmosphere is said to be unstable in the firstr'rrsc and stable in the second. If T2: Ti,that is, if the lapse rate prevailingrrr llre atmosphere is adiabatic, the stratification is said to be neutral.

A simple example of the stable stratification of fluids is provided by a layer.l water underlying a layer of oil, while the opposite (unstable) case wouldlurvc the water above the oil.

I: Lapse rate prevailingin the atmosphere

Adiabatic lapse rateII

2(h2, r;)

(l l7) lrl(ltlRl'l I l..l= l,rl,i.( r,rt(.s

1 0 A I MO:lt,l tf ntc otttct,l A t t( )Nr;

1.1.4 Molecular and Eddy ConductlonMolecular conduction is a dill'usion pft)ccss thlt cll'ccls a translbr ol'hcat. It is

achieved through the motion of individual rnoloculcs and is negligiblc insofaras atmospheric processes are concerncd. Hddy hoat conduction involves thetransfer of heat by actual movement of air in which hcat is stored.

1.1.5 Condensation and Evaporation of Water VaporThe pressure of moist air is, according to Dalton's law, equal to the sum p ofthe partial pressure e of the water vapor and that of the dry air, p - e. It hasbccn established experimentally that if the pressure e exceeds some value E,known as the saturation vapor pressure, condensation of the excess moisturewill occur, and that the saturation pressure E increases exponentially as thetcmpcrature of the moist air increases.

An elementary mass of ascending unsaturated moist air (i.e., for whiche/E < l) will experience a temperature drop that can be shown to be essentiallyequal to the dry adiabatic lapse rate. As the element ascends and its temperaturedecreases, its saturation pressure will also decrease. If the element reaches alevel at which the ratio e/E becomes unity, condensation will normally occur.Above this level, water vapor contained in the air element will continue tocondense. In the process, heat of condensation is released. This is equal to theheat that was originally required to change the phase of water from liquid tovapor, that is, the latent heat of vaporization stored in the vapor.

The heat of condensation contributes to the mechanical work involved inthe expansion of an ascending particle, which before saturation was performedonly at the expense of the internal energy. The temperature drop of the saturatedadiabatically ascending element of air is therefore slower than for dry or moistunsaturated air (Fig. 1.1.5). By furnishing energy that increases the temperature

\ to,r, {or saturation)

\diabatic lapse rate

l,'l( ;t llll,) l. l.-5. l illtt'(s ol' corttlcrtsirlion ulxrr Lrlrsc nrlc l.'l(;tlltll 1.2.1. Vcrtic:rl Irr':,:.rrr.s ()r irr e lcrrL)nrrrry nrirss .l uir.

\ lM( )t;t,nt nt( , I tytrn( )t)yNn Mt(;ti I I

ol'a parliclc: witlr n.spcr'l lo wlrat it woultl hirvc lrccrr urrtkrr tlrv arliabirtit.cottclititlns, tltc hcitl ttl cttttdcttsation hclps sul)lxrt1 corrvct.liorr ol thc 1ir t9lrighcr levcls ol'thc ltlntospltcrc. 'l'his lactor is irrrpollirrr( irr tlrc gcncsis ol'cqlaintypcs ol winds.

1.2 ATMOSPHERIC HYDRODYNAMICS

'l'hc motion of an elementary mass of air is determined by Newton's secondl:rw

DF : ma (1.2.1)

where m is the mass, a is the acceleration, and D F is the sum of forces acting'rr the elementary mass of air. It is the purpose of this section to briefly describetho forces F and some of their effects upon the motion of air.

1-2.1 The Horizontal Pressure Gradient Force('onsider an infinitesimal volume of air dx dy dz (Fie. r.2.r), and let the meanl)rcssures acting on the lower and upper face be p and p + (0pl0z) d1, respec_tivcly. In the absence of forces other than pressures, the net vertical forcerrcting on the volume dx dy dz will be -(0pl0z) dx dy da, or -)pllzperunitvolume. similarly, the net forces per unit volume acting in the "r and y directionwill be denoted -\pl0x and -0pl0y, respectively. The resultant of these forcesis called the horizontal pressure gradient and is denoted -0pl0n, where n isthc normal to some contour of constant horizontal pressure. The horizontall)rcssure gradient is the driving force that initiates the horizontal motion of air.

12 n lMolil'l ll lil(; cllt(;l/ln ll()N:i

Htglr llrcsltrtrr

An

I o,r".,,.,, ol pressurc gradrent

Low presure

FIGURE 1.2.2. Direction of prcssurc gradient fbrce.

'l'hc nct lirrce per unit mass exerted by the horizontal pressure gradient, (l/p)t)1tl\n, is oltcn referred to as the pressure gradient force (p is the air density.)

Air subjccted solely to the action of pressure gradient forces will move fromrcgions of high pressure to regions of low pressure. The direction of the pres-sure gradient force is indicated in Fig. 1.2.2, in which the isobars (lines con-tained in the same horizontal plane and connecting points of equal pressure)are also shown.

1.2.2 The Deviating Force Due to the Earth's RotationIf defined with respect to an absolute frame of reference, the motion of a particlenot subjected to the action of an external force will follow a straight line. Toan observer on the rotating earth, however, the path described by the particlewill appear curved. The deviation of the particle motion from a straight linefixed with respect to the rotating earth may be attributed to an apparent force,the Coriolis force, the vector expression of which is [1-7]

F,.:2m(v x a) (t.2.2)

where m is the mass of the particle, or is the angular velocity vector of theearth, and v is the velocity of the particle relative to a coordinate system rotatingwith the earth. Fc is perpendicular to crr and to v, is directed according to thevector multiplication (right-hand) rule, and has the magnitude 2nlc,rl lvl sin a,where a is the angle between o and v.

Let N (Fig. I .2 .3) be the norlh pole, and consider an element of air movingin a straight line in space along the direction NP. If the motion starts from Nat time t : O, at time / the particle arrives aI P, and the position of the meridianalong which the motion started is NP'. To an observer on the earth, it appearsthat the element is deflected westward by an amount P'P.

It can thus be seen that, in thc Northern Hemispherc, owing to the rotationof the earth, a wind initially dircctcd along a mcridian vcors to the right of itsinitial dircction; (lurl is, il tlirr:ctctl northwarcl il vc:crs lowurtl thc oast (bccomcsa wcstcrly wintl). ll tlilct'tctl sottlltwatd il vccrs lowrrnl llrc wcst (bccorncs an

I 1' n I M{ t!;t't il !tl( I tyl)t l( )l )vNn Mt( tli l3

I-IGURE 1.2.3. Apparcnt rnotion ol'an air particle due tothe earth's rotation.

t';rstcrly wind). In the Southern Hemisphere the reverse of these statements isllllc.

ll' the Coriolis parameter is defined as

f:2llsind (r.2.3)

wlrc:rc d is the latitude of the point considered, it follows that the Coriolis force;rcting per unit of mass in a plane (P) parallel to the surface of the earth (Fig.I 2.4) on an element of air moving in such a plane with velocity v relative totlre carth will have the magnitude

F, : mfu

f lrc values of f are given in Table 1.2. 1 as functions of latitude.

(1.2.4)

fll(;lllll,l 1.2.,1. l 'r,ntlr,lr.trl:, ol tlrt'lrltrlion vccl()r (').

14 n lM()l;l'l ll lllc ()ll l(;t,l n ll()Nl;

TABLE 1.2.I. Corirllis l)ararnclcr.Latitude

(deg)f:2asinb

(s -')l-ittitudo

(rlcg).f -- 2a sin $

(r ')05

10r520l5t( )

l'r,lo.ll

00.1211 x l0 a

0.25330.3'1750.49ttu() (rI(r4( ) /:l().1O lJ l(r\o (n/1I Oll\

1.1172 x 10-4t.1947r.2630r.3218I .37051.4087t.43631.46291.4584

-505.5()( )()-5

1015tt( )

1{-5

9o

1.2.3 The Frictionless Wind Balance

At sullicicrrtly grcat hcights, the eft'ects on the wind due to friction along theground bccome negligible, and the horizontal motion of air relative to thesurface of the earth is determined, in unaccelerated flow, by the balance amongthe pressure gradient, the Coriolis, and the centrifugal force.

The effect of the forces acting on an elementary mass of air is shown inFig. 1.2.5 (the mass is assumed to be in the Northern Hemisphere). If theparticle started to move in the direction of the pressure gradient force (denotedP), it would be deflected by the Coriolis force F.o @ig. I .2.5a). The particlewould then move in the direction of the resultant of P and F..o, shown asdirection II in Fig. 1.2.5b. The deflecting force would now become F,.6, towhich there would correspond a new direction of motion (direction III in Fig.1.2.5b). When a steady state is reached, the wind flows along the isobars asshown in Fig. 1.2.5c.

The isobars in Fig. 1.2.5 are depicted as straight, which means that in thecase shown there is no centrifugal force. However, in the more general caseof a curved isobar centrifugal forces will be involved. This case is taken upbelow.

The steady velocity for which a balance between the pressure gradient forceand the Coriolis force alone obtains is called the geostrophic wind velociry Gand is related to the pressure gradient by the equation

do/dn2u:Csin6:P:-'p

(1.2.s)

,, .,, 'Q!!'pl'(t.2.6)

I ;' n I M( ): il ,t lt ||t( il\ I rlt( )l ryNAMt(:l

High pressure

Low pressure

(b) (c)

FIGURE 1.2.5. Frictionless wind balance in geostrophic flow.

rvlrcrc P is the magnitude of the vector P,/is the Coriolis parameter, and p istlrt'lrir density.

I l' the isobars are curved (Fig. 1 .2 .6) , the force P as well as the centrifugalI'r.cc C will act on the elementary mass of air in the direction normal to ther:,.birrs, and the resulting steady wind will again flow along the isobars. Its

High pressure High pressure

ll'rr\:tion Ofwrrrrl (in theN,rr tltornI l"rrst)here)

Direction ofwind (in theNorthernHeqrisphere)

Low preisurc(cycl<xric circulirlr0il)

(n)

Itl(llllll,l 1.2.(r. lrlictiorrlt.ss rvrrrrl lr;rl;rrri r

Low pressure{anticyclonic circulation)

(h)

ttr ( y( l()ni(' lurtl lrrrlit'yr'lorrit' llow

16 n tM()t;l'l tt Ittc otti(;lll n ll()Nl;

velocity results fiom the relations

vn,f -r yl : ,"r _ dp/dnp

(1.2.7)

where, if the mass of air is in the Northern Hemisphere, the positive or thenegative sign is used according as the circulation is cyclonic (around a center6fl<tw prcssurc) or anticyclonic (around a center of high pressure), and wherer is tlrc nrrlirrs ol'crtrvalurc of thc airtra.icctory.* The velocity I/r,is calledthe.tittttlit,ttt rt,itttl tt,lttt'ilv; it is ct;ttal ltl thc geostrophic wind velocity in thep;rrtrt rrlrrr t:rst' irr wlrit'lr lltc ('tlrviltlllc tll' (hc isobars iS zerO. If the radiuS ofr'llr\';ll1!(' rs ltrvlt'. trr lllt' Nol'tltt'r-rt I lt:rrtisphcrc

'{ 'l',':: - ({)')'' (1.2.8)l';,,

lirr cyckrrric wirrtls, atrd

vr,: t{, -le)' -;#)" o 2s)

for anticyclonic winds. The sign of the radicals is given by the condition thatVc,: Owhendp/dn: 0. It follows from the expressions fot Vr, that for thesame values of r, f and dp/dn, anticyclonic winds are weaker than cyclonicwinds [1-1, p. 121].

The foregoing discussion explains Buys-Ballot's /aw, which states: If, in theNorthem Hemisphere, a person stands with his back to the wind, the highpressure will be on his right and the low pressure will be on his left. In theSouthern Hemisphere the reverse is true.

1.2.4 Effects of FrictionThe surface of the earth exerts upon the moving air a horizontal drag force,the effect of which is to retard the flow. The effect of this force upon the flowdecreases as the height above ground increases and, as indicated previously,becomes negligible above a height 6 known as the height of the boundary layerof the atmosphere. Above this height the frictionless wind balance is estab-lished, and the wind flows with the gradient wind velocity along the isobars.The atmosphere abovc thc boundary layer is called the free atmosphere (Fig.1.2."t).

*Strictly sllctrkirrl'., llrc r:rtlirrs ol ( lr v:r1lrt' ol lllc triricct()t] rtr:ry tlillt'r lirrltt lltc radius ol'curvaturcll llrc isglr:rr'.'llrc tlilltrclr,r.rrlrV lrc lreliletlctl, ltowrrvt'1. il tt r':rtt lx':tssttlttctl th:rl lhc wintl llrlwis :rllptrrx irttitlt'ly slt':trly

A I Mil'.t'l il ltt( I li I rl to| ryNn Ml( :: l7

S (fricliorr lorr:r')

Irl',rlltoil!l!'t('

(;r,r(il,nt wrnrl luvtl

High pressure

Low l)ressure(h)

,tt IItr':,1)lt('t i(. lroulrtlltt.y l:tyt.r

-layer depth)

FIGURE 1.2.7. The atmospheric boundary layer.

It is the wind regime within the boundary layer of the atmosphere that is of'lrr.ct interest to the designer of civil enginlering structures. The questions oftlrt' boundary-layer height, of the variati,cn of wind speed and direction withlr.rrht above ground, and of the turbulence structure within the boundary layer.ut' therefbre discussed in more detail in Chapter 2.

It will be noted here that unrike the gradient wind velocity, the steady_stater'irrrl velocity within the boundary layer crosses the isobars. consider a geo_''t.rphic flow (i.e., a flow in which the isobars may be assumed to be straight)'rrrtl the balance of the forces acting on particles A and, B, which move hori_z'rrrirlly within irs boundary rayer (Fig. t.z.s;. If ,4 (Fig. r.2.ga) is at a higherl('vcl than B (Fig. 1.2.8b), its speed u and (by virtue oithe relation F, : mfu)rts ('.riolis force will be largerthan those orb. tre deviation;;gr; " betweentlrt'wind direction and the isobars will therefore be smaller fir ttre highertl;rs(cr) particle. The angle o will be zero at the gradient level and will reachrts.rrraximum value os nearthe ground. In the Northern Hemisphere the windr.ltrcity in the boundary rayer may thus be represented by a spiral, as in Fig.l.).9.

High pressure

F" (Coriolis force)

l)[oction A

It()lt()rrP {Jlressrrrc r;r;rlrr.rrt lorr r.)

Low l)t{:ssrlr(:(r)

lf l( ll lltl,) l.2.ll. llrl:rrrt'c ol lor, {.., rr tlrr

18 ATMOSPHERTC ctRCUl AroNsi

FIGURE 1.2.9. Wind velocity spiral in the atmosphericboundary layer.

In the case of a cyclonic storrn (or flow around a center of low pressure),near the ground, the wind will cross the isobars toward the center. The air willthus slowly converge and ascend. If the low-level convergence exceeds high-level divergence, the mass and weight of the air column at the center of thestorm gradually increase and therefore the inward-directed pressure gradientforce decreases. As a result of such a decrease, the center of low pressure isdissipated and filling is said to occur.

In the case of an anticyclone, the wind near the ground will cross the isobarsaway from the center of high pressure. In the lower portions of a high, if low-level divergence exceeds high-level convergence, the atmosphere will tend tospread out and sink, and dissipation of the center will occur.

1.3 ATMOSPHERIC MOTIONS

Most atmospheric pmcesses can be described in terms of the quantities brieflydiscussed in the prcccding sections: wind velocity (i.e., horizontal and verticalwind), pressurc, tcmpcrature , density, and moisture. The behavior of these sixquantities is govcrncrl by six cquations: the equation of state, the first law ofthermodynanrics. tlrc ctluirtirtns <ll'continuity of mass and moisture, and thehorizontal antl vcrlicll ccprations ol'tnotion. Proviclccl that an adequate databasc cxists, llrr:sc r:r;rrirlit)ns clur bc intcgratcd to yicltl it t;tutrttitative dcscription

ol itttttosphcric t'tttttltltrttts itl s()tnc shorl tittre irltcl llrc t.ollecli6rr el'llrr-r rlirlir.'l'hc calculatctl vitlttcs ol lhc six variablcs obllirrctl by irrtcgnrtiorr can (hcrr lrr.:ttsctl as initial corttlitiotts lilr a l'urthcr inlcgnrtiorr;;11rp. 'l'his succcssivc ap-Pfilxitnation pK)(:L!ss is lhc lrasis of numcrical wclllrcl procliction tcchniquoslllirl came into bcing lirllowing the increasecl availability ol- observations-rncluding, more reccntly, observations obtained by satellites (Fig. 1.3.1)-andtlrc: clevelopment of modern electronic computers.

Atrnospheric motions may be described as superpositions of interdependentlkrws characterized by scales ranging from approximately one millimeter tollr.usands of kilometers. To analyze such motions, it is convenient to classifzllrr:rn according to their horizontal scale. In meteorology three main groups ofrrtrnospheric scales are commonly defined: microscale, mesoscale, and synopticst'irlc. According to the classification of Il-g], the synoptic scale includes mo_tions with characteristic dimensions exceeding 500 kmor so and time scalesol two days or more. The microscale includes motions with characteristic di-rrrc:nsions of less than 20 km or so and time scales of less than one hour. Thenrcsoscale is defined by dimensions and periods between those characteristicol' rnicroscale and synoptic scale.

lrlOtIRli 1.3.1. Srrtcllitc( )ccanic ;rrrtl Alrrrosplrcric

vicw ol lrrrrrrt'irrrt' lriliAiltrrtrrisl rrrl rnrr )

r;r AIM()i;l't r nt(: M()lt()Nli 1g

Se pl . 113. 1974 (coLrnosy National

20 n lMosl'l ll lll(; (;lilc(ll All()N:;

Nortlr PolcPolar easterliesPolar front

,z .z .zt/ tt t/ t/

l,'l(,llil{l,l 1..1.,1. lltt'llrrlrrr rrrt'ritliorrirl r'ircttluliott rnodel. After General Meterologylry ll l{. llyt'r:. ('opyrill,lrl l()17, l(4,1 hy llrc Mc(iraw-Hill BookCompany, Inc. Usedwrllr 1rt'nnissiort ol Mt ( ilrrw I lill lltxrk ('olrtpittty.

1.3.1 The General CirculationThe combined effects of the earth's rotation and of friction break the thermalcirculation cell of Fig. l.l .2 into a pattern that consists basically of threecirculation cells as represented in Fig. 1.3.2 [l-2]. The theoretical pattem iscompatible with the existence (at sea level) of a high pressure belt at the horselatitudes and of a low pressure belt at the polar front.

In reality, the tricellular meridional circulation model is complicated byseasonal and by geographical effects. Seasonal effects consist of variation inposition and intensity of the pressure belts and are caused by the annual marchof the sun north and south of the equator. Geographical effects are caused bythe difference in physical properties and by the uneven distribution of waterand land over the globe.

In summer, because the ocean surface warns up more slowly than the land,the air is colder over the ocean than over land. Just as in Fig. 1.1. I fluid flowsin tube 2 from the colder to the warmer tank, air near the surface will be drivenin summer by a pressure gradient force directed from the ocean toward theland. On the other hand, in winter the air is colder over land and the oceansbecome heat sources.

1.3.2 Thermally Direct Secondary Circulations: Monsoons andHurricanesSecondary circulations are said to be of the thermally clircct typc if the centersof high or low pressure (i.e., the highs or the lows) aroutttl which thcy developare formed by heating or cooling of the lower alttros;rhe rc.

Monsoons, Monsoor.rs itrc scirsorral winrls tlurt lirrrrr r'ells ol (lrc gcncral cir-c:ulalion irntl tlcvr:krp rrrtrttntl llrclrrurlly prrxlttt'etl t'ottltttr'ttlitl Irighs irt wiltlcr

I:r AIM{}!,t ,tililt(;M()Il(iNl; 2l

;rtttl ltlws itt sttttttttct. ()wirrg lrl tltc vlrsl llrtttl rturss ol llrt.Asi;rrr r'onlirrt.rrl,lll()llso()ll cllccts irl.r'(l('v('lolx'rl ttxrsl strorrgly irr Asitr. wlrcrc llrcy lurvt: tr t'oltsitlcnrhlr: inl]rrcltct' orr lltc sr:lrsonal charrgcs ol welrllrt.r l)itllcnrsi.

Hurricanes. -lr<lpiclrl cycklncs are storms that dcrivc all thcir cnorgy I'r<lrn thcllrtt:nl heat relcascd by tho condcnsation of watcr vapor and <lriginatc, gcncrally,lrt'lwccn the 5 and 20 latitude circles. Their diamctcrs arc usually of the order,rl' scvcral hundred kilometers. The depth of the atmosphere involved is of theorrlcr of ten kilometers. Hurricanes are defined as tropical cyclones with surfacewirrd vefocities exceeding about l2o kmlhr. Spacecraft views of hurricanes areslrown ln Figs. 1.3.1 and 1.3.3.

Hurricanes (known as typhoons in the Far East and cyclones in the regionol Australia and the Indian Ocean) occur most frequently during the late sum-

;1*W*d,*lrl(;llltli l.-f,1. Ilrrrlit'ilrc (illulvr ir, .,(.r.u l)v( )t t':rrrit' lrntl A( rrlrsplrt'rit' Atlrr rirrr:,1 r ;rl inrr I

llrt' Apollo crcw (cotrrtcsy Nlrliorrirl

lrl( ll ll{l,l 1..1.:1. Mcrrrr tlircctions ol' hurricane motions [1-9].

lnor ancl oarly autumn months (August-September on the Northem Hemisphere,February-March on the Southern Hemisphere), except in the Northern lndianOcean. Hurricanes normally travel as whole entities at speeds of 5 to 50 km/hr. The mean directions of hurricane motions are shown in Fig. 1.3.4. It isnoted, however, that individual hurricanes may follow unusual, indccd erratic,paths. World tropical cyclone statistics are presented in Fig. 1.3.5 ll-91. Dataon tropical cyclones reaching the United States coastline are presented in somedetail in Sect. 3.3. For detailed basic information on hurricanes, scc Il-10]and [3-62].

As seen in a vertical plane section, the structure of a hurricanc in the maturestage consists of five main regions, represented schematically in Fig. 1.3.6, inwhich approximate dimensions are also shown. Region I consists of a nrughlycircular, relatively dry core of calm or light winds, calk:cl thc cyc, aroundwhich the storm is centered. The air rises slowly near thc pcritnctcr of thc eyeand settles in its center. Region II consists of a vortox in which warm, moistair is convected at high altitudes (by the thermodynantic Incchanism discussedin Sect. 1.1) and forms tall convective clouds. Conrlcttsulion of water vaporoccurs as the moist air rises, and this results in intcnsc lirinlirll and thc rcleaseof vast amounts of latent heat. It has been estimatctl lhitt lltc cttrrtlcnsation heatenergy released by a hurricanc in one hour may bc ctltrivltleltl lo thc clcctricalenergy used in the cntirc Unitcd States in onc wct:k ll t)1. 'l'lrc 1ir l'lows outof region II into an outllow ltrycr (rcgion lll). ln lcgiorr lV llrt' llow is vortcxtlikeand settles vcry sl<lwly iltlo tllc lltlunclaty lltyct'tcgiolt V. llclow rcgion II,where strong uptlrllts i1i. l)t('sc1l , scl)itfttti()rl ol'lltt'lrottttrl;rly l;rycl-lllily occur.

According to (illrlrlutr :rrrtl llrrtlson ll l ll :rrr r'xlttcssiott ol (lrc lirl'nr

n IM():ll'l il lil(; (;llr(;t,l n ll()Nli

I tllt1t ,lt ( ',' 1,, \ /{,,, ti", rI t'

l) / I(r.3.r)

(llllotlr t,xl,rl[,r.r^V

ai

(.)

oo.

ra(.)

14

p

f,i o@>@o:-_COc -Ooz

I S R; Pf tslS-r:HSits;i:l vs.!:Eeg5 3n-1i-.-o C q V " h.:q

o -!biEao i-\i f o f,u9Qoo<jo@'io

9 *-

E 3>fa cc:- rf i

-O s

l;\

,9:@

{ l,'i r

:U@L@o-z

i; oU.o oc:.g o?I O a6i di";i1alr

; oi@; uo-O-

Ud

3=3>l'3P6 :+c:o Y!r.. F:Ib ^

tn'j

'll

at

o

I

zO

.l>

1:t:i:l?

l=l"i:l;l3l^lol:l>

;zo

:

;qoOO

crooo,o rj .cj v ;

":-ONoc>oug.EaEaso o= 9=- oo vo!'3:;t;I;6r-9f,9b'u<=<"'I I

Ohci-ri>^ 6 O

3-g a bX'6 o I qbs> € ii,;> o 6 a c^<i@ < >5lt;lri

\)

24 n lM()..il,l ll lil(; (;llt(;(,1 n ll()Nli

h (km)

tt (krrr)

lil(lllltl,l 1..1.(r. Slrrrcturc ()l' a hurricane.

in which p6r is thc pressure that is approached as the radius r + @, p0 is thepressure at the center of the hurricane eye, (pn - pdlp is assumed independentof height, and Ra is twice the radius of maximum dp/dr, is fairly representativeof typical hurricane pressure fields. If this description of the pressure gradientfield is used, the gradient wind velocity field results from the expression of thegradient wind (valid for cyclonic winds) derived in Sect. 1.2. There resultsfrom this expression that the gradient velocity reaches a maximum at a radiusof the order of R.. From this radius the velocity decreases rapidly to zero atthe center of the eye, and more slowly to the relatively small values that obtainat large distances from the center.

While the gradient wind velocity is directed along the isobars (see Fig.1.2.5c), in the boundary layer the wind velocity has a radial component directedtoward the low pressures, as was shown in Sect. l.2.It is this component thateffects the inflow of the warm moist air at the ocean surface into region II,thereby maintaining the supply of energy of the storm. Over land the dissipativeeffect of friction increases, while the supply of energy in the form of warmmoist air tends to be cut off. As a result tropical storms over land usually fillup within a few days at most.

The destructive effects of hurricanes are considerable and are due to thedirect action of the wind-which may reach peak surface velocities of 250 km/hr or more-and, usually to an even larger extent, to the massive piling up ofwater by the wind known as storm surge, together with flooding by heavyrainfall (Fig. 1 .3.7).ln 1992 Hurricane Andrew alone caused damage estimatedat more than $20 billion I l-121.

Arctic Hurricanes. An'tic' lrrrrr.icltncs tlcsignulc polirr krws with a sylnlnctricalcl6utl signittrrr.t'iurtl winrls ol ;rl lt';rst.lO rrt/s (5ll krrots), wlriclr rcgulitrly cxcccclllrr: corrvenliorlrl llrrr'slroltl lol lrrrn'it':utc lirt'cc witttls. ;rrtrl irt wlriclr lltrxcs <ll

T-6x-

I

l0

I'l'

)ll

I r A I M{ ,:,t,t il ilt(; M( ) ll( )Nl; 25

.::a'!4

:fS-*;*.J..''rEd {*$swL*il1ffi,j+i$iffi

l.'l(;uRE 1.3.7. Hurricane damage, Mississippi (courtesy National oceanic and At-rr rospheric Administration).

lrr:at at the sea surface are largely responsiblel('nance of the storm Il-131.

1.3.3 The Extratropical Cyctonesrrch circulations are produccrr cirrrcr hy rhc mcchanical action of mountainlrltrricrs on large-scale atmospltt'r'ic crrrrcrrls, ol by the interaction of air masses:rlong fionts. An examplc rll'tlltrttirgt't'rruserl by an cxtratropical storm is shownrrr Iiig. 1.3.8.

Air tnasscs arc charitclct'izcrl lry rt'lrrtrvt'ly rrrrilirlrr physical propcrties overlrolizontal tlislirnccs colrrllrllrlrlt'ln 111,.,1,,,,,.,rsiorrs ol'.ra"un, rlr contincnts.'l'lle ir physicitl PtrrPt:t'tics lttc;tcr;ttttr'rl ur tlr(' :,our'( (. regiorr arrtl nr:ty 5c rr16rlilicrlrlttl'irtg strbscrlrrcrrl ll'irvt'l ol llrr.;rrr rrr;r.,r, Arr rrr:rsst,s lrury be: r.llrssilicrl, lrc

"lt.tl

f,ii "-e- q'',lE-* k

kilr- .-- !t*;E; t#Sr

r.i*si',,.: .'4&''.,

for the structure and the main-

26 n l Mo:itllil iltc (;ilrot,l A lloNl;

FIGURE 1.3.8. Damage caused by winter storm, Fire Island, New York, March 7,1962 (courtesy National Oceanic and Atmospheric Administration).

cording to the source region, into three main groups: arctic, polar, and tropical;each of these may in turn be divided into continental and maritime. Continentalpolar air, for example, is dry and cold, whereas maritime tropical air is rnoistand warm.

Transition zones between air masses are called frontal zones. The variationof the physical properties of the atmosphere across frontal zones being fairlyrapid, the latter may be idealized as surfaces of discontinuity known as f-rontalsurfaces. The intersection of a frontal surlace with a surface of equal elevationwith respect to the sea level is called a front.

The equilibrium slope of the front between two air masses can be calculatedapproximately on the basis of simple hydrostatic considerations and variesnormally between l/50 and l/400.

A front is rclcrrcd to as a <'old.f'ront or as a warm .frrn.l (Fig. 1 .3.9) accordingas it movcs in thc tlircctiott <tl'thc wanlcr tlr coltlcr irir. Gcncrally, a warml-r<lnt rrrovcs slowly irntl is rro( rrssociir(crl wilh violclrl wt'itlhc:r conclitions. Onllrtr othcr hturtl, ir t'olrl lirrrrl ciur nrovc: rapirlly ltttrl r';tttsc s('v(:r'(: wcitlhcr. Irrc-

I.r AIM{r:il,lilIil( t\,4o111 ;11, 27

Wtttttt + -/Cold

Wrrrrr- C*""llWarnr lrcrrl slopr: Colrl lrorrl slo;tc

Ir'l(;tJlll,l 1.3.9. Warm and coltl lirrrrt skrps.

rltrcntly ahead of cold fronts squall lines develop that may be associated withlruge thunderstorms and with tornadoes. The disturbance of the temperature,vclocity, or pressure gradient field may cause wavelike perturbations on thelhrnt that propagate as waves in a continuous medium. Major disturbances may('iruse waves whose amplitudes increase with time and develop into intensevortices. The formation and development of the most intense large-scale cir-t'ulation in middle latitudes, the extratropical cyclones, is connected with suchrrrrstable waves occurring predominantly along a front. on the average, thet'xtratropical cyclones move eastward with velocities of the order of 20 km/hrirr summer and 50 km/hr in winter.

1.3.4 Local Winds'l'hc influence of small-scale local winds on the general circulation is negligible.ll.wever, their intensity may sometimes be considerable and in certain casesf.ovcrn the design of buildings or structures.

Foehn winds. Air ffowing across a mountain ridge is forced by the mountainskrpe to rise. If the air ascends to sufficiently great heights, condensation andprccipitation due to adiabatic cooling will occur on the windward side. Afterlurving thus lost most of its initial water-vapor content, the air passes over thet |cst and is forced to descend. Consequent adiabatic compression results inlrigh temperatures of the dry descending air. An example of a foehn wind issrrggested in Fig. 1.3.10.

-5"C

_,f;-l_l^ ( ,1

' ) -.,. r-3,ooo m.L\- ,,r' ) V,"::i\,( ,/

ilt&" "',))_

l,'l(ll lltl'l t..1, ll). lror.lrrr rvirrtl.

+15"C

28 AIMosPHlBtcciltculAlloNS

In the United States intense and highly turbulent winds of the fbehn typc,called chinook winds, develop on the slopes of the Rocky Mountains. In winterchinook winds are notable for bringing sudden high temperature rises, withunusually rapid dissipation of local snow.

The Bora. The adiabatic heating during the clescent of a very cold mass of1ir that has passcrl ovcr il rn<lunlain hitrricr tlr it plitloittt tltily nol hc sullicientto t'llrrrgc it irrlo it wiu'nr wintl rll'tltt'loclttt ly1rc. As tltc still crlltl air fallsgrirvrl;rlror!irllv irrlo llrt'wrrlrrrcr rcgton ou lltt'lr'r sirlc, ils polr'rrlilrl cncrgy isI ptiliFtlr-rl itrll kittr.lir. 11!lrlrgV Wttrrll nl r.rlrr.rrrt. rnlcrrsi(y tuit.y lhus bc pro-

'lu' e,l. r ltrltnr leti:+:rl lr1, g_ii,ilr t'l l5(l ,r(Xl ktrr/lrt sr'rntirlr'tl hy pcriotls ol'calm.Wlttrl;q ol tlte !rrrtrt ty|e rrr r ur llt rttr-ilri rvltr'!t' tt slcr'p slopc scpitratcs a cold

Flitlt,:Il ltlrll !l tr.taal lr!!t!i! Airrnrrp- llrr-lreil kttowtt lrorit wittds arc thosc thatrrl r ur il ial:-;rlr- erirrl f'liiilrr- uir lllr: irrlllrlir:;l crursl ol tlrc Adriatic.

Jel FJtetI Wltlcl*. I lrt- ir'l rllrt'l t'urrsisls ol irrr int'rcasc in wind intcnsity dueln tupngrirplrtr rrl r orrliplrirrlrorrs llrirl pirrtlttt't lr ('onvcrgcnce of streamlincs. TheIntrrti;rl rvirrrl ol llrr'lowct ltlrtittc Virllery irt soutltcrn France is a wcll-knownr'rirnrplt' ol ir lxrnr winrl inlcnsilictl by .yot cllbcts.

Thunderstorrns. A necessary condition for the occurrence of thundcrstomsis thc fbrmation of tall convective clouds produced by the upward motion ofwoffi, moist air. The motion may be started by thermal instability or by thepresence of mountain slopes or of a front. Thunderstorms are classified ac-cordingly as thermally convective, orographic, and frontal.

If condensation of the water vapor contained in the ascending air producesheavy precipitation, viscous drag forces exerted by the rain on the air throughwhich it falls contribute to the initiation of a strong downdraft. Part of thefalling water is evaporated in the underlying atmosphere that is thus cooledand therefore sinks. The cold downdraft spreads over the ground in the mannerof a wall jet (i.e., a flow caused by a jet impinging on a wall) and producessqually winds. This stage in the life cycle of a thunderstorm associated withstrong downdrafts usually lasts from 5 to 30 min and is called the moture stagell-141. As the energy supplied by the updraft is depleted, dissipation of thethunderstorm occurs. A schematic vertical cross section through a thunderstormcell in the mature stage is shown in Fig. I .3 . I 1 . Characteristic of thunderstormsis the sharp wind speed increase, known as Jirst gasl, which is associated withthe passage of the discontinuity zone between the cold downdraft and thesurrounding air.

Tornadoes. Tornadoes contain the most powerful of all winds t1-151. A tor-nado consists of a vortex of air, typically of the order of 300 m in diameter,that develops within a severe thunderstorm and moves with respect to theground with spceds of the order of 30-100 km/hr in a path, approximately 15krl long, clircclcrl prcdorninanlly toward the northcast. Thc maximum tangen-

\ l)rroction ol movement

\-\

FIGURE 1.3.11. section through a thundersrorm in the marure stage Il-14].tial speeds of tornadoes have been estimated to be of the orderof 350 km/hr,but the possibility that some may actuaily be considerably higher has not beennrlcd out.

Tornadoes are observed as funnel_shaped clouds (Fig. I .3.12). The tangen_(i.l speeds are probably highest at the iunnel edge uid d.op off toward theccnter and with increasing distance outside the funnel.

l .r n tM( )t;t,ilt iltc M()il()Nli 29

krn2

l|l(;tjRlJ 1.3.12- 'lirrttitrlo ltttttt('l (utllrl('ry N:rtiorr:rl()t'crrrric irntl Atrrlrsphcric Arlrnin-islrilli()lt).

30 ATMOSPHERIC CIRCULATIONS

FIGURE 1.3.13. Balance of forces in tornado vorrex.

Slttt t' {ltt' t'r'trltilirliirl lott'cs itt llrt' lolrurtlo v()t1cx l:u' cxcccrl lhc Coriolislrtlrt=',. llrt'l;tllr'r ttt;tv lrt'ttt'1',lrt'lrrl;ur(l llr('1ir;rtlrrlrl wirrrl t.t;rurlir)n (scc Sect.I .'l rri:rt lrt- rr,t illr.rr :r',

I

tlt, t' l 1,. Ltlrrrit ili, lll{. r yr ll,,ltn1rllirlirtitt llti r rillr-t nl ll!r. t Iilr=r, l, t', lltr,IrLiIIIFIrI it|ilit!' IItr' t;trIttt',

tllt ,1,

lt(1..r.2)

I rirrl | 11' I I I t, \\'lurlr r('l)r('r.t'rrl:; rlrc lon.t.s lrc(irtg on a l)lrrticlc in alilllt;lrlo \'iltlt'\, tl r:ttt lrt' rr'r'rr llritl llrt' ;rrt'sstn' ilt lt t<lrnad<l (locfcilr,ics lowardll5 ( ('lllt'l Iltr' rllllt'tt'trt't' lrt'lwt't'n llrt' plt'ssrrlc a( thc ccntcr iyxl at a f'ewIrrrrrrlrt'rl lt't'r lr,rrr rlrt' t'cnrt:r' .l rlre v()11cx .ury bc as high as 0. I ol' oneirlrrrospltt'n', or lrlxrul J(XX) llsl'.'l'.rrrrtlrr's lr:rvc rrls. hccrr rcp.r1cd, although much less fiequcntly rhan inthc Unitcrl Statcs, in Australia, wcstern Europe, India, and Japan. -l.ornadoesthat occur in Japan are known as totsumaki.s. Typical diameteri lilr {atsumakisare of the order of 50 m. Their forward speeds are of the ordcr o1'40--50 km/hr; the average length of their paths, which are directed gencrally toward thenortheast, is about 3 km and their maximum tangential .speecls arc probablyabout 200 km/hr t1-161.

The destructive effects of tornadoes on buildings are illustrate<t by Fig.1.3.14.

ADDENDUM: LOSSES DUE TO W|ND STORMS

wind storms are the largest single cause of economic and insured losses dueto natural disasters, well ahead ofearthquakes and floods [1-17, 1-lg]. In theunited States, between 1986 and 1993 hurricanes and tornadoes caused about$41 billion in insured catastrophic losses, compared with $6.1g billion for allothernatural hazards combined [1-18, p.4], hurricanes being the largest con-tributor to the losses [1-19]. In Europe, in 1990 alone, four winter stormscaused $10 billion in insured losses, and an estimated $15 billion in economiclosses [ -20 to l-221.

ru'rrrrl vt'lot r(y, r'is tlrc rurliirl rlistance;rll (l('rrsr{y, tntl tllthlr is l[c prcssure

ilrIIntNot: 31

I"IGURE 1.3.14. Tomado damage in Rochester, Indiana (courtesy prof.essor u. F.Koehler, Ball State University).

REFERENCES

l-l w' J' Humphreys, physics of the Air, McGraw-Hilr, New york, 1g40 (reprint,Dover. New york. I964).l-2 H. R. Byers, General Meteorology, McGraw_Hill, New york, 1944.l-3 G' J' Haltinerand F. L. Martin, Dynamicnr and physicar Meteororogy,McGraw-Hill, New york, 19-57.l-4 L' T' Matveev, Fhrt,st2s t..f rrtt'Atttr-,s1ilttrc,'167-513g0, u.s. Department ofcommerce, Nationar Tcc:rrrrit'rrr rr'rr.rrrrrri.' scrvicc, springfield, va.l--5 A. Miller. Mctutnil.1i.v, (,lurr.lr.s 1,, Mt.rril, (..1'rrrbus, OH, I971.l-6 M' Ncihurgcr,.t. (;. ririrr;icr.;uur W r) rr.rurcr., (/rttrcrsruntrirtlg tht A!n._s1;haric I.)nt,innncttl, W. ll. l;rt.r.trr;rrr. S;rrr l;l;urr.ist.o, 1973.

32 AtM()t;t't l nt(: (ilt(;ut n il()Nli

l-l H. Goldstcin, Chssit'ul Mcclrttttit',t, Atltlison Weslcy, Ncw Yolk, 1950.l-8 F. Fiedlerand H. A. Panof.sky, "Atrrxrsphct'ic Scalcs and Spccrral Gaps," Ilull.

Am. Meteorol. Soc., 51 (Dec. 1970), I I 14,l I 19.l-9 Hurricane, U.S. Department of Commercc, ESSA/PI 670009, 1969.l-10 R. A. Anthes, Tropical Cyclones: Their Evolution, Structure and Effects,

Monograph No. 41, Am. Meteorol. Soc., Boston, 1982.1-l I H. E. Graham and G. N. Hudson, Surface llinds Near the Center of Hurricanes

(and Other Cyclones), National Hurricane Research Project. Report No. 39,U.S. Department of Commerce, Washington, DC, 1960.

l-12 R. D. Marshall, Wind Load Provisions of the Manufactured Home Constructionurul Safety Standards: A Review and Recommendations for Improvemer?/, NIS-'l'lR -5 189, National Institute of Standards and Technology, Gaithersburg, MD,I 993.

I 1.1 S. Busingcr, "Arctic Hurricanes," Am. Scientist, 79 (1991), l8-33.l-14 'l'ltuntle rstorm, Report of the Thunderstolm Project, U.S. Department of Com-

rncrcc, Washington, DC, 1949.l-15 E. Kessler, "Tomadoes," Bull. Am. Meteorol. Soc., 5l (Oct. 1970), 926-936.1-16 H. Ishizaki et al. "Disasters Caused by Severe Local Storms in Japan," Bull.

Diaster Prev. Res. lesf., Kyoto University, 20 (March l97l),227-243.l-11 G. A. Berz, "Global Warming and the Insurance Industry," Interdisciplinary

Science Reviews, 18 (1993), 120-125.1-18 D. D. Mclean, Chairman's Report to the Annual Meeting, First Annual Meeting

of Insurance Institute for Property Loss Reduction, Seattle, October 12, 1994.1-19 A. C. Boissonade and S. K. Gunturi, "A Knowledge-based Computer System

for Financial Wind Risk Management," Proceedings, Computing in Civil En-gineering (K. Khozeimeh, ed.), Am. Soc. Civil Engineers, New York, 1994.

l-20 G. Berz and K. Conrad, "Stormy Weather: The Mounting Windst<lrm Risk andConsequences for the Insurance Industry," Ecodecision, April 1994, pp. 65-68.

I-21 Winter Storms in Europe-Analysis of 1990 Losst,s arul l;'utur( L()ss Potential,Munich Reinsurance Company, D-80791 Munich, 1993.

l-22 Windstorm-New Loss Dimensions of a Natural llultnl, Munich llcinsuranceCompany, D-80791 Munich, 1990.

CHAPTER 2

THE ATMOSPHERIC BOUNDARYLAYER

As was indicated in chapter l, the Earth's surface exerts on the moving air ahorizontal drag force, whose effect is to retard the flow. This effect is diffusedby turbulent mixing throughout a region referred to as the ot_orp,lr"rf, bound_ary layer. The depth of the boundary rayer normally ranges in the case ofneutrally stratified flows from a few hundred meters to sl"ueJ kilometers,depending upon wind intensity, roughness of terrain, una ungt" of latitude.within the boundary rayer, the wind speed increases with elev?tion; its mag-nitude at the top of the boundary layeris often referred to as the gracrient speed.Outside the boundary layer, that is, in the f."" ut_orpt;;", ;# wind flowsapproximately with the gradient speed along the isobars.This chapter is devoted to the study of aspects of atmospheric boundary-layer flow that are of interest in structural design. The theoretical and experi_mental results presented include descriptions of irean wind profileq the relationbetween wind speeds in different roughn"r, regimes, and the structure of at-rnospheric turbulence. Since the structural engineer is concerned fimarlty wittrthe effect of strong winds, unress otherwise noted it will be assumed in thelbllowing that the flow is neutrally stratified. The justification of this assump-tion.is that' in strong winds, mechanical turbulence* dominates the heat con-vection by far, so that thorough turbulent mixing tends to p.odu." neutrarstratification, .iust as in a shailow raycr .f incompressible fluid mixing tends toproduce an isothermar state. Ars., sircc wincr speeds are considerably lowerthan the specd of sound, inc<lrrr1'l'cssirririry rrury bc assumed in the study of theclynamics of thc flow.

'r'A tltrirlilitlivc rk:sclipliolr ol-llrc rrrt.clrrrrtrr.:rl ltrrlrrrh,rrrr.plrt.rr6rrrt,rr6rr is prcsclrlul i1 Sccl.,1..l.

33

34 il[ AtMr)til'ilt ilt(: rr()uNt)nny rnyt tl

2.1 GOVERNING EQUATIONS

2.1.1 Equations of Mean MotionThe motion of the atmosphere is governed by the tundamental cquations ofcontinuum mechanics that include the equation of continuity-a consequenceof the principle of mass conseryation-and the equations of balance of mo-menta, that is, Newton's second law. These equations must be supplementedby phenomenological relations, that is, empirical relations that describe thespcci(ic rcsponse to external effects of the continuous medium considered. (Inllrc cirsc ol'a lincarly elastic body, for example, the phenomenological relationst'orrsist ol'lltc: so-citllccl Hookc's law.)

ll tlrc t'r;rrrrtiorr ol'conlinuity and the equation of balance of momenta areirvcllrp.t'tl with rcsllcct to litttc:, ancl if'tcrms that can be shown to be negligible;rlt'rlroplrctl ll 1.2 21" thi: lirlkrwing cquations describing the mean motion inllrt' llrrrrrtllrty lrrye l ol tltt: rtlttursphcrc arQ obtained:

0u }tlU IV Idx 0v

AV AV dVt/- + v- t w- 0x dv 02,

ll) I 0nW+Lil; p 6x -fv*:*:o (2rr)l0n l0r.r-+ t.fU- -;r:0pdy p dz

(2.r.2)

l0o;;,r*8:o (2'l'3)

AU AV AWE* i,r+E:o (2.t.4)

where U, V, and W are the mean velocity components along the axes x, y, andz of a Cartesian system of coordinates, whose z axis is vertical; p, p,f, and gare the mean pressure, the air density, the Coriolis parameter, and the accel-eration of gravity, respectively; and r, and r,, are shear stresses in the x and ydirections, respectively. The x axis is selected, for convenience, to coincidewith the direction of the shear stress at the surface, denoted rs (Fig. 2.1.1).

It can be seen, by differentiatingEq.2.l .3 with respect to,r or y, that the

\,-..Itl( il llll,l :.1. 1 " ( 'oorrlirrrrlt' rrrts

Irovl Itl.itNti |(Jt In Il()Ni; 35

vt'tlical vltriittiott ol lltr'ltot izotttrtl l)rL:ssur('1ll=lrrlrt'rrl tlt'1x.rrrls rr;xrrr llrc hori-zorrtal dcnsity gr':rrlit'rrl. lior llrc l)urposc ol tlris it.xt, it will bc srrllicicnt klt'ottsidcrorrly lklws irr wlrit'lr llrc horizontal dc:nsily gllrtlicnl is rrogligiblc (c.g.,l):rft)tft)pic l1ows; c.g., scc l2-2 1). In this casc rlrr: lrolizorrtul prcssurc gradienttlocs not vary with lrciglrt irnd l.hus has, througlrotrl lhc lroundary layer, thesrrure magnitude as at tlro top of the boundary layor:

(2.t.s)

whcre z' is the gradient velocity, r is the radius of curvature of the isobars,:urtl n is the direction of the gradient wind (see Eq. 1.2.7).If the geostrophicrrlrproximation may be applied, it follows from Eq. 1.2.6 that

H: ,lrn,, *Y7

l0n-: : fv"pdx

wlrcre U, and Vr are the components of the geostrophic velocity G along ther and y axes.

The boundary conditions may be stated as follows: at the ground surfacelcvel the velocity vanishes, while at an elevation from the ground equal to thelroundary-layer thickness, the shear stresses vanish and the wind flows with theliurdient velocity.

2.1.2 Mean Velocity Field Closure'lir solve the equations of mean motion, it is necessary that phenomenologicalt:lations (also referred to as closure relations) be assumed defining the stressest,, and r,. A well-known assumption [2-l] is that an eddy viscosity Kand arnixing length L may be defined such that

l0n-+:-fu"pdy

(2.1.6a)

(2.1.6b)

(2.1.7a)

(2.r.7b)

(2.1.8)

AUru : pK(x. !. z) c,Z.

AVr,, : pK(x. j, z) dz.

K(x, y, z) :

'f 'lrc usc ol' ljc;s. 2.1.1 -2.1 .tl in t'orr jrrnt'ti'n with l,)qs. 2.1.1-2.1.4 is rcfcrredIo its lhc tttcatt vckrcity licltl t'krsrrrt' lrr lir;s. 2.1.7 cithcr lho ultly viscosityot'lltc ntixittg lcrrglh lir:kl rrrrrsl lrt. ,.tpr.t rlit.rl.

1.21,, v, ., l(ur:)' * (Yu,)'1"

36 ilil n lM():;t't il nt(; lt()t,Nt)nt ty

2.1.3 Mean Turbulent Field ClosureFrom the equations of balance of momentaequation may be derived (e.g., see [2-31):

fbr the mean motion, thc lirllowing

.., a /q'\l l r, oI)* u,\:, )l- liA *lalu-l3x e).,h(9.+ve.n).

r,,3V);al(2.1.e)

(2.1.10)

(2.t .1t)

(2.t.t2)

(2.1.13)

c:0

rvlrt'rt. llrt'lr;rrs irrrlit'lrlt'irvcllrging willr rcspcct to time, u, tt, w are turbulentlrlor rly llrrclrrrtliutrs irr llrr r, t,, ;: tlirct.tions, rtspcctively.

Iu I /' I r,'')"'

r:; llrr rt'srrll;rrrl llrrtlturlrrrft vt.lrxily, /,'is llrt.llrrctrurtirrg prssurc, and e is thet;tlc of ctt('tf.v rltsstlt:tltott l)('r unil rrurss. lit;rr:rliorr 2.1.9 is rclbrrccl to as thetttrltttlttt! /.ittt'tit' .'tt.'t,q.\' r'rlttrtliott rrrrtl c'xPlt'sscs llrt. blrllurcc ol'turbulcnt cnergyittlvcctiott (llle (rrttls itt lltc lit'st brlrt'kel ). pnrtlrrcliorr (thc lcrr.ns in thc secondbrackct), clillirsion, antl tlissipirtiorr. 'l'hc rrsc ol lxl. 2.1.9 and attendant phe-nomenological rclations-in c<lnjuncti<ln witlr llqs. 2. l.l-2.1.4 is rcf'crred toas the mean turbulent field closurc. Phcnorncrrokrgical dcscriptions of the quan-tities involved in Eq. 2.1 .9 havc becn attorrrptccl by various authors lz-4,2-5,2-61 . Successful predictions of boundary-laycr characteristics based on Eq.2.1.9 and various phenomenological descriptions have been reported in theliterature [2-7], although differences of opinion with regard to the relative meritsof these descriptions still exist.

In particular, the mean turbulent field closure appears to be advantageous inthe study of three-dimensional boundary-layer flows. Following [2-g] and12-91, [2-10] proposed the relations

{rf,+ r2,lt'': porq,

/p's2\ t.1..) /v\' \; * z ) : q'n''^^.to'taz 1;1(qr)r,,

L -- LaQ/6)

Tu : '''|Ul6z )Vl0z

in which ar = 0.16, 6 is the boundary-layer thickness, and e,. is the resultanlvelocity at the edgc of thc boundary layer (or thc gr:rtlicnt vclocity in atmo-sphcric filrw).

:r:' tull ANvi llll llYl't r()r rr rlirNil()ril,/()NrAr ry il{rMr}rir Nr ,t,:,il()w 37

1.O0.5

v/6FIGURE 2.1,2. Empirical functions. From J. F. Nash, .,The calculation of rhree_I)imensional rurbulent Boundary Layers in Incompressible Flow," J. Fluiel Mech.,37(1969), Cambridge University press, New york, p. 629.

In the case of the mean turbulent field closure in which Eqs. 2. r.g-2.r.r3are used, the empirical functions that have to be specified are the diffusionl'unctions a2(yl6), and the dissipation length z7(y/6). Reference 2-r0 proposeslbr these functions the form represented infrg.'Z.t.Z.

2.1.4 Second-Order Closure'fhe second-order closure consists in supplementing the equations of balancerf momenta and of continuity by the Riynolds

"qirations, which govem thebehavior of the stress tensor components and are dlrived from first principles12-111. Reynolds equations contain unknown terms, including triple velocitycorrelations, for which suitable phenomenological relation, ,rruit be sought. Torbtain such relations, the method of invariant modeling has been proposed,which is based upon the following requirements. The -od"l"d terms must: (r)cxhibit the tensor and symmetry properties of the original terms in Reynoldscquations, (2) be dimensionally correct, (3) be invariani under a Galilean trans-lbrmation, that is, a translation of the coordinate axes, (4) satis$z all the generalconservation laws [2-1 r, z-rz]. The second-order closure has been applied, forcxample, to the study of the flow structure in the boundary layer near a suddenchange of surface roughness 12-131.

2.2 MEAN VELOCITY PROFILES IN HORIZONTALLYHOMOGENEOUS FLOW

It lnay bc assurnccl that in littgc st'lrlc sl()lnl:i, within a horizontal sitc ol unilirrm(lttghncss ovcr a sullicicrrlly lltrgt lt'lt lr ;r rr'p,ion cxisls ovcr w6ich rho ll.w is

3B ilil AtM()lipltilil(; t()t,Nt)Aily tAyl il

FIGURE 2.2.1. Growth of a two-dimensional boundary layer along a flat plate.

horizontally homogeneous. The existence of horizontally homogeneous atmo-spheric flows is supported by observations and distinguishes atmosphericboundary layers from two-dimensional boundary rayers such as occur along flatplates. Indeed, it is known that in the latter case the flow in the boundary layeris decelerated by the horizontal stresses, so that the boundary-layer thicknessgrows as shown in Fig. 2.2.1 . rn atmospheric boundary layers, however, thehorizontal pressure gradient-which, below the gradient height, is only partlybalanced by the coriolis force (Fig. 1.2.8)-"re-energizes" the fluid and coun-teracts boundaryJayer growth. Horizontal homogeneity of the flow is thusmaintained 12-141.

Under equilibrium conditions, in horizontally homogeneous flow Eqs. 2.1.1and2.l .2, in which Eqs. 2.1.6 are used, become

v^ - v: !0". p.f az(2.2.ta)

(2.2.tb)

(2.2.2a)

(2.2.2b)

2.2.1 The Ekman SpiratIf in the above model the shear stresses are represented by Eqs. 2.1.7 andif,in addition, it is assumed that the eddy viscosity is constant, thJmodel obtainedis called the Ekman spiral. Equations 2.2.1 thenbecome a system with constantcoefficients. With the boundary conditions U : V: 0 for z : O and U :U* V : Vrfor z: oo, the solution of the system is

u--u:-!0"" pfaz

IU : 6Gt1 - e-oz(cos az - sin az.)l

v:4ctt2where a : (.l.l2K)v2

* e-"Z(cos az. * sin uz.)l

i':' Ml AN Vt t{'{ tiy t,il()l ll ll; lN ll()lll.1()NlAl ly ll(,Mrxil Nl ()(,li lt()w 39

Ucluations 2.2.2. wlrrclr tk:scribc tltc Ilkntittt s;rit'irl. irtl' rcl)lcric:tltctl sclterrrratically in liig. 1.2.(). 'l'lrc ap,rocrnont ol' lhr:su ctlttitliotts witlt obscrvrrlittttslrirs bccn fountl to be rrnsirtislactory, howcvol'. liot cxruttplc, whilc acc<lrdingto Eqs. 2.2.2 lhc irngle rr,, hctwcen thc surlacc strc:ss 11 antl thc geostrophicwind direction (trigs. 2.1.1 and 1.2.9) is 45o, obscrvations indicate that, inblrotropic flows, dcpcnding chiefly upon roughncss ol'tcrrain, this angle mayrrrrrge between approximately 6" and 30'. The causc of the discrepancies isthc assumption, mathematically convenient but physically incorrect, that thet,tldy viscosity does not depend on height.

2.2,2 The Turbulent Ekman LayerMcteorologists have attempted to solve Eqs. 2.2.1 using assumptions on thevlriation of eddy viscosity with height that are more plausible than the as-surnption of constancy. A survey of corresponding solutions can be found inl2 2l and 12-151.

A different type of approach was recently developed in l2-l4l in which,lrrther than resorting to a mean velocity field, closure is based on similarityt'onsiderations analogous to those used in the theory of two-dimensional bound-irry layer flows. In this approach the boundary layer is divided into two regions,ir surface layer and an outer layer. It is logical to assert that the surface shearr,, rnust depend upon the flow velocity at some small distance z from the ground,thc roughness ofthe terrain (i.e., a roughness length zo), and the density p ofthc air. Thus rs may be expressed as a function F of these quantities:

ro : F(Ui * Vi, z, zo, p) (2.2.3)

where i and j are unit vectors in the x and y directions, respectively. It isr'onvenient to write Eq. 2.2.3 in nondimensional form as

ui+vj (2.2.4)

where the quantity

U*: (2.2.s)

is known as the shear velocity ol' lhc lkrw and.ll is some function of the ratio;:/;1y. Equation 2.2.4 is a lirnrt 1vl' 1111: vve:ll-known "law of the wall" and de-scribcs thc flow in thc surlitc:g litycr.

In thc outcr laycr it can hcr sirrrillrrly irsscl'lc(l that the reduction of velocityl(t/,i + V*.)) * (Ui + lz.i)l rrl lu'rglrt ,'ttrtrsi tlcpond upon the surface shearr1y, the: hr:ight to wlrich thrr cllt't'l ol llrc witll sttt:ss hits diffused in the flow,

U4.: /, (:)\(0/

(?)"'

40 nil AtM()t;t'ilI tl(i tx)(,Nt)nt ty tAyt l

that is, thc boundary-laycr thickncss 6, arrrl thc tlursity p ol'thc air. 'l'hc cxpros-sion of this dependence in nondimensional lirrrrr is known as thc '.vckrcitydefect law":

ui+vj _ u|i + v|i(2.2.6)u4 u4

where f2 is some function to be defined.Il it is postulated that a gradual change occurs from conditions near the

gnrrrnrl to contlilions in the outer layer, it may be assumed that a region ofovcr'f ir1r e xists in which b<llh laws are valid. Let Eq. 2.2.4 be written in thelorttt

(;) (*)l (2.2.7)

f irrrrrr tlrc lirnn ol' L<trs. 2.2.6 and 2.2.7, and the condition that their right-handsidcs bc cqual in the overlap region, it follows that a multiplying factor insidethe function 11 must be equivalent to an additive quantity outside the functionf2. rn the case of the analogous two-dimensional problem, it is well-known thatthe two functions must be logarithms tz-16, 2-ril. The requirements of theproblem at hand will be satisfied if f1 and f2 are defined as [2-14]

., (;)

ui t v.i-.1 tll *.

where B and k are constants. Substituting Eqs. 2.2.g anct 2.2.9 in Eqs. 2.2.7and 2.2.6, respectively,

f{0 :1ln g'lkyi

fzG):gngr/!i+fj

i ('"; + r'.e) i

Usi + VRj , t /,-t--ttt't4 k \

u. 16^ : - ln -tt4 k 2.0

vu--4llq: k

(2.2.8)

(2.2.e)

(2.2.10)

(2.2.tt)

(2.2.12)

(2.2.t3)

Ui+ViU4

ui+vi 'i) ' . f iU4

If Eqs. 2-2.10 and2.2.l 1 are now equated in the overlap region, there result

:':, Ml nN vt tr){ ili t,n()l ll Il; lN ll()lll,/()Nlnl ly ill )M(t(it Nt ()t |; |()w

lhrrrr wlriclt thert. lirlIrws, " , t/.'

(; (nt+rn'"\ ttt\ ;,,,/ [

41

It can further be shown that the boundary-laycr thickness 6 may be expressedlts

(2.2.14)

(2.2.1s)

(2.2.16a)

2.2.15 is established

whcre c is a constant. To prove this relation, let Eqs. 2.2.1a and 2.2.1b berrrultiplied by the unit vectors j and i, respectively. From the expressions thusrrlrtained, and remembering that r, : r0, r, : O at the surface and that r, :/,, : 0 at z : 6, it follows that

^Uxd -._ f

J trt * vi - (usi + vs:)r az : !t*i

[ ,.r, (#) o, : T [,u, ot

: const a-pl

'l'hat is, Eq. 2.2.16 is verified and the validity of Eq.l),-141. Equation 2.2.14 may then be written as

(2.2.16)

where the integration is carried out over the boundary-layer depth. Since thelrrrlk of the mass transport takes place in those parts of the boundary layerwhcre Eq. 2.2.6 holds-which include the overlap part of the surface layertlrrwn presumably to a very small height-the velocity profile in Eq. 2.2.16rrrry be approximately described by Eq. 2.2.6. rf Eq.2.2.15 is now substitutedirrl<r Eq. 2.2.6 andEq.2.2.5 is used, the left-hand side of 8q.2.2.16 becomes

o:lu' . (^n- o)')''' T (2.2.17)

llquation 2.2.17 was oblairrctl irrrlt'pcrrtlcntly in [2-18] and [2-5]. The deri-vrrtion o1'12-51 is bascd <lrr tlrt'lrrrlrrrlr.rrl crrcrgy equation and the assumptionrrl'ir rnixing lcngth pnrporlionlrl to .'. 'l'lrc tlrurntities A and B arc univcrsal('()nslanls. lironr lho arr:rlysis ol olrst'rv;r(r('ns il wrrs liluncl that 4.3 < 1l < 5.3irrrtl 0 < A < 2.tl12-14.215, I l!'1. .) l{). ). )O.221.2-22,2-2'3.2-241. On(lre llitsis ol'cxpcritttt:ttls ilt lltt' u,urrl lrrrrrrr'l ;rrrtl irr lhc: irtrrroslthcn., llrc wt.ll

THE ATMOSPHERIC BOUNDAIIY LAYER

known von Kdrmin's constant is generally assumcd to bc ft = 0.4.r,coefficient c in Eq. 2.2.15 is of the order of 0.25-0.3 \Z-ZO, 2-261.

2.2.3 The Logarithmic LawEquation 2.2.1O may be written as

(2.2.18)

(l ll'l), rvlrt'rc .r rr llrt'lrt'ip.lrl rrlrovt'tlrc srrrlircc, z, is the roughness length,rrtttl l/( I tr' lltt' t!rr';ttt lvttttl s1x't'rl. lir;rurlion l.l.lli is known as thc logarithmicI it tr'

llr=r r=ttl lillr'i{rittt'l('ornluliir;rl tt'sr'irlt lr lurs cstlrblishccl thirl thc hcight abovegtrttttttl .'r ltp t{r t!lttr lr lttl ,r ,r tll nr;ry lrc:tsstrrrrt.tl lo lrr: irpproxirnatcly valid,i:; rlr'Ilirrrl lrv lltr' Ir'lirlrrrrr

It(2.):!r*n1o ?.o

i1 ,"'.,' (2.2.19)

whcrc b is a corrstlrrt. lhc onlcr ol'rrragnitutlc ol'which is 0.015-0.03 12-26,2.271. As notcd in 12-261,l.c1.2.2.19 oxprcsscs the fact, well-kn.wn fromlaboratory experiments-including cxpcriments conducted in rotating wind tun-nels [2-28, p. 148, 2-29]-that the logarithmic layer extends to some fiaction(of the order of lo%) of the boundary layer depth 6 (see Fig. 2.2.2). Figure2.2.3 [2-30] represents averages of 14 mean wind profiles (average mean speedat 9.1 m above ground u(9.1) : 5.3 m/s) measured in nearry neutral flow nearDallas, Texas. It is seen that for the profiles of Fig. 2.2.3 the logarithmic lawprovides a good description of the data up to at least 100 m elevation. This isin agreement with Eq. 2.2.19.Indeed, for U(9.1) : 5.3 m/s, zs : 0.03 m,f = 0.77 x 10-4 Qable 1.2.1), and b = 0.022, Eqs. 2.2.1g arnd'2.2.19 yieldzr = 100 m. Note in Figs. 2.2.2 and2.2.3 that the use of the logarithmic lawfor heights exceeding z7 is conservative from a structural design viewpoint.

Equation 2.2.19 may also be shown to follow from the assumption that, inthe region 0 1 z 1 21, the shear stress r, differs little from the surface stressrs (see, for example,I2-1, p. a89l), and the component Izof the velocity issmall. Integration of Eq. 2.2.la over the height z7 yields

ru : ro * ,f I', (vc - V) dz = ro * pf Vrzl (2.2.20a)

*The acttral valuc ol ,/< has in roconl ycars bccomc thc objcct ol sorlc ilchatc 12-251. Hgwcvcr,cllculalions ol inlcrcst in t:rrgirrccring upplications dcscrihcrl irr tlris lcx{ urt rxrt allcctctl signili-t'lrrrlly by lht' irtlrr:rl vlrlrrt. ol (.

l,;' Mt AN Vl l{x.ily t,t t()t iltsi tN il()nt./()NtAt ty

r) Z.3ti crn

llttYl 11 ,; Nt ()ljti ll()W 43

4.O

3.O

2.O

1.0_ o.BI,9 0.6

6 o.+att) o.2It

0.1o.08o.o6

o.04

0.5 0.75 1.0 1.25 1.5VELOCITY (m/s)

FIGURE 2.2.2. Mean wind profile asmeasured in a rotating wind tunnel12-291. Copyright @ l9j5 by D. ReidelPublishing Company.

10 1? t4 t(; 1r1 20 22 24ll/rt.

l,'l(JIllll,l 2.2.J. Avcrugc ol l,l nt(.iut \\,lt(l ;rrolrlt. rr.t.otrlt.rl rrrlrr. l)irlllrs. ,l,cxlrs. Allcr.It II 'l'lrLrillcl.:ttttl II (). l,lrP1x" "Wttrrl ;u!11 l('nrlx'rrrulr. l,r'olilc ('hlrr.rrt.lt'r.islit.s liirrrr()lrst'r'vitliorrs olr rr l,:l(X) li 'lirr.vr'r." ./ 11t1tl At,.t .l (l()1y1 y, l()() .t0(r, Alilt.r.it.;rrrMr'lt'on rlolt it'lrl Sot, it'l y.

u*= O.147 m/sze= O.OO9 1 cm

-Ulu*= ltXlnlo; zo=3 cr

o 50F<(T9.1-T320)<60F

llll n lM()l;l'l ll lilo lr()llNl)nl rY lAYl lr

lpJ vrz,l : qru

where 4 is a small number. Using Eqs. 2.2.5 and2.2.13,

:,:, Ml nN vt trI tt\ I,t i()t It1; tN lt()l il./()Nlnt t\ ilirM{)(,t Nt ()t tl; tt()w'l'Altl,lt 2.2.1. Vnlucs ol'Sur.ljrt, ltorrghlrcss l,t,rrglh (11) iurl ol.Srrll:rt.t. l)r:rgCrrclliciclrls lirr V:u.ious 'l,y;x.s ol' ,l.rrrains

45

lt is slrowrr in 12 2(rl irntl l2-3 ll that thc logarithmic law holds, for practicall)urlr(rs('s. t'vt'rr lrt'yontl lrciglrts lrt wlrich r7 is of the order of 30%.

Il. lor r'r:rrrrplt'. / l() 'r st't' '. Il - 30 m/s at l0 m above ground, zo :(l (f1 rrr (()lx'r l('rr:rirr). :rrul /r 0.02, il lillltlws then from Bqs.2.2.18 and.'.' lt) tlrrt .:1 ,l(X) rrr. lrr llrr.'t'irsc ol'stnrng winds, the validity of theIo1';1,r,1r"r'. 1;rw rrp 1o t'lt'v:rliorts rll'tltc ortlcr of 200 m has been confirmed bynrr':rsur('nr('nls rt'por1t'tl irr l2 12l lrrrd l2-331, as well as by observations at Salef J .l-lf rrrrrl ('r'rrrrlit'kl l2 .i-5 1 rrnirlyz.ccl in 12-221.

()rr:rct'orrnt rll (hc lirrito hcight of the roughness elements, the followingcnrpirical urodilication of'Eq. 2.2.18 is required 12-361. The quantity z, ratherthan dcnoting hcight above ground, is defined as

1-1R LJ (2.2.22)

where z, is the height above ground and za is a length known as the zero planedisplacement 12-371. The quantity z will be referred to as the effective height.The flow parameters ze and z./ are determined empirically and are functions ofthe nature, height, and distribution of the roughness elements [2-38]. Theroughness length z6 is a measure of the eddy size at the ground. It is suggestedin [2-33] that reasonable values of the zero plane displacement in cities maybe obtained using the formula

Type of Surlircc

Sand"Snow surfaceMown grass (-0.01 m)Low grass, steppeFallow fieldHigh grassPalmettoPine forest(mean height of trees: 15 m; one treeper l0 m2; Z+ = 12 m l2*4}l\Sparsely builrup surburbsbDensely built-up suburbs, townsbCenters of large cities,

(2.2.20b)

\2.2.21)

(2.2.22a)

(t'nr)

0.ol o.I0. I 0.60.l-l

t-42-34-to

l0-30

90-10020-4080-120

200-300

t.2,1.9t.9-2.91.9-3.43.4-5.24.1-4.75.2-7 .67 .6-13.0

28.0-30.010.5-15.425.1-3s.661.8- 1 10.4

l0tr<

rlu2*zt: fvgnk ,u4:.fB'*:Dj

I

il

il{t

tl

:lI

"Reference [2-38]."Values of eo to be used in conjunction with the assumption 2,1 : 0 [2_42].

wake effects). For this reason values of zo in built_up terrain may differ con_siderably from experiment to experiment. ihe values listed in Table2.z.l areintended for use in structural engineering calculations in .on;un.tion with theassumption Za : 0- They are based on a careful analysis oi fuii-scare data,given in 12-421.The surface drag coefficient r (Eq. 2.2.23) for windflow over water surjhces

1"pr9r upon wind speed. on the tasis of a large numbe, or -"urur.ments,the following empirical relations were proposed for the range 4 < u(r') <20 mls [2-431:

r : 5.1 x l0-4 [U(10)]046r : lo-a [7.5 + 0.67U(10)]

(2.2.24a)

(2.2.24b)

where 11 is the general roof-top level.Typical values ofzs forvarious types ofterrain, and the corresponding values

of the surface drag coelilcients (defined as

where u(10) is the mean wind speed in m/s at r0 m above the mean waterlevel. According to [2-44], tor UltO; ) 20 m/s or so l< is constanr.A more recent evaluation of existing measurements led to the expressionproposed in [2-45] for wind speeds U(10) up ro 40 m/s:

r:0.00r-s [r + exp ( tl0!!.5)l ' .0.00104 e.2.2s)I t..5. /lFor examplc. il utlO) : 20 nr/s. ir irrr.ws liirrrr l:qs. 2.2.25 ancr 2.2.23 thatx , 2.5 X l0 I ancl 7,, : 0.3-5 c.rrr. ll t.lrn lrt.vt.l.ilit:tl lhlrl crr<lrs irr the cstirnationo1'wincl spoctls cluc ttl tlnccr-lrtittlrt's :rss.t irrt'<l witlr tlillcrcrrccs lrr'.ng [Jqs.2'2'21t' 2'2'241't, antl 2.2.25 ittt' ittsil'trrlrt;rrt Atkliliorurl inlirnrrirli., .rr lhcsurlirct'tlrlrg lirr-wirrtr fl.w.vt.r'lrrr',,,.,',,,, ,. prr.r;t.rrrt.rr irr l?,r{rl :rrrtr lr rr.2 l

,r:H-?

.:I n I'| ln 110/zo) I

(2.2.23)

in which z0 is expressed in meters) are given in Table 2.2.1 [2-39,2-40,2-41,2-421. Table 2.2.I also incluclcs suggested values o1'2,' lirr built-up terrain. Thedetermination <ll'rcproscn(irtivc wintl pnrfilcs in brrill rrp tcrrain is gcncrallyrlillicLrlt on acc()unl ol lot'rrl llow irrlllrrxrgcncitics (c.9.,llrosc associlr(crl wilh

46 ltE AlMosplt tirc llouNunny InyFR

According to 12-1331, the influence on thc wavcs on thc wind prolilc appcarsto be restricted to elevations below three wave heights; in this zone wind speedsare lower than indicated by the logarithmic profile.

2.2.4 The Power LawHistorically the first representation of the mean wind profile in horizontallyhorrrogcncous lcrrain has been the powcr law, proposed in l9l6 by 12-471:

(2.2.26)

rrlrerr-,* irr;ur r'\lr(llt(=ttl rL'|t'tttlt'irl rrlxrrr rorrglrrrr.ss 0l lctrrain ancl ;*l and zszr lt=lir rlr. lir. tprltl,, irl rlt,t. f, rorrrrr I

Itt l.t .lHl il lii rrfiirltiit(=rl ( l) tlrirt lltr'powcr l:rw lroltls wilh consllrrrt cxponentrr ttp ll lltr= gtiulir'ltl lrr'igltl ri;rttrl (,t)llrirl i rtst'll rs ir lrrrrcliolt ol'rv alonc. Thefi1ril nl llrt'rit' rri:iiiittlf tlttr; lrrrIltr.r llt:tl

/-.\"l/(.',,r) {/(;,,,1 ('"' }\ r ,.'/

"l;" (;) (2.2.27)

'l-ho socott(l itssttttt;rliott lcl)r'cscnls rrr t'llt'r'l tur crrginccling sirrrplilicltion of theboundary-laycr clcptlr clcscri;rtiorr givcn by l;.t1.2.2.1-5. Vllucs ol'D and arecommendcd for dcsign l)url)oscs irr l2-4ttl ancl l2-491 arc slrown in Table2.2.2. Yalues of 6 (in opcn tr:rrairt artrl ccntcrs ol' largc citics) sintillr to thosegiven in Table 2.2.2 were pK)poscd in 193,5 by Pagon [2-.5t), p. 7441. TheASCE 7-95 Standard 12-1391 is bascd on the valucs ol' a arrtl d given in12-491. However, [2-1391 uses 3-s gust speeds instead ol'fastosr-rnilc speeds,and the power law exponents are adjusted accordingly-scc 'I'ablc 2.2.2.

Currently, the logarithmic law is regarded by meteor<lkrgists as a superiorrepresentation of strong wind profiles in the lower atmosphcrc 12-26,2-51,2-52, 2-53, 2-54, 2-551.

TABLE 2.2.2. Yalaes of 6 and c, Recommended in [2-48], 12-491, and [2-1391

Centers ofLargeCities

SuburbanCoastal Areas Open Terrain Terrain

Reference6

(m)0

(m)6

(m)o

(m)2-482-492-139*

l/ 10l/il 5

0. r6l7

t 19.5

275274274

0.28I 14.5U1

400366366

2132ll

0.40 520U3 457U5 457*lirr 3-s grrsls l,'l(lllltl,l 2.1..1. Wrrrrl lt.lot rty lrlrliles

Ml AN Vl lor 11y 1,11111 ll l:; tN ilOlll,/()NtAt ty ltilM0rit l.ll (,t |; Ilrw 47

2.2.5 Relation botwecn Wind Speeds in Dlfferent RoughnessRegimesConsidcr tw<l acliaccnl lcrrt'ltitts, cach of'unilirrrrr nrrrglrrrcss iurrl ol'srrllicirrrrtlylarge fetch. Lct tlrc nlttgltttoss lcngths fbr thc lwo lcl'ruins bc rlcnolctl by 1111

and 20, and assume that z9r ( zs. The retardation ol'thc llow hy surl'acc l'rictionwill be more effective over the rougher terrain; thcrclirrc, if the geostrophicspeed is the same over both sites, at equal elevations the mean wind speedswill be lower over the rougher site. A schematic representation of the respectivewind profiles is shown inFig.2.2.4.

The profiles of Fig. 2.2.4 suggest the following procedure for relating windspeeds in different roughness regimes. To calculate the wind speed U(z* z6)over the rougher terrain if the speed U(24, zo) is known, Eq. 2.2.27 is appliedto each profile; then the quantity G is eliminated from the two relations thusobtained, and

(2.2.28)

where cv(zo),6(ae) and ot(zo),6(261) correspond to the roughness lengths Zs andZs1, respectively. Equation2.2.28 was proposed in [2-48] and will be referredto as the power law model.

Recently, an alternative procedure has been proposed that is based on resultsof both theoretical and experimental studies I2-22l.If the speed U(zd, zs) isknown, it follows from Eq. 2.2.18

u(2,, zs): (6)*"'(?)" " (r(,,,,,0,)

48 lltt AIM()lil'ilt tit(; tr()t,Nt)nny tnyt n

where the notation of Eq. 2.2.22 is used. Applying now Eq. 2.2.29 to the twoprofiles represented in Fig. 2.2.4 and eliminating G,

luprrrtiorr 2.2..10 tlt:tcrrrrincs tho valr.rc of the friction velocity a*. Then

[r'* ('".* - o)'),.:lu' * (," ^^,-

o)'l''' u*, (2.230)

U(2,,r,:,rr)U*t:-'' 2.5 ln(zrlzor)

{/(;,,, rrr) 2.5u* lnk4O

(2.2.2e)

(2.2.31)

l'irlrurtitrrrs l.l.19, l.l. 10, rrrrtl 2.2.31 will be ref'erred to as the similarity model.As lrrrs bct'rr slrown in 12 22l, tl"rc unccrtainty with regard to the exact values

of llrc corrstiurts ,4 urrcl /i in tlq. 2.2.30 turns out to be of little consequenceinsolirr as cstinratcs ol'wind speeds in the lower atmosphere are concemed.With possible errors of the order of 3% or less, it may be assumed A : 1.4and B : 4.7. Also, the dependence of the results on z* andl is insignificantand may be neglected. For practical purposes, therefore, the ratios u*lu*1 maybe calculated simply as functions of the roughness lengths Z1y and zor. Thedependence of u*/u*1 upon Ze and zor can be represented by the relation [2-56]

(2.2.32)

However, subsequent research has shown that the similarity model must besubjected to empirical adjustments in the case of terrain for which z, ) 0.30m or so. Table 2.2.3 lists ratios u*/u*1 based on full-scale measurements,corresponding to zs1 : 0.07 m and various values ze of practical interest12-421.

The application of the similarity model will now be illustrated by a numericalexample. The data used in the example were obtained by measurernents in andnear London and were reported in [2-33]. At Heathrow, Z,t : 0.08 m, 2,,, =0, and the measured mean wind at a height above ground zsr : l0 m is U(zrr,zo) : 11.7 m/s. The mean wind U(z' Zo) at a height above ground z, : 195m is sought at the Post Office Tower in London, where zo : 2.5 m (2,7 : 0).

TABLE2.2.3. Ratios u*lu*rf<tr za1,: O.07 m and Various Values ar12-421

z r 0.0706u4 /Zo\-:t-lu *1 \zor ,/

zo (m)tt *l u a.r

0.(n5o.83

0.07I .(X)

0.30I 15

r.001.33

2.501.46

:';' Ml ANVI lr)rllt l,lt{}l ll l:; lNll()lll./()Nlnl lyll{}il/(){il Nt ()t ,:; lt()W 49

litrrrrr l')t;. 2.2.)t), rt,t O.(Xrti rrr/s. liRilrr 'l'irblt. ).).1. 11 ,711 ,,r 1.4(l; thatis, r.r,,. - l.4l rrr/s. Usltli lit;s. )..2.31, {/(2,,,1,) l-5..1,1 rrr/s. lt is notcd thatthis rcsult coincitlcs witlr tlrr. irctuirl rncasLrrctl spccrrl ll .l.ll.

If thc rncan spcc:rl rrcrul rhc l)ost officc'l'owcr ut ;,, r95 rn is calculatedusing the powcr law rrroclcl (hq.2.2.28) with thc paranrctcrs cv and 6 suggestedin 12-481, therc rcsults U(2.", z.t) : 13.4 m/s versus the measured 15.3 m/sspeed.

2.2.6 Effect of Thermal Convection on Mean Speed profiles inStrong WindsIt is of interest to estimate the extent to which the effect of thermal convectionis significant in structural engineering and extreme wind climatological calcu-latlons. To do this, we use the following expression, based on the work ofMonin and Obukhov f2-2, p. 282;2-51:

(2.2.33)

where u* : friction velocity, k : von Kdrmdn's constant, zs : roughnesslcngth, / : Monin-Obukhov function, and t : Monin-Obukhov length. Ifthe stratification is neutral, L: a, tl, : O, and Eq. 2.2.33 becomes the well-known logarithmic law (Eq. 2.2.18).

The length L is defined by the following expression [I-4, p. 281]:

U(z):T1,":-r(;))

,-8 QoT cpp

where g : acceleration of gravity (S : 9.81 mls21, T: absolute temperature,r;, : specific heat at constant pressure (co:240 callkg degree [l-4,p. 132]),p : air density (p = l.2kglm'|. and Q6 : eddy heat flux (usual orders ofrrragnitude for Qo are 10 to 60 callm2ls [l-4, p.276]).

unstable stratification. In unstable air the following expression will be used|or tklL):

l, u*(2.2.34)

(2.2.3s)r (;): I:: , - - 160 t^rT

Equati.n 2.2.35 was pn)pos(:tl irr ltcl. I15. Acc'rding to Rcl,. 2-51 , itprovidcs a vory gtxrcl lrt to cxllcrirrrcrrl;rl tllr(ir ovt'r'rrrrilirrrn tcrrain and lor0 >:ll, > -2. (N<ltc thal /, is by tlt'lirtitiorr nt'|;11'ur,' rt lltc slrlrtilicati<lrr is unstirblc.)litlrrittiorr 2.2.35 is lcprcscnlcrl irr lirp1. .t..t 5

50 t Ht A t M()lit,ilt nto ti()uNt)n ny t n yl il

FIGURE 2.2.5. Function {(z/L) for unstably stratified flow. From E. Simiu, ..Thermalconvection and Design wind Speeds," Journal of the structural Division, ASCE, l0g(July 1982), 16l t-1615.

stable stratification. In the case of stable stratification it may be assumedthat

(2.2.36)

[2-25,2-51]. The length I is defined by Eq. 2.2.34; however, empirical studiesreported in [2-58] suggest that under stable stratification conditions it may beassumed that

3.0

2.O 2.5 3.0 3.5 4.O 4.5 5.O

_zL

L = l. I x l03a3x

where a* is expressed in m/s.Table 2.2.4 lists estimated dcviations from

2.2.18) for three representativc c.lscs of interest

(2.2.37)

the logarithmic profile (Eq.in structural engineering ap-

TABLE 2.2.4. Deviation of Mcan wirrd speeds from Logarithmic profile [2-59](':rsc l " Case 2t' Case 3'

Elevation 50 rrrUnstable stratiliculion l'X,Stable stratilicltioll l'X,

200 nr_4%,

4'n,

l-5 m-4%

l5m- - LL /O

- t2%"Hourly wirrtl s;xt'rl lrt/'llorrrly wirrtl spct,tl :rt'llorrlly wirrrl sl)r'r'(l ;tt

lO nt r'lt'vlrliorr ovt'r ollt'rr lt:r':tiltlll ilr r'li \:tlI)il rrv(.t rll)(.n l(.il;Inlll ltr r'lIr.rlrrrrr 'rv(.t r'l)(.il l(.il:ilil

l5 rrr/sl.)rrr/s1 rtr/',

p lr n t M( ll ;t 'l il iltc t t,nBl,t t NCL 51

Plicitlirlnsl.'l'ltc tcsrrlls ol 'l'irhlc 2.2.4 show llrirl srrt'lr tkrvirrliorrs rnay indeedbc ncgloctccl wltctr cslirrritlirrg wind prcssuros oll slrucluros (scc Case l) orwhen reducing to it c(tltlttt()ll clcvation largcsl rrr<lltllrly or ycarly wind speedsrccorded at a wcathcl' sruti.n (scc case 2). Howcvcr, lbr wind speeds u(10).f the order of 5 m/s thc dcviations from a logarithmic profile are significant(see case 3). The lattcr conclusion is of interest fbr the design of structures,such as smoke stacks, that exhibit a significant across-wind response at lowwind speeds. This response is usually enhanced if, as in the casl in unstablystratified flow, the actual mean wind profile is closer to being uniform than8q.2.2.18.

2.3 ATMOSPHERIC TURBULENCE

Figure 2.3.1 shows that wind speeds vary randomly with time. This variationis due to the turbulence of the wind flow. Information on the features ofatmospheric turbulence is useful in structural engineering applications for threemain reasons. First, rigid structures and members are suu3e&eo to time-depen-dent loads with fluctuations due in part to atmospheric turbulence. Second,

lThe estimates were based on the assumption that in unstably stratified flows e : 50 kcal/m2 sand 7': 290"-

*e

Itl(,illl{l,l 1,.1. l= Wrrr'l .,lrr'r.rl r('(.()t(l

52 iltt AtMoril'ilt nt(; tr()t,Nt)At ty tAyt lt

flexible structures may cxhibit rcsorriurl irrrrlllilicltiorr c:llbcts inclucc:cl by voktc-ity fluctuations. Third, the aerodynanric bchavi()r ol' structurcs-and, crlrrc-spondingly, the results of tests conductcd in thc laboratory-rnay dependstrongly upon the turbulence in the air flow.

The following features of the atmospheric turbulence are of interest in var-ious applications: the turbulence intensity; the integral scales of turbulence; thespectra of turbulent velocity fluctuations; and the cross-spectra of turbulentvelocity fluctuations. Also of interest to structural designers is the dependenceof the largest wind speeds in a record upon averaging time.

2.3.1 Turbulence lntensity'l'hc sirrrplcst clcscriptor of atmospheric turbulence is the turbulence intensity.Lct u(z) dcnotc thc vclocity fluctuations parallel to the direction of the meanspccd in a t.urbulcnt flow pussing a point with elevation z (Fig. 2.3.1). Thelongitudinal turbulence intensity is defined as

ulottzI(71:u(z)

(2.3.1)

where U(z) : mean wind speed at elevation z and rl r/2 - root mean squarevalue of z.J Vertical and lateral turbulence intensity may be similarly defined.

The longitudinal turbulence fluctuations can be written as

(2.3.2)

where z* : friction velocity (see Eq. 2.2.18).It is commonly assumed that Bdoes not vary with height.+ Values of B suggested for structural design purposeson the basis of a large number of measurements are listed in Table 2.3.112-421.

The averaging time in Eqs. 2.3.1 and 2.3.2 should be equal to the durationof the strong winds in a storrn. It is commonly assumed that this duration isbetween 10 minutes and t hour.

tThe altemative notation o, : u2t/2 is also commonly used.iThis use of the notation 0 should not be confused with its use as the safety index (AppendixA3).

TABLE 2.3.1. Values of p Corresponding to Various Roughness Lengths

u2 : Bu'*

zo (m)a

0.0056.5"

0.07(r.0

0.305.25

r.004.1t5

2.504.00

"Based on mcasurcnrcnls rclxrrlctl irt l2 7t); irrxl rrsctl irr i'orrjrrrrcliorr witlr lulsScc also 12-l'12l'.

2.2.23 itntl2.2.25

i, ll n I M( )l;t't lt ttll . tt,ntttJt I N( lt 53

lrrlr cxarrrlllc, il l: l(l rrr. :1y - 0.07 nr, irrrtl l/( lO) l0 rrr/s, il lolkrwslirrnt Eqs.2.l.ltJ,2..1.1,1..1.2, and'l'ablc 2..1.1 tlrlr( thc trrlbrrlcrrcc intcnsityis /(30) : 0.162.

2.3.2 lntegral Scales of Turbulence'lhe velocity fluctuations in a flow passing a point (Fig. 2.3.1) may be consid-cred to be caused by a superposition of conceptual eddies transported by thernean wind. Each eddy is viewed as causing at that point a periodic fluctuationwith circular frequency <,s : 2rn, where n is the frequency. By analogy withlhe case of the traveling wave, we define the eddy wave length as )t : (Jln,where U : wind speed, and the eddy wave number, K : 2rl)t. The wavelcngth is a measure of eddy size.

Integral scales of turbulence are measures of the average size of the turbulentcddies of the flow. There are altogether nine integral scales of turbulence,corresponding to the three dimensions of the eddies associated with the lon-gitudinal, transverse, and vertical components of the fluctuating velocity, u, u,rrnd w. For example, Ii, Ll,, and L'; are, respectively, measures of the averagelongitudinal, transverse, and vertical size of the eddies associated with thelongitudinal velocity fluctuations (-r is the direction of the mean wind u andof the longitudinal fluctuations z).

Mathematically, 1, is defined as

u: Ru,ur(x) dx (2.3.3)

where R,,rr(x) is the cross-covariance function of the longitudinal velocity compo-rrents z1 = u(xr, !r, Zr, t) and u2 : u@r -l x, yr, 21, t), defined in a mannerrrnalogous to Eq. A2.29, / : time, and u2t/2 is the root mean square valueof u I (and a2). Note that in horizontally homogeneous flow, tj is independentolxl and y1 . Similar definitions apply to the other integral turbulence scales.

From their mathematical definition it follows that integral scales are smallil'the cross-covariance functions are rapidly decaying functions of distance,rrnd conversely. Velocity fluctuations separated by a distance considerably largerthan the integral scales are uncorrelated, and will therefore act on a structuralc:lcment at cross-purposes. For example, values of r), and Li that are smallcompared to the dimensions of a panel normal to the mean wind indicate thatthc effect of the longitudinal velocity fluctuations upon the overall wind loadingis small. However, if D" and Li, uc largc, the eddy will envelop the entirepancl, and that effect will be signilicrrrrt.

Equation 2.3.3 can be translirrrrrcrtl il it is rrssumed that the flow disturbancelrirvcls with thc vclocity L/(r) lrrrtl. tlrclt'lirlc. lhir( thc fluctuation u(x1, r -l t)rrury hc irlcntiliccl with a(,r1 tlll, r). wlrcrt./ tintc (Taylor's hypothesis).'l'hcn

#t:

54 THE AlMOSPI IFRIC R(}I'NI)NIIY IAYI II

Li, : Il,,1r) dr (2.3.4)

where R,(r) is the autocovariance function of the fluctuation a(x1, l). Thc lcngthof the record from which R,(z) is estimated should be the same as that used toestimate (J and u2 (i.e., about one hour; see Sect. 2.3.1).

Estimates of turbulence scales depend significantly upon the length and thedegree of stationarity of the record being analyzed, and usually vary widelyfrom experiment to experiment. For example, for open exposure, measuredvalues of Lj reported in [2-60] (Part 2, pp. 31 and 32) vary between 120 mand 630 m at 150.8 m elevation (the average value being 400 m); between 110rn and 690 m at 110.8 elevation (average value: 350 m); between 60 m and(r50 rn at 80.8 m elevation (average value: 300 m); between 130 m and 450rn at 50.8 m elevation (average value: 200 m); and between 60 m and 460 mat 30.U rn elevation (average value: 200 m). Data reviewed in [2-61] suggestthat Li is a decreasing function of terrain roughness. For example, the followingdata are listed in [2-61]:

Site z(m) zo(m) Ii@)

#\,:,

CardingtonRound HillBrookhaven

15 0.01 82t7 0.04-0.10 5516 1.00 36

The following empirical expression was proposed in [2-6ll for the heightrange z: 10-240 m:

Il, : Cz^ (2.3.s)

where C and m are given in Fig. 2.3.2 and z is the elevation (t) and z inmeters). The application of F,q.2.3.5 to the data just listcd yields, approxi-mately, the values 1: l5O m (Cardington), 140-120 rrr (lbund Hill), 70 m(Brookhaven), which are about twice as high as the mcasrrrctl values.

According to 12-611 the integral scales L) and I.i, lrcr, rcspcctively, aboutone-third and one-half the integral scale Ij as givcn by lirg. 2.3.-5. However,according to 12-621, a better estimate of L) is obtuinrrtl lirrrrr lhc cxprcssion

L', : 0'2Li, (2.3.6)

It has been suggested that

I \' = 1-"rt 5

(z in rr-rctcrs) l2 l.'l()1. 'l'lrt' cxplcssiorr

/,1, 0."[.:

(2.3.1)

(2..1.t|1

:, ll n lM(,:it't il ilt(. il/ntlt,l l N(;r 55

0.001 0.01 0.1 1.O 10zq (meters)

l''l(;tlRE 2.3.2. values of c and m as funcrions of zo [2-611. Reprinted with permis-',r.rr l'rom J. counihan, "Adiabatic Atmospheric Boundary Layers: A Review and;\rr:rlysis of Data from the Period 1880-1972," Atmospheric Environment, 9 (1975),li/l 905, Pergamon Press.

\virs proposed in 12-641 and confirmed by subsequent measurements, as indi-,;rlctl in 12-611.

2"3.3 Spectra of Longitudinal Velocity Fluctuationsfhe Energy cascade. It was mentioned in Sect. 2.3.2 that the turbulentvchrcity fluctuations may be considered to be caused by a superposition of.rltlics, each characteized by a periodic motion of circular frequency a : 2rnt.r flt u wave number K :2rl\,, where x is the wave length). The total kinetic('n('r'gy of the turbulent motion may, correspondingly, be regarded as a sum of, 'rrtlibutions by each of the eddies of the flow. The function E(K) representingtlrt' tlcpcndence upon wave number of these energy contributions is defined astlrt' cncrgy spectrum of the turbulent motion.

ll'thc equations of motion of the turbulent flow are suitably transformed, it,;rn bc shown that the inertial terms in these equations are associated withtr;rrsl'cr of energy from larger eddies to smaller ones, while the viscous terms,rrtount fbr energy dissipation 12-63]. The latter is effected mostly by the',rrr:rllcst cddies in which the shear deformations, and therefore the viscous',lr('srics, arc large. In the absencc of sources of energy, the kinetic energy oftlrt' ltrr-bulcnt motion will decrclsc thirt is, the turbulence will decay-fasterrl tlrc viscosity eff'ccts are largc, rrr.rc skrwly if'these effects are small.

Molc prcc:iscly, in thc lallsl t'rrst' rlrt' tlt'r'lry tirr-rc is long if compared to thelrr'riotls ol'thc cclclics irr llrc hi1',lr wlrvr. rrrrrrrlrt:r rangc. Thc energy of these,,ltlit's rrriry thcrclirrc hc consitlt'tt'tl lo lrt' rrpploxirrrtrlcly stcacly. This can onlylrt'lltt't'trsc il thc: c:rrc:rgy le.tl inlo llrr.trr llrrorrl',lr ilrertiirl lr-utrsl'cr l'nrrrr lhc largcrltklit's is lrlrltrrrct:rl by lhc (.ncllly rlr:,r,1|;rlr.tl lltrorrglr visr'ous rlli'cts.'l'ltc srrurll

llll n lM()i'il'lll lll(; ll()(,Nl)nl lY lnYl ll

eddy motion is thcn dctcrrninccl solcly by lhc nrtc ol crrcrgy lrarrslcr'(or'. ctlrriv-alently, by the rate of energy dissipation, dcnotcd t (scc [Jq. 2.1.9) antl by thcviscosity. The assumption that this is the case is known as Kolmog,orov's.firsthypothesis. It follows from this assumption that, since small eddy motion isdependent solely upon internal parameters of the flow, it is independent ofextemal conditions such as boundaries and that, therefore, local isotropy-theabsence of preferred directions of small eddy motion-obtains.

It may further be assumed that the energy dissipation is produced almost inits crrtircly by (hc vcry smallest eddies of the flow. Thus, at the lower end ofllu' lrilihcl wrrvc rrrrrrrbcr rlngc t<l which Kolmogorov's first hypothesis applies,tlrt' rrrllrrt'rrr't'ol (lrc viscosily is srrrirll. ln this subrange, known asthe inertialtttl,t,rtt,t:r'.llrt't'rltly rnol iort ttury lrt'lrssuntctl to bc independent of viscosity,;ur,l llrrr:, rlt'lt'rrrrint'tl solcly lry (lrt' r'trtc ol'cnerrjy transfer (which, in turn, ist't1rr;rl lo llrr'r;rlt'()l ('lr('rl'.y rlrssiplrtiorr). linrrrr this assumption, known as Kol-nttt,tltt,(tt".\.\('tttn(l ltvltttlltt',ti,t, il lollows tlrlrt lr rclirtion involving E(r$ and e

Itolrls lot :;ttlltt tt'nlly lrrglr A:

l'll',(K ). (. r | 0

whorc /j(/() is llrc cncrgy pcr ruril wavc nunll)or.

(2.3.9)

The clirncnsions ol'thc quantitics within brackots in Eq. 2.3.9 are[L3T 2f,[L-r], and ILtT '1, respectively. From climensional consiclcrations (see Sect.7.1) it follows immediately that

E(K) : ar62t3Y-s/3 (2.3. r0)

in which rz1 is a universal constant. On account of the isotropy, the expressionof the longitudinal velocity fluctuation spectrum* [which will be denoted S(K)]is, to within a constant, similar to Eq. 2.3.10. Thus

S(rK) : aezt3 K-s/3 (2.3.t1)

in which it has been established by measurements that a : 0.5 12-211.

Spectra in the lnertial Subrange. Measurements carried out in the surfacelayer of the atmosphere confirrn the assumption that in horizontally homoge-neous, neutrally stratified flow the energy production (see Eq. 2.1.9) is ap-proximately balanced by the energy dissipation [2-3]. The expression of thisbalance may be written as

ro dU(z)p dz.

'r'A tldrrilctl tlistrrssion 1)l :il)('( lrr is plt'st'rrlt'tl irr Allllcrrtlix A)

(2.3.12)

w I rt'rrr

l-{/(:)-.a,1, lrr'O lr)

ll lrqs. 2.2.5,2.3.12, and 2.3.13 are used.

"1*e : kz

Srrlrstituting Eq.2.3.14 into Eq. 2.3.1l, if it is assumed that

2rnK:-u(z)

tlrcrc results

, .r n t Milr;t'ilt ilt{ , il,ttltt,il N(;t 57

(2.3.t3)

nS(z,n) ^^_^_)/,- --- : 0.28t--''u4 I

rvlrcrc the nondimensional quantityt

^nzt: uun known as the Monin (or similarity) coordinate, and

(2.3.14)

(2.3.ts)

(2.3.16)

(2.3.17)

S(2, n) dn : S(2, K) dK (2.3 . 1 8)

f 'tlrurtion 2.3.15 implies the validity of raylor's hypothesis (see Sect. 2.3.2).'l'hc left member of Eq. 2.3.16 is called the reducetl spectrum of the lon-1'rtrrtlinal velocity fluctuations and is seen to be a function of height. Althoughrrrtlividual samples may deviate considerably from the predicted values, Eq.' ]. 16 is, on the average, a very good representation of spectra in the high-I r trlrrcncy range [2-5 1, 2-52, 2-53, 2-&, 2-65, 2-67] and may, for engineeringl)rrl)oscs, be conservatively assumed to be valid for f > O.Z [2_64, p.27,; (t] , 2-691. As in the case of thc logarithmic law, for high wind speedi such;r:r iuc irssutlcd in structural dcsigrr (<ll'the order of 2o mls, say or more), it isr,':rsorr:rblc t() apply F,q.2.3.1(r tlrRrrrglrout the height range of interest to the'.1 nlt'lurll cnginccr.

Iltls ttst'ol lllt'st:ttttl:tttl ltolrrli()rt / slrrrrtlrl n.t lx tr)rlu.,r'(l rvillr its previous rrsc:rs llrt.(.1yri9lisl ';il ;[ [('l('r.

58 ilil n tM()rit,t I nt(i il()t]Nt)nny tnyt il

Spectra in the LOwer-Frequency Range. 'l'lrc krwcr-li-c(luoncy ltttgcr is tlc-fined between n : 0 and thc lowcr cnd ol'thc incrliul subrangc. As rtotccl in[2-511, 12-52], and [2-65], in the lower-f-requency rangc sirrilarity brcaks downand the spectra cannot be described by a universal relation. Howevcr, dcscrip-tions that are useful for engineering purposes may be obtained by noting that:

l. The value of the spectra for n : 0 is

s(0) : +,ft], (2.3.te)

wlrt'r'e rr' is lhc nlcan squilrc valuc of the longitudinal fluctuations, U istlrt'rrrt';rrr vckrc'ily, untl /,) is thc longitudinal integral scale.f F;q.2.3.19Iollows lirrrrr lir;s. 2..1.4 ancl A2.25.

l. 'l'he tlclivltivc ol'S(rr) wilh rcspect to n vanishes atn :0. (This followsliirrrr lit;. 42.25.)

3. 'l'hc spcctrurn S(n) is lnonotonically decreasing.4. 'l'hc spcctrum S(n) is continuous at the lower end of the inertial subrange

with the curve S(n) given by 8q.2.3.16.5. The area under the spectral curve in the lower-frequency range is equal

to the mean square value of the longitudinal velocity fluctuations (Eq.2.3.2) less the area under the spectral curve S(n) represented by Eq.2.3.16. (This follows from Eq. A2.15.)

Two comments on lower frequency spectra are in order. First, as in thecase of the mean speed U, the mean square value u2, and the integral scaleLj, estimates of spectra in the lower-frequency range depend upon the lengthof record being used. For consistency, the length of the record from which S(n)is estimated must be the same as that for (J, u2, and I). As indicated in Sects.2.3.1 and 2.3.2, for structural engineering purposes this lcngth should be equalto the duration of the strong winds in a typical storm. Corrmonly this is as-sumed to be I hour, although record lengths as low as l0 rninutes are used bysome workers. The l-hour period beyond which winds in a typical storm maybe assumed to become relatively weak is sometimes rcf'crrccl to as the "spectralgap" (or quiescent period) in a conventional reprcscntation of wind activitycorresponding to a continuous range of periods, including daily, monthly, sea-sonal, yearly, and secular periodicities [2-681. Spcctra ol' longitudinal windspeed fluctuations for periods longer than about I lrtlttr ctlrrcspond to meso-meteorological flow pattems. Thcy wcrc tcntativcly Ittotlclctl by Van der Hoven

rBy virluc ol (he tlelinitiorr ol (lrr's1x't'trirl tlt'rrsily, lit1. 2.1. l() rrrrplics ir vltrisltittgly snrall, rathcrthan it lirritc. t'orrlrilruliorrs ol llrrrlrr:rlirr1l totttlxrnt'nls willt zt'lr ltt'tlucrtty t() lllc tlrolln squatcvirlttc ol lltt' lltttltt;tlrotts

U

:r n tM()l ;t,t il ttlL ililillt,t I N(:l 59

ll l'l ll' wlttl ttolr'tl lltt' t'ris(t'rtcc ol's1rt:cllrrl lx.;rks irl pt'r'itxls ol'lrlrout 4 clays.lilut'lrr:rlions willr;x.r'irxls lorrgcr Lharr tlrtlsc tylricrrl ol'llrc: spcctral gop ir"tlrsrt:grrnlcd in slnrclru'irl crrgirrccring rnotlcls. 'l'lris irlkrws thc usc o1'Eq.2.3.19:rrrrl ilcttts 2 to 5 trllrvcr as c()n.tponcnts 9l' a rcirs.,rlrflc rnicnrmeteoiologicalrrrtxlcl all<lwing thc cslitttittion of longitudinal spcctra fbr periods shorter than;rlxrut I hour.

A sccond comment pertains to the relation between the frequency zps11 &tr'lriclr the curve ns(n) reaches a maximum and the integral scale ri. As shownrrr l2-(rll, the assumption has been used in the literature that

rxlu" 2t flpeak

f lrrr.vsvsl, it was pointed out in 12-731 that the estimation of Il based onrrrt';rsrrrcd values of u and npeal can be in error several fold, owing to the',('rsitivity of Il to the assumptions conceming the spectral shape between n

. o irnd_ n. : ,peak.This shape is in general unknown and, thereiore, so is therll;rtionship between npear and Il,.

Expressions for the Spectrum used for structurar Design pur-,roses. The curve

t'l nS(2, n), u'*

2o0f

(2.3.20)

(2.3.21)

(2.3.22)

rrlrrrsc lirrm was proposed in [2-66], approximates very closely Eq.2.3.16 intlr. rrrcftial subrange (zis the height above ground, n isihe trequency inHertz,rr , ;rrtl.f'are given by Eqs. 2.2.18 and 2.3.r7, respectively). Ii .un t" verifiedrlr;r( lir1. 2.3.21implies that

,/ : 6u2*

rrlrtlr, lilr built-up terrain (zo > 0.30 m, see Table 2.3.1), may result in an,,\('l('stirnation of structural response of the order of 5%. Requirements pre_rrrrrr5;fy listed pertaining to the value of .{n) and ds(n)ldn at-n :0 are not''.rrr:;lit'tl. However, this is inconsequential as far as the design of most land-lr.r:'t'rl structures is concerned, since their fundamental frequeicies of vibration;r(' rriuillly higher than the frequcncy corrcsponding to tire lower end of therrrt rtr:rl subrange. Therefore, pr<lvitlcrl that Eq. 2.3.22 is satisfied, the response,'l :;ut'll structurcs does not dcpcrrrl sig,rrilit.rrrrtly upon the shape of the spectrumrrr llrt' lowor l-rcqucncy rangc (st.t' St.t.l. (). l.-l).

Ilrc tlcvcl.pr'cnt of'l'\. 2..1.)l ll 701 wrrs rrr.rivutccl by criticisrn of thelrrlllv*1rt* cxprcssion, pnr;rost.tl irr l.r'lll;rrrtl rrst,tl irr rhc Natignal Building('rrrlt' ol ('unutll l2-721:

60 Tt tE nTMOSt'Ht-llto tx)(,Nt)ntiy tnyt il

ns(z.. nt -r'l- :4.0ui ''" tl * "'ro"(2.3.23)

in which x : l2OOnlU(l0); n is expressed in Hertz and U(10) is the meanwind speed, in meters per second, at z : l0 m. Equation2.3.23 was obtainedby averaging results of measurements obtained at various heights above groundand does not, therefore, reflect the dependence of spectra on height. In theirbscncc ol' rnodcls capable of describing this dependence-such models wererrrrf y tlr:vckrpcrl subscqucntly in the 1960s-Eq. 2.3.23 and similar expressionsProlxrst'tl in llrt' litcr-lrlurc havc pnrviclcd useful first approximations of thelonpiiiurlirr;rl trllrrrlt'lrct: r.il)(:ctrit in lhc atl'nospheric boundary layer. It is notedllr:rt llrt' th'Pt'nrlt'rrt'r' ol's;rectllr rln hcight is clcarly suggested by data publishedrrr lJ 7ll (lrip. l.1..lir).

As rrrt'rrliolrr'tl t'trlliu', llrc spccttrl tlistribution in the lower-frequency rangelrlrs littlt' irrllrrcrree on brriltlirrg tcsponsc; however, the magnitude of the tur-lrrrlcrrl llrrclrrrtiorr r'orrrponcnts at licquencies cqual, or close, to the naturall'r'cclucncics ol'u tall structurc rnay affect its response very significantly. It isthcrolorc ol'intcrcst to comparc thc higher-frequency components inEq.2.3.23to those of Eq. 2.3. l6 (or, equivalently , Eq. 2 .3 .21). Such a comparison showsthat Eq. 2.3.23 may overestimate the longitudinal spectra of turbulence in thehigher-frequency range by as much as 100-4fi)%, as can be seen in Table2.3.2 and Fig. 2.3.3b.

It is also noted that Eq. 2.3.23 yields z2 : 6u2*, and that it implies S(0): 0, or U:0 (see Eq. 2.3.19), which is physically not possible [2-3].The von Kiirmrin spectrum [2-1341

nS(n)a-

U-*

[' _'

(2.3.24)

was proposed before the development of Eq. 2.3.16. Equation 2.3.24 satisfiesthe conditions S(n) + O and dS(n)ldn : O for n : 0. However, for Eq. 2.3.24to be consistent with Eq. 2.3.16, it can easily be shown that it would benecessary to have Ii = 0.303t22, which does not appear to be the case in theatmosphere. That Eq. 2.3.24 is, in general, not consistent with Eq. 2.3.16 canbe explained physically by the fact, discussed earlier in connection with Kol-mogorov's hypotheses, that the higher-frequency spectrum is independent ofthe large-scale features of the turbulence that determine Lj. Equation 2.3.24is not used in applications where the magnitude of the higher-frequency com-ponents of the longitudinal velocity fluctuations is of interest. However, it canbe used in applications in which the effect of the low-frcqucncy componentcould bc irnporlant, suclt irs thc analysis of structurcs with vcry long nalural

48q'U,,r(+)l

:1 N-'l -6l-r5

' .1 A l lv'l( ', il,ill ttl( il,1il il,1 1 N(.1 6l

0.002 0.005 0.01

Wave number -lL- cycles/meteru(r2)(a)

(b)

lfl(;UltE 2.3-3- (u) Longitudinal turbulcncc spcctra measured at Sale, Australia (basedrrrr 20 rccords)12-ill. Frorn A. G. I)lvcrrporl, "'t'hc Spectrum of Horizontal GustinessNcrrr thc Gr<rund in High Winils,'' ettrrt. .l . lilt.1rrl Mcteontl. Soc., g7 (1961):202.1/r; ('ornp:uison ol'spcctra givcrr by litls. J..1.21 rrrrtl 2.3.23. From E. simiu, ,,windSltct'trir rrrxl I)ynurric Akrngwintl ltcsPorrst'," .l . ,\rr.rrt.l)ir,., ASCE 100 (1974): lg97_t9 t0

Eq.2.3.21

U(10): 30 m/s, eo = 0.08 m

n (cycles/s)

62 lltE AtMOS|'l tLtilo t]()tjND^lty tAyt il

TABLE 2.3.2. Yafues of nS(n)lu?* lirr 7., : l).0t1 nr an<l {/(10) : 30 rn/s l2-7(ll

z:100m z:300mnCycles per

Second(l)

Eq.2.3.16or 2.3.21

(3)

Eq.2.3.16or 2.3.21

(s)

All Valuesof z,

8q.2.3.23(6)

f(4)

f(2)

0.10.20.-51.0

0.2550.4501.1252.250

0.70o.430.240.15

0.5861.1722.9305.860

0.370.230.130.08

t.4l0.980.54o.34

pclirxls ol' vibnrlion (c.g., corrrpliant off.shore platforms, which have motionswith lrcrirxls or irlrotrt 50 io 120 s). A modified form of the von Kdrmiinril)r:ctnur, bascrl orr lirs( principlcs and reflecting the variation of the spectrumwitlr lrc:ight irbovc gnrun(|, wils rcccntly proposed by Harris l2-l4}].

ljor thc purposc ol' studying thc sensitivity of tall building response to changesin the valuc ol'various parameters determining spectral shape, an alternativeexpression fbr the spectrum, consistent with Eq. 2.3.16, was proposed int2-lo). This expression depends upon the parameter 0 and an additional pa-rameter allowing the modification of the shape of the lower-frequency part ofthe spectrum, and is subject to the constraint imposed by Eq. 2.3.2. A similarexpression was developed in [2-74] to study the sensitivity of compliant struc-tures to changes in the values of the parameters B and Ij, and to changes inthe shape of the lower-frequency portion of the spectrum consistent with Eq.2.3.2. The expression of [2-74] is

.f < .f^ (2.3.25a)

f^<f<f, (2.3.25b)

f--f, (2.3.25c)

where a* and/are given by Eqs.2.2.18 and 2.3.17, n is expressed in Hertz,z is the height above the surface (in the case of flow over the ocean, the heightabove the mean water level), f, is the lower limit of the inerlial subrange (f,- 0.2),f-is a parameter allowing changes in the shape of the spectral curveforf < f,, and

ariQ)0

( o,f + b,72 + d,13I

nS(2, n) j---- :\ cz * azf t brf,u'x I

I o.ze.s ''''

At:z

8r : 0.26.f ,2tl

(2.3.zsd)

(2.3.25c)

!,,,,) '''5r6lL,-.1',1' , 2(.1 ,,, .lil

tt2 : *2b2f,n

23 n lMr,!.t'ilt ilt(; il,lilitit tN(;t

i 2!",,(.1', -- .l',,,1 r .1,( 1., * zt,,,,l tnfr.

, -2lo,f^ o I,,, - I*l.f - Pt + b2(f^ -f ,'Ib,:t-t.sf.dlt'z:0t-azf,-brf?

(2.3.2st)(2.3.259)

(2.3.zsh)

(2.3.2s1)

(2.3.2sj)

llquations 2.3.25 are plotted in Fig. 2.3.4 for k : 0.4, zo : 0.001266 m,I : 35 m, U(35) :45 m/s (u*: 1.76 m/s), B : 6.0, f,:0.22, U: l8Orn, andJ, : 0.07. Also plotted in Fig. 2.3.4 isF;q.2.3.23 (intemrpted lines).

Eqs.2.3.25

'..r*/-ur rtrt

r1llz)

l,'l(;tJl{lt 2.3.4. Spcctrl ol' hrrrgitrrrlirrrrl vt'locil.y lluctr.ralions (Eqs. 2.3.25).

I

I

o0 0.150.0500

64 ilil n tM():;t'ilt nt(i lr()t,Nt)nl ty lAyt I

Unlikc l:q.2.3.24, Uqs. 2.3.25 atc cotrsis(ctrt with Ilt1. 2.3.1(r. lkrwt'vo', tlrc:ydo not satisfy the requirement dS(n)ldn : 0 lirr rr : 0. 'l'his rcquilotrtr:llt couldbe satisfied by modifying Eq. 2.3.25a in thc imrncdiatc vicinity ol'a : 0.However, such a modification is not necessary in practice since its efl'ect onresults of engineering calculations would be negligible. Finally, it is seen inFig. 2.3.4 that Eq. 2.3.23 significantly underestimates the spectral ordinatesat very low frequencies. This is due to the fact, noted earlier, that Eq. 2.2.23implies that Il : g.

Finally, we mention the spectrum proposed by Hanis in 1968 12-1371:

(2.3.26)

wlrt'rt r I.lt(X)rrl{/(10). Likc F.q. 2.3.23, Eq. 2.3.26 does not reflect thevlrrilrtion ol'thcr spcrclnurr with height above ground. However, it has over Eq.2.3.23 tlte: rrtlvirrrtagc that it irnplies a nonzero integral scale of turbulence lf: 10001 U(r.)l IJ(I0)l/0 (in rneters).

2.3.4 Cross-Spectra of Longitudinal Velocity FluctuationsThe cross-spectruml' of two continuous records is a measure of the degree towhich the two records are correlated and is defined as

S'j,,r(r, n) : lf,,,r(r, n) + iSf,,r(r, n) (2.3.21a)

in which i: Jl. The real and imaginary pafts in Eq. 2.3.21a are knownas the co-spectrum and the quadrature spectrum, respectively. The subscriptsu1 and u2 indicate that the two records are taken a.t points M1 and M2, thedistance between which is denoted by r.

The coherence function is defined as [2-751

Q(r, n): [Coh(r, n)]2 : cf,,ur{r, n) + ql,,,r(r, n) (2.3.27b)

where

nS(n\ x.-:4.0-u'* -'" (2 + x2)s'6

) 15f,,,,{r, n\12c;tu2tr. nt : se:;)se, n)

) 1sf,,1r. r)12Qituttr.r) : ik:;)Nru n)

(2.3.27c)

(2.3.27d)

In Eqs. 2.3.27c and d, S(21, n) and S(22, n) are the spectra of the longitudinalvelocity fluctuations at points M1 and M2.

lA tlctailctl rlistrrssion ol tlrss spt'tlt:r is ptcst:ttlttl irr Allpt:rulix A,?.

:| n I M{ }t;t't ti til( I t,ililt,t I N(:t 65

'l'lrc lirlklwirlg ('xl)t('\\t('rr lot llrt's(luiur txrl ol llrr't'olrt,n'nt'c lirrrcli6rr (alsokltowtt its tlltn'()w lt:urtl t.trrss t'orr-crlirliorr) wiri l)ll)l)()ri(,rl irr lJ-7(rl:

Clrrh(r, n) : (, t (2.3.28)

wlrcrc

I

lil

ll

i:r r'. altemativ ely [2-7 6],

nlc1,k, - zr)2 + C1Jy, - yr)tlt,,u(t0) (2.3.2e)

.f: (2.3.30)

Irr llqs. 2.3.29 and 2.3.3O, !r, !2, and 21, Z2 are the coordinates of points M,,A /

' . the line M 1, M2 is assumed to be perpendicular to the direction of the meanrvirrcl, u(10) is the wind velocity at 7 : 10 m, and the exponential decayt otrllicients C, C, (or C1r, Cy) are determined experimentally.

ln homogeneous turbulence the quadrature spectrum vanishls [2-64]. rnthe;rlrrursphere it appears that the ratio of quadrature spectrum to co-spectrum is:.rrurll and that the square root of the coherence function may therefore be:rss.rned, for engineering purposes, to be approximately equal to the reducedt rr spectrum cu,ur. on the basis of wind tunnel measurements, it has been:;rrggcsted in 12-771 that it is reasonable to assume in engineering calculations

S'u,ur(r, n) : Stt2(zr, n)Stt22r, n1"-i (2.3.31)

rvlrc,rcf is defined by Eq. 2.3.30 and Cr: 10, C, : 16.* It appears, however,tlr:rl the exponential decay coefficients C, C, (or Cv, Cr), iatherthan beingrrrtlcpendent of roughness, are generally larger for iougir surface conditions.rrt'lr as urban areas than for smooth surfaces l2-el. Moieover full-scale mea_',rrrr:rnents indicate that the exponential decay coefficients depend on height;rlxrvc ground and, quite strongly, on wind speed, as shown in f,igs. 2.3.5 ind; \.6 12-60, 2-781. The dependence of the exponential decay coeftcients uponrvirrtl speed is illustrated in Figs. 2.3.7aand2.3.7b, which represent Eq. z.i.zs

'A rrrtrtlified modcl of the spatial structure of turbulence proposed in t2-147) eliminates thel'lhrwirrg two drawbacks of Eq. 2.3.31. First, Eq. 2.3.31 does not allow for negative values oftlrt'lttl'lrttloncc co-spcctrum, regardless of spatial separation. For homogeneous turbulence thisrIr|lics lltat' contrary to its definition, thc mcrn ol'lhc lluctuating longitudinal velocity component'hrs ttttl vitnish. Sccond' 8q.2.3.31 inrplics lurgc colrclalionsirlthe low-frequency components' r'r'tr il lltc scpitlitlion is largc. Thc rrtort rt':rlrstit rrlrxlcl pnrposcd in 12-All may result in atr'(lrr( li()f) ol lltc calcttlirtctl rcsoniutl ltslxrrs(' ol slt'lrrlt'r slntclrtros by as rnuch 1ts 25%. Sce also| .' t,lr{, 2 t491.

66 THE ATMOSPHFIIIC f]Ot]NI)AIIY IAYI II

0102030405060u(10) (m/s)

Irl(;llltl,l 2.J.-5. Virliation ol-cxponential decay coefficient C,,, with wind speed (openIrllrrirr) l.l 781.

and nlcasurctl valucs ol'thc square root of the coherence function for records(takcn at points of cqual clevation) with U(10) : 20.8 m/s (Cr:u : 3.5) andU(10) : 35.2 mls (Cr, : 8.8) t2-601. The dependence of the exponentialdecay coeflicients upon terrain roughness, height above ground, and wind speedis insufficiently documented and therefore represents a source of uncerlainty instructural engineering calculations.

It was pointed out in Sect. 2.3.2 that relatively large uncertainties remainconcerning the integral scales of turbulence. In view of the close physical

r 0.0

8.0

6.0

4.0C,,

2.O

0

F'I(;URI,l 2.3.6. Virrirrtiorr ol (', with wincl spcctl anrl hciglrl (opcrr lclr:rirr) l2 Tlll

o

(:ry 3.5t/(10) 20.8 m/s

ao

;,:t n lM(lt;t,t il lll(. lUl ilIlt tN(:t 67

0.1 0.2 0.3 0.4nll yr-!ztl/ul'tol

i -.'C,lt, - ',1I - (/t:\ (2.3.32)

with cr : 3.0 ovcr warcr.rrtl (', (r.0 rvr:r r.nd. A thcorctical approachtcptlrlccl in l2-t'tt)l suggosts thirt tlrr krrrliitrrrlirrirl cohcrcrrcc clcpcntls up1;n thc

-. 0.6

5€ 0.4O

-* o.o|,I

IE o.aoO

o.20.1

nlly*y2ll/u11o)

Run 'l 1 7llv,I v,]l(meters) |

I o I o I rz Il"lolssll:l;l:: I

(b)

FIGURE 2.3.7. Measured values of Coh(l lr _ !zl, n) t2_6O1.

rclationship between turbulence cross-spectra and integral scales, similar un-certainties can be expected concerning the exponential decay coefficients.Nevertheless, results of recent research quoted in lz-sol ,ugg".i that the valueL. = t0 is acceptable or even conservative from a structural design viewpoint.A similar conclusion regarding the value Cy = 16 follows froil 1Z_St1, ac_:":d:g^,g,ylich. C, is a funcion of the raiio ly, - yrltz, as shown in Fig.2.3.8. Additional research into the vertical and lateral coherence of the lon_gitudinal velocity fluctuations is reported in [2-62] and, [2-g2,2-g3,2-g4,2-85, 2-86, 2-871.

In some applications the longitudinal (along-wind) coherence of the longi-tudinal velocity fluctuations is of interest. According to [2-gg], the longitudinalcoherence between the fluctuations at two points M,1x, , y, zj una M2(x2, y, z)can be expressed by Eq. 2.3.28, where

9oo

Run 1 18

68 lHf n tMosit,ilt llt(; tlotJNt)ntly tAyr lt

50

40

cy 30

20

0.t2345lv1-vrltz

l"l(,lllltf,l 2..1.t1. l)clrcntlcncc 1rl'(i,. upon ly, - yrllz according to [2-g01. copyrightr, ) l()t{ l by l). ltcirlcl l,rrlrlislring ('ornpany.

Irrrlrrrlcrrcc ilrtcnsity /(1), thc distance lr, - rrl, and integral scale He), asslrowrr in lrig" 2.3.9.

2.3-5 spectra and cross-spectra of vertical and Lateral velocityFluctuationsIt is shown in [2-31 that the spectra of vertical fluctuations up to about 50 mmay be estimated by the formula

10

o

cJ

NI

X

co()

o

FIGURE 2.3.9. t,ongiruclinala : l(2.)lx, - .r.,11.;(r) 12 8el.

12nllru

cohcrcncc as a lunction o| nl.i,l IJ f ur thrcc valucs o1.('opyrigh( () 1979 hy I). ltcitlcl l)rrhlishing (-orrrpirny.

:) ll A lM(l:;l,t il iltl . il,t il tut t N( il 69

rr,\,,,(l:, rr) _ .1..1(r/'u)r, I I l(ll"' (2..j.-1.j1

(2.3.3s)

According to n)casurcnrcnts rcportccl in [2-60, 2 ttOl, thc cn)ss-spectrum ofvcrlical fluctuations ilt lwo points M1 and M2, ol'clcvution z may be expressedils

S.,-(Ay, n) : S.(2, 11\s-8navtu171 e3.34)in which Ay is the horizontal distance between the points M1 and. M2.

The spectrum of the lateral velocity fluctuations may be written as

nS,(n) l5f-;T:(r+riJ-s'

where c(r) is a coeflicient that depends on t and z, is the longitudinal turbulentlluctuation. If Eqs. 2.2.18 and 2.3.2 are substituted into nq. Z.Z.Ze .

0t '' r(tl \tt2.5 tnetzu) /

'l'he form of Eq. 2.3 .35 was proposed in [2-661 . Equations 2.3 .33 and 2.3 .35 ,irr which the parameter/is given by Eq. 2.3.17, are consistent with the re-tluirement that, in the higher frequency range, the ratio of the vertical andlrrteral to the longitudinal spectra is equal to 413 [2-651.

Cross-spectra of lateral velocity fluctuations can tentatively be assumed tobc given by an expression similar toEq.2.3.3l , with exponential decay coef-licients lower by about 33% rhan those used in Eq. 2.3.31 [z-go, 2-go]. Al-lcrnative expressions for the spectra of vertical and lateral velocity fluctuaiions,based on a modified von Kdrmdn formulation which takes into account thevariation of spectra with height, were proposed in [2-140].

2.3.6 Dependence of Wind Speeds on Averaging TimeIt fbllows from the definition of the mean value that mean wind speeds dependupon the averaging time. As the length of the averaging interval decreases, thernaximum mean speed corresponding to that length increases. The relationbctween the wind speed averaged over / seconds, u,(z), and the hourly speed,L/3u1p(z), may be written as

U,(z) : Uzr,cnk) + cG)71n (2.3.36)

(2.3.37)

'l'hc cocllii:icnt ('(1) is clclcrtttittetl on llrc lrlrsis <ll's(atistical stuclics of windsPcctl rccorcls. llcsrrlts tll'sur'lt slu(licH w('t(' rclx)tlcrtl l-ly I)urst l2-9 ll anrl arc

U,(2,) : U,,,,n,(.) (l

70 ilil n tMofit't il til(: n()t,Nt)nily tAyt il

100r (s)

1 0,000

l"l(illl{l'l 2.-1,10. It:rrio o| pnrbablc maximum speed averaged over period r to thatluvcragctl ovct' ()llc lxrur l2-921.

plotted in Fig. 2.3.10, which corresponds to open terrain conditions (zo = 0.05m) and an elevation z : l0 m. values of c(t) consistent with Fig. 2.3.10 arelisted in Table 2.3.3.Experimental results presented in 12-931suggest that Eq. 2.3.36 is appli-

cable, with the values of the coefficient c(t) of rable 2.3.3, to wind ,p""0,over terrains with roughness lengths of up to Zs : 2.50 m.

Mean speeds used in the design of tall buildings are hourly averages, whileinformation on wind intensities is currently provided in terms of fastest milewind speeds at about l0 m above ground in open terrain. Fastest mile windspeeds are averaged over the time required for thi passage over the anemometerof a volume of air with a horizontal length of one mile. From this definition itfollows that for the fastest mile u|the averaging time in seconds is r : 3600/UJ, w^here Uyis given in miles per hour. For eiample, if UJ.: 90 mph, thent :40 s and the corresponding hourly mean is, from Fig. i.z.to,9ofl.2g =70 mph (31 m/s). A recent study 12-1441 essentially.o-nfi.-, the validity ofFig. 2.3.10. For hurricane winds see Sect. 2.4.3.

2.4 HORIZONTALLY NONHOMOGENEOUS FLOWS

Horizontal nonhomogeneities of atmospheric flows may be ascribed either toconditions at the Earth's surface (e.g., changes in surface roughness, topo-

TABLE 2.3.3. Coefficient c(t)llt:(r) 3.00

l0 202.32 2 (X)

30 -50r .71 r .35

100 200 3(x)1.02 0.70 0.54

600 1000 36000.3(r 0. l(r 0.(X)

1.5

81.45] t'.

1.2

l_1

1.0

:,,1 ll()1il/()NtAt lY N()Nll{}M()(it Nt ()U:i lt{)w:; 71

glirllhic lcirttttcs ol llrc tt'rr;rirr) or'1o tlrc nrclcorrlogit'lrl rurlrrr.t'ol tlrc llow (asirr tltc cuso ol'ttrrpit'irl tyt'krrrcs rlr ol'thurtrlr-:rstorrrrs). Wlrilc thc structure ofllrrizonlally honrogcrrr:orrs llows is basically wcll rrrrtlclsltxrcl, rcsults obtainedin thc study ol'horizorrtully rxrrrhonrogcncous llows urc lo a large extent stillirrcornplctc or tcntalivc. Sornc of these rcsults arc, ncvortheless, of interest tolhc designer and will thcrcfbre be discussed hcrcin.

2.4.1 Flow near a Change in Surface Roughnessln the case dealt with in the preceding sections, of a horizontally homogeneouslkrw, it is assumed that the surface roughness is uniform over an infinite plane.In reality, a site is limited in size; the flow near its boundaries is thereforerrll'ccted by the surface roughness of adjoining sites.

Useful information on the flow structure in the transition zones may beohtained by considering the simple case of an abrupt roughness change along:r line perpendicular to the direction of the mean flow [2-13, 2-94,2-95, 2-96,) 9l ,2-981 (Fig. 2.4.1). Upwind of the discontinuity, the flow is horizontallylromogeneous and, near the ground, governed by the parameters Zs1 &\d u.a1.l)ownwind of the discontinuity, the flow will be disturbed over a height h(x).'f 'his height, known as the depth of the internal boundary layer, increases with(ho distance x until the entire flow adjusts to the roughness length zs2 of thelcrrain downwind of the discontinuity.

If the investigation is limited to the lower portion of the boundary layer, itrrray be assumed that the flow is two-dimensional. For steady flow, and ne-glccting the pressure gradient force-the effect of which was shown to beinsignificant [2-98]-the equations of continuity and of balance of momentanray be written as

ua+w9!:taJ0x 0z pAzAU AW-+-:00x 0z

(2.4.1)

(2.4.2)

Since Eqs. 2.4.1 and2.4.2 contain three unknowns, a third equation is requiredIo close the system. In the solution of [2-96] the mean turbulent field closurewas used (F,q.2.1 .9), which, for two-dimensional flow and with phenome-rrolrrgical relations similar to those proposed in 12-91 and [2-10] (see Eqs.l. I . l0-2. l. 13) takes the form

II---J1125

--{ " fii,{czo1, u 11

l,'l(Jtll{lt 2.4.1. l;low zorrt's rlowrrwrrrl ol rr t'lurrrgc irr loLrghncss ol'lcr-rlin.

llll n tM()lit'l il til(: tl()t,Nt)nny tn yl n

in which l, is the mixing length.In horizontally homogeneous flow, the validity in the surface layer of the

Iogarithmic law implies the following expression for the mixing length [2-l]:L : kz e.4.4)

([ o l) Ii,ll.wirrg M.nin 12-991, ir is assumed in [2-96] that in Eq. 2.4.3rlrt' srunt' cxPressiorr lirr /. holils ncar the ground throughout the flow, inituding(lrc rlistrrrlrt'tl lkrw tlowrrwind ol' thc discontinuity.

'l'lrc lrorrntliuy corrtliliorrs lilr Eqs. 2.4.1-2.4.3 are

;, ,t ll( )lit,/( )N tAl I y N( )Nt t( )M( )( ,t Nt ( )llt; I I ( )wl; 73

describccl irr lclrrrs ol llrt'I;u;rnr(:l(:r's io2, u*2. For practical purposes it may beassumed that ( l) tlrc Plrlilt' t'ollt'sporrcling to these parameters is completelyestablished at distanccs ol rrrorc llrrrn -5 krn downward from the roughnesschange, (2) for a distancc dowrrwirrtl ol' thc roughness change of less than 500m the profile is the same as upwind ol'the discontinuity, and (3) in the interval500 m < "r < 5 km the profile is logarithmic below line AB, with zero speedat the ground surface, and a speed at elevation x/12.5 equal to the speed atthat elevation upwind of the roughness change t2-421.

A more "exact" model of the internal boundary layer growth is

(2.4.10)

where zs, is the larger of zs, and zo, [2-100]. Equation z.4.lo was based on theanalysis of a considerable number of data and holds for both smooth-to-roughand rough-to-smooth transition. It is approximately valid for values h(x) <0.26 where 6 is the boundary layer depth. For additional references on flowsnear a change in surface roughness, see [2-100] and [2-138].

2.4.2 Wind Flow over Hillswind tunnel investigations of simulated flows over ramps and escarpments arereported in [2-101, 2-102,2-1031. For open terrain conditions, ratios (u2lu)2at various stations given in t2-1011 are represented in Figs. 2.4.2 and,2.4.3.(uz and u1 denote wind speeds at height z above ground downwind and upwindof the ramp, respectively.) Measurements of l2-lo2l tend to corroborate theseresults. The results of [2-101] and 12-1021also suggest that for ramps withslopes of about 2O% to 35%, the ratios (J2l()1 are, for practical purposes,independent of slope. However, for a ramp with a l0% slope, the ratios (U2- U)lUl are only about one-half as large as in the case of a 2O% slope[2-101]. More detailed wind tunnel measurements of ratios u2lu1 for escarp-

U A(rlpt w dtrlpl0.t6 0x -o.to a,

_ritll , I (rtp il(rtptfitlt\p oz az \o.ro az I az )

(rlol3/2- ', -n- L :0 (2'4'3)

h(x) : o.28zo,(*)'-

I

I

'l

i

i

ii

ii

l

i

L

I

I

I

I

I

l

L

l

ri

IJ : 2.5u*t ln LZot

w:0r : puz*t

U:Ow:o

1,"-,

(2.4.s)

(2.4.6)

(2.4.7)

(2.4.8)

(2.4.9)

l., : olo o,. (#),,,1')0;z:zozi)

(see Eqs. 2.1.7 -2.1.8).Equations 2.4.1-2.4.3 with the boundary conditions, Eqs.2.4.5_2.4.9, werc

solved numerically in [2-96) for various values of the parameter m : ln(261/zoz). In the case of the smooth-to-rough transition, the calculations indicate thatthree regions may be distinguished downwind of the discontinuity (Fig. 2.a.D.In region I (above line AB, approximately defined by a slope or t : iz.5;, thevelocity is essentially equal to the velocity upwind of the discontinuity. Thisresult is consistent with conclusions reached independently by other authors12-941 and 12-95). In region III (below line AC, defined by a slope of aboutl: 100) it may be assumed, at least very roughly, that the flow is adjusted tothe new roughness conditions, that is, is determined by the same parametersZoz, u*z that would control the flow if the roughness length were everywhereZs2. rn region II, as the distance downwind from the discontinuity increases,the velocity profiles deviate increasingly from the profile given by Eq.2.4.5and the turbulcnc:c cncrgy varics graclually l'rom linc AB, whtre it is prcsunrablyncarly lhc sillllc:ts tlpwitttl ol'lhc tliscontirrrriiy, lo lirrc,4(1, wlrcrc it lrrtry bc l'l(;tll{l,l 2.4.2. Wirrrl lrtoltlt:, r)\'( r ;ul (':i( iul)nr('nl ll lOll

74 iltf ArM(xil,ilf nto t(tt,Nt)Any tAyt tt

FIGURE 2.4.3. Wind profiles over an escarpment t2_l0ll.

ments with 25%, 50%, and loo% slopes and for a cliff, as well as measure-ments of the root mean square of the longitudinal turbulence fluctuations, arereported in [2-103]. The ratios urlu, of t2-1031 are similar to those of Fig.2.4.2, except at low elevations (about 5 m above ground) where they are largerby about20%.

Results of theoretical and numerical studies of wind flows over hills havebeen reported in [2-lM, 2-105,2-106,2-lo1 ,2-108]. For a hill with maximumheight ft, a longitudinal scale L(L >> /r) and a profile hf(xlL), wheref(xlL)< 1 (Fig. 2.4.4), the following resulr was obtained in [2-1041:

U,_-_:: .l *ulho ln2(Llz$tr(o)(", z) (2.4.11)

in which U2 is the wind speed at (x, a), U1 is the wind speed at (x : -@, z),ze is the roughness length, x is the horizontal distance (see Fig. 2.4.4), z is theheight above surface of the hill at the point considered, il(0) is the approximatevalue of a dimensionless quantity representing the perturbatlon to the upwindvelocity due to the presence of the hill,

Lln(llz) rLn(zll) + ln(//zo)l

li'l(Jllltl,l 2,4.4. l,rolilc: ol l low lrill.

'4 ll()|il,/oNlnl lY N()Nl t()Ml ,{it Nt (}(,li |()wl;

lnd

(2.4.t2)

(2.4.13)

(The quantity / is the thickness of the intemal boundary layer created by thechange in surface shear stress as the air flows over the hill. This internalboundary layer is similar to that caused by changes in terrain roughness.) Forany hill symmetric about x : Q, ;(0) can be expressed in terms of Kelvinlunctions as shown in [2-104]. In the particular case

t f" .l 't.ttt,\lt.\lt ,l,, J ,, elt,l

/x\ Ir\z) : T +1ld

I /L\""//.r : g \;/

(2.4.t4)

in which L is the horizontal distance from the top of the hill to the point atwhich the height is half the maximum height ft, the quantity o : l. Values ofl?(0) corresponding to the profile 2.4.14 are represented in Fig. 2.4.5 atxlL:0 (top of the hill), xlL: -0.5 and xlL:0.5, for llzo: 1gz, Llzo:2.1 tlOa (curves A), llz(): 10a, Llzo : 3.2 x 105 (curves B), and llzo : 1gs,Llzo:3.6 x 106 (curves C). Values of t(U2 * U)lUl (L/ft) calculated inl2-l04l are listed in Table 2.4.1. The analysis and results of [2-104] are valid

- 01lL

l,'l(;tll{U 2.4.5. Vulucs ol r?("'f ;low Ovcr a Low llill." Qrtrtrt

0 o4 08;(.))_>. -0.2-0.1 6 iio)e, " )=ou

l;rortr l' S .l;rt'ksotr:rtttl .1 . ('. ll. lltltl. "'l'urbulcrrrt.lt,ut llrtvtl ll,'1,'r,trtl . ,lttt.. ll)l ( 197.5). 929 ().55,

76 llt n tM()sl,lFilto n()(,Nt)Ally tAyt n

TAIII-|t 2.4.1. Values of l(U, - Il)lIltl(l,lltl il 'l'op ol'lliilzll zull : 10 ' ,.t/l - ft) 4 ;,,// - l0 5

0.00.10.30.61.0t.52.1

2.092.462.332.202.O81.9'7l .88

1.872.132.071.971.871.791.73

t.721.92l .851.78t.721.66t.62

lirl hills in rural tcrrain (zo : 0.03 m) with 0.1 < Z < l0 km and with ratioshll, > l(1,,/1,)0 '. For cxample, if a6 : 0.025 rr7, L :500 m, and h : 25 m,tlrcrr, lirrrrr |tq.2.4.13,2.111 l: 1.0 x l0 3, to which there corresponds, from'l'ablc2.4.l, UzlUt: l.l2atzll:0.1 (or z = 2.5 m). Thetheorybecomeslcss accuratc in rough terrain (zo : 0.5 m), the actual speeds U2 being lowerthan thosc givcn by Table 2.4.1.

For flow over escarpments (Fig. 2.4.6), the following relation is derived in[2- lOe]:

!:=r+!!1n(LtzdhUt L 4r ln(zlzi "'(zlLi)2+U+(x/L)12klL\2+.1-(xtL)12 (2.4.rs)

in which notations similar to those of Eq. 2.4.1r are used. It is suggested in12-1091that Eq. 2-4.15 may be applied to flow over escarpments with t <<5 km and with slopes as large as 202" or so. For example, if L : 250 m,h:50 m, and zo:0.025 m, for x: L andz: l0mthe ratio U2lU,:1.19. According to [2-1031, Eq. 2.4.15 provides useful indications of the trendsof the variation of u2lu1 with x and z, rather than dependable quantitativeresults.

Full-scale and wind tunnel measurements of flows over two- and three-dimensional hills and over embankments are reported in [2-l0g] (which extendsthe analytical approach of [2-1041ro three-dimensional hills), and [2-tlo,2-lll,2-112, 2-113,2-114, 2-115,2-l4zl. As noted in [2-ll0], estimates obtainedindependently in l2-lo4l, [2-105], and [2-106] agree well with each other andwith the full-scale measurements of [2-110].

l,'l(Jlllll,) 2.4.(r. lrlow ovL:t' cscitrpnlcllts noltrlions l2 1091.

\l IYN()Nl l()M()(il Nt t)t,i; lt()Wl; 77

A sirtrplc: ttterlltrxl lot r'itlt'ulrrlirrg wirrtl spr:ctl irrt.r't.:rscs ( "spectl-ups") lirrbuildings l<lcalctl ()ll lw() tlitttt'ttsiotral ridgcs or cscarl)nrcnls or orr axysinrrnctrichills is includccl irr llrc AS('lj 7-9-5 Stancl:rrcl 12-l'3gl iintl, in corrrputcrized fbrm,as parl of thc cliskcilc "l)cvckrpmcntal cornpu(cr-llascrl Vcision of ASCE7-95 Standard Provisi'ns krr wind Loads" [17--51 appcnded to this book.

2.4.3 The Hurricane Boundary LayerThe horizontal inhomogeneity of a hurricane wind flow over a uniform, hori-zontal surface is associated with the variation of the pressure gradient withdistance from the centerof the storm (see Eq. 1.3.1). tn aerivinfthe logarith-mic description of the mean velocity profilei near the ground ftq. Z.Z.fS; itwas assumed that the flow in the free atmosphere is geostrophic lSect. 2.2).This assumption does not hold in the region of highest winds of the maturehurricane; the question therefore arises as to wheiher or not Eq. 2.2.1g isapplicable in this region.

Several analytical solutions of the hurricane boundary{ayer problem havebeen atempted so far [2-116,2-117,2-llg,2-llg, z-l2iJ], att oi wnicn apptyto steady, axisymmetric mean flows. The solutions of [2-116] through p lirjlare based on the assumption that the eddy viscosity is constant, and they .unnoitherefore provide a reliable detailed description of the flow near the ground. Aconsiderably more realistic modeling of the turbulence effects is used inl2-l2ol, in which the equations of motion and continuity are supplemented bythe turbulence closure relations discussed in Sect. 2.1 (Eqs. 2.1.g-2.1. 13). Th;system of equations thus obtained-in which the expre.ssion for the pressuregradient field given by Eq. 1.3.1 was used-was solved numerically assumingvalues of the surface roughness of 0.002 m to 0.90 m, differences between thehigh pressure in the far field and the low pressure at the storm center of 60 mbto 140 mb, and radii at which the gradient wind has a maximum value of 30km to 50 km. According to 12-1201, in the lowest 400 m of the boundary layerthe mean wind profiles differ only insignificantly from the logarithmic profilesdescribed by Eq. 2.2.18.

As Table 2.4.2 shows, for decaying hurricanes the increase of mean wind

TABLE 2.4.2. Yariation of wind speeds with Height in Hurricanes caror andEdna

Heightabove Ground

(m)Carol Edna

Mean Max. l-rnin Mean Max. 1-minI 1.322.945.7

r08.2t25.(l

t4.s18.124.129.1

22.8l(). I

t58l.t tl

r t.8

20.3

259

17.0

25.9

30 rJ

78 IltF ATMospHEntc frot,ND^ny iAyt n

spceds with hcight in appnrxintalc ircconlirrrcc wilh tho logarithnric l1w w1sdocumented in 1954 following thc passagc ovor lJnxrkhaven National Labo-ratory of hurricanes Carol and Edna 12-121, p. 461.

More recently [2-122] reported extensive observations of mean wind speedsrecorded at elevations from 9. I m to 390 m during the passage of four decayingtropical cyclones over northwestern Australia. The mean wind profiles were inmost cases irregular, and as noted in [2-1221, a wind speed maximum wasoften observed at 60-200 m. Nevertheless, the profiles corresponding to thelargest l0-min wind speed observed during each storm at 9.1 m were by andlarge consistent with the logarithmic law and a roughness length of 1 to 4 cm,as can be seen in Table 2.4.3, in which the only significant anomaly is thespeed observed during cyclone Karen at 59.7 m elevation.

whether the logarithmic profile holds in the case of mature hurricanes re-mains an open question. Implicit in the provisions of the 1975 Southern Build-ing code 12-1231is the assumption that hurricane wind profiles are considerablyflatter than would be indicated by the logarithmic law. To date there is noconclusive evidence that this is the case. Since design wind speeds specifiedin building codes correspond to an elevation of l0 m or so, the use of thisassumption in the design of tall structures might be imprudent.

According to a study of tropical storm and hurricane records peak gustfactors for hurricane speeds are about lo% higher than indicated in Fig. z.3.lofor extratropical storms t2-1351. The conclusions of 12-1351(see also Iz-1391,p. 155) were based on the analysis of about 12 records.

Reference !2-l4ll contains information on gust factors, longitudinal turbu-lence intensities, scales and spectra in typhoon Mireille, that traveled overomura bay (Nagasaki) and passed directly over the anemometer placed at100-m elevation on a tall building at the shoreline. The gust factors were foundto decrease as the mean speed increased. The turbulence intensity during theperiod of the strongest 10-min wind (about 25 mls) was over 25%. Thetur-bulence scale was estimated to be 780 m during that period and 2g0 m duringthe 10-min period preceding it. The von Kdrm6n spectrum (Eq.2.3.24), with

TABLE 2.4.3. l0-min Speeds at Various Elevations Corresponding toMaximum 10-min Speed at 9.1 m during Four Tropical Cyclones

Wind Speed (m/s)Heightabove Ground

(m)Beryl

(12173 12:00)Trixie Beverly Karen

(2175; 18:30) (3175:2r:O0) (3177; t90O)9.1

59.719t.4279.2390. r

32.539.541

.57..5

30.55l43.548.s4ti..5

21

32

36

222831

34

Noto: Nurrrbcrs ilr pirrcrrlltcscs irulit'irtc tltc rrronth, ycar, trrrrl lrorrr ((iM'l') f,'l(;tlRl,l 2.4.7 , I lrrrricrrrrr. wirul sllcctl reconl.

HoFtzoNTAt I y N( )Nt t( )M( x lt Nt ()t,s I I ow$ 79

thc lurbulonco scitlc:s.ittsl irrtlicatotl, rnatchctl thc rncirsul'ctl spcctra wcll, cxceptlirr thc rangc ol'abrlut 0.o25 to 0. l5 Hz, whcrc it undcrcstimate<l the measuredspectra by as much as l0]ol, fbr certain frequencies. Finally, according tol2-l4ll, surface wincl spccds in the eye can be comparable with or higher thanthe estimated speeds at the gradient height level; see also comments followingEq. 3.3.7 and Ref. 13-791.

Two more notes on hurricane winds are in order. First, in the immediateproximity of the eye, flow separation occurs and the boundaryJayer assump-tions break down (see Sect. 1.3). The implications of this phenomenon to thedesigner are not yet well understood. Second, as the hurricane moves inland,filling occurs (see Sect. 1.3) and the maximum winds tend to decrease. Em-pirical descriptions of the wind intensity reduction as a function of distancefrom the coastline were proposed in 12-1241, [2-1251, and [2-l5l]. Accordingto [2-1251, the ratios of peak gusts at 50 km, 100 km, and 150 km inland topeak gusts at the coastline are, approximately, 0.90, 0.80, and 0.70, respec-tively. See also [3-57] to t3-601.

A hurricane wind speed record, which clearly indicates the passage of theeye, is shown in Fig. 2.4.7. The nonstationary character of the record of Fig.2.4.7 is noteworthy, as is the contrast to Fig. 2.3.1. For techniques to char-acteize turbulent fluctuations for nonstationary records, which are typical ofhurricanes but characterize other storms as well, see [A2-14] to [AZ-241.

2.4.4 Thunderstorm WindsThe cold air flow which, in a thunderstorm, spreads horizontally over theground was compared in Sect. 1.3 to a wall jet. Just as in the case of the walljet, the surface friction retards the spreading flow, which may thus be expectedto be similar, near the ground, to an ordinary boundary layer [2-126, 2-127,2-1281.

of particular interest to the designer is the so-calledfrst gust (or gustfront),that is, the wind occurring in a thunderstorm that exhibits a considerable andrelatively rapid change of speed and direction (Fig. 2.a.8). Following l}-l}9land [2-130], the wind speed increase and the time interval during which this

MDIIT l0 Plrl

I lll n I M( ): it ,l ll lll( I ll( )l,Nl )/\t ty I n., I I I

c. l77m f. 444m

b. 90m e. 355m

a. 45m d.266mFIGURE 2.4.8. Thunderstom wind speeds recorded simultaneously at six elevationsfrom 45 m to 444 m above ground near Oklahoma City (courtesy of National SevereStorms Laboratory, National Oceanic and Atmospheric Administration).

increase takes place will be referred to as the gust size A,V and the gust lengthAt, respcctively. Depending upon thunclerstorm intcnsity, thc gr-rst sizc rnuyvary appnrxirtr:t{cly l'roltt S ltt/s to 30 rtr/s, whilc lhc gtrst lengtlr nuly rangcll'otn ir l'cw rrrirrulcs (o ?0 rrrirrrrlt's ()r-s().

'l'lrc llrtlllrlt'r'slotttt wiltrl tt'trtllls n';xrr'lt'rl irr l.) | lOl r;rr1'1'r':;t ilrrrl rlrlrirrl'tltt'

n lMr '"t ilI t{tl . tt()t ,Nt)nl ty lnyl it tllt( t:;{)il ()|t nlJ lt()w Bl

itttc:rvitl A/: (l) rr;t lo l(X) rrr :lltovt' gtrrrrrrtl wirrtls slrt't'tls v:rry wrllr lrt:iglrt irraccortlancc witlr llrt'1o1,:rrrlltrrric lrrw, urrtl (l)) irlxrvr' l(X) lrr llrt'vtrr-ilrti1ll glwincl spccds witlr hc:iF,lrt is Ircgligiblc. ('t'his is rerrsolrirlrl-y t'orrrplrliblc with therccords of Fig. 2.4.8.) Ntl rolation betwccn wirrtl spcctls in tlillorcnt rclughnessregimes, based on a rational model of the thunclcrstorrrr wind lklw, has beenderived so far. To convert thunderstorm wind spccds rccordcd over open terraininto wind speeds over built-up terrain, the samc procedure is used in currentpractice that is applied to extratropical cyclone winds (Eq. 2.2.26), even thoughthe notions of gradient height and gradient speed have no meaning in the caseof thunderstorms. whether or not this practice is acceptable for structural en-gineering purposes is a question that merits investigation, particularly if it isrecalled that, according to [2-131], about one-third of the extreme wind speedsrecorded in the United States are associated with thunderstoms.

2.5 ATMOSPHERIC BOUNDARY LAYER EFFECTS ON OCEANFLOW

If suflicient data on wind flows over the ocean are available, it is possible tomodel the ocean waves induced by those flows. Such modeling is referred toas hindcasting. A vast literature on this topic is available (e.g., see [14-34,p.5201).

Mechanisms by which kinetic energy is transmitted from the atmosphericboundary layer to the ocean water are exceedingly complex. In some appli-cations one may assume, however, the existence of uniform surface shears ata hypothetical horizontal ocean/atmosphere interface (see Sects. 2.2.2 and2.2.3). one among many instances were this assumption is used is the recentmodeling of wind-induced along-shore ocean currents over bottom topographycharacterized by comrgations normal to the shore. It was shown in [2-145] thatthe equations of motion of the wind-induced ocean motions can be representedapproximately by the equations

a* : AyH(x, y, z) -t eg{x, t)

di : - u* H(x. y.;) * cg2())

2 : eg(x, z, t) (2.s.D

where e is small, x is a basic along-shorc specd, y is proportional to the out-ol-phase component of a strcarn lirnction lilr rnoli<tn due to the topography, zis thc cnergy-cnstropy, ancl

,(r = ,:r t r1y I t\tl.,tj, ,.\'. ,(r ,. 1,, I {r l)1r,, I tltll (l-5 l)

82 lltt AIM(xitlll tito tlot,Nt)nny tAvt n

H(x, y, z) : j.y" + zx t ,l(r,lil r:).*' 1t'r1l;xn, ,1; = I tD'l(2.s.3)

and where 6 is the amplitude of the bottom topography comrgations, er is afriction coefficient related to the eddy viscosity of the ocean flow, and €ze ander(t) arc, respectively, the steady and fluctuating wind stress at the oceansurface. The wind stress fluctuations of interest in this problem correspond tothe very low frequencies studied in [2-1431. Reflecting the effect of the bottomcomrgations, Eqs. 2.5.1 form a bistable system capable of chaotic behavioreven if the fluctuating wind stress is assumed to be harmonic, as was done in12-t451.

The model used in [2-145] becomes more realistic if it is assumed that thewind speed fluctuations are random, rather than harmonic. Using chaotic dy-namics techniques developed in [6-101], and van der Hoven's results on thespectra of low-frequency wind speed fluctuations [2-143], the case of randomwind excitation was studied in [2-146]. Among other results, [2-146] providesestimates of lower bounds for the probability that during a specified time in-terval, the amplitudes of the wind-induced fluctuating currents do not exceeda safe threshold associated with the barrier of the unforced system's doublepotential well.

REFERENCES

2-l H. Schlichting, Boundary Layer Theory, McGraw-Hill, New york, 1960.2-2 L. T. Matveev, Physics of the Atmosphere, TT 67-5 1380, National rechnical

Information Service, Springfield, VA, 1967.2-3 J. L. Lumley and H. A. Panofsky, The stucture oJ'Atmospheric Turbulence,

Wiley, New York, 1964.2-4 H. A. Panofsky and J. A. Dutton, Atmospheric Turbulencc: Models and Meth-

ods for Engineering Applications, Wiley, New York, 1984.2-5 A. s. Monin and A. M. Yaglom, statistical Fluid Mechanics: Mechanics of

Turbulence, Vol. 1, MIT Press, Cambridge, 1971.2-6 J. F. Nash and V. C. Patel, Three-Dimensional Turbulent Boundary lnyers,

S.B.C. Technical Books, Atlanta, 1972.2-1 Computation of rurbulent Boundary rrtyers, proceedings of the AFOSR-IFp

Stanford Conference, Vols. I and 2, Stanford Univ., 1968.2-8 A. A. Townsend, The Structure of Turbulent Shear Flow, Cambridge Univ.

Press. Cambridge, 1955.2-9 P. Bradshaw, D. H. Ferris and N. P. Atwell, "Calculation of Boundary Layer

Developmcnt Using thc Turbulent Energy Equation," J. Ftuid Mech., 2811967;.59.1 6l(r.

2-10 J. li. N:rslr, "'l'ltc ('rrlctrl:rliott ol''l'hrce-l)irncnsional 'l'urbulcnl Bourxlluy l,ly-crs itt lnt'ontplt'ssilrlt. lilow," .1. l;lttitl Mtclt.,.l7 (l()69), 625 M2.

nt I I nt N(it l; 83

2ll (1. dul). l)orurltlsorr, "('lrlculatiort ol"l'ullrrrk:rrl Slrclrr'likrws lirr Atrnosphcricancl Vollcx Moliorrs," AlAA.l ., 10, I (Jan. 191')1,4 12.

2-12 J. L. l,urrrlcy rrrrrl li. Khajch-Nouri, "(lrnr1'rrrt:rlionll Mrxlcling of TurbulentTransport," Adv. Oct4thys., l8A, Acadotnic, Ncw York (1914), 169-192.

2-13 K. S. Rao, J. C. Wyngaard and O. R. Cotd, "The Structure of the Two-Dimensional lnternal Boundary Layer over a Sudden Change of Surface Rough-ness," "/. Atmos. Sci., 31, 3 (April 1974),738-746.

2-14 G. T. Csanady, "On the Resistance Law of a Turbulent Ekman Layer," J.Atmos. Sci., 24 (Sept. 1967),467-471.

2-15 S. R. Hanna, Characteristics of Winds and Turbulence in the Planetary Bound-ary Inyer, Technical Memorandum No. ERLTM-ARL 8, ESSA Research Lab-oratories, Oak Ridge, TN, 1969.

2-16 G. B. Schubauer and C. M. Tchen, Turbulent Flow, Pinceton Univ. Press,Princeton, NJ, 1961.

2-17 C. B. Millikan, "A Critical Discussion of the Turbulent Flows in Channelsand Circular Tubes," in Proceedings of the Fifth International Congress ofApplied Mechanics, Cambridge, MA, 1938.

2-18 A. E. Gill, "Similarity Theory and Geostrophic Adjustment," J. Royal Me-teorol. Soc., 94 (1968), 586-588.

2-19 R. H. Clarke, "Observational Studies in the Atmospheric Boundary Layer,"J. Royal Meteorol. Soc., 96 (1970), 9l-114.

2-20 E. J. Plate, Aerodynamic Characteristics of Atmospheric Boundary ktyers,U.S. Atomic Energy Commission Critical Review Series, 1972.

2-21 H. Tennekes and J. L. Lumley, A First Course in Turbulence, MIT Press,Cambridge, 1972.

2-22 E. Simiu, "Logarithmic Profiles and Design Wind Speeds," J. Eng. Mech.Div., ASCE,99, No. EM5, Proc, Paper 10100 (Oct. 1973), 1073-1083.

2-23 E. L. Deacon, "Geostrophic Drag Coefficients," Bound Inyer Meteorol., 5,3 (July 1913),321-340.

2-24 A. K. Blackadar (penonal communication, Feb. 1976).2-25 J. A. Businger et al. "Flux Profile Relationships in the Atmospheric Surface

Layer," J. Atmos. Sci., 28 (1971), 788-794.2-26 H. Tennekes, "The Logarithmic Wind Profile," J. Atmos. Scl.,30 (1973),

234-238.2-27 J. Wyngaard, "Notes on Surlace Layer Turbulence," in Proceedings of the

American Meteorological Society Workshop on Micrometeorology, Boston,1972.

2-28 D. R. Caldwell, C. W. Van Atta, and K. N. Helland, "A Laboratory Studyofthe Turbulent Ekman Layer," Geophys. Fluid Dyn.,3 (1912),125-160.

2-29 G. C. Howroyd and P. R. Slawson, The Characteristics of a Laboratory Pro-duced Turbulent Ekman l,aycr," Iilnnd- Ityer Meteorol., 8 (1975), 201-219.

2-30 R. H. Thuillier ancl V. O. L:rp1x'. "Wirll :rntl 'fcmperature Profile Character-istics l'rorn C)bservations orr rr l,lO0 li'lowcr," .1. Appl. Meteorol..,3 (Junet9(A),299 306.

2-31 A. K. lllackatlaruntl Il.'l'crutt'Lcs.'Asyrrrplolit Sirrrilulity in Ncutnrl Banrtnrpicllotrttllrt'y l.lryct's," .l . tllrrt,t,r. ,\', r 15 (Nov l(X)li), lOl5 lO2O.

ilil n tM(): :t't il lll( . lr()t,Ntr/\ny tn itll

l .l-l l). M. ('ltr'1, 1. ('. 1';rllrt'll lrrrrl ll A l':rtt,rlr;1.y. 'l'rrrlilcs ol Wrrul :rrul lt.rrrf)L:ritltllL: l.nrltt 'lowcls tlvt'r lloilrol'tnt'orrs lt'rr:tr'r,.1 . tlltttl,s.,\,i., -\0 l.lrrly1912t.188 194.

2-33 N. Cl. Halliwcll, "Wintl ()vcr l.otttlotr. it I'nx'rulings ol thc 'l'hird Intcrnrttional Confcrcnrc oLt Wind lil.li'r'r,s ott lhtiltlirt,qs und Strudurc.r, Tokyo, 197 l,Saikon, 1972, pp.23 32.

2-34 E. L. Deacon, "Gust Variation with Hcight up to 150 m," J. Royal Meteorol.Soc., 91 (1955), 562-513.

2-35 R. I. Harris, "Measurements of Wind Structure at Heights up to 598 ft AboveG round Level , ' ' in Proceedings of the Symposium on Wind ElTects on Buildingsand Structures, Loughborough University of Technology, Leicestershire, U.K.,l96rJ.

2.16 O. G. Sutton, Atmospheric TurbuLence, Methuen, London and Wiley, NewYork. 1960.

2 l1 P. S. Jackson, "On the displacement height in the logarithmic velocity profile,".t. F.luid Mcch., lrr (1981), 15-25.

2 3u A. C. Chamberlain, "Roughness Length of Sea, Sand, and Snow," Bound.Inyer Meteorol., 25 (1983), 405-409.

2-39 G. E. Daniels, (ed.), Tenestrial Environment (Climatic) Criteria Guidelinesfttr Use in Space VehicLe Development, l97l Revision, NASA Technical Mem-orandum TM X-64589, George C. Marshall Space Flight Center, Ala. (1993Revision edited by D. L. Johnson.)

2-40 H. R. Oliver, "Wind Profiles in and Above a Forest Canopy," J. Royal Me-teorol Soc.,97 (1971),548 553.

2-41 P. Duch6ne-Marullaz, "Full-Scale Measurements of Atmospheric Turbulencein a Suburban Area," in Proceedings of the Fourth International Conferenceon Wind Effects on Buildings and Other Structures, London, 1975, CambridgeUniv. Press. Cambridge, 1976, pp.23 31 .

2-42 J. Bidtry, C. Sacr6, and E. Simiu, "Mean Wind Pr<rfiles and Changes of TerrainRoughness," J. Struct. Diy., ASCE, 104 (Oct. 1978), 1585-1593.

2-43 S. D. Smith and E. G. Banke, "Variation of the Sea Surfacc Drag Coefficientwith Wind Speed," J. Royal Meteorol. Soc., l0l (1975), 665 673.

2-44 J. Wu, "Wind Stress and Surlace Roughness at Air-Watcr lnterflace," J. Geo-

2-45ph1,s. Res., 74 (1969) 444-445.J. Amorocho and J. J. deVries, "A New Evaluation ol'thc Wind Stress Coef-ficient over Water Surfaces," J. Geophys. Res., 85 ( l9U0), 433-442.

2-46 J. R. Garratt, "Review of Drag Coellicients over Oceans and Continents,"Mrnthly Wearher Rev., 105 (1977),915-929.

2-4'7 G. Hellman, "Uber die Bewegung der Luft in den unterstcn Schichten derAtmosphire, " Meteorol. Z. , 34 (1916) 273.

2-48 A. G. Davenport, "The Relationship of Wind Structure to Wind Loading,' inProceedings ol the Symposium on Wind Ellects on Buiklings and Structures,Vol. l. National Physical l-ahoratory. Tcddington, U.K., Her Majesty's Sta-tioncry Ollict'. l.ortkrn. l9(r5, pyr.53 102.

2-49 Arru'ritttrt Nttliotrrtl ,\lttrtrhtnl A.5ll. I l9lt2, Mitrirrrtrtrr I)asigrt ltxul.s .litr llrtilditr,q.s trtttl ()llrr't ,\ltrtr'lrn.r'.r, Atttt'tit':rn N:rliornl Sllrnrlrutls llrslilrrlt'. lrrt., NcwYotk. l()i'i.)

lil I I ilt N(:t :

2 50 W. W. l':t1',,rtt, Wrrrrl Vclotily ilr llt'llrliorr to llt.ry'lrt Alxrvt'(ilrrlttl." /tir1.3.Nry',s /lr'r',. I l.l (M;rv l() 15)" /.ll 7,1.5.

2-51 li. l)lrstltrill, "Wirrtl Slrttt'ltttc irr thc: Alrttosplrt'rit' l}rrtrulruy Laycr." in A Dist'ussi.ott rttr At<ltitr'ttuntl A<nxl_vtrutrtics, l'ltil.'l'nut.s. lilty. Stx.. London, A269( 197 l ). 4 1() zl5(r

2-52 F. Pasquill, "Sonrc Aspccts of Boundary l,aycr'[)cscription," J. Royal Mete-orol. Soc.,98 (1912),469 494.

2-53 P. R. Owen, "Buildings in the Wind," J. Royal Meteorol. Soc.,97 (1974),396 413.

2-54 H. C. Shellard, "The Estimation of Design Wind Speeds, in Prot:eedings ofthe Symposium on Wind Effects on Buildings and Structures, Vol. l, NationalPhysical Laboratory, Teddington, U.K., HerMajesty's Stationery Office, Lon-don, 1965, pp. 30-51.

2-55 N. Sisserwine, P. Tattleman, D. D. Grantham and I. I. Gringorten, ExtremeWind Speeds, Gustiness and Variations with Height for MIL-STD210B. Tech-nical Report No. AFCRL-TR-0560, Air Force Cambridge Research Labora-tories, 1973.

2-56 J. Bi6try, personal communication, July 1976.2^57 S. Schotz and H- A. Panof.sky, "Wind Characteristics at the Boulder Atmo-

spheric Observatory," Bound. Inyer Meteorol., 19, (1980), 155-164.2-58 A. Venkatram, "Estimating the Monin-Oubkhov Length in the Stable Boundary

Layer for Dispersion Calculations," Bound. lnyer Meteorol., 19 (1980), 481485.

2-59 E. Simiu, "Thermal Convection and Design Wind Speeds," J. Struct. Div.,ASCE, 108 (July 1982), t67t-t675.

2-60 M. Shiotani, Structure of Gusts in High Winds, Parts l-4, The physical SciencesLaboratory, Nikon University, Furabashi, Chiba, Japan, 1967-1971.

2-61 J. Counihan, "Adiabatic Atmospheric Boundary Layers: A Review and Anal-ysis of Data from the Period 1880-1972," Atmos. Environ., g (19j5),8jl-905.

2-62 P. Duch6ne-Marullaz, "Effect of High Roughness on the Characteristics ofTurbulence in the Case of Strong Winds, Proceedings Filih Inlernational Con-ference on Wind Engineering, Vol. l, Pergamon Press, 1980.

2-63 J. O. Hinze, Turbulence,2d ed., McGraw-Hill, New York, 1975.2-64 H. W. Teunissen, Characteristics of the Mectn Wind and Turbulence in the

Planetary Boundary Inyer, Review No. 32, Institute for Aerospace Studies,University of Toronto, 1970.

2-65 N. E. Busch and H. A. Panofsky, "Recent Spectra of Atmospheric Turbu-lence," J. Royal Metartrol. Stx'., 94 (1968), 132 148.

2-66 J. C. Kaimal et al., "Spcctritl ('llrnrctcristics of Surlace-Layer Turbulence,"J. Royal Metertrol. &x.. 9lt ( l()7.1). 56] .5U9.

2-67 I. A. Singcr. N. E. llLlstlr :rrrtl .l A lirittttl't. "'l'hc Micnrrnctconrkrgy ol the'l.urbulcnt Flow lricltl ilt lltt'Alttt,':;plu'rr' llrtrrlllrry Surlircc I-aycr." in Pn>t'ccdings tt lltt lrtltrrrttlirtttrtl l('sr'ttr,lt .\,'rrtittttr ttt Wirrt! l,),llt,t't.s ort l)ttiltlin,q.sultl ,t;lrtt(tttr(,r, ()llltwlr. Vol |. Ilrrrlr'r:,rly ol I'olrrrlo Itlt'ss, lirrrrrlo, l()(rll,pp. .557 5()4.

tl6 ttlt AIM():;t't iltrt{.lourJr)nnyrnyr l

I (rtt li. liit'tllt'r rrrrtl ll. A. l':rtt,rlsl'y. t\lrrror,lrlrt'rrt Stlrles :rtul SPt'tlr:rl ( ilrPs."llull. Arrr. Mcltttntl. Jor'.. 51, Ll (lrtt l()/O). lll.l lll().

2-69 G. H. Ficlttl alt(l (;. li. McVr'lril. "l.ottlirltttlirt:rl rrrrtl Litlclal Spt:clnr ol 'l'ttrbulencc in thc Atrnosphcric llorrrul;u.y Liry('r'rrl llrc Kcnncrly Spacc (lcntcr, ./.AppL. Metcor. (Scpt. 1970), -5 I ().]

2-70 E. Simiu, "Wind Spectra and I)ynarrric Akrngwirrd Rcsponsc," J. Struct. Dir.,ASCE, 100, No. ST9, Proc. Papcr lOltl5 (Scpt. 1974), 1897-1910.

2-11 A. G. Davenport, "The Spectrum of' Horizontal Gustincss Near thc Ground inHigh Winds," J. Royal Meteorol. Soc., 87 (1961), 194-211.

2-72 Canadian Structural Design Manual, Supplement No. 4 to the National Build-ing Code of Canada, Associate Committee on the National Building Code andNational Research Council of Canada, Ottawa, 1971.

2-13 F. Pasquill and H. E. Butler, "A Note on Determining the Scale of Turbu-lence," J. Royal Meteorol. Soc., 90 (1964),19-84.

214 E. Simiu and S. D. Leigh, Turbulent Wind Effects on Tension Leg PlatformSurge, Building Science Series BSS 151, National Bureau of Standards, Wash-ington, DC, March 1983.

2-15 H. A. Panofsky and I. A. Singer, "Vertical Structure of Turbulence," J. RoyalMeteorol. Soc., 9l (1965), 339-344.

2-76 A. G. Davenport, "The Dependence of Wind Load upon Meteorological Pa-rameters," in Proceedings of the International Research Seminar on WindEIJ'eos on Buildings and Structures, University of Toronto Press, Toronto,1968, pp. 19-82.

2-71 B. J. Vickery, "On the Reliability of Gust Loading Factors," in Proceeding,sof the Technical Meeting Concerning, Wind Loads on Buildings and Structures,National Bureau of Standards, Building Science Series 30, Washington, DC,1970, pp. 93-104.

2-'78 M. Shiotani and Y. Iwatani. "Conelations of Wind Velocities in Relation tothe Gust Loadings," in Proceedings of the Third International Conf'erence onWind Elfects on Buildings and Stuctures, Tokyo, 1971, Saikon, Tokyo, 1972,pp.57-67.

2-79 S. SethuRaman, "Structure of Turbulence overWaterduring High Winds," -/.Appl. Meteorol., L8, (1979), 324-328.

2-8O L. Kristensen and N. O. Jensen, Lateral Coherence in Isotropic Turbulcncc andin the Natural Wind," Bound. Layer Meterol., 17 (1979),353-373.

2-81 L. Kristensen, H. A. Panofsky, and S. D. Smith, "Lateral Coherence of Lon-gitudinal Wind Components in Strong Winds," Bound. lnyer Meteorol., 2L(le8l). le9-205.

2-82 C. F. Ropelewski, H. Tennekes, and H. A. Panofsky, "Horizontal Coherenceof Wind Fluctuations,' Bound. Inyer Meteorol., 5 (1975),353-363.

2-83 S. Berman and C. R. Steams, "Near-Earth Turbulence and Coherence Mea-surements at Aberdeen Proving Ground, Md.," Bound. lnyer Meteorol., ll(1977),485 506.J. Kunrlu antl Il, Iloylc:s. "Fuflhcr Consitlcrrtion ol'lhc llcighl l)cpcntk:ncc oltlrc l{rrol ('oltt'r't'trt't' irr llrc Nllrrrrrl Wirrtl." Ilttiltl. l'.ttvit,,rr l.l (l()711). 175I tt'1

nt IIIu t{(.1 r, a7

I t'i-5 It. l'1 . llllrl , A N'lt 'tt Vr'tlicrtl ('oltt'r'r'ttct ol Wurrl Mt';rsrlr'tl irr lrn (Jr-banIlorrrttl:ry l.;r-yt'r. llttuurl. lrtt,tr Mtlttttttl ., t) 1l()/-\). 1.17.

2-tJ6 G. Il.. Sle:gerr rrrxl ll . l., l'lurrpc, "Vcrliclrl ('ohcrcncc in thc AtmosphericB<runtlary l,uycl," I'nx'ttlittgs F-ifih ltilcntutiorutl Conlcrence on Wind Engi-nccri.ng, Vul. I, l)clgarrxrrr, New York, 1980.

2-111 H. A. Panof.sky and'1. Mizuno, "Horizontal Coherence and Pasquill's Beta,"Bound. lnyer Meteorol., 9 (1915),247-256.

2-88 H. A. Panofsky et al., "Two-Point Velocity Statistics over Lake Ontario,"Bound. ktyer Meteorol., 7 (1974), 309-321.

2-89 L. Kristensen, "On Longitudinal Spectral Coherence," Bound. Inyer Mete-orol., 16 (1979), 145-153.

2-9O A. K. Blackadar, H. A. Panofsky and F. Fiedler, Investigation of the TurbulentWind Field Below 500 Feet Altitude at the Eastern Te.st Range, F/orida, NASACR-2438, National Aeronautics and Space Administration, Washington, DC,19'74.

2-91 C. S. Durst, "Wind Speeds Over Short Periods of Time," Meteorol. Mag.,89, (1960), l8l-186.

2-92 I. Yellozzi and E. Cohen, "Dynamic Response of Tall Flexible Structures toWind Loading," Proceedings, Technical Meeting Conceming Wind Loads onBuildings and Structures, Building Science Series BSS 30, National Bureau ofStandards, Washington, DC, 1970, pp. 115-128.

2-93 P. Sachs, Wind Forces in Engineering 2d ed., Pergamon, New York, 1978.2-94 H. A. Panofsky and A. A. Townsend, "Change of Terrain Roughness and the

Wind Profile," J. Royal Meteorol. Soc., 90 (1964), 147 155.2-95 P. A. Taylor, "On Wind and Shear Stress Profiles above a Change in Surface

Roughness," J. Royal Meteorol. Soc., 95 (1969), 77-91.2-96 E. W. Peterson, "Modification of Mean Flow and Turbulent Energy by a

Change in Surface Roughness under Conditions of Neutral Stability," J. RoyalMeteorol. Soc., 95 (1969), 561-515.

2-97 R. A. Antonia and R. E. Luxton, "The Response of a Turbulent BoundaryLayer to a Step Change in Surface Roughness, Part I, Smooth to Rough," -/.Fluid Mech. , 48, (1971), 121-161.

2-98 C. C. Shir, "A Numerical Computation of Air Flow over a Sudden Change ofSurface Roughness," J. Atmos. Sci., 29 (March 1972), 304-310.

2-99 A. S. Monin, "Smoke Propagation in the Surface Layer of the Atmosphere,in Adv. Geophys., 6, Academic, New York (1959), 331-343.

2 100 D. H. Wood, "Intemal Boundary-Layer Growth Following a Step Change inSurface Roughness," Bound. ktyer Meteorol., 22 (1982), 241-244.

2-l0l B. G. De Bray, "Atmospheric Shear Flows over Ramps and Escarpments,"Ind. Aerodyn. Abstr.,4, 5 (Scpt. Oct. 1973) I 4.

2-102 C. Sacri, Influcncc d'unt ctllirtc .sur lu vitcsse du vent dan,s la couche limitetlc sutfut'e, Centrc Scicr)tilit;rrc t't lt'clurirluc du Bitiment, Nantes, France,t913.

I 103 A..l . llowcn lntl l). l.irttllt'v. "A Wrrrrl lrrrrrrt'l lrrvt'sligrrtion ol'lhc Winrl Spccditlttl 'l'ttrltttlcttcc ('ltltntt lt'rslrr :, ( 'lo:.r' lo llrt ( irrrrrrtl ovt'r' Virriotrs lisr'trrprrrclrl,Slr:r1rcs," liltruttl. ltt\\'t i'lt'tt't'rr,l l.t, tlt, l/1. .t"t1 .t71,

2-81

BB ilil ntMo: ;t,t lll(:l()uNt)Altyl/\\ttl

'2-104 I'. S..lltt'ksott ltrltl .l ('. ll. llrrrrt. "'lurlrrilcrrl l;low ovcr it l.()w llil. .l lirttttlMclrontl. Jar'., l0l (1975), ().)t) ()\'l

2-105 W. Frost, J. IL. Matrs, rttttl (i. ll. l'it'lrl. "A lllrntlary l,aycl Anulysis ol'Atmospheric Motion Ovcra Scrrri lilliplit'rrl Srrllrrcc Obstruction." l*tuntl. 2tvcrMeteorol., 7 (1974), 165-184.

2-106 P' A. Taylor, "Numerical Studics ol'Ncutrally Stratitied Planetary BounclaryLayer Flow Above Gentle Topography," Bountl. Layer Meteorol., 12 (1977),37-60.

2-lo7 H. Ngrstrud, "wind Flow over Low Arbitrary Hills, " Bound. Inyer Meteorol. ,23 (1982) rt5-124.

2- 108 P. J. Mason and R. I. sykes, "Flow over an Isolated Hill of Moderate Slope, "J. Royal Meteorol. Soc., 105 (1979), 383-395.

2 lo9 P. S. Jackson, "A Theory for Flow over Escarpments," in proceerlings oJ theFourth International Conference on Wind Effects on Buildings aml Structures,London, 1975, Cambridge Univ. press, Cambridge, 1976, pp. 33 40.

2-ll0 N. o. Jensen and E. V. Peterson, "on the Escarpment wind profile, " J. RoyalMeteorol. Soc., lO4 (1978), jl9-j28.

2-lll F. Bradley, "An Experimental Study of the profiles of wind Speed, ShearingStress and rurbulence at the crest of a Large Hil1," "/. Royal Meteorol. soc.,106 (1980), 101-123.

2-ll2 R. E. Britter, J. C. R. Hunt, and K. J. Richard, ..Air Flow Over a Two_Dimensional Hill. studies of Velocity Speed-up, Roughness Ell'ects and rur-bulence," J. Royal Meteorol. Soc., l07, (1981), 9l-110.

2-ll3 G. J. Jenkins et al., "Measurements of the Flow Structure arouncl Ailsa Craig,a Steep, Three-Dimensional Isolated Hill," "/. Royal Meteorol. soc., r07 (l9gl),833 851.

2-ll4 c. Sacr6, "An Experimental study of the Air Flow over a Hill in the Atmo-spheric Boundary Layer," Bound. Loyer Meteorot., 17 (lgjg),3g1_401.

2-ll5 T. Hauf and G. Neumann-Hauf, "Turbulent wind Fkrw over an Embank-ment," Bound. ktyer Meteorol., 24 (1982),351 369.

2-116 S. L. Rosenthal, A Theoretical Analysis of the Fie kl ol'Motiptt in the HurricuneBoundary zrzyer, National Hurricane Research pro.iccr, Ilcp.rl No. 56, u.s.Department of Commerce, Washington, DC, 1962.

2-ll7 R. K. smith, "The surface Boundary Layer of a Hurricanc,' 'l't,llus,20 (196g),413-484.

2-ll8 G. F. carrier, A. L. Hammoncl and o. D. George, "A M.dcl of the MatureHurricane," J. Fluid Mech., 47 (1971), 145-1j0.

2-ll9 E. Simiu, "variation of Mean winds in Hurricanes," J. Eng. Mech. Div.,ASCE, 100, No. EM4, Proc. paper 10692 (Aug. 1974) 833_837.

2-120 E. Simiu, V. c. Patel, and J. F. Nash, "Mean wind proliles in Hurricanes.,'J. Eng. Mech. Div., ASCE, 102, No. EM2, proc. papcr 12044 (April 1976),265-213.

2-l2l Survey of Meleorologicttl Frtttors Partincnt to Rt'tlut,tirn of Lpss sf'Lifc undPropcrty in Hurricutrt' Siluttlitn.r, Nutionul Flurricanc Rcscarch prgjcct Rcp6r1No' 5, tJ.S. l)cP:trltttt'ttl ol ('onttttcrcc. Wclrlhcr Ilurcur.r, WasIirrgl1trr, IX',March l()57

r.r( .t r, Bg

I lll l(..1. Wilsorr. "( lr;rr:rtlt.risl i(.s ()l lll(.srrlrr.lorrtl l.:ryt.l Wrrrrl Slrut.lrue irr'l'r.oplclrl ('.yt lorr,'s,' llrrrt'lrrr ol Mclt'onrhrl.ly" l)r.lrt . ol St.it.lrt.t. irlrtl 'l'cchnology,Mcllrottlttc, I'lt'Jr:tli'tl lirr Ittlcrttationirl ('orrli'rr'rrr't'on'llo;lrcul (lyclones, Perth,Austlllilr. Nov- l()7().

2-123 Srnthcn Stt,tthtrd Buildittg Ci,rlr, llirrrrirrgltrrrr, Ala., 1965, p. l2_5.2 124 w. Malkin, Filling ond Intensity Changt:; irr Hurricanes over Lanj, National

Hurricane Research Pnrject, Report No. 34, u.s. Department of Commerce,Washington, DC, 1959.

2-125 J. L. Goldman and r. Ushiyima, "Decrease in Maximum Hurricane windsafterLandfall ,".1. Struct. Div., ASCE, 100, No. STI, proc. paper 10295 (Jan.t974), t29 t4l.

2-126 M. B. Glauert, "The Wall Jet," J. Fluid Mech., f (1956) 625.2-127 P. Bakke, "An Experimenral Investigarion of a wall |et, J. Fluid Mech.,2

(.t957) 467.2-128 J. Bumham and M. J. Colmer, On lnrge Rapid Wintl Ftuctuations Wich Occur

when the wind Had Previously Been Light , Technical Report No. 69261 , RoyalAircraft Establishment, Famborough, U.K., 1969.

2-129 M. J. colmer, "on the Character of rhunclerstorm Gust-Fronts," TechnicalReport Aero 1316, Royal Aircraft Establishment, Famborough, U.K., 1971.

2-l3o R. w. Sinclair, R. A. Anthes, andH. A. panofsky, variationof the LowLevelwinds During the Passing of a Thunderstorm Gust Franr, NASA ContractorReport No. CR-2289, 1913.

2-l3l H. c. s. Thom, "New Distributions of Extreme wind speeds in the UnitedStates, .r. Struct. Div., ASCE, No. ST7, proc paper 603g (July 196g), l7g7_1801 .

2-132 S. D. Smith, "wind Stress and Heat Flux over the ocean in Gale ForceWinds," J. Phys. Oceanography, l0 (May l98O),'709-726.

2-133 L. Knigermeyer, M. Gninewald, and M. Dunckel, "The Influence of SeaWaves on the Wind Profile," Bound. lnyer Meteorot.,14(lg7g),403 414.

2-134 T. von Kiirmein, "Progress in the Statistical rheory of rurbulence ," proc. Nat.Acad. Sci., Washington, DC (1948), 530-539.

2-135 w. R. Krayer and R. D. Marshall, "Gust Factors Applied to Hurricane winds,"Bull. Am. Meteorol. Soc., 73 (1992), 613-6lj.

2-136 J. Bidtry, Personal communication. 1981.2-137 R. I. Harris, "The Nature of wind, in The Modern Design of wirul-sensitive

Structures, Construction Industry Research ancl Information Association. Lon-don, U.K., 197 l.

2-138 D. M. Deaves, "computations of wind Flow over changes in surface Rough-ness," J. Wind Eng. Ind. Aerdyn., 7 (1981), 65-94.

2-139 ASCE 7-95 standard, Minimum Desil4rt Loads for Buildings arul other struc-,ure.s, American Socicty o1'Civil E,nginccrs, New york, 1995.

2-140 R. L Harris, "Sorttc litrflhcr 'l'lrorrglrls on thc Spcctrum of Gustiness in StrongWinds," J. Wind l,)n,q. ltttl. .'lt,trnlltr. ll. (1990). 461 461 .

2-l4l Y. Tamura, K. Shilrrlulrr, :rrrrl lr.. llilri, "Wirrtl l{csponsc o{'a 'lirwcr ('l'yphtxrnObscrvation al thc N:tglrs:rki llrrr:, li'rr llrsr'lr l)orrrlorcrr)," .1. Witrtl lin.4. ltrt!.Acnrl. . -50 ( l9().1). .1O() l lti

90 lltf n tMOSI,I l nt(; nouNl)nny tAyt n

2-142 A. I). Pcrcira, M. ('. (i. Silvrr, l). X. Vtr'1,',irs, ruttl A. (i. I-opcs, "Wilul 'l'LrnnclSirnulatiorr ol'thc likrw arourttl 'l'wo I)irrcrrsionul l-lills," .1. Wirul lhtg. lnd.Aerod., 38 (1991), lO9-122.

2-143 I. Van de Hoven, "Powcr Spcc:(rurrr ol' Wintl Vclocity Fluctuations in theFrequency Range fiom 0.0007 to 9(X) (lyclcs pcr Hour, " J . Meteor. , 14 (1957),1254-t255.

2-144 J. Ashcroft, "The Relationship betwccn the Gust Ratio, Terrain Roughness,Gust Duration and the Hourly Mean Speed," J. Wind Eng. Ind. Aerod.,53(r994). 33 l-355.

2-145 J. S. Allen, R. M. Samelson, and P. A. Newberger, "Chaos in an Model ofForced Quasi-geostrophic Flow over Topography: An Application of Melni-kov's Method," J. Fluid Mech., 226 (1991), 5ll*547.

2-146 E. Simiu, "Melnikov Process for Stochastically Perturbed Slowly Varying Os-cillators: Application to a Model of Wind-Driven Coastal Currents," J. AppliedMech., ASME 63 (June 1996),429-435.

2-141 S. Krenk, "Wind Field Coherence and Dynamic Wind Forces," Proceedings,IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics (A. Naess,ed.), Trondheim, Norway, July 1995.

2-148 J. Mann, "The Spatial Structure of Neutral Atmospheric Surface-Layer Tur-bulence," J. Fluid Mech.,273 (1994), l4l-168.

2-149 J. Mann, "Fourier Simulation of a Non-Isotropic Wind Field Model,"' inStructural Safety and Reliability, (G. Schueller, M. Shinozuka and J. Yao,eds.), pp. 1669-1614, Bakkema, Rotterdam, 1994.

2-150 J. Blessman, O vento na engenharia estrutural, Editora da Universidade, Uni-versidade Federal de Rio Grande do Sul, Porto Alegre, Brasil, 1996.

2-151 J. Kaplan and M. DeMaria, J. Appl. Met.,34 (Nov. 1995), 2499*2512.

CHAPTER 3

EXTREME WIND CLIMATOLOGY

Climatology may be defined as a set of probabilistic statements on long-termweather conditions. The branch of climatology that specializes in the study ofwinds is referred to as wind climatology. Wind climatology provides the de-signer and the code writer with information on the extreme winds that mightaffect a structure during its lifetime.* Such information is required for makingrational decisions on the magnitude of the wind loads to be used in design.

This chapter is devoted to a review of problems involved in the descriptionof the wind climate for structural design purposes and in the development ofcriteria for the definition of design wind speeds. Procedures for estimatingextreme winds are presented, and the uncertainties inherent in these proceduresare discussed. Some of the material included herein is heavily dependent uponprobabilistic and statistical notions and tools. These are presented in some detailin Appendix A1.

The reliability of climatological statements based on the analysis of extremewind speed data is clearly dependent upon the quality of the data. This topicis discussed in Sect. 3.1. The question of the prediction of extreme wind speedsin well-behaved wind climates and in hurricane-prone regions is dealt with inSects. 3.2 and 3.3, respectively. The dependence of extreme wind speeds upondirection is discussed in Sect. 3.4. Information on the frequency of occurrenceof tornado winds of various intensities in the United States is presented in Sect.3.5.

In the United States surfacc wirxl spoocls rcpofted by the Weather Servicehave traditionally been exprossc(l itt tttilcs pcr hour (l mph : O.447 m/s). In

*Wincls othcr than thoso ol'inlcrcsl lirrn lr slrr( lurrl sirlr'ly vicwpoittt will bc rlcalt wilh in Chaptort5.

91

I X llil Ml wlNI) ( il tMn t( )l ( x iY

Itrtrriclrttc-rollr(ctl wot'k, lt:rrgllts ltt' lrt'rlrrrrrlly t'xplt'ssul irt rlrrrlrt'rrl rrrlt's (Inuli - l.l5 rnilc). Iior corrve:rricrrcc, wlrt'r't'lrpPnrplirrtc:, lhcsc trnils will rrlsobe used herein.

3.1 WIND SPEED DATA

To provide useful information on the windspeed data recorded at that location mustmicrometeorologically homogeneous set.

climate at a given location, windbe reliable and must constitute a

3.1.1 Reliability of Wind Speed Data

Wind speed data may be considered to be reliable if:

I . The instrumentation used for obtaining the data (i.e. , the sensor and therecording system) may be assumed to have performed adequately and wasproperly calibrated. If it can be determined that the calibration was not ade-quate, the data must be adjusted-whenever the information needed for thatpurpose is available.

Example The following information is excerpted from [3-l] regarding the5-minute winds given in the original U.S. Weather Bureau records taken before1932 "Up to 31 December 1921 , all recorded wind speeds were the uncor-rected readings of 4-cup anemometers. From 1928 through 193 1, all speedsfrom the older 4-cup anemometers were corrected to agree with the readingsof the 3-cup instruments, then being introduced, readings from which were notcorrected to true speeds. From I January 1932 onward all readings, whetherfrom 3- or 4-cup anemometers, were already corrected to true speed in theoriginal records." Official U.S. Weather Bureau instructions fbr the correctionof 3- and 4-cup anemometer readings are given in Table 3.1.1, which is ex-cerpted from [3-2], and whose use will now be illustrated. At Williston, N.D.,the original readings of the maximum 5-minute wind in 1922 and 1930 onrecord at the National Oceanic and Atmospheric Administration are 56 mphand 37 mph, respectively. Using the corrections of Table 3 . I . I , the true speeds(according to U.S. Weather Bureau calibrations) are 56 - l2 - 44 mph and37 - 2 : 35 mph, respectively.

2. The sensor was exposed in such a way that it was not influenced by localflow effects due to the proximity of an obstruction (e.g., building top, orinstrument support). For most U.S. weather stations, the existence of such anobstruction during the period of record is noted, in principle, in Local Cli-matological Data Summary sheets (LCD Summaries) issued by the Environ-mental Data Scrvicc ol'thc National C)ccanic and Atmospheric Admlnistrationl3-31.

3. 'l'hc: lrlrrxrsplrt'r'ic slrlrtilit'irtion lrray bc: assurrretl lo havc hccn rrcrrtr:rl. 'l'his

:t I wlNt):it ,l lt) l)n tn

'l'Alll,lt J.l.l. ('orn.t.liorrs lo lrrrlie:rlc<t Wirrrl Sllctrls l-l-21Spt't'tls lrrtliclrlc:tl

By 3-Cup "S" 1'ypcAnemometer, mph

[1928-l93l"l

By 4-Cup Anemometer,mph [Up to 31 Dec,

1g2l"l

Corrections inWhole Miles

per Hourob-rct7-2627-3536,4445-5253-6162-707 t-7980-8788-9697-t05

106-1 14tt5-122123-132r33-139140-149150-r57158-166t6t 174175-184r85-192193-2(n

ob-g9-12

l3-16t7-202t-2425-2829-3233-3637-3940-4344-4748-5 I52-5455-5859-6263-6566-6970-7374-7778-808l-8485-8889-9r92-9596-99

100- 103104-106107-t l0I I l-114I l5-1 l71 l8-l2l122-125126-128129-132r 33- 136t37 -140l4t-t43

+l0

-l-J-4-5-6-l-8-9

-10-11-12-13-14-15-16-17-18-t9-20-21

1a

-23-24-25-26

a1

-28-29*30-31*32- J-t-34- 3-5

"Reference [3- l].'Movement of anemometer cups obst.rvt.rl

94 I x tltl Ml wlNt) ( il tM/\ t( )l ( )( iy

llsstlllll)ti()ll is itcccp(itblc lirl witrtl spcctls;rt lO rrr;rlrovt'glorrrrtl irr opcll lcll-irirrin cxccss ol'2-5 rnph rlr so (scc Scct. 2.2.-5).

3.1.2 Micrometeorological Homogeneity of Wind Speed DataA set of wind speed data is referred to herein as micrometeorologically ho-mogeneous if all the data belonging to the set may be considered to have beenobtained under identical or equivalent micrometeorological conditions.

These conditions are determined by the following factors, which wiil bebrictly cliscussed below:

. Avcrirgirrg tirnc (i.c., whcther highest gust, fastest mile, one-minute av-t'r:rgc, livc nlinulc avcragc, etc., was recorded).

o Ilciglrl irl'rovc gnrurrtl.. l{orrglrrrc:ss ol' surnlunding tcrrain (exposure).

I . Avantging 'l'ime. If various averaging times have been used during thepcriod o1'rccord, the data must be adjusted to a common averaging time. Thiscan be done by using Eq. 2.3.31 and Tables 2.2.1 and2.3.3, or Fig. 2.3.10.(For hurricane wind speeds, see also Sect. 2.4.3.)

Data averaged over short time intervals, such as highest gusts or fastestmiles, may in certain cases be affected by stronger than usual local turbulenceeffects, and thus provide a somewhat distorted picture of the intensity of themean winds. In principle, it is desirable, therefore, that the data used for thedescription of the wind climate be averages over relatively long periods, sayfive minutes or so. However, owing to the current data collection policy of theu.S. National weather Service and the availability of 3-s gust speed data at alarge number of stations in the United States, the ASCE 7-95 Standard lz-1391uses 3-s gust speeds at lO-m elevation as basic wind speeds.*

2. Height above Ground. If during the period of record thc erevation of theanemometer has been changed, the data must be adjusted to a common elevationas follows: Let the roughness length and the zero plane displaccment be denotedby z6 and 27, respectively (zo and za are parameters that clefinc the roughnessof terrain; see Sect. 2.2). For strong winds (i.e., with speecls cxceeding l0m/s or so), the relation between the mean speeds u(ar) and u(z) over horizontalterrain of uniform roughness at elevation z1 and Z2 above ground, respectively,can be written as

xThe National Weather Service and the Federal Aviation Administration are currently imple-menting the Automated Surface Observing System (ASOS). It is anticipated that by the year 2000there will be 1700 ASOS units in operation. The ASOS anemomctcr reading is sensed once asecond. Every fivc sccttnds it rlnning ilvcnlgL- is eomputcd, which is rclorrctl kr as a (igsl. A2-rninute running avcrilgc ol thc 5 s irvcrllgcs is irlso conrprrlctl lrrrtl is rrst'il rrs rr rrrcirsrrrc ol thcprcvailirrg winrl.'l'ltt'st'tltoiccs rtl:rvt'nrgirrg lirrrc wcle rlct'lrrcrl t() l)('nr()sl rrsr.lrrl lor:rvilrlionpurJxrscs l.l (rl{

1 .

Equation 3. l. I lirllows clircctly from Eqs. 2.2.18 ;ttd 2.2.22. For open terrainZa : O, and the values ol' the roughness lcngth 2,, can bc taken from Table2.2.1. The power law (Eq. 2.2.26 and Table 2.2.2) may be used in lieu ofEq. 3.1.1. As noted in sect.2.2.3, considerable uncertainties subsist withregard to the values of the roughness parameters in built-up terrain. Goodjudgment and experience are required to keep the errors inherent in the sub-jective estimation of the roughness parameters within reasonable bounds. It isclearly advisable to investigate in individual cases the effect of such possibleerrors upon the predictions of extreme wind speeds.

3. Roughness of surrounding Terrain In many cases anemometer locationshave been changed during the period of record, for example, from a town toa neighboring airport station. The corresponding records can, in principle, beadjusted to a common terrain roughness by using the similarity model (Eqs.2.2.29 and 2.2.31 and rable 2.2.3) described in Secr. 2.2. As indicated insect.2.4.1 , this model may be assurned to be applicable in horizontal terrainif at each station the terrain roughness is reasonably uniform over a distancefrom the anemometer of about 100 times the anemometer elevation. In terrainin which sheltering effects by small-scale obstacles are present, the data maybe adjusted by using a procedure presented in [3-4].

A situation commonly encountered in practice is one in which, while theanemometer may not have been moved, the roughness of the terrain surroundingthe anemometer has changed significantly over the years as a result of extensiveland development. In such situations the adjustment of the data to a commonroughness may pose insurmountable problems, unless detailed information onthe phases of the land development is available.

Anemometer elevation and location changes are listed for most u.S. weatherstations in Local Climatological Data Summaries t3-31. wind climatologicalinformation for various locations around the world is available in [3-77 , 3-78].

3.2 ESTIMATION OF EXTREME WIND SPEEDS INWELL-BEHAVED CLIMATES

Infrequent winds (e.g., hurricanes) that are meteorologically distinct from andconsiderably stronger than the usual annual extremes are referred to herein asextraordinary winds. climates in which extraordinary winds may not be ex-pected to occur are ref'erred to us wr'l/ ltehuvctl. In such climates it is reasonableto assume that each of thc tlrrt:r in rr st'r.it's ol'thc largest annual wind speedscontributcs to the dcscripliott ol'llrc plrb;rbilis( it'l'rr.rhlrvirlrof the extreme winds.A statistical analysis ol'sttclt rt st'tir's r'lrrr llrt'rt'lir.t' bc cxpcclcd t<l yicld uscfulprctlictions ol' l<lng*lorrrr wirrtl t.xllt.rrrr.:,

'l'htrs, in tt wcll-hchltvtrtl t'lirrrrrlt',:r( :nrv 1'rvcrr st;rliorr ir rirrrtkrn virrilrblt'rrlry

;llMAll()ll rrl r 'ilttl Ml

I l(.'.tl( /(;'. ,)

wlNl r:;t,t tll:; tN wt lt ttt ltnvt t) (,t lMn il:

lnf(;, ;',,111:ul

lrrl(i, :'.,111;'.qyl

r95

(3. r. r)

96 I x tilt N/lt wtNl) (.1 lMn l( )l { )( 'Y

bc tlclirrr,tl, wlticlr corrsisls ol tlrc l:rrIr':;l yt':rrly wtrrtl sllcctl. ll llrr'sllr(rorr ts

9pc lirr wIich wiltrl rccortls ()vcl lt rrrrtttlrt't ol tttttset'tllit'c yoilrs ittt'lrvltil:rble:,thcn thc cuntulativc dislributiolr lirrrclion (('ll1'1 ol'tlris ritrlclottt vllriitlllr-: tttaybe estimated to charactcrizc thr: pnrblbilistit' bcltitvi<lr of thc largcst annualwind speeds. The basic clesign wintl spectl is thcn dclincd as the speed corre-sponding to a specified value p ol' thc Cl)lr or, cquivalently, to a specifiedmean recurTence interval N.* R wind cttrrcsponding to an /V-year mean re-currence interval is commonly referred to as the N-year wind.

This section is devoted to the question of estimating (1) the CDF of thelargest annual speeds and (2) errors inherent in the wind speed predictions.Such errors include, in addition to those associated with the quality of the data(sec Sect. 3.1), modeling errors and sampling errors. Modeling errors are dueto an inadequate choice of the probabilistic model itself. Sampling errors area consequence of the limited size of the samples from which the distributionparametcrs are estimated and become, in theory, vanishingly small as the sam-ple size increases indefinitely.

3.2.1 Probabilistic Modeling of Largest Yearly Wind Speeds

Extreme wind speeds inferred from any given sample of wind speed data de-pend on the type of distribution on which the inferences are based. For largemean recurrence intervals (".g., N > 50 years) estimates based on the as-

sumption that a Type II distribution is valid are higher than correspondingestimates obtained by using a Type I distribution, while estimates based on areverse Weibull distribution wittL tail length parameter ^Y < 15 , say, are lower. f

According to [3-5], extreme winds in well-behaved climates may be assumedto be best modeled by a Type II distribution with p : 0 and 7 : 9. However,subsequent research has shown that this assumption is not borne out by analysesof extreme wind speed data f3-6, 3-1 , 3-91. In [3-6] , 37 year-series of 5-minutelargest yearly speeds measured at stations with well-behaved climates wereruUi""t"A to the probability plot correlation coefficient test (see Sect. A1.6) todetermine the tail length parameter of the best fitting distribution of the largestvalues. Of these series, 72% were best fit by Type I distributions or by TypeII distributions with "v : 13 (which differ insignificantly from the Type Idistribution); ll% by Type II distributions with 7 < ^Y < 13; and 17% byType II distribution with 2 = I I 7. Virtually the same percentages wereobtained in [3-7] from the analysis of sets of 3'7 data generated by the MonteCarlo simulation from a population with a Type I distribution. On the otherhand, the analysis of sets generated by Monte Carlo simulation from a TypeII distribution with tail length parameter 7 : 9 led to percentages differing

*Rccall that lV: tlf f ' 7r) (scc Appcntlix Al,Irq. Al-).lDifl'crcnccr hctwccrr spet:tls cslirrrirlctl ott lltc basis ol T'ypc II tlistributions rntl lhc -l'ypc I

tlisll'ibrrlion ilttrt'rrst.lrs.y tlcclclrsr's. l)illi'rcrrt'cs hclwcctt s;rt:ctls lt:tsctl on lhc'l'ypt I tlisllilrtlliott:uxl rcvt'rse Wt'ilrrrll rlislrilrttliotrs rtlso irtt tt':tse its 1 tlt't'rrl:tst's

l:lllMnllillt ot t"lltt Mt WtNt):;t,t tlrl ; lNt Wt lttililnvt tr(.t tMAilti 97

sigrrilicirnlly ltttltt lltost' r onr'sporrtlirrg to tlrc irclrrirl wirrtl spcc:tl tltrlir. ()rr thcbasis ol'thcsc t'csrllls il r'rrrr lrt'conliclcntly statcd that in woll-bchavccl clirlatescxtrcmc wintl spr:ctls lrlt' rrrotle lctl rn<lre realistically by thc Type I than by theTypc II distributiorr with ^y ,., 9. This conclusion was reinforced by studiesreported in [3-9], in which tcchniques similar to those of [3-7] were used inconjunction with wind speed data at one hundred U.s. weather stations listedin [3-9].

As indicated earlier, the Type I distribution results in lower estimates of theextreme wind speeds than the Type II distribution with 7 : 9. An interestingresult obtained in [,{1-36] is that at most stations in the United States even theType I distribution appears to be an unduly severe model of the wind speedscorresponding to large mean recurrence intervals; at these stations a better fitto the data is obtained by reverse weibull distributions (see also end of Ap-pendix Al). Thus, structural reliability calculations based on the assumptionthat the Type I distribution holds are in most cases conservative [Al-36]. Forthis reason we will assume in this section that the Type I distribution modelholds. The degree of conservation inherent in this assumption is generallymodest for basic (5O-year) speeds, but it can be very significant for wind speedscorresponding to nominal ultimate wind loads, i.e., 5O-year wind loads mul-tiplied by a wind load factor (see Sect. A.3.3).

3.2.2 Estimation of and Confidence lntervals for the N-year Wind:Numerical ExampleIt is shown in Sect. Al.7 that, given a set of data with a Type I extreme valueunderlying distribution, several techniques can be used to estimate the param-eters of the distribution and, hence, the value of the variate corresponding toa given mean recurrence interval.* However, inherent in these estimates aresampling errors. A measure of the magnitude of the latter can be obtained bycalculating confidence intervals for the quantity being estimated, that is, inter-vals of which it can be stated-with a specified confidence that the statementis correct-that they contain the true, unknown value of that quantity. Tech-niques that can be used to estimate the N-year wind, and confidence intervalsfor the N-year wind, are discussed in some detail in Sect. A1.7. one of thesetechniques is presented and illustrated below.

Using the approximation -ln[-ln(1 - l/N)] = ln N, it follows from Eq.Al .74 (which is based on the method of moments) that the estimated valueDp of the N-year wind u1y is

0N = X I 0.78(lnN - 0.577)s (3.2.1)

where X and s are, respcctivcly, rlrrr sirrrrplc rncarn and the sample standarddeviation of the largest ycarly wilrtl spt'erls lirr (hc pcriod of recorcl.

*InAppendixAlthisvalucistlcrxrh.rl lry(i,q711,rllrr.rr./r I l/N:rrxl Nisllrt, 1rt.ir1 t1.(.rtrronccinlcrval, In this chaptcr lhc trottrliolr (i,( I l/N I r,. r,, rrr,r.rl

IXIlilMl wlNl)(.1 lMnl()l ()(iY

As Prt.vipttsly rurlctl, irtltr.:tcrr( iil lltt't':-lrlrItlt':; ol l'1u 11t'q s:tttlpllttl', t'ltot:.. lllirllows l.nrrrr I')t1s' n I 76 lrrrtl Al'/o (wlritlr:tlt'lrltst'tl ott lltc: tttcllrotl ol ttttrllcllls) llrirt thc staurlanl tlovialiorr ol tltt'srtttplirtg ert'tlt's in thc cslilrr:rtiolt olr',y cln hc writtcn as

.\1)0r^) -- 0.781 l.& + l.46tln ru - O.SZtl I l.l(ln rV - O'S;Zt'l'' ;,/n(3.2.2)

wlrt't'c rt is thc samPle size.

Example At Great Falls, Montana, the largest yearly fastest-mile wind speeds

rrt l0 rrr above ground during the period 1944-1971 (sample sizen :34) were

l3-t)l:

51, 65, 62, 58, &, 65, 59, 65, 59, 60, 64, 65,'73, 60, 61, 50, 74

60, 66, 55, 51, 60, 55, 60, 51,51,62,51,54,52,59,56,52' 49

(mph). The sample mean and the sample standard deviation for these data are

X:'5g mph and s : 6.41mph. From Eqs. 3.2.1 and3.2.2 it follows thatfor N : 50 years and N : 1000 Years'

lso = 76 mph SD(i5o) : 3'7 mph|wn = 91 mPh SD(irooo) = 6'4 mph

As shown in Sect. A1.7, the probabilities that u vl is contained in the intervals0v1 + SD(01y), 0n t zsD(|il, and 0p + 3SD(0,'v) are approximately 68%'gi%, ana 99%, respectively. These intervals are referred to as the 68%, 95%,and 99% confidenci intervals for u7, and are shown fbr the 34-year GreatFalls sample in row 1 of Table 3.2.1.

It is also shown in Sect. A1.7 that the width of the confidence intervals can

TABLE 3.2.1. confidence Intervals for the N-year wind at Great Falls

I i;llMn ll()t\t ()t I i: ilil [It wtNt) :;l'l I t)t; lN wt il ilt ilAV' t) (]t lMn il ti gg

bc rctlLtcctl il it lttott't'lltr'rt'ttl t'slitturtor is uscd; lurwcvrrr', llrc intcrvals cannolbc trarnrwcr lltirtt tlrrst' ohlrrirrr'tl by using the Crarrcr,llacl (C.R.) lower bouncl(Eq.Al.77). tt<tr tlrr: (irt:;rt Irlrlls sltnple, the confidence intervals based on thelatte r are shown in linc (2) ol 'l'ablc 3.2.1. lt is seen that the differences betweenthe results of lines (l) and (2) ol"rable 3.2.1are small.'rhis is consistent withthe conclusion of Sect. Al.7 that the e{iiciency of the method of moments (Eq.3.2.1) is generally adequate for structural design purposes.

In Table 3.2.1 the errors in the estimation of the 50-year wind are of theorder of lo% at the 95% confidence level. Since the wind pressures are pro-portional to the wind speeds (see Chapter 4), the corresponding errors in theestimation of the pressures are of the order of 2O%. An altemative approachto accounting for sampling errors, which applies the theorem of total prob-ability, is suggested in [3-51].

To reduce sampling errors, [3-8] resorted to the consolidation of recordsfrom different stations, thereby creating "superstations" with large samplesizes. This approach, if valid, would be quite attractive: for example, theinformation yielded by a "superstation" consolidating 2o-year records takenat 100 stations would be equivalent to information yielded by a 2000-yearrecord. However, according to [A1-15], statistical tests did not validate the"superstation" concept for extreme wind analyses based on peak gust speeds.

For that concept to be valid, the population distributions for the stationrecords being consolidated would have to be identical, and in addition thoserecords would have to be mutually uncorrelated. In general, the first of theseconditions cannot be assumed a priori to be true. Second, if the records consistof peak gust speeds, the observed lack of correlation between records taken atdifferent stations may be spurious; that is, it would be likely to occur even ifthe corresponding mean wind speed records were well correlated. The apparentlack of correlation would be an artifact due to the strong random variability ofthe ratios between gust speeds and mean speeds. For these reasons the "su-perstation" concept may yield inadequate results. In our opinion this is likelyto be reflected in the quality of the wind speed map specified by the ASCE7-95 Standard [9-5], which is based largely on the "superstation" conceptapplied to peak gust speed records.

3-2.3 Methods for Estimating the Extreme speeds at Locations withlnsufficient Largest Yearly Wind Speed DataThere are about one hundred u.S. weather stations for which reliable andrelatively long wind speed records arc available (e.g., records overperiods of20 years or more). Some of thcsc sl:rlions c()vcr arcas of tens of thousands ofsquare miles, over which-lilr rrrc(t:onrkrgical rcasons or owing to topographiceffects-the extreme wincl clirrrrrtt' is rrol rrt't'r,ssirrily uniform. Thcre arisestherefore in practice thc prohlcrrr ol t'slirrrrrlirrl'('xll('tn(: wintl spcccls at variouslocations where long-tcntt t'ccottls ol llrc l:rr1'r':rl yt'lrll_y witrtl spcctl tlutl tlo 191exisf.

Confidencc level 687 95%

Mean recunenceinterval, N (years) s0 1000 50 1000 50 1000

(1) Estimated bymethod ofmoments 76 + 3.1 9l + 6.4 16 + 1.4 9l + 12.8 76 + I l.l 9l + 19.2

(2) Estinratcd usingC.R. lowcr bounil 76 I .l.l ()l Il5.o9l I5.0 '76 r 6.2 9l I lO.O 7(r I (). \

lo0 txiltt Mt wtNt)ct tMnt()t ((iy

Estimates Of Extreme Wind Speerls lrr a Marirrc EnvirOnmenl. l{t'lcrt'rtt't'1.1-l ll lists ilrrcer rrrcllrorls tlrir( ,iu1', ur lrrrrrt rplt', :rvltilitblc (o clt'l'y ottt sttt'ltcstitnatcs lilr lrrarinc cnvirorrrttcrtls wltt'tt' lltt' cxltctttc spcccls itte itssot'iltlctlwith extratnrpical strlrtrrs. 'l'hc lirst rrrt'llrrxl rnlrkcs usc ttl'climattlltlgical inlor-mation on various paramctcrs ol' thc s(or.rrr iuttl ol' physical modcls rclatingthose parameters to the surface wincl spcctls. lt is shown in Sect. 3.3 that sucha method can be applied to estimatc oxlt'cttlc wind speeds in hurricane-proneregions. However, as noted in [3-lll, owing to the complexity of the surfacewind patterns in extratropical storms, the usefulness of this method appears tobe uncertain in regions where such storms are dominant.

A second method listed in [3-11] is the use of objective analysis schemes.These consist of (1) an initial guess at the surface wind on a regular grid, (2)an automated procedure for screening wind reports from ships to eliminateerroneous readings, and (3) a procedure for correcting the initial guess on thebasis of the usable set of ship reports, which involves relations among thesurface wind speeds, sea-level pressures, and air and sea temperatures. Detailson objective analysis schemes and of errors culrently inherent in such schemes(which may range from lO% to 30%) are given in [3-11].

The third method listed in [3-11] is referred to as direct kinematic analysis.The method, which involves subjective judgment by experienced analysts, con-sists in synthesizing discrete meteorological observations to obtain a continuousfield represented in terms of streamlines and isotachs. Objective or kinematicanalyses applied to a sufficient number of strong storms make it possible toprovide estimates of extreme winds that may occur at any one location. Asindicated in [3-11], one of the major diliculties in conducting such analysesis that much of the vast store of existing data is currently not accessible inreadily usable form.

Estimation of Extreme Wind Speeds from Short-Term Records. A prac-tical procedure for estimating extreme wind speeds at locations where long-term data are not available is described in [3-12]. The method, whose appli-cability was tested for a large number of U.S. weathcr stations, makes itpossible to infer the probabilistic behavior of extreme winds from data con-sisting of the largest monthly wind speeds recorded over a period of three yearsor longer. Estimates based on the monthly speeds, denoted by 0n.., are ob-tained by rewriting Eq. A1.74 as follows:

0N.^ = x. + o.zAPn(l2N) - 0.5771s* (3.2.3)

where X. and sa are, respectively, the sample mean and the sample standarddeviation of the largest monthly wind speed data, and N : mean recurrenceinterval in years.

The standard dcviatitln tll'thc sanlpling error in thc cstirlation tll'I'x7.,,, isobtaincd from l'it1s" Al .76 irrrtl A I .70 as

r:' Ir;ilMAil()N ()t I . |ltt N/,1t wtNt) l;t 'l It)ti IN wt |t tlt ltAVt t) (.t tMn il:

,S/)( /',,r,.,,, ) o /ttl l.(4 I 1.4()llrr( llN) O..5771

I r.rlrn(t2N) - o.sztl'l',, h:, (3.2.4)

i

ir

where n-: sample size.

Example At Great Falls, the sample mean and the sample standard deviationof the largest monthly fastest-mile wind speeds at l0 m above ground for theperiod September 1968 through August l97lx (sample size n*:36) aret^:42mph, s^:6.96 mph. From Eqs.3.2.3 and3.2.4, the estimates forN: 50 years and N : 1000 years are

0s0.. : 74 mph SD(0s0,) = 6.23 mph

iuxn,^: 90 mph SD(irooo..) : 8.85 mph

It is seen that the estimated speeds based on the set of 36 largest monthly dataare only slightly lower than those obtained from the set of 34 largest yearlyspeeds (|so:76 mph and 01es0 : 91 mph; see Sect. 3.2.2); however, thesampling errors are larger.

Similar calculations carried out for 67 sets of records taken at 36 stationsare reported in [3-12], where it was found that the differences iso,^ - lso,where 056 is the 50 year wind speed estimated from long-term largest yearlydata, were less than sD(0s0.) in 66% of the cases and less than twice the valueof sD(05s,-) in 95% of the cases. This remarkable result, confirmed by addi-tional calculations reported in [3-13], indicates that the estimates based onlargest monthly wind speeds recorded over three years or more provide a usefuldescription of the extreme wind speeds in regions with a well-behaved windclimate.

Inferences concerning the probabilistic model of the extreme wind climatehave also been attempted from data consisting of largest daily wind speeds13-121, or of wind speeds measured at one-hour intervals 13-141. one problemthat arises in this respect is that data recorded on two successive duys u..generally strongly correlated. A second and more serious problem is that thedaily (or hourly) data reflect a large number of events (e.g., moming breezes)that are altogether unrelated meteorologically to the storms associated with theextreme winds. These events can be viewed as noise that obscures the infor-mation relevant to the description ol'thc cxtreme wind climate. Indeed, it wasverified in [3-12] that estimatcs ol' cxtr-cnrc winds based on daily data differsignificantly from estimatcs obllrirrt:tl lor krrrg,lcrrrt rccords of largest yearlyspeeds. This conclusion is a.litrtittri ltut'lirl irrlc'r'ctrc:cs based 9n hourly data.

*For the actual data. scc thc l,octrl ('liru:rlolo1,r, .rl I ):rt.r ',rrrrrrr;rrit.s li1. tltr. ycitrs l()()ll l.)7.1: "'--=\,;' \

T

'liI

1O2 I x I ill Mt wtNl) (;t tMn t( )t ( x iy

l,lstirrlrlcs ()l'L:xirL:nlL: wirttl spct'tls lr;r:;t'rl orr st'ls ol tllrllr itt cxccss ol s1r't'rltcrl{hrcshokls (scc cncl ol'Sccl . A I.7) lrrt' rt'llrlcrl irr l.t l0l lirt' sltorl r('('()11ls.

It was shown in Scct. 2.4.4 tltal lhrrrttlt'r.slot'rrr wirttls havo I'caiures tlrirt rlillcrmarkedly fiom thosc ol'othcr typcs ol'wirrtl. (ic:ncrally, extremc wind spocdsare analyzed without separating thundcrslonr tlata lrcm the other extremes.Whether it would be useful to extract thcsc tlata l'rorn the mixed sets and analyzethem separately-despite the difficulties this would entail-is still being debatedt3-s01.

3.3 ESTIMATION OF EXTREME WIND SPEEDS INHURRICANE-PRONE REGIONS

We now consider the prediction of extreme winds in climates characterized bythe occurrence of hurricanes. It was suggested in Sect. 3.2 that in a well-behaved wind climate each of the data in a series of the largest yearly speedscontributes to the description of the probabilistic behavior of the extreme winds.However, in a hurricane-prone region most of the speeds in a series of thelargest yearly winds are considerably lower than the extreme speeds associatedwith hurricanes; they may therefore be irrelevant from a structural safety pointof view. This situation is illustrated by the plot of Fig. 3.3.1, which shows theS-minute largest speeds recorded at Corpus Christi, Texas, between l9l2 and1948 [3-6]. It may then be argued that in hurricane-prone regions the series ofthe largest yearly speeds cannot provide useful statistical information on windsof interest to the structural designer, much in the same way as the populationof a first-grade classroom-which might include a teacher-is of little use in astatistical study of the height of adults. That this is the case is suggested below.

The abscissa in Fig. 3.3.1 represents the reduced variate

t / l\lv:-tnl-tnlr-:ll" | \ N/lwhere N is the mean recurrence interval. In virtue of Eqs. Al .43 and A1.45,a Type I extreme value cumulative distribution function would be representedin Fig. 3.3.1 by a straight line, whose intercept and slope would be equal tothe distribution parameters p and o, respectively. To the extent that the pop-ulation of largest yearly speeds would be described by a Type I distribution,the actual data would then approximately fit a straight line. In Fig. 3.3.1 thisis roughly the case as far as the winds of less than hurricane force are concemed.However, if-as in Fig. 3.3.l-the hurricane-force winds are included in theset being analyzed, clearly the fit of a Type I distribution to the data is extremelypoor.

A bettcr fit can bc obtainccl il-a Typc II distribution with a snrall valuc ol'thc tail lcngth pirrrrrrtclcr is usctl. Howcvcr, As slrowrr in 1.1 (rl. 1ln:1lir'1i1;11r,i,r1'cxlrclnc wintls irr ltttl-rit'lrltt'plottc tcgiotts b:rscrl ort'l'ypt'll tlisl r.ilrrrliorrs lrlt'irr

t--Ft--FatsHia-!!txxx,FtsFa__xt FFrxF!a

-Frl-FrttsFFtHrHlrF- xxx I FxH I xFx | -HH I xFx

ooooooooooooCOooooooooooooooOaFrEOFF

IIaIIIIIIII

IIIIIIIIIIIa

IIIIIItIa

IIIIIIjIIIaIIIIIItIIIIaIIIIIa

IIIIII

I

tIooIooo?n6

I

tF olF ot0 at0I Ztto

ItJqI ou dr td xI t<oI FOJrNu-I r!I FIO .JI 6F Ut€ra <'ctN a ;t-nr>(-Jtuat=rdt!tN6lo-:r+UHotd*lnloa!a!

-Ytn r YIN . HtNqI F^l6- al No oH .. tr toI iO !-r €F dI OdI OUI iuI €C :r ou =I aFt .z trrHfil

HoOOru+I IIF AldI lrI FFF tI O- d ' :ao u =I (F 9lI NOI rJtoo =I l>tI FoI Hr -:I Jo O-I FGlE.>i6

=r cr -OldG:E-or uu 9I ox ,liI >u glF U .I rJL It o< uI NFu !

=UJ J:-H C, \O-5 =.ruo-!A

JULl

Jl@G@FOUd

103

IztI

ooooo0ooooooooo

oooots-oo

IoII

l0,l r,(rIilMt wtNt){]tl\/nt()tI)(iY

1t,,.r , .i',( s tlnloalisiic. l'ot cxlttttpl,', ltltrtrl' r'rr{ lr ;l tlistrillrrtiorl trt tllt' l()l'llir!r, r,,ol.tl ol'thc llrrgcst yclu-ly slx'('(l:,;rl ('orlrrr:, ('lrlisti wotrltl yicltl, lirr'(lrcr trr':rtr'rl 1000-ycar wind, a valu('()l l()fr0 rrrplr rr lirlicttlous rcsult l.l ()1.

',{ uous difficulties als() arisc il rrrixr"rl lirt't lrt'1 pnrbirbility distributions arr'ii ,,1 lt 51. lndeed, sincc hurriciltlcs ill(' l:rl('('v('ll(s, thc number of hurricaner,,r rr'plcrl cyclone) wind spccd cllttlt irr lt t'e,.'ttltl ol'thc largest yearly winds,,i,.1 1v1'l at any one stati()n is small (t:.g., irr lrig. 3.3.1 only two of the datar,l'r(':iL:nt hurricane wincl spccds). 'l'hcrclirrc thc confidence intervals for the, tr!('nrc wind preclictions arc, in gcncral, unacccptably wide (e.g., for N :l{x) years, of the ordcr of irgo(l + 0.6) at the 68% confidence level; see

1r l5l). It is for this rcason that the 50-year fastest-mile wind estimated in1l 5l for Corpus Christi on the basis of a mixed Fr6chet distribution is only/(r mph at 30 ft above ground in open terrain. This value appears to be severelyIow; indeed, in the pcriod 1916-1970 Corpus Christi was hit by three devas-tating hurricanes 13- l6l with fastest-mile winds of up to 120 mph at 23 ft aboveground in open terrain (see Corpus Christi 1970 Local Climatological DataAnnual Summary).

Because this series of the largest yearly winds does not appear to provide asuitable basis fbr predicting hurricane wind speeds, alternative bases for suchpredictions have been proposed in the literature, which are now briefly dis-cussed.

3.3.1 Procedure Based on the Maximum Average Monthly Speed

In this procedure, proposed in [3-17], it is assumed that the behavior of theextreme winds is described by the cumulative distribution function

F(u) :p,"^p[-(;) "] * (1 - p7)exp[ (;) '] (3 3 1)

where zr is the wind speed, p7 is the probability of an annual extreme windbcing produced by a tropical storm, and o is a scale parametcr. The parameter,rr, determined in [3-17] as an empirical function of the mean number oftropical storm passages per year through a five-degree longitude-latitude square,is represented in Fig. 3.3.2.The parameter o is given in Fig. 3.3.3 as a functionol' the maximum of the average monthly wind speeds recorded at the stationconcerned over a reasonably long period (e.g., ten years or so).

The application of this procedure is illustrated in three cases: West Palmllcach (Florida), Boston (Massachusetts), and Columbia (Missouri), for whichpt = 0.43, Pt': 0.72, andPr: O, respectively (Fig. 3'3.2). At West Palmllcach, thc maximum of the average monthly speeds in the period 1952-1974(olrtainccl lnrrn thc Local Climatological Data Summaries) was 13.9 mph at 30It lrhovc gnlrntl. linrnr lrig.3.3.3, o - 5l mph. Thcrcfbre

o n ,.xll (;;) -'

| , ,,r, ,.-,,1 (; )

Lr l :'llMn ll()N ()l I 1lilt Ailt wtNt, :,t 't t lr:, ll! lIrlilili t\|t t,lr{ r,il r ttMAll .. l(}1r

FIGURE 3.3.2. Probability p, of an annual extreme wind being produced by a tropicalstorm. From H. C. S. Thom, "Toward a Universal Climatological Extreme WindDistribution," in Proceedings, lnternational Research Seminar on Wind ElTects onBuildings and Structures, Vol. I, p.682. Copyright, Canada, 1968, University ofToronto Press.

o 2 4 6 8 t0 12 14 l(i l8 20 22 24 26 2830 32 34Maxirnrrrrr nr()nlllly,rvcr,r;r. wirrtl speed (mph)

FIGURE 3.3.3. Scalc panunet('r'o. l;rrrr ll (' S.'l'lrolrr, "Toward a UniversalClimablogical Extrcnrc Wirrtl l)islrilrrrlr.rr." nt l'trtt t't'tlirr,q.t. lntcrnational ResearchSctninar on Wintl lillccls otr llriltlirrl':, ;urtl Strrr, trrcs. Vol. l, p. 6112. Copyright,('rulttllt. l9(113, Univcrsily ol lolonlo l'r, ,.',

-+-

0.17 o:11

lt( t'\'| (r i2)

106 I x I nl Ml wlNl r ( rl lMn l( )l ( )(;Y

Recalling that N llll /,ilrv)1" i( lolkrws lrrrrr Ilt1. .1.-1.2 tlrlrl tlrt't'sltrrrtt(t'tlN-year wincls lirr N 50" l(x), irrrrl l(x)o y('iilri itrt: /'5{) , 102 rrrPh, t'1,x,

120 mph, and t/l(xx) : l9ti trph, rcsllc:t'livt:ly.At Boston, the highcst of thc avcrirgc rttrtrrlhly spccds rccordcd bctwccn

1950,1914 was 18.8 mph at 30 fi abovc grouncl in opcn terrain. To this valucthcrc corresponds o :63.6 mph. With Pr - 0.12, it follows from Eq. 3.3.1that thc extreme wind estimates are zr5s : 106 mph and z/rur : I 19 mph. It isnorcrl that the estimates presented in [3-5| otc l/5s : 88 mph and ales : 93rrrph, that is, considerably lower than those based on Eq. 3.3.1.

Al Cblumbia, Missouri, the probability of occurrence of hurricanes is nilirntl lit1. 3.3.1 becomes

F(u) :".0[ (;) '] (3.3.3)

'l'hc maximum of the monthly wind speeds recorded between 195 l-1974 wasl-5.7 mph at 30 ft above ground in open terrain so that o : 51 .O mph and zr.e: 88 mph, 2,100 : 95 mph and z1ooo : 123 mph. It is noted that the estimatedextremes of [3-51 are lower, that is, u5o : 70 mph and 1]roo : 85 mph. Theextreme speeds at Columbia were also estimated assuming the validity of theType I distribution, with parameters inferred from the l95l 1974 series of thelargest yearly speeds at 30 ft above ground in open terrain. The results thusobtained were t/56 : 66 mph, zroo : 69 mph, and ?1s00 : 8l mph, versus z'so

: 88 mph, urut : 95 mph, and zr1ee,l : 123 mph, as estimated on the basisof Eq. 3.3.1 with the attendant assumptions of [3-17].

Among these assumptions is the relation implicit in Eq. 3.3.1 and Figs.3.3.2 and 3.3.3 between maximum average monthly speed and the extremewind speeds. No fundamental meteorological grounds are offered in [3-17] orelsewhere in the literature for this relation which, frorn the evidence availableso far, does not appear to be justified.

3.3.2 Procedure Based on Climatological and Physical Models ofHurricanesTo illustrate the principle of this procedure, an estimate will be made of theprobability that hurricane winds in excess of 155 mph will occur at any onespecific site on the Texas coast. The following information will be used in theestimation:

o Average number per year of hurricanes with-wind speeds in excess of 155mph moving inland in the United States, t iss. According to the NationalWeather Service, there have been two such hurricancs in thc past 75 ycarsor so, the Labor Day Florida Kcys hurricanc in 1935 ittrtl ltttrricittlc Cltln-illc in 1969 l3-181. A rcasonablc ostirnittc is tlrt'rr ll,tt - 2ltrrlricttrros/(75 ycirrs) 0.027 htrrr/yc:irr.

l:l ll:llMn ll()N l)l I "llrl [Il Wllil):,1'l ll)i; lN llllllltl{ nl]l l'lt{rl.ll r |M/\il:, llll

r Avt'r'rrl.:t'rrrrrrlx'r lrr v('irr ol lrutt'it':rttt's nr()vlrl' rttl;rrrrl trr tlrt'I lrrilctl Sllttc:s,1r,,,.'l'lris(ilr:ur{r(ytrrrlrt't'slitturtcrlI'nrruliil'.. t.t..[:r,, ll(rlrtrtricuttcs/(63 ycrrrs) I .i"i,l lrrrr r'lyeltr.

o Avcragc rrrrrrrlrt'r'lx'r'y(:ll ol'all hun'icattcs trrovittg irrllrrrtl irr 'l-cxas, a7.From Fig.'J.J.4. ttr = 2J hurricancs/((r-1 yclr's) = O.43 hurr/year.

o Average width ol'area swept by winds in cxccss ol'155 mph in onehurricane, I'lz. According to [3-20], thc path of destruction of the LaborDay Florida Keys hurricane was 35 40 miles wide. It will be assumedconservatively that winds in excess of 155 mph affected a width W : 30miles of that path. In the case of huricane Camille, it appears that it maybe assumed conservatively W : 20 miles [3-211. A reasonable value tobe used in the calculations is then W : (30 + 20)12 : 25 miles.

It will be assumed that the average number per year of hurricanes with speedsin excess of 155 mph moving inland in Texas is

'-t ,t;ttU1

(Implicit in Eq. 3.3.4 is the assumption that the probability distribution of thehurricane intensities, given that a hurricane has occurred, is the same through-out the U.S. Gulf and Atlantic coasts.) The length of the (smoothed) Texascoast being about 375 miles, the probability sought is

t55U7

P(u > 155 mphl : ,'r" : :0.00042' 375

(3.3.4)

(3.3. s)

that is, approximately ll25OO per year.The estimate just presented has several significant weaknesses. First, the

errors in the estimate of utfs could conceivably be large, the estimate beingbased on a 7S-year-long record containing just two relevant data. Second, theassumption that the rate of arrival of hurricanes is uniformly distributed overthe length of the Texas coastline overestimates the probability of hurricanestrikes over the coastline segment adjacent to the Mexican border (by about25%), and underestimates that probability (by about25%) nearerthe Louisianaborder (Figs. 3.3.5 and 3.3.6). Third, the reliability of the estimate of er7 isdifficult to ascertain. Indeed, according to Fig. 3.3.6, u7: 1.6/(100 x l0)entries/yearlnmi of coast x 0.53 huricanes/entry x 330 nmi of coast :0.28hurr/year, versus 0.43 hurr/ycar, as obtained from the data of Fig. 3.3.4. (Thisdiscrepancy is possibly thrc to lltc courtlirrg ol'ccrtain tropical cyclones ashurricanes in [3-l9lr';. lirrullr. tlrc esl inurlcs ol'14/:rrc largcly subjccfive, since

'r'Rcl'crcncc l3 l9l w:rs lcvisctl irr l()/li.rrrrl r', ul)(l.rl(,l,rnrrr.rllr lry llrt'N:rtionlrl llrrlritlrrrr'('crrlcrl3 -521. Arlditiorlrl ittlirttt:rliott otr Norllr r\ll,rrrlr, lrrrrrr,,ur". r', ,r\r;rilirl)lc irr l1 -5 11 lrrtrl. on llrpt',irr I I 5'1 1. lirr irtlirtrrtrliort tttt Wr':,lcttt Norllr l'.r, rlr, lr{rlr! .rl ( \r lor( s. rt't' I I 551

lrirrlrrrrl I

i

L

I

d>ao\--g 5".'=o-.: -. cn, g2\o

9 -PE!adgtool E.c.F !

^ilo =t:(/oU(E>, = ,-j

dotr.:!d

trE --^O

o9"iLF*dJ9Yrc H=o 5;h 5

.r-.i " O

_* boU

<. aJ -C.o!s?e

e4A__ a0;llE@

rl o-'x

lrrrtl lrtrtl lrrrrl 'ir r r r'l'

zJ4zU_Eg

o>UOk =o(J(rU=or2 -<-Ezo&-rrllo(/)r':tzz 64- trlo zzuJ<9U>EKE.UUFFzzUUOOtlc

itEiP

I:q

c.S;:\ (t)TE.\(r\908'r Eiin9()trltdErIOr?rHt%n

Ao,--^r=jfA^t:11\\

qmt'aJ.\

&\'p^,/e )try iJ/ls{ t1| _-*1 )-V

lrrlnO

.' u/ 't !' '.1

; l'jl!t,l ,1,":,:- -1,, '.'i;:'l

-,

109108

tll

Nurrrlr:r ol crrtrrr..;/l(X) yrr;rrt/lO ntnt of coaslONNto rr rt, O A

9s)99 5qo!Ratio (hurricanes/total srorms)

Nbo

su,4 s.)tlsr.rJl.lr:.il:rl.t.lu0l.)A.l lu.)t(l().tl U() Uotll:tU.t()llil.l() slsll(1 .)(ll u() Jlls ll lllsJrlrlr(lr:(l().r(l l)Ur/r\.rlt.).t lxJ.l() ifuillolx)rU or.ll ()l r-lrr:().r(l(lu,)All.).).llJ l)lilr JArsuitlr).r(l-rrr()J v 'spaads pully'r ouea!ilnH 6u//|€tu//ls3 Jo, arnpacotd opec auow

'uorleturlse eql ur pe^Io^ur serlureuecun eql lo cillJrdInJ8urueew ,{1pcrs,(qd 'Juelo u JeJo feqt 'peeds flqtuoru eSeJa^e rununxurueql uo pesEq lepotu eql e{rlun'1eq1 sr s'€'€ puE ?'€'€ 'sbgJo ernlBeJ Injosrr rl

'oslv 'seleJ esuBJnsur Jo suorluurrxoJdde lsJg JoJ 'eldl'uexe JoJ 'suorSer ouo.l(l-auBclunq ur seJnlcruls Jo eJnlreJ Jo ,{lrlrquqoJd eqt 1o seleturlse ssoJS SurluilrJoJ esn etuos Jo eq ,{?lu peurllno lsnl qrBoJddu oql sesselr)lee,$ eseql etrdso( I

'lle^\ se l4 Jo uorleurlsc crll ullu€JuruSrs eq eJoJeJoql ,{uu ro.ue aql :plou aql ur uo{Bl aJo,r s}uetueJnsuour oU

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t

JFtsI {A

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0a.n'3=

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=yd 5

=! ir5

J.9 - oxe.DD@

-o 36 l./ YoI

\o+a; -J--ur!) 6-3b r*^' a6 ry=-J

:'m r

*-/(-F-,- )oz.-\\

\,1*

I@ I@

ltil rrr dl

^o_o

=.? o!1 fii3aoac3goo

o3

Lake Charles, La.+

St. Marks, Fla.

Ft. Myers, Fla.

Miami, Fla.

Cape Hatteras, N.C

0rr Alx)t()lvv\It:) (tNtM tv\t IIIx I

112 I x tttl Ml wlNl) (

tlcvcl6pol in l-3,2-1 l.'l'his ir1-rprxrclr wrrs srrbsctlrrcrr(ly irplllit'tl irr l-l )'ll' wllercextreme wind speecls associatccl witlr lrtrrricrrncs wLlro cstirrratctl ott tltc bltsis ol'the climatological and physical modcls dcscribcd bclow'

Climatological Models

1. The hurricane frequency of occurrence is modeled by a Poisson processwith a constant rate.

2. The probability distribution of the pressure difference between center andpcriphcry of the storm, Ap,,"*, is lognormal. To eliminate values of Ap.,*jrulgctl, in thc light of historical data, to be unrealistically high, thetlislribrrtion is ccnsorcd stl that AP,."* ( 101.6 mm (4'00 in) of mercury

l.\2.]1.(Ntl(r:tlrirtA/).''.*:l0l.6mmcor.respondstothelowestatmo_sgrlrcle l)rL:ssurc cvcr rccorclcd worldwide t3-25].) Theoretical studies ap-pcru. (o cottlirttl this bound l3-61].

l. 'l'hc p(rbability distribution of the radius of maximum wind speeds, R,is krgnorrnal. This clistribution is censored so that 8 km < R < 100 kmto avoid unrealistically "tight" or "broad" storms 13-231'

The average correlation coefficient between R and Ap-,^ is about -0'3'(see 13-221, pp. 68 and 69.) All other climatological characteristics ofhurricanes are statistically independent.The probability distribution of the speed of translation, s, is normal. Thisdistribution is censored so that 2kmlhr < r < 65 km/hr 13-231'

The probability distributions of the distance between any specified pointon the coast and the hurricane crossing point along the coast (or on aline normal to the coast) are curves matching the historical data. Separatecurves are defined for entering, exiting, upcoast heading, and downcoastheading storms.For entering storms the probability distribution of the direction of stormtranslation is a curve matching the historical data. For exiting upcoastheading and downcoast heading curyes the distributions are uniform be-

tween 130" of the mean directions of storm translation. In all cases thestorm path is assumed to be a straight line.

Physical Models

l. The maximum gradient speed is given by Eq' 1'2'8 in which r : R' and

in which it is assumed that

- dAP,,t,,,

13-261. This rclation corresponcls to thc avcrage of data observed duringthe 1949 hurricane that crossed Lake okecchobee, Florida [3-27, 3-2g].whether Eq- 3.3.7 can be assumed to be generally valid is uncertain.For example, according to [2-l4ll, during typhoon Mireille observedsurface wind speeds over ground (which are lower than over the ocean)were comparable in the region of the eye with estimated winds at thegradient height. However, according to [3-79], measurements indicatedthat the 10-m level sustained surface winds over water were generallywithin 55% to 85% of the winds measured by reconnaissance aircraft at500 m to 1800 m. Reference 13-791also suggests that the logarithmiclaw appears to be valid up to about 1g0o m but that at aboul 3000-melevation winds may be less than at the surface.Let the center of the storm be denoted by o, and consider a line oMthat makes an angle of ll5'clockwise with the direction of motion ofthe storm. The l0-minute wind speed at l0 m above the ocean surfaceat a distance r from o along line oM is denoted by u(10, r). The ratioU(10, r)lU(10, R) is assumed to depend on r as shown in Fig. 3.3.713-261. Let the angle between a line oN and line oM be denoted by 0.

I li;llN,4nll()N()l l!llli fint WtNt):it't tt)l; tNil( ,tilil(nfJt I.t t.,rJt 1.t tMnil:, ll3

'l'ltc tttltxilttrrrrr rvrrrtl sl)(.(.(l:tl l0 r1 lrlrpvt'(lrt.6t.t.:rrr srrr.lltt.c, itvctirgctlovcr l0 lrrirrrrlt's, rr p,rv(.n by lhc clnpiriclrl rcllrliorr

t/( 10, R) : 0.865/",(/t) I 0..5,r (3.3.7)

t,l I 0,,)

rrr(;rjrr,r-r..1.7. rr;rrr',, lil';li , ttrt1111li) |r .r(,1

')

-1_

4.

6.

7.

50

40

30

EEd

l0I1

6

5

4dpdn

whr-'l'c rv is ohl:tinctl lirr crrrpiricirl tllrtrr l3-2(r. .1 2ltl

(3.3.6)

I l4 I X i lll Ml wlNl) ( ,l l[/n l( )l ( )( '|Y

l'lrc lO tttittrttc wirrtl sllectl l/1 lO,(.rN is givcn by Lhc cxPlcssion l'l

U(10, r,0) : {/(lo' r') I ,' cos (/)

. ,l,lrt. witul vckrcity vector has a cornptlncnl. t.lirccted toward thc center oflltt.s{tll'llt'().Theanglebetweenthatvcctorandthetangenttothecirclett'rr(r'tt'rl rrt O varies'iinearly between 0o and 10" in the region O I r' /i :rrrtl l)ctween 10" and 25" in the region R < r < l'zR' and isr'r1n;rl l,r .15" in the region r > l'2R13-261'

' llr,' :,lorttt clccaj results from a decrease with time of the difference

lrt lrl,r'r'il l)tcssure at the center and pressure at the periphery of the storm

rr ,r( ( ()r(llrtlce with the relation

r'. //) trl :t tlis(lttlt't' r' llrttt () irlolrg lirrc2(rl:

(3.3.tt)

Lp(t) : APn,"* - 0.02[1 + sin {lr (3.3.e)

ir:r r:iilMn il()u ()r rrrr I\,4r wll.il,:ir ,r rr): ; rN ilr,rrrr{ Ar]r ll(,r!r (1Mn 1:, r15

l,cl llrt'lllrlxrlrilily llr:tl llrt'wttttl spt't'rl irr rrrry orrt.slorrrr is lcss tlriyr sgl.rrcvttltlc' tr, bc tlt:ltolt'tl lry /',, l'lrt'pnrblrbility llr:rl llrt. lrililrest wirrtl {/ irr rr st1;rnrsis lcss lltart ll t':rrr lrt.wttltr.n:rs

(3.3.11)

(3.3.rZa)

(3.3.tzb)

(3.3.t2c)

I''(U < uln) : 1"',', (3.3.10)

The probability that U < u in z years is denored by F(U I u, r). The totalprobability theorem (Eq. Al.5) yields

F(U < u, r) : i^ ofu < uln)p(n, r)n:0

where p(n, r) denotes the probability that n storrns will occur in r years,Assuming thatp(n, r) is a Poisson process (Eq. A1.34), Eq. 3.3.11 becomes

F(U < u, r:) : ! o' (\t)'" ^"n:O ' nl

- ')"3txt41'-(4- r:0 nt_ --)tnl - 1,1-(

where \ is the annual rate of occurrence of hurricanes in the area of interestfor the site being considered. For z : r, F(u 1 u, r) is the probability ofoccurrence of wind speeds less than u in any one year.

consider now the wind speed, ui.The probability that u I u,in any onestorm is

rrlrr rr. / . travel time in hours, Ap(f) and Ap"'o* are given in inches ofrrr'r{ rrry. and @ : angle between coast and storm track (0 < d < 180')'llrr', rrttrtlcl is consistent with measurements repotled in 12-1241'

(rllrr'rt'tluctionofwindspeedsduetoincreasedsurfacefrictionoverlandr" r'rv('n by the t u'irollu'(10) : 0'85' where u/(10) and u'(10).rtr.lltcl0-minutespeedsatl0-melevationoverlandandoverwater,rr':,lrcetively.ltcanbeverifiedthatthemodeldevelopedforextratropical'.r,,,,rrs (Flqs.2.2.29 ancl 2'2'31 ' and Table 2'2'3) would yield a some-

rvlurt smaller ratio Ul( l0)/U''(10)'/ llrt: clependence of wind speeds upon averaging time is modeled as in

Sr'ct.2.3.6.

Notc that physical models proposed in [3-28] are in some cases slightly

rir,rtliliccl wittr respect to the corresponding models o1' l3-26].lrstirnates of the probabilities of occurrence of hurricane wind speeds were

rrtrrrrirrcd inl3-24lby assuming each of the areas adjoining 56 mileposts (Fig'r 1.5) to be hit by m : 1000"hurricanes. The climatological characteristics ofllrc hurricanes were cletermined by Monte Carlo simulation from the respective

lrKrbabilistic models as fitted to historical data. For each of the rz hurricanes'

thc climatological characteristics used in conjunction with the physical models

dcscribedearlierdefineawindfieldwhichdependsuponthepositionofthehurricane.Toeachpositionofthehurricanewithrespecttothesiteofinteresttherecorrespondsawindspeedatthatsite.Windspeedscausedbyahurricaneat the site are calcurateJ ro, u sufficiently large number of such po-sitions. The

largcst among these speeds is the maximum wind speed caused by the hurricane

atthcsitc.Asctill.mspccdsisthusobtained,whichisusedasthcbasicsetol'rllrlu lirr tlrt: cslirrurtion ol'tlrc prtlbilbility ol'()cctrffcllcc ol'ltttrricrrrtc wind

sPt't.tls, 'l'lrt'st' sPr'ctls :tR' tltttkt'tl lry rtrilgtriltttlc 'l'lrc i (lr srrt:rllt'sl spt't'tl ilt lt

s..'t ,,1 lr wiltrl spt't'rl:; is tlt'ttolt'tl lry tt,.

(3.3.13)

F(.U < ui, l) : e ^(t i/m+t) (3.3.t4)

For each of the mileposts in Fig. 3.3.5, estimates of hurricane wind speedscorresponding to various probabilities of occurrence (or mean recurrence in-tervals) were obtained in t3-24l both at the coastline and at various distancesinland lrom rhe coastlinc.

Results of a rccent stucly inrlicrrlt' llr:rt lrrrrr.it'irrrc wincl speed data obtainedby simulati<ln and ftrrnring lhc brrsis ol llrt't'stirrr:rtr.s ol'13-241 rnay be clcscribeclby thc rcvcrsc Wcitrull rlistt'ibrrtiorr l l /ll 'llr;rr tlistr.ilrution lurs lirnitcrl uppcrtail and is consistcnl willr llrt':rsslrnlll rln tlr;rt lrrrrlt.lrrre witttl sPct:tls irrchrltltttlltl. Ilcl'crt:ncc 1,1 7l l lllrvi(l(':i 1rl()llrjrtrr)!r {)1 :rt r'r'ssirr1,, llr1;st' tl:rllr (llrt'rllt(lr is ltlso ltvirilltltlt'irr ll.i ()l). lr:; rr,,.ll tr.. ( r)!nIrttr.r l)r1)t,tiuns lor llrt.(.sllu:tlt()lt

_tA -_"' m*1Thus

I l6 I X llll Ml wlNl) { )l lMn l( )l (l( tY

()l tllc t(lvcrst: Wcilrttll tlislt'ilrrrtiorr l)ill;lllrt'l("ls:tttrl ol ltttttit'ltttt'witxl sPtetls

witlr vltt'iotts lllcan rcctlrt-ctlcc irttcl-vltls'llsti.urtctl hurricanc win..l ,gr.,,1, tlillr'r lnrrrr tlrc "tt'rtc" s1.rc:ctls owittg ltt

.5se'vuti.., pnlhabilistil ltx)clolillg, lllysiclrl lrrtxlclittg' irrttl sarrtllliug crrors

(i.t'.'clrrtrscluctothelimitcclsiz,ctll.(Itctllttlrsirttrplc.sbr:ins.u'"..])ljtlrwindspt'etls with lltcan .""u""nt" intervals tll'thc ordcr rll'50 ycars' it was shown

rrr I I ]01 rhirt thc *tunJura deviation ()f thc sanrpling errors, duc t<l thc limited

:;t.r .l t.lirrrrt.r.gical Jata availaule (about 100 years), is about l0% of the

,1.,1 ,,r':rr,'.1 s1,".,1*' it is about 15% fot 2000-year speeds'

I'r.t.t.tlrrrt.s ti,' "*ti,niting-r,u.ri.un" wind speeds-that-are similar conceptually

r. lt .)\l rvt'rt' ,r.,u"r.rf"J-in i: i1]^ana tS-:Zl for the purpose of studying

lnil | r, ;ril(' 1't'ttt'trtlccl *uu"'' A simplified methoJ for estimating hurricane wind

..1,,,,1', \\'it:' l)l()l)()scd in [3-56]'

Mrrrlr.' (--irrlo Simulations Based on the Shapiro Hurricane' Boundary'

I:ryt.t Mtt<Ial- 'l'hc #;;il; boundary-laver flow in a translating hurricane

\\';,'. .,l,r,r('\trrtrtlocl i"i;-;ii Lt uling empirical physical models^based on his-

r.rr,;rl ,lrrr:rt.l.gicat Jata,-as lndicated earlier in ittit section. Shapiro [3-57]

1|1'r,r.lr11ir'tI ,..' ..pp.o*i*ut" model of the hurricane boundary-layer flow based

orr ,r ,,rrrr1,lrlr,..t ,olution of the Navier-stokes equations' complemented by the

l()ll{,\\rnr' :rssrtttrptions: the boundary-layer height 6 and the eddy viscosity are

(..r.,r,,rr (;, I m; f ='i;1dt;t; '), uid th" frictional drag due to the

Irrrr',1;rltott vt'ltrcity i, "qtluf to the square of the flow velocity times an empirical

trrt t'rr , r,r.llicient rrr"il"J* rin"u.iy with flow velocity. Rather than obtaining

tlrr' lrrllv rrottlinear solution of the equations of motion' a simplified approach

w;1.'tt''t..lwltcreinthevelocitywaswrittenaSthesumofanaxisymmetrictetmlttt,lllrt.lllsttwot",m,ofuseriesreflectingtheflowasymmetryduetohurricanelriilr:,l.rlr()il. The facf that the series is truncated af'ter the first two terms is

t':,trrrr,rlt'tl by Shapiro to result in an approximation of the fully nonlinear so-

Irrrrrrrr lrr within uAoiZS% t3-57, p' f q96t' This errorcstirnatc doesnot include

nr,r,l, ltttg crrors; "u"" if 'f'L fully nonlinear solutions wcrc available' the Sha-

1,rr,, rrurtlcl would ";;;; uut" to iescribe the detailed s(ructure of the boundary

!,rv.r. cspecially near the eye wall [3-57' p' 19951'

liesrrlts of ri-ufution' Uu'"d on ih" Shapiro modcl wcrc reported in [3-58]'

rvlrrth used Shapiro's truncated series approach' The rcsults of the simulations

tlrllt:r from those of 13-241 inone main respect 13-591' Owing to a new and'

ilr ()ur opinion, *"diil. nuing rate model that is less conservative than its

toLrnterpart in Ref. t'iOl,the! yield.lower wind speeds inland' Otherwise' in

spitc of the use "t ln" Sit"pit mode-l'-they yield wind speeds-comparable to

those of 13-241. Although Ref' t3-581 rnual use of data and models from

13-601, whereas f3;4f ;?; basecl tn [3122] (i'e'' an earlier version of [3-60])'

this is not a Source of significant dill.erences. E,stimated wind speeds obtained

in 13_581 are lisrccl;; i;;i;3.3.1, which atso shows cstimatcs fronr [3-24],

t3-29l.arrtl|3-71|.Ntltc(Iur{(lrr:clllllparistlnstll'|3_5ttIllclwcclrtltt:cstitrratcs (,

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I I B I x tltl Ml wllll) (,1 IMA l{ )l { )( iY

<ll'13 5ttlirrrtl 1.1 241irrc irrvirlitl 0wirr1', lo lrrr irrtorrsislt'rtt'y itt lltt'll:tltslirtlttlttiottsof hourly spccds iukr lirstc:st rrrilc spt'ctls. 'l'ltcse Ir'ltttslitt'tttttlirttts lttt brtsctl itt

[3-58] on the moclcl ol'12-1351 lirr (lre e:slirrur(t's ol l.] -5ltl, bLt( ott lht: tlill'crcntmodel of [2-91] for the cstimatcs ol'1.1-241. lior 50-yc1r spcctls, tlillcronccsbetween [3-24] and [3-58] excecd 10"1, lol trrilcposts 2(n. 3(X), ancl l3(X). For2000-year winds the differences excccd lO'/,, l'<tr milcposts 700 and 1700. Atl0 m above open teffain, the hourly spced corrcsponding to the largcst 2000-year estimate of [3-71] is about 47 mls at l0 m over water the cstimate of thelargest 2000-year hourly speed would be about 41 x 1.2 : 56.4 mls'

Load Factors. It is shown in [3-71] that the wind load factor $,u : 1.3

spccified in the ASCE, 7-95 Standard would in most cases coffespond for wind-scnsitive structures to nominal ultimate wind loads with mean recurrence in-tcrvals of, roughly, 500 years or less. For the other sets of estimates of Table3.3.1, the load factor d' : 1.3 would in many cases coffespond to nominalultimate wind loads with even shorter mean recuffence intervals' These resultsare reflected in the average estimated ratios of 2000-year speeds to 5o-yearspeeds, which are about I .3, 1.4,1.45, and 1.46 for the sets based on t3-7ll'13-241, I3-2g1, and [3-58], respectively, so the squares of these values are

uUo"f t.Z, 1.g5,2.1, and 2.15, respectively. The results of [3-71] and Table3.3.1 therefore indicate that for wind-sensitive structures, the wind load factorfor hurricane wind speeds should be larger than 1.3, even if hurricane designwind speeds are multiplied by a factor of 1.05, as is done in the ASCE 7-95Standaid t}-t3gl. For additional details, see [3-80], tA3-3ll and Sect. A3.3'

Estimates of Hurricane/Tropicat Cyclone Wind Speeds for Various Lo'cations Outside the U.S. Estimates of hurricane/tropical cyclone wind speedsbased on models similar to those of 13-241 were repofted lor French overseasdepartments and territories in [3-63] and are summarizcd in Table 3.3.2.

Estimated standard deviations of sampling errors in m/s were 2.2 (3.4),2.5(4.2),2.0 (4.0), 1.6 (2.5) and 3.9 (12-3) for 50-yr (1000-vr) speeds at Gua-daloupe, Martinique, R6union, New Caledonia, and Tahiti, respectively'

Estimates of hurricane speeds are reported in [3-64] for the Eastem Car-ribean, Jamaica, and Belize and in [3-65] for the Northern Australia Coast.For information on westem Norlh Pacific tropical cyclones, see

t3-ssl.

Saffir-Simpson Scare. The National Hurricane Center, the National WeatherService, and emergency management departments use a classification of hur-ricanes into five categories (Table 3.3.3). The central pressure and wind speedportion of the classification was proposed by H. Saffir in 1970, while the stormsurge portion was added subsequently by R. Simpson. Thc avcraging timc,height above grouncl, and surfacc cxposurc (i.c., whcllrcr ()l)cll lcll'llilt or wlttcr)

lil l:;,Mn '()N

{)t I r.,llA,lt WtNt) :it,t tt)ti lN ,'t.'( A,t t,t tr', (.t tMn '

:, l1g'l'Alll,l,l -1.-].2- l'lslirrrrlrrl wirrl s;x'r'rrs (r(!r'irr s;x.r.rr irr r0 rrr ()vcr.irrt.()rcarr)!.r-6rlMcanrcturnperiod(yrs) Martiniquc

sr.Barthilcrrry (iuatlcloupc New

Ildunion Caledonia Tahiti3034394752

2550

100500

r000

32374t4952

32

404750

2935394851

3840434850

3840434849

for the wind speeds are not specified in the classification. For this reason storrneffects and evacuation requirements for the various categories are described forthe use of, among others, emergency management personner 13-661, rather thanstructural engineers or building code officiils.

Mixed Distributions. Hurricane-prone regions are also subjected to winds notassociated with hurricanes (or tropical cycrones), whose effects can be ac-counted for by developing mixed distributions of hurricane and nonhurricanewind speeds. Since the occurrence of hurricane winds and the occurrence ofnonhurricane winds are independent events, it is possible to write

F(.U < u) : F11(J <. u)F7s11(J I u) (3.3.1s)

where F(u ( rz) is the probability that the wind speeds u associated with anystorm are less than u in any one year, and Fs(U ( z) ancl Fuu(J I u) arethe probabilities that hurricane speeds and nonhurricane wincl'ri""a, are lessthal a in any one year.-The probability Fs is determined as shown previouslyin this section' The probabitity F,vi7 is determined as shown in sect. 3.2.

TABLE 3.3.3. The Saffir-Simpson Scale

Category DescriptionMean WindSpeeds (m/s)

Storm Surge(m)

North AtlanticExamples

1

2345

MinimalModerateExtensiveExtremeCatastrophic

33-4243-4950-.5f1.59 #)

( ilctrlcrlltlur (r()

t.2-1.6t.7-2.52.6 3.83.9- -5.-5( i r clrlcrllr:rrr.5.-5

Agnes 1972Cleo 1964Betsy 1965David 1979Camille 1969

l2O {xllll Ml wlNlr (.

('irlculirliorrs tcptttlt:tl irr 1.1 l':l lsll(rw thlr( lltt'Prrrlrirbility /'lll 'r) r\ vir'

lulrlly tltc rurn" r,li I,'tr(/ < tt) lirl'lttcltlt lc('llll('ll('r'ilt(t'l'vltls N ' 5O yt:1r's'

Fo. N : 20 ycars, cstiptl(ccl wirrtl spce:cls tlr:rt irrcltttlc: tllt: cllcc( ttl'Itotllttlrricltltcwincls cxceed the esl"imatccl hurricailc wiltcl spcctls lly al-tout 5%' Nrlto that

these conclusions are not neccssarily appliclrblc ntlrth tll'Capc Hattcras' whcrc

nonhurricanewindsmaycontroltheclcsigna(coflainlocatitlnsl3_33|.

3.4 WIND DIRECTIONALITY

Witrtl cllbcts on various structures and components depend not only on the

nrilgrritudo of the wind speeds but on the associated wind directions as well'lr.i this rcason, knowledge of continuous joint probability distributions of ex-

trcrne wind speeds and directions would be useful for design and code devel-

0pmentpu,po,",.However,sofarnocrediblemodelsforsuchdistributionshave been proposed in the literature'

In the abience of such models, wind effects and their probability distributionsmay be estimated in well-behaved wind climates on the basis of informationconsisting of largest yearly wind speed data recorded for each octant overperiods oT ZO y"uit, tuy, oilong"r (see Sects' 8' l '2 and 8' l '3)' Such data have

teen published for a number of U.S. weather stations in [3-341. Summary

statistics of largest yearly wind data recorded at Sheridan, Wyoming, in the

period lg58-1l:i'7 (iee 1'aUte 8.1.2) are shown in Fig. 3.4.1. h is seen that in

ihi, "u*" wincls blowing from the noftheast are considerably weaker than north-

west or southwest winds.As shown in Sect. 8.1.3, there are important practical applications in which

information is needed on the univariate probability distributions of the largest

yearly wind speeds associated with each of the principal compass directions,una on the correlation coemcients for the largest yearly winds blowing from

any two directions. In well-behaved climates the largcst yearly wind speeds

ioi ony given direction are in most cases-though not always-adequately fittedUV fVpJI distributions of the largest values. As indicated in [8-141, the cor-

retatlon between wind speeds occurring in any two of the eight principal com-pass directions is in -ort .ur"r weak. For example, the estimated correlationcoeflrcients between wind speeds from directions i and i(i,i : 1,2, " ' ' 8)

are shown for Sheridan, wyoming, in Table 3.4.1. These values are fairlytypical. However, there are stations where the values of the correlation coef-

dci"nt, are higher (e.g., Detroit, Michigan, where 8 of the 28 estimated values

are larger than 0.45, although none exceeds about 0'6)'An important practical p-ut"- faced by the designer is obtaining the largest

yearly wind ,p""d dutu for each of the eight principal compass directions at

io.ution, not covered in [3-341 . There are two such sources of data- One sourcc

consists of the original LrnpLrblishecl rccorcls storecl hy thc Nlr(irlnlrl Ot:canic and

Atrn0sphcric Arlrilinisf nrtiorr (NOAA). Ob(aining ittttl e xlrltt'lilrg thc rrcctlctl

rl:lltr lhltil tlrttst.rct'()ttls is lrollr irtcortvcrricnl rriltl littlr'('()llsllllllllI A st't'otttl

r ,t wllll, i,il lt I :ilr )Nn I il v

FIGURE 3.4.1. Summary statistics of largest yearly wind speeds by direction at Sher-idan, Wyomine (1958,19j1).

source consists of published Local Climatological Data summaries issuedmonthly by NoAA. Directionar largest yearly speeds in the published datadiffer in a few cases from the corresponding speeds in the original records.The reason for these differences is that the published data consists of (l) thelargest daily speed for every day ofthe year and (2) the direction for that speed.

TABLE 3.4.1. Estimated correlation Coefficients for Directional wind speedsin Sheridan, WyomingDirection

I I -0.05 0.352 1 0.013t45 symmetrrc

61lt

0.12 0.16-0.1-5 -0.340.04 0.34

| 0 17I

*0.031 -0.22 0.070.03 0.07 -0.010.10 -o.12 -0.430.03 -0.16 0.400.01 -0.16 -0.41t 0.20 0.01| 0.32

I

20 30 40 50 50

122 IXiltt Mt WtNt )(jtMnt{)l (x;Y

Cgnsidcr, lirr cxartrplc, lIc ctrse wlrr'tt'in lr givr'tt ycltl lltt'lrrr'g.est prrlrlisllt'tlspeeds fbr wincls bckrwing l'nrrrr lhc r)()11lr lrxl lltc c:ltst arc 70 tttplt ittttl (r-5 rltPlr,

respectively. It is conceivablc that on thc salnc clay (ha( thc n<lrtlr witttl occurrccl'the winds blowing from the east were 69 rnph. Thc highcst wind spccd f'rom

the east would not be reflected in the published data'An exhaustive study of original and published data listed in 13-341 for 24

stations showed conclusively that the extreme wind speed estimates based onpublished data differ insignificantly (by about 3% or less) from those based onihe original clata. It is, therefore, appropriate to base structural engineeringcalculat"ions on the largest yearly directional fastest-mile wind speeds obtainedl'rorn l-ocal Climatological Data summaries t8-14]. A novel probabilistic ap-pnxrch to thc modeling of directional extreme wind speeds, in which the ex-trcnrc valuc distribution parameters are functions of direction, and which ac-

counts lirr the correlation among extremes across directions, was reported in

I 3-67 l.In hurricane-prone regions estimates of hurricane wind effects can be carried

out on the basis of hurricane wind speed data generated by Monte Carlo sim-ulation for each of 16 directions, as shown in sects. 3.3 and 8. 1 .3 (Eqs. 8. 1 .21-8.1.23). Such data-used in 13-241 fot estimating extreme hurricane windsblowing from any direction-are listed on tape in [8-9] (see also t3-7ll) for 56mileposts (Fig. 3.3.5).

3.5 PROBABILITIES OF OCCURRENCE OF TORNADO WINDS

consider an area ,40, say, a one-degree longitude-latitude square, and let the

tornado frequency in that area (i.e., the average number of tornado occulTencesper year) be- denoted by D. The probability that a tornado will strike a particularlocation during one year is assumed to be

aP(S):t-Ao

(3.s.1)

where c is the average individual tomado area. In certain applications, forexample, the design of nuclear power plants, rather than the probability P(S)'it is oi interest to esrimate the probability P(S, Z0) that a tornado with maximumwind speeds higher than some specified value tr/' will strike a location in anyone year. This probability can be written as

P(5, vi : P(viP(s) (3.s.2)

where P(Z) is the probability that the maximum wind spccd in a tornaclo willbc higher than 2,,.

tlsiirlu(cs 6l'p1rl'rlrbililics /'(S) in thc Urtitc:tl Slrrlt's rttt' sllowtt irl lril', 1.5' l.

r,, l,ll()ltnllll llll :; ()t ()(;(:tlilt il t.t(.t o; 11rt thlnt )() wtNt): ; t23

FIGURE 3.5.1. Tornado strike probability within 5-degree squares in thc contiguousUnited States (units are l0 5 probability per year) [3-351.

which is taken from 13-351. Figure 3.5.1 is based on Eq. 3.5.1 in which D wasestimated from l3-year frequency data, a : 2.82 sq. miles (as estimated in[3-36] for the state of Iowa), and,46 : 4780 cos S, where @ is the latitude atthe center of the one-degree square considered. Estimated probabilities p(zn)are shown in Fig. 3.5.2, also taken from 13-351. These estimates are basedupon observations of 1612 tornadoes during l97l and 1972, and the rating(largely subjective) of these tornadoes according to an intensity scale proposedin[3-371.* It is noted that in estimating the probabilities of Fig. 3.5.2 it wasassumed that tornado path areas are the same throughout the contiguous UnitedStates.

The maximum speed of the tornado corresponding to a specified probabilityof occurrence can be estimated using Figs. 3.5.1 and 3.5.2. According tot3-351, "in order to adequately prorect public health and safety, the determi-nation of the design basis tomado is based on the premise that the probabilityof occurence of a tornado that exceeds the Design Basis Tornado (DBT) shouldbe on the order of 10-7 per year per nuclear power plant." The requiredprobability P(Ze) is then determined from the relation

P(r/o)P(S) : t0-7 (3.s.3)

*According to this scale tomadoes may bc dividcd into the following classes: F0 (maximum windspeed <72 mph), Fl ('73-112 mph). F2 (lll 157 rnph). F3 (l5tt 206 rnph), F4 (207-260 rnph).F-5 (261-318 mph), and F6 (3 19 3ll0 rnphy

124 I X llll Ml wtNl) ( ;t tMn l( )l ( x iY

It too=9flat73oz;5

4

o.l o.2 0.5 I 2 5 lo 20 3040506070 80 90 95 9899PERCENT PROBABILITY

l'l(;llltl'l -1.5.2. Percent probability of exceeding ordinate value ol'the wind speed [3-lsl

wlrt't' lhc value of P(S) for the location considered is taken f'rom Fig. 3.5.1.'l'lrc wincl speed corresponding to the probability P(lze) so determined is thenrrlrtrrirrccl l'rom Fig. 3.5.2. The average tornado intensity with a l0-7 probabilityP('r ycar for each 5-degree square in the contiguous United States, based onlitl. 3.-5.3 and Figs. 3.5.1 and 3.5.2, is shown in Fig. 3.5.3 13-35].

lior nuclear power plant design purposes, the contiguous United States aretlivided, in [3-35], into three tornado intensity regions shown in Fig. 3.5.4.'l'hc corresponding tornado winds are given in Table 3.5.1.

Thc pressure drop due to the passage of tomadoes can be estimated fromtlrc ccluation f<rr thc cyclostrophic wind. Using the relation Vr, : drldt, Eq.I ..1.2 crrn hc wrillcrr lrs

1,, t,t t(rt rnilll llll :; ()t (xt(;lItl il N(.t ()t t()t tN/\t )() wtNt]:i r25

FIGURE 3.5.3. Calculated tomado wind speed by 5-clegree squares for l0 7 proba-bility per year [3-351.

wherep is the pressure, / is the time, 2,, is the translationar speed, p is the airdensity, R- is the radius of maximum rotational wind speed, and z, is themaximum tangential wind speed* t3-351. Assuming R- is typically 150 ft forintense tornadoes and that Vt = Vrur, Eq.3.5.4, in which the parameters ofTable 3.5.1 are used, yields approximately the values of rable z.s.z 1z-2s1.Following the development in [3-35] of the estimates summarized in Tables3.5.1 and 3.5.2, vaious attempts to improve the probabilistic and physicaldescription of tomado winds have been reported [3-3g, 3-3g, 3-40, 3-41 ,3-42, 3-43,3-44,3-45, 3-46, 3-4i1. Using as a point of departure tornado riskmaps presented in [3-46], a regionalization of tomado risks that divides thecontiguous United states into four areas was proposed in [3-45] (see also [3-44,p. 4801. Regional tomado occurrence rate (per mi2 per year) were estimated in[3-45] from a29-year (1950-1978) data bank maintained by the National SevereStorms Forecast Center and comprising about 20,000 reported tornadoes. Theseregional occurrence rates are corrected in [3-43] and [3-45] to account for:

l. Failure to record tomado intensity, which affects about lo% of the totalnumber of reported tornadocs. J'his corrcction is based on the assumptionthat unrated tornadocs ntay bc apporlionccl anrong the various intensitycategories according lo lltc tcporlctl tor-rriukr licquencies lor those cate-gories.

r'llrt'nrlational spcctl (,,, is tltc tlsttll:rtl ol tlrc tirrrllr'rrtrrl;rrrrl rirrli:rl vt'ftx,itit.s.

FOR ENTIRE CONTIGUOUS

FoR ALL STATES wEsT oF 1O5O w LoNGITUDE

( 1.5 -l)

oo

oz

\'$rrNot

N€iN

:t,, I'il{ )ilnltil ilil :i ()t

l'Alll,lj J.5.1. llcgirlu:rl'lirrrratkr Wirrds

( x;cunt u N( .t ( )t t( )nNn t ,( ) wtNt)ti 127

MaxinturttSpeed (,,,,*

Region (mph)

RotationalSpeed tr/,.,

(mph)

TranslationalSpeed 2,,

(mph)

Radius ofMaximumRotational

Wind Speed R,"(f0

Iilm

290240190

150150150

360300240

706050

Temporal variations in tomado reporting efficiency. The number of re-ported annual tomado occurrences in the United States has increased fromabout 250 in 1950 to 850 rn 1979. The growing trend in the number ofreported tornadoes during this period has been ascribed to a correspond-ing increase in population density. An explicit relation to this effect hasbeen proposed in [3-47]. Corrections accounting for tornado reportingefficiencies were effected in [3-45] by averaging the 1971-1978, 1970-1918, 1969-1978, and 1950-1978 data and assuming that the true oc-currence rates are equal to the largest of these estimates.Possible errors in the rating of tornado intensities on the basis of observeddamage. The reason for the occunence of such errors is that maximumtornado winds are in practice not measured, but inferred, largely on thebasis of professional judgment, from observations of damage to build-ings, signs, and so forth [3-42].Inhomogeneous distribution along the tornado path of buildings and var-ious other objects susceptible of being damaged. In the possible absenceof such objects over the portions of the tornado path where the windsare highest-or even over the entire tornado path-the rating of the tor-nado is bound to be in error. The effect of corrections for such errors isto increase the estimated probability of occurrence of tomadoes withhigher intensities.Variation of tornado intensity along the tornado path. Accounting to thisfactor results in smaller estimated risks of high tornado winds than wouldbe the case if the maximum tornado winds (by which tomado intensities

TABLE 3.5.2. Regional Pressure Drops and Pressure Drop Rate

Total Pressure Drop Rate of Pressure Drop(psi/s)Region (psi)

Iilu

2.

1_

4.

c.tc.)

bol.)

o

cn

oF$laar)

Ftrr

5.

3.02.25r.5

2.O1.20.6

124 I x I tll Ml wlNl) (.1 lMn l( )l ( x iY

arc ftrlc(l) wu'c unili)r'rtt lrlortpl lltc t'rtlttt'p:rllt. ('ot't'ccl itttrs clli'c'lctl inl3-451, bascd upon lhc irrtalysis ol rhrt'rrrrcn(ctl lorrrarkrcs, lctl to riskreductions by a llctor ol'abou( livc lirr li4 (orrratlocs antl about tcn lirrF6 tornadoes.

The corrections for the factors listed involvc subjective judgments that maybe formalized by Bayesian techniques (see Eq. Al.6). In [3-45] the corectedrates of occurrence differ insignificantly from the uncorrected (prior) rates, withthe following exceptions. For the three areas of the regionalization map pro-posed in [3-451 in which the most intense tomadoes recorded in the period1950 1978 were F5, it was estimated in [3-45] that rates of occurrence of F6tornadoes, rather than being zero, are about 1/20 times the rate of occurrenceof F5 tomadoes. For the fourth area of that map, in which the most intensetornadoes recorded in the same period were F4, it was estimated that thecorrected rate of occurrence of F4 tomadoes is about six times the uncorrectedrate, and that the rate of occuffence of F5 tornadoes, rather than being zero,is 1l2O times the corrected rate of occurrence of F4 tomadoes.

Reference [3-43] suggests that the velocity ranges associated in [3-37] withthe tornado ratings F 1 through F6 (see p. I I 1) are overconservative by amountsvarying from about 5% for Fl tornadoes to about 2O% or more for F6 torna-does. The wind speed reductions proposed in [3-43] are used in [3-45] as abasis for suggesting a reduction of the 360 mph, 300 mph, and 240 mph windspeeds, specified in [3-35] for regions I, II, and III of Fig. 3.5.4, to 300 mph,225 mph, and 200 mph, respectively. In the authors' opinion, the argumentsadduced in 13-431 in favor of such reduclions are tentative, in some instancesat least. For example, to support the contention that the maximum wind speedsin a tornado classified as F3 are lower than the values proposed in [3-37],[3-43] interprets a 133-167 mph estimate of the velocity causing the collapseof a chimney during the Xenia, Ohio, tornado of 3 April 1974 13-42, p. 17151simply as a 133 mph estimate 13-43, p. 16251. On the other hand, it should benoted that the estimates of [3-35] and [3-37] are also tentative.

A position that is to some extent a compromise between [3-35] and [3-45]was adopted in the American National Standard ANSI/ANS-2.3-1983 [3-481,which divides the contiguous United States into three zones, denoted as 1,2,and 3. Zones I and 2 cover, approximately, region I of Fig. 3.5.4, while zone3 covers approximately regions II and III. Table 3.5.3 lists the maximumtomado wind speeds Z-o^, the translational wind speeds Vr,, the radius of themaximum wind speed R-, and the maximum atmospheric pressure drop po,given in [3-48] for tomadoes corresponding to various probabilities of exceed-ance.

According to [3-72, p. D-ll, the ANSI/ANS-2.3.1983 Standard was notapproved by the Nuclear Regulatory Commission. Efforts to develop an im-proved standard are under way. Reference [3-73] is an overview of recentdevelopments conccrning dcsign critcria fbr tomadoes. It notes that ncw Nu-clcar Rcgrrlat<lry Corrrrrrission c:ri(cria ckr not clclinc tornaclo clcsign crileri:r on

trt l lt N(;t l; 129

'l'Alll,ll -1.5.J. Slrrrrrlitltl 'l!rt'rtirtlo ('har:rclerislics (llxlraclcrl li-6r1 ArlcricanNali'nal slanrl.rrl ANsl/ANs-2.J-l9tt-] with per'rissirrr 'l'thc publisher, ilreAmerican Nucletr Socill.y)Probability ol'

Exceedance, per Ycar ZoneV rrru

(mph)V,,

(mph)Rnro*(f0 (psi)

107

106

l0-5

320250180260200140200150100

70554057453245JJ25

540435320453355253355270185

t.961.350.701.460.85o.4l0.850.47o.20

I2JI231

23

a probabilistic basis, although_the design parameters it accepts for new nuclearreactordesigns are in the 10 6 range 13-74,3-151. Reference [3-76] is a studyof tornado climatology in the contiguous United States based on the NationalSevere Storms Forcast Center's tomado data base for the period January l,1954, through December 31, 1983. Strike probabilities were estimated in[3-76] on the basis of expected tomado areas, conditional probabilities of tor-nado intensities were based on affected area, rather than on number of occur-rences, the intensity distribution was based on a weibull model, and designwind speeds were based on regional intensity distributions. wind speed ioobtained were 50 to 100 mph lower than the estimates of [3-35], and tornadodesign basis wind speeds suggested in [3-76] are 200 mph and 330 mph,respectively, for the United States west and east of the Rocky Mountains. Foradditional information on tornadoes, see also [3-69] and t3_701.It was noted in [3-40] that probabilities of a target being hit by a tornadowind in excess of any specified threshold depend upon the iize olthat target.This topic is analyzed in detail inf3-44,3-451, where the estimates are basedupon statistics of tomado intensities, path lengths, and path widths on the onehand, and on the geometric characteristics of the target on the other. It issuggested in [3-49] that tornado wind loads dominate the design of most trans-mission lines over 10 miles in length over wide areas of the united States.

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3-63 D. Delaunay, Vents efiremes dfrs awr cyclones tropicaux dans les DOM-TOM,Cahier 2078, Centre Scient. et Techn. du BAtiment, 4 av. rect. poincar6, paris16, France, May 1988.

3-& A. G. Davenport, P. N. Georgiou, and D. Surry, Hurricane Wind Risk Studyfor the Eastern Carribean, Jamaica and Belize with Special Consideration ofthe Effects of ropogrctphy, Eng.science Res. Report Blwr-ss3l, Universityof Western Ontario, London, Ontario, Canada, 1985.

3-65 L. Gomes and B. J. Vickery, on the Prediction of rropical Cyclone Gust speeclsAlong the Nonhern Australia coasr, Res. Report R27g, school of civil Eng.,University of Sidney, 1976.

3-66 H. Saffir, "Florida's Approach to Hurricane-Resistant Design and Construc-tion, " "/. Wind Eng. Ind. Aerodyn. , 32 (1989), 221-230.

3-67 S. G. coles and D. walshaw, "Directional Modeling of Extreme wind Speeds,"J. Appl. Stat.,33 (1994),139-158.

3-68 ASOS Tool Box, Surface Observation Modcmization Oflice, 8455 Colesville Rd.,Silver Spring, MD 20910, Junc 199.5.

3-69 T. P. Grazula, "Significant'lirr.irtl.c:s," .l " Wind llng. Ind. Aerodyn.,36(1990),l3l-15r.

3-70 L. A. Twisdale antl P..l . Vicke'y, "l,lxr'errc wrntl ltisk Assessnrcnr," pp.46-5509, Pnbabilistic Strttt trutrl h,llt ltrtrrir..s lltttrrlltntk, ('. Srrrrtl:rrirr:r jln, (ctl.)Chaprnan lntl lllrll, Nrw York, lt)t)5.

134 txttuMt wrNr)ct tMAl()t (xly

.l-7 I li. sirrrirr, N. A. lltrckcrr, rrntl 'l'. M. wlrrrlt'rr, 1,.):;tintttrt',t ttf'llurrir'ttrtr,witttlSlttul.s lt.v th<, 'l'tttkl; rn'tr 'l'ltn'sltttltl' 19.1,'11,,r1, NIS'l' 'l'cchnicirl Note l4l(r,Natitlnal lnstitutc ol'Slarrtlarrl irrrrl 't'ct.llroLrtriy. (i;rithcrsburg, MI), l()()6.

3-72 DOB' Stanrlanl l(120-94, LJtritctl Stirlcs l)r:p:rr(rncnt ol' Encrgy, (icnrrankrwn,MD. 1994.

3 73 J. D. Stevenson and Y. Zhao, "Mrxlcrrr l)csigrr ol'Nuclear and Other PotentiallyHazardous Facilities," Nuclcar S'zfi,l.y (in prcss).

114 U.S. Nuclear Regulatory commission, "Final sal'cty Evaluation Report Relatedt. the Certification of the Advanced Boiling water Design," NUREG-1503,V.l. l, July 1994, National Technical Information Service, Springfield, VA2216t.

i /5 I l.s. Nuclear Regulatory commission, "Final safety Evaluation Report Relatedr. rhc ccrtification fo the System 80+ Design Docket No. 52-002," NUREG-l;162, Vol. I, August, 1994, National Technical Information Service, Spring_licltl. VA 22161.

| ](t .l . V. Ramsdell and G. L. Andrews, Tornado Climatology of the Contiguouslltritul states, NUREG/CR-4461 pNL-5697, May 1986, National rechnicallrrlirruration Service, Springfield, Y A 22161.

| 71 stnt(tural Engineering, Loads-Design Manual 2.2, NAVFAC DM 2.2, Navylrar:ilitics Engineering command, 200 Stovall St., Alexandria, virginia 22332,198I .

I7tt 'l'ccltnical Manual-structural Design Criteria, Loads, Army TN 5-g09-1, AirIirrcc AFM 88-3, Chap. l, Departments of the Army and Air Force, 1992.

\-l() M. D. Powell and P. G. Black, "The Relationship of Hurricane Reconnaissancet"light-Level wind Measurements to winds Measured by NoAA's oceanic plat-lirrms," Int. J. Wind Eng. Ind. Aerod.,36, (1990), 381-392.

l-ttO E. Simiu, Discussion of "Prediction of Hurricane windspeeds in the u.S." byP. J. Vickery and L. A. Twisdale, submitted toJ. Struc. Eng. (Apil 1996).

PART B

WIND LOADS AND THEIR EFFECTSON STRUCTURES

I FUNDAMENTALS

CHAPTER 4

BLUFF.BODY AERODYNAMICS

The subject of aerodynamics covers a very wide range. of necessity, therefore,only a few highlights can be emphasized in the present chapter. The fieldreceived its great initial impulse from the efforts in the early twentieth centuryto achieve heavier-than-air flight. Since that time it has continually receivedstrong contributions from a great variety of aerospace studies, and from thesustained, intensive development of machines with internal flows, such as jetengines, pumps, and turbines.

In addition, interesting new advances in applications of aerodynamics tocivil engineering structures have occurred in the last three decades. Dealing asthey do with the natural wind, these applications of aerodynamics are limitedmainly to relatively low-speed, incompressible flow phenomena. In this appli-cation, aerodynamics is also closely associated with meteorology and concemedin particular with turbulent flows in the boundary layer of the earth's atmo-sphere.

Besides a primary concem with the mean velocity of the wind, two aspectsof these turbulent flows are of interest to the structural engineer: the state ofturbulence of the natural wind approaching a structure and the local or "sig-nature" turbulence provoked in the wind by the strucrure itsell. Since moirstructures in civil engineering present bluff forms to the wind, emphasis isplaced, in wind engineering, upon bluff-body aerodynamics. This fact, char-acteristic of a new situation not emphasized as strongly in aeronautical andother previous studies, has occasioned new research on the details of, flclweffects around bluff forms typical of such structures as buildings, towcrs, anclbridges. In this context, interest centers particularly on details ol'thc clcvcl-opment of body pressures by the givcn flows.

As pointed out by Roshk<l in a rcccnt rcvic:w, "llrc: pnlblcrrr ol' blrrll lxxly

r35

136 nt lI I il()l )Y nl ll()l)YNnMl(ll

ll1;w rclrpritrs irlrrrosI errlilcly irr llrt't'trpttit';rl. rlt'st'r'iptive tt'ltltlt ol kttowlt'tlgc"l4 tttil. Altlxrr.rgh ctx)rnl()us lrtlvlrnt'r:s lurve lrt't'tt lttttl lttc bcittg rrtitrlc itl coltt-ltutali6nal lluicl clynarrrics (C-Irl)), so llrt'resrtlts ltitvc boctr Itltltle:sl irr tllc sllc-cializ,cd branch ol-CFD known ils ('onrl)ulirliontrl wind cnginccrirrg (CWts)-Lllrl)/CWE can proviclc qualitativc rcsrrlts ott wittcl llow lor pcdcstrian comlbrtpufposcs (Chapter l5), although cvcn irr this casc no delinitive validationsitppcar to be available [4-89]. Howcvcr, lilr structural engineering purposes,0wing t() the computational problcms arising in large Reynolds number, tur-lrulcn(, separated flows (Sect. 4.3), current methods are inadequate and/orlrnrhihitivcly expensive. For details on the current status of CW-E, see [4-90ro 4 92, 4-951.

ln this chapter a few basic theoretical principles and experimental facts are

r-r:vicwed that lay a foundation for the study of wind engineering'

4.1 GOVERNING EQUATIONS

4.1.1 Equations of Motion and ContinuityConsider a fixed elemental volume dV in a fluid. The vector velocity* of thefluid is commonly expressed bY

u:ui+/j+wk (4.1.1)

where i, j, k are unit vector components along the usual three fixed rectangularcoordinate axes x, y, 3. For compactness of notation let x, y, Z be replacedrespectively by x1, x2, x3, I't, u, wby u1, tt2, tt3, and the unit vectors i, j, k byi1, i2, ij so that Eq. 4.1.1 may be rewritten

3t : '?' ''i"

(4'l'2)i

The force acting on the fluid contained irt the volume dll consists of twoparts. The first part, referred to as the body lforce and caused by some forceheld, such as gravity, will be denoted byFp dV, where p is the fluid density.The second part is due to the net action on the fluid of the internal stressesoa(i, i : 7,2,3). For example, the contribution to this action of the normalstress 01 (see Fig.4.1.1) is

-otr dxz ar, + (o,, + # *,) *, dr, -- ai dxl cLrt dvl

dxt

FIGURE 4.1.1. Forces on an elementary volume of fluid

It can similarly be shown that the net force component in the i direction dueto the action of all the stresses o,; is

a,, dxadx.

,t I (i{'vt iltltN(, t(Jl,Ail()Nl; 137

t.,,* ff4,,tdxrdx.

), uo o,1:r 0x1

Du, I a",,A p dV : Fip dV * ,",i,0,

Da:a+ Lu-Dt 0t i:r '0*i

(4.1.4)

Denoting the components of F by F,(i : 1,2,3), the force balance equations,given by Newton's second law, are

where the operator DlDt, known as the substantial or the material deivative,is defined as follows:

(t : 1 ,2,3) (4. r.s)

(4.1.6)

Since Eq. 4.1.5 is true for all volume elements, the factor dV may be dividedout of Eq. 4.1.5 and the equations of motion, in component form, of a fluidparticle can be written as

(4.1.1)

Various forms of this basic equation can be derived depending upon the natureof the forces d and stresses o4 acting upon the fluid particle.

Before examining these particular cases, it will be useful to recall the prin-ciple of mass conservation. This principle states that the rate of increase of thefluid mass contained within a fixed closed surface must be equal to the differ-ence between the rates of influx to and effiux from the volume enc.loscd by thatsuriace. The equations of continuity can then be shown to be [4- | . 4-21:

Du, .l a",i,' : rF, *,1,#, (i : 1.2.3)

: oot' ,rvdr'(4. 1.3)

tlrr :rpltlicirliorrs wlrr:rt' llrclt.t'xisls lt sirtglt' inllx)t'lirtll tttt':tlt llttw vt'lot ily lti lotttltltttir'rl lry vlrri:rblcc()n)l)on('nls. lltt'nrt':rrr llow is ollctt lirkt'tt:ts ltt'irr1g irt lltt" t rlit't'tliott. tltllt vclotrlv (l(:,il'lllll((lrrs l/. fltc tcslx'r'livc r. \',.'(()llll)()ll('ttl:i lltt'tt lx'irt11 rlcsrlltt:rlr'rl ltr l/ I tt I' tt

IIrl

i)(Nri\/lr

opi)r

(-1.I n)

*138 Bl Ut I tr()t )y At n()t)yN^t\,4t(;r;

l'or lttt ittcottt;ltcssiltltr lltritl whe tt'irr rro r'lr;rrrg,r. irr tlt'lrsily /) ('(:(:ut.s, llris rr.rlrrt.t'sItr

.', arr.L..', 0 (4.1.9)i r 0.r,

4.1.2 The Navier-Stokes Equationstjnlikc a solid, a fluid under static conditions is incapable of suppor-ting anyslcady-state stresses other than normal pressure. In dynamic situations, on theothcr hand, it may support shear in a time-dependent manner. Most often, inlltricl-mechanical applications, it has been adequate to assume then that thestrcsscs involved are either normal pressures or ascribable to viscosity only.lrluicls with internal shear stress proportional to the rate of change of velocitywith distance normal to that velocity are termed viscous or Newtonian. Forcxurnple the shear stress ol2 in the simple two-dimensional flow pictured inlrig. 4.1.2 is expressed as

outotz: lt *whcrc the proportionality factor is defined as the fluid viscosity.*

rlFiGURE 4.1.2. Linear velocity increase with distance from a wall.

r'l'hc units ol viscosity arc

liler. lcngth lorcc timc' ilr'('it vt'kre ily lcngth r

'l'ypitirl v:rlrrcs ol lr lirl t'x;rnrplt' lirr:rir:uttl wlt(cr irl 20,, :rrc

ltuu 0 ()(X)ltt p/t.rrr s, 111,,, O.Ol 1ilt,rtr s

(irttttttllt ttttil: trtr'/r,rrr'r, r!lrr,lr. I lrrisr. I 11lt.rtr s (].()(]lO') llrl r/lt

'l I (i()Vl llNlN(i l(Jlln ll()Nri -l 39

lirrllhcr, by rlivitlirrg tltc wlrolc s(rtss /r'rt,srrl o,, lt( it lltlitl ptlint itt(o prcssttrcstrcss (6r'sirrrply l)ft,,t,\ur(, tltirl is, rtorttutl sltcss) 1t ltttd tltvitrtorll' s(tcss, dclincdas

(i, j : l, 2, 3) (4.1.1 l)

where

(4.1.t2)

du : 2r' (r, _ iu, -i, ,--)

I ( 6u, 8ri\eii:i\a*-i,)

(4.1. l0)

" fr. i:jur:Lo. i+jthe following breakdown of stress oii can be obtained:

(4.1.13)

@.r.4)oij : -pbij + 2r, ("i- +r, -i, "--)Using this form of stress for a Newtonian fluid results in the equations of

motionx ,

Dui _,_op.ig\r*/ r : \). 0"fi : oF, 0x, 7:r dx; ( (", - 16,i ^4,'^r)\

(4'l'15)

Equations 4.1.15 (i, j : 1,2,3) are the well-known NQvier-Stokes equa-tions. If Flq.4.1 .12 is used, and if the viscosity p may be considered to beionstant throughout the fluid, then Eqs. 4.1.15 become

3

Z @uol\xo)k:10*,

Further simplification occurs in the case of an incompressible fluid, that is, onefor which Eq. 4.1 .9 holds. Equations 4.1 .16 can then be written in the vectorform

,o#: oF,- y-, (,; W. ) (4 116)

: pF - ; P ii -r tt > f it I tlX; i I (txt

aI.-1

lcngllr tirrrc

DuPDt

ilior :r rrrort'tlt'liriletl ir(('()tlrrl s('('. lirt t'rittttltle, l'l ll. l'l .ll, l'l ll' or l'l'll

A.t.t7t

*140 ilr Ur I il()t)y nt il()t)yNnMtcl;

4.1.3 Bernoulli's EquationlJor a lluirl that, in aclditi<ln to hcirrg irrerrrrprcssiblc, is irryl,rr.irl (/r 0) irrrtlis actcd upon by ncgligiblc bocly lirlccs, llt1. 4.1.17 rctluccs to

(4. l. l8)

ll'the coordinate axes are so oriented thatJl corresponds to the direction ofrnotion, and if the flow is steady, it follows immediately from the integrationol' [lqs. 4. I . 18 that

;r :, llowlNA(:llllvl l)l'Alll v(|l lllxll()w t4l

Du i, al, .e Dr : -,?,; ''

jl"l'*4:consr

Ja//I/srnenulrrurs \ \,,/

-('dr

-\-u

ll ovory point of a streamline. Equation 4.1.19 is a specialtltt'rtrcm and is most commonly written as

(4.1.19)

form of Bernoulli's

lpu' + p: const (4.1.20)

wlrcrc a is the flow velocity along a streamline. The quantity |puz has therlirncnsions of pressure and is referred to as lhe dynamic pressure.'l'his important equation is widely used to interpret the ie-lation betweeni)rcssurc and velocity in atmospheric and wind tunnel flows. Detailed commentson llcrnoulli's equation and its applicability in fluid flows-including flows inwhich viscosity is present-are provided in Sect. 3.5 of [4-3].

4..2 FLOW IN A CURVED PATH..VORTEX FLOW

('orrsicler a two-dimensional flow u"rr")lio locally concentric streamlines atlistance dr apart and having radius of curvature r (Fig. 4.2.1). For the flowto nraintain its curved path, it must experience an acceleration toward the centertrl'curvature of the streamlines of amountu2lr, where z is here used to designatetlrc local tangential velocity of the flow. Let the pressure acting on the fluide:lcrrrcnt under consideration be denoted by p. The pressure differential fromonc strcamline to the next along r, which is responsible for this acceleration,is r/7r. The equation of motion for the fluid element is then

rlp dA : p d.r dl llrwltcrc trr is tltc llrritl tlcnsily lu'rd dA is thc arca <lf thc clcntcrrt in ir pllrrrt: rrtlrrrralto llter plitltc ol'llrt'ligrrlc. 'l'lris rclulion inrlicirlcs lhlrt lltc l)rrsriur'(. t.lurrrge rror.rnirll() lllc sll'ctttttlitres ol'lt r'tttvt'tl llrtw irr tlrc: :rbscrnt'e ol irrry ollrt.r' lort.r.s rs

Bernoulli's equation (4.1.20) then permits calculation of the pressure along a

curved path of such a streamline flow.In particular, one may consider the case wherein the flow is completely

circular and the value of p6 in Eq. 4.1.20 is the same on all streamlines. Thisis the case of vortex.flow. Differentiation of Eq. 4.1.20 yields

.drdp : pu';

du doPui+,rr:owhich, when combined with Eq. 4.2.1, yields

FICURE 4.2.1. Flow in a curved Path

(4.2.1)

(4.2.2)

(4.2.3)

Equation 4.2.3 can then be integrated to yield

ur:C:const (4.2.4)

'I'his simplc law sta(cs lirr irrr inconrprcssihle, inviscid fluicl thc thcrllcticlrl(hypcrb1;lic) rclation bclwct'rr positiorurl t'rrtlius r antl tangcnliltl vclrtcily rr irt lt.lit't'rrtrtc.r.

du drur

142 lil ul I il( )t )y n t tt( )t )vNn Mtcii

Itt lttt ;tcltt:rl lit't' votlcx, ltowcvt'r. llrt' r'llct l:r ol vrscosily il'(. l)t(',.r(.nl ilswt'll.'l'llcy llltvc ttol llcctt ittt'ltttlt'rl rtr llrt'srrrrplt tlt'r'iv:rlion irlrovc.'l'lrcst.willItitvt', itt l)ilrt, lllc cll'cc( ol'"lockirrg," s()rn('l)()rtr()rr ol lhc lluitl (ltclrr llrt.t'r'lrlg.)logclltcr alttl citusittg il to ft)lllc lrs lr liliitl lrtxl-y irrslt:lrtl ol'as thc pclli.t.l llrritlrlcscrilrtxl by l-q. 4.2.4. T'hus locirlly, neru'thc t'crrlcr ol'a I'rcc vgrtcx. llrcvclocity u intrtu:sc,t with radius, whcrclrs ucconling to Eq.4.2.4 il tlt,crttt,scswitlr irrcrctrsing r. This latter condition lctually hokls outwarcl fnrrn a tnu1sitilntr',t\i.rt irr which rz attains its maximum valuc. 'l'hc value of a in such a regionrs rlt'Perrtlcnt on the values of the fluid viscosity and of the total angular mo-nrcnlunr rrl'thc vortex. Figure 4.2.2 lllustrates qualitatively the pressure anclvt'krt'ity rclrrti<lns that hold in a free vortex occurring in a real fluid. It shouldlrt'rrrtt'rl tlrirt thc free vortex here described differs from the forced or con-ttttttttt'rl rrlTr'.r that may develop in a fluid held in a rotating container.'l'lrr' llcc vortex is of interest in many flows that occur in engineering ap-lrlrt';rtirrrs. lior cxample, atmospheric flows along the curved isobars of thervt'rrf lrcr nrrl) ilrc described by generalizations of Eq.4.2.1 . These have beenrlt'strilrt'rl irr scct. 1.2, where additional Coriolis forces have been included.

.l :l lt{ )l lNl )n I lY I AYI I l" n lll ) :;l l'l\l tn I l( )ll 143

4.3 BOUNDARY I AYERS AND SEPARATION

'l'lrc lrngc: ol vist'o:;rly v;rlrrt's lo bc lirtrtttl itrrl()rt1l vrtriotts lluids is vory grcat.'l'ltc viscosity ol lrir lrt rurlrrrirl rrrclorlrolrlgicitl 1tl'cssrtl't:s ancl {ctnpcraturcs how-t'vcr has a r-clltively srrlrll virluc. Noncthclcss, in stlrrtc circumstances this smallviscosity plays irrr irrrprlllnt nrlc. An imporlanL rnanif'estation of the viscouscll'ccts of air occut's in tlro lbrmation of boundary layers.

Consider an air llow over and along a stationary smooth surface. It is ancxpcrimental fact that the air in contact with the surface adheres to it. Thiscituses a retardation of the air motion in a layer near the Surface referred to aS

tlrc boundary layer. Within the boundary layer the velocity of the air increasesl'rom zero at the surface (no slip) to its full value, which corresponds to thecxtemal (as opposed to boundary layer) flow t2-11. A boundary-layer velocityprofile is depicted in Fig. 4.3.1.

Air, since it has mass, evidences inertial effects according to Newton'ssccond law (or, more specifically, the Navier-Stokes equations). The two mostinfluential effects in an air flow are then viscous and inertial, and the relationol'these to each other becomes an index of the type of flow characteristics orPhenomena that may be expected to occur. This index can be expressed as anondimensional parameter G", the. Reynolds number, which is a measure ofthc ratio of inertial to viscous forces. For example, consider a volume of fluidwith a typical surface dimension L. Then, by Bernoulli's theorem, the netpressure p - po caused by fluid flow at velocity U, which is of the order oft, p[J2, creates inertial forces on the fluid element enclosed by that volume which

Ip

HEIGHT

VELOCITY

.1.,!. L I \ lr( .rl lrrunrl;trv l;tyt't vt'lot'ily lllirlilt'.

u =C/r

lll(illlll'l 'l.l.f- I'r.ssrrrt' :rrrtl vt'krt iry rlisrrilruri.rr rr ;r 'rirrr'\ lr.r' Itl(llllll,l

144 ilt rJr I il( )t )y l\t il( )t )yNt\Mt(

irr'('()l llr('()r(lt't ()l /){/ /. . ()rr lltt'ollrr't lr;rrr,l. llrt'visr'otts stt't'sst's otr llrr"t'lr'tnt'rtlIrr.t'ol lltt'onlcl ol lrllll,, sr> visto:;rly rt'l:rlt'rl l()r'('(':j itr(r ol tltt: otrlt'r rtl 1r.llll,' /,'. 'l'lre lirlio ol'irtctlilrl (o vistorrs lolt t s rs tlrt'rr ol'lho rlttlcr ol'

,l :l li()t ,Nt)nnY tnYt ilr; ANt) ::t I'nt tn |()N 145

FIGURE 4.3.2. Flow separation at corner of obstacle.

l)oint,* the wake will contain the effects of vortex formation. Depending upontlrc magnitude of the Reynolds number, the flow willbe turbulent to a greateror lesser extent. Many turbulent flows may thus be typically viewed as wakelkrws in which upstream objects have already "stirred" the flow in some suchrnanner as has been described. Turbulence can bc caused by means other thanthc stirring mechanisms mentioned above (e.g., by thermally induced convec-tion), but for the majority of flows of importance to wind engineering, turbu-lcnce can be considered to be initiated mechanically, as described. Thus, forr:xample, trees, buildings, or telrain upstream of a given point play an importantrole in developing the turbulence of the wind observed in the atmosphericlroundary layer at that point. Descriptions of turbulence in the natural wind aregiven in Sect. 2.3.

When turbulence is present, one turbulent layer of the fluid tends to produceIurbulent motions in adjacent layers, as, for example, in a wake or boundaryltyer. This takes place through transfer of momentum from one layer to an-other. A similar phenomenon occurs in the absence of turbulence when alirrninar, as opposed to turbulent, boundary layer is created. The differencebctween a laminar and a turbulent boundary layer is that, in the former, thetransfer of momentum occurs at the molecular rather than the macroscopicscale. The fluid viscosity p is in fact the result of such molecular transfers ofnx)rnentum. As noted in Sect. 2. I in the context of atmospheric flows, turbulentlxrundary layers may be viewed as being governed by an equivalent kinematicviscosity callcd eddy viscosity, whose value reflects the large momentum trans-lcrs induced by turbulcncc.

(4.3.1)

wlre:rc: y : pl p is called the kinematic yi^r<'o,ril_y.* (See also Sect. 7.1.) Thus,wlrcrr 61. is large, inertial effects preilominate; when it is small, the viscoust'llccts are the stronger ones. It is noted that the concept of Reynolds numberrs. irr rclaticln to the boundaries affecting a flow, a very local thing; that is, thest'lt'c(ion of the representative length Z for the calculation of G" depends upontlrr' irrtcrcst ol'the investigator in local details. Thus a flow over a given objectrrr;ry tlcvckrp a wide variety of Reynolds numbers, depending upon the partic-rrl;rr rcgion focused on for study. When discussing the whole flow that envelopsrr llivcrr lxily, it is usual to select for the length -L some overall representativetlirrrrnsion ol that body.

Iirrrrrtlrrry-layer separation occurs if fluid particles in the boundary layer aresrrllicit:rrtly dccelerated by inertial forces that the flow near the surface becomesn'vt'r'sctl. 'fhcse deceleration effects occur as a result of the presence in thellow ol'arlvcrse pressure gradients. Such severe adverse pressure gradients ast'rrrr lrr: llrocluced, for example, by the flow over the comer of a bluff bodyI't'nt'r'irlly cause flow separation. Through processes that are not well under-strxxl, thc scparation layers generate discrete vofiices, which are shed into thewrrkc llow bchind the bluffbody (Fig. 4.3.2). Such vortices can cause extremelylril',lr srrctions ncar separation points such as comers or eaves.

Iilows ol-practical interest have Reynolds numbers ranging from nearly zeroto irs lrigh as 108 or lOe. Steadily increasing the Reynolds number of the flow()v('r irr) obstacle generally produces a widely varying sequence of flow phe-n()rncnlr lirr which the Reynolds number provides a convenient index, as isst't'rr. lirr cxample, in Sect. 4.4.

ll'" as is true in most cases, the flow over a body has separated at some

''l vpical valucs of kinenratic viscosity for air and watcr are. respectively:

/,i, : 0. 150 crn2/s at 20'C

r,".u : 0.01 crn2/s at 20'C

A t orrrnror rrrril lirr kincrrratic viscosity is thc .stote:

I stokc - I crrr/s - 0.001764 ltrls

A rrrclrrl :rplrlrlirrt:rlc lorrrttrl:r lor lltL'l{t:ynoltls rtrrrrrbcr in:rit'lrl lrllrul lO"(':urtl ;rlrrrosplrcrit'lr( r;sllr( rs (r/ (X)() l// . r|lrt'ri {/ is in Ittt'lr'r's l)r'r s('('on(l :rrrtl /. in nr('l('rs l lrs lrr'r'rrrrrt s (r.) }O l//.lor l/ rrr lt/:, :lrrl / rr l1-( l

olJ L(lle : 'pUL ll

p( )l , I!:lLv

llrt lltt' t'rrsr' ol lrirlirrls (x ( lrrr( r( ('tlrc lrtxly. in ;rttottlrrtttt rvillr llrr'tr'tltttr' tlt;t1', lry ttttlttts ol llcrttttt'lttr

ol llrl,.,r'lrrr.rlrorr r:, rr:rr;rlly tk'sirt'tl lts l:rlc:rs possilrlt':rlorrl',,rlnr "l , t'nlr'rllrr,' Irr':,r,lr( rlisttilrttlions 1o iltt li':rsr' lill :rrrrlIorrr

146 n UI I Ir( )t )y n t tr( )t ,yNn Mt( il

4.4 WAKE AND VORTEX FORMATIONS IN TWO-DIMENSIONALFLOW

lrt thc lilllowing discussion, lho lklw is assrrrrrctl (o bc smooth (laur"inlr) arrdtwo-rlirncnsional, that is, indcpcntlcnl ol'lhc c<xrrclinatc normal to tlrc planc olvicwing. Consider a two-dimensional llow around the sharp-edgccl flat platcshrrwn in Fig. 4.4.1 . At a very low Rcynolds number (e.g., ULlu: 0.3,whcrc L is the dimension of the plate across the flow), the flow turns the sharpcorncr and follows both front and rear contours of the plate (Fig.4.4.la). Atrr slightly higher Reynolds number (&" = l0) obtained by merely increasingtlrc llow velocity over the same plate, the flow separates at the corners anclt'rt:alcs two large, symmetric vortices behind the plate that remain attached t<l(lrt: back of the plate (Fig. 4.4.1b). At increased Reynolds number (G" = 250)thc syrnmetrical vortices are broken and replaced by cyclically altemating vor-tit'cs that form by tums at the top and bottom edges and are swept downstream( l;ig. 4.4. I c). A full cycle of this phenomenon is defined as the activity betwecnllrc occurrence of some instantaneous flow configuration about the body andllrc ncxt identical configuration. At still higher Reynolds numbers, say Ge 2It){)t) (Fig. 4.4.1d), the inertia forces predominate; large distinct vortices havelittlc possibility of forming and, instead, a generally turbulent wake is formedIlchintl thc plate, its two outer defining edges forming a "shear layer" con-sisting ol'a long series of smaller vortices that accommodate the wake regionlo thc udjacent smooth flow region. Overall, these results dramatically illustratc

rDe e Q.3

(o)

lll(;llltl,l.l.,l.la. lihrw pirsl lr slt:u1r t'tllicrl pl;rtt.(11,. -- O I

.l 'l Wnlll nl.Jl'V()l lllXl0liMnll()Nl; tl.l tW(llrlMl il:;l{)t!At |()W 147

(b)

FIGURE 4.4.1b. Flow past a sharp-edged plate Ge = 10.

(, )

l,'l(illllll,l 4..1.lr'. l;kru |1r;l ir "lrirrI r'rl1'r'r; ,t1,,,. ,t,, _) 5()

t4B lll l,l I lt()t)Y At n()t )YNnMt|1

FIGURE 4.4.1d. Flow past a sharp-edged plate Ge > 1000.

thc changes in the flow with Reynolds number, proceeding from predominantlyviscous effects to predominantly inertial effects.

Next the renowned case of two-dimensional flow about a circular cylinder(rig. 4.4.2) is briefly examined. A number of flow situations can be createdby increasing the flow velocity, each situation being identified by a specificllcynolds number range. At extremely low values of Reynolds number (G" =l) thc flow (assumed laminar as it approaches) remains attached to the cylinderthrrrrgh<rut its complete periphery, as shown in Fig. 4.4.2a. At G" = 20, thelklw lirnn rcnrains symmctrical but flow separation occurs and large wakectklics urc lirnnctl which rcsidc ncar the downstream sudace of thc cylinclcr,rrs srrggcstt:tl irr Iiig. 4.4.2b. lror 30 < (R" < -5000, al(crnuting vor.ticcs arcslretl liirrll tltt't'ylilrrlcr rttrtl lirt'nr ir clcar "vorlL:x lr;ril.' rlowrrslreirrrr.'l'hisltltt'ttolttt'ttott wirs litsl rt'1xrt1ctl by llt<rlrltl l4-.5 1 irtrtl vorr Klrrrrr;ur l.l O; 11r;*.

Qe=l

(o)

,l .t w^l,.t nt.lt I v(,t illx l()l tMnll()N:; lN tw()trlMl Nt;l{)Nnt ll()w 149

qe =20(b)

@VON KARMAN VORTEX TRAIL

30 39" S 5OOO

cuuuS.z{S ZU(JUUU 4"2ZOOAOO

(d) (e)

l|l(;uRE 4.4.2. (a) Flow past circular cylinder (Re = l. (b) Flow past circular cylinderill" = 20. (c) Flow past circular 30 < G" < 5000. (d) Flow past circurar cylinder.5(XX) < G" < 200,000. (e) Flow past circular cylinder G" > 200,000.

1.1 .2c). The finer details ol this striking occurrence are still not fully under-slrxrtl, and the process cor.rlirrrrcs to be the focus of many studies, both exper-irncrrtrrl and thcoretical 14-21 l. llchintl thc cylinder there is establishcd a staggcrctl, stablc arrangctllcltl ol'vorlit'cs lhlrt rrrovcs <tffdownstream at a vcltritystttltcwltitt lcss thltrr tlurl ol lltr'sttt.lirrrrtrlirrg llrritl . In this rangc ol'llc:yrroltlstttuttbcr lltc w:rkc llow is lr.litlrvcly snrtxrllr irnrl rcgrrlirr aplttl I'r-onr lltc vorlit'csrlllclttsclvt:s. liigrrrt:.tr.,4..1 tlt'1rrr'ts llrt'slrt':rrrrlirrcs ol tlrc wlrkc llow lx'ltiltl ir

(c)

WAKE

5OOOs.4.3 2OOOOO

150 ilr ur r tt()t)Y AI lt()l)YNAMlcl;

l,'l(;tjRli 4.4.3. Vortex trail in water tunnel. Courtesy of the National Aeronauticallislablishrncnt, National Research Council of Canada.

circular cylinder in a water tunnel [4-7] within the above-mentioned Ge range.'l'hc lkrw in this photograph was made visible by the emission of dye from thet'y I irttlcr.

As ltcynolds number further increases into the range 5000 < G" < 200 000,(hc irttlchcd flow upstream of the separation point is laminar. In the separatedlLrw (hrcc-dirncnsional patterns are observed, and transition to turbulent flow(x'('uls irr thc wake-farther downstream from the cylinder for the lower Rey-rxrltls rrrrnrbcrs and nearer the cylinder surface as the Reynolds numbers increasel.l l()l l,or thc largest Reynolds numbers in this range, the cylinder wakerrrrtlcr-gocs transition to turbulence immediately after separation, and a turbulentwrrkc is pnrtluced between the separated shear layers (Fig. 4.4.2d).

Ifcyonrl 61" = 2OO 000 (Fig. 4.4.2e) the wake narrows appreciably (givinglisc lo lcss drag; see p. 158).

( )thcl bluff bodies, notably triangles, squares, rectangles, and other regular;rrrrl ir.rcgrrlar prisms, give rise to analogous vortex-shedding phenomena.

'l'lrc prorllrnccd regularity of such wake effects was firQt"rgp-pged,by,$trquhall.l ttl wlxr pointccl oul that the vortex-shedding phenomenon is describable intcrrrrs ol':r rror.trlirncnsional number (the Strouhal number):

NDs-?i

wlrerc N, is tlrr. lit't1rrt'rrr'y ol lirll cyclcs ol'votlcx slterrltlilrg. /) is it t'ltltlrtt:lct'islictlirrrt'nsiort ol llrc lrtxly lrlojt't'terl ott lr pltrttrr tttll'ttnl ltt lltr'tttr'lttt llow vclocily,

't .l w^t,t Ailll V(lt iltx I()llMn il()Ni; tN tw(,t)tMt N:;t()NAt il()w r5t

r-ffi--r-.-

, o.2llI

o.1

107106

. "eff@f 3 3 ""etrt e R"6 STxf t ^3O Smootho k/D=0.0003^ k/D=o.oo12o k/D:0.0101

105

REYNOLDS NUMBER

l"l(;uRE 4.4.4. Relation between the Strouhal number and Reynolds number for cir-t'trlar cylinder. From W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, ,.Experi-nrcnts on Flow Past Rough circular cylinders at Large Reynolds Numbers," J. wrutI,)rg. Ind. Aerod., 49 (1993), 351-348.

irnd u is the velocity of the oncoming flow, assumed laminar. The number Stakes on different characteristic constant values depending upon the cross-sec-tional shape of the prism being enveloped by the flow. Figure 4.4.4 t4-g6lshows the relation of 3 to G" for a circular cylinder in the range 10s < G" <107. The values of Fig. 4.4.4 were inferred from the unsteady pressure mea-sured in smooth flow at about 90 degrees from the front stagnation point.('oherent vortex shedding was noted to disappear at Reynolds numbers beyond4 x l0s, and contrary to results reported by some observers and summarizedin l4-9], there was no increase of the Strouhal number to values near 0.5. Table4-4.1 [4-l0l also lists a number of values of s for different cross-sectionalshapes for Reynolds numbers in the clear vortex-shedding range, the approach-ing flow being laminar.

A certain amount of debate continues on the question of whether or notlrcriodic vortex shedding gan still be exhibited at extremely large Reynoldsrrurnbers, say, G" >> 108. If one substitutes an effective eddy viscosity (seescct. 2.2) for the actual kinematic viscosity of the fluid, it is conceivable thatir ncw Reynolds number range can be calculated in which altemating vortexshcdding from extremely largc bluff ob.jects can once more be forecast. In thiswly thc occasionally ohscrvul lrttgc vorlcx trails in ocean currents downstreamol' islands may possibly lrc tt't'ottt'ilt'tl with srn:rllcr-scalc expcrimcntal <lbscr-vlttiotts. Irigttrc 4.4..5, lttl ittslltttt'c, is lr lcpnltlrrcl ion ol'a satcllitc photogrlphl4 Ill ol'tt voflox tllril irt tltt' itlrrroslrltt'tt' rn;rtlc visiblc by ckrutl Jlrc:sctrctr irr,tltc votliccrs slttxl lhrrrt lltt' nrrtttttl;rrn l)('irh ()l ( irrrrrltrlrrpc lsltrrrtl ovr'r' 12(X) rrr

152 ill liltlt()t )Y nl ll()l )YNnMl(.1,

I'Altl,l,l 4.4.1. Slroulral Nurrrlx'r' lirr rr Vrrrit'l.y ol' Slritpr:s

Profile dimensrons, Value 0l Prolile drmensions,in mm

Val0e ol:/t= 2.O

0.120r= 1.0

r25{l_l___T{rz.s{

l*so--l

o.147

0.1 37

_-_>,=0.5

ilH['r]

0.120

r= 1.0

r2s{[-l-+-t2.5c<-T

1..-so-t

0.150

1.0

f.-.-l0.144

0.145

0.t42

/ 0.147

r= 1.5

rzs{f-.1[-*-]

0.145 $-t_[ru!

0.13 1

II

0.134

/ 0.137

r= 1.0

L-uo-t

o 140 E_l t_l*rs*zs.l.zs!

0.12 I

0.153 0.143

125{l-JL_*-J

0.145r= l.O

0.135

0.168

+

G0.156

0.160

I

,t0.145

Cylinder 1l8OO..*r. tttOO Q

0.200T-lLl

l-. .'lo.1 l4

t 0.145

\i1,r.r,.. liRrril .'wirrrl lirr.t.t's orr Stlrrt.lrrR's," '/izrrr.r. AS('li, l2(r (l(x'l), ll.t,l ll()11 (r - lo IIIZ

.l .t w^t, I /\lillv{)l illxl(}ltMnll()Nt,tt.t tw(rtrtfi,,iltl.,t{rlll\l ttow

.r..\li<{ffCi .,t: *a\\."{f\ q ". ,r*' .,* };1* ,, ,r".*

;,,} P

*;w^"htffi

i$:i. ..

&d'

i;:-le . ..,

..,,.s, '

.-ig" q&:

:.l i!,i,,i.1;*6,'.:i,:-' .,,-'all$::,"i.'

I''IGURE 4.4.5. Satellite photo of cloud vortices downstream of Guadalupe Island (offItlla Califomia) [4-l l]. Courtesy of the National Aeronautics and Space Administra-lion.

high off the Pacific coast of Mexico. The photograph spans some 250 km.Assuming, as in [2-1171, an effective value of (kinematic) eddy viscosity an= -50 m2ls, a full-scale Reynolds number of the order of 1010 for the phenom-cnon (based on u = 1.5 x l0 5 m'ls; would be reduced to an effective valueol'((R")"s = 3000, which falls well within the laminar vortex-shedding range.Assuming the island to be about 20 km long, the distance between successivepcriodic voftex centers is roughly 55 km. Further, assuming a Strouhal numberlirr thc island peak as S : 0.12, a mean wind velocity of U :30 m/s, andrrrr cfl'cctive island clirnrrrrsion ol'/) = 6000 m yields the vortex-shedding frc-(lucncy

0 l.)( t())(il()()

I fi4 ill t,t I il( )l)Y n t il( )l )YNn Ml( ii

l;.

"';ffit; *:.."s,ffit

lrl(ltll{lt 4.4.6. Satcllite photo of Jan Mayen Island (Arctic Ocean). From Weather,1I. l0 (Oct. 1916), 346.

wlrich in turn gives a shedding period of Z : 1/N" : 166l s. Employing S :l/7'yiclcls a calculated vortex separation of

S : 30 x 166l : 50.000 m : 50km

rr rlislurrcc consistcnt with rough fireasurement of the photograph. Anothcr in-tcrcs(ing photograph ol'largc-scalc vol'tcx shedding is prcscntcd in Fig. 4.4.6l.l t2;.''

WIrcrr contlilions:rlc srrclr thirl ir rlistinct voflcx triril is l)rescn( in lhc wrrkc,lt llow ct1)sli()v('r-:rll ol (lrc llrtly occttrs llltl hrts it ('()nllx)n('rrl rrolrnirl lo lltc

lll lurs lx'trr lrroul'lrl lo llrr' ,rlllrtlrott ol lltt :tttllllrs llltl lt sitttll;r |rrrlrlr rtr r', ltr';rlr'rl rrr l.l i{)l

.l ', llll(.1:;()t'l lW()l)tMl tl',lol lnl .lllllr lrlll/\l lrrl lM

t\I

(-.._-- - -\)-

-_*7

\-'

I l(.llltl,l 4.4.7. lillcct ol'splittcr plate on flow behind a circular cylinder [4-13'I r' I I

,r1,rr,r;11 11i11g llow dircction. Thus it becomes possible to inhibit the establish-nr rr ol ir v()flcx trail by placing a "splitter plate" in the near wake of the,.i rir'l;rlurtt l'rocly, as first pointed out in [4-13]. (See Fig. 4.4.7.) The action ofrlrr.. pl:rtt'is to prevent the flow crossover and thus to quiet the entire wakell,,s' t.)tr:rlitativcly, the presence of the plate has the same type of effect asi, rr1'tlrt'rrirrg thc body in the stream direction and causing it to approach, to.,,irr, ;rppnrxinration, the form of a symmetrical airfoil. Following this type of,rtri,r();r('lr it can bc seen that elongated bodies, oriented with theirlong dimen-r,,rr l';urrllcl to the main flow, tend to elicit relatively narrow wakes, many

r.. rtlr,rrt lrpprcciablc voftex production.ll llows irbout square and rectangular prisms are compared (Fig. 4.4.8), the

,|il:il(' rs sccn (at reasonably high G") to produce flow separation followed by,r n'rrlt'. (rrrbulcnt wake, whereas the more elongated rectangular form (de-1,, r.lrr11, orr lcngth-to-width ratio) may exhibit separation at leading comers thatr. l,rllrrwt'rl rl<lwnstream by flow reattachment and finally, once more, by flow,p,u;rtr()rr rrt thc trailing edge. Thus it is seen that not only does the bluffface

,,1 rlrr'lrotly prcscnted to the fluid affect the resulting wake, but the streamwisei, rrlllr rrrrtl gcncral form of the body also play important roles in the wake form.irr',lr.rr1r tlis(inction to the casc of Fig. 4.4.8b, if the rectangle is placed withrr. l{)rr)'. tlirrrcnsion normal (o thc llow, the wake exhibits a strong voftex-i,,,l,lrrr1, i'lnractcristic, lirlkrwctl at highcr G" by a turbulent wake not unlike

rl,.rr ;rr11f111'gtl by thc sltiu'1'r ctlgctl llirt platc (see Figs. 4.4.|c and4.4.ld).

,I II PRESSURE, LIFT, DRAG, AND MOMENT EFFECTS ON! WO I)IMENSIONAL STRUCTURN I FORMS

! rlrrrt" ,l.5.l srrl',1qesls l s('( l11 )rl ol .r lrlrrll lrotly ltttttr'tst'tl itt lt llrlw ol vclot'ilyit llrt. ll6w will tlt'vt.l91r lpt:rl ;'t,..,.,rrr{.', /, r'\,('l llrr' lrrxly irr ttcr'olltrilt'c willr!l, r il{}illli':; t't;tIt(iottl

156 tlt t.I I il( )t)Y At il( )t )YNAM|(:ii

.=.t.t^"t

(b)

lr'l(;Ljllli 4.4.8. Flow separation and wake regions of square and rectangular cylinders.

ATTACHMENT

t,'l{;llltl,l 4.5.1. l.ill irntl tlritg rtn irn itrhil!rt!y ltltt!l hrtly (4.5..s )

,t h il | c ri oN two DtMl NtiloNAl riffflll:lllll^l | ()llMli 157

wltetc tlrc cottslirttl lroltls irkrrrg a strcarnline irrrtl l/ rrl)rrsonts thc volocity ontltc strctttrline itt thc itnrttcrcliatc vicinity ol'tlre hxly (i.c., irrrrrrccliatcly outsidellrc llrundary llyer tltal lirrrrrs on its surlacc). 'l'hcr intcgration of the pressuresovcr thc body surlitce rcsulls in a nct fbrcc and a lnolnont. The components oftlrc lirrcc in thc along-llow and across-flow dircctions are referred to as dragilnd li.li, respectivcly. 'l'hc drag, lift, and moment are quite obviously affectedlry llrth the shapc ol'thc body and the Reynolds number.

'l'hc body may, lirr cxample, be contoured with the express purpose ofrrrininrizi'ng drag and rnaximizing lift, resulting in an airfoil-like shape. Again,;rs in rnany civil engineering applications, the shape of the body may not be;rrucnable to such special adjustment; its form will most likely have been fixedlry other design objectives than purely aerodynamic ones. Nevertheless, thelill, clrag, and moment developed by the fluid flows about the structure willrt'rrrlin of strong interest because these are effects that must be designed against.

ll is usual to refer all pressures measured at a structural surface to the meanrlynrunic pressure )pU2 of the far upstream wind or the free-stream wind atrorrrc distance from the structure (e.g., at a point well above it out of thelrrrrrrrrlary layer). Thus nondimensional pressure cofficients Co are defined by

t pl l) t p r.'orrsl

P-PoLr: Ei,

FLL, --' iPu'B

Fnt t, : t_pu)n

ttt( 'tt : 1,,, ,: ,r:

(4.5.l)

(4.s.2)

(4.s.3)

(4.s.4)

wlrcrc U is the mean value of the reference wind and p - ps represents theprl'ssure difference between local and far upstream pressure p6. Such nondi-rrrcrrsional forms enable the transfer of model experimental results to full scale,irrrtl the establishment of reference values for cataloguing the aerodynamic;rro;rcflies of given geometric forms.

Arralogously, the net wind-pressure forces (per unit of span) F1 and Fp intlrc lili and drag direction, respectively, can be rendered dimensionless andr'rlrrcssed in terms of lift and drag cofficients Cy and Cp as

rvlte tc /J is somc typic:al t'cli't't'ttt'c tlittrcrtsiorr ol'lhc s(ructure. For the net flow-tttrlttt'e:cl lnolncnt M lltcr crtrlcsPotttlittg cot'llir'icrrl is

Su br; rit ical

158 tlt Ut I tK)t)Y nt ll()l)YNnMl(

106

Reynolds number,4e

FIGURE 4.5.2. Evolution of mean drag coemcient with Reynolds number for a cir-cular cylinder. After L. R. Wooton and C. Scruton, "Aerodynamic Stability," in TheModern Design of Wind-Sensitive Structures, Construction Industry Research and In-firrmaticrn Association, London, 1971, pp.65-81 and 14-221.*

When the flow is fluctuating as a consequence of oncoming turbulence,vortex-associated flow changes, or signature (body-induced) turbulence, theabove quantities become time dependent. In such cases, when time-varyinglitrccs ancl moments occur, mean values of force coefficients aS well aS spectralclcnsity clistributions of these quantities are required for their fuller description.'(Note that in two-dimensional flow L1 , Fp, ?fid M represent correspondingvalucs per unit of dimension normal to the plane of observation. In three-dimensional cases, correct dimensionality is preserved by including an addi-tional factor B in the denominator of each expression.)

Retuming to the prism of circular cross section in smooth flow, the variationol' its mean drag coefficient Cpmay be represented as in Fig. 4.5.2, where theclcpendence on Reynolds number is shown. Note particularly how Cp dropssharply in the rang6 of about 2 x l}s S Ge < 5 x 10s. This region of sharpclrop is called the critical region and corresponds to a condition wherein thef ransition from laminar to turbulent flow occurs in the boundary layer that formson the surface of the cylinder. The turbulent mixing that thus takes place intlrc boundary layer helps transport fluid with higher momentum toward thcsurlace of the cylinder. Separation then occurs much farther back and the wakec()nscquently narrows, finally producing a value of the time-averaged C, thatis only about { of its highest value. As G" increases into the supercritical andthcn ihe transcritical range (G" = 4 X 101. CD increascs once more butrcnrains much l<lwcr than its subcritical values.

l,llcccrrl tl:rtu lrl ll(rl slrow llr:rl tlrt.tlrirg c0cllicicttls itt thc lt'giorr 5 l{)' (ll, ' lO/ :rtt'stttltllct'lly ltllrttl l5%' llllrrr lltost'tttrlitltltrl irr lrig 4'5 2 rtlt lltt: hitsis ol t';ttllr': tltlrttttt;tltrtttSct' Appr'rrtlir A.)

aUt1.oO+ooObrrN'o

107

n,, til l(,t:i ()N lw()l)lMt t!!;l(lt,tAt !iililt( illt t^t t()t tMl; l59

o" "o" , ,oaorra' l2o" l8o"

l,'l(;tlRE 4.5.3. Influence of Reynolds number on pressure distribution over a circular, ylirrtlcr (after [4-22]).

lrigure 4.5.3 depicts a typical distribution of the mean pressure coefficient;rlrorrt the circular cylinder in smooth flow as a function of angular position.I'lrt' rcsults are evidently sensitive to Reynolds number.*

'l'lrc drag coefficient of an elongated rectangular-section body in smooth flowtlrrg. 4.5.4) [4-14,4-231 is also a function of the narrowness of its wake, butrlrc krwer limit of wake width is approximately the full width of the body. Then,rrkc width at somewhat lower G." is much greater than the body width, andtlrrs is accompanied by higher cp;then, when flow reattachment to the bodylx'1iins to occur, the drag cocfficient drops. This is a function mainly of ther'lrrngirtion blh of the brxly, irs shown in the figure. Flow in the critical regionl:' ilccompanied by turhulcrrcer, irncl thcrcfirrc this region is shown as a shadedlr;urtl ol'possible valucs in lrig. 4..5.4.

liigrrrc 4.5.5 14-1.5 1 illrrsrrrrrcs tlrt' t'volrrliorr with Rcynolds number of therrrt':ttt tlrag cocflicicttt ol'it stlttirtt irt srrroollr llow tlrrrirrg successivc rlodifica-Itotts ol'ils corncrs. Nolrr llrirt otrly llrt.slttlP t'orrrt'r't'tl stlrrar-o cxhihits practicirlly

rllrt'lrcsstttr:s ('on('sl)on(ling k) /l ll" irrrrl ll ll"i{1" ;rrt lt.lt.trt.tl l() itri llt(.plt.ssrrrt.trl llrt.',lrr;',rlrliorr lloirrl :riltl lltt lltsc l)ti'r{ilr', rt rlx r livrll'

4r.61 xlOs

.4c=l.lxlO'

160 nt t,t I lt()t)y At n()l)yNAMt(]ri

0L0

FIGURE 4.5.4. Effect14-231.

24of afterbody upon

6

drag of a

o

rectangularb/h

cylinder 14-14),

unchanging drag with change of Reynolds number. This is simply accountedfor by the early separation ofthe flow at the upstream corners and the shortnessof the afterbody that practically precludes the possibility of flow reattachment,whereas squares with rounded corners tend to possess the same kind of criticalregion for the drag coefficient as seen earlier for the circular cylinder. Notealso, in the case of the circular section, the dependence of the drag upon theroughness of the cylinder surface. This dependence was studied in detail in14-241. (See also Sect. 11.1.1.)

Because of such effects, certain features of the flow in tests over wind tunnelmodels can be expected to be independent of the Reynolds number, while othersmay be quite sensitive to it. Thus it can be argued that cerlain Reynolds-number-insensitive flow phenomena may be encountered in tests in which thellow will always break cleanly away at the same identifiable points. certaintypes of bodies such as the circular cylinder offer extended regions of possiblellow separation in which the location of the actual separation points dependsrupon Reynolds number. with such bodies the entire structure of the flow willhc highly Reynolds-number-sensitive (see Secr. 7.3.2).

l'or cxtremely low Reynolds numbers the drag coefficient increases greatlyirs ir rcsult of viscous effects. This is illustrated in Fig. 4.5.6 t4-I41, which1fL:;)icts Cpfor circularand square flat plates for 10-2 < G., < 107. (Analogouscllccts on lift and moment do not necessarily follow, though some distortionis vcry likcly.)

sincc thc prcssurc dil'l'crcnces across a sharp-cornered square vary with time,llte soctional lili crrcllicicnt will also be a function of time: C1.: C1,Q). Figure4.5.1 14- l6l illustra(cs tl.rc spcctral density of c7. plottcrl as a lirnction ofrrll/{/, whcrc rr is l'r'c:t;ucrrcy in Hz, B is thc dimcnsion ol'rhc: sitkr ol'tlrc scprarc,ittttl l/ is ttlcitll (tttr"()trring vt:krcily (irssurnccl to bcconslirrrl llrlorrg.lrorrl tlte lc:gion

4h rIil(lii ()N twot)tMl Nl;l()Nnt 1;ililt(;ililtnt t()ilMli 161

2.2a--r-- il r --r--Tt-T- r --lt ----'!- [-1r,4 f , t, =oozt' , , ,, , l'"_lI8J- r r rr r ___r rr r

1.2

0.8

0.4

---) u--D]n

(h)

r/h = 0.167

r/h:0.5(circular section)

8105 2 8106 2 8l 07Ee

- sanded surface k)

--- Smooth surfaceFIGURE 4.5.5. Influence of Reynolds number, comer radius, and surface roughnesson drag coefficient, square to circular cylinders (r is the corner radius; k is the grainsize of sand). After [4-t51.x

o.tto-2 lo-l

9ttFIGURE 4.5.6. Typical rlr:rg coc{licicnt as a lirnction o1'Reynolds number [4-141.

*Motc tcccnl tllta lor circtrlitr cylittrlcts l,l tl(rl rrrt in gcncrirl tprrirliltrlivc itgrcclllcnt wilh tllrsc glIriP,.4.5.5 btrl intlicatc lltirl lirl l;tti,ic l{r'ynrltlr rurrrlx'rs thc cylirxk:r rrurglrrrr":ss brings irlxlrt irsonrcwhitl slr'ongcr irrcrcitsc irt tllrg

co

l07to6t05t04t03t02

k1h = 0.007I'Fk/h: O.OO2

h:0.001 \--'

t62 lil ul I Ir lt)Y n t il( ltlYNAMtcl

50

20

10

2

1

0.5

o.2

0.05

0.02

0.01

0.005

0.002

0.02 0.05 0.100.20 0.50 1.00 2.00NB/U

l"lGURE 4.5.7. Spectrum of lift fluctuations on a square-section cylinder for flownonnal to a face (G" : l0s). From B. J. Vickery, "Fluctuating Lift and Drag on al,.ng Cylinder of Square cross-Section in a Smooth and in a Turbulent Flow," ,/.lluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.

ol'lkrw under consideration). In both smooth and turbulent flow, a high spectral1rt'rrk occurs at the Strouhal number nBlU : 0.12.

'l'his is clcar evidence of periodic voftex shedding. For any given bluff body,tlris shodding is not a purely sinusoidal phenomenon, as seen from the spreadt. rrtlrcr I'rcqucncics ol'thc spcctral peak in Fig. 4.5.7; however, a good firstlrprpIrrxirna(ion lo tho lili lirrcc pcr unit span occurring at the peak Strouhalrrrrrrrbcr is givcn by

N

aNpNd!

f,lcn

lr, t,pU)B(-, sin <,:t

I

J

llolisilllltblr/lirii

/'f tr./,t \'\.rt"# \:t,

,ff tpj'dtTurbulent (

streamSmooth -+_o+stream

('1..5.6)

1,, tt|(.11; ()N lw()t)lMl N:it{)l.JAl f;lltll(jillilnt t()t tM:i t63

t4

Eao

k;OtoLJ

o(ooFlFOlJLLo]-rrlahUoOa(r

SMOOTH STREAMU+

o5"1o"15o20"25"30"35"40"45"ANGLE OF ATTACK, o

lfl(;URE 4.5.8. Variation of the coefficient of fluating normal force, C1y_.. with angleol attack for a rectangular prism. From B. J. Vickery, "Fluctuating Lift and Drag onrr Long Cylinder of Square Cross-Section in a Smooth and in a Turbulent Flow," "/.Iluid Mech.,25 (1966), Cambridge Univ. Press, New York, pp. 481-494.

where Cl is a mean lift coefficient that depends on the particular cross sectionshape and a : 2rn, r? satisfying the Strouhal relation.

The root mean square (rms) value of the fluctuating normal force coefficient(ry,,,,. on the square section is shown in Fig. 4.5.8 t4-161 as a function of angleol'attack a with respect to the mean wind direction. Here the turbulencex issccn to lower the highest normal force below, and to raise the lowest normallirrce slightly above, the respective laminar values.

Figure 4.5.9 l4-ll presents two photographs of flow over proposed bridgetlcck sectional forms as visualized in a water tunnel flow containing fine alu-rrrinum particles. Figure 4.5.9a shows a section that produces severe flowscparation; Fi9.4.5.9b portrays the flow-smoothing effect of a modified sectionproviding lower lift and drag.

Rcf'erence [4-10] prescnts mcan values of cp and c. obtained under laminarllow conditions for a largc rrurnbcr ol-scctional shapes common in construction,irs takcn lnrm il2-21ntl 14 lltl; scc'l'ablc 4.5.1.l4-171, an<l 14-621.

r'lltc ltttlrttlcttec clt;tt;tt'lt'rislits itt lltt trlr'rrrrrrrt ol lir1l .l 'r lJ wcrc tlrc lirllowirrli: klrpitrrtlil:rlsr';rlt' l.:l/J. lirlt'r'lrl st:rlt' O.,l/1. lrrrlrrrlt nr ! ltl( u.,tl\, lll'i,

CNr..{r/2pua)u

164 ilt t,t I 80t)Y nt tt()t)YNnMt(;1;

FIGURE 4.5.9a. Visualization of water flow over a model bridge deck section. Cour-tesy of the National Aeronautical Establishment, National Research Council of Canada.

FIGURE 4.5.9h. Visualization ol'watcr flow ovcr a paI1itlly slrcirrrrlirrctl rrrtxlcl lrlitlgcdcck scction. (-ottrlcsy ol (lrt' Nrtlionul Acntnitulicitl lislitblislrrrrt'ttl , Nitliorrrrl l(cscirrclr('outrt'il ol ('lrturtLt.

Prtlth arrrl w[r(l rlilFr lh)il

*M-r.cD cL

2.O3 0

-[r,96 -2.01 o

-I 2.O4 0

-D#l1.81 0

------- L- 2.O 0.3

-l 1.83 2.O7

_L r.99 -0.09

+lJ 1.62 - 0,48

-lF 2.O1 o

nllr+ll

ilhlll

Iit(;t:; ()N tw()l)lMl Nlit()Nnt :;ilIt,(;lUllAl l()ltMt; 1€5

'l'Alll,ltl 4.5.1. 'l'wrl l)irtttttsiottul l)rug urttl l,ilI ('rnllicicrrls lirr SlructrlrulSlurpes

\rrlrrr'. Iilorrr .'Wirrtl lirrtt's ott Sltttr lttrr': /irrrrr A5('lr. ll(r ( l()(rl ), I l.l.l I l()t{ rrrrtl I ll Jl

166 Bt Ut I not)y Al tr()t)yNAMt(;ri

'l'hc rosults ol"l'ablc4.5.1 are irppliclrblc lo nrcnrbors with luryc lrspr:cl urtio(ratio of length to width) \, or lo rrtcntbcrs with ond platcs (abutrncnts). Formembers with small aspcct ratio (c.g.,

^ < l0) and no cnd platcs (abutmcnts),

end flow effects are significant, and thc drag cocflicients are smaller than inTable 4.5.1 (see Sect. 4.6.2). The drag coeflicients are also modified by thepresence of turbulence in the oncoming flow. Experiments have shown that inmost cases of interest in practice these modifications are small Il2-2, 12-51.For this reason wind tunnel tests aimed at measuring aerodynamic forces ortrussed frameworks with sharp-edged members are to this day conducted insmooth flow [12-1, 12-6]. Note, however, that in some cases the effect ofturbulence on the drag force can be significant. For members with rectangularcross section, this effect depends upon (l) the ratio blhbetween the sides ofthe cross section and (2) the turbulence in the oncoming flow. If the ratio blhis small, no flow reattachment occurs following separation at the front corners.

5

(b)

FIGURE 4.5.10. Separation layers in smooth flow (solid line) and in turbulent flow(intemrpted line). After A. Laneville, I. S. Gatshore, and G. V. Parkinson, "AnExplanation of Sonrc Ell'ccts ol'Turbulence on Blufl'Bodics," /)rrcclrlirg.r, l,ourthInternational Conl'crcttcc, Wirrtl llllccts on Buildings antl Slnrclrrn's, ('irrrrbritlgc t)niv.Prcss, Carnhritlgc, l()77.

!--Hrgher ,/drae f

tf, tlt(;l;()N twot)tMt Nlit()NAt l;ltlt t(;lt,ttnt t('ttM!; 167

l)cpcntlirrg ttlxrtt ils tttlt'ttsily, thc) turbulcnccr cirrr crrlrirrrcc llrc llow (,ulti1n1r(.nlin thc wlrkc ittttl, lltcrcliut. (:itusc stK)n8,cr sucliorrs lrrrtl lirrgcl tlrirtl (1,'ig.4.5.10a). ll'thc ntlio /r//r is sullicicntly largc, tlrer turhulcrrcrr ciur t'rrrrst. lLrwrcattachmcnt which wottltl rrot have occurrcrl irr srrrtxrlh llow arrtl llrrrs nsrrllin reduced drag (lrig. 4.5.lob) 14-25,4-26|. A bcaurilul visualizltion o| r|cllow around a body with rectangular cross scction (blh :0.4; srn<xrtlr lkrw,Re : 200) is shown in Fig. 4.5.n [4-87] and may be compared, qualitativcly,with the smooth flow case depicted in Fig. 4.5.100-see also [4-94]. -thedependence of the drag coefficient upon turbulence intensity is shown for tworatios blh in Fig. 4.5.12* 14-261. Additional studies on turbulence effects ondrag and lift of sharp-edged bodies are reported, in [4-271, 14-281, and [4-85].The effect of turbulence in the case of bodies with rounded shapes is, essen-tially, to reduce the Reynolds numberat which the critical region (Fig.4.5.2)sets in. This is shown in [4-291, which includes, in addition, information onthe fluctuating lift and drag forces on a rigid cylinder due to vortex sheddingand to turbulence in the oncoming flow (see also t4-301).

For a recent, wide-ranging review of turbulence effects on bluff-body aero-dynamics, see [4-87]. Reference [4-14] is compendium of drag effects thatcontains limited data obtained in smooth flow on models of buildings andstructures.

|I'IGURE 4.5.11. Flow around rectangular cylinder (b/h - O.4, G." : 200). From Y.Nakamura, "Bluff-Body Aerodynamics and Turbulence," ./. Wind Eng. Ind. Aerod.,49 (1993). 6s-18.

t'Notc llral lir Itlh - l, (), irs ohl:tittcrl in l,l 2{rl lol srrrrxrllr lkrw <lillors by ll-xlrt l0%, lirrrrr tlrcvitlttc lislcxl in 'l'irhlc 4..5.1. l)illctt'rr('cs ol llri$ ottlcr or l:rlgt'r' irrc c()null()n t:vt:rr lirr rr.strlls olsinrplt: wirrtl lunnrl {(:sls.

168 lrl t,l I lr()l)Y nl lt()l)YNAMI{;l

CD

048121620 _y,u., enl

FIGURE 4.5.12. Dependence of drag coefficient upon turbulence intensity. After A.Laneville, I. S. Gartshore, and G. V. Parkinson, "An Explanation of Some Effects ofTurbulence on Bluff Bodies," Proceedings, Fourth Intemational Conference, WindEffects on Buildings and Structures, Cambridge Univ. Press, Cambridge, 1977.

4.6 REPRESENTATIVE FLOW EFFECTS IN THREE DIMENSIONS

Most flows have a three-dimensional character, principally as a result of theircontact with boundaries. For example, if a hypothetical laminar flow consistingof an air mass displaced uniformly as a single unit encounters an object, it willbe diverted in several directions. Also the passage of such a flow along a surfacesets up boundary-layer velocity gradients. Three-dimensionality is clearly in-herent in turbulent flows.

Although the general equations for fluid flow remain available for applica-tion, few flow problems in three dimensions have been satisfactorily solved ina purely analytical fashion because of the considerable complexities involved.As a result, most three-dimensional studies rely partially or wholly upon ex-periment. Therefore, this section is mainly concerned with broad aspects ofthree-dimensional flows, with conditions of testing, and with some represen-tative results obtained by test.

4.6.1 Cases Retaining Two-Dimensional Flow FeaturesThe success of the two-dimensional flow models discussed in the previoussection has in a few cases been considerable because sorlc actual flows retaincertain two-dimcnsional t'catures, at least to a first approxinrllion. Consiclcr,forexample, lhc casc ol'a long nld ol'squarc cnlss scclirttt itt:rtt lrit llow withunilorm nroarr vcrkrcily norrrurl (o onc lircc:. lixccgrl rtt'rrr llrr't'rrrls ol llrc nxl,

40

o(E

FzUJotrLr-UJoozot-JUJ(E(Eo(J

o.

o.4

't F nt t,nt 1;t Nln ltvt I l()w I lil (;t:; ||J ilillt I t)tMt N:;t()Nli 169

r1"/D

I,'IGURE 4.6.1. Spanwise correlation of the fluctuating pressure difference across thet'cnter line of a long square-section cylinder for flow normal to a face (G" : 105;.lirrm B. J. Vickery, "Fluctuating Lift and Drag on a Long Cylinder of Square Cross-Scction in a Smooth and in a Turbulent Flow," J. Fluid Mech., 25 (1966), CambridgeOniv. Press, New York, pp. 481-494.

llrc mean flow may, in this case, be considered for practical purposes as two-tlimensional. However, the effects associated with flow fluctuations are notitlcntical in different strips, the differences between events that take place atrrny given time increasing with separation distance. This is shown in Fig.4.6.1l4-16] for the pressure difference between centerlines of top and bottom facesol'the rod under both laminar and turbulent approaching flow.* It is observedtlrat the three-dimensionality of the flow manifests itself through spanwise lossol'correlation R7s between pressure differences (measured respectively between;xrints,4 and A' at section.4 and points B and B' at section B), this correlationkrss being strongly accentuated when turbulence is present in the oncomingllow. From this example one may infer that fluctuating phenomena, includingvortcx shedding, cannot nonnally be expected to be altogether uniform alonglhc cntire length of a cylinclrical botly, cvcn if the flow has uniform mean speed:rntl thc body is gcrlrnctricirlly trrtilirrrn.

ln practicc, rncan llow t'orrtliliorrs rrpwirrtl ol'tall slcndcr structurcs arc usu-rrlly no( unilorrrr, ls trssrrrrrctl irr (lrt'sirrrplr'r.('ilri('r tliscrrsscrl ahovo; inrlcctl, in

r'l'lrt'lrrrlrrrlt'rttt't'lr;uittlt'lislir's lvr'tr'lltr'',;trrrr';r" ttt llrr'r'r1x'tttttr'rrt ol liil',.'1.5.11.

170 ilt t,t I il()l)y nt il()t )yNnMr(:l

thc atlttospltcric ltrlLtlttl:rty lltyt:t'tlrt'rrrt'rrrr llow vclocily itrcreirscs witlr lrciglrr.Also certain tall structurcs (c.g., sllreks):ur lt()t gcorrrctrically unilirrrrr.'l'lrcscimportant features-in addition to thc incitlcnl lurbLrlcncc-furthcr dccrcasc thecoherence of vortices shed in thc wako ol'structurcs.

4.6.2 Structures in Three-Dimensional Flows: Case StudiesThe complexities of wind flow introduced by the geometries of typical struc-tures and by the characteristics of the terrain and obstacles upstream emphasizethe need to carry out detailed studies of wind pressures experimentally usingwind tunnel models and simulation. In order to give some idea of the type ofresults so obtained and to emphasize the important roles of the boundary layervelocity profile and of the turbulence in such results, a few examples are citedbclow.

wind flows about buildings are prime examples of three-dimensional flowsthat cannot be described acceptably by two-dimensional models. Ftgure 4.6.2il5-l ll suggests such a situation. Here a tall model building in a wind flow ispreccded by a lower building. This latter trips off a vortex in the space betweenbuildings. Air descending close to the windward wall flows through openingsbeneath the building at ground level. Regions ofaccelerated flow are producedaround vertical and horizontal corners of the building. In the areas of vortex-flow, through-flow, and corner streams, many design problems are presentedby the special characteristics of the locally accelerated flow. (See Sect. 15.3and p. 188.)

A few examples are now shown of differences between drag or pressurecoefficients measured in a uniform and in a boundary layer flow. The existenceof such differences was first pointed out by Flachsbart in 1932 t4-311.we consider first the case of a rectangular plate normal to the wind in a

l"l(;tJltlt 4.(r.2. Mrrirr Icit(tttcs ol lhe llow rrnrrrn<l :r lrrll lrrriLlirrll rrrrrlt.l Il5 III

,l ri lll l'lll ',1 Nlnllvl ll()Wllll{il:;llJ llllll I l)l[/l N:;l{)Nl; lll

'l'Altl,l,l 4.(r.1. l)rrg ('rx'llil'irrrls l'rrr a llccllrrgrrlul l'lrrlt'Nolrrrirl lo Wirrrl irrSrrrrxrlh l,'low l.l-lll, l! 2l

l{cctarrgular Platc in Nonnal Wind"

Itrrelurtgttllrr l)lirtcon (inrrrnrl

(Standing on I-ongSidc)

Aspcct ratio I .0 2.O -5.0(',, 1.18 1.19 l.2O10. 20. 40. oo

1.23 1.48 I .66 I .981.0 10. oo

l.l0 1.20 1.20I'l'hc values listed in [4-10] were taken from [2-2]. Some of these values were incorrectlytrrrnscribed in [4-10] and therefore differ from those shown in this table.

snrooth flow. The drag coefficients depend strongly upon aspect ratio and uponwhether the plate is held in midair, as in the case of a tralfic sign, or standson the ground, as in the case of a free-standing wall; see Table 4.6.1. Forrcctangular plates on the ground, the drag coefficients of Table 4.6.1 are rea-sonably consistent with mean drag coefficients obtained in boundary layer flow14-931. Reference [4-93] contains additional results on free-standing walls,rncluding pressures in the presence of a building upwind or downwind fromtlrc wall.

Note that the aerodynamic force normal to the plate is not necessarily largestwhen the yaw angle a (Fig. 4.6.3) is zero. For a plate with aspect ratio X :5, the dependence of the aerodynamic force normal to the face of the plateupon cv is shown in Fig. 4.6.3. It is seen that for ot : 4Oo the aerodynamiclirrce is larger by about 15% than in the case cv : 0o. A similar, thoughsornewhat smaller, increase was reported in ll2-21for a plate girder with aspectrrrtio X = 10.

The effect of turbulence on a square plate normal to the flow was studied in14 251, where drag coefficients were measured for both smooth flow and tur-bulent flow with 8.3% turbulence intensity and 7 .6 cm longitudinal turbulencescale; see Table 4.6.2.

Note that the drag coefficients measured in smooth flow differ slightly among

1.2

08(',, 0 4

0

o4

lr'l(;llltl,l 4.6.J. lX!l)(:nrlt'rrt't'ol tlt:r1'. tocllrr'rctrl lot pl;tlt'willt itspccl ntlio \ 5 ttpotttlirccliorr ol lrolizon(rrl wirul I l.) .'l

172 lll Ul I ll()l)Y nl lr()l)YNAMI(:i;

'l'Alll,lil 4.(r.2. lh'ug ('rx'llicitrrls l'rrrSr;uarr: l)latc Nrlrrttrtl lo lhe Mr.un lr'krwl4-2sl

Plate Sizc(cm) Sntrxrlh 'l'urbulcnt

(',,

5.08 x 5.0810.16 x 10.1615.24 x 15.2420.32 x 20.32

t.121.09l.1ll. l5

t.26L22t.20I.t8

themselves and from the value of Table 4.6.1 (CD : l.l8). Note also that asthe ratio between the longitudinal scale of turbulence and the dimension of thcplate decreases, the influence of the turbulence on the magnitude of the dragcoemcient becomes smaller. These results are further discussed in Sect. 7.3.3.

Figure 4.6.4a shows a model used for measurements reported by Flachsbartin 1932 t4-311. The measurements were conducted in both smooth and shear(boundary-layer) flow (Figs. 4.6.4b and c). The measured mean pressure coef'-ficients Q, referred to the free-stream velocity, are shown in Fig. 4.6.4d forsmooth flow and Fig. 4.6.4e for boundary-layer flow (interrupted and solidlines represent pressures and suctions, respectively). It is seen that the differ-ences between the results obtained in the two types of flow are significant.Similar results were subsequently obtained in 14-321 and [4-33].

Figure 4.6.5a depicts mean flow patterns around a vertical wall of height-to-width ratio I : 1 with uniform approaching flow. Figure 4.6.5b depicts thesame situation in boundary{ayer flow. Figures 4.6.6a and 4.6.6b display thepressure coefficients developed on the faces of a cube resting on a horizontalsurface (due to flow normal to one face) first in uniform flow, then in a bound-aryJayer flow. Figures 4.6.7a and 4.6.7b present similar results for a tallbuilding. It is noted that in Figs. 4.6.5b,4.6.6b and4.6.7b the pressure coef-ficients are referred to the free stream velocity t4-201.

Loads on structural parts (e.g., cladding) are determined by the algebraicsum of the extemal and intemal pressures acting on these parts. In the idealcase of a hermetically sealed building, the internal pressure is not affected bythe external wind flow (Fig. 4.6.8a). If the building has an opening on thcwindward (leeward) side and is otherwise sealed, the wind flow will create apositive (negative) internal pressure, as shown in Fig. 4.6.8b (Fig. 4.6.8c).

In most cases the opening or porosity distribution over the building envelopcis not known, and intemal pressures could be either positive or negative (Fig.4.6.8d). Building standards (e.g., [2-491) specify intemal prcssurc cocflicicntsgenerally believed to be conseryative fbr use in design. lrrvcstiglrtiorrs into thcmagnitude of intcrnal prcssurcs and of thcir dcpcntlurt'c orr tirrrt' lrrt' rc;xrr1e:tlin 14-521 to l4--571, which contuins adclitional rclcrctrccs.

,l ri lll l'lll iil Nln llvl ll()W lllll lri ll! llllll I lrlMl t..l!;l()l.l!i 113

Wrrrrl

---d>

Wtrrrl

('lllo Io 2.O

l, llJlllll,,.l.(r.4. Srrrnnurry ol rrrrxh'l l(':,1:, rr :;rrlxrllr rrtrl lrottntiary-laycr llow. F'nrtl\\'rrrlrlrtrt'k lrrrl gcsclrloss('nr' llr(l nllr'rrt ( i('lr;ru(l('." lry ( ). liluchsbafl, in Iirgtltri,t.tt

,1, t lt't!\l\',tttrtti.st'lrttr Vt'r,tttt'lrttit.tt,tlt .it (;tttttut:('tr, lV l,it'li'rrrrrg, 1,. l'rrrrllll, rttttl A.llt t., (r'rls. ), Vcllirg vott ll . ( )lrlt'trlrotttp, l\ltttttr lt irrr,l ll,'rlirt, l().J2.

zE

174 Bt t,t I B()l)Y nl lr()l)YNnMl(i:;

r.0 0AP"p-ifr"

FIGURE 4.6.5a. Flow pattern and center line pressure distribution of a wall of height-to-width ratio I : 1 in a constant velocity field. From W. D. Baines, "Effects of VelocityDistribution on Wind Loads and Flow Patterns on Buildings," Proceedings, Sympos-ium No. 16, Wind Effects on Buildings and structures, held at the National PhysicalLaboratory, England, in 1963, published by HMSO London in 1965.

4.7 THE RELATION OF TIME-VARYING FORCES TO WINDVELOCITY IN TURBULENT FLOW

For a given body immcrscd in a wind flow it is of intcrcsl (o crtttvc:tl irrlilrntationon vclocity lluctualions into inlirnnation on prcssttrcs ovt't lltt'lrotly or onrcsullanl lirrcc:s iurtl nl()nrc:nls. Sinc:c: tttrtst rr:itl lltlws lttr' sttlltt it'lttly t'oltllllcx

!-H

r'0

4l llll lillAlloll()l llMl v^l tYlN(it()t t(:t t{ r wt[]D vt I { '(

.ltY 175

lrlGURE 4.6.5b. Flow pattern and center line pressure distribution over a wall ofhcight-to-width ratio 1 : I in a boundary-layer velocity field. From W. D. Baines, "Ef-lccts of Velocity Distribution on Wind Loads and Flow Pattems on Buildings," Pro-,'rulings, Symposium No. 16, Wind Effects on Buildings and Structures, held at theNirtional Physical Laboratory, England, in 1963, published by HMSO London in 1965.

tlurt analytical calculation ol'such rcsults is not possible, it is usual to employlirrrnulas l'eaturing unknown cocllicicnls that may be evaluated by experiment.

4.7.1 Drag Forces'l'lrc nct tlt'itg lirrcc c()nsisls ol lltr rt'sttll;url ()v('r'ir givcrrt botly surlircc ol'trllt'orrr1'roncnls ol'clcrrtcntlrl lirtcr':; llurl ;rrr';rlrlqnr'tl witlr llrt'tlnrg, orirlorrg wintl,

176 tJttJt I tlot)Y AFllol)yNAMtct

--.5 5

-.6 0)

--_ =-.-.80--......-...- -.2=-_.70- )

__--s__--

\--__

--.65 -

- -.60 --/

-\--.80-at---

--.,

J

_-.70r'(_

'.(.

rilI,, 99

l/rlf*'no

(a)FIGURE 4.6.6a. Pressure distributions on the faces of a cube in a constant velocityfield. From w. D. Baines, "Effects of velocity Distribution on wind Loads and FlowPatterns on Buildings," Proceedings, Symposium No. 16, Wind Effects on Buildingsand Structures, held at the National Physical Laboratory, England, in 1963, publishedby HMSO London in 1965.

direction. The time-varying drag FoQ) on a body completely enveloped by aflow is conventionally given by the formula

Fugy : [pu2gynzc,, (4.7.t)

whcre B is l lypicirl lrorly rlirncnsion irnrl (i7, is lhr.: rrsrrirl tlrirll t.rx'llicicrrl

ilttAil()N ()t ltMl

0.2 0

vnllYlN(i l()t t(;l :; t() 11y151; vt t(x;t ly 177

-0 ,0 0./0

(b)

lfl(luRE 4.6.6b. Pressure distributions on the faces of a cube in a boundary-layer't'l.city field. From w. D. Baines, "Effects of velocity Distribution on wind Loadsrrrrtl lrlow Pattems on Buildings," Proceedings, Symposium No. r6, wind Effects onlhrildings and structures, held at the National physical Laboratory, England, in 1963,grrrblished by HMSO London in 1965.

Irr Eq. 4.7.1 a seconcl tcrrrr .l' rlrc lirrrrr pn\au1rlldtlc. is often included,1r;ulicularly if the lluitl irr tprcstiorr is rclirrivcly rlcnsc, for examplc, as in thctrtse: ol'walcr; (),,, is an cnrpitit':rl "virlrlrl nritss" erx'llicicnt intcn(lcd lo ucc()111Ior cll'ccls linkctl trl thc lltrirl itt t t'lt'tirlron. At lrrrrlly llrc cocflicicrrl (1,, trppr:trr.sIo bc ttsr.:l'trl itt citsc:s wlrt'n'itt lltr'llttirl ruirrr urvolvt'rl is tr;.r1'rrcciirblc rclir(ivc to

4 / lilt

{l*,no

178

5

llr t,t I l1()l )Y nl ll()l)YNnMl(;i

. -0.9

z

f *'"0

Fro nt Bock

(a)

FIGURE 4.6.7a. Pressure distributions over the sides and top of a tall building modelin a constant velocity field. From W. D. Baines, "Effects of Velocity Distribution onWind Loads and Flow Pattems on Buildings," Proceedings, Symposium No. 16, WindEffects on Buildings and Structures, held at the National Physical Laboratory, England,in 1963, published by HMSO London in 1965.

std?

4 / llll llt tn it()N ()t ItMt v^l tytN{i t()t t(jt j; t() wtNt) vt t(x;ilv l7g

(b)

lrlGURE 4.6.7b. Pressure distributions over the sides and the top of a tall buildingrrurdel in boundary-layer vclocity ficld. From w. D. Baines, "Effects of velocityl)istributi<rn on Wind Loatls irrrtl lilow P:rltcrns on Buildings," Proceedings, Sympos-ittttt No. 16, Wind E,ll'ccts orr lhriltlings irntl Stnrcturcs, held at the National Physicall.rrboratory, Iinglancl, in l()(r.1. prrlrlislrt.tl lry IIMS() Lonckrn in 1965.

f *'"0

.6

S idc

-0',1 to-0.49

Bock

180 Bl tJt I lr( )l)Y n I ll()l)YNn Mloii

(o) HERMETIC BUILDING (b) WINDWARD OPENING

WIND

(c) sucrroN oPENTNG,r, ?t_.il't:iro:,X:*.

FIGURE 4.6.8. Mean internal pressures in buildings with various opening distributions. From H. Liu and P. J. Saathoff, "Intemal Pressure and Building Safety," ./.

Struct. Div., ASCE, f08 (1982), 223*2234.

the body mass. One can then visualize it as specifying a hypothetical mass

which, given the acceleration dUldt, accounts for the net force due to all thcvariously accelerated fluid elements in the entire flow around the body' In mostflows of interest in wind engineering. however, the entire term containing C.',,,

contributes only a negligible part to Fp. For this reason it is usually neglectctlin this context, and it is not retained in what follows.

A three-dimensional flow will have three components, U(t)' V(t), and W(l),in three mutually perpendicular directions. In the neutrally stratified flows* ol'strong interest to wind engineering the mean wind velocity -U is horizontal,and the wind then can be represented as the sum of mean and fluctuatingcomponents.

U(t):D + uO)

V(t) : u(t)

W(t) : w(t)

(4.7.2t

the means of u, u, and w being zero.One may then express drag in the horizontal direction by means of Fq.4'7 'l

with U(r) as in Eq. 4.7.2. In general, when time-varying vclocities arc lltttsintroduced, the imperfect spatial correlation of thc vckrcily llucttutti(tns Itrtlslalso be considered. Howcvcr, hcrc it is first assutnctl lhtrt lhc lrtxly in tlttcsliolt

*Sco Clt:rptcr l, 11.(), irtttl ('hitplt'l 2' 1l .l.l

4 / illl ilt tAil()N ()t ltMt VAnytN(i tr)lu F.i trr 6111.11, vt t(x.ilv 1Bl

r'':'rrllrt'icttlly stttitll t'ottt;trtretl lo llrc crlntlttliort rlrslrrrrt'e.s ol llre lltrgllirlrotrs 1,r'.;ttttl tt', srl lltitl, lirt tlrt'l)ttrl)()scs ol'lhc: lllrrblt'trr rrl lr;rrrrl, llrcsr. lirller rrriry lte,,rttrltlt'lctl kr bc ptrr'lirt'll-y corlclulcrl. Sirrec irr llrt. lupilr wilrtls rrsulrlly ol'grllrlcslrrl{'r('st lo wintl crrgirrcclirrg u(tllU ftrrcly cxr.t't.tls o.l, r' nriry gcnuirlly bcnt';rl,'r.',r.,,, witlr srrritll c:r'nrr yiclding

I"n(t) - F,, 1 plJulrTll(',,

rllrt'rt' thc stoarly arrcl tlro lluctuating parts of thc clrag lirrce are, respectively,

F,,: lpo2cotu, t u\Ol (4.7.4a)

:l!ltl

Fo: pUu(t)yz'Co @.7.4b)

I'rrrrrr litl. 4.1 .4b it is seen that Fp(t) varies directly as u(t). This is true ro alrrr,r ;rpPtrxirnation only, since observation of physical flows reveals that cpirrirv ilscll'also vary as a function of the frequency components of ,r(/).

Irr orrlcr to cxamine the statistical characteristics of Fb@, it is useful to, ,n"r(l('r' ils spcctral density Sro@).one first calculates its autocovariance func-tirn (s('(' Appcndix A2., Eq. A2.21):

(4.1.3\

(4.1.s)

(4.7.6)

(4.7 .7)

(4.1 .8)

1fr/2111,,(r) -' ,l'i f ) ,,rnb{,lnptr + 11 dt

I: prurFilcrpe4t + n

: FbG)Fb(t + r)

rrlllr t'

S6,(n) : t J* Rp,{r;cos 2trnr dr

r.'\1r1rt'rrrlix A2, Eq. A2.20), it follows that

.\'1,;,(rr) : p2O2 F C2rS,1n1

f trr rrlrrrlr rlris through hy (t,p(/; lll)'' yie:ltls rhc spectral density sc, of the fluc-tn,rtrrl' rlr':tg cocllicicnl :

,!,,,{rr) 't. ;, 't;l'l'

llrr" t'rlttitliott will l.ltr vitlitl ovr't lltl t;rn1:t',r! lrt'rlrrt.rrt'it's ol',\',,(l) prrrvitletl lrllr llr'r l\ rr'ntitin ;x'rli.ctl.y cot!rlirlr.rl in .ui!iuuri.{l ;rlrovt'_ llowcver, ltt.r.ttttst. in

182 ttt ut I tt()l)Y Al li()l )YNAMlcl

practice this assurrrptiorr tkre:s rrol lroltl, il is rrsturl (o ittcltttlr-: itrt lttlitrsllttctttfactorto preservc thc validity ol'l')t1. 4.7.t'|.'l'his is tlonc by writirrg (4.7.11) as

(4.1.e)

where the newly introduced factor y21r1 is termed the aerodynamic admittance*of the body in question and represents a modifying adjustment (fbr an actualbody) of the ideal case of a body enveloped by turbulence with full spatialcorrelation. This modification brings the drag coefficient spectrum into align-ment with actual conditions.

Thc aerodynamic admittance is a function of body shape and dimensionsancl of the characteristics of the turbulence. For a given body it is thus atiequency-dependent function. Figure 4.7 .l l4-3slsuggests the manner in which12(n) varies for a square flat plate placed normal to a turbulent flow withuniform mean speed. The decrease of the aerodynamic admittance with in-creasing frequency corresponds to the fact that the smaller turbulent eddieshave shorter wavelengths; thus those eddies with higher frequencies will sufferloss of coherence more rapidly than do the large eddies. References [4-36] and

[4-37] appear to be among the earliest to have introduced and used aerodynamicadmittance concepts in buffeting problems.

4.7.2 Relation of Wind Pressures over Slender Buildings to WindVelocitiesThe type of arguments employed in Sect. 4.7.1 in relation to total drag forcesis now applied to the case of a high-rise building of rectangular plan form,with the horizontal wind blowing normal to one face. In this instance, thealong-wind structural motion is dependent on the windward and leeward pres-sure distribution in a manner that is conceptually simple.

The pressure acting at a point Q of elevation z on the surface of such a bodyin a steady flow of velocity U(z) may be expressed as

p(Q) : ip(I2tz'tCo<Qt (4.7.10)

where p is fluid density and Co is the appropriate pressure coefficient at thispoint.

In the case of unsteady flow U(z) : U(<) + u(2, r) the pressure may beapproximated by

p(Q, r) - pQ) + p',(Q, r) (4.1.1t)

*Thc ttso ol'tltis letttt irr wirrtl trttgitttrctittg is ltlt cxtcttsitllt ol lls otif itt:tl ttsc itt :rt'trtttltttlicitl

conlcxts l4 .141.

- ,\..(rr )56r(r) - 4Ci, :';',;' \ t(r)

4I llll ilt lAil{)| (}t ilMt vnl tylN(i l()l t( t!, lil wtt.Jtt vt t(t{.ilv

to-3 lo-2 to-r I to

^B/Ul.'l(lllRIl 4.7.1. Aerodynamic admittance of a square plate in turbulent flow. After p.W llcarman, "Wind Loads on Structures in Turbulent Flow," in The Modern Design,'1 ll'irul-Sensitive Stuctures, Construction Industry Research and Information Associ-,rrrrrrr, l-ondon, 1971, pp. 42-48. By permission of the Director of the National physicall;rl)orirtory, U.K., and the Director of the Construction Industry Research and Infor-rrr.rtiorr Association, U.K.

rvlrt'rc 2 and p' have the following values:

(4.7.12)

UJozFF

=oo=zoo(EUJ

I -rrlr+-s!lpe) : rpcptetu'k I It_(z)l

p'(Q) : ) oc,Q)u'alz 4! . okfi{n) ro, ,r,,

rvlrcre overbars indicate the mean values.A bricf numerical example is in order here. In the atmosphere u21z1t/2 =

' 'jrr,r, rrnd U(z) :2.5u*lnl(z - 2.,1)lz1l (see Eqs. 2.2.18 and.2.3.2 and Table

,,,' ', Forcxample, il'1,, - 0.03 rrr, ?.,1 = O, and U1 l0) :40 m/s atz : 50

rl'(':)rt't.t () ()llt

',o lltitl lltc cltrrt ilr rrr-:glct'lirrli lltt' trorrl!rrr':rr lt'rrrr rrr litl .1.7. ll is lcss lhln ),X,.

184 lll ut I u()t)y nt n()t)YNnMt(;li

Morc gcncrally ancl anuklgotrsly lo llrr'rllrrg tcsttlls lrlrcatly tliscttssetl irt Scct.4.'7.1, calc'J,lations reportcd in l4-.ltil itrtlit'irtt: tlrirt (hc lirlkrwing tclations arcsatisfactory, with insignificant erK)r, lirr 7r lttttl 7r':

If, as in the case of many buildings, the horizontal dimensions of the bodyare small compared to the scale of turbulence, it is reasonable to assume thatthc fluctuating pressures affecting along-wind response-which consist entirelyol'thosc on the windward and leeward faces-may be given by

pQ) : )pD'Q)c,,(Q)

p'(Q, t) : pOQ)u(Z, t)Cp(Q)

p,(e) : pU(z)u(2, t)C,(e*)

p,(e) : pry1a1u(2, t)C{e)

(4.1.t4)

(4.1.1s)

(4.7.16)

(4.',l .r1)

(4.1 .18)

(4.7.1e)

where Q", and Q1 are points on the windward and leeward faces, respectively,and where

c,(Q*): #+C/Q):ffi

where z is the elevation of point Q, or Q7. As discussed in Chapter 9, it isusual in current procedures for estimating along-wind building response toassume that Eqs. 4.7 .16 and 4.7 .17 are valid regardless of the ratio of buildingtransverse dimensions to the scale of turbulence. This point is brought up againlater in Sects. 4.7.3 and 4.7.4.

In calculating along-wind structural response (see Chapter 5) information isrequired on the spatial correlation of pressures applied at any two points Q1and Q2. Such information is supplied by the co-spectra of fluctuating pressures(quadrature spectra being assumed negligible). Assuming the validity of Eqs.4.1 .16 and 4 .7 .17 , the co-spectra take the form

S,',i,(Q,, Qz, n) : co(Q)cp(Qr)p'DQ)u(z)sc,,,,(Qr, Qz, n) (4'7 '20)

That is, the co-spectra of the pressures are proportional to the co-spectra of thefluctuating longitudinal wind components in the undisturbed oncoming flow atthe elevations of the two points. The pressure coefficient C,IQi) rcpresentswindward or lccwarcl vatlucs dcpcnding upon whcthcr lhc poirrl Q, is on thewindwarcl <lr lccwitnl sitlc.

4/ lllr nt tAill )ft()t ilMt vnttytN(it()t ll:t t; t(lwll.lt )vt t()(]ily tB5

'l'ltc co spt't'tnrrrr,\f,,,,, rrury lrc cxllrcssctl irr llrc lollowilrl' 1,,t"t'

,\'j,,,,,t1t,. Q,. rr) ,tf,,,,tr'. tttN(ttl (4.1.21t

whcrc S,f,,,r(r, n) is lltr: ('() sl)octrum <lf thc lorrgitutlirrll vckrcity lluctuati()ns .ltlxrints Q1 and Q2 (Qj bcing thc pro.jection ol'Q, orr u planc. nonnal to the mcanwind direction, that contains Q), and r is thc clistancc bctween Q1 and Qi.'l'hc f'unction N(n) is ref'crrcd to as the along-wind cnrss-correlation coefficient.

11' B1 and Qz are contained in the same vertical plane normal to the meanwind (i.e., if their along-wind separation is zero), then N(n) = l. For nonzerorrlong-wind separation, an expression of N(n) is given in Sect. 4.7 .4. In thet'rtsc Q1 = Qz,

so,(Qr, d : c3(Q)p',O2k)s,(2,) (4.7.22)

4.7.3 Pressure Fluctuations on the Windward Face of Bluff Bodiesn theory of turbulent flow around two-dimensional bluff bodies has been de-vclcrped in 14-391, which has subsequently been applied in [4-40] and [4-4llto the study of surface pressures generated by turbulent velocity fluctuations.'l'hc theory is based, essentially, on the following assumptions: (l) the turbu-Icnce intensity is of the same order of magnitude as, or lower than, the tur-lrrrlence intensity typical of atmospheric flows, (2) the body is sufficiently long{lrat end effects may be neglected, (3) in the flow region upwind of the body:rrry velocity fluctuations induced by wake flow are statistically independent ofthc velocity fluctuations caused by the oncoming turbulence and so the lattert'irn be studied separately. Fundamental to the approach of [4-391 is a gener-rrlization of "rapid-distortion theory" which allows the linearization of theer;uations describing the turbulent motion near the upstream face of the body.'I'his linearization follows from the assumption that the changes in the meanllow associated with the presence of the body distort the turbulence sufficientlyr:rpidly so that, during the distortion process, the nonlinear inertial transfer ofcncrgy between eddies of different sizes is negligible.

The equations for the turbulent nondimensionalized vorticity vector a,)i G :| . 2, 3) are then [4-39]

D6, +Dt .i :t

\fi,cdr^ (i:1,2,3)" dxj(4.7.23)

wlrcrc l; (i : 1,2,3) is tlrc norrtlirncnsionalizcd velocity fluctuation vector'I'hc nondimensionalizccl prcssiur'('7)' is givcn by

lF' ittr,=ldl; ilt )i / rr t"" , ,, t"4\

r r \ I t)\t ' ilt,/(i r. 2. 3) (4.1.24\

186 lil t,t I tt()t)y nt n()t )yNnMtcli

wherc Oi (i : l,2, 3) is lhc trotttlitttt'rtsionirlizcrl rrrcirn vclocily vc(.[or, wlroscfield is approximately irrotatiotrirl.* 'l'lrt' lrorrrrtlirry conclitions arc csscntiallythe following: (l) at large distanccs lirrrrr (hc b<ily, thc velocitics approachtheir values in the undisturbed flow ancl (2) in thc immediate vicinity of theupwind surface of the body, the velocitics at, cach point are perpendicular tothe outward normal from that surface.

Calculations carried out on the basis of the above equations and boundaryconditions suggest, for example, that whenever alL" < I (where a is the typicalhorizontal dimension of body and I is the longitudinal turbulence scale), Eq.4.7.22 is applicable, on the windward face, up to frequencies r, = 0.15 Ula,where U is the mean speed of the undisturbed flow 14-411. For higher fre-quencies the pressure spectra decay more rapidly than the velocity spectra sothat, for structural design purposes, Eq. 4.7.22 is conservative. That this is thecase appears to be confirmed by experimental results reported in t4-421,t4-431, and [4-44].r Calculations also suggest that the smallei eddies are "piledup" against the upward face of the body and therefore that the coherence ofthe high-frequency pressure fluctuations is somewhat greater than the coherenceof the high frequency velocity fluctuations in the undisturbed flow. In structuralengineering computations this "piling-up" effect can be taken into account bychoosing appropriately small values of the exponential decay coefficients inEq.2.3.29 or 2.3.30.

4.7.4 Pressure Fluctuations on the Leeward Face of Bluff BodiesAccording to Eqs. 4.7.16 through 4.7.19, the ratios of the pressures on theleeward face to the pressures on the windward face are the same for bothfluctuating and mean pressures. Results of full-scale measurements suggest,however, that the pressure fluctuations on the leeward side are less strong thanindicated by Eq. 4.7 .17 (e.g. , see Fig. 4 .7 .2 taken from t4-501). It is reasonableto assume, therefore, that the use in design of Eq. 4.7.17 is conservative froma structural safety viewpoint.

Also of interest for design purposes is the question of the extent to whichpressures on the windward side of a building are correlated to the pressures onthe leeward side. It is intuitively clear that this correlation cannot be perfect.The correlation will be greater for eddies with large wave lengths-which canbe thought of as enveloping the body in the same manner as the mean flow-and will decay as the wave lengths decrease. This dependence can be expressed

*An irrotational flow is one in which the components

,,:# y,,,,_y,_ _*, ".:y,_y,are all zer<r.lAccording to l4-(r71, ltowt:vcr, ul any givon Ircqucncy thc prcssun':rrrrl vchx rlv rpt.r'trr lurve lhcsanrc skrpc.

4t lilt ilt tnit()N()t ilMt vnnylN(il()l t(:l :, t()wtNttvt t(x]ty t87

0.0 25.o 50.0 7s.0 100.0 125.0 1s0.0Time (s)

lfl(;uRE 4.7.2. Yaiations with time of wind pressure on the windward and on thek'cward wall of a building. After w. A. Dalgliesh, "statistical rreatment of peak Gust,rrr Oladding." J. Struct. Div., ASCE, 97 (1971),2173-218j.

lr.y choosing an appropriate expression forthe function N(n) in Eq.4.7.2l.Inl.l-45| an expression for this function has been proposed of the form

od

c;

N(n)

t

ll: E- '1(r - e-'El

15.4nA,x

(4.1.2s)

(4.',|.26)U

wlrcre u is the mean wind speed at elevation (213)H; Ar is the smallest of thetlirrrensions B, H, and D; B is the width; F1 is the height, and D is the depth.l'the prismatic body. Full-scale and wind tunnel measurements reported inl,t-461, [4-471, t4-48], and [4-49] suggest that this expression is adequate forpnrctical use.

4.7.5 Peak Local Wind Loads'I'hc adequate design of roof members, roofing, cladding, and other elementssrrsccptible to failure due to the local action of wind (e.g., solar collectorsl'1 601) is of foremost impoftance for reasons of both safety and economy. Itis thcrcfbre desirable that wincl-inch.rccil loads on such elements be ascertained:rs roalistically as possiblc l4-591.'l'hc clcrncnts potcntially ittvolvctl irt lirilurcs duc to local wind loads aretrstttrlly rclativcly rigitl so llurl llrt'tlyrr:urric lrrrrplilication ol'thc rcsponsc isltcgligiblc. Thc witrcl krlrtl rrclirrl'. ort lrrt clt'rrrt'nl is tlrr:rr cqr.ral to llrc sulrr, ovcrlltt'crttirc itrcit ol'thc e:lctttcttl. ol llrc inslrrrrllrrrt'orrs l)11:ssutcsi intlrrccrrl by wirrtl.I)trring cvcly sl()rttt llris loittl tt':tr'ltr's;r pritlr. lltt'clcrrrr:nl coltccrrrrctl, tirrtl its

- - .ti;q fP- -' - ^ -' -/zv-- -

188 uttl I B()DY nl not)YNAM|(;li

connections, must bc closignccl lo srrsllirr lhe llcuk wind load attaincd cluringthe N-year storm, whcrc N is thc lttcan rccut'rotlcc intcrval of the dcsign windspeed specified for that element.

The total wind force acting on an element such as a roof member or a curtainwall could, in principle, be measured directly. However, the experimental set-ups required for such measurements are prohibitively expensive and impracti-cal. For this reason forces acting over an element have recently been measuredby devices that automatically add pressures occurring simultaneously at severalpoints of the element, weighted by the respective tributary areas. In particular,such techniques have been used at the University of Westem Ontario to measurewincl l<rads <rn models of low-rise structures 14-71, 4-72, 4-'13, 4-74, 4-75,4-16,4-771. These measurements, as well as results of full scale tests, [4-51,4-1t1,4-79,4-801 have been used to develop new design load provisions forrnain f-rames and for parts and portions of low-rise buildings that have beenrecently incorporated in various standards, including [2-1391; see also Sect.9.5. Local pressures can have strongly non-Gaussian distributions, especiallyat comers and edges; see [,4.2-13].

4.8 SECONDARY WIND FLOW EFFECTS

In addition to the wind loads caused by the direct action upon the structure ofthe wind flow, it is of interest in certain situations to examine secondary effectsproduced by wind, such as the blowing of roofing gravel [4-58,9-63], and thedrifting of snow. Systematic studies of these effects have been reported in14-63,4-&, 4-65,4-66, 4-81, 4-82, 4-83, 4-841.

Mention is also made of wind action as a factor that influences the energyconsumption of buildings by increasing air infiltration. It is shown in [a-68]that energy losses due to wind-induced air infiltration can be reduced signifi-cantly by the sheltering effect oftrees acting as wind breaks; the energy savingsthus achieved may in certain cases be as high as l5%. The results of [4-68]were obtained in wind tunnel tests and were subsequently confirmed by full-scale measurements [4-69].

ADDENDUM

For the sake of its historical interest, we reproduce here a note by Count Buffondescribing the flow changes occurring upwind of a tower, for which it offers acharming (if scientifically no longer tenable) explanation. A translation of thenote follows.

On Reflected WindI must rcporl hcrc lrrr obsr:rvlliorr which it sccnls to Ittt'ltits t'st'rt1tt'tl lltt'ltllcttiionol'plrysic'ists, cvt'rr llrorrglr cvt'r'yorrc is in ir posiliott lo vt'ttly rl. ll st't'rrts llutt

Atlt)t Nt){ tM l Bo

rcllcctctl witttl is slroufl('r llrrrrr rlit'ccl wirrtl, irrrrl llrt'rrron'ri() lrs ()lrc is cklscr totlrc obslaclc lhrrl rclle't'ts rl. I lravc cxpc:rir-:rrcctl llris ir rrrrrrrhcr-ol'tirncs byirppnlaching l l()wcl'tlurt is alrrxrst l(X) lbcl lrigh irrrtl is situatccl at lhc nortlrt'nrl ol'tny ganlcn in Monlb:rrtl. Whcn a stnng wintl l'rkrws lhrrrr tlro s<lutlr, upIo thirty stcps lirrur llrc towcl onc fbcls stnrngly pushetl, alicr which thcrc is:rn interval ol' livc ol six stcps where one coascs to bo pushcd and where thcwind, which is rcllcctcrl by thc tower is, so to spcak, in equilibrium with thetlircct wind. Aficr this thc closcr one approachcs tho tower, the more the windre llccted by it is violcnt. lt pushes you back much more strongly than the directwind pushed you forward. The cause of this effect, which is a general one andt'rrrr be experienced against all large buildings, against sheer cliffs, and so forth,rs not difficult to find. The air in the direct wind acts only with its ordinarys1rccd and mass; in the reflected wind, the speed is slightly lower, but the massis considerably increased by the compression that the air suffers against theobstacle that reflects it, and as the momentum of any motion is composed ofthc speed multiplied by the mass, this momentum is considerably larger afterI I rc compression than before. It is a mass of ordinary air that pushes you in thelilst case, and it is a mass of air that is once or twice as dense that pushes youb;rck in the second case.

d I'Hitloire Nanrelle. t t t6 Suppldment

ADDITIONSA l'Article gui a pour titre: Des

Vents riglis, page zz4.I.

Sur Ie Vent riflichi, pege 2+2.TJ r': oors rapporter ici une obfervation.;,ri rne paroir avoit 6chappC I l'attcncion'l<'s Phyficiens , guoique tour le rnondeli'it en drat de la viLifer ; c'eft que levc'rr riflichi eft plus violent que Ie ventrlrrcdt, & d'autant plus qu'on eft plus1,,rt's de I'obftacle qui le rerrvoie. J'enri llir nonrbre de fois I'expirience , en.rl,prochant d'une tour qui a prls de, cnr pieds de hauteur & qt-ri fe trouvelirtrdc au nord , ) l'extrCrnirC de rnoni.rrrlin, I Montbard ', lbrfqrr'il lirrrfflcrrrr grand venr du nridi, on lc lcrrr lirr-r('nrent pouflc iulqu') trcnte p:rs rlt' l:rr,"'r i ry,ris <1uoi , il y a trn irrrcrv.rllc ,h.tirrtg t-ru lix pas I oi l'on cclli rl'trre

ooufG & or) Ie veur, qtri eft r6flichi parla tour, fair , pour ainfi dire , iquilibreavec Ie vent direCt ; aprls cela, plus onapproche de Ia tour & plus Ie vent quien eft riflechieft violenr, il vous repou(een arriire avec beaucoup plus de forceque le venr direC! ne vous poufibit enavant, La caufe de cer efter qui eft gi-n6ral , & dorrt orr peut faire l'ipreuvecontre tous les grands bntinrens, conrreles collines coupies ) plornb , &c. n'eftpas difficile I rrouver, L'air dans le ventdire& n'asir qu€ par fa vireffe & [anra(fe ordirraire ; dans Ie vent rdfichi,la vite(G eft un peu dirninude , nrais Iarna{Ie eft confiddrablemenr augmerrriepar la cornprefiion que I'air fo'rftiecontre l'obftacle qui Ie rdflCchir ; &comnre Ia guantird de tout rrrouvelrlenteft compolie de Ia vire(G nrulripli6epar Ia nraflb , cette quantird eft bien plusgrande apris Ia courprefiion gu'supara-vanr. C'eft unc nra(G d'air ordinaire, quivous pouffb dans le prenrier cas, & c'eftttrrc rnafG d'air une ou deux fois plus

lt:;:t" , U"t vous repcu(G dans le lccond

I rrr'silrrilc ()l n()le {)rl rcllcclcrl wiilrl l,trtttt llt\lt,ilr Nttlutrllt', (ir,ttr,tttlt tt l\trticuliin,, ('(ilttriltutt l(.\lilrtlttt'.t tlr Itt Nttlurt, Itrrr M. lc lrtttlr rlr llrllltt, Vol I l. lttlcrtrlirrtl rlrr .l:rrrlirt t^l tlrr (':rltitrel rlu l{oi,rk' l'Atrtrlcrilir liritrrr,'irist', rk' r'clle rli'r :ir t.ilr.r r=lr lo[rr' lrIrrrt'rrrr'. A l'irris, l)r' l,'lrrrPrrrrrrreItoyrrlc, I //ll

190 tJt t,r I tKltlY nt ti()t )YNnMt(;:;

REFERENCES

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ilt IItil tJ( I,; lgI

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't-36 H. W. Liepmann, "On the Application of Statistical Conceprs to the BuffetingProblem," J. Aero. Sci., 19, 12 (1952),793-800, 822.

'I'31 A. G. Davenport, "Buffeting of a Suspension Bridge by Storm Winds," -/.Struct. Div., ASCE, 88, No. 5T6 (1962), 233-2&.

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l.l9 J. C. R. Hunt, "A'l'hcoly ol 'l'urbulcttl Iilows Round Two-Dimensional Bod-ics," ./. Fluid Mcclt., (rl, I'rrr1 .l (l()7 l). (r2.5 7(Xr.

I '10 J. C. R. Htrnl, "A'l'ltcoty ol lilttt'ltt;rlirrli l'r.cssrucs orr Illull'llorlics in'l'rrrbtrlclrtlil<rws," in Itnx't'ulirt,ti.t rtl lltt' ll l l'..lII l..lllll ,\t'rrtltolitrttr ttrt lilol' lrulrttt,tl Slrtrt'luntl Viltnttitnt,r, l(lrtlsrttltr'. Wr'r,l ( it'rrrr;rrrv, l()'/1. li. N:rrrrlirsclrct (c(1.). ,Sl)t.illgt'r' Vclllrg, lle llirr, l()/,1. pp lrr() -'O l

192 nt tJil D()t )y At n()l )yNnMt(;l

4-41 J. C. lt. Hurrt, "'l'ulhulcrrt Velocities Nr:r'llttl lilttcluirlittg Srrrllrce I'rcssrrrcson Structures in'l'ulhulcnl Wintls," irr I'nxvnlitrg.s tl tlrt Iirurtlr lnttnuttiorrttlConference on Wind F)ffccts ttrt lluiltlirr,qs tuul Strudur(s, l,undon, 1975, Carn-bridge Univ. Press. Cambridgc, 1976, pp. -109 320.

4-42 B. J. Vickery and K. H. Kao, "Drag or Along-Wind Rcsponsc ol' SlcnderStructures," J. Struct. Div., ASCE,98, No. STl, Proc. Paper 8635 (1912),21-36.

4-43 P. W. Bearman, "Some Measurements of the Distortion of Turbulence Ap-proaching a Two-Dimensional Body, J. Fluid Mech.,53, Part 3 (19722),451-461.

4-44 R. D. Marshall, Surface Pressure Fluctuations Near an Axisymmetic StagnationPr.,lnl, NBS Technical Note No. 563, National Bureau of Standards, Washington,DC. r97l .

4-45 J. Vcllozzi and E. Cohen, "Gust Response Factors," J. Struct. Div., ASCE,94, No. 5T6, Proc. Paper 5980 (1968), 1295-1313.

4-46 R. Lam Put, "Dynamic Response of a Tall Building to Random Wind Loads,"Pnx'ccdings of the Third International Conference on Wind Effects on Buildingsarul Structures, Tokyo, Japan, 1971, Saikon, Tokyo, 1972, pp. 429-440.

4-47 H. van Koten, "The Fluctuating Wind Pressures on the Cladding and Inside aBuilding," in Symposium on Full-Scale Measurements of Wind Effects on TallBuildings, University of Western Ontario, London, Canada, 1973.

4-48 J. D. Holmes, "Pressure Fluctuations on a Large Building and Along-WindStructural Loading," J. Ind. Aerodyn., l, | (1975), 249-278.

4-49 K. H. Kao, Measurements of Pressure-Velocity Correlation on a RectangularPrism in Turbulent F/ow, Report No. BLWT-20, University of Westem Ontario,London, Canada, 1970.

4-50 W. A. Dalgliesh, "Statistical Treatment of Peak Gust on Cladding," J. Struct.Div., ASCE, 97, No. ST9, Proc. Paper 836 (1911),2173-2187.

4-51 R. D. Marshall, "A Study of Wind Pressures on a Single-Family Dwelling inModel and Full-Scale," J. Ind. Aerodyn., 1,2 (1975), 177-199.

4-52 G. A. Euteneuer, "Druckansteig im Inneren von Gebduden bei Windeinfall,"Der Bauingenieur, 45 (19'70), 214-216.

4-53 J. D. Holmes, "Mean and Fluctuating Internal Pressures Induced by Wind," inWind Engineering, Proceedings of the Fifth International Conference, Fort Col-lins, CO, July 1979, J. E. Cermak (ed.), Vol. 1, Pergamon Press, Oxford, 1980.

4-54 H. Liu and P. J. Saathoff, "Internal Pressure and Building Safety," J. Struct.Div., ASCE, 108 (19822), 2223-2234.

4-55 P. J. Saathoff and H. Liu, "Intemal Pressure of Multi-Room Buildings," -/.Eng. Mechs. Dlv., ASCE, f09 (1983), 908-919.

4-56 R. L Harris, "The Propagation of Internal Pressures in Buildings," J. WindEng. Ind. Aerodyn.,34 (1990), 169-184.

4-57 B. J. Vickery, "Intemal Pressures and Interactions with the Building Envelope,"J. Wind Eng. Ind. Aerodyn., 53 (1994), 125-144.

4-58 J. E. Minor and L. W. Beason, "Window Glass lrtilrrrcs irr Wirrtlstornrs," J.Struct. Div., ASCE,, 102, No. STl, Proc. Papcr lll3.l4 (l()7()). 1,17 l6O.

4-59 "Hanc<rc:k (illrss l}clrkagc: A ('ornbination ol' littrrs'f " l'.rr.t,ittct't itt,q Ncu.,lRtutnl (Miry l()76), ().

lll lllll l\l(,1 :, 103

,l 60 ll. |. Mclk'rur. "Wrrrrl l,orrrl lilli'c(s on l;lll l'irrlr Sol;rt ('ollt't lots." .l ,\trttr'1.l,.lttg.,lll{lttxry. l,l t l.5l

,1 .61 "Ncw Appnxrtlrt's lo l)t'silr,rr Against Wirxl At'lion." irr A. (i. l)rrvcrrlxrr( (lil.),Ctnrsc No/r,.r', 'l'lrr' llorrrrtluly Laycr Wirrtl 'l'rrrrrcl l.lrlxrlrrloty. tlnivclsily ol'Wcstorn Orrtirrio, l,oruLrt, ('anacla, 197L

1-62 J. Blcssllun. Atnnlintinrirt dus Con,slrucor),s, Iililorit tlrr tJnivcrsiditdc, PontrAlegre, Brasil, l9tl.l.

,t-63 R. J. Kind arrtl l{. 1,. Wardlaw, Design o.l'Iilxtlir4ts A14,uinst Gravel Bktw-ofJ',Repoft No. 1.5544, National Research Council ol'Canada, Ottawa, 1976.

4-64 R. J. Kind, "A Critical Examination of the Requirements fbr Model Simulationof Wind-Induced Erosion/Deposition Phenomena Such as Snow Drifting," ,4r-mos. Environ, f0 (1976), 219-227.

4-65 C. Mateescu and H. Popescu, Accumulations de neige sur les constructions,Etude explrimentale sur mod?les, Annales de 1'Institut Technique du Bdtimentet des Travaux Publics, S6rie EM, Pais, 1974.

4-66 J. Wianecki, Banc d'essais d'accumulation de la neige due au vent, Annales deI'Institut Technique du B6timent et des Travaux Publics, Sdrie EM, Paris,1976.

4-61 W. Z. Sadeh and J. E. Cermak, 'lTurbulence Effect on Wall Pressure Fluctua-tions," J. Eng. Mech. Div., ASCE, 98, No. EM6, Proc. Paper 9445 (1912),t356*1319.

4-68 G. E. Mattingly and E. F. Peten, "Wind and Trees: Air lnfiltration Effects onEnergy and Housing," J. Ind. Aerodyn.,2 (1977), l-9.G. E. Mattingly (personal communication, April 1977).H. P. Pao and T. W. Kao, "On Vortex Trails Over Oceans," Atmos. Sci.,Meteorological Society of the Republic of China, Taiwan, 3 (1976), 28-38.D. Surry and T. Stathopoulos, "An Experimental Approach to the EconomicalMeasurement of Spatially Averaged Wind Loads," J. Ind. Aerodyn., 2, 4 (Jan.1978), pp. 385-397.

4-72 A. G. Davenport, D. Surry, and T. Stathopoulos, 'Wind Loads on Low RiseBuildings: Final Report of Phases I and II-Parts 1 and 2," BWLT Report SS8-1977, The Univenity of Westem Ontario, London, Ontario, Canada, Nov.,t917.

4-73 A. G. Davenport, D. Surry, and T. Stathopoulos, "Wind Loads on Low RiseBuildings: Final Report of Phase III-Parts I and 2," BWLT-SS4-1978. TheUniversity of Western Ontario, London, Ontario, Canada, July, 1978.

4-74 L. Apperley, D. Surry, T. Stathopoulos, and A. G. Davenporl, "ComparativeMeasurements of Wind Pressure on a Model of a Full-Scale Experimental Houseat Aylesbury, England," .1. Ind. Aerodyn., 4 (1979),207-228.

4-15 T. Stathopoulos, "PDIi ol'Wind Pressures on Low-Rise Buildings," J. Struct.Dlv., ASCE, 106 (l9lJ0). ()7.1 990.

4-76 T. Stathopoulos, l). Strrr.y,:rtttl A. (i. l)avcnport, "E,fl'ective Wind Loads onFlat R<xrl.s." .l . Strutl. /)ir', AS('lr..l 107 (l9ltl),281 298.

4-11 'f'. Statl-roptrukrs, "Witul l,o;rrls ort lrrtvt's ol l.ow lltriltlings," .l . Strul. l)ir'.AS('li, 107 (l9t'l l). l1).! I l()t'l

tl 18 K. .l . liutorr trrrtl .l . l{. M;rVtr,'. Iltr' Mr';r:.rut'rnr'nl ()l Wintl l)ttssutt' ott 'l'wrtStory llottscs :rt Aylrslrrrty." .l lrnl '1'.'tt\lttt . | (l()75), (r7 l0().

4-694-70

4-7 |

194 ut ut I l()t )y nt ll()t )yNnMt(it;

4-19 J. l). llolrncs, Witttl ltttrl.s ()n 1,.,lr'lli,st'Iiltiltlirt.q,t A llryit'w, ('Sll((), l)ivisiorrof Building Rcscarch, Iliglrctt, Viclorirr, Austlrrliir, I91i3.

4-80 T. Stathopoulos, "Wind Loacls orr l.ow ltisc lluildings: A Rcvicw ol'thc Statcof the Art," Eng. Struct., 6 (l9tJ4). ll9 l3-5.

4-81 J. T. Templin and W. R. Schricvcr, "l.oarls duc to Drified Snow," J. Struct.Div., ASCE, r08 (1982), t9t6-t925

4-82 J. D. Iversen, "Small-Scale Modeling of Snow-Drift Phenomena," in WindTunnel Modeling for Civil Engineering Applications, T. A. Reinhold (Ed.),Cambridge Univ. Press, Cambridge, 1982.

4-83 R. J. Kind and R. L. Wardlaw, "Failure Mechanisms of Loose-Laid Roof-Insulation Systems (High-Rise Buildings)," J. Wind Eng. Ind. Aerodyn., 9(1982), 32s-341.

4-84 R. J. Kind and R. L. Wardlaw, "Wind Tunnel Tests on Loose-Laid RoofingSystems for Flat Roofs, " Proceedings , Second International Symposium on Roof-ing Technology, National Bureau of Standards, Gaithersburg, MD, Sept. 1985.

4-85 I. S. Gartshore, "Some Effects of Upstream Turbulence on the Unsteady LiftForces Imposed on Prismatic Two Dimensional Bodies," J. Fluids Eng.,106(1984), 418,424.

4-86 W. C. L. Shih, C. Wang, D. Coles, and A. Roshko, "Experiments on FlowPast Rough Circular Cylinders at Large Reynolds Numbers," J. Wind Eng. Ind.Aerodyn., 49 (1993), 351-368.

4-87 Y. Nakamura, "Bluff-Body Aerodynamics and Turbulence," -/. Wind Eng. Ind.Aerodyn., 49 (1993), 65-78.

4-88 A. Roshko, "Perspectives on Bluff Body Aerodynamics," J. Wind Eng. Ind.Aerodyn., 49 (1993), 79-100.

4-89 A. Gadilhe, L. Janvier, and G. Barnaud, "Numerical and experimental modelingof the three-dimensional turbulent wind flow through an urban square, ' ' J . WindEng. Ind. Aerodyn., 46-47 (1993),755-763.

4-90 S. Murakami (ed.), "Cunent Status of Computational Wind Engineering," -/.Wind Eng. Ind. Aerodyn, 35 (1990), l-318.

4-91 S. Murakami et al. (eds.), "Computational Wind Engineering," J. Wind Eng.Ind. Aerodyn., 46-47 (1993), l-912.

4-92 D. Laurence and J.-D. Mattei, "Current State of Computational Bluff BodyAerodynamics," J. Wind Eng. Ind. Aerodyn.,49 (1993),23,44.

4-93 C. W. Letchford and J. D. Holmes, "Wind Loads on Free-standing Walls inTurbulent Boundary Layers," J. Wind Eng. Ind. Aerodyn., 5l (1994), l-21.

4-94 Y. Nakamura and Y. Ohya, "Vortex Shedding from Square Prisms in Smoothand Turbulent Flows," J. Fluid Mech., f64 (1986), 77-89.

4-95 A. Baskaran and T. Stathopoulos, "Prediction of Wind Effects on BuildingsUsing Computational Methods-Review of the State of the Art," Canadian J.Civil Eng., 2l (1994),805-822.

CHAPTER 5

STRUCTURAL DYNAMICS

Structural dynamics is the discipline concemed with the study of structuralrcsponse to time-dependent loads. This chapter reviews certain elementary re-sults of structural dynamics theory and derives expressions for the response ofstructures subiected to distributed stationary random loads. These results arethen applied in the particular case where the loads are induced by wind toobtain expressions for the along-wind response, including deflections and ac-cclerations. Several of the results obtained will also be useful in other appli-cations occurring throughout the text.

5.1 THE SINGLE.DEGREE-OF-FREEDOM LINEAR SYSTEM

('onsider the system represented in Fig. 5.1.1 consisting of a single mass ttt('oncentrated at point B and of the member /8 assumed to have negligiblenrass. The displacement x(r) of the mass nr is opposed by (1) a restoring forcesLrpplied by the member /B and (2) a damping force due to the internal frictionthat develops within the system during its motion. It is assumed that the re-storing force is linear, that is, proportional to the displacement x(r), and thatrlrc damping is viscous, that is, proportional to the velocity dxldt. It followstltcn from Newton's secon(l law lhat the motion of the mass is described bytlrc cquation

,,ri'I (\ I A.r I'll) (s. r. r)

wlrcre: /'i/) is tlrc lirtrc tk'gx'ttr['rtl l(]jr(l :r('ltnl', ott (ltt: tttttss, I is thc sllrirrgt.6rrslllrl (11r lIc slilllrcss) ol llrt'rrrt'rrrlrrr ,'lll, r'ts kttowlt its lltt: cocllit'it'ttl ol

r95

196

FIGURE 5.1.1. Schematic of a single-degree-of-freedom system.

viscous damping, and the dot denotes differentiation with respect to time. It iscommon to write Eq. 5.1.1 in the form

X -f 2(,(2zrn)* -l (2rn1)2x : F(t)m

(s.1.2)

where

-l.trrrurt I

,l'[-tm

-I Ik' 2r\lm.CSr * ,

2,,/km

(s.1.3)

(s.1.4)

are known as the natural frequency and the damping ratio of the system, re-spectively.* The quantity zJtcm is known as the critical damping coefficientand can be shown to be the value of the damping coefficient beyond which thefree motion of the system is nonoscillatory. The damping ratio is expressed asa percentage of the critical damping.

5.1.1 Response to a Harmonic LoadIt can be easily verified t5-ll that if

F(/) : Fo cos 2Tnt (s. r .s)

xThe quantity 22"11, is tclcrlctl to as lltc natural circular I'rcquoncy rrrrtl is t.orrrnrrrly rlcrrolctl hy@1.

!i r llll lilN(il l l)t (int t ()t 1ilil t,{)M ltNt Ail r;yl;ltM 197

llrc slcatly-stlrle solrrliorr ol lit1. .5. 1 .2 is

r(/) ItuItQt) c<ts(2rnt $) (.s.1.6)

wlrcrc

d:tan , 2(t(nln)| -hk$ (s.t .7)

(s.1.8)

(s.1.10)

the action of a load0, that is, to a load

H(n) :

'f'lrc quantity F(n) is known as the mechanical magnification factor or me-t'lrttnical admittance function of the system with parameters m, n1, and (r.

Similarly the steady-state response to the load

F(t) : Fs sin ZTnt (s.1.9)

rrray be written as

4r2 nlm{lt - (nl n,12fiJt1r;7y

x(t) : FoII(n) sin(2rnt - g)

5.1.2 Response to an Arbitrary LoadLct the system described by Eq. 5.1.2 be subjected toct;ual to the unit impulse function 6(r) acting at time / :rlcfined as follows (see Fig. 5.1.2):

6(o:0 forr + 0)FLI t

lllJ, bl)dt:t )(5.1.1r)

Il(;lll{1,: 5.1.2. tlnil irrrprrlsc lillctiorr.

t98 st n[,o I ulln L t)YNAMloti

FIGURE 5.1.3. Load F(r).

The response of the system to the load 6(r) is a function of time and is denotedG(/).

An arbitrary load F(r) (Fig. 5.1.3) may be described as a sum of elementalimpulses of magnitude F(r') dr' each acting at time z'. By virtue of the linearityof the system, the response at time / to each such impulse is G(l - r')F(r')dr' and the total response at time / is

(5.t.12)

where the limits of the integral indicate that all the elemental impulses thathave acted before time / have been taken into account. With the change ofvariable r: t - r',F,9.5.1.12 becomes

LetF(t) - F6 cos 2rnt.It follows then from Eqs. 5.1.6 and 5.1.13 that

x(ty : J' * o,, - r')F(r') dr'

x(t) : J- *rou - r) dr

H(n) cost : f Qz) cos Ztrnr dr

and

(s. 1. 13)

(s.1. 14)

/y'(rr) sin q5 : I G(r) sin 2rtt'r tlrIir

()

Using now Iit1s.5.l.l4 rrntl 5.1.15, it is possible lo wlilc

(s. I . ls)

II(n) and G(r):

H21n1 :

,, I il[ litN(il t I)t (iltil ()l Iltl Il)r]il/ lll.ll All :;Yr;lltrrt 100

r"["/1r(rr) crrs'rl J,, .\,, ,t,t1)c.s 2zutr1(i( r,) t'.,s )trttt, tlt, tlt; (.5.1.1(r)

l"'r"''H)tr) sinro - ],, J,, ,t, r,\ sin 2rnr,G( r.,) sirr 2rttr; tlr, dr, (5.1.l7)

I'ltc addition of Eqs. 5.1.16 and 5.1.17 yields thc firllowing relation between

f-Jn Gr')G(ru) cos 2rn(r1 - 12) dr1dr2 (5.1.18)

5.1.3 Response to a Stationary Random Load'l'hc case is now examined in which the load ,F(r) is generated by a stationaryl)()cess with spectral density Sflrz). The expression for the spectral density oftlrc response S"(n) can be derived using Eqs. A2.20, A2.21, and 5.1.13:

f-,S,(n) : t ) _&(r)cos 2rnr dr

:2[- 1,,, tf'/2 'lJ - I .* V ) ,'rx(tlx1

-t r) dt lcos

2rnr dr

: , I f ',- 1- ['" ,, I I er1)F(t - 11) dr1J--(r,*T J-r,z lJo

x I tr,,r,, * r - rt\ clrtl]ro, znm a,Jo -l)

: t J; *',, [j, *", [J-- ^",' * 11 * 12)

' cos 2trnr arlarrlar,

: 2 [ I cO)c(rr)cos 2trn(r1 - 12) dr1 dr2-Jo Jo t

J;

r-J _ *o,r I rt - r2lcos2rn(r I rr - r) d(r * 11 - 12)

, J,, j,, (i(rr)(i(r,)si n 2ntt(r1 - r.) dr1 dr2

J ,,r,r,t, ,, r,)srrrlrrr(r I rr r;)tl(r I 11 r;)(5.l.lt)l

200 silrt,(;ttlnAl t)YNnMlcli

where, in thc last stcp, thc itlcrttity

cos2rnr = cos Zrnl(r I rt r:)-(r1 -r)l (s. r.20)

is used.From Eqs. A2.20, A2.23 and 5.1.18, there follows

Sln) : nzg1so1nS (s.1.21)

This result is extremely useful in applications. See also [5-l] to [5-4].

5.2 THE MULTI.DEGREE OF FREEDOM LINEAR SYSTEM

5.2.1 Natural Modes and Frequencies of a Continuously DistributedStructureIt may be regarded as an experimental fact that a continuously distributed elasticstructure with low damping, when excited by a sinusoidal force, will vibratein resonance at certain sharply defined characteristic frequencies. Associatedwith each such resonant or natural frequency, there will also be a characteristic,or modal, form of vibration amplitude distributed throughout the structure. Suchforms are called the normal modes of the structure. For example, Fig. 5.2.1depicts the first four normal modes of a vertical cantilever beam.

These characteristic deflection modes and associated frequencies are prop-erties ofthe structure, independent ofthe loads, and represent very fundamentaldynamical evidences of its internally distributed inertial and stiffness proper-ties.* In fact, the set of normal modes may be regarded as a fundamental setof special deflection forms by means of which any general deflection of thestructure may be expressed.

Thus, if z is a running coordinate (e.g., height) of a structure, the modaldeflection forms of lateral ("{ direction) vibration may be written as x; (z), where

FIGUR Fint fbur normal modes of a cantilcvcr bcam

*Dctails <ln pnlccrlrrtt:s lirrtlclt:r'rrrirring, rtortttirl ntotlcs antl ttitlttt:tl ltt'r1ttr'tttit'r tttrty lrt' lirtttltl. lilrcx:rtnplc, irr l5 ll ot l5 11.

TE 5.2-1.

11 Mt 1 l lll (i1l I ()l llg ltlr)M ttfJt At t r:yr;ltM 2Ol

r. 2" l. 'l'ltcrr :ttry tlclle:ctiott r(:, /) lllily lrt'exlttt'sst'tl :ts lltt' sttttt

whcrc the coeflicients t,(r) inclicate what fructiort ol'caclt ntotlc r,(z) ctltcrs (ltcp,ivcn deflection pattern. The coefficients {r(l) arc callod Lhc gcntruliT.ed ut'ttttlinat€s of the system.

An important property of the normal modes xi(z) is their mutual orthogonalitywith respect to mass weighting, by which is meant that

I,r.,.- t'(t)"' (z)m(z) dz : o (i + i)

rKE : ; J.r.,., .,.,[*k, t)]2 dz 6-2.3)

ll thc system is vibrating in the single resonance modex;(z), then

r(2, /) : )",t.)t,t,l

*(2, t) : xiQ)EiQ)

:.o thaf the kinetic energy becomes

KE - +Mtt?

wlrt'rc

M,

A./, is kttowtr lts lltr.: ,qlrtr'rrr/t

15.2. I )

(s.2.4)

(s.2.s)

(5.2.2)

wlrore rn(z) is the mass of the structure per unit length.Since the system is actually continuously distributed but responds at each

rrl its resonant frequencies like a single vibrating entity (or single degree ofln:cdom), it becomes very useful and convenient to liken continuous systemsto single-degree-of-freedom systems. It is helpful in this context to use the('rrorgy approach. The kinetic energy of a single mass M is )M*', where i isrrs vclocity of displacement. We now seek the corresponding energy for therlistributed system.

'f'he lateral displacement being x(2, t), the elemental kinetic energy at point.is

im(z)lx(2, t)21 dz

'l'lrc kinetic energy (KE) of the whole system is therefore

I

\ t r,(: )l'rrr(.:) r/r (5.2.6).l..1 .r, r,,

,',1 trnttt ol lltt' rvslt'trr ilt lllt: itlt ttorlttltl lttrxlt'.ii

l,!

2O2 liltr,otunnr t)yNAMtoli

ln this scnsc a cttntinttous sys(qnr vibrlrtirrg irr irny orrc ol'ils rlrnrrirl nroclcsmay be viewed as thclugh it worc sintply ir sirrglc{ogrcc<rl-l'rccdr)tl systotnwith a mass M; and velocity {,.

5.2.2 General Expression of the ResponseConsider a structure for which it may be assumed that the displacement in thedirection x is the same for all points in the structure that have the same coor-dinate z (Fig. 5.2.2).It can be shown [5-2] that if the damping ratio is smallthe generalized coordinates €i(r) satisfy the equations

whcrc (,, n,, and M, are the damping ratio, the natural frequency, and thegcncralizcd mass (Eq. 5.2.6) in the ith mode. The quantity QiG) is known asLhc gcncruliz.ed force in the ith mode and has the expression

{,(r) t 2(,(2rrt,)fr(/) + 12rni)2tilt):T

ei(t) : \i ,u,t)x{z) dz (s.2.8)

where .F1 is the height of the structure, and pk, t) is the time-dependent loadper unit of length acting on the system. It is seen that each of the equations5.2.7 is of exactly the same form as the equation of motion of the single-degree-of-freedom system Eq. 5. 1.2.

If the load nQ, t) is such that

nk, t) : F(t)6(:z * z) (s.2.9)

where 6(z - zt) is defined in a manner similar to Eqs. 5.l.ll, that is, if thestructure is subjected to a concentrated force F(r) acting at a point of coordinate

FIGURE 5.2.2. Schcrnatic ol'a slcndcrstructure.

H

.11. llrt: gcrrcrirliztrl krrt't' (),(l) will

Q,u) o'1"],,

=. xik.)

It lirllows fromt txrrdinate z is

(illt I ()l lnt I l,()M ltr.ll Alr !;Yr;llM 203

I 4,,

lt(:,, l).t,(:,1 tl:.

bcr

I.F(t) (5.2. r0)

5.2.3 Response to a Harmonic Load

ll tr concentrated load

F(t) : Fs cos 2rnt (s.2. 1 1)

rs rrcting on the structure at a point of coordinate z1,by virtue of Eq. 5.2.10tlrt: generalized force in the ith mode will be

Q,(t) : F11xi(z)cos 2rnt (s.2.12)

rrrrtl the steady-state solutions of Eqs. 5.2.7 wlll be similar to the solution 5.1.6ol liq. 5.1.5:

t,(t) : Foxik)Hi@)cos (Zrnt - $') (5.2.13)

wlrcrc

Hi(n) :+rznl u,{11 - (nlni1212 + 4y,2tnl n,\'\"'

, 2f i@lni)o, : tan

-_

r| - (nln;)'

F,q. 5.2.1 that the response of the structure

x(2, t): Fo I xik)xik)Hi(n)cos(2rnt - $)

lt is convenient to write Eq. 5.2.16 in the form

x(2, t) : FoH(2,, 21, n)cosf2rnt - 6(2, zt, n)l

rvlrcrc, as fbllows immcrliittcly l'rrrn lrqs. A2.4t and N:2.4b,

(s.2.t4)

(s.2.ts)

at a point of

(s.2.t6)

(s.2.11)

I r,(:)r;(z1rff,f,,l*in,l, | |(.s.2.

tt(t..7.r, lr) : Il ] r.,,.,,,,::r)r/,(rr)t.,s,/,, I I

ItJ)

. r L;.t;(z).r,(11)//,{rr)silt <15,

o(z' z1' n) : lan {x1k.'lx1k.1)H,(rr)cos <},

l5.2.lt);

(s.2.22)

Similarly, the steady-state response at a point of coordinate z to a concen-trated load

F(t): Fssin2rnt (5.2.20)

acting on the structure at a point of coordinate Z1 elr' be written as

x(2., t) : FyII(2, 21, n)sinl2rnt - 6(2, \, n)l 6.2.21)

5.2.4 Response to a Concentrated Stationary Random Load[,ct thc rcsponse at a point of coordinate z to a concentrated unit impulsiveload 6(t) acting at time / : 0 at a point of coordinate z1 be denoted G{2, 4,r). Following the same reasoning that led to Eq. 5.1.13, the response x(2, t)of the structure at a point of coordinate z to an arbitrary load F(r) acting at apoint of coordinate Z1 czn be expressed as

x(:2, t) : I, ou, zt, t)F(t - r) dr

Note the complete similarity of Eqs. 5 .2.11 , 5 .2.21 , and 5 .2.22 to Eqs. 5. 1 .6,5.1.10, and 5.1.13, respectively. Therefore, by following the same steps thatled to Eq. 5.1.21, there results

S,(2, 21, n) : H2(2, 21, n)Sp(n) (s.2.23)

where S,(2, zr, n) is the spectral density of the displacement x(2, t), the me-chanical admittance function H(2, q, n) is given by Eq. 5.2.18, and Sln) isthe spectral density of the force F(r).

5.2.5 Response to Two Concentrated Stationary Random LoadsLet x(2, /) now denote the response of the structure at a point of coordinate zto the action of two stationary random loads F1(l) and Fr(t) acting at points ofcoordinates Z1 and 22, respectively. The autocovariance of the response can bewritten as

t l'''R,(2, z) - lim .,. \ .r(2, r)x(z. t -t r) dt

t'ql J t':

lt : il il Mt,l ll l)l (illl I (ll

l':',: I I";, I J,, (i(:.,r. r,\t,1u

+ J," r;t., ?.2.r1)[,2(t - r,ldr,l

Iilt I t)()M ItNt At I f ;Yt;il M

t r\ tlt r

I {'-x I I GQ, q, r)F{t -t r - r) dr2IJ0('-

|+ I G\2. zz, r)F2Q -t r - r) dr2 | drJo ' 'lf- I r- I: I Gz,21.zlt| \ G{,z.zr,r)Rp,(r * rr - r2'sdr2ldr1Jo lJo l

r- I f- r+ Jo ctz. zz.rzllJo *.. z2'r)Rp,(r t rr - r27dr-,ldr,

I

f- [l- |+ I CAz.z1.r'll \ G1z.zz.r2)Rp,p,(r I rr - r2ldr2ldr,Jo 'lJo 'l("- I f- |+ I G(2. 22. ryt | \ Cl,z.21. r2lRprp,(r * rr - 'r2\ dr2ldr,Jo LJo l

(s.2.24)

wlrcre the definition of the cross-covariance function (Eq. A2.29) was used.'l'he spectral density of the displacement x(2, t) is

l"-S,(2, n) : 2 \ &(2, r)cos 2zrnr dr''J-

: 2 [ &(2, r)cos 2trnl(r i rr - rz) * (rt - r)l- J_-' d(r + rr - rz) (5.2.25)

l.t'r Eq. 5.2.24 be substituted into Eq. 5.2.25. Using the relations

H(2., z,i, n)cos 4,k..:.i. trt J,] ",..Z;.7)coS 2trnr dr (5.2.26)

II(2., z.i. rr)sirr y'r(.l, .1,. rt) [ '

,r, ,'. .:,, r)sin 2trnr tlr (5.2.21\.t ,,

206 sillt,ott,nnl r)yNnMtori

(which arc siurilar to llqs. 5.1.14 irrrtl 5.l.l5) rrrrtl

H(2, zr, n)H(2, 22, n)coslg(2, z.t, n) - 6Q., zz, n)l

: f f G{z' zt' r)G(z' z2' r2)cos Ztrn(rv - 12) dt1 dt2

(s.2.28)

H(2, zr, n)H(2, 22, n)sinfg(a, Zr, ft) - Qk, zz, n)f

- [- [- /:t- ,: .1,, J,, ,,., 21, r)G{2, Zz, rz) sin2rn(r1 - :,2) dr1 dr2

(s.2.29)

wlrich carr bc derived immediately fiom Eqs. 5.2.26 and 5.2.21 , and followingthc stcps that led to Eq. 5.1 .Zl, there results

S*(2, n) : H2(2, 4, n)Sp,(n) r H'(2, 22, n)Spr(n)

+ 2H(2, 21, n)H(2, 22, n)

' {S$,r,(nlcosl6k, q, n) - 6k, zz, n)l

+ Sf;,o,(n)sinf6(2, zr, n) - 6k, zz, n)l\ (5.2.30)

where S"(2, n) is the spectral density of the displacement at a point of coordinatez, H(2, Z;, n) are the mechanical admittance functions defined as in Eq. 5.2.18,Sa(n) is the spectral density of the force F,(r), and Scr,rr(n), Sf,p, are the co-spectrum and the quadrature spectrum of the forces F1(r) and Fr(r) defined asin Eqs. A2.33 and A2.34, respectively.

5.2.6 Effect of the Cross-Correlation of the Loads upon theMagnitude of the ResponseConsider two random stationary loads F1(r) and Fz(r) acting at points of coor-dinates Zr and 22, respectively, and such that Ft(t) : Ft(t) at all times. Bydefinition, in this case the cross-correlation equals the autocorrelation, SF,r, :S",(,e), and Sp,p, : 0 (Eqs. A2.21 and A2.29, A2.20 and A2.33, A2.23 andA2.34). The loads F1(r) and Fr(t) are said to be perfectly correlated. Thespectral density of the response to the two loads can then be written as (Eq.s.2.30)

S,(2, n) : {U'(r, zt, n) + H2(2, zz, n) + 2H(2.,24, n)H(7.,7.t, 11)

' cosld(2, 2.1, tt) - $k, z.z, n)llS7,,(rr)

In thc parlicullrr casc wltctt ;' r.,,

(s .2.3 r )

nI Mll il l)l {illl I ()l l lll l l,r )M IllJl All :iY:;ll [/l 2O1

(5.1..]l )

('tllsitlcr rrpw llre t'rrst'wlrclt'tlrc loatls /"r(l) itrrtl /',(l) irrc strch thlrt llrc:ir, nrss covariipllce: /11,,1,,( r ) 0. 'l'hcn, by Llqs- A2.'1.1 irrrtl A2'14'

sj.,,,'{n) : sf,,,u(rr) - tt (s.2.33)

;rrrrl, ilthe statistical pnrpcrlics of the loads arc thc same, that is' if Sp,(n) :'\1,(rr),

S,(2, n) - [H'(2, zt, n) + H2(2, zz, n)]Sp,(n) (s.2.34)

ttr. tl' zt : Zz,

S,(2, n) : 2H2(2, 21, n)Se,(n) (5.2.3s)

I'lr(' spectrum of the structural response to the action of the uncorrelated loadsr:, tlrus seen to be only one half as large as in the case of the perfectly correlatedlo;s115.

5.2.7 Distributed Stationary Random LoadsI lr(' spectral density of the response to a distributed stationary random loadrrury be obtained by generalizing Eq. 5.2.30 to the case where an infiniterrrrrrrbcr of elemental loads rather than two concentrated loads are acting on the.,tnrcture. Thus, if the load is distributed overan areaA, and if it is noted thatrrr lhc absence of torsion the mechanical admittance functions are independentll tlro across-wind coordinate y, the spectral density of the along-wind fluc-trrrrting deflection may be written as

ttS,(2. n) : I I Il(2. 21. n)H(z.72. 1)' J,t Jt

x {Sfio;@)cos[d(2, Zr, n) - Q(2, zz, n)f

+ Sf;;oi(n)sinI6k, zr, n) - 6Q, zz, n)l\ dAt dAz

(s.2.36)

rVlrt'r'er /)i anJ pi denotc prL:sstlr(rs ircting at points of coordinates 11, Z1 and y2,,, n'sllcctivcly.

It r':ut bc vcriliccl that llirrlr lir1. 5.2.-l(r lltc:rc lirlltlws*

rll1,rrsitr1,, li1s.5.2.lllillxl 5.1.1(r.'1 ,' l.l :tllrl 'r'lt.A.r,lrr;rrrtl A].4/r. lrot':rtlclivlrliolrol lil', .t l/ irr l('nns ()l cotttplt'x v;tti:tlrlt'r, r,r'1 1'r 't. l t)ll

208 l-;lru(;l(,nAt t)yNAMtoti

s,(2.r) : -L ) > rr(r)'\/(:)vr\(r "/ l6ra 7 7 ,,i ,,i ltt,u,r_" {lr - (ntn,)zl2 + 4yl@tn;2\{t - (ntn)212 + +yllntn)2}

" ll ' - (;)'l l' - (;)'l + 4rci::,JI^ln*,,,,,*/,,,

x sfi,;1n1 dA, dA2 * fr r, :,1, - (;)' I - z r, I,[, - (;)' ]]

' J, J, x1Q)x1Q,)sfi,,{n)dA1dA,) 6.2.3i)

If the damping is small and the resonant peaks are well separated, the cross-terms in Eq. 5.2.37 become negligible and

*? tO \ ^ I n *,rr,rr,,rr) s'oio;(r) dA I dAz

S,(z, r) : I (s.2.38)I6ranlul {11 - (ntn)212 + +y!1zntn,12l

5.3 EXAMPLE: ALONG-WIND RESPONSE

To illustrate the application of the material presented in this chapter, the caseof the along-wind response of tall buildings will be dealt with below.

5.3.1 Mean ResponseIf in Eq. 5.2.8 the load p per unit of length is independent of time, the cor-responding along-wind deflection, which will be denoted by x(z), results im-mediately from Eqs. 5.2.1 and 5.2.7:

:'l{ Pktx'(zl dzXlZr:./t -

X[Z)i r'niM;

, : \: x?k)m(z) az

(5.3.1)

where

(s.3.2)

and p denotes thc linro-irrvarian( loacl.As indicatctl in ('hirlttcl'4, lhc rttcan wincl loatl aclirrg orr ir lrrriltlilrg ol'wiclth

l,:| lXnMl'l I nl ('lltiWlrllr lltril,r,Nl;l

,4',z- |//lt"I

FIGURE 5.3.1. Schematic view of a building.

/J tliig. 5.3.1) may be written as

p(z):ipG.+ C)BU2(Z) (5.3.3)

rvlrt'rc p is the air density, c, and c7 are the values, averaged over the buildingrvrtlth, of the mean pressure coefficient on the windward face and suction, .t'llicient on the leeward face, respectively, and U(z) is the mean speed atrlt'vir(it>n z in the undisturbed oncoming flow. Equation 5.3.1 then becomes

II

ll

I

r-.-

(s.3.4)

5"3-2 Fluctuating Response to Wind: Deflections and AccelerationsA:; irrtlicatcd in Chapter4, thc co-spcctrum of the pressures at point Mr, M2of, ootlinatcs (yr, zr), (.y:, :.,), rcsl'rcctivcly, may be written as

Sl,ii,,t(n) ='S,11'(::,' rrl,f/,1'1;',, rt)('olt(.1',,.v.r, zr . z.z, n)N(n) (-5.3.-5)

,tlr,.'r'c ,S,1/'(l;, rr) is thc slrr't'ltirl rlt'rrrrly ol llrr' plt'ssrrrcs irl poinl M,(i - 1,2):rrrtl ('olt(.)r1,.)r-r, 11, 1.,. /l) :ttrrl N( l) ;ttr'lltr':rr'r'oss wintl lrrrtl llrc irlolrg-wintl. r.rss coltclirliorr cocllit'it'rrl, rcrpr't lrvt'ly lly rlt'lrrrrrorr, il hotlr M, ir,td M, irrt'

_ I ^ _ \- l{ IJ2<zlx,(zt dzx(z) : ,p(c, + ct)B l\j4*; *,1r1

p(y,z,t) dA

210 stttu(;tt,liAl l)YNnMlo:i

on the same-windward or lccwatd--lirco ol' tlrc stl-ucrlutc, N(rr) = I . 'l'hcexpression for Sr(z; , n) is, approximatcly,

so,(zi, n) : o'czu2{z)s,(zi, rz) (s.3.6)

where C : C. or C : Cr according as M; is on the windward or on the leewardface, and Su(zi, n) is the spectral density ofthe longitudinal velocity fluctuationsat elevation z; in the undisturbed oncoming flow (l : 1,2). Equation 5.2.38thus becomes

.. p) r, xitzl tc'z, + 2C*C,Nln) + cil.\,(:, rr) : l6rrl ? ri *i@

. f f f J. xi(z)xiez)u(z)u(z)Stt2(2,)s',''(zr)

x Coh(y1 , !2, Zr, Zz, n) dy, dyz dz1 dz2 (5.3.7)

The mean square value of the fluctuating along-wind deflection is (Eq. A2.15)

f-oitzl : )o S,tz. n) dn (5.3.8)

From Eq. A2.l6b it follows that the mean square value of the along-windacceleration is

olk) : t./ J; nas,(2, n) dn (s.3.e)

The expected value of the largest peak occurring in the time interval Z is,in the case of the fluctuating deflection,

r.u*(Z) : K,(z)o,(z) (s.3.10)

where, as indicated in Appendix A2 (Eqs. A2.38 and A2.43), the peak factor&(z) is, approximately,

K,(d : [2ln v^{ilTlt'2 -l 0.571f2ln v,(z)Tfit2

(s.3.1r)

",(z): I

lff n2s,(2., n) dn

and

J,T S,(z' rr) rln l"' (.5.3. l2)

z-

l, ll I XAMI 'l I Al I rr.j( i WlNl ) I ll lil 'oNl;l 21 1

Siltrilirlly tlte lirr'1icsl lrcitk ol lhc: itlottg witttl irt'r'clclrrliorr is, lrppnrxinrirtr.ly.

(.s..r. t .t )

wlrcrc

Kr(z) : 12ln vr(7)Tltt2 + o.51712 ln v1(2.)Tltt2

(5.3.14)

;trttl

(s.3.1s)

ll is convenient for computational purposes to rewrite the fluctuating re-\t)onsc in terms of nondimensional quantities in the form

I Jff ,os,(2. n1 dn lt'2u{z): lff7e,*-)d,,)

o,(z) pBH Jg(z)H fl16 4r'

u,(z)H : J(z)u* Jzk)

orQ)H pBH - .) : -

Jt\Z)u; tlts

ulz)H _ Jzk)t4* Jzk)

(s.3.16)

(s.3.r7)

(s.3.18)

(5.3.19)

(s.3.20)

(s.3.21)

(s.3.22)

r5 I )lr

rvlrt:ro rxg is the mass of the building per unit height at some specified elevation,L is the friction velocity (see Chapter 2)-or any suitably chosen referencevt'locity-and

rr(z):l+1#)' (r: o, t,z,3)

- f' m(ZlM' : )'*;tzt '"' dz

.)/

ui,' Y

tt, I Ill+

212 rit nt,(i I t,n^t t)YNn Mt(;ri

6f,(i) :Y,,(f) :

- U(Z\U(Z) : --:----:U4

l'"'1-. : J,, It' ..blt.llY,,(.ll 'tl

-nHf:- U4

tr - (ftf,)')z + lzf,(iti)1z

J, J, J: J. [,, + 2c,c,N (y) + c?)

x x,(Z 1tx,(z )) o(z tt otz,t lt# N+* l"

(s.3.24)

(s.3.2s)

(s.3.26)

(s.3.21)

(s.3.28)

(s.3.30)

5.3.3 Total Fluctuating Response to Wind as a Sum of Backgroundand Resonant ContributionsConsider a single-degree-of-freedom linearly elastic system with mass m, nat-ural frequency n1, and damping ratio f1 . Let this system be subjected to theaction of a forcing function with a spectrum S(n) such that

S(n) : So (n = 0) (5.3.29)

where 56 is a constant. The mean square value of the response can be writtenAS

c-"i : so Jn lHtn'112 dn

where

lH(n)l' : :--The quantity lH(")l'is an analytic function; therefore, the integral in Eq. 5.3.30can be evaluated by means of complex integration or integral tables to yield(see [5-3], p. 501)

, I ittr"' -' 17ni,,1,,,t Aa 'sl' (5.3.32\

(5..1.17 )

Lll IxAMl,l I Ai (,H(iwlr'JtIIilt.t,oNt;t 213

FIGURE 5.3.2. Spectral densities S(n)lH(n)|2, S(n,)lH(n)12, and S(n).

ll'the damping ratio f1 is small, the bulk of the contributions to the totalv:rlrrc ol is due to the "resonant" portion. with reference to Fig. 5.3.2itmaylrr' observed that if S(n) is not constant, a fair approximation to the integral

r-"i: I sstlntnll2dn" Jo

ronsists of two contributions:

oj: o|, + of,,

rvlrcrc

l'lr:rl is, il'lllr) l/i(rr)12

^@,lo,', : Jn S{n,ylHtnllz dn

, f-oi : .1,,

S(nl dn

Ar, Az, ancl A1 lrrt. llrt. trlclrs urrtlcr lhcancl S(l), rcsltr,rclivcly (lri,l.5..1 .1). thcrr

.'1,:.'1,1.'1,

(5.3.33)

(s.3.34)

(s.3.3s)

(s.3.36)

curvcs S1n lH1n1l2,

I

214 rirnt,(:tt,nAt t)YNnMtcl;

The intcgral ol'tr,q. -5.3.:15 is givtn by lx1. .5..1..12, with

S| =- .5(rrr) (s.3.38)

and the integral in Eq. 5.3.36 can be obtaincd if the function S(n) is specified.In the case of atmospheric turbulence this may be assumed for structural en-gineering purposes to be a decaying function as suggested in Fig. 5.3.2. Henceit may be concluded that

f- | [('- Tvtr II strtln(r)12 dn - --;-r-l \ s@) dn + - slnl) | (s.3.39)Jo lor ntm- | Jo +( r I

The first and the second terms of the sum in Eq. 5.3.39 are usually referredto as the background part and the resonant part of the response, respectively.

The above relation can similarly be applied to Eq. 5.3.24:

wherefl : nlHlu*.To verify the extent to which the approximation involved in Eq. 5.3.40 is

acceptable, numerical calculations were carried out for a large number of casescorresponding to a wide range of typical buildings and terrain roughness con-ditions. The calculations showed that the approximation is of the order of l%.It was also verified that for L : 1,2,3 the background term may be neglected,and therefore

It is convenient to define the quantities G and G as follows:

I f-* : 4 J, v',t|t dI

": &*^,"'n^'If the notation

(s.3.40)

(L : 1,2,3) (5.3.41)

(s.3.42)

(s.3.43)

(.-Irrt: )n OilI)f"Yrl.f ) df

= J; f"Y,,(i) di + #,ii'Ytr(Jr\

rrrr : ff,fl'r,,tf,l

1,,( f ,) cl,,+zc,,,c,N(n)t ('iis r-tsctl, llrc c;rrirtrlily lll rlrtry llr-r wt'illcrrt lts

Y,,( l,\ (5.3.44)

(';, I l(',,,(jN(rr1) I ('i'n[1((

",, I c'l )'r 4i ,

nr il ilr l{{.t !, 215

l',,t 7 I 1.s.1.-15)

Lt'l tltc mcan spcctl l/ in litl. -5.3.2li bc rcprcscrrlctl by lhc logirlitlrrrric llrw.l'ltt'zcro planc clisp-llcclttcttt r,7 will thcn hc a par:irrrrctcr in thc cx1'lrossion lirrrli Il'thc quantity 63 is dolincd as

- I l-* : -t; J' Y"(f )l''' odi

rt ('irn be verified that, approximately,

/ _\G:{l-g'l G\ H/

ltN(n\: i-;f (t - e 2t)

rvlrt:rc C, and C, are known as exponential decay parameters,

, _ 15.4nLxt : 6. (s.3.s0)

tt UGn, and A"r is the smallest of the dimensions B, H, and, D.

(s.3.46)

(s.3.47)

l,,tlrurtions 5.3.42 through 5.3.47 may be used for the computation of along-rvrrrtl response.

lrinally, recall that expressions for S,(2, n) in Eq. 5.3.27 are found in Chapter.' rrnd that, as indicated in Chapters 2 and 4, it is reasonable to assume

Coh(y', !2, Zr, Zz, n)

) + U(z)l (s.3.48)

(s.3.4e)

: "^e [-

,J-)+ c10t - yr)zltt

Z1

Zz

u(;iNIC?QI

REFERENCES

i I W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, prentice-Hall, En-glcw<xrd Cliffs, NJ, l9(r4..l . I). Robson, An Inlnxlut'tirtrr ttt liltrrtlottr Vihnttion, Elsevier, New York, 1964.1,. Mcinrvitch, Arutl.vtit'ttl Mt'lltttl,r itt I'iltnttiorr:s, Mac:rnillan, Collicr-Macnrillan('irilarla, l.l(1.,'li)r'()nl(), l()(r7.

i 'f 'f'.'l'. Srxrng ltlttl M. (iligotrrr, l{tttthutr l'tlrtttlittrt.t rtl Mrtlutttitttl tttttl Slnrt'trtntl,\'l'.r/.'r,r,r, l)r'crr(ir't' Illrll, lirrlllr'rvlrxl ('lrll',, Nl, ltt().1

1.11l

CHAPTER 6

AEROELASTIC PHENOMENA

A body immersed in a flow is subjected to surface pressures induced by thatflow. If there is turbulence in the incident flow, this will be the source of time-dependent surface pressures. Such stresses are also caused by flow fluctuationsinitiated by the body itself.

Further, if the body moves or deforms appreciably under the induced surfaceforces, these deflections, changing as they do the boundary conditions of theflow, will affect the fluid forces which in turn will influence the deflections.Aeroelasticity is the discipline concemed with the study of phenomena whereinaerodynamic forces and structural motions interact significantly.

An aerodynamic instability car, be a phenomenon occurring wholly withinthe flow alone, as when a trail of vortices or a rapidly diverging wake is shedfrom a fixed body. But if a body in a fluid flow deflects under some force andthe initial deflection gives rise to succeeding deflections of oscillatory and/ordivergent character, an aeroelastic instability is said to be produced. A purelyaerodynamic instability such as vortex shedding may occasion structural de-flection as well, initiating a phenomenon having aeroelastic character. Allaeroelastic instabilities involve aerodynamic forces that act upon the body asa consequence of its motion. Such forces are termed self-excited.

The purpose of this chapter is to discuss fundamental aspects of aeroelasticphenomena that need to be taken into account in the design of certain structuralmembers, towers, stacks, tall buildings, suspended-span bridges, cable roofs,piping systems, and power lines. Not all of these phenomena are presentlycompletely understood. Indeed, only a few theoretical forrnulations from firstprinciples exist for modcling acrodynamic firrces on oscillirlirrg lrrxlic's. In nrostinvestigations, crnpirical ntotlcls arc sct up in which llrc r'sst'rtt't'ol'tlrc: itcr<l-dynamics rrrrrs( l-rc cortlribtr(ctl by c:xpcrirttcrtl .'l'lrc t'ottt'slrorttlirrl':rrr:rlylicitl

216

I vrrl rllx t;l ll l)l)lN(i nNl, llll ltx tr [] I'1il t!()Ml N()N 211

rttrxk:ls trstrirlly iltr'ltttlc itsl cn()uglr;latlrtte(crs lo ttrirlt'lr llrt'slnrrrllt'sl olrst'r'vt'rlIt':rlrtros ol'lltt: plrt'rtonr('nit. Srrr'lr rrrotlcrls lrt'llrus nurrlrurlly tlt'sc'r'iptivc. lrrrlrro( cxplanatory itr llrc serrsc ol' r'cvcaling blrsir' plrysit'itl t'trrrst's; srrlrllr: lrulrulx)flllnl dctails ol'lltt':tt'ltutl llrritl-structttt'c inlt't'irt'lit)n nrily irr t'cllrrirr t'lrscslrr lcli unattcndctl.

lirnpirical modcls rrtay only bc uscd lirr tho prcrliction ol'acnrclastic oll'cctsrl thc ranges of thc govcrning nondimcnsional paranrctcrs in the modcl arc, Lrsc: to those of the prototype. Most commonly, it is thc Rcynolds number oftlrt'llrototype that is not realized in the model. As a result, uncertainties subsistrrr irrtcrpretation of model test results. (See also Chapter 7.)

Most of the empirical models described in this chapter apply to situationstlr:r( nray be considered, at least approximately, as two-dimensional. In practice,tlrn'c dimensional effects are present, owing to any numberof factors such as:lhrw adjustments near the ends of finite cylinders; spanwise variations, either,,1 thc body cross section (e.g., for tapered stacks) or of the body deformation;rr,rrnrnifbrm mean flows; or imperfect spatial coherence of the incident turbu-Irrrt:c t)r of the vorticity shed in the wake of the body. Information on three-,lrrrronsional effects is in most cases scarce and must be obtained from windtrrrrncl experiments.

'l'hc topics dealt with in this chapter include vortex shedding and the asso-r r;rlcrl lock-in phenomena, across-wind galloping, wake galloping, torsional,lrvt'rgence, flutter, and buffeting response in the presence ofself-excited forces.

{i.1 VORTEX SHEDDING AND THE LOCK.IN PHENOMENON

lr wrrs seen in Sect. 4.4that under certain conditions a fixed bluffbody sheds,rltcrrrating vortices whose primary frequency N" is, according to the Strouhalrt'lltt ion.

N^D" :sU

(6.1. l)

rvlrcrc S depends upon body geometry and the Reynolds number, D is the,r, rrrss-wind dimension of the body, and U is the mean velocity of the uniformllow in which the body is immersed. The frequency N" is also that of the netl,r rnrrtry forces acting transversely to the direction of U while the primarylrt't;rrcncy of net forces acting in the flow direction will be 2N". Actually theru'l lirrcc vector defined by thc intcgral of instantaneous pressures over a givenI'lrrll bocly will vary in magnitutlc lrntl rlircction with time in a fairly complexrrr;rurcr dcpcnding upon rlc:tirilctl lllrly gc()luotry and Reynolds number of thellow. Only thc l'rcqucncics ol its prirrt'i1xrl lurnrrlnics urc givcn by N, and 2N,.

ll tltc brxly thll ins(igirtt's llrt'vorlt'r slrt'rhlirrg is cllslicully supp<lrtctl or il'It rs subjcc( lrl lrtcrtl c()rtl()ur rlclottu;rltrrtt. rl will tlt'llcct wlrolly or locirlly lrnrl,lry lrtis uc'tirttt, irtllttcttcc llrr' lot;rl llon, Nol rrlrrry ol tlrt' lirll lrrrl.lt' ol' possi

218 ntllol lnt;ll(; I'l ll N()MI Nn

bilitics latcnt irr this situation lurvt'lrt't'rr slutlit'tl in tlctiril. l)cliltrttitblt: stcclshells have givcn risc l.o so-callctl ovrrlling oscillations l6-ll unck:r thcsc ctln-ditions. Many examples of cross-wintl rigitl-cttnttlur <lscillalions havc bccnnoted; and in water flows impoftant along-lkrw dcflections have been observed[6-2,6-3,6-4].

Unless otherwise noted, it will be assumed in this section that the structureis a cylinder with a rigid surface, the oncoming flow has uniform mean velocity,the deflections of the body are the same throughout its length, the body iselastically sprung and possesses mechanical damping in the across-wind direc-tion, and it is rigidly constrained in the along-wind direction. Under the actionof the vortices shed in its wake the cylinder will be driven periodically, butthis driving will elicit only small response unless the Strouhal frequency ofalternating pressures approaches the natural across-flow mechanical frequencyof the cylinder. Near this frequency greater body movement is elicited, andthe body begins to interact strongly with the flow. It is experimentally observedat this point that the body mechanical frequency controls the vortex-sheddingphenomenon even when variations in flow velocity displace the nominal Strou-hal frequency away from the natural mechanical frequency by a few percent.This control of the phenomenon by the mechanical forces is commonly knownas lock-in. In dynamical systems theory this phenomenon is referred to assynchronization. Observations show that during lock-in the amplitude of theoscillations attains some fraction, rarely exceeding half, of the across-winddimension of the body. The effect of lock-in upon vortex shedding is repre-sented in Fig. 6.1. 1, which shows that in the lock-in region the vortex-sheddingfrequency is constant rather than being a linear function of wind velocity, assuggested by Eq. 6.1.1 (and as it in fact is outside the lock-in region).

No completely successful analytical method has yet been developed, startingfrom basic flow principles, to represent the full range of response behavior of

F requency

F low velocity

FIGURIT 6.1.1. I,lvolution ol'vorlcx-shcdcling licqucncy willr wintl vclocity ovcr clas-tic slnrcturL:.

cfo

Ef

z

of

f

o

I \/(rl illx :;l ll l)l )lN(i nNl) llll trt{ tr [.J I't il u{)Mt Nt)N 219

;r ltlrrll cllrsiit'lrrttl.y rrrrtlt'r llrt';rtlitrt ol vollcx slrctltlirtl, 1 ll llrs, irtstelrtl, lrecrtlorrrtl rcirsontrbly lirrrllrrl lo lrrrrltl errrpirit'irl rtrxlt'ls:rrrtl trr:rlt lr llrr'ir'Pcllirlrrrirrrt't'l() r(:irlily hy ir jrttlit'iorrs t'lurit'c ol'pitrirtttclers. ltclt'rt'rrt'cs l(r ll lo l6.t.ll.l{r tt7l, ancl l(l-9 ll to l(r ()5

1 plovitlc irrt ovcrvit'w ol'sorrrc ol'lltc rcet'rrt li(cr.rrlrrrt'rrr llris arca.

6.1.1 Analytical Models of Vortex-lnduced ResponseAssurnc first that the circular cylinder dealt with above is fixed not only in therrhrrrg wind direction but in the across-wind direction as well. In this case art':rsonable first approximation to the across-wind force perunit span acting ontlrt' cylinder is

F : )pUzDCl5 sin c,r"/ (6.1.2)

rvlrcrc c,r" :2rN, N" satisfies the Strouhal relation (Eq. 6.1.1), and C15is thelrlt c<refficient. (For a circular cylinder and Reynolds number 4O I G.. I 3" 105, in a uniform smooth flow C15 = 0.6 16-4, p.721.

An important feature of this across-wind force, however, is that it is im-perl'cctly correlated along the cylinder span. When the cylinder is allowed tooscillate, this simple expression for the forcing function F is inadequate forrwo rcasons. First, the across-wind force increases with oscillation amplituderrrrtil a limiting amplitude is reached. Second, the spanwise correlation of the:r,'ross-wind force also increases, as indicated in Fig. 6.1 .2. Let y denote the;r. lrss-wind displacement of a cylinder of unit length for which the effect oftlrc imperfect spanwise force correlation is not explicitly accounted for.f Theltluation of motion of the cylinder can be written as

m1) -l cy + l{y : 5(y, i, y, t) (6.1.3)

rvlrcrc ru is the cylinder mass, c its mechanical damping constant, k its spring:,tillhcss, and $ its fluid-induced forcing function perunit span, which may be,lt'pcndent on displacement y and its time derivatives ) and j; as well as onIililc.

Much effort has been spent on finding by empirical means a suitable expres-:,rorr lirr $ in Eq. 6.1.3 that fits the experimentally observed facts. The com-1'lt'xity of such an expression will depend on the detail and completeness withrvlrich the experimental facts are observed, on the one hand, and on the needsto bc rnet by the subsequcnt predictions from the model, on the other.

rltctt'rtl slutlios in contputlrlion:rl llttitl rlVrr:rrrrir's lr:rvt't'x:urrinctl a limited numberofsuch cases| {r ()l.l

| .

rllrrs t'lli:tl is irttrrtrrtlctl lirt r'trtpirrtrrlly rr I l() .'l (rtr';rlso ('lr;rl)t('r lO, lit1s. 10.2,(r, 10.2.7, andll).1 l(r).

220 Al llol lAlill(: I't lt N()Mt Nn

,orO ..rO

.-*--'lti*-.#.to

68SEPARATION ,/D

FIGURE 6.1.2. The effect of increasing the oscillation amplitude al2 of a circularcylinder of diameter D on the correlation between pressures at points separated bydistance r along a generator: (a) smooth flow; (b) flow with turbulence intensity l1%.Reynolds number = 2 x 104. 1After t13-951.)

Among the many empirical analytical models of vortex-induced oscillationare a number that recognizethe near-sinusoidal response ofthe cylinder at eachof two prominent frequencies-the Strouhal and the natural frequency of thestructure. The response in each of these two simultaneously gives rise to abeating oscillation when the velocity of the cross flow is not precisely at thelock-in value. Figures 6.1 .3a, b, c depict some illustrative experimental resultsfor deflection response ofan elastically supported circular cylinder before lock-in, at lock-in, and after lock-in, respectively, together with the correspondingdisplacement spectra, where f, , f, are the Strouhal and natural structural fre-quencies, respectively.

A considerable variety of empirical analytical models have been devised torepresent the vortex-lnduced response of bluff cylinders t6-951. one particularaspect of the phenomcnon itsclf that has been notcd is that thc wakc of thebluffbody, composctl ol'a "strcct" ol'altcnratcly shcrl vollit'cs, slrows itspcctsttl'a scparatc "oscillitlot'," couplctl in u lirirly c<lrrrplt.x nr;uur(.1 to llrt'irrilirrting

2.89F.< .r,Jurt.40ioa.z

z9FJlrlcEo(J

06

.o5

SE PA RAT ION ,/D

ri I v()t iltx :il il t)t)tN(i ANI) lilt tlx;t( tN t,ilt N()Mt N()N 221

(a)

s(0YID

s(f) .'rrl'rl00500

240

200

160

120

80

40

0

o.0B

0.00

.o.oB

0 004

YID

0.000

o 004

(b)

6

5

S(0 4

5

2

t

0

t(c) f

lrl(;URE 6.1.3. Across-flow oscillations y/D of elastically supported circular cylinder:(rr) before lock-in; (b) at lock-in; (c) after lock-in. (After [6-931.)

rrrcchanical body. Another characteristic of vortex-induced oscillation is that,wlrile self-excited, it never proceeds to divergent amplit,fdes but enters a limit.yclc of relatively modest level. I

Numerous qualitative or semi-quantitative attempts have been made to setrrp associated, descriptive mathcmatical models, in particular several so-calledtorrplcd oscillator models govcrnccl by two differential equations, one for theslrrrcture and another frlr ils wlktr. whilc such efforts have not been unreward-rng, it is oftcn thc casc lhlrt lhc rnos( irrrlxrr(ant cngineering need is to be able{rr lirrccast the largast suslrtittt'rl r'('sl)()nsc lrrrrplilutlc ol'the structure alone, thatrs, lhal which <lccr.rrs al lock irr

A tttorc lirrritctl silrgk: rlcp,rn'rtl ltt't'thrttt tnorlt'l is tlrcn <llicn usclirl. ll.cf'-('11'nct' l(r-9(lI lurs suggcslctl:

222 nnr()r rnritt(; t,t tt NoMt NA

where D is a frontal dimension of the structure, K : DalU, and <o satisfiesthe Strouhal relation

:2rS (6. r.5)

ln this rnodcl, which obviously exhibits aspects of a Van der Pol oscillator,Y., t, Y2. ancl Cy, arc parameters, functions of K, that are to be fitted to obser-vations. Various cxploitations of this model may occur. In particular, aspectsof nonlincar, scll-lirniting amplitude are inherent in it. in agreement with similarefl'ects witncssed with vortex-induced oscillation. In effect, the model allowsfor linear, fluid-instigated "negative damping" at low amplitudes, an effectreversed at higher amplitudes. At lock-in o J @r and Y2 = O, C1 : 0, sinceat lock-in the last two terms are found to be small compared to the termreflecting the aerodynamic damping effects. Then Y1 and e remain to be deter-mined from experimental observations.

At steady amplitudes the average energy dissipation per cycle is zero, sothat

l'lo^y, - prrDYl (, - ,*) lr' dt : o (6 r 6)Jr L ,, D'/ 1"

where c,rZ : 2zr. Assuming that y behaves practically sinusoidally,

) : )o cos cdt (6.1.1)

leads to the results

mIi + 2r,ry+,?y I - )ou'n l r,,n, (' - . #) L+ Y2(K) { + c,,1x1sin (c,,/ + d)] (6.1.4)

aDi

\',, *:,vtr

{'.lrqdl"

(6.1.8)

(6.1.9)\inr'at : ,vrLo

Then (6.1.6) yields the steady amplitude solution

(6. r. r0)

i I V()l rllx l;l ll l)t)lN(i ANI) illl t()|l!lr.l ttilt NttMt N()N 223

wltt'rc S is tltc Sllrtrlutl rrrrrrrlrt.l=1tr.l.l) irrrtl,f,, lllc,!r',r,/r,/r nrtnrlu't tlr.liur.rl lrs

l'tttt" ,,r' t(r'l ll)

'l'lrc rnodel is usclirl irr prctlicting prototypc aclion lirrrrr thc bchavior ol'labo-Iirlory tests.

A process by which thc parameters 1, and c may be cvaluated from a modelIt'st will be describcd. If, at lock-in velocity, the mechanical model is displacedIrr irn initial, higher amplitude ! : Ao and then released, it will undergo atlccaying response (Fig. 6. 1.4) until this latter levels out at the steady-statevlluc ye given by Eq. 6.1.10. It can be shown [6-94] that this devolution ofrrrrrplitude is describable by the form

y(/) =D

lolDlr - ((Ai - yblA'd exp (-ayf,Ut/4D3)lt/2

pDzYtOt:-(2m

(6.1.12)

(6.1.13)

rn which

'fhe value of cv is determined from the model test as follows: Defining R,AolA, where .4n is the amplitude of y at n cycles after the release, cv may

lrc cvaluated as

lll(;tJlllt (t.1.4. l)cciryirrg oscillrtlirrrt lo:lt'ruly ;.lirtt'ol lrlrrll, clrrslic:rlly sprrrrrg rrrrxlclrrrrlcr vort0x klck-irr.

224 nt n()t tnl;llo l'l llN()Ml NA

4ri/)' . 1,,t,i,, lt;vil I(Y - ,,y,1 l" I ll,, u,'l I

so that Y1 and e are given bY

,,: #1"#+ r6rrsl

2ma' : ody,

(6. r.14)

(6. l. ls)

(6.1. r6)

liurploying an analytical model of this type for a circular cylinder the max-irrruur arnplitudcs ol Fig. 6.1.5 were obtained. On the same figure (dashedcurvc) an cmpirical fbrmula of Griffin, Skop, and Ramberg [6-33] is plotted.This lonnula, for circular cylinders, has the form

ll + 0.43(8rr'sts..)lt tt t6. l.l7t

6.1.2 An Empirical Model Developed for the Estimation of theResponse of Chimneys and Towers

A model derived in effect from Eq. 6.1.4 was developed in [6-88] for appli-cation to the design of chimneys and towers with circular cross-section. It isnoted in [6-88] that the product pUzYrlXl of Eq. 6.1.4 is considerably lessthan mal, so that in practice the term Yz(K)ylD may be ignored. It is alsonoted in t6-881 that in the case of alandom motion, the term ,yzlDt of Eq.6.1.4 may be replaced by the ratio y2l(XD)2, where \ is a coefficient whosephysical significance is discussed subsequently. The term

)o:D

r.29

0. l0

YJD

0.05

o ExperimenlEq.6.1.17

-\o'\q,

o\_ \o--o_....--o-o o

l .5 2.O 2.5 5.0 5.5 4 .0

Scruton numberFIGURE 6. 1.5. Maxililurn arnplitudcs vcrsus Scnttott llttttthct'(lrlicr l(rr.)31).

.00.00

I

I Vrll illx :il ll l)l)lN(i ANI) llll lilr:ti tH t,t il il()Mt u()N 225

| / r'r l^,

/,1/ /))'r(A) \1, tr, ) ttol lrc1. 6. 1.4 is wrillcn itt l6 ttl'll itt thc lirrrrr

2a11,r)'K,,"(#) I ,J;,'l ,

r.vlrcrc K'6(U/U,,) is an acrodynamic coefficient, and U,., : alDl(2rS). The;rlxrvc term is equated to the product -2mlo<'s1, where f, is defined as therrt'lrdynamic damping ratio, which may thus be written as

r": -+r.,(*o,)l -,*il (6. l . l8)

11'.,r y2l12 : \D the aerodynamic damping vanishes, so the structure no longert'rpcriences any aeroelastic effects causing the response to increase. The coef-lrcicnt \ may thus be interpreted as the ratio between the limiting rms valueol (hc aeroelastic response and the diameter D.) The total damping ratio of the:'vslcm is then

(,:(iJ" (6.1.19)

rvhcre f is the structural damping ratio. The aeroelastic effects are, in effect,rntrrduced in the equation of motion simply by substituting into that equationtlrc total damping ratio f, for the structural damping ratio f.'l'he validity of this simple approach was verified in [6-88] by numerical:trrrlies and by comparisons with experimental results reported in [6-39]. Figure(r 1.6 shows the dependence of the measured response 4*. : y't''lD upon thert'tluced wind speed 2rUlalD for various struclural damping ratios f. Figuretr. 1.7 shows calculated versus measured ratios y'^^'.lD for various values of thel)rrameter K, : m(l pd, where yill'" ir the rms response corresponding to therrrost unfavorable reduced wind speed. Three regimes are noted in Fig. 6. 1.7,,,'r'rcsponding, respectively, to (l) vibrations whose character is largely due totlrt: random nature of the forces associated with vortex shedding (forced vibra-tiorr regime), (2) a transition zone, and (3) self-induced vibrations (lock-inrt'girnc). Vibrations typical of these three regimes are shown in Fig. 6.1.8.Notc that the ratios of peak to rms response are about 4.0 in the forced vibrationrcginrc, an<I about r.D in th. Iock-in rcgime.

llascd on inferenccs I'nrnr cxpcrirrrcntirl clata available in the literature,Itr tttll prop<lscd curvcs rcprcscnling ( l) tlrcr tlcpcnclcncc of K,,9,,,,- upon theItt'yrr<rlrls ntttnbcr 61": IJI)l t', wltt't't'4,,,,,,,,,, rlt:notcs thc maxitnutn valut: ofK,,r(l.llIJ,,) in snuxrlh lklw (lri1l. (r. l.()). rrrrrl (l) tlrc tlcpcnrlcncc ol'thc ratio

Anlol lAt;ll0 I'lll NoMl NA

72rU't D

FIGURE 6.1.6. The response of a model stack of circular section for different valuesof structural damping (Ge subcritical). From L. R. Wooton, "The Oscillations of LargeCircular Stacks in Wind," Proc. Inst. Civ. Eng., 43 (1969),573-598.

Koo(UlU-)1Ko0',- upon UIU,, for smooth flow and flows with various turbu-lence intensities #tt2lU Gig. 6.1.10).

For a vertical structure experiencing random motions described by the re-lation

89tO

[6-89] proposes the following expression for the total damping in the ith mode:

Vk):zt?v?rrtI

(6.1.20)

(6.1.2t)

(6.1.22)

(r;:lifl";

u, :-#lrr,, -r,*)Il, K,u(z) [#l' y?(z) az

E .06FulolFJ(L

otrjc)loUJE .O2

Kri :1t,i ,112,1 dz.

(6.1.23)

fi r v(lilil x ril ll l)lllN(i ANI) llll l()r,h tN t,l I N()Mt N()N 227

o Experimental [6-39](:itc:600,OOO)Calculated

" Lock-inRegime

"Transition"Regime

"Forced Vibration"Regime

0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0Ks

l,'l(;[JRE 6.1.7. Measured and estimated response in smooth flow. From B. J. Vickery;rrrtl Ii. L Basu, "Across-Wind Vibrations of Structures of Circular Cross-Section. PartI l)cvclopment of a Mathematical Model for Two-Dimensional Conditions," J. WindI rt,q. lnd. Aerodyn.,12 (1983), 49-73.

I

t--*I

\tri x,,,,12.1Y!121 dz1(r,:--' \ I',\ .v itz.l dz.

(6.r.24)

rvlrr:rc C and L,; are thc structrrnrl irrttl tltc rrcnlclynarnic damping in the ith modeol vihralion, rcspcctivoly, /),, is llrt'tlirrttrelet'rrl clcvation z : 0, D(1) is thcrlilurrclcr at clcvation l, /r is llrr'lrr'ip,lrl ol lhc slnrclul'o, nr,,i is lhc cquivalcntnliris por unit lcngilr irt (ltc itlt rrotlc ol vrl)!irlion, tle:lirtcrl as

K,/ri,,, r

Ks/Kao = I

228 At n( )t l A:i il(: t ,t il N( )Mt Nn

{.5

Y0D

0.1

0or fTL) ul-,f

.n. I

IKr/K"u t ,

FIGURE 6.1.8. Simulated displacement histories for low, moderate, and high struc-tural damping. From B. J. Vickery and R. L Basu, "Across-Wind Vibrations of Struc-tures of Circular Cross-Section. Part I. Development of a Mathematical Model forTwo-Dimensional Conditions," ,/. Wind Eng. Ind. Aerodyn.,12 (1983), 49-13.

Mrch No. > 0.15o

z' o/' Nr.h No < o t5

Rt YNOr t)s Nti!ilil R

FIGURE 6.1.9. Experimcntal data, and sr.rggestcd dcpcntlcncc ol (,,,,,,,,, rrpon Rcy-nolds numbcr. Fnrrrr Il..l. Vickcry anrl R. l. Basu, "Ac:nrss Wirrrl Vilrllrtiorrs ol'Stnrc-turcs ol' Circular ('trrss Scclirlt. l)iu1 I. I)cvclopnrcnl ol :r Mrllrr'rrurllt lrl Mtrlcl lirr'f-wtr-l)itttcttsiottirl ('olrrliliorrs." ./. Witttl l,.lrr.t:. ltttl. ,'lt,nult'rt . l: (1,)ttl). .l() 71.

ti r v( )ilil x 1;1il I)t)tN(i ANt) ilil I ( x.h tN t,lI N( )Mt N( )N

; l'.'

lt

229

0.8 0.9 1.0 1.1 1.2 1.3 r.4 1.5 1.6 t.7u u",

I,'l(;URE 6.1,10. Dependence of ratio Kuol Koo-o* upon ratio (JlU.,for various turbu-lencc intensities. From B. J. vickery and R. I. Basu, "Across-wind vibrations ofslrlrctures of Circular Cross-Section. Part L Development of a Mathematical Modellrrr Two-Dimensional Conditions," l. Wind Eng. Ind. Aerodyn., 12 (1983), pp..l() 73.

(6.1.25)

;nrLl Ml is the generalized mass in the ith mode. Equations 6. l.2l to 6.1.24 arelr;tscd on the assumption that aeroelastic effects occurring at various elevations;rrc linearly superposable.

For the relatively small values of the response that are acceptable for chim-r('ys and stacks, the estimated response depends weakly upon the assumedvrrltre of \. It is suggested in [6-89] that the value \ = 0.4 is reasonable forrrsr: in estimates of the response of concrete chimneys.

6.1.3 Experiments on the Lock-in Phenomenon in Turbulent Flowl;nrrn tests in turbulent flow with Rcynolds numbers of about 75,000 on a 200-rrrrrr diameter cylindrical oscilllr(or with lincar springs, statistics of interest forl;rligttc studics were obtaincrl irt l(r ()()l on klck-in l'rcqucncy intervals and across-wirrtl oscillations during irnrl lrlicl krt'k in. l;igrrrc 6. l.ll shows time historiesol'( l) wind spcctl lluc:tulrliorrs lrrrtl (l) lrr'lrss wirrtl rrulli<lns that cxhibit irrcg-rrl;rl' lockotl-in rlscillittiort cpisrxlt'r. l,t'l llrc lowr'r'lrrrtl rrppcr cntl ol'lho lock-inlrr'tlrrcrrcy irrlcrvlrl bc tlcrrolt'rl 1',t.rrt.rrr'lrlly lry..l trrul /1, r't:sltcclivcly.'l'lre: pr.ob,:rlrilislic bclutviot'ol',4 lrrttl /l w;rs lorrrrrl rrr ltr t)t); to rlilli'r'lrt't.otliltg lo wlrr.llrt.r.

0.8Kd0

230 n t n()t t n li ilo t't tt N( )Mt NA

0 100 200 300 400 s00 600Time [s]

FIGURE 6.1.11. Time histories of longitudinal wind speeds and across-wind displace-ments [6-99].

(l) dVldt is positive when the longitudinal turbulent velocity Z crosses thelower threshold A and when it crosses the upper threshold B (in this case thenotations A = At andB = 81 are used); (2) dVldtis negativewhen ZcrossesB end when it crosses ,4 (in this case we denote B = Bt and A = A); (3)dVldt is positive when Z crosses ,4 (which is again denoted by Ai, changessign during lock-in, and is negative as Z again crosses ,4 (which in this caseis denoted by A); and (4) dVldt is negative when Zcrosses B (which is denotedby Br), changes sign during lock-in, and is positive as Zagain crosses B (whichin this case is denoted by Br). Scatter plots of crossing limits are shown in Fig.6.1.12, where the abscissa is normalized with respect to the Strouhal number,which was about 0. 175. Stochastic properties of successive lock-in intervalswere found to be independent. For additional details, see [6-99].

6.2 ACROSS-WIND GALLOPING

Galloping is an instability typical of slender structures having special cross-sectional shapes such as, for example, rectangular or "D" sections or theeffective sections of some ice-coated power line cables. Under certain condi-tions that are defined later herein, these structures can exhibit large-amplitudeoscillations in the direction normal to the flow (one to ten or even many moreacross-wind dimensions of the section) at frequencies that are much lower thanthose of vortex shedding from the same section. A classical example of thistype of instability is the acnrss-wind large-amplitudc galloping ol' powcr lineconductor cablcs that havc rcccivcd a coating ol'icc untk:r contlilions ol'l'rcczingrain.

67E

--oo t

!.943=10EE8(D^EO.=4o_EC

0

0.30

ozs I

< o.2o 1o o.ts -l

o'o -]o.os -{

0.30t! 0.20(D

0.40

0.10

0.00

0.85 0.90 0.95Ar

1.00 1.05

0.80 0.90 1.00At

l.oo 1.10Br

1.OO 1.10 1.20Br

1.20

1.00

0.95

A,0.90

0.85

1.25

1.20

1.15

Bt 1'101.05

1.00

0.95

FIGURE 6.1.12. Scauer plots of lock-in frequency inrerval limits [6_99].

Early and clarifying analyses of the galloping problem appeared in [6-40],16-411, and[6-42]. References [6-43] to [6-50] have dealt with the problem aslu nonlinear phenomenon. In across-wind galloping the relative angle of attackof the wind to the structural cross section depends directly on the across-windvclocity of the structure. Experience has proved that knowledge of the meanlift and drag coefficients of the cross section obtained wder static conditionsrrs functions of angle of attack suffices as a basis upon which to build a satis-lactory analytical description of the galloping phenomenon. Galloping is thusgoverned especially by quasi-steady forces.

As in the case of the vortex-induced oscillation, the phenomenon will beconceived of, and dealt with analytically, as two-dimensional in nature. Furthertluestions related to galloping response are discussed in t6-461 to [6-50]. Astudy of a system of two elastically coupled square galloping bars that cancxhibit chaotic motions, reported in detail in [6-100], is summarized in Sect.6.2.2.

6.2.1 Analytical Formulation of the Galloping problemllclirrc presenting thc basic unirlyticirl lirrrnulation, it is of interest to note someol'thc rccognizcd litcraturc lhlrl lrr.lrts lhc gall<lping phenomenon. Rcf'crcnccl6-4131 rcvicws tho siatc ol lltc lttl rrtttl prescrrls a compact analysis 9l'tlrcpnrhlcrrr. lt irlso poinls orrl tlrt.t.:lly lurtl lrirslt.t'orrll.ibulions ol'(iltrrrul l(r_401Ittttl l)t:tt lllrrttlg l6-4 1,6 42 1. l{clt'rt'rrt't's l(r.l llrrntl l(r 4t)lcorrslilrrtt.irrrl.rorlirrrl

232 Atlr()LtAlitt(; t'ltt NoMl Nn

FIGURE 6.2.1. Lift and drag on a fixedbluff object.

contributions particularly toward clarifying the nonlinear questions related tothe aerodynamics. Reference [6-50] offers a critical discussion of existing an-alytical models of galloping.

Consider a section of a prismatic body in a smooth oncoming flow (Fig.6.2.1). Assume that the body is fued (i.e., experiences no motion, oscillatoryor otherwise) and that the angle of attack of the flow velocity U. is a. Belowis obtained an expression for the force coefficient in the y direction. First, thecomponent of the mean drag (mean force in the direction of U,) can be writtenAS

D(e) : )pulncrla\

while the mean lift (mean force in the direction normal to U,) is

L(oi : |puI,ncr1a1

The projection of these components on the direction y is then

F,(cY) : -D(o)sin a - L(cv)cos cv

If {(cv) is written in the alternative form

Fn(cv) : )pttzBco,1o)

where

U : U,cos a

it follows from Eqs. 6.2.3 and 6.2.4 that

(6.2.1)

(6.2.2)

(6.2.3)

(6.2.4)

(6.2.s)

Ce,(cv) : -tcr(a) * Co(cy)tan alsec cy (6.2.6)

The case is now c<lnsiclcrcd in which thc samc hody tt,stillrtlr',r irt lltc ac()ss-wind direction .y in a lkrw witlr vclocity l/ (F'ig. 6.2.2|.'l'ltc tnirgnitttrlr-r ol'thc

ri :, n(:ll( )lil: wll.Jl I I iAl I I )l'|lN( i 233

FIGURI.I (t.2.2. Elfcctive angle of attackon an oscillating bluff object.

rclative velocity of the flow with respect to the moving body is denoted by U,lrnd can be written as

U,: (U2 + i\t,tl'he angle of attack, denoted by a, is

(6.2.7)

(6.2.8)

If the body has mass ru per unit length, is elastically sprung, and has linearrrrcchanical damping, its equation of motion can be written in the usual form

mly+2lt}1_alyl:F, (6.2.e)

whcre f is the damping ratio and c,r1 the natural circular frequency, and where/'',, denotes the aerodynamic force acting on the body. It is assumed that therrroan aerodynamic lift and drag coefficients C1(cv) and Cp(cv) for the oscillatinglrrrdy and for the fixed body are the same so that Fy(cr) is given by Eq. 6.2.4whcre Cp,(a) is given by F,q. 6.2.6.

Let us first consider the case of incipient (small) motion, that is, the conditionirr the vicinity of i : 0 wherein

(x=

l,or lhis condition

l,' -. oF'l u' i)rv 1," ,,(6.2.t0)

wlriclr lcarls (o cxalrrinutiott ol lltt'lttr'tot r/(),,/r/rv lirrrrrtl uptln tlill'crcrtliation ollu1. (r.2.(r (o lutvc tltt: vltltle' ltl rv O.

arctan lu

n'=0U

234 nl ltot lnrill(] I't il NoMt NA

(6.2. r r)

Thus for small motion the equation o1-Inotion takcs the form

mly+2(rJ+rlyl (6.2.12)

Considering the aerodynamic (right-hand) side of the equation as a contri-bution to overall system damping, the net damping coefficient of the system is

2m(a1 * (6.2.t3)

where, by analogy to the first term of the left-hand side, which is known asmechanical damping, the second term is referred to as aerodynamic damping.From the well-known theory of the linear single-degree-of-freedom oscillatorwith viscous damping it follows that the system tends toward oscillatory sta-bility if d ) 0 and toward instability if d < O. Since f, the mechanical dampingratio, is usually positive, instability will occur only if

(6.2.14)

This is the well-known Glaueft-Den Hartog criterion, a necessary conditionfor incipient galloping instability (a sufficient one being d < 0). It is clearfrom Eq. 6.2.14 that circular cylinders, for which dCylda = 0 because of theirsymmetry, cannot gallop.

To summarize the problem to this point, the initial tendency of a slenderprismatic structure toward galloping instability can be assessed by evaluatingits time-averaged section lift and drag coefficients and assessing the sign of theexpression dCl,lda * Cpat a:0.

For many problems of wind engineering this initial assessment suffices todescribe possibilities of incipient instability relative to galloping. For example,Fig.6.2.3 16-51,6-521depicts the lift and drag coefficients for an octagonalpost structure having a region of wind approach angle (-5' ( cy ( 5") wherethe structure is susceptible to galloping according to the Den Hartog criterion.To pursue the problem further, however, and describe the galloping action indetail requires full development of Cp, in powers of ylU. Reference [6-48]suggests an abbreviated power series with several odd powers of ylU and withan appropriately signed second-power term to smooth the fit:

tl('1 ,l,/t 1,, ,,

("1,,: , ,',,),,

: -)pu'n (# . ,"),+

)oun (* . ,,),: o

(9*.,) <o\ dd /o

^,(L) - ^,(t)' fr - ,,, (;)' . ,, (.;)' ^,(,r)'cr, :(().2. l-5)

ri ;, n(:ltt):;:;wltnt) (i^t t0t 'tN(i 235

dlu [ -' t-^ Y

TJ'

5Uo

lrl(;URE 6.2.3. Force coefficients on an octagonal cylinder (G,e: 1.2 x 106) [6-52].

ll the dependence of CTand Cp upon a is known, the coefficients.4, through.'1 , can be evaluated as follows. First, Cp, is plotted against tan cv. Since tan cv

ilu, CF, can then be approximated by the above polynomial using either aIt'ast squares fit or some other technique as desired. Reference [6-48] appliestlrc method of Kryloff and Bogoliubotr [6-53] to the solution of the resultingrurnlinear equation, postulating as a first response approximation:

!: a cos(c,11/ * @)

y : -aor sin(o1r * S)

(6.2.16a)

(6.2.16b)

whcre a and S are considered to be slowly varying functions of time. Threelrrrsic types of curv'es Cp" as functions of a and the corresponding gallopingr1'sponse amplitudes a as functions of reduced velocity UID<,:1 are identified(scc Fig. 6.2.4). The only possible oscillatory motions are those with ampli-trrtlcs a traced in full lines in Fig. 6.2.4.If the speed increases fromUoto U2tl;ig. 6.2.4a), the amplitudc of'thc rcsponse is likely to jump from the lowert() tllc upper branch of thc solitl curvc. Il thc specd decreases from U2 to Ustlrc .jump occurs fr<rm (hc ul)l)cr l() llrc: lowcr curvc.

l{clbrcncc [6-491 discusscs llrc loilx)r)sc ol'eklnglrlctl thrcc-climcnsional bod-It's by ttsc ol'thc sccliotltl llrt'oty otttlirtt'rl rrlxrvc lrutl nrcntions tho cll'cct oflkrw turbulcncc r.rl)on llre g:rllopirrlt. lt is rrolcrl llurl ttrlbulcrrcc clrrr llirrrslirrrnslt'luly ost'illitliotts ittlo ttttslr'irtly on('s, r('(lu( t'llrt'rrr;rllrrilrrtlc ol'(ltt'irt'txl-yluutrir'

CORNIR RAI]IUS =O O5D

236 ntn()LtnsilC l,ilt N()Mt NA

U1 u2 u/D(4

FIGURE 6.2.4. Three basic types of lateral force coefficients and the correspondinggalloping response amplitudes tr. From M. Novak, "Galloping Oscillations of PrismaticStructures," J. Eng. Mech. Div., ASCE, 98 (1972),27-46.

damping, and in ceftain cases, depending upon its scale and intensity, destroythe necessary conditions for galloping. Under certain conditions of an initialtriggering disturbance larger than the steady-state amplitude, certain sectionscan experience galloping at much lower velocities than those required in smoothflow. Finally, it is noted in [6-49] that galloping oscillations also depend uponthe extent to which the mean angle of attack varies as a function of the mag-nitude of the wind drag.

The closely similar problem of a long flexible beam free to deflect in bothalong-wind and across-wind directions is analyzed in [6-54]. Reference [6-55]discusses the effect of incident wind skewed to the long axis of a gallopingbody. For information on galloping tendencies of stranded cables, see [6-901.

6.2.2 Galloping of Two Elastically Coupled Square BarsReference [6-1001 dcscribcs an cxpcritncnt concluclcrl irr ir wrrlt'r'lrrrrrrt:l on lhcbehavior ol a syslcnt ol'two cl:rstically rcstraincrl arrtl t'orrplt'rl :rlrrrrrinirrrrr :;(lurrrc

uo

U/D'{U.,I

{':l WAI\I {inl l()l'lN(i 23'7

FIGURE 6.2.5. Schematic of double galloping oscilla-tor.

liius with sides ft1 - hz:6.35 mm and lengths 0.215 m. The spring constantsr','rt',1.1 : 56 N/m, kz -- 78 N/m, and kn : 145 N/m (Fig. 6.2.5). To preventilr:;pllccments due to drag, the bar ends were attached to fixed points by thinrvrrcs with lengths r : 400 mm. The bars were observed to gallop in phase,lrrrt t'xriept for relatively low flow speeds 4 this oscillatory form alternated inrrrrlrrcrlictable, chaotic fashion with a second oscillatory form wherein the twol';rrs g,alloped with higher frequency in opposite phases (Figs. 6.2.6a, b). Theurt':rrr cxit time of the system from the region of phase space corresponding totlrr in-phase oscillations decreased as the flow speed increased.

( )rrc conclusion of the study concerns basic limitations of empirical fluid-i'l;r:rtic rnodels. As is shown in earlier sections and elsewhere in this text, suchrrrotlt'ls can be adequate for some applications. However, it should be remem-I'r'rt'tl that the relatively small number of empirical fluid-elastic parameters thatrlt lrnr: the models may not be capable of reflecting in sufficient detail the, , ,rnplcxities of what is after all an infinitely dimensional fluid-structure system,,1,'r;r'r'ihcd by a Navier-Stokes equation whose boundary conditions are depen-rl'rrl rrpon the solution of the system itself. Therefore, unless its range ofr;rlrtlity is carefully circumscribed, an empirical fluid-elastic model is boundr,' lrt'inadequate as a predictive tool. We referthe readerto [6-100] fordetails,rrr tlrt' rnodeling problem for this case study and similar cases.

l'lrc laboratory observations just summarized gave rise to the development,,1 ;r rrurthcmatical theory of chaotic motions (i.e., motions that are apparentlyr;rn(lonr and exhibit sensitivity to initial conditions) applicable to nonlinearrrrrrltistublc systems subjected to excitation by noise (see [6-10l]). For an ap-t,lr,;rliorr of the theory to thc problem of wind-induced along-shore currentsrr\r'r ir c()rrugated ocean botltlrrt, scc Scct. 2.5.

I' II WAKE GALLOPING

llrt't':rst. is rurw coltsitlcrt'tl ol lwo t ylttttlcrs, onr'ol wlrit'lr is locitictl upsllcitln,,1 lltt'otlrct'. Unrkrr ecrlitirr t'olrlrltotu; lltc rlowttsllt'lrttr cylitttlt:t'rrtlty l-rc sttll-

k1

h1

krz

h2

k2

238 nFH()t lnl;il(: l,l ll N()Ml Nn

Time (s)

(b)

FIGURE 6.2.6. (a) Observed time history of displacement y'; (b) observed time his-tory of displacements y, (solid line) and y2 (interrupted line). From E. Simiu and G.R. Cook, "Empirical Fluidelastic Models and Chaotic Galloping: A Case Study," J'Sound Vibr., 154 (1992\, 45-66.

jected to galloping oscillations induced by the turbulent wake of the upstreamcylinder. This has proved to be the case, for examplc, fbr powcr transmissionline cables grouped in so-callcd buncllcs, that is, lirr grottJrs ol' ctlncluctorsconsisting of two, lirtrr, six, cighl, or rrrorc panrlltrl r'irlrles scpltntlctl hy rrrc-chanical spucol's irr tlrc rlirccti()rr lliursvL:l'sc to tlrcil splrtt. (l;illrtn'(r.l.l tlc:picts

li :l Wn 11l ( in I I 1)l 'll..l( i ?39

FIGURE 6.3.1. Spacer in four-bundle power line.

.r :il)rccr in a four-cable bundle of a power line.) With the spacers in place, itr', tlrc cable region between them that is most susceptible to wake galloping! 'rn(litions since cable freedom of motion is greatest there.

Wake galloping may occur only under conditions where the frequencies ofr,' lx)nse of the downstream cylinder are low compared to its vortex-sheddinglrt'rprcncies and to those of the cylinder located upstream. Just as with thelrlrt'rrrrmenon treated in Sect. 6.2, wake galloping is governed by parameterstlr;rl tlcscribe mean (rather than instantaneous) aerodynamic phenomena and canlr,' rrrcasured when the body is fixed.

'l'hc wake of the upstream cylinder may be pictured as suggested in Fig.t, 1 2. Investigating this wake with a "probe" consisting of the downstream, ylirrder itself reveals a distribution of along- and across-wind forces (Fig.t' I -1) acting on this cylinder as a consequence of its particular locations in theurrkc. One important finding is that the across-wind wake forces have a ten-,L'rrt'y to center the downstream cylinder, that is, draw it toward the wake

UPSTREAMCYLINDER

lll(;Illll,l (r.-1.2. Srr'(ion:rl l',('()nlr'lry. t ylirrtL'r:, irr w:rkt'P,lrlloPirrg Pltt:rtotttctlott

240 Al n()t tnlilt(; l'ilt N()Ml NA

LIFT

FIGURE 6.3.3. Qualtitative sketch of the distributions of mean velocity, drag, andlili on a circular cylinder in the wake of another.

centerline, contrary to the possible intuitive expectation that, since the outerflow beyond the wake edges is faster, by Bernoulli's principle it should tendto pull the downstream cylinder outward, away from the wake center.

An explanation has been sought for this apparent anomaly, which may ten-tatively be ascribed to numerous criss-crossings of the flow field inside thewake by time-varying local jets of fluid that have strong components directedinward toward the center. These jets, or local fluid velocities, would tend tocreate repetitive drag forces directed, on the average, toward the wake center.This view of the phenomenon has been supported to some degree by flowvisualization studies in a water tunnel t6-561. As indicated in Fig. 6.3.3, thecentering lift is strongest at about a quarter of the total wake width outwardfrom the centerline.

When the downstream cylinder located a few diameters of the upstreambody behind this latter is displaced-for any reason-into approximately theouter quarter of the wake (see Fig. 6.3.2), it enters a region of gallopinginstability. ln this region a galloping motion will begin, growing in amplitudeuntil an apparent limit cycle is reached. This motion consists of large oscilla-tions in an elliptical orbit with the long ellipse axis oriented approximatelyalong the main flow direction. The direction of the elliptical orbit is such thatthe cylinder moves downstream near the outer edges of the wake and upstreamnearer the center of the wake, or clockwise above the centerline in Fig. 6.3.3and counterclockwise below it. These directions coincide with the intuitiveassessment that net drag forces will be higher in the outer, faster portion of thewake and lower in its interior. References [6-56] to [6-65] cover various aspectsof the wake galloping phenomenon. An oscilloscope trace of a developing wakegalloping orbit is shown in Fig. 6 .3 .4 [6-52]. For a useful review of interferenceand proximity effects, see [9-ll.

6.3.1 Analysis of the Wake Galloping Phenomenon

Thc phcnonrcnon is lrrrirlyzrtl lrs il'i(s l.xrsic irtgrctliettls wt'tt'lwo tlittrettsiottitl,:ts wlri tkrrrc irr llrt' pn't'r'rlirrll set'tiotts. ('ortsitlt'l lwo t ylltttlt'ts 1l;i11. (r.1 .5).

lr:l Wnl,t rinl l{tl,[!{t 241

ft,'n-q{nffiffi

. Wul*"**-*"-*.*rtt"*f#

l'l(;llltlt 6.3.4. Amplirude trace of a wake galloping orbit [6-52]. courtesy of thef l,rtr.rrrrl Aeronautical Establishment, National Research council of Canada.

"rrt' windward, producing a wake, and one leeward, within that wake. Thel.t'wrrfd cylinder will be assumed to be elastically sprung in both horizontal'rrr.l vcrtical directions about some position (x, y), where X, yare along-wind;rrr,l rrcross-wind coordinates conveniently centered on the windward cylinder.

l'lrc cquations of motion tbrthe leeward cylindermay be stated in terms oftlrt't'xcursions (x, y) of that cylinder away from (X, y):

mt+d,*IKux*K,ry:F"mli+dry*Kr"x+Knny:f,

* lrt'rt: /r? is the mass per unit span (normal to the figure) of the leeward cylinder;,/,. r/,, are respective damping constants; K,,(r, s : x, y) are direct and cross_,,rrPling spring constants restraining the motion of the leeward cylinder; andI , , /,',, are the net X- and I-force components.Ncxl, if c, and c, are defined as the steady average force coefficients referred

r'r lrt't: stream dynamic pressure )p(l')that apply to the cylinder located at pointr \ )'). then it can be shown that the incipient forces in -r and y directions maylrr' 1'r111psssd as [6-65.|

(6.3. la)

(6.3.1b)

U

Y

EQUILIBRIUMPOStTtON

IYPICAL LIMITINGORBIT

;';rl lopirrli lrrlrlysrs.

ffiffi=ffiil:tWffi ilI

r-@--

242 nLn()t tnlitt(; t'l ll N()Ml Nn

(6.3.2r)

(6.3.2b)

where U is the free upstream velocity and U, is the average wake velocity inthe udirection at (x, Y), and D is the projected across-wind dimension of thecylinclcr. Expressions similar to Eqs. 6.3.2 were first developed in [6-58] and16--591. Values of C", C' and their derivatives are obtained by direct mea-sutrnlont ol'timc-avcraged values in wind tunnel model studies. Cases of in-lcrcs( havc conccrncd smooth circular cylinders and the rougher surfaces ofs(randccl wire cables.

Analytical solution of the problem, in which the forces given by Eqs. 6.3.2are clcarly self-excited only, proceeds by assigning values

r;" -- )pr./2D[(a# , .. "rt'n) , ,', ;, - ,.', ,),, ]

r, : tp'zDt(X . * ft,) - * t-,r, t)

x : xoe)\t

Y : Yoert

(6.3.3a)

(6.3.3b)

tox andy in Eqs. 6.3.1 and 6.3.2 and setting the determinant of coefficientsof Eqs. 6.3.1 equal to zero. It follows from Eqs. 6.3.3 thatthe solutions X areunstable if trr > 0 in the calculated value of form

\:Xr*i\z(where i : J=), since they then contain a diverging exponential factor. Suchsolutions are then sought for the parameters associated with a number of pointsX, Y.

The agreement between the theory and experiment has been found to besatisfactory, as seen in Fig. 6.3.6 16-56], where the curves indicate points atwhich marginally unstable solutions (i.e., where \ : iXz) are found. Forthese

THEORY- EXPERIMENT

x/D

-5Y/D

FIGURIt 6.3.6. Mclsurctl :rrrtl prcclictccl stability hottrttlrttit's lirl wrrke grrlloping

l6 561.

li ,l lr rl l:,1( lll^t lrlvt il( it l.l( .t 24:l

',r'lrrlions, tltc olllit l\(/), \'(/)l rrriry llc ctrlt.rrllrlt.rl. ll tlr.slr,tl. lry rrsirrli litls.{i I | ;11111 (.1.1.f .

As irr o(hcr trcnlclirsl ic pltc:norrrcrtir, tltrr slr.ut.ltrlrl Plrrirrrrt,lt.rs t.xt,r.l slrorrgr.rllll)l ovcrlhcchar-nctoristicsol'wakcglrllopirrg. lrrlr:rrl it'ulirr', irrt.irrrylrrgrrrrtrrr,rtlt'l stuclics thc valucs ol'thc spring consllurls K,,(r., ,r .r, .1,) r.t.t;uir.e lxrlItr'ttl:tt itttcntion. This is cspccially truc irr tlrc rcllrcscrr(irliorr ol'tlrc ucligrr 6l',;rlrlt's, a sub-jcct that has received rnuch attcrrtiorr l(r-(r0, 6-(12, (r-(r-5 1 but is,'rrl:;rrlc thc scope <lf the present discussion.

{i 4 TORSIONAL DIVERGENCE

llrt' phcn<lmenon of torsional divergence was at first most closely associatedrrrrlr :rircraft wings and their susceptibility to twisting offat some excessive air',g','t'tl. 'lir fbrm a conceptual picture of what occurs in such a situation, consider;r tlrrrr uirfbil, or any other analogous structure, such as a bridge deck (Fig.(r 'l l). Under the effect of wind, the structure will be subjected to, and willir( r l() rcsist, a drag force, a lift force, and a twisting moment. As the windi.hrt ity increases, the twisting moment in particular increases also. This intrrrrr (wists the structure further, but this condition may also, by increasing the,llctlivc angle of attack of the wind relative to the structure, further increasetlr. twisting moment, which then demands additional reactive moment fromrlrr' :;lnrcture. Finally, a velocity is reached at which the magnitude of the wind-rrr,lrrt'r'rl moment, together with the tendency for twist to demand additional'.rrr( lurlll reaction, creates an unstable condition and the structure twists toil':,rrrrc(ion. The problem is one of stability, quite analogous in a structural",'rr,(' to column buckling. Just as column buckling occurs when a critical, r 'lrrrrn load is reached, torsional divergence occurs at some critical divergencer.l't'iry of the wind. The phenomenon depends upon structural flexibility andtlrr' rrIrnncr in which the aerodynamic moments develop with twist; it does notrL 1rt'rtrl upon ultimate structural strength.

lrr tlrc case of thin airfoils, the aerodynamic twisting moment increases withrrr, rt'irsccl angle of attack. In other, more complex structures, it may be that

I I Ali|(iAX I'J

I"l(illlllt) (r.'1.1. (it'otttt'lty irtt{l l)it!;trttr'l('t', l'r l.r:.i,rr:rl rlrvt.rllt'rrt.t.prrrlrlt.rn

244 At ti()t ln:;ilo t,l il N()Mt Nn

the acnldynarnic twisting nl()nrcnt rhrtrs ttot lirllow tlris sirrrplc lr.:ttrlettcy. As itresult such structures may not lirllow tlrc pirltcrrr clcscribcd ahovc; in lirct,depending upon the relation bctwccn acnltlynarrric rnomcnt and anglc of attack,some structures may be immune to torsional divcrgcnce. Finally, it should benoted that in most cases of practical interest in civil engineering the criticaldivergence velocities are extremely high, well beyond the range of velocitiesnormally considered in design.

6.4.1 Analytical Modeling of Torsional DivergenceTo analyze the torsional divergence phenomenon, consider, as in Fig. 6.4.1,the section of a structure that can rotate against a torsional spring about somepivot point (or elastic center). Let the spring constant and the angle of rotationbe denoted by k" and a, respectively.

Assuming that the mean wind velocity is U and that the deck width is B,the aerodynamic moment per unit span can be written as

u": |pUzB'Cr(o) (6.4.1)

where Cy(a) is the aerodynamic moment coefficient about the twist axis. Anexample of the dependence of Cy upon cv in the case of an open truss bridgedeck is shown inFig. 6.4.2.

At zero angle of attack the value of this moment is

M,(O) : )pU2BzCro 6.4.2)

where Cye : CyQ). Fora small change in c away from a : O, Mo may begiven to first approximation by

(6.4.3)

o (DEGREES)

FIGURE 6.4.2, Monrcrrt cocflic:icnt lirr a blull'stnrclurc irs ir littttlion ol'irnglc olattack.

M,:;pu'Elr^ * "]dCrldo l":n

li4 l()l t!;lrtNAt

lrrlrrlrlittg lhc lrcrrttlyrlrrrrit'lo llrc irrlc:r'nirl slnrt.lrrr.lrl rrrorrrt.nllt()lt

l)lvf n( it N(:l 245

It'irrls lo llrt' t't1ur

\pU)ll1cr,, * (t,izrfyl A,,rv 1(r..t..{ )

rvlrcrc

-t aculLMo - , Ida lo=o

(6.4.s)

llrc divergence problem is summarized (in this two-dimensional description)lry liq. 6.4.4. We now examine its solution.

l)cfine )t : )pu'zE. Equation 6.4.4 then becomes

(k"-)\C'Mo)o:XCys

l'lrc solution of Eq. 6.4.6 for cv approaches infinity (diverges) for the value

\Crok" - >\Cho

.k"A:- C,,

l lris therefore defines the critical divergence velocity:

t,pl l) d {'n,1,r) - k,,rr

(6.4.6)

(6.4.7)

(6.4.e)

(6.4.8)

'l'hc problem may readily be generalized to three dimensions, but this isrt'scrvcd fora specific application in Chapter 13 (Sect. 13.1.2).It should alsolrt' noted that the problem considered here is that of incipient instability only.ll rrurre complex structural action with increasing velocity occurs (due to arrrrrrc complex curve of Cy vs. a, for example, than that shown in Fig. 6.4.2),tlrc rlivcrgence problem can be solved by a systematic solution of the relation

Iol tttty titngc ol'vclocilics rlt'sitt'rl= 'l'lrc prlsrril ol this pr<lblcrrr is l-rcyorrrl lhc:rirrr ol' this scction-

246 n I I t()t:t Ati I l(; I 'l ll N( )Ml Nn

6.5 FLUTTER

One of the earliest aeroelastic oscillaliorrs lo bc rcc<lgnizcd was thc lluttcr ol'airfoils. The term "flutter" has been variously uscd; recently, htlwcvcr, thisuse has become more restricted. The most common present uses of the termemploy additional qualifying terms, for example, classical flutter, stall flutter,single-degree-of-freedom flutter, and panel flutter. All of these terms wereoriginally employed in aerospace applications, but some have carried over towind engineering.

Classical flutter oiginally applied to thin airfoils. The term also finds ap-plication today to suspended-span bridge decks. It implies an aeroelastic phe-nomenon in which two degrees of freedom of a structure, rotation and verticaltranslation, couple together in a flow-driven, unstable oscillation. Coupling ofthe two degrees of freedom-indispensable for thin airfoil flutter under normalstructural circumstances-has come to be the identifying sign for classical flut-ter.

Stall flutter is a single-degree-of-freedom oscillation of airloils in torsiondriven by the nonlinear characteristics of the lift in the vicinity of the stall, orloss-of-lift condition. This phenomenon can also occur with structures havingbroad surfaces that can stall depending on the angle of approaching wind. So-called "stop-sign-flutter," the torsional oscillation of traffic stop-signs abouttorsionally weak posts, is an example in a nonaeronautical area.

Single-degree-of-freedom may include stall flutter, but may simply be as-sociated with systems undergoing strongly separated flows. Bluff, unstream-lined bodies are typical examples. Prominent among these are the decks ofsuspended-span bridges, which can in various instances exhibit single-degreetorsional instability. These are discussed in more detail in Chapter 8.

Panel flutter is a sustained oscillation of panels-typically the sides of largerockets-caused by the high-speed passage of ait along the panel. The mostprominent cases have been in supersonic flow regimes and so have not appearedin the usual wind engineering context. Flutter of taut canvas covers and flagflutter are, however, phenomena related to panel flutter.

It is likely that, in its detail, flutter in practically all cases involves nonlinearaerodynamics. It has been possible in a number of instances, however, to treatthe problem successfully by linear analytical approaches. The main reasons forthis are two: First, the supporting structure is usually treatable as linearly elasticand its actions dominate the form of the response, which is usually an expo-nentially modified sinusoidal oscillation. Second, it is the incipient or startingcondition, which may be treated as having only small amplitude, that separatesthe stable and unstable regimes. These two main features enable a flutter anal-ysis to be based on the standard stability considerations of linear elastic systems.

It is characteristic of flutter as a typical self-excited oscillation that a struc-tural system by means of its deflections and their timc clcrivativcs taps <lll'energy from the wind flow. lf'thc system is givcn an irritiirl tlisltrrbancc, itsmotion will cithcr rlccay rlr tlivcrgc (i.r:., its oscillllirlns will lrt'tlrtrrt;rt:tl rtr will

r;f, tlilil|t 241

1'row itttlcrlittilr.:ly) itct'onlrrrg lo wlrcllrcl llrt'cnctJ'.y ol rrrolton t,xlr;rclt.rl lrolntlrt'llow is lcss llr:rtt ot'cxccctls tlrc crterrgy tlissipirlt.rl lry llrr'sysl(.nr llrrorrg,lrrrr,'.'lt:ttticltl dalnping. 'l'hc thcorctical clivitlirrg lirrt. lrt.lwr't.rr llrt' tlct':ryirrg :rrrrlrlrvt'lgcttt cascs, nalllcly, sustaincd sinusoitlal ost'illirliorr, is tlrcrr r.r.rt.1lgrrizctl :rstlr,' t'ritical fluttcr condition.

lrr llro treatment of flutter, in the prcscnt wirrrl cngirrccrirrg corrlcxl, orrly, l,r:;sicll lluttcr and single-degree-of-fiecdorn llr-rttcr will bo cliscussccl.

|i 5"1 Equation of Motion for an Airfoil or a Bridge Deck('r,rrsitlcr a section of an airfoil or a bridge deck (Fig. 6.5.1) subjected to the;rr rr()n ol'a smooth oncoming flow. The section is assumed to have two degreesr'l lr.trtf<rm: bending displacement and twist denoted by h and cv, respectively..\ rrnil span of the system has mass tn, mass moment of inertia 1, static un-lr;rlrurcc s (equal to the product of mass m and a distance, ab which separatestlrt' t'r'rr(or of mass from the elastic center),* vertical and torsional restoringlirr(('s characteized by spring constant c1"and c", and coefficients of viscous,l;rrrr1rirrg c1,&ndco. withthesedefinitionstheequationsofmotioncanbewrittenlt, ()(r. 6-67J

mi+sa+coh+Cph:LtS1;+ld]-coarCoa:Mo

(6.5.1a)

(6.s.1b)

0rB4

)*-

l,'l( jl lltl,l (r.-5.1. Ntrt:rtions

'Nllt llr;tl willr lr lixt'tl sign (onv(.nliotr, ,\ trr:ry lx. ;rl,rlrvr. {}t n(.},,:tliv(.tlc;.rltlilrg 9rr llrc l6clt(i9rtrl,r\\,;il(l 0r ttll) 0l lltt'et'ttlt'l ol rrltss $tllr rr..,;r,tI lo llrr.t.l;rsltr.r.t,rtlt.t.

248 nl lt()l lnlill(; l'l I

where L1, and M,, arc:, rcspcc(ivcly, (lrc st:ll-cxcitul lrcrorlytrirntic lili irntl ttrrl-ment about the rotation axis pcr uttil sprttt. l)csignating by ru thc radius ol'gyration of the body about the centcr o1' rotation and using notations sinrilar trlthose of Sect. 5.1, Eqs. 6.5.1 become

mlli + ad t 2(6a6h + af,t4 : ro (6.5.2a)

(6.s.2b)+ & + 2l,a.a :Mo

whcrc f,,, f,, arc damping ratios-to-critical,licc;ucncics in h and a degrees of freedom,

and c,.r6, @q are the natural circularrespectively, defined by

(6.5.3a)

(6. s.3b)

In the case of bridge decks that are symmetrical, the center of mass lies inthe vertical plane of the centerline. In this case a : 0. Usually the rotationaxis lies in this plane also, though it may be at some vertical distance from thecenter of mass. In the case of bridges with arched decks the effective rotationaxis may lie well below this center. When accounting for the dynamics of thedeck, the mass moment of inertia 1 is calculated above the effective rotationaxis and hence is typically, even for a uniform deck, a quantity that variesacross the span. Actual determination of the effective rotation axis is a structuralproblem outside the scope of the present discussion.

6.5.2 Aerodynamic Lift and Moment

In the case of thin airfoils in incompressible flow, Theodorsen [6-66] showedfrom basic principles of potential flow theory that the expressions for Lp andMo are linear in h and a and their first and second derivatives. The coefficientsin these expressions, referred to as aerodynamic cofficients, are defined interms of two theoretical functions F(ft) and C,(k) 16-661, where k : balU isthe reduced frequency, b is the half-chord of the airfoil, U is the flow velocity,and r,r is the circular frequency of oscillation. The complex function C(k) otwhich F(k) and G(k) are the real and imaginary parts, respectively, is knownas Theodorsen's circulation function (Fig. 6.5.2). For aircraft flight regimesin all velocity ranges, wide research has developed further analytical expres-sions for all necessary aerodynamic coefficients. There exists a vast literaturcon the subject, to which [6-67] to [6-701 and [6-951 arc usclirl in(rocluctions.Attention is confinccl hcrc lo thc low-spccd incomprcssiblc lkrw n'giltrc.

+ r3rlf a..Il-hlr;

,cn@n: -m

,Co,": i

I r,ilt il 249

300

*=+l''l(JURE 6.5.2. Real and imaginary parts of the Theodorsen circulatory function c(K)tt(K) + iG(K).

'fhe theo,retical expressions for sinusoidally oscillating lift z and moment Morr :r flat plate airfoil are, respectively:

Lr: -pb2((Jra t rti - rbait) - 2rpC(k) lUcx + h + n1| - Ocj(6.s.4)

Mo: -pbz {"(l - euba + rb2([ + a2)ix - arbi]+ 2pUb2r(\ + a)C(k) [Ua + h + U1] - dc1 (6. s. s)

wlrcrc

c(k):F(k)+ic(k) (6.s.6)

I' balu is the reduced frequency, <,r is the oscillation circular frequency, brs crlual to Bl2, B is thc chorcl of the airfoil, p is the air density, u is therr;rlrnrach laminar flow vckrcity , ttlt is thc distance from the midciord to ther.lirtion point, ancl rr antl /r rrrt.. n's1'lcctivcly, angular rotation and verticaltlislrlaccrncnt, l6-661, l6-671. 'l'lrt' firrrcri.rrs /'Ik), G(k), are shown in Fig.(r.5.2.

lior blull'objccls ol'wirrtl (.nl1ut('(.trrr1, irplrlit':rliotrs, it has n<ll l<t tlirte ltcerr

*J

25O AERoflntillo l'lll N()Ml NA

possible to develop cxprcssi<lns lilr thc ircrotlytuttttic cocllicicrr(s stltrting I'rtltttLasic fluid-flow principles. Howevcr, it has bccn shown in [6-7 ll that litr srnall

oscillations the self-excited lift and moment on a bluff body may bc trcatcd as

linear in the structural displacement and rotation and their first two derivatives,and that it is possible to measure the aerodynamic coelficients by means ofspecially designed wind tunnel tests. Such tests indicate that just as in the case

oi the airfoil the aerodynamic coefficients of a bluff body are functions of the

reduced velocity.Various forms for the linear expressions for L1, and Mohave been employed.

Thc classical theoretical (and some experimental) work has used complex num-bcr lirrms based on the representation of the flutter oscillation as having the

complcx fbrm ei''. However, in the wind engineering practice developed totlatc in the Unitcd States real forms have been employed. Below are statedcommonly uscd lincarized forms of this type [6-71]:

Ln:

(6. s.8)

where additional terms in h are included and the reduced frequency K is definedas*

lpu'nfraftrl Lr+ xuitxrui * xznl(x)a + K'Hf *)(6.s.1)

M,: )p(JzP[*fto Lr+ rc$rxrui * xz,s,t(x)a + K'4*]

Ba B(2rn\K: U: U

B is the chord, deck width, or along-wind dimension of the structure, U is theuniform approach velocity of the wind, and <o is the circular frequency ofoscillatioL(i is the frequency of oscillation). In Eqs. 6.5.7 and 6.5.8 terms inii, Ahave been omitted as being of negligible importance in wind engineering.(ln aeronautical practice terms in ti and it but not h areretained.) The coeffi-cients II,t and Af (l : 1, 2, 3) ate nondimensional functions of K' Thequantities a, hlu, and BalU are effective angles of attack and therefore alsonondimensional. The typical term in Eqs. 6.5.7 and 6.5.8 can be viewed as

following the classical pattem of expressions for aerodynamic lift force perunit span, such as

*The reduccd frcqucncics li, usotl in ucnrnuutical practicc, and K, ttscrl ilt witttl ctlgittt:crirrg' dillctin that ft is dcfinccl in tcrrrrs 0l tlrr: hull.chortl lt - Ill2, whctcits li)r l(':ls()lls ol toltvt'trit'ttt'c K is

tlolinctl in lontts ol thc lirll cllrll lJ, its irt li1. (r'5.().

(6.s.e)

lol surall anglc ol'attack a. Formally, (cnns suclr lrs K// jr' or' A'il I lrrc lhrrsrrturlogtrus to lift coefiicient derivatives tlC1,ldu.'l'lrcsc lcrnrs slrorrltl bc re lcrlctlItt as vnrr|loral derivatives, however, and thcy go ovol' into stoacly-stalc rlcriv-;rlivos, such as dC1.ldu, only for K - 0 (zero liocluorrcy). Irnrrn an cxpcrintcntalpoint of view this means that the aerodynamic cocllicicnts of Eqs. 6.5.7, 6.5.8, :ur be measured only if the body is in an oscillatory state, whereas dCylda isohlrrined under static conditions (i.e., with the body fixed; see Sect. 6.2). Thel:rt'trrrs K or K2 preceding Hf and,4f could just as well be included with theselrr(lcr in a total coelficient of some other designation if desired, but the evolutionol lhc theory [6-71] has identified them as nondimensional factors. References| {r 7 I I through 16-771discuss various experimental techniques used in the United /Stir(cs, Japan, and France for obtaining the nonstationary aerodynamic (flutter) '

tlt'r'ivatives. In France, through usage at ONERA (Office National d'Etudes etrlt' llccherches A6rospatiales), the following alternate forms [i3-45] have beent rst'tl:

L t,1t(t'I)('t =

t,ttll'll'l(,' ,,rl,v

h1- m'l , -l m'6a

DA

|]r, IIiltiltt 251

((r..5.lOl

(6.s.11)

(6.s.12)

Lr" :

Mu:

.ur'postulated:

K)Hf:*2rk']:-2rKF

o:**ki,a+ol::.\D@ e/"/ h-rpu'b \0, t *../ h-rpU'b' \*I U

d\+m'i-l@/

rvlrcrc the coefficients kL, k';, etc., have come to be called the "Kiissner coef-lrt'icnts. "

ln real terms, the following equivalences among the coefficients of the aboverxlrcssions may be verified, when oscillations

h : hoei''ioiOt : d6€

(6.s.13)

(6.s.14)

K)rr! -K)H{ :

h')rt.l'-,

-rKl 4c /t \ I-rk'i: ; lt,o*r\;-')r) (6516r

(6.s.1s)

(6.-5. r7)

(6.5. lr()

-trki,: -*1r,, (j ,,) ,,n t "ry:

zr ,l l(;l]rkl. A lll I" I | ^l

AEROELASTIC PHENOMENA

KzAf : -rm'l

K2,1,* : -n ^'l'2

K2A{ : t *t

: trKF (i . ,) (6.s. re)

:;1-;t -,o (". ;) . *o (,, - i)](6.s.20)

:;lt (* . *) . ,. (" .;) + KG("' - ;)l(6.s.2t)

KzA; : -trm'o -;l+ + 2KG(, . ;)] (6.s.22)

In case a : 0 (common for bridges, though not usual for airfoils in aircraft),the above equations reduce to

(6.s.23)

(6.s.24)

(6.s.2s)

(6.s.26)

(6.s.27)

(6.5.28)

(6.s.29)

(6.s.30)

K'H7: -2rKF

KzHt : +lt *f, * ,)K'H{:-"lro-TlK,Htr:;"I'-T]x,ef : | rcn

K2AY:;l;-"-T)rc2t{:llt*"_{91- 2L32 4)x''qf : I wo1

Sample experimental values of the coefficients r1f and Af for streamlinedbridge sections are shown in Fig..6.5.3, where forpurposes of comparison theanalogous coefficients Hf and,ef rcr a thin airfoil are also given.

L,l0,80400

-04

543

i0I

5

2

1

0

a 2 4 6 B rA1214l.i /nB

a 2 4 6 I 101214U/nB

x

0/

_B

-12-16-20

4

2

0

-2

20

-?4b

_B

-10

I

D2 4 6 8 tD1214U /nB

-0. 0

-0. 4

? og-L.2

1.6 a 2 4 6 B 10 12 14

U/nB

XX

w\^

0 2 4 6 B t01214lllnB

a 2 4 6 I 101214 0 2 4 6 I ID1214U/nB Il/nB^ NORI\IANDY } TSURUM I''+_- CREAT BELT _€_- AIRFOIL ( EXPERIMENIAI I

(a)

lrlGURE 6.5.3a. Aerodynamic coefficients trf and Af for a thin airfoil (i : 1,2,3)rrrrcl three streamlined box decks (i : l, 2,3,4) shown in Fig. 6.5.3b. After [13-109].

8.5.3 Solution of the Flutter Equationsllccause of $gge_pendgncg 9{ thg aerodynamic terms up,gn K, the analyticalrlrlution .ql-lbg.-fluJte"f problem becomes more involved ihan ttre compai?btestubility solutions where quasi-steady aerodynamics holds. under K-dependentcrunditions, a typical solution method is as follows. A value of Kis choien andthc values of r1f and Af conesponcling to rhut value"iie obtained from plotsol'these experimental functione, It is then nssumccl that h and cv have soluiionslrnrportional to €i'r which arc inserted lnt() Eqs, 6.5"2,6.5.7, and 6.5.g, The

.!

fM. rr llr /

,!^\r.."- .

{

\*

tli!

-:t l,-

.4 [_0

254 Al ll()t lAiiilo I'l il N()Mt Nn

TSURUMI FAIRIA/AY BRIDGE

31000

GREAT BELT EAST BRIDGE

AIRFOIL(b)

FIGURE 6.5.3b. Box decks for three bridges (dimensions in millimeters), and airfoil.After [13-109].

determinant of coefficients of the amplitudes of h and o is then set equal tozero as the basic stability condition. This constitutes in fact u .ornpl"" quarticequation in the unkno*n flutt", frequency c,r, which must then be solved. Thesolution obtained will. in general. be of the lorm o : .,r * rc,.r2 with u2 * 0.and will therefore represent either a decaying (r,lz > 0) or a divergent (co2 (0) oscillation. A new value of K is then chosen and the procedureis ,"p"ut"duntil the solution_is-purely (or very nearly) imaginary, tirat is, until <,r, = Q,so thal Q = @r. To that solution lhere corresponds th! flulter condition at realfrequency co,. Let l(, be the value of K for which @ : @t Thc critical fluttervelocity is then

:]6000 -.-....-

NORMANDY BRIDGE

ri', IIt,tilil 25b

Il.,l,Il. . 1{r.5. tl )'A

Ilcrc:ausc of its intsrost in applicatiorrs, u usclirl vlu'itrrrl on thc solrrlion orrllrrrctl abovc is skctchcd bclow.* Let

Uts:-B

l,r' rr n<lndimensional time (or distance). Noting that

o:T:#f,:, )'vB

lrtlrrrrlions 6.5.2 and 6.5.4 can be reduced to

It" - h' .h oB2 l h,tr | 2(nKnE + Ki;: Tlorr ; i KHla, + x2H!a

,t" + 2loKoa' + K2.u : KAta' + K2Ala

rvlrcrc K6 : BallU, Ko : BaolU.l'osing now the solution forms

4l *nrL +II B

(6.s.32)

(6.s.33)

(6.5.35a)

(6.s.3sb)

!:bri't -honix'B B. Ba : &oei(t'* O) : uoei't : o4eiKt

+ x'nr L]- Bl(6.5.34a)

+ K,4 *f(6.s.34b)

Ki-4(iK2Hf+x'uttl\'' m -'lBoB2 .l+'- K'HT

lcvg : 0 (6.5.36a)

lrr;rrirtions 6.5.7 take the form

I

| -,r' + 2ifhKhK +I

f oB2- l; iK'HI

il lrrlrkt' ltitctitlt wirtgs, britlgc tlt:t'ks ttt:ry cx;r'ricrrt t' sigrrilic:rrrl swly (rnotion akrng thc dircction,,1 llt('(lriUl lirrcc).'l'lris is txrl litkctt ittlo irtcorrrrt rrr tlris st.r'liorr, lrrrl scc lit;s. 13.1.43.

256 nLnoLIn:;ltC t,ilt N()Mt Nn

l-+ eKzA{ + FAi'nd :,,q!-4r,n1l*o-u-i'^ - r -l

Deflning an unknown X as

I K:,

(6.5.36b)

(6.s.37)

and setting the determinant of Eqs. 6.5.36 equal to zer-o results in a complexrpolyndriiial in Xof degree four. This breaiii down inio-*two real equations,assuming that X is always real at the flutter condition. These two equations are

' solved successivcly fbr different assumed values of K, and their roots X areplotted as functions of K. At the point (X,, K,) where the two plots cross,.theflutter condition is identified [6-66, 6-671.

The flutter problem as treated above is seen to be a semi-inverse oroblemr srqle-the aerodynamic coefficients are functions of the solutioir fiEffil;-unOI a range of frequency parameters K m11st therefore be used to survey the solution, region.

Altemate methods are also available, though they are beyond the scope ofi the prbsent discussion. One of the more important of these approaches involvesthe use ol aerodynamic indicial funcrions [6-671 ro 16-701 and t6-78. 6-79].Such functions, derivable from the coefficients H! and,4,I , represent the re-sponse of the bluff section to a step change in angle of attack. They also permitrepresentation of transient response problems under the general hypothesis thatlinear superposition of effects remains valid. Reference [6-80] makes use ofindividual response functions in predicting bridge response under natural wind(see also Sect. 6.6). In general, the use of such functions gives rise to moreinvolved calculations than the stability determinant method sketched above.Avoidance of the more general indicial function approach is justified in thosecases where structural frequencies and natural modes are not greatly altered bythe aerodynamic forces.

- / the fluttei equations and the nature of the flutter phenomenon in the case ofbridges as opposed to that of airfoils. In the flutter of airfoils under normalstructural conditions (center of mass not excessively far aft of the rotation point)it is impossible for single-degree-of-freedom flutter to occur since both degreesh and u are individuilly positively dampedx (i.e., Hf and A! are negative for

*Because of the formal similarity between the mechanical damping terms in the left-hancl si6csof Eqs. 6.5.2 and the terms containing the coefficients af and;f in eqs. 6.-5.7 and 6.5.g, tholatterare referred to as aerodynamic damping tcrms. The diflbronccs 2(j,o,,rrr \pUr@)XUl anl2(.u,,1 - |oUt{n'1X'l'{{U {,/) arc rclcrrccl to as nct (or total) tlarrrpinll irr tlrr.tlrlrslirtionirl untl tlrt:rotational mrxlc, rcspcclivcly. (Scc also Scc(. (r.2.l.)

(,x--{n1

(i {i l}t.,t ll llN(i lil :;l'oN:,1 tN illl I'1il l;l NCI ()l nl ll(tl lA!ill(. I'lll Nr}Ml }!n 257

;rll valucs ol'K). 'l'his is tlrc birsic rcirsort wlty t'lirssit'rrl ;rirloil llrtllr'r', il rrrrtlvrlrorr it occurs, rrrrrst irrvolvc couplcrl llcctkrttts; llrirl ts. ll nnrsl lrr'ir torttltliottrrr which it is mainly thc coupling (not tltcr tlrrrrrlrirrg) lct'ttts llrrl liovt'nt lltt'I('sl)OnSC.

On theotherhand, as shown in [6-7 Il, ccrtairt ty;lcs ol'struclurc (c:.g., sotttc:opcrn-truss s_uspension bridge decks) exhibit,rl-j (torsiorrirl tlarrrping) cocf licicrrlstlur( change s!gn-from negative to positive with aclvancing values ol'rcduccdwirrcl velJcity*UlnB (where n : itZn). As a rcsult whether or not coupling "'lt ocllicients exist, single-degree torsional motion becomes unstable and drives;r sclf'-excited flutter due to its net negative damping. Thus purely single-degreelf trttcr, or "single-degree-driven" flutter, can exist for cases where Af evolves.rs tlcscribed above.

'l'he flutter of three-dimensional structures is essentially based on the two-tlrrrrcnsional theory presented above and is discussed in Chapter 13.

6.6 BUFFETING RESPONSE IN THE PRESENCE OFAEROELASTIC PHENOMENA

Itull'cting is defined as the unsteady loading of a structure by velocity fluctua-Irorrs in the oncoming flow. If these velocity fluctuations are clearly associatedrvith the turbulence shed in the wake of an upstream body, the unsteady loading ,

rr lclbrred to as wake buffeting. Effective analytical models of the wake buf- 1

It'tirrg phenomenon do not currently exist in the wind engineering field. On theorlrcr hand, notable contributions [6-82] to [6-85] have been made to the prob-.k'rn ol the bu.ffe-tjgrg of_linelike stru_qjllres by atmospheric turbulence. Many ofllrt' icleas employed below can be traced to origins in these references.

'l'hc problem dealt with in this section is that of buffeting by incident tur-lrrrlt:nce that develops in an atmospheric flow over relatively homogeneousrr'r ririn-open, suburban, or urban. For such turbulence it_is possible, in certainr :rscs, to

"Sp-q_gl _!hg response to buffeting forces for .bo,th those structures thatrkr not andihose that do exhibit aeroelastic interaction"with the wind forces.'i'r'tion 5.3' deals with aerodynamic loadings that are independent of structuralrrrotion. However, structures like slender towers or the decks of suspended-,,1r;ur hridges, which exhibit aeroelastic effects, are also of considerable interestrrr prlctical applications. The present section is concerned principally with ther{'slx)nse of such linelike structures.

ti-6-1 Aerodynamic Forces on Linelike Structures('()nsi(lcra linclikc structurl:. willt sprutwise: r'rxlrrlinatcx, that is being buffetedlry rrtrrrosphcric turbuloncc. ll tlrc ost'illirliorrs ol'thc stnrcturc in each respondingrrrorlt' itrc srrrall, il rrriry bt: irssrrrrrt'tl tlr;rl tlrt' rrcrotlyttatnic bchavior of thc',lnr('llrrLr is linciu. 'l'hc: ltcrorlyn;uruc lort't's t'orrsisl ol'rr strpcrposition <ll'(l)

258 At n( )t t Al; nc I'l il N( )Mt NA

self'-excited lirrccs ol'llrc tyllc tleralt witlr irr Sct't. (r.-5 irrrtl (2) brrlli:lirrg lirrcc:sinduced by the incidcnt turbulcncr:.

Bufteting Forces. For turbulence intensities typical of winds in thc atnro-spheric boundary layer, and for turbulence components with fiequcncics thatare of interest in practice, it 1ryy*.pe .assumed that the squares ancl products ol'the velocity fluctuations u. u. and, w are negligible with respect to the squarcof the mean velocity U and that the force coefficients Cp , C1., and, C11o arcindependent of frequency in the'?nge considered. As a result expressions forthe buffeting forces based on quasi-steady theory are acceptable, so that forscction ,r of the span the buffeting drag, lift, and aerodynimic moment (secFig. 6.6.1) can be written as

u(x- t\l+2U

* , u(r. tlfulM(tl I lr l I tt(v t\ I sr |

(6'6' lb)

Fih : lr^*, + cD(ao) #ll' *, ryfl. *1,=,,*f(6.6.1c)

where B is a typical body dimension such as deck width, ,4 is the across-windarea per unit length projected on the plane normal to the mean wind speed u,

D(t)6NTE

-L(r)FNTE

: c,r(oi f: c.r*o [r *lelI dal- 'd:q0

(6.6.1a)

**",a]ry

Th

Lft)I

-a-Ir.o.

c.m. = MASS CENTEROF SECTION

r.o. =EFFECTIVE ROTATTONAXIS OF SECTION

FIGURE 6.6. l. Bull'cting r.rccs 'n sccri.n .r'. li'clik. srnr(.rrrc.

r;{; l}t,l llllN(i lilt;l'ol.l:,r ltJ llll l,l rl til N(;l ()l Al ll(,ttA'iil{ I'1ilill)Mt flA 2lio

r rs lltc tlistrtttcc ol lltc tlt't'k nutss ccrltcr lo lltt't'llt'tlrvr'rol:tltorr;rrt:.. l/ I

r.{/)iul(l rr(/):rrt: tlrc wintl sPct:tl corttlxrttt:rtls irt llrt';rlorq;' wrrrrl ;rrrtl llrt'vt'r'l rt';rlrlnt't'(iotts, l-cspoclivcly,'r'iuttl rvly is lltc rrtcirrr ltrrglc ol itllirck rrrrtlcr wrrrtl irtlrorr.lrr li,t1s. 6.6. lb artcl (r.(r. lc thc dintcnsiottlrrss rtrlio rr'(l)/l/ rr'prr':i('nls :tn ;rr111rrl:rr'llrrt lrlrtion l'rom thc mcan anglc o1y. ln lic;s. (r.(r. l , lhc tplrrrtity l l I )rt(tll I llr', olrtrrined by squaring the sum ll + u(t)ll.l Ilrrrtl negkrc(irrg lltcr st;trirrrr ol ils,,,'. on(l tctm, as shown in Sect. 4.7.

Self-excited Forces. lt was indicated in Scct. 6.-5 that fbr a body oscillatingu,rllr circular frequency o in both the vertical displacernent and the torsionalrrrrrtlcs, the self-excited lift and moment L1, and Mo may be expressed as inl r1s. 6.5.7 and 6.5.8.

Since the random buffeting load action on a structure may be viewed as a'.rrptrrposition of elemental harmonic loads (see Appendix A2), the vibrationsnl llrat structure may, conespondingly, be viewed as a superposition of har-rrronic responses induced by these loads. Each such oscillation induces, in tum,.rrr clcmental self-excited load expressible by Eqs. 6.5.7 and 6.5.8.i

{i.6.2 Buffeting Response of a Suspension Bridgetl'()r nlany types of bridge deck sections the aerodynamic coupling coefficientsrrr l,)qs. 6.5.7 and 6.5.8 may be disregarded in first approximation as havingrun()r or negligible influence, so the vertical and torsional motions of a straightlrrtlgc may be taken as uncoupled. The aerodynamic coupling coefficients are,,1 socondary importance particularly in those cases of common occurrencervlrt:rcin single-degree torsional instability is manifest (i.e., where,4f changes',r1in with increasing UlnB).

lixpressions forthe bridge response will now be sought following a proce-,lrrlr closely parallel to that employed in Chapter 5 to study along-wind re-',lx)nsc. Here, however, the effect of aerodynamic self-excitation terms will bet:rkt:n into account in addition to the aerodynamic buffeting forces.

'lirrsion will be dealt with first. Consider a full bridge for which the torsional

rl (luirtions 6.6. lb and 6.6.1c are written assuming that the linelike structure is horizontal (e.g.,.r I'ritlgc). In the case of a vertical structure (e.g., a tower), the vertical velocity component w(r)rrrrrsl bc rcplaced by the lateral velocity component z/(t).'\n crluivalent altemative fomulation is to employ the aerodynamic indicial function approach

l(' /ll. 6 79, 6-80, 6-971 wherein the frequency-dependent information contained in the self-, \r'ilrli()n acrodynamic coefficients n! and,4f is first converted into time-dependent indicial,r,'trxlyruttttic lunctions and the aerodynamic forces are then expressed in terms of an integral overtlrr' plxluct of an indicial lunction and thc structural motion. This approach, typically employed!r l,ttsl ttsponsc stuclics lirr aircrrrli, usually lcarls to cxplicit time-history calculations, but these,rrr'ltvttitlctl in thc prcscrrt con(cx1. llctc lirrtc tlcpcntlcnl lilrrnulations will be transformed into'.ptt lr:rl, ol lrcrlrrcrrcy tlcpcrrtlcrrl, tlcscriptiorrs ol rt.sporrsc :rrnplituclcs.Iltts ptrrbk:rtt is llclrtctl rrrolt: y1t'lrt'r:rlly ilt Sr'tt. I i.1.,1.

260 n f nol Asi I t(: I 'l u N( )Mt NA

response at any spanwisc soclion .rterms of generalized coordinates as

a(tr, t)

is rv(.r', l). 'l'hc losp()nse ctrrr bc writtcn in

: \ ai(x)p,(t) rc.6.2)

where pr(t) are the corresponding time-dependent generalized coordinates ol.the problem and o;(-r) are the torsional vibration modis. The equation of motionof the deck section x is

I(x)ir(x, t) * c,(x)a(x, t) * k"(x)a(x, t) : JfL(x, t) (6.6.3)

whcrc /(r) is thc local mass moment of inertia of the deck about the ell-ectiver.l.ti'n axis ancl r',,(x) and k,,(x) are, respectively, the effective structural damp-ing ancl stillhcss .l'thc sccti'n. To bring the generalized coordinates into thepnrblcm, Itq. 6.6.2 is used fbr .'(x, r) in Eq. 6.6.3. The result is then multipliedthrough by <r;(x) and integrated over the full span Z, yielding

Iipi(t) * 2(o,(2rno)bie) + (2rn,,)2p,(t)l : M,,(t)

where { is the generalized inertia

n : J, I@)a?@) dx

(i+j)

(6.6.4)

(6.6.s)

(6.6.6)

(6.6.7)

f-, and ndi are, respectively, the damping ratio and the natural frequency (Hz)in the ith torsion mode and Mo, isthe generalized force. Implicit use has beenmade of the orthogonality relation

ft)o l(x)a{x)a1(x) dx : O

The generalized force M.,(t) has the form

Mo, : J, *o, t)a,(x) dx

The attention of the reader is drawn at this point to the similarity between Eqs.6.6.2-6 -6 -5 and Eqs. 5 -2. r, 5.2.6, 5.2.7, and 5.2. g. Both sets iepict the usuarmodal approach to a dynamics problem in a continuous structure.In the present context the distributed moment per unit span will have both

self-excited and active, time-dependent components, the fomer associated withthe motion and the latter a function of the gust velocity c.rrrprlrcnrs in thcatmospheric flow passing <lvcr r.hc structure. Thc scll'-c,xr.it",l ..,,,,,1.,.,,rr:nrs (scc

t,r;

r;(; ltt,l ll llN(i lll i;l'()rl:;t rN llll I'lrl l;l N(;l ()l At lr()t tA!,il{ t,t il il{tMt NA 2Bl

(r.5.tt) will hc irssrrrrrctl lo lirke: tltr.: lirrrt+ (willr ,.1 I ttt

(6.6.11)

(6.6. r3)

( (r. (r. ll )

rvlrt'tc K : 2rnBl U while the time-dcpcndont gusl conllillution will lx: rlrrdorrrtr,cc lrq. 6.6.1c).

lk:lirrc applying the full random gust ntorncnt, lct a singlc sinusoidal com-;xrrrr,:rrl of amplitude Mo and frequency n bc applied at spanwise section x :r, 'l'lrcn the applied distribution moment is

Jlt(x, t) : M.(K) I Mg6(x - x1)cos 2rnt (6.6.e)

rvlrt'rc 6(x - x1) is the Dirac delta function (see Eq. 5.1.11), so the generalizedI.r('c, Eq. 6.6.7, becomes

M,,(K) )ou'n'lnnrror"r", , *',.rfrxr,, I

pLMo, : )nlU"tXl + Mg6(x - x,)cos 2rntlu,(x) dx (6.6.10)

E 4@ [^t t;t")*,f"1 a, : 4 Gijpj(t)j Jo " r

rvlrcr.c G;7 : I3 oioi dx and

I, ,U - xt)oq@) dx : a;(x1) (6.6.12)

l'lrt' lirst occurs in M.(K) and the second occurs in the single sinusoidal com-I rottr'ltt.

Sirrce the modes ai(x) are dimensionless and of arbitrary scale, it is conve-rrrt'rrt to normalize them arbitrarily, for example, setting

I f'; Jn aitxl d"x : I

I r:,t' ol' a(x, t) from Eq. 6.6.2 in Eqs. 6.6.8 and 6.6.10 implies thar calculationu rll bc required of factors having the form

'Wrntl lunncl tests performed by thc wlilcrs havc tcndcd to indicate that the destabilizing effect''l llrt scll oxcitcd ftrrces acting on a srtslrt:rrsiorr britlgc dcck is somewhat reduced by the presence,,1 lrrtlrttk:ttcc in thc incidcnt llow. 'l'hc rrst' in cirlcrrl:rlions ol'acrodynamic coeflicients H,t andl' ,rbl:tinql unclcr sm<xrtlr lkrw totttliliorrs is llrt'rclirrr lhorrghl hcrc to bc conscrvative. Modelr \lrliltl('ltls l(r-ll(rl crrtployirtg 1t:r'lrrrirprcs ol rrrrrLrrrr :rrr:rlysis ltavc shctl lurthcr light on thc cli'ect,'l lrrtl)ttl('tlco ttpott lltc vrtlttcs ol //,+ :url .1,' lirrll t'x;rLllrliorrs ol'lhc cllcct ol'lrpplr)priatcly'',,r1|rl ltlrlrttlt'ttt't: tttt lltc llrrllertlt'tivitltvt':, ol lr;1111'1' rlt'r'lrs rr'rrr:rilr lo lrc titlrictl orrt.

262 Arnotlnt; n(; I't l N()Mt NA

one may then note that G;; : L, but llurr. irr gcncral, lilr i *.i, rlrc valucs <ll'Gi1 ar1.much less than L.It will bc assurnccl hcro that Gii U +.i ) is nogligihlc,which is reasonable for bridges in which /(x) is approximately constant acK)ssthe span, as can be seen from Eq. 6.6.6. The net value of the generalized fbrccM*, then is

+ KzAtK)p,] + uoo,1*)cos 2trnt

"r,: n'-, - ,'l* nfrfrj|

T-,: *1r,,,., *t@nfEquations 6.6.16 and 6.6.17 introduce the effect of the aerodynamic self-excited forces into the response at frequency n.

Equation 6.6.15 (i : 1,2,3, .. .) is similarin form to8q.5.2.7 forwhichthe generalized force is given by Eq. 5.2.12.In Chapter 5 the system definedby Eq. 5.2.7 is analyzed under distributed random loading, leading to Eq.5.2.38. Completely analogous steps hold here, yielding the following resuitfor the spectrum of torsional response:

Mo,= p[JzB2Ll*ffnff(6.6.14)

Equation 6.6.4, which describes the motion of the ith mode, may then bewritten with use of Eq. 6.6.14 as

Iilpi|) + 2t,",(2Tfi,)b,@ + (2ili.,)2p,(t)l: Msai(x)cos2rnt (6.6.15)

wherc new cfl-ective fiequency rio, and, damping i", have been introduced suchthat

S,(x, n) = Ii

(6.6.16)

(6.6.17)

(6.6.18)

where sfa,^a'(n) is the co-spectrum of the buffeting moments M1 and M2 perunit span which act, respectively, at the coordinates x, and x2.

Equation 6.6.1c describes the applied aerodynamic moment per unit spandue to steady wind and gust components. In this equation, the moment anddrag coefficients Cyand Cp are functions of the mean twist angle as(x) at thespanwise section x, and the velocity components u and u are also functions ofx and time. For convenience the following notation is introduced:

Cyllus(x)l = Clalas(x) + C,,1u,,{.01 lrt,

"?ti [3 [! a;@)ai@)Sfr,u,@) dxr dxzt6rafi,t! {ft - (ntn")'fi4,qUfi

;tttt I

{; (; lJ(,t II ilN(i ilt :;t,()N:,t tN ilil I'tit l;l NCI ()t nl li()t In:;lt(; t'ilt N()Mt Nn 2ti3

"i':1,,,,(,(\) ('ialrvlv( r )l

l'lrrrs, rcf'erring to Eqs. A2.29 and A2.33 (Appontlix A2), tlrc nrorrrcnl cosl)cc-tnun between sections x1 and x2 may be writtcn

s fi ,,,{ n ) : | ), u, r, l' lo

r,,t *,,( r I tlC 1a s I .*s( r,rl \!'t 2C yefas(x )lC'M[ag(x)t 8#

-t 2C TaBfas(x)lC yfc,s(x 1), fi#

+ c',a[as(x )]c .ala.@)l 8#)

f-o'1x; : )n s*{x. nldn

liv:rluation of CTap and C'7a at values 11 and -r2 requires knowledge of the meanrlt'llcction distribution os(x) over the span. This can be obtained by a static.trrrly of the type discussed in Sect. 6.4.1 or may be described in terms of thelorsional vibration modes by the expression

os(x) : 2ltlpu'n'cr4to!"r)lcvi(xr) dx' - ,..'i 4T-n;,1,- o''(x\ (6'6'22\

rvlrich is a result derived from Eq. 6.6.4by neglecting all time-dependent terms.f lrr: solution of Eq. 6.6.22 for a given wind velocity u requires an iterative:rppnlach, starting conveniently with ao : 0.

lnEq.6.6.21 the co-spectra Srt*r(n) and lfi.,@) are negative in value and.rlrlrrcciably smaller in magnitude than lf,,,r(r) and Sfi,,(n); they may conser-vrrt ivcly be neglected.

'l'hc root mean square of the fluctuating torsional response at section x is

((r (r.lO)

(6.6.21)

(6.6.23t

I',"rrk values of the fluctuating torsional response may be obtained by following',tt'ps sirnilar to those of scct. -5.3. Mcthods of calculation relative to the quan-lrlrcs rlrcntioned abovc arc tliscrrssr.:tl in ('haptcr 13.

Il'tlro vcrtical (bcncling) r'eslx)nsc ol tlrc brirlgc is written as

h(.r.r) )J/r,{r)r7,u) (t l,2....)(6.6. t9)16 6.241

264 AEnofl Asltc t,lt N()Mt NA

where /z;(x) are the verlical bcnding rrroclcs ol'vibration arrtl r7; 1rc lhc gcncr-alized coordinates for these modes, thcn, by a proccss cor.nplctcly arraklgousto that described above for torsion, the spcctrum o1'the vertical rosponsc canbe shown to be

n?@ 13 [! h,@,1h,(xr)s?.,r,(n) ctxr ctxz (6.6.2s)Sn(x, n) = i r6ran|,tvt! {[t - (ntnp;212 + +yf,1ntrr;t]

where

m1x1hl1x1 dx

is the generalized inertia, m(x) being the deck section mass per unit length,n1r, the natural frequency* in the ith mode, and ;,,, the aerodynamically influ-enced system damping defined by

nLM:1,

(6.6.26)

where K : 2rBnlU and f1,, is the mechanical damping ratio in the ith mode.The co-spectrum of the time-dependent lift forces z1 and Lzper unit length ofspan, which act respectively at span points xr andx2, is (from Eq. 6.6.lb)

: . pB2L .,. n(n : (n, - ,M UfK) i,

sf,,,(n) : l* rr, 4' l+c,too{,1)l c1[cve(x)] 6#

t 2 C Tfc's(x 1)lCL6[c.s(x), fi#

* 2C yfag(x2)lC Ln[cxor, )t fi#-t C;s[c.s(x 1)]C Lr1oq(x), fi#)

where

(6.6.27)

A+ U Cpfuo@)l (6.6.28)C'yefas(x)l : dC,l

El":*0r.,

*There is no aerodynamic inllucncc in this casc upon the natural Irctlucrrcy, owirrg to lhc assumcrlabsence in thc basic rrxxlcl ol tcrnts in l.

{;t; ltt,lllllN(illliil'()l'Jl;l lt'J llll I'llllilN(:l ()l nllll)lln:,ll( l'lllt'l{rlvlltl^ 265

Ilrt' rrrcarr, nlciul s(luiu(' irrttl pclk vcrlicirl r'(:slxrrsr'b r'rrtt llrt'tt lrt' t rtlt'til;rlt'rl. irsrv:rs irrtlica(crl lrlrove lirr lolsirlrtrrl lL:sponsc, by lollownrll rlt';rs srtttillt lo lltrtst'ol Sc:c:t.5.3.'lo calculatc |hc ulong-winrl rcsponsc, contpletcly rulrloplorrs pttter'tlttres tollrosc abovc arc uscd, thc basic lbrcing lirrtcliorr bcirtg tltc tlnrg lrs givcrt by1,.t1. (r.6. la; a knowledge of along-wind vibra(iou rtttttlcs is irlso rctlrritetl.

6.6.3 Outline of the General Buffeting Response Problem ofLinelike Structuresl.r't the across-wind bending and torsional modes of a symmetrical lineliker,lrrrcturex be representedby hi@) and a;("r) as in Sect. 6.6.2, so that sectionaltlt'llcctions h and q (Fig. 6.5.1) under dynamic excitation are

h(x,t):Vh{x)q{t)

a(x,t):lo.{x)p{t)

(6.6.29a)

(6.6.2eb)

Analogously to previous formulations (Sect. 6.6.2) the equations of motion(rrrtrchanically uncoupled about the centerline) become

Mi[Qi + 216(2rnn)ei -t (2rn1,)zqil :

Ii[Fi + 2(*,(2rn.,)b, + (Zrn.,)2pil :

r.vlrcre S(x, r) and 5lt(x, /) are, respectively, the lift and moment per unit span:rl scction r of the span.

ln order to obtain the necessary system admittance functions, J(x, l) and:)lt(.r, /) are alternately specified in the following manners. For lift-associated;rr llrrittances,

S,(x, t) : Lt t l"ei2""'67x - xrl

5lL(x' t) : M"

I ior rnoment-associated adtnillitnccs,

tlror rrnsyrrrrrrctrical slrlrclurcs (,t / 0, lirl rr r 1), llr(' lr'('irlnl('nl is anakrgous, with aorodynamiclort'c rrrrtl ll)on)cnl rclcrtctl 1o lltt' t'litsltr' ;trtr

I, "O, t)hi@) dx (6.6.30a)

J. **, t).,i@) dx (6.6.30b)

266 At n()t lnl;lc I'lr NoMr Nn

ril, '] -:'1r,,

, M,,,,i,,,,,,61x - x,1

where L1, and M. are the self-excited aerodynamic lifi and momcnt per unitspan given by Eqs. 6.5.4.

Modified equations of motion (6.6.30) can then be written that are similarto Eq. 6.6.15 though now coupled by the presence of the full set of unsteadymotion derivatives proportional to HI and Af . From these equations, aero-dynamically modified mechanical admittances can be calculated analogously toprevious results, but now for two coupled equations. The results, representing( l) the across-wind deflection due to a concentrated harmonic lift at section x1,(2) the torsional deflection due to a concentrated harmonic lift at x1, (3) theacross-wind deflection due to a concentrated harmonic moment atxy, and (4)the torsional deflection due to a concentrated harmonic moment at x1, may bedesignated, respectively, Hnr(x, xr, /t), Hot(x, xr, n), HnuQ, x1, n), and Ho1a(x,xt, n).

Assuming now that the structure is subjected to a distributed buffeting liftL(x, t) and moment M(x, t) as defined by Eqs. 6.6.lb and 6.6.lc, the spectraof across-wind bending and torsional response can be calculated by integratingelemental effects. Designating by 51,p, Sr,u, 5u,4, and S1a,7a, the cross-spectracorresponding respectively to the lifts and moments at x1 and x2 as suggestedby their subscripts, the following typical expression for vertical response spec-trum S6(x, n) is obtained:

fL fLs{x' n) : J. J" lHft6' x1' n\Hv(x' x2' n)s7,1,(n)

+ Hfr@, x1, n)H61fx, x2, n)Sy,yr(n)

+ HftAx, x1, n)H1,1(x., xy, n)S11a,yr(n)

+ Htt{x, x1, n)H1,1fx, x2 n)S7a,7ar(n)l dxr dxz

where F1* denotes the complex conjugate of F1.It should be remarked that both the mean speed of the flow and the values

of lift and moment may, in the above expressions, be a function of x. In thatcase modal orthogonality relations can no longer be used (e.g., as was donein Eq. 6.6.14), and the expressions forthe modified admittances become moreelaborate; however, the attendant calculations can be conveniently programmedfor electronic computers.

Possible applications of the expressions for the response of linelike structuresdealt with here include the calculation of the responses of tall prismoidal build-ings with strong torsional motions, and those of tall towcrs antl suspcndcd-spanbridges.

ltl llltl fl{:l !, 26/

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6-89 R. I. Basu, and B. J. Vickery, "Across-Wind Vibrations of Structures of Cir-cular Cross-Section, Part 2, Development of a Mathematical Model for FullScale Application," "/. Wind Eng. Ind. Aerodyn., 12 (1983), 15-97.

6-90 D. J. B. Richards, "Aerodynamic Properties of the Severn Crossing Conduc-tor," Proceedings of the Symposium on Wind Efects on Buldings and Strut''tures, Yol.II, National Physical Laboratory, Teddington, U.K., Her Majesty'sStationery Office, London 1965, pp. 688-765.

6-91 O. M. Griffin and R. A. Skop, "The Vortex-Induced Oscillations of Struc-tures," J. Sound Vib., 4 (1976),303-305.

6-92 K. Y. R. Billah, "A Study of Vortex-Induced Vibration," Doctoral disserta-tion, Princeton University, Princeton, (1989).

6-93 I. Goswami, R. H. Scanlan, and N. P. Jones, "Vorlex-Induced Vibrations ol'Circular Cylinders. I: Experimental Data; II: New Model," J. Eng. Mech.,r19 (1993), 2210-2302.

6-94 F. Ehsan and R. H. Scanlan, "Vortex-Induced Vibration of Flexible Bridges,"J. Eng. Mech., ff6 (1990), 1392-l4ll.

6-95 E. H. Dowell (ed.), A Modern Course in Aeroelasticirlr (Chapter 6: "Aero-elastic Problems of Civil Engineering Structures"), Kluwer, Dordrecht, 1995.

6-96 E. Simiu and R. H. Scanlan, Wind Effects on Structures, 2d ed., Wiley, NewYork. 1986.

6-97 R. H. Scanlan, "Problematics in Formulation of Wind-Force Models for BridgeDecks," J. Eng. Mech., ff9 (1993), 1353-1375.

6-98 S. Murakami, A. Mochida, and S. Skamoto, "CFD Analysis of Wind-StructureInteraction for Oscillating Square Cylinder," in Wind Engineering, Proceed-ings, Ninth International Conference, Eastern New Delhi, Wiley, New York,pp. 671-682, 1995.

6-99 C. F. Christensen and O. Ditlevsen, "Fatigue Damage from Random VibrationPulse Process of Tubular Structural Elements Subjected to Wind," in Proceed-ings, Third International Conference on Stochastic Structural Dynamics, SanJuan, Puerto Rico, Jan. 15-18, 1995.

6-100 E. Simiu and G. R. Cook, "Empirical Fluidelastic Models and Chaotic Gal-loping: A Case Study," J. Sound Vibration,154 (1992),45'66.

6-101 M. Frey and E. Simiu, "Noise-Induced Chaos and Phase Space Flux," PhysicaD,63 (1993),321-340.

CHAPTER 7

WIND TUNNELS

,\ltlrough the science of theoretical fluid mechanics is well developed and, orrrputational methods are experiencing rapid growth, it remains necessary toI't'rlirrrn physical experiments to gain needed insights into many complex effects;r:sociatcd with fluid flow. This is the case in the well-established field of,rt'rorrirutics, for which wind tunnels were first developed, and, to an even1'rt"rrtcr extent, in the practical study of buildings, structures, and machines that'.1;rnrl in the earth's near-surface atmospheric layer.

lior the most part such structures have been designed for other purposes than;,r,rvitling minimal resistance to the air moving about them. They have there-lort'. in recent decades, been the focus of what is termed bluff-body aerody-rr,rrrrics. In such aerodynamics there is much emphasis on flows around sharp, r)nlcrs, on separated flows, and so forth. These situations are among the mostrrr'()n(lite when it comes to both theoretical and computational methods. Thervrrrl tunnel is thus naturally resorted to as an investigative tool in this context.

I'ypically the full-scale bluff body is immersed in a turbulent atmosphericllrrw lrlachsbart determined as early as 1932 (see Sect.4.6.2 and Fig. 4.6.4)tlr;rl sirnulations of the aerodynamic behavior of buildings should be conductedrrr rl,irrtl tunnel flows with characteristics similar to those of the natural wind.t 'rrrlcrrtly, the vast majority of tests are carried out in wind tunnels that simulate,rtrrrosphcric flows. (In some instances tcsts in smooth flow are still accepted,lrrr 1'x'1111p1s, in the case ol'lrurssccl fiameworks-see Sect. 4.5 and Chapter| ' ol lirr prcliminary invcstiglrliorrs ol'thc gcrlrnetric shape of bridge deck',, t liorr rrrorlcls. H<twcvcr, (lrcst' insltrrrt'cs iu'(' tlrc cxccption rather than therrrlt').'l'ltr:lc is thcrolillc il iilri)rl', inl('t'sl irr glrinirrg a knowlcclgc-firr latcr','Ptrttltteliort in lltc wintl luturt'l ol llrt'rlrlrrrc ol wintl llows irr thc ctrrlh's

273

274 wrNl) tt,NNt t:i

boundary laycr; "l"argct" charuc(clistit's to be tltrlllic:alctl in (hc witttl tttltttolare acquired from meteorological ilrvcrstiliirliolts ol' tho attnospltcric boundarylayer (see Chapter 2 and l7- ll to l7-41).

Simulation occurs at reduced getltnctric scalc fbr obvious reasons oi ccon-omy and convenience. The question of scale then opens up the whole area tll'physical similitude and the necessary underlying theory, which places emphasis

on'u ,"t of dimensionless numbers and/or similarity criteria applicable to bothflow and test models of structures placed in it. With characteristics of the targetflow and scale factors for similitude established, it soon becomes apparent thatcertain of the model criteria established for similarity cannot in fact be satisfiedunder typical, everyday test conditions. The wind tunnel modeler is thus

launched upon a series of inevitable compromises that render his task complex,revealing ii as an art of both perfotmance and interpretation rather than an exactscience.

A basic discussion of similarity criteria is presented in Sect. 7.1. Windtunnels usecl in civil engineering applications are briefly described in Sect' 7'2,which also includcs comments on some difficulties in achieving similarity be-

tween wincl tunnel and atmospheric flows. Section 7.3 is devoted to scalingproblems, insotar as they affect the aerodynamic and aeroelastic behavior ofihe models to be tested, and to the question of wind tunnel blockage. Section7.4 reviews some attempts to validate results of wind tunnel tests by compar-isons with full-scale *"uru."-.nts. Information on general wind tunnel testingrechniques is provided in [7-5] to [7-10]. Reference [7-11] is a useful com-pendium on wind tunnel modeling for civil engineering applications and in-

"lud"r, in particular, useful information on modern wind tunnel instrumenta-

tion.

7.1 BASIC SIMILARITY REQUIREMENTS

In analyzing any problem-more particularly one that is expected to be studiedexperimentally-it is usual to identify a set of governing dimensionless param-eters. These parameters are in certain cases obtained by first writing the partialdifferential equations that describe the physical system at hand. These equationsare then rendered dimensionless by dividing each of the key variables by a

reference value having corresponding dimension. When the process is com-pleted, a number of dimensionless groups emerge as factors goveming thephysical behavior of the system. Maintaining the values of such groups intacti-- on" situation (prototype) to another (model) will automatically ensuresimilarity. In the case of fluid flow, this process involves the conservationequationi for mass, momentum, and energy, together with the equation of statcof ttr" fluid. These are written and converted to dimensionless form in thcmanner describccl. In thc prcscnt chapter, however, an ltllcrnitlivc: ltnd simplcrmethod fbr arriving al lho rliutonsionlcss gft)ups will srrllit'c. 'l'lris is rt tliltrcn-

/ I lt^til( j l;lMll Alllli nt {rtiutf trr,l tltf ; 215

',r,rttrrl ltlUtlysis ltirsCtl ott lr sct ol'physicirl l)iuilntclt' th it:i:,unt(.(l (! l,n(ttt lo ;111,.,'1tlr,' wintl tunncl llow.

I I "1 Dimensional Analysisl'rt t'oltcrctcness, lct it be assumed that thc lilrcc /,'tlcvclo;x'rl sontcwlrr.r't.on;r lrotly itnmersed in a flowing fluid is a lirrrcliorr only ol'tlrrr lirlkrwirrg sixl';il;ililctcrs: density p, flow velocity U, sonrc typical dirrrcnsion 1), sonrc I'rc-rlur'rr('y rr, fluid viscosity p, and gravitational acceleration g. One writes

n ! p"rfn'rut'g( (7.1.1)

l'lrt'rt' rv, . . . , f are exponents to be determined. There are three basic quan-trtr('ri: nrass M, length L, and time T, to which all of the above parameters arerlrrirt'rrsionally related. Writing the dimensional equivalent of each of the quan-rllrr's in Eq.7.l. I results in the following dimensional equality:

(7 .1.2)

Ir',rrr which the following three independent equations are obtained by equating,,,r rt's;xlnding exponents:

M: 1:o*eL: 1:-3cv+B+-y-e+fT: _2:_p _6_e_Zf (7 .1.3)

l lrt'sc cquations may now be solved for any three of the exponents in termsrrl llrt' r'crnairring three; for example,

cv:1-e0:2-€-6-2f7:2-e*6+f (7 .t.4)

irlrr'rrt't' it is seen that

p ! pt eg2-e-b 2fD2-E+6+(16rert (7.1.s)

Y! t (y)"(n)',.' (;)'(#) (#)'

( 'i; )' ( ,,"",,) (?i )'p' !-- ,l tt tr (7 .1.6)

276 wtNf) r UNNII rl

From this it fbllows that thc dirncnsionlcss lirrcc cocllicicnl* I;lpIJ)l)) is afunction of the dimensionless numbers DnlIl , p"l pIlD, and Dglu).

The dimensionless numbers mentioned are of coursc alrcady wcll known influid mechanics. For example, when n is the frequency n, of vorlex shcddingfrom a bluff obiective of cross-sectional dimension D, then

is the well-known Strouhal number. When n is n-, a characteristic mechanicalfrcqucncy associated with a structure, then Dn^lU is termed the reduced fre-qu(n(y relative to a steady flow past the structure of velocity U; its reciprocalUln,,,D is the associated reduced velocity. The group nzlU-where z is heightabovc ground, n rcprcsents a frequency associated with a component of variablcwind vcl<rcity, and U is mean wind velocity-is a dimensionless frequency.f'(called thc Monin coordinate) often used as abscissa in depicting wind velocityspectra (see Eq. 2.3.17). Further, if n is replaced by the circular frequency./.: 2<,r sin @, which is the Coriolis parameter (where c,r is the rotational speedof the earth in radians/second and S is the latitude-see Eq. 1.2.3), then thequantity

(7.1.8)

is called the Rossby number.The group y.l pUD is the reciprocal of the well-known Reynolds number

Dn-.:sU

oUD UD6l€-'l.t' v

(1 .1.7)

(7.1.9)

UDfr

:GO

which is sometimes more specifically called the molecular Reynolds numberwhen z : pl p is the kinematic molecular viscosity of the fluid. In some ap-plications (see Sects. 2.1 .2 and 4.4), a turbulent Reynolds number may bcemployed in which z is replaced by /tu.b, an "eddy" or "turbulence" kinematicviscosity. It is tentatively suggested, in [7-12] that in the atmosphere such aviscosity has an order of magnitude near the ground given by

!6t5 a U4lg

*Typically coefficients of lift force F. and drag force F, are written

F, F,,c, - iirr. c ': iil;where lpl/) is tccognizcrl rrs lltc tlynlrtttic l)rcssurc lirrrr llrt'llcrrrorrlli r'r1r:rtiorr (str

(7. 1. l0)

lul. ,1. 1.20)

/ t ilntit(] lil[]4il Aililr ilt r,lt ,iltl Ml r!||; Z7.l

tvltclc tt., is lher shcirr, or'll'it'l iorr, vcrlocily, lrrrl .t,, ts lltr slr.lir(.c rlrrp,lrrrt.ssIt'rrgllr (scc'l'ablc: 2.2.l). Notc lhirt lit1. 7. l. lO yit.ltls t.orrslrlcrrtbly lowt.r.virlucsllr:rrr lh<rsc suggcslctl (irlso lcrrrlativcly) irr 12 ll7l.

lrinally, thc rocipr<lcal ol'thc group /),q/l/'' is t'rllcrl lft' lirotttlt' ttrtttrltt,t.:

'l'hus simple analysis reveals the several dirnensionless groups that play keyr,lt:s in wind tunnel similitude, particularly in aiding the transfer of resultslrorn experimental model to full-scale prototype.'l'hough it is not directly pertinent to the present discussion, it is worth|.irrting out here that, were thermal effects to be included in the above analysis,tlrrr:c additional commonly occurring dimensionless numbers would "-".g",rurrrtcly:

I'randtl number:

(7.1.t2)

lickert number:

u.l5t:-D,,<

(7 .t .t t)

(7.1.13)

go : FCPK

",:+(#)

nu'Coo

l{ichardson number:

(7 .1.14)

r'lrcrc Q is specific heat at constant pressure, K is thermal conductivity, andt/ is absolute temperature. Note that the Richardson number consists of a di-ilr('nsionless temperature divided by a Froude number; G; plays an impor"tantr'k: in thermally induced convection in the atmosphere. Because this ihapterr:r t'oncemed principally with mechanical effects, the last three numbers are not,'rrrphasized in what follows.

7.1.2 Basic Scaling ConsiderationsIt will bc rccognizcd in lhe: cottsirlt'r';rliorr ol'tlirrrt.nsionlcss numbcrs ab6vc thutrut tlislincti<ln is lnarlc ils l() s()ut1'(' or.or.i1,.lr ol'lr givcn paritnrclcr: il cirrr ltcllrritl' s(ructural, <trolltcr. l;ot'cxirttrplt', :r lt'rr1illr, llt.rlucrrt'y, rlcrrsity, or vr'l9t.il-yttut,y lrtr itssociltlcltl willr rlrrt'r'lt;tt;tt'lr'u:.l rt ol llrt' llrritl or slrtrclul.r, irrurrt,r.sctl irr

I

i

278 wlNl) lt,NNl lli

it. This implics IhaI rutirts anr()ng strch rluirrr(i(ir:ri nlr.rsl ltc tttititttltittctl ctltts(itll(from prototype to model. For cxatttplc, il' p,, antl p/ arc thc ilcrtsity ol' lhcstructure and of the fluid, respectively, thcn

(7. l . ls)

where the subscripts m and p refer respectively to model and prototype. Sinccthis holds as well for geometric ratios and geometric shapes in general, itimplies that all model shapes must be geometrically similar to prototype shapesand that, for example, vibrational modal shapes of prototype structure must bcmaintained in the corresponding model. Likewise frequencies from all sourcesmust bear the samc ratios to each other in model as in prototype' Further, sinccoscillatory deflcctions must maintain proper proportionality from prototype tomodel, dirncnsionlcss damping ratios that affect such deflections must remainthe same in prototype and model.

There now may be examined a typical set of scaling factors together withthe process by which they are set. Three such factors may be arbitrarily chosen.The first might be an arbitrary length scale:

(7.1.16)

set, for example, by comparison of model size to prototype size. (It will beseen subsequently what particular considerations enter into the setting of a

length scale when turbulence is involved.) A second choice might be a con-venient velocity scale

(fi). : (",,),

_D^'Dp

-P-Pp

U^x,, : -:r, UI'(1 .t.17)

set perhaps by available wind tunnel speeds compared to expected natural windspeeds, and a third might be a density scale

(7.r.18)

usually forced upon the experimentalist by fixed circumstances (e.g., testingin airof the same density as that surrounding the prototype, whence Xo : l).

Given the fundamental exigencies of mass, length, and time, the three fixcdscale choices, once madc, condition all others in conscqucncc ol'the requirc-ment that the dimcnsionlcss groups maintain their conslitrtcy l'trlttt l)K)totypc tomodel and vicc vcrs1. 'l'hrrs, lilr cxanrplc, thc rcducctl I't'etlrrcrtt'y rctlttirctrtcn(

/ t llnl;t(: ritMil Attil r nt rrr fiil Mt NI:, 2'l.J

,r'lr- llrc li'cquency l;t'ulc )r,, l'<tr ull pcrtincrrt lcsl lr.t't1ut.rrt'it.s

(:'i,'),, (''i,') (/.1.1())

(1.t.20)

(7 .1.22)

,\u

\ilr()s('tcciprocal is the time scale X7..It rrury be emphasized at this point in this illustrative discussion that \r, \r,,

,rrrrl tr,, lrave been fixed either arbitrarily or in consequence of some unavoidablei rrt rrrilstilnce. we now inquire as to the consequence of invoking Froude num-l','r sirrrilitude, requiring

(7 .t.21)(#)^:(#),

rt lrt'tt' L" is the gravitational scale factor. In most instances gravitational effectsrlu.,t lro considered to be the same in model and prototype, so \, : 1, whence

\r: J\. (t .1.23)

r'lrt'rr liroude scaling is respected, this may contradict an original choice forAr lrr rn<lst cases it is convenient to accede to Froude number scaling, adjusting,t, ,rlt'orclingly, whence frequency scaling takes the value

\r:l/ JI" (7 .1.24)

\rr'rrtiorr kr gravitational effects may be required for certain structures (e.g.,',rr',Pr'rrsi()rr bridges) or for certain cases where convective air motions are im-l,'rrl:ur(. As noted above, the latter are disregarded in the present discussion.\\'t' rrow rnay examine the effect of invoking Reynolds number scaling:

r2' :lx.\,

("',',,") (+) (7 .1.2s)

ll Irolo(y1.lr-: ltncl tttotlcl ltlc lrollt itt irit rrrrrlt'r'lrlrrxrspltc:ric conditi<lns, Rcynoldsrrrunlrt'r scllirrg rcrlrrilc:s sirrrlrl-y llrirl A1tr1 I or'

280 wrNr) rt,NNt ll

^r, l/^/ t].1.26)

which is, in general, in sharp conllict wilh olhcr rcquircr.ncnts sct ahovc, ftrrexample with:

x/: Jt (t .1.21)

in the case of Froude scaling. Thus Reynolds number scaling is seen to beincompatible with the prior setting of length and velocity scales unless testingis undertaken at full scale X. : 1.

Another view of the same effect is that, for example, under Froude scaling,Reynolds number scaling is hugely distorted:

\*":9+:\r,\r-\jrz(Ge),

To illustrate, if X/. : 1/300, then

/ I \t'' Ix'":(:oo) :r,*

(7 .1.28)

(7 .1.29)

indicating a tcst Rcynolds number less than one five-thousandth of G" for theprototype. It is notcd that some aeronautical testing achieves Reynolds numberscloser to prototype values by using rarefield or compressed fluids, or fluidswith lower kinematic viscosity than air, such as freon. A further recent stageinvolves use of gases at cryogenic temperatures t7-131.

Rossby number scaling also proves to be intractable under most circum-stances, since an equivalent Coriolis acceleration effect (as represented byl.)cannot practically be realized to the frequency scale Xn mentioned above. Suchan effect would require some means for imparting lateral acceleration to theflow, which is not easily achieved, 12-281,12-291, V-141.

Thus normal wind tunnel testing in air under standard gravity and atmo-spheric conditions typically entails fundamental scale violations of the Reynoldsand of the Rossby number.

7.2 WIND TUNNEL SIMULATIONS OF ATMOSPHERIC FLOWS

To achieve similarity between the model and the prototype, it is desirable toreproduce at the requisite scale the characteristics of the atmospheric flowsexpected to affect the structure of concern (see Sects. 4.6 and 4.7). Thcsccharacteristics have been outlined in Chapter 2.They inclucic (l) thc'variationof the mean wind spccd with height, (2) the variation ol' lrrrlrrrlt'ncc intcnsiticsand integral scalcs with hcight, and (3) thc spcc(rrr irrttl r'r'oss slx'('tr:r ol-tur-bulcncc in tho lrlrlng wirttl, rrctrtss-wirttl, ltntl vcrticltl rlitr't ltotts

Arcr'prs ,., -,,,,,,,,,,,. "',:. ,;:.;" ;,,,:" ,.' ;,, ;,.::, :,':,;,::.',,, ,,.-,ol lhc lrottntlitly laycI l.ylre, lcw, il'irrry, lrrlxrr:rloly invr.sligltigrrs havc hccrrrlt'vtltcd t<l thc sirtttllltliott ol'rlownsl<lpc wintls, lrrrllicirrr., crylwalls, trlrnacl'cs,

;rrrtl thundcrstorllls. (Ntltc, lrowcver, thc tcn(ativc sirrrrrllrtion ol'tornado-inclucctllorccs in J7-151.)'l'unnels used fbr civil engineering purposcs havc cnrss sections that rarclyt'xcccd 3 m x 3 m. (A notabre exceptirn is thc g m x g m tunnel of theNrrli.nal Research Councir, ottawa, canacra.) Three types of wind tunnels havelrt'.rr used for simulating atmospheric flows. They arl referred to as long tun_,r'ls, short tunnels, and tunnels with active devices, and are described in sects.I )'1., 7 -2.2, and 7.2.3, respectively. Sections 7.2.4 and 7.2.5 comment ontlrc possible effects of violating the Reynolds and Rossby number similarityrt't;uirements upon the simulation of flow turbulence.

7.2.1 Long Wind Tunnelslrr lrng wind tunnels ([7-16,7-17]) a boundary layer with a typical depth ofo 5 to I m develops naturally over a rough floor oi the order oi)o to 30 m inIt'rrgth (Figs. 7 .2.r u-541, 7.2.2, 7.2.3). The depth of the boundary layer canlrt' increased by placing at the test section entrance passive devices or tn" typ",rlt'scribed in Sect. 7.2.2. s,ch an artificial increase may be necessary, partic-rlrr'ly in simulations of flow over the ocean or over terrain with low or moderater'rrghness. The height of most tunnels may be adjusted to increase slightlyrvrtlr position downstream. The purpose of such an adjustment is to achieve a/('11) pressure gradient streamwise, which would otherwise not obtain, owingtrr c11c.tt losses associated with flow friction at the walls and with internallrrt'lion due to turbulence.

Atmospheric turbulence simulations in long wind tunnels are probably thelr.sl that can be achieved in the present state of the art. However, even whenl';rssivc devices such as spires are not used, similitude between the turbulencerrr thc laboratory flow and in the atmosphere is generally not achieved (seet'r'rs- 7.2.5 and7.3.1). The rack of similitude becomes stronger if, for ex_,rrrrlrlc, spires are employed (see Sect. j.Z.Z).

7 "2.2 Short Wind TunnelsIrrrrrcls used foraeronautical purposes are usually designed fortesting in smoothll,vv ilnd therefore need not have long test seciions. tutuny such tJnnels havelrr't'rr coflvefted for use in civir cngineering applications by adding, at the test

"r't rirrr cntrance, passivc dcviccs, such as grids, barrierr, r"n."rlund spires,tlr:rr gc:ncrate a thick bounrlirry lrrygr.. 'l'lrr: il<xrr.f the test section, which isrr"rr;rlly <ln thc order tll'5 rtr lorrli, is t'ovt'n'tl wilh nlr.rghncss clcrlcrrts (l;ig.1 ' "Il Vitritltts typcs. slrrrPes, irrrrl r'rrrrlrirrirliorrs ,rl';lrrssivc rlcviccs l1rvt, lrct.rr',rrlilicslctl arrtl corrlrrcnl(:(l ulx)n rrr |/ l{rl to l7 .}51.

*

N(,1|

\ .rril{i,irnr.Ir.

ri, \ ,, !' l,' l() ' ilr lil,tl,.

, ={l \ "r,\ t |'r:t I't'\ l--l l-J =-J I-H E IJW)\tt,uttl-

20 15 10{--Dirtance from leadinq edqe of

1) ittUC;21jtrL bltr:- c--:g'E)4

=.yOO,'= o.= ..14,.u:-11 1)

'- cz\ .)cF<Ea>.-- ?-:a- ,I

--i

.d,()c

5- -,: -.t*lv, ':l.d x f I

!3 r,)et-tii .. '/,

-o .X n,

-vtt=0 u-'trtr^ E--o=1 '=Yu',=ut,*ui

. al^l -.F\e

dc ijI ;,i ilf i r,lr -llrE'i

Fc

F

0.Eocl

\;.aoti=:a.Ea

c0

Eiiu

!ila

'a

nY

,c

L)

;a

Oa

jca-

Eell mouth

I f(;uRE 7.2.2. Devclopment of boundary layer in a long wind tunnel. After A. G.l):rvcnport and N. Isyumov. "The Application of the Boundary-Layer wincl runnel torlrt' Prcdiction of wind Loading," in proceedings. Intemational Research Seminar onwirrtl Effects on Building and Structures, Vor. l, p.221. copyright, canada, 196g,I lrriv. of Toronto Press.

l'l( jl lltl,l 7.2.-1. 1);rsln';rrl vit'tv ol ;r lorr;, ,r.,,,,; trlrrrr.l (r orllr.sy llrrlrrllrry l.trVt.l WilrrlI rrnr'l I .rlrrrl:rloly. I lrt. I Irrtvr.r,.rl\ ,,1 \\, ..t, rn ( )ill.rrr,r,

2$:l

ilwlNt) iltNNt I

FIGURE 7.2.4. Spire and roughness arrays in a short wind tunnel (courtesy of thcNational Aeronautical Establishment, National Research Council of Canada).

Reference [7 -26] proposes the following procedure for the design of spireswith the configuration of Fig. 7.2.5:*

1. Select the desired boundaryJayer depth, d.2. Select the desired shape of mean velocity profile defined by the power

law exponent, a (Eq. 2.2.26).3. Obtain the height h of the spires from the relation

1.396h:- l*al2 (1.2.t)

4. obtain the width of the spire base from Fig. 7.2.6, in which rl is rhcheight of the tunnel test section.

xThe base length of thc triangular splittcr plalc in Fig. 7.2.5 is /r/,1. 'l'trc l;rtt.r;rl sp:rt'irrg bclwcclthe spires is h12. ln praclicc, thc witlth ol lltc lunncl ncctl rrol lrt' trrr intr.1lr:rl rrrrrltiplt' ol /r/2.

Wlljl ' ltll\lNl l:;lMlll nll()N:i{ll AlMrr',1'lll lll( llrrw', 1'lllr

FIGURE 7.2.5. A proposed spire configura-tion. From H. P. A. H. Irwin, "The Designof Spires forWind Simulation," J. Wind Eng.Ind. Aerodyn., 7 (1981), 361-366.

'l'he desired mean wind profile occurs at a distance 6ft downstream from theslrires. According to [7-26], the wind tunnel floor downwind of the spiressluruld be covered with roughness elements, for example, cubes with height ksrrch rhat (11-261to [7-28]),

f,'l(lIJRlt, 7.2.6. Graph fbr obtaining spirclrrrsc wirllh. Iinrrn H. P. A. H. Irwin, "'l'lrcl)r's11'1q 1;l Sllircs lirr Winrl Sirrrtrllliort," ./.ll irt,l l,.rr,t1 Ittrl. .4rntl.vrr., 7 ( 191{ 1 L l(r I

It't'

f : *o [(3) '''(?) - ' "''[(a) + 'z

0s] ] (7 22)

il286 wrNr) il,NNl r:;

whcrc 1l is lhc sltrrcirrg ol llrc nrtrg.lrrrt'ss t'lt'rrrt'rrls lrrrtl

C,:o116l '|v I' ll+rvl (1.2.3')

Equation 7.2.2 is valid in the range 30 < 6D2lkt < 2000.A study of the dependence of flow features upon the type of passive deviccs

being used was recently presented in U-171. Figure 7.2.1 ll-l1l shows thcmean velocity, longitudinal turbulence intensity, and vertical turbulence inten-sity profiles at (l) 6.1 m and (2) 18.3 m downwind of the test section entrancc,fbr flows obtained by using three different types of spires, the wind tunnel floorbe ing covered by staggered 1.27 cm cubes spaced 5.08 cm apart. In Fig.7 .2.7the boundary-layer thickncss 6, the mean wind speed U at elevation 6, and thcpower law exponent rr (Eq. 2.2.26) are denoted by deha, Uinf, and EXP,respectively. It may bc assumed that the mean flow with exponent cy : 0.16at station x : 6.1 rn, and the mean flow with exponent a : 0.29 at station ,r: 18.3 m, are approximately representative of open terrain and suburban terrainconditions, respectively (see Table 2.2.2).

Some modelers adopt a geometric scale equal to the ratio between the bound-ary-layer thickness measured in the laboratory and the value 6 of Table 2.2.2,even though the latter is nominal, rather than physically significant (see Eq.2.2.15 and Sect. 2.2.4). If this geometric scaling criterion is used for thosimulations of Fig. 7.2.1 , the geometric scales are found to be 0.751215 :1136l for the flow with a : 0.16. and l/400 forthe flow with a : 0.29. Thcrespective longitudinal turbulence intensities at 50 rn above ground are about0.07 and 0.15, versus about 0.15 and 0.225, as obtained from Eqs. 2.2.18,2.3.1 ,2.3.2, and Table 2.3.1. As expected, the discrepancy between the lon-gitudinal turbulence intensity in the wind tunnel and the "target" value in thcatmosphere is more severe at the station x : 6.1 m, which would corresponclto the fetch available in a short tunnel.

Figure 1.2.8 ll-l1l shows spectra of the longitudinal velocity fluctuationsmeasured at station x : 18.3 m and elevation z/6 : 0.05 in the three flowsdescribed in Fig. 7.2.7b. Forthe flow with a :0.29, it isseen in Fig. 7.2.1tthat at the nondimensional frequency nzlU(z): 0.8, nS(n)luz = 0.05, versus0.06, as obtained from Eqs. 2.3.2,2.3.16, and Table 2.3.1. Unlike the tur-bulence intensity, the higher-frequency spectrum measured in the wind tunnclis in this instance relatively close to the "target" value.*

The results of []-l7l and of other studies (e.9., U-241, 17-531) indicate that,regardless of the type of passive devices being used, simulations in short wintltunnels generally do not achieve similitude between the turbulence in the lab-oratory and in the atmospheric flow.

i'lnfilrrrrirliorr on irrlepnrl st:rlcs lirr llrc wirrtl lrrrrncl llows is rrol rt lrrrlcrl tt l / I ll

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^* - !:.8€'oo-E*avr\6 2 >\v!Ft^

18.h8^t:!E.Ud=-^ o ^^o[j E fo4! o-i1 <.vJ,

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< /. sv -'r' Yg.i

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4

o

t'l(;tJl{lt 7.2.11. Spcctra of longitudinal velocity fluctuations measured at 18.3 rrrdownwind ol spires. Reprinted with permission from J. E. Cermak, "Physical Mocl-eling of the Atmospheric Boundary Layer (ABL) in Long Boundary-Layer Wind Tun-nels (BLWT)," inWindTunnel ModelingJbr Engineering Applications,T. A. Reinhold(ed.), Cambridge University Press, Cambridge, 1982.

7.2.3 Tunnels with Active DevicesIn tunnels equipped with jets (Fig. 1 .2.9) it is possible, within certain limits,to vary the mean velocity profile and the flow turbulence independently of eachother [7-29,7-301. Such tunnels are relatively expensive and do not necessarilyresult in superior flow simulations. However, they may be useful for basicstudies in which the effect of varying some flow characteristics independentlyof the others can be studied in detail.

Active cascades of moving airfoils (Fig. 7 .2.10) have been recently designedwith a view to creating, and simulating effects of, large-scale turbulence overbridge deck section models l7-31, 7-321.

7.2.4 Reynolds Number and Turbulent Flow SimulationIt is suggested in [7-33, p.204,7-34, p. 266, andT-35, p.290], that Reynoldsnumbers of turbulent flows obtained in the laboratory downwind of square meshgrids may in some cases be too small to give rise to a turbulence spectrunrhaving an inertial subrange. It is further suggested [7-35] that the Reynoldsnumber based on eddy size should be the order of 105 to ensure existencc ol'this subrange. Applying analogous rcasoning to a devclopcrl lurbulcnt houndarylayerof depth, say,0.5 rn, in which the intcgral scalc lcrrg,llr /,) (rr rrrcusurc ol'typical eddy sizc) is :rbotrt 0. 125 rn, u Ilcynolils rrtrrrrlrcr ;rl :r vt'locily ol' lrm/s may bc culcrrlirlt'tl

l jllll' lrllltll I :;lMl,l n ll()N:i ()l n IM(1: ;l,l ll ltl(. ll()W:,

FIGURE 7.2.9. Upstream view of rhe test scction and jets of the 1.20 x 1.70 mclosed-circuit jet tunnel. University of Toronto Institute for Aerospace Studies (courtesyI)r. H. W. Teunissen).

Itl( il lltl,l 7.2.111. Mt't'lurrtit;rllv tlr rvcn :rrr l.rl r ;r:;t :rtlrsirrrrrl:rliorr l7 l-?1.

wtNt) tUNflt t :

gol

:'!| I

D

EXPo .26

^ .2O

o29

soot csto r . aoo6 t6 . w6a

lol low l'rctlrrcncy lurbulencc

292 wrND ruNNr tr

Thus typical boundaryJayer simulations of the kind discussed may be expectcclto develop velocity spectra with satisfactory inertial subranges, though at lowervelocities and turbulence integral scales they may be borderline.

7.2.5 Rossby Numbers and Turbulent Flow SimulationFailure of Rossby number equivalence in typical test circumstances is due tothe difficulty of scaling the coriolis parameter f, above its automatically achievedfull-scale value. Rotating wind tunnels (12-28, 2-29D, or tunnels with porouswalls and acnrss-wind suction imparting lateral acceleration to the flow [7-17]arc currcntly not used in civil engineering applications. An investigation intothe effect ol'the Rossby number on boundary-layer flow is therefore in order.

It was shown in Sect. 2.2 that the approximate depth of the atmosphericboundary layer may be expressed as

ur .:. l2(o. r25 )6le : ": .-10'u 1..5 x l0

a = o.zsT (7.2.4)

where z* : U(h)l{2.5Ln(hlzd\, U(h) is the mean speed at a reference heighrh, zois the roughness length, andf, is the Coriolis parameter. Equation7.2.4can also be written in a form that emphasizes the dependence of the atmosphericboundary-layer depth 6 upon Rossby number:

6:cGo (1.2.s)

where c = 0.25h1{2.5ln(hlzi} and G.o : U(h)lhf,. The boundary-layer depth6 is seen to be an increasing function of wind speed. For high wind speedssuch as are of interest to the structural designer, it follows from Eq. 7 .2.5 that6 is of the order of several kilometers. For example, if z6 : 0.05 m (openterrain), U(10) : 25 m/sec, andf,: lO-a sec-l(corresponding to an angleof latitude 6 = 45o, see Table 1.2.1), then 6 = 5 km. The region of interestto the structural designer, that is, the lowest few hundred meters of the at-mospheric boundary layer, is thus seen to amount to about one-tenth or less ofthe full atmospheric boundary-layer depth.

As noted in [7-36], both in the atmosphere and in the laboratory the meanvelocity profile is very nearly logarithmic over the region consisting of thelower one-tenth of the boundary layer or so (see Sect.2.2 and Fig. l.Z.1l).Moreover, measurements suggest that in this region thc turhLrlcnt cnergy pro-duction is approximatcly balanccd by the energy dissipiriiorr (ltig.7.2.l2 ancl17-371) so that thc lottgitLrclinal vclocity spcctrurn in llrt'int.11 i:rl srrl'rllrngc rnaybc cxprcssccl in txrrrtlinrr:nsiontrl lilrrrr lrs

/:' wtNl) il,NNl t:ilMt,l All()Nr;()l nlM{r'.l,llt lilr 'lrrw',

2q:l

/////t Area of experimental data

0.05 0.1 0.2 0.5 1

6

t, l(;t.lRE 7.2.11. Logarithmic plot of velocity distributions in turbulent boundary lay-r'rs ()vcr plates. After F. H. Clauser, "The Turbulent Boundary Layer," Advancesl1t1t!. Mech., 4 (1956), Academic Press, New York, p. 9.

EIils 8.-lJI

ns(2, n) : 0.26 f -r,,Ux(2.3.t6)

i

wlrcrc n is the frequency, z is the height, and f : nzlU(z) (see Sect.2.3).lrtluirtion 2.3.16 is not valid in the upper region of the boundary layer wheretlrt'cnergy production differs significantly from the energy dissipation (see Fig.t).t2).

('onsider now a long wind tunnel in which the boundary layer developsrrrrtrrrally over a rough floor and in which the boundary-layer depth is of the,rrrlcr of I m (Fig. 1.2.2). Assume that the height of the building being testedrs 200 m and that the model scale is 1/400. Since, as was shown above, ther,'gion of the atmospheric boundary layer over which the logarithmic law holdsrs (rrnder strong wind conditions) a few hundred meters high, it is reasonable{() irssume that Eq. 2.3.16 is valid throughout the building height. However,rrr tlrc laboratory similarity theory suggests that Eq. 2.3.16 can only be appliedrr;r lo a height of approximately 0.1 m from the wind tunnel floor, to whichtlrt'rc would correspond a full-scale height of just 40 m above ground.

A schematic representation ol' thc situation just described is given in Fig./ I 13, which shows thc bourttllrry llrycr that dcvclops in a long wind tunnel(lrrll lirrc), and thc atmosphcric lrourrtllrry lrrycr rcrluccd to model scalc (br<lkcnlrnc). 'l'ho lowcr onc-tcntlr antl llrc orrlt't rtirtc lenllts ol'thc b<luntlitry laycr irrctk'rrolctl hy /,,,,, ancl (),,,,, rcsptrt'livcly. lot lltt'witttl lrtttttcl llow, itrrtl by /,,, lttltl(1,, respr:clivcly, lor llrc lrtrrrrrsplrt'rrt' llow. ll t'lrn lrc sccu itt lrig. 7.2.l.l tlrtrl

294 WINIJ I UNNFI Si

0 0.1 0.2 0.3 o.4 0.52 0.6 0.7 0.8

6

FIGURE 7.2.12. Energy balance in a turbulent boundary layer.send., The Structure of Turbulent Shear Flow, Cambridge Univ.1956, p. 234.

I

Iv _,0c.G

After A. A. Town-Press, New York,

,li, ,t(llir--l-

Atmospheric boundary layer(reduced to model scale)

FIGURE 7.2.13. Lowcr and outcr rcgi.ns of the bouncr,ly lirye'i'rrrr:and in thc atmosphcrc.

wintl tunncl

/3 WIND IUNNFt lilMt,l All{,N ()l Al llol)YNnMl(: nNl) Al ll(}l ln:illtr lll llnvl()l I 295

ovcr tttost <ll'thc wirttl lrrrtrtcl lrorrttrlary-llrycr tlcpllr llrc irlrrrosplrt'rrt' llow rrr llrt'lrtwor laycr L,, is sittrulalctl by tltc llow in lltc otrtcl r('p,ron (),t tt wlrt'lr,rrccxrrding to similarity thcory, L.,q.2.3.16 woultl rrol hc t'x1rr'r'tctl lo lroltl.

7.3 WIND TUNNEL SIMULATION OFAEROELASTIC BEHAVIOR OF BLUFF

AERODYNAMIC ANDBODIES

'l'his section considers some practical aspects of the dependence of the aero-rlynamic and aeroelastic response of wind tunnel models upon the turbulencet'lraracteristics and the Reynolds number of the flow. It also briefly discussesrrc:rodynamic distortions due to blockage effects.

7.3.1 Effect of Turbulence Characteristics of the Flow'l'hc details of the dependence of the aerodynamic and aeroelastic behavior oflrtilies upon the turbulence characteristics of the flow are not fully understood.llowever, it is clear that for the effects of turbulence on the model to be similarto those on the prototype (i.e., in order forthe turbulent eddies to envelop orotherwise affect the body or part thereof in a similar way in the atmosphererrnd in the laboratory), it is necessary that the ratio between some typical lengtht'haracterizing the turbulence and some characteristic dimension of the body bethc same in both situations.

lt is convenient to adopt the integral scale Lj (see Sect. 2.3.2) as the char-rrcteristic length of turbulence. The geometric scale factor of the simulation,l), : D^lDo, should then be given by

Dr:w (7 .3.1)

where (Lj)o and (Il)^ are, respectively, an estimate of the integral scale thatrrbtains in the atmosphere at some representative elevation (see Sect. 2.3.2),irrrd the integral scale measured in the wind tunnel flow at the correspondingclcvation above the tunnel floor. The application of Eq. 7.3.1 is discussed is| 7-381.

Equation 7.3.1 is violated in many instances because of the difficulty ofrrchieving sufficiently large integral scales in the laboratory, particularly in shortwind tunnels (see Sect. 7.2.2). However, even if Eq.7.3.1 is nominally sat-isliod, it should be recalled that integral scales are poorly known and can varylhrm measurement to measurcmcnt by a factor of five or even ten (see Sect.).3.2). thus, the assumed valuc ol'lhc ratio (lj)-lUi), can differ significantlylnlrn its actual value.

An atlompt to asscss crnrrs rhrt' lo llrt' irrrl'lcll'cct sirnulation <ll'thc intcgrulscirlc <ll'turhuloncc is rcporlr:tl in l7 lt)l lirr. llrt' l)rcsslrrcs at vlrriorrs lxrinls ol'

il296 wtNt) iltNNl l:

1.0 m

' 4.9 m 24.4 m

FIGURE 7.3.1. Schematic view of building with pressure taps (After I7-391).

the building represented schematically in Fig. 7.3.1.In the investigation ofU-391 the integral scale was not varied independently of the other flow features.Rather, the wind tunnel boundary-layer flow was kept unchanged while thedimensions of the model were increased. It was estimated that the integral scaleIiinthe wind tunnel was equal to about l/500 times a nominal integral scalcjudged to be typical of atmospheric flows. Measurements were made on l/500,ll25o, and l/100 models of the same building. Ratios between the peak, mean,and rms pressures measured at several points on the 1/100 and 1/250 models,and the corresponding pressures on the l/500 model, are listed in Table 7.3.1.

It is seen that in some instances the influence of the model size upon thctest results is significant (e.g., for the peak pressures at tap 29, 1/100 scale,or tap 1 I l, l/100 scale and 1/250 scale). Note also that the pattern of variationof the ratios of Table 7.3.1 is irregular. This may be due, at least in paft, rtrthe fact that by changing the height of the model by the factors 2.5 and 5, thcturbulence intensities at the elevation of the points under consideration alsochanged.

The effect of turbulcncc features upon the modcling ol'lrcrorlyrr:rrrric hohavirlris discusscd in 17 4t)l to l7 421, Accorcling ro 17,40. 7 .lll. rlrt. nrle ol' thcintcgral (urbttlcttc'c: scitlc ilt wirttl (unncl sirrrrrlirliorr is rrrrrror tl rro( 1t'pligi$t:.

II

.l

I

I

/:r wtNl) iltNNl I i,lMt,l n iii )t.t I I nl lt()l lyNnMt(] Ail! ) nl |{ )t tl\"il, nt |/\\/ti )tt ?1lI

l Alll,l,l 7.-1.1. l{:rlios ol'l't:th, Mt:ut, :tntl l{NlS I'r'rsrurr.r orr l/l(ll) ;rrrrl l/lll)Nlrxltls lo ()orrrspolrrlirrg lDrt.ssrrrcs orr l/SlX) Morlel"

ti t(x) I/"rl|l't'ltk Mr'.rrr nr:,lAl' Pcak Mcan nls

30.5

.',)" 1.34(0.93)t'/t' 0.90(0.97)(,t' 1.00( I .02)

I I r' 0.69(0.75)t/' 0.84(0.83)(rlli' 1.05(1.07)

r.6-5(0.19)r.r6(r.48)1.67( 1.90)r.00(0.43)0.96(0.93)l.40( 1.40)

1.09(0.51)0.62(0.es)l.l3(1.90)0.60(0.67)0.83(0.91)0.80(0.67)

O.(XXO. /-l)o.'/t{( Lo I )0 u4(0.tt I )

0.63(0.7rJ)0.90(0.95)0.83(0.90)

L.)(r( .f .,1O1

o.()i,i( l.o/)L20(0 n I)r.00(0.7-5)0.81(0.79)r.07(0.97)

LO t(() f{t{)( ).()l(o.l.iti )

1.0,1(0.71{)0..53(0.57)0.83(0.e 1)0.73(0.81)

'Nrrrttbers not between parentheses comespond to open exposure. Numbers between parentheses, rrrrcspofld to built-up exposure.'Srrc(ions.

| 'r c ssu res.

;rntl lluctuating pressures associated with separated flows can be properly sim-rrlrrtcd even if only the small scale turbulence is correctly reproduced. Thisr'vould require (l) the correct reproduction of the longitudinal and lateral tur-lrrrlcnce intensity, and (2) the use of sufficiently large models. Thus, as is the( rsc for the prototype, higher-frequency components of the longitudinal veloc-rly spectrum that affect the separated flow would be contained within the inertialrrrbrange; see Sect. 7.2.5, Eq. 2.3.16.

/.3.2 Reynolds Number EffectsSlrarp comers tend to cause immediate flow separation, independently of thelicynolds number of the flow. For this reason it is generally assumed that iftlrc flow is adequately simulated, pressures on rectangular and other sharp-r'or'flered structures are adequately reproduced in the wind tunnel. However,lrlrrllbodies with long afterbody extensions downstream may exhibit flow reat-t:rchment, which does depend on the Reynolds number. Such circumstancesrrny affect the values of the across-wind forces experienced by the body. Fewlrrll scale supporting data on this topic are available to date. Note also that iftlrr: details of a scale model require extremely small dimensions (as, for ex-:rrrrple, in modeling the members of a truss structure at a scale of 1/500 orlrt'low) it may be that the drag coelicient applicable to such a member can berrrrtluly influenced (raised) by Reynolds number effects. Figure 4.5.6 bears outtlris tcndcncy.

lrt thc case of bodies wilh curvc:cl surfaces, Reynolds number deficiencies,;rrr llavc significant cll'ccts. 'l'lris is sirrrply illustratcd by the evolution of bothrttr'lttt tlntg crlcflicicnt untl Slrotrlr;rl rtrrrrrlrt'r'lir rr cir-cular cylindrical section as:r lirrrction ol' llcyrrolrls trrrrrrlrt'r (st'r' liilrs. .1..1.,1 :rrrtl 4.5.2).

As irrtlic:rlctl ilr ('lrirplt'1 ,1 , lltt':r,'torlyn;unr( l)('lritviorol'srrch b<xlic:s rlcpcnrls,rlt wltt'lltt't (ltt'lrotttttl:tt'y l:ryt'ls olt llrr', rrrvt'tl :,rul:rtt's;u('llrrrrinlrror (lllrrlitrlly

a111

a6

t68

r'l )q

47a

298 wrNt) tuNNt l

or fully) turbulcnt. Sincc bounclary laycrs occurring ll lrigh llcylroltls rrurrrbcrsare turbulent, it is logical t() attcmpt tho rcpr<lduction ol'lull-scalc ll<lws ar<lunrlsmooth cylinders by changing laminar boundary layers into turbulcnt oncs.This can be done by providing the surfacc with roughness elemcnts (scc14-151, U-431 to [7-46], and Fig. 4.5.5). Ir is suggested in 17-441rhar rhcthickness e of the roughness elements should satisfy the relations

400

l0-2

whcro U is thc rncan wind speed, a is the kinematic viscosity (u = 1.5 xl0 s m2lsec in air), ancl D is the characteristic transverse dimension of themodel.

For exarnplc, in the case of the DMA tower (Fig. 15.3.22), the roughnesswas achieved by fixing onto the surface of the 1/200 model thirty-two equi-distant vertical wires. Three sets of experiments are reported in[7-44] in whichthe surface of the cylinder was (1) smooth, (2) provided with 0.6-mm wires(elD - 7 x 10-3;, and (3) provided with l-mm wires, respectively. It wasfound that the highest mean and peak pressures were more than twice as highon the smooth model than on the models provided with wires. The differencesbetween pressures on the model with 0.6 mm and the model with l-mm wireswere small. The influence of the roughness on the magnitude of the meanpressures at 2O m (full-scale) below the top of the building is shown in Fig.7.3.2 in which the mean pressure coefficient 4 is defined as follows:

Ln: jPul

where p is the measured mean pressure, p, is the static reference pressure, U,is the mean speed at top of the building , and p is the air density.

Approaches of the type described above were found to yield acceptableresults in cases not involving aeroelastic motions. However, if aeroelastic ef-fects are present, wind tunnel tests in which such approaches are used canprovide an utterly misleading picture of the behavior of the prototype (seeChapter 10).

7.3.3 Wind Tunnel BlockageA body placed in a wind tunnel will paftially obstruct thc passagc of air, causingthe flow to accclcratc, This cffect is refbrred to as bkrcklrgr:. ll'thc blockagc issubstantial, thc ll<lw anttttttl lhc nrodcl , and thc rnork:l's rrcrrrtl,yrtrnric.hchuvi11r,arc no longcr rcprr-:srrrrlirlivc: ol' prrrlolypc c<lntliliorrs.

UeU

eD

/it wtNt) tt,NNt I l;tMl,t n lt()N ()t Al ll( )l)YNn Mtc n Nt , n I il( ,t I n !;t t(

lirrroollr rrrorirl

lr! ItAVtilI I ?!lgl

Morlcl:; willr 0.U nrrr attrlwrtlr 1 rnrrr wires

-2-10+1ep

l,'l(iURE 7.3.2. Influence of model surface roughness on pressure distribution [7-44].

Corrections for blockage depend upon the body shape, the nature of therrt:rodynamic effect of concern (i.e., whether drag, lift, Strouhal number, andso fbrth), the characteristics of the wind tunnel flow, and the relative body/wirrd tunnel dimensions. Basic studies on blockage are summaized in 17-471rolT-491and in [7-50], which also contains a bibliography on this topic.

It is concluded in [7-50] that, in the case of drag, the following approximaterclation may be used for the great majority of model configurations in all flows,rrrcluding boundary-layer flows:

CD

I + KSIC (7.3.2)

wlrcrc Cp, is the corrected drag coefficient, Cp is the drag coelficient measuredrrr thc wind tunnel, S is the refercnce area for the drag coeffrcients Cp, and Cp,rurtl C is the wind tunnel cnrss-scctional area. The ratio S/C is referred to aslrkrckagc ratio.

'l'hc c<lcflicicnt K has bccn rlclcrrttirtc:tl only for a limitcd numbcr ol'situir-liotts. lior oxalnplc, in thc clrse ol'ir lrlrt'with a rcc(angular cK)ss sccli()n sl):utrrirrg thc crrtirc hcight ol'a wirttl lrrrrrrt'l willr rrornirrirlly snroolh lkrw, A wlrstlt'ltrrtttittctl lo tlcpcrttl ttpon lltc rirlto ttll, irs slrowrr in lrig. 7.-1.,1 (rr lrrrrl /r lrlt'

WIND TUNNELS

a/bFIGURE 7.3.3. Blockage correction factor K for two-dimensional prism with along-wind dimension a and across-wind dimension b in nominally smooth flow [7-49].

the dimensions of the along-wind and across-wind sides of the rectangular crosssection, respectively).

The effect of turbulence on blockage by flat plates was studied in [7-48] forflows with uniform mean speed, and was found to be negligible in most situ-ations. On the other hand, it is stated in [7-50] that this effect can be significant.Thus, according to [7-50], turbulence does not increase the drag on a squareplate, as concluded in 17-251 (see Table 4.6.2). Rather, the increased dragreported in 14-251 was only apparent, and it was the blockage effects that wereaffected differently by various turbulence levels. For a basic study of blockageeffects on bluff-body aerodynamics, see t7-551.

Despite such ongoing debates and various continuing uncerlainties. it maybe assumed that for blockage ratios of 2% the blockage corrections are likelyto be about 5%, and that the magnitude of the blockage correction is propor-tional to the blockage ratio [7-50].

7.4 VALIDATION OF WIND TUNNEL TESTING

Despite the numerous full-scale measurements reportcd in lhc litcraturc, thenumber of dependablc comparisons between modcl anrl plototypc rcsults rc-mains relativcly srnall. liigurcs 7.4.1 m,J 7.4.2 slrrrw ir cotttllrlisorr hctwccn

l:i 'eH.9"= = Fr '1, allrtilr€trd- Li1- ut*-4

H'P.,=r d g eCA

" 0E9.:,i 5 E H3,- -\, !3 '= x tsc!LAW.

ts 5 Or:5\OE o I Oy'r{\O

oe- d,:; : ts-q-c ch

-=6a

= o q) =

4,^'- h..oo9il= -a E E =o.-ad @drntrHov

X C 3;F.r.tzlExV -5-R'i E FF S69:^ .. E,H O Ui,; lteE =s= * c,.U.p * 60?\cr9 ts=sFO.o,5l-iNFll\

EEbgCF, "&>v o:: 9i.F toroo o XFlr IB E F-fr:8e E6e Bx-,65

o :\d d) @rE E;T"E ]oVA >

" :..H'o!' -ad E != 3E<5SiJ>es=Z '8:-.-=r\F- tr Y=.r I r :.5€ rL oi,, * cJ o=P:,A E itiliNFz-o

sdt_t:

lll03 lSnsslud t^isws

60l

lll0l lSnsslSd NVlw

<-zr--o--"-1ll-

:"e601l:l:,j; ]]: ::']1:j:.*

z-6-

<-zI-----1l.l-

O-z*6-

301

302 wrND rUNNr rs;

wind tunncl and l'ull-sclrlo nrcasurcnrt.rrls ol prt'ssurcs on llter ('olrrrtrcrcc ('orrt'ttower (Fig. 15.3.17). '['hc winil tttttncl vrrlrrui wcrc providccl at lhc rlcsign stagcand are represented by opcn circlos. 'l'hc solitl lincs join avcftlgc valucs ol'estimates derived from actual observations ol'prcssurc dillbrences on thc builcl-ing; the shaded areas indicate the standarcl clcviation of the full-scale estirnatcsl7-5 U. It is seen that the agreement between model and full-scale measurementsof the mean pressures is satisfactory.* However, it appears from Figs. 7.4. I

and 1 .4.2 that local fluctuating pressures attributable to vortex shedding (fluc,tuating lift) differ at some points significantly in the wind tunnel from thcpressures on the prototype.

Further data for this building are available in [7-51] and [7-52]. Figure 7.4.3shows acceleration spectra obtained from full-scale measurements and fromtests on a model of the building with seven lumped mass levels. It is seen thatin this case the model tests tended to underestimate the response in the inter-mediate-frequency range but appear to be adequate at the low and high endsof the spectrum.

Model/full-scale comparisons for pressures on low-rise buildings are alsoreported in 14-74,7-4Ol to f7-42,7-53,7-561. According to V-561, compari-

_t10'

fi-2o.2 0.6

FREQUENCY (Hz)FIGURE 7.4.3. Full-scale and model north-south acceleration spectra, CommerccCourt Building. Reprinted with permission from E. A. Dalgliesh, "Comparison ol'Model and Full-Scale Tests of the Commerce Court Building in Toronto," in WindTunnel Modeling for Civil Engineering Applications, T. A. Reinhold (ed.). CambridgcUniv. Press, Cambridgc, U.K., 1982.

*In Figs. 7.4. I and'7.4.2 tltc ltlthrcviatiott IIMSM clcrxrtcs r'(x)l nrr'irr \(lllir('virlu('itl)()rl lll('rnoan l7-5 I l.

10J

rc2

rc1

rc0

oNN

6

EE

az.uloJE.FC)IJ(LaE=(L

0.0

/4 VAI ll)All()N ()l Wllllt ltllltll I ll 'iilf lrr ilo;l

1 00 200 J00 400Azimuth, degrees.

I

I fuli Scole

- CSU

a UWo (rough exp,)o UWO (smooth exp,)

ril'll+lrtl,{" A,lr

" i"* 'tl e.

Lti

E 1.2o--u U.al

0,4

Lo 2-OO-O

O1.0

1 O0 2OO J00Azimuth, degrees-

I

I Full Sco e

- CSU

o UWo (rough exp.)o UWO (smooth exp,)

(h)(a)

l'l(;Illtlt 7.4.4. wind pressure coefficients on the Texas Tech Experimental Build-rrl' lirll-scale and wind tunnel measurements: (a) wall pressures; (b) corner roof pres-'.rtrr's. Iinrm W. H. Tieleman "Problems Associated with Flow Modelling Proceduresl,u l,ow-Rise Structures," J. Wind. Eng. Ind. Aerodyn., 4l-44 (1992), 923,934.

'.rrrrs bctween full-scale and wind tunnel measurements on low-rise gable-roofI'rrrltlings suggest that the wind tunnel does not model accurately the flow'.i'|':illrlion on the windward roof, so roof pressures often differ significantly intlrt' rrrrltlcl fiom the prototypc. Sirnilar discrepancies occur between pressuresnr,'rrsurr(l 0n models of diflcrcrr( scirlcs; scc Fig. 7.3.1 and Table 7.3.1. Figurc/'1.'l shows rncasuremcnts ()n ir lirll sclrlc:'l'cxas Tcch University cxpcrirncnlarI'rrrltlirtg ittttl Coloratlo Slirlc: []nivcrsity lrrxl I lnivcrsity of Wcslcrn Orrtirrio wintlIttttttt'l ttttttlcls rll'tlritl hrrilrlirrg. Wrttrl ltttrrrt'l nre:rsulcnlcn(s lrrcl scrr:rr lo llc;rtr't'lrlltl'rlc lilr llrc w:tll ltrcsstrtt's ltttl t;tttlt'irr;rrlcrqrrlrtc lirr lhc nrol r'()lnr.t.

304 wtND I(JNNI I ri

However, accorcling lo l7-41 l, a corrsitlcllrlllt: irrrpnlvcrttcnl ol tlrc wirrtl lunncrlmodeling of roof corncr and ttthcr prcssul'cs can bc achicvcd by placing srnallspires directly upstream of the modcl lo sitnulatc correctly thc turbulencc in-tensities, as well as the spectral densities at a I'requcncy l}U(h)lB, lbr botltthe longitudinal and lateral turbulence (U : mean wind speed, h : buildingheight, B : characteristic dimension equal to h for low-rise buildings and ttlthe least horizontal building dimension for tall buildings).

REFERENCES

7-l D. A. Haugen (ed.), Workshop on Micrometeorology, American MeteorologicalSociety, Boston, MA, 1973.

1-2 Charat:lcristics fi'Windspced in the Lower I'ayers of the Atmosphere near theGround: Strong Wintl,r (Ncutral Atmosphere), ESDU Data Item No. 72026'Engincering Scicnccs Data Unit, London, 1972.

1-3 Charude ristics 0.f'Almlsphcric Turbulence near the Ground, ESDU Data ItemsNos. 74030, 14031,75001, Engineering Sciences Data Unit, London, 1974.1975.

7-4 J. Counihan, "Adiabatic Atmospheric Boundary Layers: A Review and Analysisof Data from the Period 1880-19722," Atmos. Environ., 9 (1975), 871-905.

7-5 A. Pope and J. J. Harper, Low-Speed Wind Tunnel Testing, Wiley, New York,t966.

7-6 S. M. Gorlin and I. I. Slezinger, WindTunnels andTheir Instrumentation, IsraelProgram for Scientific Translations, Jerusalem, 1966'

1-7 R. C. Pankhurst and D. W. Holder, Wind Tunnel Technique, Putnam, London,1968.

1-8 E. Ower and R. C. Pankhurst, The Measurement of Air Flow, 4th ed. , Pergamon,Oxford. 1969.

1-9 P. Bradshaw, An Introduction to Turbulence and Its Measurement, Pergamon,Oxford, 1971.

7-10 W. Merzkirch. Flow Visurtlization, Academic, New York, 1974.

7-ll T. A. Reinhold (ed.), Wind Tunnel Modeling for Civil Engineering Applications,Proceedings of International Workshop, Gaithersburg, MD, April 1982, Cam-bridge Univ. Press, Cambridge, 1982.

1-12 R. Britter, "Modeling Flow over Complex Terrain and Implications for Detcr-mining the Extent of Adjacent Terrain to be Modeled," Wind Tunnel Modelingfor Civil Engineering, Applications, T. A. Reinhold (ed.), Cambridge Univ.Press, Cambridge, pp. 186-196.

1-13 High Reynolds Number Research, D. D. Baals (ed.), NASA CP-2O09 (19'17)Proceedings of Workshop, Langley Research Center, Hampton, VA, Oct' 197(r.

7-14 D. R. Caldwell and C. W. Van Atta, "Ekman Bountlary l.aycr Instabilitics,"J. Fluid Math..44, P'arr l (Oct. 1970), 19-95-

7-15 M. C. Jischkc and Il. I). l,ight, "l-aboratory Sirrrulrrtiott ol 'l'olttiulic Wintl l,oirtlson a Cylirrtlr-iell Slrrrt'lrrlc," in Witul l'.)r,q,ittt'rrirt.ti, I'totr'r'rlittg,s ol- tlrt: lrilill

ttl

/ l|,1

/ ltl

/ .'o

I .'l

/ .'l

| .,,1

I .'(t

I ttl

/ r()

/il

ilt lllil N(;t l; 305

Itllctrtltliottttl ('ottlr'lcltr.'e. lrorl ('ollirts, ('olorirtlr..lrrly lrl/()..1 .li. ('crrrurk (ctl.),Vol.2, l)l). 104() lO.5(), Pclgrrrnorr ltress, ()xlonl, lt)ll0.A. (1. I)avcnporl :rrrtl N. lsyrrrrrov, "'l'hc Alrlrlrt'rliorr ol rlrr: I]ourrdary-Laycrwintl rtrnncl to thc Prcdiction ol'wirrtl l,.rulirrg." rn l'nx.ccrlings of'rhe Inter-tttttitrral Rcscur<'h Serninar on wirul l'.,lli'rt,t ttrt llttiltlittg,,; ttnd Structures, univ.ol"li)()nto Prcss, Toronto, t96lt, pp. 20 1 2.10..l . B. Ccrrnak, "Physical Moclcling ol'thc Atrrrosphcric Boundary Layer in Longllrundary-Laycr wind runncls," in wintl 'lfunnel Modeling for civil EnginceringApplicutitn.s, T. A. Rcinhold (Ed.), Carnbridgc Univ. Press, Cambridge, 1982,1tp.97-125..1. Counihan, "An Improved Method of Simulating an Atmospheric Boundaryl.ayer in a Wind Tunnel," Atmos. Environ., 3 (1969), 197-214..l . Counihan, "Simulation of an Adiabatic Urban Boundary Layer in a Wind'l'unnel," Atmos. Environ., 7 (1913), 673-689.N. J. Cook, "On Simulating the Lower Third of the Urban Adiabatic BoundaryLayer in a Wind Tunnel," Atmos. Environ.,7 (1913),691-705.N. J. Cook, "A Boundary-Layer Wind Tunnel for Building Aerodynamics," ./.lnd. Aerodyn., I (1975), 3-12.N. J. Cook, "Wind-Tunnel Simulation of the Adiabatic Atmospheric Boundaryl.ayerby Roughness, Barrier, and Mixing-Device Methods," J. Ind. Aerodyn.,3 (1978), 157*t76.N. M. Standen, A Spire Array for Generating Thick Turbulent Shear Lctyers forNutural Wind Simulation in Wind Tunnels, Report No. LTR-LA-94, NationalAcronautical Establishment, National Research Council, Ottaw a, 197 2..l . A. Peterka and J. E. Cermak, Simulation of Atmospheric Flows in Short Wind'l'unnel Test Sections, Fluid Mechanics Program Research Report, Colorado StatetJniversity, 1974..l . c. R. Hunt and H. Fernholz, "wind runnel Simulation of the Atmosphericll.undary Layer: A Report on Euromech 50," "/. Ftuid Mech., 70, part 3 (Aug.te7s),543-559.l{. P. A. H. Irwin, "The Design of Spires for Wind Simulation,,' J. Wincl Eng.lnd. Aerodyn. T (1981), 361-366.l. S. Gartshore, A relationship between roughness geometry and velocity profileslrape .fbr turbulent boundary /ayer.r, National Research council of canada, NAEl{cp. LTR-LA-140 (Oct. 1973).l{. A' wooding, E. F. Bradley and J. K. Marshall, "Drag due to regular arrays.l'nrughness elements of varying geometry," Bourul. lnyer Meteorot., s (1973),28.5 308.ll. w. Teunissen, "Simulation of the planetary Boundary Layer in a Multiple-.lct Wind Tunnel," Atmos. Environ., 9 (1975), 145-1i,4.ll. M. Nagib, M. V. M.rk.vin..l. T. yung, andJ. Tan-atichat,.,On Modeling.l Atrnospheric Surtircc l,uycrs by lhc countcr-Jet Technique," AIAA Jourruil,14, No. 2 (1976). ltl-5 l()0ll. llicnkicwict,. J.lt,. ('clnr:rk. .l . I'crcr.krr, irntl ll. H. Scanlan, "Activc Mrxlcl-irrg ol' Lurgc-Scalc 'l'rlrlrrrlt.rrr't'." .l ll'irrtl 1,,'tr,q. Ittl. Acnxlvtt. ll ( lgul),'I(t5 4'/(t

il306 wtNt) ll,NNl l:i

j-32 .l . Ij. ('crrrrak. l]. lJicltkiewit'2,;rrrtl .l . l't'1t'tIrt. .'|lit't'Mrulrlirt,q, rtf lirrlutlr'rtr't'Jir Wintl'l'uttru,! Stutlit,s ol'tlritl.4r fVItr!cl,s.ltclxrrl No. lrllWn/Rl) 13l/l':llJ lictleral Highway Adrninistratiott, Mcl,cart. Vir.. Iicbrrrary I913-1.

1-33 J. O. Hinze, Turbulence, McGraw-Hill' Ncw York' l9-59'

7-34 H. Rouse (Ed.), Advanced MechuniL:s rtl'Fluid:;, Wiley, Ncw York' l9(r-5'

1-35 H. Tennekes and J. L. Lumley, A First coursc in Turbulcnrc, Mll' Prcss.Cambridge. 1972.

7-36 H. Tennekes, "The Logarithmic Wind Profile," J. Atmos. Sci., 30 (1913),234238.

1-37 J. L. Lumley and H. A. Panofsky, The sutface rf Atmospheric Turbulencc,Wiley, New York, 1964.

7-38 N. J. Cook, "Dctermination of the Model Scale Factor in Wind-Tunnel Simu-fation of the Acliabatic Atnrospheric Boundary Layer," J. Ind' Aerodyn',22(t911-18),311 321.

1 l9 A. G. Davcnporl. l). Surry. T. Stathopoulos, Wind Loads onLow Rise Buildings,liinal Rcport ol' l)hrscs I and ll, BLWT-SS8-1977, University of westem on-tario. l,otttlolt, Canada, 1977.

'l-40 W. H.'l'iclcrlan, "Pnrblcms Associated with Flow Modelling Procedures 1or'

Low-Risc Structures," J. Wind Eng. Ind. Aerodyn., 4l-44 (1992),923-934'l-41 W. H. Tieleman, "Pressures on Surface-Mounted Prisms: The Effects of Inci-

dent Turbulence," J. Wind. Eng. Ind. Aerodyn., 49, (1993),289 300'1-42 D. Surry, "Consequences of Distortions in the Flow Including Mismatching

Scales and Intensities of Turbulence," in Wind Tunnel Modeling for Civil En'gineering Applications, T. A. Reinhold (ed.), Cambridge Univ' Press, Cam-bridge, 1982, pp. 137-185.

7-43 E. Szechenyi, "supercritical Reynolds Number Simulation for Two-DimensionalFlow Over Circular Cylinders," J. Fluid Mech., 70, Part 3 (August 197 5), 529542.

7-44 J. Gandemer, G. Bamaud, and J. Bi6try, Etude de lct tour D.M.A. Partie l,Etutle des e.fforts dfis au vent sur les faqades, Centre Scientifique et Techniqucdu Bitintcnt, Nantes, France, 1975'

7-45 B. J. Vickery, "The Aeroelastic Modeling of chimneys and Towers," in wintlTunnel Motleting .for Civil Engineering Applications, T. A. Reinhold (Ed.).Cambridge Univ. Press, Cambridge, 1982, pp. 408-428.

7-46 O. Giivcn, C. Farell, and V. C. Patel, "surface-Roughness Effects on the MearrFlow Past Circular Cylinders," J. Fluid Mech',96 (1980)' 673-701'

1-47 P. Sachs, wind Forces in Engineering,2d ed., Pergamon Press, oxford, 1971t.

7-48 V. J. Modi and S. El-Sherbiny, "Wall Confinement Effects on Bluff Bodics irr

Turbulent Flows," Proc. 4th International Conference ctn Wind Effects on Buildings and Structures, Heathrow, U.K. (1975)' pp' 121-130'

7-4g J. Courchesne and A. Laneville, "A Comparison of Correction Methods Usctlin the Evaluation of Drag Coelficient Measurements ftrr Two-dimensional Rcctangular cylinders," ASME Wintcr Meeting, Papcr No. 79 WA/FE3 (19'79).

j-5O W. H. Melbournc, "Wincl Tunncl Blockagc Flll'ccts irrrtl ('otrt'l:tliotts," inWitulTunnt'l Mrxltlirr,q litr ('ivil l,)rgirttaritr,q Altltlit'tttion.s, 'l'. A lit'irrlroltl (etl.), ('irrtrbritlgc tJnivcrsily l)tt'ss, (';trrtbt'itlgc' l()ll2' pp. l()/ ) l('

(r:, 30// 5l W. n l)lrlPltt':;lr. '('{rrl):uisorr ol Motlt'l/l,rrll Srz,. lir..rlr. Wrrrrl l,rr.ssrrrt,s .rr :rlliglr ltist' llrultlrrrlt," .l Irttl. ..lttt,rlrvr., | ( l{)l.r). .)i (,()I \)' W- A. l)lrlp,lit'slt, "('ottt;rrttisott ol Mtxlt'l ;rrrrl l,ull Sr:rlt. lt.sls ol llrc (,orrrrrcrc.t:(lourt IJuiltlirrg irr'lonrlto," in Wirtrl l'rutttt.! ll!,,,lr,ltrr11 fitr.(,ivi! 1,.)tt,qitrtt,rittg

Al4sl.iuttiotts, 'l'. A. l{oinho[l (otl.), (,irrrrlrritll,t. t trriv,.r.sity l)rcss, Carrrhrirlgc:,l9tl2. pp. .575-589./ 5-l I{. D. Marshall, "A Study .l'wintl I'r1'ssrr1's 'rr :r sirrglc-Family Dwelling inM<del and Full Scale," .l . lnd. Acnxlrvr., 1,2 (ocr. l9'5), llj_19g.l5'+ R. D. Marshall, "wind'r'urrncrs Applictr t. wind Engineering in Japan," -/.Struct. Eng., ll0 (1984), tZ03 122t./ -5-5 H. utsunomiya, F- Nagao, y. Ueno, ancl M. Noda, "Basic Study of Blockage

Effecrs on Bluff Bodies," J. wind. Eng. Intr. Aerocryn., 49 (lggi),247-256./ 5() G. M. Richardson and D. Surry, "comparisons of wind-Tunnel and Full_Scale

Suriace Pressure Measurements on Low-Rise pitched-Roof Buildings, " J. wincl.Eng. Ind. Aerodyn., 38 (1991), 249_256.

;

CHAPTER 8

WIND DIRECTIONALITY EFFECTS

Wind effects on structural members depend upon direction for climatological,aerodynamic, and structural reasons. The extreme wind climate at any one siteis, in general, nonuniform with respect to direction owing to basic atmospherit.circulation pattems and/or the presence of local obstructions. Aerodynamicbehavior depends upon direction for most structural members; examples rangcfrom cladding to bridges. The dependence upon direction of the structuralresponse of a member subjected to a given aerodynamic load can be simplyillustrated in the case of a circular flagpole in horizontally homogeneous terrairr,anchored to its foundation by four bolts located at the comers of a square bascplate. For any given wind speed. the uplilt lorce on the anchor bolts is grcalcrby a factor of V2 when the wind direction is parallel to the diagonal of thcbase plate than when it is parallel to one of the sides.

This chapter describes procedures for estimating probability distributions olIargest yearly wind effects which account for the dependence upon direction olthe extreme wind climate and of the aerodynamic and structural behavior ol'the member. Also described in this chapter are procedures for estimating prolrabiiities of failure and safety indices for members sensitive to wind directionality effects.

8.1 PROCEDURES FOR ESTIMATING PROBABILITYDISTRIBUTIONS OF LARGEST YEARLY WIND EFFECTS

Three such proccdurcs arc currcntly availablc;. Thc lirst pnrt't'tlrrr.c is htsctl orrthe theory o['s(ationirry nrnclorn pn)ccsscs I13-1, lt-2. u ll l( is slrowrr in Scr't.8.1.1 that in (hc prcserrl st:tlc ol'lhc rrll tlris proct'rlrrn'is rrol :,rrlr'rl lir.pllrcliclrl

308

ltI l,lt()rItrilltt ,t{)ilt:;llMA|N(,I,il()llntilt jl, t,1,.|ililil1t0ll,. .l(l!)

ust'. A st:t'ontl 1rt.tt'rlrrrt' tlilt ttsst'tl irr sct'1. li. I "r. l:;;rllrr()l)r:rlr. l{rr r1rr.,1r..,11,11,tl clrttklirtg lurrl olltt.r rrrt.lrrlrr'rs lrol srrlrjt't'lt'tl (o:.r1,1111sq;rrrl rl1,rr.rrlrr, ,rrrrlrlllgr'lttitltl <lr :totrlclttslit' cllt'c'ls. ll rrlilizcs (l) t'xlrt'rrrt. wnr(l :,lr('i.(l rl;rl;r rt.r lrr1.rloI cslitrtatccl lilI circlr ol tlrc ti (rlr l(r) plirrt'iplrl ( (]rnl]:r:i:. tllt.t lr.rr:;, ;rrrrl 1.,;;rt'rodynarlic clata, llascd ott wincl tunncl lcsls, ()n llr('(l('lx.rrrlt.rrt.t.rrpon tlrrct lrorrol thc wind cflcct being ctlnsidcrcd.'l'hc wirrtl spt't'tl :rrrrl lr.'lxly,r,,,r,,,.tllrlrr ;rrt.rrsccl to create timc series of cxtrclnc wirrtl crllL't'ls, liirrrr wlrit'lr il rs lxrssilrlc {rrt'slitnate a univariate probability distribu(ion ol tlrc: llu-gcsl wirrrl cllcct, irs wgll;rs lhe requisite design loading (c.g., thc winrl kracl with a -50-ycar nlcan rc-('rrrrcnce interval). The practical application clf this proceclurc-par-ticularly fbrt'lirclding design-is simple and straightfirrward. The third p.o""dur", discussedrrr Sect. 8.1.3, utilizes the eight univariate probability distributions of the largestve:arly wind speeds recorded for the principal compass directions [g-2, a-11,:rrrtl the fact that the time series of the largest yearly winds blowing fromtlillbrent directions have as a rule weak mutual correlations lsee Sect. 3.+;.'l'lris procedure can be applied to any type of structure or structural member,rrrcluding structures or members subjected to wind-induced aerodynamic am,lrlilication or aeroelastic effects.

8.1.1 Procedure Based on Theory of Random processeslr this procedure the mean wind velocity is regarded as a stationary two-tlirrrcnsional random vector process, u(r), with speed u(r) and direction d(r).lirrilure is assumed ro occur tf u(0) > g@) (i.e., the curve g(d) is the failurelr.undary in the velocity space; see sect. A3.1.2). Forexample, if the relationlrctween the wind effect Q@) and the wind speed u blowing from direction 0

Q@ : h(iluz@)

thcn the boundary g(0) has the expression

(8.1.1)

s(0) : t n lt/zttLnot I

(8. r.2)

Iwlrcre R is the limit state (e.g., the wind pressure causing failure of a claddingprrncl).

'fhc mean rate at which the vector u(r) crosses the boundary g(0) in therrrrtward direction is denoted by u,rand may be referred to as the mean failurerrrrc. If the values of u1| arc srrrall, failure is a rare event and its probabilityrrury bc assumed to be .l' lhc l).iss.n typc. The probability that in the timerrrlcrvatl 7'no luilurc will oct'tl.(i.t'.. lhc: llnrblrbility that the velocity vectorwill not cnlss llrc liriltrrc lrrttutrl:rry,r;(//) irr llrt'ou(wirnl direction) can be written

t'lll. tl t '1't (li. L.l)

t310 wlNt) t)ilil(itl()Nnl llY llll(.1:i

(Eqs. A1.34, A2.39) so ihlrt tlrc pnrbtrbilily ol lrrilrrrr': tltrrirrg tirrrc 7'is

4 : t (, ,1!,t (tt.1.4)

The fact that there is no failure during the time interval Z means that thc largcstwind effect Q occurring during that interval is less than R. Thus Eq' 8.1..1yields the cumulative probability distribution of Q corresponding to the valtrc

Q: R:

Fo(Q<R):g-'nr (8. I .-s l

whcre zp is a function of R. In particular, if Z: I year, Eq. 8.1.5 representsthc cumulative distribution function of the largest yearly wind effect, that is,thc probability that thc largcst actual wind effect in any one year is less than lspcci(icd wind effect R.

Thc mcan outcrossing raLc vp may be obtained by using Eq. A2.47, whichis valid r,rnclcr thc assumption that the random process is stationary. If polarcoordinatcs arc uscd, it fbllows from Eq. A2.47 that

ll I I'llr)r I lrlllll :, l ()ll l:;llMn llu(i l'ltl,lll\ltll ll, Irl',iltlill,ilillj

l:rlgt'.'l'ltt'sc rlist'tt'p:trrt'rr's tltrr lrt':rlllilrtrlr'tl lo llrt'rr,,r'r,l rrrrrl,,1rr.r.rl rl.rl;r;rssocitttctl lo lr l:rrgt' t'xlcrrt willt ltrt'lt'onrlop,rtlrl Plrt'rrurrrr.u;r {r. l, rronlntt,lrtcczcs) thltl itrc ttttrcl:rlctl to tltc slt'ott1'. willls ol lrr(t'rt.,,l lt .,lltr lul;rl ,lr.,,r;'rrAs ntllccl in Scc:t.3.2.3, such n:ctlttls crut lrlrvrtlt':r rrrr:,lr':rrlnrl', lr.l,r:, l'r .,1.r

tislical inf'crcnccs ctlnccrning cxlrcnlc: wirrrl spt'r'tls l,irr tlrr:, r(':r,on, rnr1,..,..rt'liablc cstimates of the tcrrns lifflIl,,lllt0l ,r](//)l rrrrtl .l ,, ,ltt.l,ll :ur. rrurtlt. orrtlrc basis of data pertaining to stft)ng wirttls, llrt'rrrt'llrtxl n'vit'wt.rl irr llris st.r.liorrr';ulnot be used with confidcncc lirr stnrc(unrl tlcsigrr l)utl)()srrs.

B-1.2 Procedure Based on the Time Series of the Largest YearlyWind Effects'l'lris procedure is applicable if the wind effect Q@) can be described by an('\pression of the form

Q(0) : )pC@)C,,(0)U'(h, 0) (8. r.8)

rvlrcre p : air density, C(0) : coefficient transforming wind load into windcll'ccts (if Q(0) is a wind pressure or suction, C(0) : l), Cp(0) : aerodynamict'rrcfficient corresponding to wind blowing from direction 0, and U(h, 0) :rncan wind speed corresponding to the direction d at the reference height lr:rbove ground. It is assumed that the influence coefficient C(0) is independent,tl' U(h,0). It is also assumed that the coeflicient Q,(0) is independent of or,rrrly weakly dependent upon U(h,0). These assumptions exclude from consid-t'ration members subjected to significant dynamic amplification or aeroelastict'll'ccts.

'Ihe details of the procedure discussed in this section differ according towhether the region being considered has a well-behaved wind climate or cant'xperience hurricane winds. The two cases are therefore treated separately.

Struclures in Well-Behaved Wind Climates. Let U,(h,0) denote the largestvrrlue of U(h, 0) during year i. The largest wind effect Q; during that year istlrc largest of the values Qi(0) obtained by substituting U,(h,0) for U(h,01inlrc1. 8.1.8:

e, : )p max[C(0)Co(ilU?(h, 0)] (8.1.e)

Notc that in conventional engineering practice, wind directionality effects arerrtrl taken into account; that is, the largest yearly wind effect calculated for,lcsign purposes, Ol""', ir assumed to be given by

,o: I" ,Etr 1it,(')lu(,) : s(0)lfu,ots(01, ,l It(8.1.6)

[8-1, 8-2], where U, : derivative with respect to time of the projection of thcvelocity vector U(0) on the normal to the boundary g@), U(0) : wind speed,

fu.o : joint probability density function of wind speed and direction, and Efi': average of the positive values of U,, given that U(0) : g(0).

Attempts to evaluate the mean rate vp have been reported in [8-1] and

[8-3], where in addition to the assumption of stationarity of the wind velocityprocess, the assumption that U, and U are statistically independent was used'SO

Etrtu"@)lu(0) : s(o)l : Efftu,@)l (8.1.7)

The quantity EfflU,(:0)l can be estimated fiom spectra of wind velocities onthe basis of Eq. A2.l6a. Under the assumption of stationarity, these spectnrand the probability density fu.oof Eq. 8.1.6 can be estimated from continuotrswind velocity records or from comparable types of records, such as wintlvelocities recorded at one- or three-hour intervals [8-1, 8-3]. Once Eff[U,,([t)landfu., are obtained, Eqs. 8.1.5 and 8.1.6 can be used to estimate the ctrmulative distribution function of the largest yearly wind effect.

In [8-3] largest ycarly wind loads estimated by thc procctlrrrc tlcscribcd irrthis section wcrc conlparcrl in a nurnbcr ol cascs witlt llrrgt'st ycrrr-ly lo:rtlsobtained on (hc bltsis ttl'itc(tutl trtcasttrcttlcnts. 'l'ltc rlistn'Plttrt'it's lrt'lwt:cll tltt'rcspcctivc cunrrrllrlivt' tlis(t'ilrtrliolt lirnc(iorts wt. tt' lotttttl lo lrt' ttltltt t't'pllrlrly

Onlrrrr ]/) frurxl('(//)(),((/)lnrt;x1U!{h, 011

l,1t(',r,,.,r1 l,' {l t I

(8.1.10)

(lJ. r . r r)

312 wlNt) t)lll oil()Nnt ly I l tct:i

whercU;(h) - maxp[ U;(/r,0)l dcnotcs llrc lurgcst uttnttal wittcl spcc:cl tcgarcllcssof direction, and

C*o^ : max[C(0)C,,(0)J0

(8. r. r2)

Example Consider a 100-m tall building located in an urban environment, forwhich it was estimated in [8-5] that U(h,0)lUJ(0) : 1.39, where Uy(0) :f-astest-mile wind speed at l0 m above ground in open terrain, and lr : 3lttm. The largest yearly fastest-mile wind speeds Uf.(0) recorded during a givenycar in the region being considered are listed in row I of Table 8.1.1. Thcr.ncasured peak suction coefficients Cp(O) reported in [8-5] for a cladding panellocated at 94-m elevation near a conrer of the building are listed in row 2 ol'Tablc 8.1.1. Thc corrcsponding suctions, Q(0), calculated by Eq. 8.1.9 inwhich C(g) = I ancl p : 1.25 kg/m3, are listed in row 3 of Table 8.1.1, andrepresent thc largest suctions induced in the cladding panel by winds blowingfrom the cight directions of the compass during the year being considered.

It is seen in Table 8.1.1 that the largest suction induced during the yearol'concem by winds blowing from any direction is

Q1 : maxlQ(0)l0

: 703 Pa (8.1.13)

If wind directionality effects were not taken into account, it would follow fromEq. 8.1.11 that Ol" : ) x 1.25 x 3.33 x (1.39 x 31.3)2 :3939 Pa. It isseen that in this example the value obtained by ignoring wind directionalityeffects is considerably higher than the actual value of the largest yearly windsuction, Qt : 1O3 Pa. Note, however, that this would not have been the caschad the directional distribution of the wind speeds and/or of the suction coel'-ficients been relatively uniform, or had the directions corresponding to thcmaximum values of Ct,@) and Uy.1(0) coincided.

If extreme wind speed dara Ui(h,0j) (j : 1,2, ... , 8 or 1,2, ... , 16)are available for a su{licient number n of consecutive years (:e.9., m - 20),a set of largest yearly load data, Qi, (i : 1,2, ... , m) can be calculated by

TABLE 8.1.1. Largest Yearly Wind Speeds, Suction Coefficients, and LargestYearlv Sucfions

DiTectionNNEESESSWWNW

2.J

u1.{:0)(mis)ct,(0)Q,@)(l'a)

t2.5 8.9 22.3 10.3 22.3

0.51 0.61 0.(r(r106 ll(r .)()(r

22.8 3 r .3

l.12 0.2,1/o i 284

0.07 l.(Xrt3 tot

r0.3

426

IIrt lltl :, I()l l l:;lllvln llN(i I'l t()lrnlilt ll, ttr',ililnililr,il', :ll:l

rrsing I,)t1. lJ.l.(). linltr llrt'st'tllrtlr it is lxrssrlrlt- lo r.:,llnt;tlr.llrc l11.1,l 11111111,

rlrsllibtttiott ol'Q trntl v:rriorrs stirlistics tlrirl lrriry lrt.rr:t'rl lul .1...,,,,,, l)lulx)s(.5()ttc sttch slalistic is tlre wintl krlrtl Q,y col't1'slxrrrtlrnli lo ilry nr(':ur r('(ulr('n(('rrrlcrvtrl N.

It is convonicnt lirr ctlntpulatiottal prtrpost's lo tlt'lirrt' llrc rlrr;rrrlily, rt'lt.r'lt'tlto lrs cquivalcnt wind spccd,

(8.1. l4)

rvlrcrc C,-o^ is defined as in Eq. 8.1.12. The largest yearly equivalent wind:;pccd during year i is

, t \ nurx,,l Q({/)l }r"'' - | ]pc),,,,,* J

(8.1. ls)

Stltistical analyses of sets of data, Ueq,i, reported in [8-6] have revealed thattlrt' probability distributions of the largest annual equivalent wind speeds mayI't' assumed for design purposes to be Extreme Value Type I. This assumptionrs trsually conservative. Note that if the equivalent wind speeds have an ExtremeVrrlue Type I distribution, the distribution of Q is not Extreme Value Type I.

I-ct U.op denote the equivalent wind speed corresponding to a N-year meanr('currence interval. Assuming that the distribution of U"q is Extreme Valuel'ype I, it is possible to write (see Eq. 3.2.1)

u.q,i:l+r,*,*)"

U"qN = a"o * b"o ln N

a.q:7.q - o.45s"o

b.o : 0.78s.0

(8.1.16)

wlrore

(8.1.17)

(8.1.18)

;rntl X.o and s"o : sample mean and sample standard deviation of U.0.,. Thewincl effect Qlr corresponding to the mean recurrence interval N can then bervrittcn (Eqs. 8.1.15 and 8.1. i6) as

Qn = ipC,-u^(a.q I b.o ln N;2 (8.1.19)

It is clcar that failure to takc wind directionality into account (i.e., the use oflrrrgcst ycarly wind effects oslirrrirtctl by Eq. 8.1.10, rather than by Eq. 8.1.9)wottltl rcsult in some citscs itt trrrreirlisticirlly inllatccl cstimates of the wind loadtoltcsptlncling to an N-ycrrl rlr(':ur rr'( unt'rtt'c irrlcrvtrl. This is shown in thclollowirrg cxltrttplc, prcse:ttletl irt tlr'l;rrl ()n :r('r'()lnrt ol'thc plrcticul ilrrl-lol'lutrccol sttclt crtlculrrliotts irr clirtklirrp', 1,,1;r:rs tlt'srlirr

*314 wtNt) t)ilil (:il()Nnt ily I ll t( t,,

Example'l'hc largcst ycurly Iirs(csl rrrrlt'wrrrtl spt'r'tls lrl l0 rn irlrovt'glorurtl irropen teffain, U7, clbtaincrl lirrrrr rt't'oltls (rrkt'n rrl Slrcritlirrr, Wyorrrirrg, irr llrt'period 1958-1911 arc listcd in rnph irr 'l':rlrlt' 13.1.2. (Surrrrrrary sttrtistics lirrthese data are shown in Fig. 3.4.1 .) Wc scr:k thc -50-ycar wincl-inrluccrl suc(ion,QN:so, on the cladding panel of the prcvious cxample, lor which thc acrodynamic coemcients are given in row 2 ol"lablc 8.1.1, and the estimat.cd ratioU{h,0)lUJ(0) is approximately equal to 1.39. From Eqs. 8.1.9 (in which C.(//): l) and 8.1.15, it follows that the largest yearly equivalent wind spcctlsduring the period 1958-1977 have the values shown in Table 8.1.2, where thccorresponding sample mean {o and sample standard deviation s"u are alsoshown. From Eqs. 8.1.16 through 8.1.19, h:so:974 Pa (20.3 psf).

If the load is calculated without taking wind directionality into account, thcnominal 5O-year load is (Eq. 8.1.11)

(8. 1.20)

TABLE 8.1.2. Largest Yearly Fastest-Mile Wind Speeds, Ur, Loads, Q,, andEquiralent Speeds. {,,;.,.t

Largest Annual Fastest-Mile Wind Speed at 10m above Ground in Open Terrain (mph)"'"

Year SW NW Q,(Pascals) U"u.i(m/s)

(:l l)l,l ll l; l()l I llillMn llN(, I'il(ltiAilil il r trl..iltilrt lil()N:; 315

wltct'c {f lr -50-yclrr-lirs(cst-ltrilc wirrtl slx'r'tl (rrr rrr/:,) t'strrrurlrrtl l.nrrrr thc sctol'largcst ycarly spcctls rcgartllcss ol tlircclrorr. lirorrr llrt: tllrttr ol-'l'ablc 8.1.2,I/1so:74.25 ntph (33.2 nr/s), irrrtl Qfi"",,, ,l,l,lo I,l (92.5 psf), versus the:rctual -50 ycar load, 0t s, -,914 l'rr (20..1 lrsl ).

Directional largcst yearly lirstcst-ruilc wirrrl spccd data at a number of weatherstations in the United states arc availablc in [8-7]. Similar data that may berrsed for design purposes can also be obtained fiom monthly Local climato-Iogical Data summarics publishcd by the National oceanic and AtmosphericAdministration (see Sect. 3.4).

structures in Hurricane-Prone Regions. In hurricane-prone regions the loadrlata used for inferences conceming design loads are not yearly maxima. Rather,lhey are associated with hurricanes, which occur at irregular intervals. Therrpproach used in this case is the following. A large number m of hurricanes isgcnerated by Monte Carlo simulation on the basis of climatological informationtrn hurricane storms, as shown in Sect. 3.3. For each hurricane the load e; andthe corresponding equivalent wind speed U"o,; (see Eqs. 8.1.9 and 8.1.15) arethen obtained. Following exactly the same steps used in Sect. 3.3.2, the cu-rnulative distribution functions of the largest load and of the largest equivalentwind speed occurring in any one year are found to be

NEFo(Q < Qi): Fu"r(U,,t< U.q.i)

_ - Xll-i,ttr{ t)l (8.1.21)1958 28t959 411960 361961 251962 221963 3rt9& 221965 33t966 361967 441968 361969 28t970 28t97t 331972 23t9l3 281974 241915 221916 31t97l 44

20252tl823t4l5312ll4l9l6l3l5l92328222420

18.413.916.318.618.7t7.619.418.514.215.014.5t4.917.013.516.3t4.2t] .lll.t317.615.1

10339955371872864378270841946443845959838455341965328964t35 t.\

23 50 23 50 sl 7019 29 25 40 38 6016 34 26 43 4s 6030 36 21 47 38 6022 36 16 41 52 6023 33 36 63 48 s'718 34 19 54 54 6020 33 t7 66 43 5519 34 14 51 39 6I16 40 36 51 4t 6219 3s 2t 39 40 4715 36 22 53 34 6620 35 37 61 37 s322 31 22 49 31 4126 36 37 55 44 47t9 32 15 46 39 64t9 37 25 57 49 5619 27 28 39 33 st28 33 38 47 33 4719 40 36 34 44 s6

where \ is the annual occurrence rate of hurricanes in the area of interest forthe site being considered. Continuous probability distribution curves, F(e <17) and F(U"q I u"o), that best fit Eq. 8.1 .21 (e.g., reverse Weibull or ExtremeValue Type I distributions) can be estimated by using standard statistical tech-rriques. Note that the mean recurrence interval of the load e, and of the equiv-:rlcnt wind speed U"o , is

I _ e-Atl i/(m+ t)l (8.1.22)

A similar approach is reported in [8-4].A computer program for estimating hurricane-induced wind loads in accor-

tlirnce with the procedure outlined here is described briefly in [8-8], and isrrvailable on tape in [8-91 . Stored in the program are hurricane wind speeclst'orrcsponding to the 16 compass rlircctions at 56 mileposts located atdistanccsol -50 nar-rlical rnilcs along thc: (irrll'rrnrl Atlunlic coasts (see also [3-7 ll). Thcscspcctls worc <lbtaincd 1'rolrr t)(X) Irrrlrit'lrrrt'wintl licltls gcncratecl hy Morrlc ('trllosirrrrrlir(iorr lrI cach lllils:1.rosl , rrs tlt.sr.rilrt'rl in St.t.l,3.3.2, unrl wcr.e rrst'tl irrll"i l0l rrntl irt llrc tlcvt:lo1'rlttt'nl ol llrt'rvrrtl :;;x't'rl rrr;rp irrclu<lc:tl ilr llrt'Arrrt.rit.;rrrNlrliortrrl SllrrttLrrtl n.58. I l()li.) lt{ I ll

Notc: X.., - 16.11 rn/s; ,r,.,, 2.1 I nr/s."l rrplr - O,447 nr/s./'V:rlrrt:s irr ilrrlits:rrc l:ul'r'\t y(''lrly rvirrrl spt't'rls lirrrn;rll rlirctlrorr,

316 wtNt) I)iltt (;il()NAt ily I IIIcr

In cases whcrc it is.iutlgctl tlr:rl tlrt'lltrb:rlrili(y tlistlibrr(iorr ol lhc lirrgeslyearly loads, Fy(Q I q), ntay bc irlleclcrl by lxrtlr hurricanc urrtl rllnhrrrricrrncwinds, the following expression should bc uscd:

Fo(Q < 0 : Fa"(Qn ( 4)Fo*"(QNu < 4) (8. r.23)

where Fq^(Qs < 4) : cumulative distribution function of hurricane-induculwind loads, Qs, estimated as shown in this section, and Fp*"(Orvu < Q) =cumulative distribution function of loads induced by nonhurricane winds, cs-timated as shown in the previous section (see also Sect. 3.3.2).

8.1.3 Procedure Based on the Univariati Probability Distributions ofthe Largest Yearly Wind Speeds Recorded for Each of the PrincipalCompass DirectionsA simplc pnrccdurc is now presented that may be applied to any type ol'structurc, including structures subjected to aerodynamic amplification of aero-elastic effects.

It was pointed out in Section 3.4 that the correlation between extreme winclspeeds occurring in any two directions is generally weak. As shown in Ap-pendix Al (Eqs. A1.64), two uncorrelated variables having a joint ExtremoValue Type I distribution are statistically independent. It can be shown thalstatistical independence also holds for any number of uncorrelated variablcswhose joint distribution is of an extreme value typelAl-241 . In practice it cantherefore be assumed that the largest yearly winds blowing from the eightprincipal compass directions are statistically independent. The cumulative prob-ability distributions of the largest yearly wind effect may thus be written as

Fo(Q <R) : Prob(ut 1 r'r, u2 1 ut2, ..., us < uL) (8.1.24a)

: Prob(rrr < a{) Prob(t,, < ut21... Prob(u3 < r.,/3) g.l.24b)

where erj is the wind speed from direction i causing the occurrence of the wintleffect R [8-14]. Note that if the wind speeds occurring in all directions wcrc:perfectly correlated, then

Fo(Q < R) : Prob(ar < utr) (8.1.2s)

where I < k < 8. Equation 8.1.25 indicates that Eq. 8.1.24b is conservativcfrom a structural design viewpoint. Bonferroni techniques applicable to bivar'iate extreme value distributions were used in [8-15| to cstimatc bounds lirrprobabilities Fo(Q < R). The estimates showcd that in clinrirlcs rx)( pK)nc (ohurricane occurrenccs, Eq. 8. 1 .24b typically ovcrcsliuurlcs rrnrr;rl l:rilurc prrrl'labilities by a lactol lrl'lcss than two. Sirrrilar (cclrrritlrrr's rrury lrr'rrpplicrl tohurricanc wirrtls" lirl t'xrrrrrplc, rrsing cs(ittt:rlctl tlittt liorr;rl tl:rl;r :rv:rillrlrlc irtllt 9l or 1.1 7ll

llllllll:. nlllr',r\il iy it\ll,l{.1:; ,lll

8.2 ESTIMATION OF FAILURE PROBABILITIES AND SAFETYINDICES FOR MEMBERS SENSITIVE TO WIND DIRECTIONALITYEFFECTS

'lo detemrine whether a mcmbcr scnsitivc lo winrl tlircctiorlrlily cllccts is rrr'ccptable from a safbty point o1' vicw, ils rrorrrinirl lailLrrc probability (or itssafety index) is compared to that ol' a rtrcrrrbcr.juclgcd to be acccptablc. 'l'hcrnember is then redesigned as nccdcd until thc rcsult of this comparison issatisfactory. An application of this rcliability-based approach to the design ofglass cladding for a tall building is presented in Chapter 9. This section de-scribes procedures for estimating nominal failure probabilities and safety in-tlices required for the application of this approach.

8.2.1 Estimation of Failure ProbabilitiesConsider a member whose resistance is R, and denote by Q the largest loadcffect acting on the member during any one year. Failure occurs for any pairof values R, Q such that R - Q < 0. In most applications R and Q may berussumed to be independent, so the probability of failure in any one year canhc written as

(8.2. r)

(Eq. ,{1.21), where Fa : cumulative distribution function of R, and fq :probability density function of Q. The function/q is related by Eq. Al.ll tolhe cumulative distribution function, Fg, estimated as shown in Sects. 8.1.2rrnd 8.1.3. The probability of failure during the n-year lifetime of the structuret'an be obtained from Eqs. 8.2.1 and A1.31. The probability of failure soobtained is conditional upon a given set of values of the random parameters(lrat determine the functions R, Q, Fn, and fq. Conditional probabilities ofllilure can be useful in certain applications in which the objective is to assesstqualitatively the relative reliabilities of various members.

Unconditional failure probabilities can be estimated by using an expressionsirnilar to Eq. A3. I, provided that (1) the probability distributions of the variousrrrrdom parameters that determine R, Q, Fa, and fpare known and (2) sucht'stimates are not computationally prohibitive. In a number of situations ofpractical interest it is in principle possible to use reliability-based design meth-rrrls that employ the safety index as a measure of structural reliability. Never-tlrclcss, difficulties pertaining to thc choice of the target safety index for at leastsorrrc situaticlns remain unsolvr.rtl.

8.2.2 Estimation of Safety lnclices'l'lrt'ltursl t'orttlttttttly ttst'tl s;rlt'ly ur(l('\ ()rr rvlri,lr rt'li:rlrilily t';rlt'rrl:rliotrs irrt'lr;rst'rl lr:ts lltt' t'xPn'ssiort

(.-P, : Jn Fn@tfs@) dq

318 wlNl) t)ililcil()Nnt ily tiltct

u(h,0)

l1'l(JtJlllt tt.2.l. Wincl dircction, 0, and angle of orientation of structure, a.

lr:'r:;ilMn|ol.|'rt tnilt,ilt I't t()iln||illilt :,At.lt ,,.r\t tt, Ilt rt{t,

itl tt:rltts tll'tttt';tlts. t'ot'lltt'lt'ltls ol vltt'iltlirltt, ,lilrtl t ont'l.rltotr t ot'llrr rr.1l., 6l llrt.lrlttlorrrv:rriirblcsl,(i l,l,..,,a)irlrrl .\,(/ ,u I l.rti I .'. rtl.lrighcr-tlrclcr tcrtns irr tlre:sc cxprcssions lrt.ilrp rrt.1ilt.r.{t.rl l lior r.r;rrrrlrlr,:, ol :,rit.lrclrlculations, soc uqs. A3:26 n3.2tt irrtl lll lll)r lrr rno:.r rlt.:,r1,11 :,rru;rtr()n:,tlrlrt involvc wind action. R arrcl Zn errrt lx't'slinurlt.rl rrrlt'pt.rrtlt.rrtly ol (),,;rrrtlvo,, I

Sfructures with Specitied Orientation. 'l'lre t'rrrrrrrlirtivc: tlistributiel lirlc-lion Fy"o,, of the largest lil'ctirnc c:(luivalont wirrd spcccl 4u, ili

F t t ",,,,(u ",))

: lF u"u(u.,)1" (8.2. s)

f ru,,,rr,

(lrq. A.3.2), where n : lifetime in years and Fu.,t: cumulative distributionlirnction of the largest yearly equivalent wind speed u"o obtained as shown insoct. 8.1.2. From the distribution Fy"u,, it is possible to estimate the mean u"on:rnd the standard deviation J(a"q,).

Associated with the largest lifetime equivalent wind speed u.q,, is the largestlil'ctime wind effect Qn (see Eq. 8.1.14):

(8.2.2)

(see Eq. A3.29), where R and vp: mean value and coelficient of variationof the limit state, and Q, and vg,, : mean value and coefficient of variation ol'the largest lifetime load effect.

we consider here only members that do not experience significant dynamicamplification or aeroelastic motions [8-13]. Load effects for such members canbe described by Eq. 8.1.8. Expressions for Q, and V9, are first developed ftrrthe case of structures with a specified orientation angle, a (Fig. 8.2.1). Wcalso consider the case where gravity loads, in addition to wind loads, arcpresent. we then treat the case of structures with unknown orientation, whichis of interest for the development of building code provisions on wind loads.

In general,

Q,: A, + Q',

: itB + p') (e,^o^ + c,^^*) 1O"o, + U!r,)2

^ lnn-nQ,u: vTii;trwhcre the bars and primes indicate mean values and deviations from the mean,rcspectively. we assume that the values p and c^"" used in calculating u"0,,(lrc1. 8.1.15) are the mean values of these two variables. The following ap-prrrximate relations follow from Eq. 8.2.6:

I Vtr"o,,)

vL",,,

(8.2.6)

(8.2.7)

(8.2.8)

Q, = lpC,,'u^A?r,,(l

vb,, = vl + vzr^.^, +

R:R(Xr,X2,...,X.)Q, : Q,(X,+t, Xm+2, . . ., X,)

tThe cxprcssitlns lilr /? ltlttl Q, ttt:ry tortliritt lr rrrrrrrbcr ol toltrrrrorr v;rrr;rlrlr':,. llr;rl is. .ysornc vafrrcs r -: ltr :rrrtl i - trt I l.

where, for example, the random variables XiG : 1,2, . . . , m) may denotcmemberdimensions and material strength, and X1(j : m * l, m I 2, . . .,n) may represent aerodynamic and micrometeorological parameters. * Equations8.2.3 and 8.2.4 can be expanded in Taylor series; approximate expressions forthe mean values and coefficients of variation of R and e,, can lhcn bc obtaincrl

'l'hc coefficient of variation, 26,,,"*, reflects the variabilities of the influencet trclhcients c(0) and of the aerodynamic coefficients Co@). For example, theinllucnce coefficients transforming pressure on cladding into maximum tensileslrcss in the panel depend upon the panel thickness, which, forthe same nom-rnirl thickness, may actually vary somewhat from panel to panel. Aerodynamicr'ocllicients for any given structure can-and usually do-vary from experimentlo cxperiment. It is possible to writersirtrPlcr rnanipulations are possiblc in ceftain instances; see Eq. 8.2.6 and subsequent deriva-lr()lls-llris is lnlc r:von in (hc casc ttl gl;tss cl:rrklirrg, which experiences latigue under wind loading,ttrl wltosc sllctlSllr is inllttcrrtt'tl l)y lll('nirlurc ol thc wincl prcssure fluctuations. As shown in( ltrrPlt'r (), lhis ittlltrcncc crttt lrr' ittr'orlxrr;rlt'tl irr llrc cxprcssion ol thc load Q, so values of R an6ll t;ttt lrt'trsctl ilt lit;. IJ.2.2 tlr:rt (()l('rlrrrrl kr rrrrrvt'rrliorlrl l():l(ling pa(tcrns indcpcndent ofthelrtnt ltislory ol lltr':rt'lrr:rl wilrrl lo:rrlr {lol t.rrltrIlt'. l(, (()usl:utl lo:rtls ol (r0 s rlrrnrlion).

(8.2.3)

(8.2.4)

X, lin

*320 wlNl) l)llil oll()Nnl llY llll(.1:;

vi,_,, t ,' r l'r ,( t{.2.u;

where V6 and Vg,, are the coeflicicnts ol' vitr-ilrtittlt ol' C:({/) antl Q,(0)-'r'We now derive expressions for U.u, and Vu",t,, lirr thc casc whcrc it nray bc

assumed that the data (J"r,1(Eq. 8.1.15) are best fittcd by an E,xtrernc ValttcType I distribution. As mentioned previously, this assumption is gcncrrtllyconservative. From Eqs. A3.8-A3.9 it follows that

U"q, : X"o + 0.78s"q ln n

s(a.',) : s.o

whcrc X,.,, ancl s,.,, arc dcfincd fbllowing Eq. 8. I . 18 and n : lifetime of structurcin ycurs.

'lir cstirttirlc Vrt,,r,, wc considcr the relation

O ",r, :,,,c2c-,coc"D![f (8.2.t2)

where U"u,, and 7!!,," : estimated and true (but unknown) mean value of thclargest lifetime equivalent wind speed, and c1, c2, c3, c4, c, : coefficients witllmean equal to unity that reflect, respectively, (l) errors in the measurement ol'the fastest-mile wind speeds over open terrain, (2) errors in the transformatiollof the fastest-mile wind speeds over open terrain into mean wind speeds at l0m above ground in open terrain, U0(10), (3) errors in the transformation ol'U0(10) into mean wind speeds at 10 m above ground near the building sitc,U(10), (4) enors in the transformation of U(10) into mean wind speeds at thcelevation, h, near the building site, u(h), and (5) sampling errors in the esti-mation of (J.q, due to the limited size m of the sample of data U"r.iQ : 1,2,. .. , m). The coefficient of variation Vu.r, can therefore be written as

vru.u,: vL, + vl, + vf,. + ,i^ * *. lff (8.2.13)

whete V,, (j : 1,2,3, 4) are the coefiicients of variation of c;, and s, ''standard deviation of sampling error in the estimation of U.on.

Approximate estimates of the sampling errors s, can be obtained by notingfrom Eqs. 8.2.10 and 8.1.16-8.1.18 that, fot n:50 years, Ueqn:5gyr =U"qru:soy' so that the respective sampling errors are approximately the sanrcfor these two quantities. From F;q.3.2.2 it then follows that

4tt = *-

t",

where ru : size of data sample U"u., (i : l, 2, . . ., nt).

+Equations tl.2-7 lrrxl ll.2.ll:rrt: trpploxirrurtc hccitttsc (l) lcrrrrs rtl rtrtllt lttl'lrtr llt;rlt lw(l illt'ncglcctctl,:rlrtl (2) it is;rssrrrrrul llurt llrt'vlrriirbililit:s ol lltt'llttkrtr, llt,tl rIlcrtrtlttt't,,,,,, lrrrtl l/,.,,,,

,h'1x'rrl lrt l'lif iltlv tllr'lt rlltt r ltoll

Example lisl irrrtrtt. tlrt' srrlt.ty rlrrlt'x, /i (llt1. l{ .r..)). lor /r' /.5 psl'1}.590 l)u),V4 - O.2), p 1.2.5 k11/rrr\. (, 0.0.5, (),,,,, t.|.1. f( U, Vcr,:0.i,X.u,, 16..17 rrr/s.,.r",, l.l I rrr/s, irrrtl rrr 20 (scr. ,l'irblc

lJ. 1.2), 1,, : V,,: V,, : V,., : O.05, rt . -50 ycars.From Eqs. 8.2. l l, ti.2. 10, 8.2.t4, 8.2. 13, 8.2.9, 8.2.8, anc| 8.2.7, s(u"en)- 2.ll.,mls, U4t,,:22.61 mls, J., : 1.89 rn/s, Vu,,tn: O.tOt, Zc.^,u* :0. I,

Vo,, : 0.3+0, Q, : lO92 Pa. From Eq. 8.2.2, p :'2.94.case where Gravity Loads Are present. Equations g.2.7 and g.2.g areapplicable if the gravity loads acting on the member may be neglected (as inthe case of cladding panels subjected to wind roads). However, if"(l) the effectof the gravity load is significant and, (2) the most unfavorable load combinationoccurs when the wind load reaches its largest lifetime value while the gravityload has an "arbitrary-point-in-time," rather than an extreme, value.* then

Q,:Qao-o + G

v;: ' vba"-o-+ YL

f r + --c )' /, * o'"o\'\ Qac=o/ \' C I

8:' t:;ltM/\il(irl o, tAilUlu I'tt()ilnillt ililr. nNt r.,/\t tty tNl)t{:t :t :12 I

(8.2.1s)

(8.2.16)

(8.2. r0)

(8.2. l r)

(8.2.14t

l!"1" O"tg: g ?nd.Vo,l.":s : me?n value and coefficient of variarion of largestlil'etime wind load esrimared by 8q.8.2.7 and g.2.g. respecrively. and 6indv6 : mean value and coefficient of variation of "arbitrary-poinrin{me"gravity load, respectively.

structures with unknown orientation. A procedure is now presented forr:stimating safety indices for members of structures whose orientation is notknown. such a procedure can be useful for the development of building codeprovisions on wind loads.

The unknown orientation of the structure can be considered as one amongscveral uncertain factors that determine member reliability (member resistance,:rcrodynamic coefficients, influence coemcients, etc.). To the extent that struc-trrre orientation is included as a random variable in a properly conducted reli-rrbility analysis, the reliability of members in a structure sampled at randomwill be acceptable regardless of structure orientation, just as the reliability of:rrry properly designed steel member will be acceptable even though, owing totlrc variability of the steel strength, actual yield stresses might be lower in somet':rscs than the average yield stress.

'l-he mean value and thc variance of the largest lifetime loads acting on therrcrnber undcr consideration, avcragcd over all possible angles of orientationrv ol'thc structurc to which llrc rrrcrrrbcr hclongs (Fig. g.2.li are

l"a,, I ql,,,,r)/(,r) r/rv

.lo

'Sct' tlisr'rrssiorr irr A1r1x'rrrlir A l lqllrrs 1rl, I tl .\ I ,)

(8.2.11)

.t322 wtNt) t)ilil (;il()Nnl ilY I ttlcr:;

(Q.,, - Q,,)'' - 10,,(rv) 0,,(,v)l.r l'(u) tlu (n.1. llt)

where Q,@) : largest lifetime wind load acting on the rncmbcr givcn that tlrcangle of orientation of the structure is cv, and/(cy) : probability density functiorrof structure orientation in the region being considered. It is reasonablc to lssume that a is uniformly distributed, that is,/(cy) : ll2r. Other assurnptitutscan be made, as necessary, if predominant structure orientations are known toexist.

In addition it is assumed that wind speed data are available from 8 compasstlircclions. (ll-data arc available from 16 directions, the number 8 in the equatirrrrs that lirlkrw rnust bc changed into 16.) Using Eqs. 8.2.11 and 8.2.lti itcarr bc slrowrr alicr soutc algcbra that

lil Iilrl ilr I

n5

ll. V.'l ty1i11vrrsorr, l) Srrrry, irrrtl A. (i. l);rvt'rrporl. "l'r,',1r, lrrrll Wrrrrl lrrrlrrrt'rlllcslxltse irrlltrrit':urt'Zorrcs,".l .,\urttt. /)rr',AS('l'., l(lltlrcr l()l(r),.rIII2350.J. A. Pctcrk:r anrl .l . lr. (lcnturk, llitul Iirtttrtl Strt,lt',,1 .ltl.ntttt ()111,'t'lilttlrltrt,tl.Fluid Mcchanics and Wincl lingirrccriltg l'trr1r,r':rrrr. ('ollt'1it' ol llrrl'rr11'1'1i111i. 1',,1orado Statc Univcrsity, Ft. (lollirrs, Nov. l()7t3.E. Simiu and J. J. Fillibcn, "Winrl l)ircclion lrllccls on ('lutltling lrrl .Slluc(ululLoads, " Eng. Struct., 3 (July l9t3 I ). ltt I ltt(r.M. E. Changery, E. Dumitriu-Valcca, and E. Simiu, Directional Extreme WindSpeed Datafor the Design ofBuildings and Other Structures, Building SciencesSeries BSS 160, National Bureau of Standards, Washington, DC, March 1984.E. Simiu and M. E. Batts, "Wind-lnduced Cladding Loads in Hurricane-ProneRegions," J. Struct. Eng.,109 (Jan. 1983), 262-266.

rJ-7

l't 9 Hurricane-Induced Wind Loads, Computer Program, Accession No. PB82132259, National Technical Information Service, Springfield, VA, 1982.

tt-10 M. E. Batts, L. R. Russell, and E. Sirniu, "Hurricane Wind Speeds in theUnited States," J. Struct. Dlv., ASCE, 106 (Oct. 1980), 2001-2016.

t't I I American National Standard A58.1, Building Code Requirements for MinimumDesign Loads, American National Standards Institute, New York, 1982.

U-12 K. Rojiani and Y. K. Wen, "Reliability of Steel Buildings Under Winds," -/.Struct. Div., ASCE, 107 (Jan. 1981),203-221.

t{-13 E. Simiu, "Aerodynamic Coelficients and Risk-Consistent Design," J. Struct.Eng., l09 (May 1983), 1278-1289.

ll 14 E. Simiu, E. Hendrickson, W. Nolan, I. Olkin, and C. Spiegelman, "Multi-variate Distributions of Directional Wind Speeds," J. Struct. Ezg., lll (April1985),939-943.

lJ 15 E. Simiu, S. D. Leigh, and W. A. Nolan, "Environmental Load Direction andReliability Bounds," J. Struc. Eng., ll2 (1986), ll99 1203.

I.:'|,1 ,+

lJ6

r.t-8

Q,:

(vn,, = ftl +

t (t.8.jPcrn,* i,,?, U.-.,,,ta, )2 [ I + vL.,,,(d,)t I ,t.r. 'r,

stf, + v2r,^^.y* t::, u"qJ.,Jo [t + 6 v'r.u,,(o,)l

- L* t:: , u,,,(o,)2 tr

[*t:: , u*,(o,)2 rt +

+ Vrr"",,@,\l j I)

v2u,u,1"'ril j

18-131. In Eqs. 8.2.19 and 8.2.20, U"c,(o,) and V11.r,(a,) are the mean valucand the coefficient of variation of the largest lifetime equivalent wind spectlUrr,(o,), estimated for the structure with angle of orientation cYr as in Eqs.8.2.10 and 8.2.13. Use of Eqs. 8.2.19 and 8.2.20 inEq. 8.2.2 yields the saf'ctyindex of the member being considered in the case where the orientation of thcstructure is unknown. The case where gravity loads are present is treated in ir

manner entirely similar to that shown for structures with specified orientation.

REFERENCES

A. G. Davenport, "The Prediction of Risk under Wind Loading," ProcecdinglZnd International Conference on Structural Safety and Reliability, Munich, Scpt .

1977, pp. 51 l-538.Y. K. Wen, "Wind Dircction and Structural Reliahility." .l . Srrrtct. ling.,lllt)lApril 1983r. I028- r04L

8-l

8-2

(8.2.20,

Y. K. Wcn, "Winrl l)ircction utrtl Stntclttritl llcli:rlrilitv llll0 (l9tt4). 125.1 1264,

8-3 .l .,\trtrt't . l,.rr,q.,

*''w \ a\

PART B

WIND LOADS AND THEIR EFFECTSON STRUCTURES

II APPLICATIONS TO DESIGN

CHAPTER 9

BUILDINGS: WIND LOADS,STRUCTURAL RESPONSE, ANDDESIGN OF CLADDING ANDROOFING

'l'ltc design of buildings is based on estimates of (l) overall wind effects, whichrrrtrst be taken into account in the design of the structure, and (2) local windr'llbcts, which govem the design of components (e.g., purlins) and cladding.lrr gcneral, the aerodynamic information needed to estimate overall as well askrcal wind effects cannot be determined from first principles and must be ob-Irrincd from wind tunnel tests. However, for a number of common situationsllrc acrodynamic information is already available, and procedures for estimatingslructural response which incorporate that information may be employed. Thisis lhc case for tall buildings that (1) have geometric shapes that are not unusualrrcrodynamically or structurally and (2) are not subjected to strong interferencee ll'ccts caused by the presence of neighboring structures. As an approximategrridc it may be assumed that if the distance between two buildings exceeds$ix to eight times the average of the horizontal dimension of the buildings,rrrutual interference effects will be negligible for practical purposes. For morerrlincd guidelines and an excellent compendium of information and referencesrrrt interference effects, see [9-1]. It is noted in [9-21 that a square buildingkratcd in urban terrain near a building with similar geometry and dimensionswill perform satisfactorily, regardless of the relative position of the two build-rngs, if it is designed to withstand the loads (including the across-wind loads)il would experience in the absence of the neighboring structure. See also [9-3,,) 41.

'l'his chapter is divided into six sections. Sections 9.1,9.2, and 9.3 discuss,rcspcctivcly, methods for cstirnating thc along-wind, across-wind, and torsionallcsponsc of flcxible buildings unullbctcd hy intorlbrcnce effects. (Buildings arclclbrrccl lo as .flexible il' thcy cxl)r'ricn('r! sigttilicant clynermic arnplification cl:It'cts cluc to thc acnrdynatnic lottrl lltu'tttutirttts. A rough critcrion is put lirrth

327

328 lllrll l)lN(ili wtNI) t()nl r:;, :,ilil,r ilJt!r\t ilt .,1,{)t]:,t ANt) l)l :;t(.N {}t lror}t il!(,

by the ASCE 7 9-5 Stlrntllrtl l() 51, wlrrt'lr tk'lirrt's lr lrrriltlirrg rrs llt'xrlrlt'il tlrt.ratio between its hcight ancl Ic:rst lrolizorrlrrl tlirrrerrsion is lirlgcr lllrrr lirru'. orits fundamental natural frequcncy ol'vibrlliorr is lcss thun I llz.) l)yrrirrrrrr.amplification effects influence the structural krads ancl can crcatc two kirrtls olserviceability problems: (1) occupant discomfort due to cxccssivc buiklirrg :rt,celerations (see Sect. 15.1) and (2) nonstructural damage due to cxccssivc sr.rydrift. To avoid such damage, some designers limit story clrifi scvercry, c.g. r,l/600 at the design wind speed; see [9-73]. The serviceability problerrs .urybe solved by increasing the structure's stiffness, but in many instancos rrrreconomical complementary solution is to use damping devices. These arc tliscussed in Sect. 9.4. Section 9.5 is concerned with overall and local wincl kxrtlron low-rise buildings, that is, buildings with relatively low height which, owirrgto their relative rigidity, do not normally exhibit dynamic amplification cllct'rsCladding and roofing clesign for wind loads are discussed in Sect. 9.6.

Note that Ihc vast rnajority of available results based on wind tunnel tcs(irr1tor analytical turbulcncc rnodcling were obtained under the assumption that tlrr,atmospheric flow is stationary. In reality some flows, including hurricane llows,are highly nonstationary. Some efforts to study nonstationary flow effects lrrvt.been reported recently; see [A2-14] and [A2-15].

9.1 ALONG-WIND RESPONSE

Until the 1960s drag forces used in structural design calculations were spccilictlon the basis of climatological, meteorological, and aerodynamic consideratiorrsalone, independently of the mechanical properties of the structure, that is, olits mass distribution, flexibility, and damping. It was subsequently recognizt'trthat for modern tall structures-which are more flexible, lower in danrpirrg,and lighter in weight than their predecessors-the natural frequencies of viblrtion may be in the same range as the average frequencies of occurrencc .lpowerful gusts and that therefore large resonant motions induced by wind rrurvoccur and must be taken into account in design.

The resonant amplification of structural response to forces inducecl by rrtmospheric turbulence was first studied by Liepmann in a classic paper orr tlrrbuffeting problem published in 1952 [9-6]. The application of Lieprnlrrrr.:,concepts to civil engineering structures required the development of nrtxk.lrrepresenting the turbulent wind flow near the ground. Such models wcrc l)r(lposed in 196l by Davenport [9-7], who developed on their basis a proccrltrt,for estimating along-wind tall building response [9-81 . vellozzi an,J ('olrt'rrdeveloped a modified procedure, in which, in contrast to l9-81, it is rccognizr.rlthat the fluctuating pressures on the windward face <ll'lr hrriklirrg urc rx)l lx.rfectly correlated to those acting on the lccwarcl lircc l() ()l 'l'lris inrpcr.lr.rrcorrelation is accountccl lor in [9-91 by u r-crlrrcliorr llrttor. llowt'ver'. it lr:r:,been shown that tlwirrg l() thc way irr which llris l:rt'lor.is rrlrplrr.rl. (lrr'llnrt.t'rlurr.of [9-91 untlcrcs(irttlrlt's llrt'r'esorr:rrrt trlrrplificlrtiorr t'llt.tt l,) l()1, l,) Ill. A procctlttrc lilrcslirtr:rlirr1l:rlorr1l wirrrl n'slxrrrst'lrlrst'rl r.r;:rr-rrlr.rlly on l() i^il lr:rs lrt.r.rr

r'r:,r 329

"', lutlt'tl ilt (ltt' (':rrr:rtlr;rr Slrrrr.lrrrlrl l)t.st1',rr M;rrrrr:rl l() l.ll. Vit.kcl.y srrbst:rlu('nlly tlcvclopt'tl :r Ilott'rlrur'slrrrllu lo llr:rl ol l() fil (lurl :rllows" lrowcvcr,l.r lll()lL'llr:xrltility willr n'spt't'l lo llre clroir't'ol ecrl:rirr rrrclconlkrgical param-,lr'rs lt) l3l. An ttltcrl'tt:ttivc rrpptrr:rr'lr is rrst'rl in l() zl-3 1, which utilizcs cquations,,1 crltrilihriullt anlonll ltolizolrtlrl lirrucs trl crrclr llrxrr.

lrr llrc proccdurcs ol'19-l2l lrrrtl l9-l.ll i( is ussurned that the characteristics,,1 rlre turbulcnce do n()t viuy wilh hc:ight abovc ground. Actually, accordingr,' tlrc rcsults clf moclcrn rnctconlkrgical research, the energy of the turbulentllrrtltnttionS that causc rcsonant oscillations in tall buildings decreases signifi-,.rrrlly at higherelevations (see Sect. 2.3.3). Computerprograms forcalculating.rl.rrs 1ryip4 response, in which this decrease is taken into account and which'll,rw therefore more economical designs, have been deveioped independentlyiir l() l4l to [9-16].

()rr the basis of [9-14] and 19-161, simple procedures were deveroped inl't I 7l and [9-18] that account for the dependence of turbulent fluctuations onir,'r1llrl, and on whose basis rapid manual calculations of the arong-wind re-'tx)nsc can be performed. The procedure of [9-18] is easy to use, and it is

, ()nsistent with specifications in which the mean wind profile is represented bytlrt' krgarithmic law. we include it in this chapter for users of such specifica-rr'rrs. ThiS procedure also applies to elevated structures, such as signs whoselrotloln side does not reach to ground level.

'l'he commentary to the ASCE 7-95 Standard [9-l] includes a procedure.r,l;rptcd from [9-18] by A. Kareem [9-19], which accommodates wind climate.rrrrl wind profile information expressed in terms of 3-s basic wind speeds andrlr(' I)ower law, respectively. In addition to being compatible with the format.rrrtl rcquirements of the ASCE, 7-95 standard, Kareem's version has over the1rr.r'cdure of [9-18] the advantage of added flexibility with respect to the choice,'l thc fundamental modal shape. It is available, in interactive computerized1,rp111, 2s part of the diskette "Developmental computer-based version of ASCE/().5 Standard Provisions for Wind Loads" [17-5] appended to this book.*

All the procedures mentioned above are based on the assumption that, aroundrlr. structure, the terrain is approximately horizontal and that its roughness is,r':rsonably uniform over a sufficiently large fetch. In practice it may be nec-, ',:jiuy to adjust the results obtained on the basis of this assumption by takingnrt() llccount the effect upon the flow of changes in the terrain roughness upwind,'l tlrc structure (see Sect. 2.4.1).1 If the topography of the surrounding terrain, ' urrusual, or if the building is strongly affected by the flow in the wake of

I or lruildings in lrurricanc prrltc '('!'.i()ns it ir irnprrn:rn( to verify that thc convcrsion factor liom

t',.r1 lirrsl l() lllcan lrtturly tttcrtrt slx'('(l us('(l t'xplicitly or irrrplicitly in thc calculation pnrccclurcl'|{ (oll\i\l('ttl wi{lt lltc cottvtitsiott l:rr'lor rrsr'rl irr llrr'Sl:rrrrlirnl to ()blllill tlcsign pcak gusl spcc(lslr'rtrl 1|.'1,1r;t\r'titl'('(l r)V\'t lr,il)'( t lilI, Ilt, t\,t1.,l ,rr lrtliltlirrgs lot:tlttl ort lwrt rlrttrcrr"rorr:rl r,l1', r, rrrrrl t:,r rr;rrrrt rrls :rrrrl orr lrxisynrrttt.llit. lrills, ;rrrrr;rlctttt'lltrxl lilt:rltul;tlitrl'rnr.trrr rrrrrl ,.1,rrrl rrr,!r..r,.i.,( .,1r.(.(l 1l)s")isilrr.lrllgrl 1t llt(.nS(,lt/r)'r $l;p1111;1111 l() \l (\('( ScrI .).l ). ('lt,ll)1, r I / rrrrl r!r..l.r'llr. I)r.r't.lplrrrrr.lrltrl ( llrrlrrrlr.l lr.r.,r.rl

\ ( ri l(,ll (tl AS('lr / r)'r Sl:ttrl.ttrl I'tor r',rrrr'. l,,r \\ rr,,l I ,'.',1. I I / ',11 .r1r;r.rrrlr.rl to lll., lr,,,l.

330 lJt,ilt)tN{il; wtNt) t()nt ): ,. t;illl,r.l jltnt ltt :,t ,{)t,l:;t nNt) t)t r,t{;t.J ()t n()()r rrJ(, *large neighboring buildings, arralylicrrl pnrcr:tlrur:s becorrc irrirlrplrt'rrlrle llrrlwind tunnel testing is necessary.

Another assumption common to all thc abovc-rncntionccl pnlccrlurcs is llr:rlthe mean wind is normal to the building face under considcration. Wincl tunrrcltests suggest that, in cases commonly encountered in tall-building clcsign pr-lt.tice, to this assumption there correspond the highest values of the along-wirrrlresponse [9-2, 9-20). In the case of a square building, the peak along-wirrrlresponse decreases continuously as a function of mean wind direction, frorl irmaximum value that corresponds to the case where the direction is normal loa building facc to about 0.8 times that value when the direction is parallcl torr tlirrgorxrl l9-21.'l'hc gcncrltl li'rttttcwork ttl'lhc aklng-wind response problem is presented inSt't'l ().1.1. scctiorr ().1.?. tlcscribcs thc procedure developed in [9-lg] lilcslirrrlrtirrg tlrc rrlorrg wirrrl rcsponsc ol'prisrnatic, oralmost prismatic, structurc:slirr whiclr it nrly bc lssurncd thal (l) the fundamental vibration mode shapc islrl.rpnrxinraloly a straight line and (2) the contribution to the response of thesccond and higher vibration modes is negligible. Also described in Sect. 9. 1.2is a procedure for estimating the along-wind response of point structures, thatis, structures that may be viewed as consisting of a single mass concentratctlat a height H (e.9., water towers) 19-181. In the procedures described in Sect.9.1.2, referred to here as simplified, all computations can be carried out manually. If the shape of the fundamental vibration mode deviates strongly from irstraight line, or if the contribution of higher vibration modes is significant, theuse of a computer program is required as indicated in sect. 9.1.3. In Sect.9.1.4, results of numerical calculations are used to discuss some of the ap-proximations and errors inherent in the models being used.

9.1.1 Basic Relations, Equivalent Static Wind LoadsThe total along-wind deflection may be viewed as a sum of two parts: the mearrdeflection, induced by the mean wind, and the fluctuating deflection, inducctlby the wind gustiness. The maximum along-wind deflection of the structure ulelevalion z may thus be written as

X-o-(z) : i(2.) * x,,o*(z) (9.1.1)

where x(z) is the mean deflection, and x-o^(z) is the maximum fluctuatingdeflection in the direction of the mean wind. It is convenient to express ,r,,,,,"(:)in the form

r,."*(Z) : K,(2.)o,(z) (9.t.2t

where o,(z) is the root mcan square value of the fluctr"rating tlcllcction rrnd K,(;)is the peak lactor, lho vltlr-rc tll'which is usually irlxrrrt l to,1. Sirrril:trly tlrr.maximurn along-wirxl lrccclctlrliorr lrury hc cxprcsst'tl ;rs

,t I At (,ll(,willt ) l{l :,1 .ot.l:,t :l:ll

r,,,,,,(.') A,(.:)rr,(.:) (9.1.-l)

rrlrr'tt'o,(;) is llre lixrt nr(':ur s(luiuLr vlrlrrt'ol tlrc:rlolrg wincl accclerations andA,(.:) is u pcak lirctol', thc virlrrc ol'wlrit'lr is rrsrr:rlly ubout 4.'l'lrc gus( tosp()lric lirctor is rlclirrctl lrs

(i(z) - I t*"'l(z)i(z)

I lr( rnaximum along-wind deflection can then be written as

(e.t.4)

X,,u*(z) : G{z)i(z) (9. 1.5)

It rs convenient to define an equivalent static wind load that would induce inrl,('structure along-wind deflections equal to those caused by the gusty wind.It l.llows from Eq. 9.1.5 and the assumed linearity of the structure that the,,lrrivalent static wind load is equal to the product of the gust response factori'r lhc mean wind load.

l'lrc general expression for the mean deflection x(3) is given by Eq. 5.3.1.llrt' lluctuating deflections and accelerations as well as the respective peakr,rr rrrs (Eqs. 9.1.2 and 9.1.3) are obtained from Eqs. 5.3.8 through 5.3.15,iii wlrich the general expression for the quantity s"(2, n; (the spectral density,,1 tlrc along-wind fluctuating deflections) is given by Eq. 5.2.37.It followslr.rrr thcqe equations that the calculated deflections and accelerations dependrr1r,n the properties of the structure, that is, its dimensions, mass distribution,rr.rlrrrirl fiequencies, damping ratios, and modal shapes, and upon the assumediii,':rn and fluctuating wind loads.

', L2 A Simplified Procedure for Estimating Along-Wind Responsel ,rllrrwing [9-171, a procedure for calculating along-wind response is now pre-,r'rrlctl, applicable to prismatic, or almost prismatic, structures for which it mayl', rrssumed that (l) the shape of the fundamental mode of vibration is linear.rrrrl (2) the response to wind loading is dominated by the fundamental mode.I lr. lirst of these assumptions is acceptable in a large number of situations oflr:rt'rical interest such as in the case of typical multistory framed structuresr, 1'.. sce 19-21, p. 4281 or [9-22, pp. 60 and 242]).'the second assumption,rrll gcncrally hold if the ratios of natural frequencies in the second and higheri,r,rtlt's to the fundamental frequency are sufficiently large (see Sect.9.l.4).\1.'. givcn in this scction is a pnrcedure for estimating the along-wind response,'l lxrirrl slnrcturt:s, that is, structures that may be viewcd approximately as

, ,'rrsisting ol'a sirnplc rrlrss M c<lnccntralctl lrl tr hcighl l/.ll;tsic Assumptions. 'l'ltc procctlttrc pn'st'rrlt'tl in llris st.t'liorr is blrst'tl on llrt.l.llr r1yi11;' :rssrtrtrPl iolrs.

332 tit [l t)tN(i:i wlNt)t()At):i.:;ililt(.ilt|lAt ilt :;l ,()N:it nNl)l)l :;l(,N()t tt(xrt tN(i ilI . 'l'hc bchavior ol' llrr: slnrctrrrt' is lirtr'lu ly t'llrstic.2. The f'undarrrontal ttrotlc ol'vibt'lrtiort is rr lirrcirr lirrrctiorr ol lrciglrt rrlxrvt'

ground, that is, x{z) : 7.111.

3. The contribution of the second and highcr vibration uxrclcs l<l thc rcsl'rorrst.is negligible.

4. The mean velocity profile is described by the relation

!rt nl ollt iwllltrttt ,,t,|il"t l3J

t:tsc tll wcltlctl slt't'l slltt'ks, ol r'ct'l:utt ptt'sltt's:;t'tl .,l lt( lur(.r., ut ol :,llt( {lt(':,ol lltt: li:urtctl (trlrc lylx' l() ll, 9 l.ll. lrr irtltlilrolr lo llrt. trrt.t lr;rrl,:rl rl,rrrrPnr1,.lltc ltct'tlrlyruuttic tlirrtllirrg nrily, irr Plint'iPlt'. ;rl:;' lrt' r;rIr'rr rrr' ;rt r,lrrr 'llrt.:rt'trrtlynatrtic tllrrtrpirrg, wlticlr rrriry ltt'lp r-t'tlutt'lltt'nr;rl,rrlrrtlt.ol llrr.r{.r()n:urloscillati<ltts, is associltlctl willr cltrrrtgt's irr (lre rt'l:rlrvt' vclot rly ol llrr. :rrr rvlllrr(^sl)cct t<l thc builclirrg as thc little r oscilllr(t's ;rlrorrl l(s nr('lut rlt'lorrrrt.tl lx)stlt()n.Its clctennination is vcly uncorlllin, irrrtl il is tlrt'n'lirrt';l'rrtlt.rrl to rrt'glccl it irrst nrctural calculations.

According to [9-691, darnping nrtios llrvc signilicant statistical variability,r.vith coefficients of variation ol'about 0.4 to 0.8, depending upon buildingrypc; mean damping ratios increase with vibration amplitude in accordancewith a power law with exponent ll9 to lllo measurements indicate that 5- to.]O-story buildings tend to have roughly 60% larger mean damping than building.vcr 20 stories high, presumably because energy dissipation by the foundationslurs a smaller relative contribution to the damping of taller buildings; on thelrirsis of limited observations, it appears that for buildings with more than 20slories, concrete buildings exhibit only about 3o% more damping than steellrrrildings.

To reduce occupant discomfort due to wind-induced building accelerations(scct. 15.1.1), the damping inherent in the building may be augmented throughthc use of dampers (Sect. 9.4).

[errain Foughness Parameters, Zs, 26. The variation of mean wind speedwith height is determined by two parameters, the roughness length ze and thezcro plane displacement z7 (Eq.9.1.6). The roughness length may be inter-1r'cted physically as a measure of the turbulent eddy size at the ground level.Values of zo suggested for structural design purposes are given in Table 9.1.1(scc Sect 2.2.4).

ln densely built-up cities (or in forests) rhe buildings (or trees) obsrruct thellrw near the ground; the mean flow thus begins to develop above an elevationrr:lbrred to as the zero plane displacement and slightly lower than the averagelrcight of the surrounding buildings (or trees). For design purposes the zeroPllne displacement may be assumed to be zero in coastal and open terrain and,il'the values of zo of Table 9.1.1 are used, in built-up terrains as well.

'l'AIILE 9.1.1. Suggested values of Roughness Lengths zn for various Types of'l'crrain

Towns,Sparsely DenselyBuilt-up Built-upSuburbsb Suburhs/'

u(z) : 2.5uxlnZ - Za z >- za * l0Zo

l0U(.2):2.5uxln- z < z7-l lO211

(e. 1.6)

(9.t.1\

(lrr lils. (). l.(r lrntl t).1.'7, i., i1y, and 2,7 ?rc expressed in meters.)'l'hc rrsc ol'thc logarilhrtric pnrfile above elevation (z,r * l0) meters implic:s

thc assunrl'rtion ol' horizontal homogeneity of the flow. This assumption mlynot hold ovcr rcgions neara change in surface roughness, as indicated in Secl.2.4. However, in such regions Eq. 9.1.6-with suitable values of the pararncters u*, Ze, and ZaTma! be used to obtain reasonable upper and lower boundsforthe value ofthe response. Equation 9.1.7 is used, conservatively, on ac-count of the uncertainty with regard to the actual nature of the flow near irbuilding for z < z7 f l0 or so.

5. The mean velocity U(z) in Eqs. 9.1.6 and 9.1.7 is averaged over a periotlof one hour.

6. The longitudinal velocity fluctuations are described by Eq. 2.3.2 andTable2.3.1, and by Eq. 2.3.16.

7. The mean and the fluctuating pressures are described by Eqs. 5.3.3 antl5.3.6, respectively. The expressions for the mean response are thereforcgiven by Eqs. 5.3.4 and 5.3.2, and those for the fluctuating response hyEqs. 5.3.7 through 5.3.15 (or the equivalent expressions in nondimen-sional form, Eqs. 5.3.16 through 5.3.28).

8. The spatial cross-correlations of the fluctuating pressures in the across-wind and along-wind directions are described by Eqs. 5.3.48 and 5.3.49.rspectively.

Response Parameters. A brief discussion is now presented of some of thcstructural, micrometeorological, and aerodynamic parameters involved in tlrcestimation of along-wind response with a view to assisting the structural clcsigner in their interpretation and selection.

Damping Ratio, (,,. Suggested valucs fbr mechanical <lirrrrpirrg nrlios ol'stc:cland reinforcecl concrclc I'nrntcs arc 0.01 and 0.02, leslx'ctiv('ly l() Ill l,owcl'values ol'lhc Itrcchitttit:rrl tlrurtpirrg rrriry llrvc lo bc rrst'tl. lor ('\iunl)l(', in lll('

l'ypcol'

It'r'r'lin .. Coastal"'/' Olrcn',,(rtr\ 0.(X)-5 0.01 o.03 0. r0 0.20 0 40 0.80- L20 2.00,3.00

"ApPlitirblt'1o srru('lurrs tlirt'tlly t'xlxrst'rl 1. wirrtls blowirrg lirrrrr opcn watcr.''V:tlrrt's ol :t,, 1o lrt'ust'tl irr t.orrjrrrrt.lrorr willr llrt ;rssttrrrpliott :t,, O.

Centersof LargeCities/'

334 tttrtt t)tN(iti wtNI)lo^t l;.::lltt r{.il|rnt ilt iit '()N:;t nNt)t)t :;t(iN1)t n()()t tN(i

Exponential Decay Parameters, C, C..'l'lrt' n:rrnrw llurtl spirlilrl cr()ss ('()rrelation of the fluctuating prcssurcs in tlrr: lrcross wirrrl tlircctiorr (llt|. 5 .1 '16;is a measure of the extent to which prcssurcs trpplicrl al dillcrcnl poirrts ol'tlrt'same building face act coherently or at cross-purposes. 'l'hc srrrallcr tlrc vlrlrrt'sof the parameters C, and C, in the expression fbr the cross-corrclation lhc rnon'coherent will be the action of such pressures and, therefore, thc largcr llrt.response.

On the basis of wind tunnel tests, it has been suggested that it is reasonahlt.to assume Cy: 16 and C. : l0 [9-13]. The procedures presented in thissection are based on these values. However, as indicated in Sect. 2.3.4, fullscale measurements do not always confirrn this assumption. As shown in Sccl.9.1.4, the effect upon the total along-wind response of changes in the valucsof Cu and C of as much as 30% to 4O% is, in general, relatively small (of tlreorder of 5%-10"/"). However, the effect of such changes upon the accelerationsmay be considerable. (See also footnote following Eq. 2.3.31.)

Friction velocity, u*. The friction (or shear) velocity a* is a measure of thewind intensity over terrain of given roughness. If the mean wind at a specificrlreference height above ground za is known, u*. can be obtained by using Et1.9.1.6:

U*: U(zn) (9.l.ltt2.5ln[(zn - z)lzo]

In meteorological work, the reference height most commonly used in zn .,

l0 m.In designing tall buildings it is reasonable to use mean wind speeds averagctl

over a period of one hour. In this chapter the symbol u will denote hourlymean speeds. If mean wind speeds (J' are specified that are averaged ovcr'periods t different from one hour, the mean winds averaged over one hour canbe obtained by using Fig.2.3.10. For convenience, the information includcrlin Fig. 2.3.10 is summarized in Table 9.1.2. (Forbuildings in hurricane-proncregions, see also first footnote of Sect. 9.1, Sect. 2.4.3, and [9-5, p. 155].)

For values of / not included in Table 9.1 .2,linear interpolation is permissible. If the wind speeds are given in terms of fastest-miles u7, the averagingtime in seconds is given by

r : 36OOlUr. (9. l.tll

TABLE 9.1.2. Approximate Ratios of Probable Maximum Speed Averaged ovclPeriod / to That Averaged over One Hour (at l0 m atrove Ground in OpenTerrain)

rt ! Al 0il{ . Wltjt I nt ,t ,r )t.|,t 3:15

l .\lll,l,l '). 1..1. ltirlios it ,ltt ,, lot \':n.iolls Sur-lirr.r. ltoulihrrcss ( ,:rlt,gol.ir.s

I r,ln., rl

l, l:rilr ('olrsl:rl

0. tts

Syr:r r:;t'lyllrrrlt rr;rStrlrrrrlrs

l l.5

Iorvrts,I )crrst'lylitrilt -u1r

Su hr r rbs

Clcntcrsol' Largc

Citics( )lx il

l.(x) I .33 l.45

As indicated in chaptcr 2, thc retardation of the flow due to increased terainr,r111'l111gss causes thc mean speeds over built-up terrain to be lower-for any,'r\'('il large-scale storm-than the mean speeds at equal elevations over openr{ rririn. Since wind climatological information is commonly provided in terms,'l wind speeds measured over open terrain (generally at airport weather sta-rr.rs), the problem arises of converting this information into wind speeds ap-f irtrrlrlc to a built-up environment. In Sect. 2.2.5 this problem was shown tol'{' solved as fbllows. Let u*1, 201 denote the friction velocity and roughnessl, nr',lh over open terrain, and let z* denote the friction velocity over terrainrr rrlr roughness length 20. For the surface roughness categories of Table 9.1.1,.r;r;rrrrximate ratios u*lu*1can be obtained from Table g.1.3. once z* is known,l/( .:) can be calculated by using Eq. 9.1.6.

lturation of storm, r. This parameter appears in Eqs. 5.3.1r and 5.3.14, whichrrrrlrcats in effect that the expected peak values of the fluctuations will be higherrl llrc: duration of the storm increases. The assumed storm duration is implicit1,1,1''"

ut" of design mean speeds averaged over one hour, that is, I: 3600

Mr:an Pressure and suction coefficients, c*, cr. The mean pressure and'.rrt'tion coefficients are functions of the shape of the structure (see chapter4).Ir rlrc case of tall buildings with a rectangular shape in plan, it may be assumed(,, 0.8, C/:0.5, andCp: C*,+ Ct:1.3.A"tt:an square value of rurbulent velocity Ftuctuations. The ratio, p, be-rr't't:rr the mean square value of the longitudinal velocity fluctuations,7, andrlrt' square of the friction velocity, u?* @g. 2.3.2) depends upon surface rough-r('ss. ils shown in Table 9.1.4.

i i\lll,lJ 9.1.4. Approximate Ratio p : it"r* for Various Surface Roughness(':rlrgories

i ypeol

Icrlrirr

ti

SparselyBuilt-upSuburbs

52s

Dcnse lyIluilt-upSubu rbs

4. t{.5

t(s)

lo06030l0 200

l.ll

5(X) I (XX) 3(,(X) Ccntersol'Largc

('itics.l (x)

Ctxtslal(r 5o

( )pcn().(X)u'lu l.-53 1.41 L12 l2tt I24 l. llJ lo/ tor l.()()

336 aUlLDlNos: wtNt) ro^l)li, lirlrt,(;ilrrnr nr rl,()Nlir , nNr) r)l ,,;r(iN ()r n()()r rN(i

Expressions for the Along-Wind Response. Using thc basic ussunr;'rtiorrslisted earlier in this section and relations given in Scct. 5.3, rcsults ol'nurncricalintegrations were closely fitted in [9-18] by simple functions, and cxprcssionsfor the along-wind response were developed that are listed in Table 9. 1.5 lirrbuildings with a nearly linear fundamental mode shape (Fig. 9.1.1), and inTable 9.1.6 for point structures (Fig. 9.I.2).

In Tables 9.1.5 and 9.1.6, h and H are the vertical dimensions shown inFigs. 9.1.1 and 9.1 .2, b : across-wind dimension of structure, d : along-wind dimension of structure, Zs : roughness length (see Table 9.1.1),70:zero plane displacement (for practical calculations it may be assumed that 2.,1: 0), nr : natural frequency of vibration in fundamental mode of vibration,ux =. f'riction vclocity, Cp: drag coefficient (Co: C* + C), C*and C1 :avcrrgc prcssurc coefficient of windward and leeward face of building, re-spcctivcly, M : L<ttal mass of structure with dimensions b, h, and d in Fig.9.1.2, z. : hcight above ground, M(z) : mass of building per unit height,poQ) : bulk mass o1'building per unit volume, f1 : damping ratio, p : massof air per unit volume, 0 : coefficient given in Table 9.1.4, T: duration ol'storm (Z : 3600 s), x : mean displacement at top of structure, 6 : gustresponse factor, X-u* : peak displacement at top of structure, oj : rrns ac-celeration at top of structure, and iu^ : peak acceleration at top of structure.

TABLE 9.1.5. Equations for Estimating the Along-Wind Response of Buildings with rrrApproximately Linear f,'undamental Modal Shape t9-l8l

/ ,),\ r'--l)Q:z(t-3)ln" "-l\ h'/ Zs(2) J : 0.18Q1

(3) G : 6ltQ2| + 0.26bth

- n,h(4) .f' : Lu+

t1(5lClxt :*-Z*r(l-cL)

(6) x, : 1232L4' Qh(7) N(h: c@)@ c'?Drc) : c?" + 2c*ctN(f) + c?(9) x, : 3.55 [0

(10) M(z) : bdp1,Q.)

(il) G : o.5e q ( ?1" cf,r./,r" -. r, \./, / cl,' ( ix,t-\--:' I r .1.()\ ( l,let t!,lltl

- M,(2rn,l2 \ 6Mr(2rnr)W#(f *u,)"

(tz) M, : # f, Me)22 ctz

(13) q* : )out*

fl4)t: u'*=Mr(2rn,)'

Cobhq*_ rJ

(15) o,

n6)u. =,,( o \''.v, qr ,,r \tpG/Ol + G/(17) K, : [1.175 1- 2ln(u,T)ltt)(18)G-r*&?(19) X*.. : Gieol o, Cubhq* o,,tMl(21) K\. =. I l. l7-s + 2 ln(nit')ltt'QD x,,,.,. K,o,

1f I n l ttu( ; wlt{ I l lt :;t rol.Jt;t '.1:l l

FIGURE 9.1.1. Schematic of rallbuilding with rectangular cross section.

f :rlrlcs 9.1.5 and 9.1.6 are in principle applicable only if nlhlU(h) > 0.1 andu,lllU(H) = 0. 1, respectively, which is the case for most structures. In prac-trtt'. they may be used even if these conditions are not met, in which case ther,'srrlts obtained will be slightly conservative.x

Nttrnerical Example Consider a building with h : 200 m; b : 35 m; d :f') ln; nr : 0.175 Hz h : 0.01; po : 20Okg/m3; C. : 0.8; Cr : 0.5; and('t, 1.3. The building is located in a townf (zo = | m, see Table 9.1.1). Itr', :rssumed p : 1.25 kglm3, and the fastest-mile wind speed at l0 m abovel,rrrrrrrd in open terrain (zo : g.Ot m, see Table 9.1.1) is UdlO) :78 mph.

Iirorn Eq. 9.1.9, the averaging time forthe fastest-mile wind speed is r =l(r s, and from Table 9.l.2theratio Ua6lU = 1.25, that is, the hourly wind'.lrcerl at l0 m above ground in open terrain is U1(10) = (7811.25) mph ='/ 13 rn/s and u*1 : 27.81[2.5 ln(10/0.07)] :2.24 m/s (Eq. 9.1.8). FromI ;rlrlc 9. 1.3, u*lu*1 : 1.33, and a* : 2.98 m/s. Then, refering to the equation,,r 'lrrlrlc 9.1.5, Q:9.60 (Eq. l); J :71.83 (8q.2); G :591 (Eq.3);,fr

tt.74 (Eq.4); x, :2.63 (Eq.6); N(,fr) : 0.31 (Eqs. 7 and 5); C'zDJ.(f) :I I I (llq. 8);x: : 4.34 (8q.9), M(z) : 245,OOO kg (Eq. l0); G : 353 (Eq.

' \ ptott'tltttt: sitttilltr to -l'ablc ().1..5. irlso birst'rl on tlrt' wolk ol' 19-lul but adapttrl lo rrsc willr1 ,1'tlsl spt'etls ltrttl tltc ltowt'r l:tw rvirrtl s;xt'rl prolilr', is irrt'lrrtlctl irr llrc ('ontrrrt.rrl:rry 1o lltr'Ui('li 7 ()5 Slrtlttl:ttrl l() 5l :rrul is ;tv:rl;rlrlt irr llrr'rlr:,kt'llt ol I I / 5l rrlrlrt'ntlt'tl kr llris lxxrL'llt:,;tsstttttt'tlllIIlllttlel:tiltlrttlllrrl'rr,t,1r,,rrr,,1,,11,,y11,.ovr'l;rrlisl;trir.t,lpwirttlplirllt.lrsl l.)\/r

r'., , Scr'l .1..1. I ).

:t&*

wnYl

€Io\

a)L

a

()q

o

Ib0

b0

ardL

a

rd\o

o\rdtl

3

338

S

s.,!l+'lPcet- l\5 3 sdl- l+ c c\ <l-_ €51^ i i ;.r'\ "lh - lX + dlr'< € +*t -Nt -tve*t

='$ ilSSlS -le i Y s$lss$+qrt-vt!_!l--.' vt\ t+ s - ll L)l - llil l<"_" tt ll ll ll ll 6 ll ll 6>-$ '* d J q u tf d g,xF

a.t ca + ra) \o F- oo o, o .i a.lC.l N c.lv v!

65!JUG 3+U^t

'c; ,ql ^

-bt u\' u ulru'sI c'ri' + o- l'-i

- l.\ s l\ =r's l\ -s l\)--_| ,Slq J. U,.<f q,l;la-\lr;

_r ! \O v ll \O O e{l x -'5 n \o s{l s ,, 6 l,G 6i r oirr jj__ ll ll ll"5"':.-q< U \ =U q d €

$ n \O F-go O\ O3g

fn\l.\'sco(\o

I

\.s\ac.lo+

\nO+p-i

\

\5c.)

vlI\I H,61

llttA,\

ci

1l I Al oNo wtNt) tit litroNt-it 339

.-Y FIGURE 9.1.2. Schematic of point srructure.

ll); M1 : 16,333,300 kg (Eq. 72); q*: 5.55 kg/m/s2 (Eq. l3); x : 0.1g4m (Eq. l4); o,:0.074 m (Eq. l5); u,:0.114 s-'(Eq. 16); & : 3.63 (Eq.l7); G :2.46 (Eq. 18); Xpu" : 0.452 m (E^q. 19); dj : 0.058 mls2 1eq. ZOj;Kt : 3.75 (Eq.2l); and X.u* : 0.218 mts2 1yq.2ZS.

9.1.3 Computer Programs for Estimating Along-Wind ResponseFor certain structures the assumption that the contribution to the response ofthe higher modes can be neglected may not be realistic. Also, it miy be ofinterest in cerlain situations to employ micrometeorological and aerodynamicrnodels different from those incorporated in the procedures of rable 9.1.5 orl9-5]. In such cases, in lieu of those procedures, a computerprogram must beused to estimate the along-wind response. The computation of the responseamounts essentially to the evaluation of the integrals in Eqs. 5.3.1, 5.3.2, and5.3.7 through 5.3.15. computer programs have been developed in which suit-able numerical integration schemes are used and in which the specified struc-lural, micrometeorological, and aerodynamic information is incorporated asinput or in specialized subroutines. A computer program developed by theNational Bureau of Standards is available on tape in [9-14].

9.1.4 Approximations and Errors in Estimation of the Along-windResponseIrl this section estimates basctl orr nurncrical calculations are presentcd of errorsitssociatcd with uncertaintios rcgirnlirrg ce:rtirirr lbllurcs ancl paramctcr valucsol'lhc lnoclcls cmployccl. 'l'her t'rr['rrlirtiorrs werc r,irrlicrl oul lirr lhrcc typicll

h

\

^1

340 tstJtLDtN(i!i: WlNl) l()nl):;, i;lllt,(;ll,llnl lll lil'()Nl;l , nNl) l)l :;l(iN ()l li()()l lN(i

TABLE 9.1.7. Description rll'lluil<lings St'ltclcrl its (last Stutlits

Building iDHB

(m)llt

(Hz)l) r,

kg/rrrr

I23

buildings selected as case studies and described in Table 9. 1.7. The wind speedat l0 m above ground in open terrain (zo : 0.07 m) was assumed to be Uv :7-5 mph, wherc U7 is the fastest-mile of wind.

Contribution of the Higher Vibration Modes to the Response. The rootmean squarc ol'thc lluctuating deflections and accelerations were calculated forbuilclings I ancl 2 in open and town exposure. The assumed modal shapes inthe first three modes are similar to those represented in Fig. 5.2.1. The dampingratios were assumed to be fr : h: f: : 0.01. Calculations were carried outseparately for the casas n2ln1 : 1.2, n3ln1 : 1.5 and n2ln1 :2.5, ry|ry : 5.The contributions of the higher (i.e., of the second and third) modes of vibrationto the response are listed in Table 9.1.8. The contribution of the cross-modeproduct was also included in Table 9.1.8. This contribution represented abouthalf of the amounts shown in columns 1 and 5 and was altogether negligiblein all other cases.

lnfluence upon Calculated Response of the Deviation trom a StraightLine of Fundamental Modal Shape. A convenient means for estimating theinfluence upon response of the fundamental modal shape is provided by theexpression

36515045

606045

454545

0.100.201.00

0.010.0r0.01

150r50150

o,_l*^yI2aOx l*7-la- (9.1.10)

derived by Vickery [9-13] on the basis of the assumptions that the power law(Eq. 2.2.26) holds and that the fundamental modal shape is described as fol-lows:

x(z) (9.1.il)

where .y is a constant. In Eq. 9.1 .10, o" is the rms of the fluctuating deflections,x is the mean deflection, Q is a function of geometrical, mechanical, ancl

environmental parametcrs, independent of "y. It may bc assumcd, roughly, thatd can vary bctwccn 0. l0 krr ()pcn cxposurc and 0.40 lilr ccrttlcrrs ol'largc citics.It Ioll11ws thcn l!rlrr l;.q. 9.1.10 thal lirr ry =. 0. l0 tlrt't'rtlt'ttlittctl rittitls o,/.r

: (;)'

u*QM.l

UT

(-) -9\O

crr)

--.:Oa()+o:/

(.) c.)

ONOJ

(.) i

ti ,a,

Iil

-:-

"J "?

il lt

{is\

n-6lil

tl

E

.l "?

il ll

s\n_Is\

F

C)roo.X

IJ]

a)a

O

a6J

u!

I

f*

C!

(a

0)z

q4)

aLo!

Q

0)lr

€o\rdFl

F

bo

FA

I (t)

oc!

oo O\

tr-:f

*N

h@

* a..l

34t

342 tluilt)tN(i:; wtNt) l()nl ):;. i;lltll(.lllllnl lll i;l '{)N:;l nNl) l)l :;l(iN ()l ll(){)l lN(;

calculatcd assulning ? : 0-5 irrrtl 7 1.5 tlilll'r by llxrtrl l%' ll'orrr llr:rlcalculated assuming 7 : I (i.c., a lirroar llntlarrrcn(al rrurtlitl slttPc). l'ot'rv0.4, the corresponding diff'erences arc about 3'/,,. lL is thus scctl lhll ttttltlcr:tlt'deviations from a straight line of the fundamental rnodal shapc havc rtrr irtsig.

nificant effect upon the calculated ratio o,/x.

tnfluence upon Calculated Response of Errors in the Estimation of theRoughness Length. To estimate the magnitude of the error associated witlrunccrtainties regarding the actual value of the roughness length, the responst'<rl'builclings 1,2, and 3 was calculated for coastal, open, suburban, centcr ol'lowr.r, irntl con(cr of large city cxposures. The zero plane displacement was irlIrll crrscs irssrrrrrod t<l hc zcro. Thc calculations showed that the sensitivity olllrrr lc:srrlts (o cvcn llrrgc crrors in thc estimation of the roughness lengths (e.g.,50%,) is lrtlcrlbly srrrall (abovc l0%).

Spectra in the Lower-Freguency Range and Along-Wind Response. llwas shown in Sect. 2.3.3 that no universal relation exists describing the shapt'of the spectral curve in the lower-frequency range and that this shape appeanto vary strongly between sites and between atmosphere and laboratory. 'l\testimate the effect of this variation, the response of buildings 1, 2, and 3 (sccTable 9.1.7) was calculated for open terrain and town exposures, using lrrexpression for the lower-frequency portion of the spectrum of the longitudinirlvelocity fluctuations that depends upon a parameterJ,, as in Eqs. 2.3.25. Ratios[X-o*]y',,/[X."*]0.c,: of the peak response calculated by assuming various valucsJ, to the peak response based on the value l, : 0.03 are listed in Table 9.1.9.

The results of Table 9.1.9 show that the dependence of the peak responscon the shape of the longitudinal spectrum in the low frequency range is rclittively small, particularly for taller buildings.

It is also noted that as indicated by Eq. 5.3.41 the influence of the spectrrrlcurve shape in the lower frequency range upon the value of the accelerati<lltsis negligible.

Across-Wind Correlation of the Pressures and Along-Wind Re-sponse. It was noted in Sects. 2.3 and 9.1.3 that uncertainties subsist willrregard to the actual values in the atmosphere of the exponential decay cocllicients C, and C.. It is therefore of interest to estimate the errors in the calculalt'rl

TABLE 9.1.9. Ratios [X",o*1y,,/[X",uJo.or

Building I Building 2 Building .l

Exposure Opcn 'l'own

,ri' n(.ttrl,t.wllJlr lil:,1 ,(tl{:,t 343

,rloltp. wilttl l('slx)lls(' llt;tl t or,'sP,rrrrl lo lxrssilrlt' r.uors ln lltt' v:rlrrt's ol lltcscl)irtilttlctct's. 'l'lrt' lrlonli lvrrrtl lt.:;1xlr:;t' ol lruiltllrrl,s l. .1, lurrl .! ilt oPerr trrrtll()wil cxl)()sufL:s w;rs llrt'rt'lorr'r':rlr'rrl;rlt'tl st'p:rr:rlt.l-y lor'(' lo. (',, l(r (cascIt, lilr Cl - 4, (', (r..1 (t'rrst. (r). lrrrrl lil. lirrrr. ilrlel,rnc:tlitrlc cascs irr which( ' (".wcrc asstllllt)tl t'itlrel tottsl;rrrl llrlrrrglroul tlrc licqucncy rangc (casc 4)'rr lo havc ltlwcr vltlttcs ltl low lt'ctqrrcnt'ics:rrrtl lrighcr valucs near ancJ beyondtlrt' lirndamcntal li'cqLrctrcc rr, (cust:s 2, -3, urrtl -5).('hangcs in the valucs <ll'C,, anrl (' in thc lowcr-f-requency range were foundtrr lravc little efltct on thc rcsponsc (cascs I ,2, and 3). lf for frequencies near:rrrtl beyond the tundamental fiequency the values of these parameters are c,6.3, Cr: l0 (cases 4 and -5), the total response is approximately 5% toIt)% higher than if C. : 10, q : 16 (cases 1,2, and 3); however, rhe:r,'t'clcrations increase in rhe case of the taller buildings by 20% to 4o%. If c,

4, ct: 6.4-a situation that may be encountered in moderate winds such.rs occur during full-scale measurements of tall building response-then ther()tirl response is about l0% to 2o% higher than in the case c. : lo, cv :| (r. while the accelerations of the taller buitdings are higher by 30% b aon .I'lrc significant dependence of the exponentiar decay coefficients upon wind',1rt:cd reflected in Figs. 2.3.5 and2.3.6, and the sensitivity of the along-wind;rt'cclerations to variations in the values of these coefficients suggest that cautionr:; in order in the interpretation of full-scale building acceleration measurements;rntl the extrapolation of results based on such measurements to design situa-I l( )nS.

11.2 ACROSS-WIND RESPONSE

I .ll buildings are bluff (as opposed to streamlined) bodies that cause the flowl. 1;nd".ro separation, rather than follow the body contour. Depending upon( ()nclitions discussed for certain classical cases in chapter +, tie wake flowtlrrrs created behind the building exhibits various degrees of periodicity, ranginglrrrn virtually periodic with a single frequency to fully turbulent. In each oftlr('sc cases' at any given instant, the wake flow is asymmetrical (e.g., Fig.I 1.3). The across-wind response is due principally to this asymmetry, althoughtlrt' lateral turbulent fluctuations in the oncoming flow may also contribute tot lrt' :rcross-wind lorces.

lixpressions based on first principles for estimating the across-wind response,l tall buildings do not currently exist. However, empirical information ob-t:rilrcd fkrm wind tunnel measurements is available concerning the across-windr( sl)onsc of tall buildings not sub.jcctctl to inteference effects, and expressionslr;rsctl t)n such infbrmation hrrve bc:cr11 1111lp1;1;cd in thc literaturc. Dilltrcntt \l)lcrisions arc applicablc ltccolrlirrl', (o wlrcrhcr ()r nol thc rrns vrrluc ol'(hc:rttrtss wintl tlscillltlirllts irt (lrt'tip ol (lrt'lrrrilrlinll. (,\, t:xcocrls tr r'ri(it';rl vrrlrrt''r',,- ll'o, ) o,,,,, l0ck-irr clli't ls l)( ( ()rlr(' :,r1,rrilrr';rrrl. lrtttl llte :tt'lrrss wirrrl lolrrls.rrr<l rtst'ill;rlitltts ittt'tt'itsr' rts llrr' \\'rn{l r,l)( ('rl:, r!rr'r(.:l\(. Slrlr'lgli's slrpllll lrt.'lt'si1'1;1'1; so llr:rl lot'k irr t'llt't'l:; rl,r r,,r rrr( ur (lurrrr;, llrt.rr :rnlii.ip;rlt.rl lrlr.

i

l

i'

:0.01: 0.10

0. l9

l.(x)0.9r3o.t) l

I .(X)097O (Xr

l.(x)0.91O.(Xr

Towrr

l(x)o (,,1

o()I

( )pcrr

l(x)o ()\o()I

'l'own

l(x)0.() I

o 8/

Opcn

f,,,f,,,f,,,

344 llult-ulNGS: WINU tOADS. 5i1tll,(:ltlllAl lll r;l'()NSt, ANIJ DESI(JN Ol ll(X)l lN(i

For square tall buildings, oxpcrinlclll;i rr.rpoflctl in l9-20, p. ttll antl l9-241suggest that it is conservative to assutttc ilrat

? = o.ott (oPen terrain, zo = o.o7 m)

T = O.ort (suburban terrain, zs = 1 m)

trs!b = 0.045 (city center, zo = 2.5 m)

where b : horizontal across-wind dimension of building. It is emphasized thutthese ratios are largely tentalive.

Structures for Which 6, 1 6ycr. Several expressions for estimating o.v areavailable in the literature. In all these expressions, the wind is assumed to blowfrom the most unfavorable directions (in the case of a square building, normulto a building face). Vickery [9-251proposed the expression

gyo),(h)(e.2.1)

where or(ft) : rrns of across-wind oscillations at top of structure, gy : Pcakfactor expressing the ratio of the peak response to rms response (8, = 4.0), h: height of building, ,4 : cross-sectional area of building, U(h) : mean windspeed at the top of the structute , fl1 : fundamental frequency of vibration, f1: damping ratio, p : air density, pr : bulk mass of building per unit volumc,n and C : constants determined empirically from wind tunnel measurementi(n :3.5, C: 0.0006 + 0.00025). The rms of the accelerations at the top ofthe structure, or(h), can be estimated by using Eq. 9.2.1 and the relation

or(h) : (Zrn)2or(h) (e.2.21

Equation 9.2.7 is based upon measurements of the response of building modcl$with a linp4r fundamental modal shape and with geometric shapes, slenderncsnraios JA/h, densities, and dampingratios shown in Fig. 9.2.1. h is notcd in[9-25] that the use of Eq. 9.2.1 should be restricted to buildings with charac.teristics that do not differ drastically from those shown in Fig. 9.2. 1.

The Supplement No. 4 to the National Building Codc of Canada l9-l2lproposed an expression that may be written in thc fonrt

^l u(t lf' t p: 'l;lil f,,;

Io,,(ttt = ttiltxttt')

ti, i;,,0.(x)5e

JA

rt(ttl l,'tt Jlulltt' I

++

@a@

aaA (worst

direction)aVA

mv

1).1' A(;llOt;liwlNt) nt lil\)Nlit 345

JA:,h 4.2

r/A lh7JA:,h 4.1

JA:,h 3.4

FIG[]RE 9.2.1. Characteristics of modelstested in the wind tunnel [9-251.

(e.2.4)

Pn = 2OOkg/m3f = 0.0t

where b : across-wind and d : along-wind dimension of the structure. It canlrc seen that Eq. 9.2.3 is similar toEq.9.2.2 (where o, is given by Eq. g.2.r)trxcept that the exponent n : 3.5 is replaced by n :3.3, and the coefficient(' : 0.0006 + 0.00025 is replaced by c - 0.0006. Equation 9.2.3 is basedon measurements on models similar to those described in connection with Eq.e.2.1.

we note that unlike [9-25] and (9-121, which do nor differentiare amongIruildings with the shapes shown in Fig. 9.2.r, [9-74] and [9-g1] report windIrrrrnel test results according to which buildings with square cross section haveir rnuch greater across-wind response than circular buildings or square buildingswith chamfered comers. Reference [9-81] contains detailed results on winde ll'ccts and their dependence on wind direction for buildings of square crossscction with and without chamfered corners or spanwise openings.

Expressions in which measured modal force spectra are used follow froml;.t1. 5.3.32.If Se(n) : across-wind modal force, and the notation

orl n,b I ngefnptU(h)l'lu&rl- ppuruWis uscd, Eq. 5.3.32 becomes

o,(h) = ;r:i::,, ,, n,,',r,r,1, r,u)ttt)hh0.2,31 f ().2.s\

llt,ll lllNcs wlNl) l()nl):l, iilllllclt,llnl lll :;l'()Nlil , nNll l)l l;l(iN ()l ll()()l lN(i

lf it is assumed that thc nrass is unilirlrrrly tlis(ributccl ovcr llrc bLriltlirlg hciglrl,that the building has a square shapc in plan, ancl that tho llntlantcrrt:tl rtt<xlrlshape in linear, then

M, : \o6bzh (9.2.tt1

(Eq. 5.2.6), and

I utn 12

a,(h) :0.0337 I ,e I

Noto thrrt thc quantitics I implicit in Eqs. 9.2.1 and 9.2.3 are, respectively,

ol--bYQn \r

(9.2.1)

(9.2.8\

and

i : 14.45 r r.8o)'o-' I #l''

i = 4.45 x ro 3l#)" (9.2.e)

Values of f based on measurements reported for square building models itl19-241, 19-261,19-211, and [9-28] are listed in Table 9.2.1. Also included inTable 9.2.1 are values of f given by Eqs. 9.2.8 and 9-2-9.

Table 9.2.1 shows that for any given nlbl(t(h), f is a function of terrainexposure and the slenderness ratio b/h. For example, the peak values of 7 firrurban terrain appear to increase by a factor of approximately two if b/h clc"

creases from l/3 to 1/9. Also, according to data from t9-241, 19-271, antlt9-281, the peak values of lincrease as the terrain becomes smoother. This is

the case because in rougher terrain the turbulence intensity is higher, which irt

tum causes the across-wind force to have a less peaked spectral density (Fig.9.2.2), as well as a decreased coherence in the spanwise direction. Note tha(there are significant discrepancies among values oi i obtained by various rcsearchers. For example, for urban terrain, nPlU(h) : 0.105, and b/h '118.33, f : O.tO according to 19-261, versus t : O.ZZ on the basis of dalilfrom 19-271; for urban terrain, nlblU(h) : 0.105 and b/h : ll4, i : O. ttzccording to [9-281, versus i : O.l5 on the basis of data from [9-27]. Dilf'crences between values of lgiven by Eqs. 9.2.8 and9.2.9 and those from l()241, [9-26], t9-271, and [9-28] are also relatively large in several instances.

Numerical studies [9-29] show that, as in the case 0f along-wind responsc.the contribution to the total building deflections ancl itccclcntlions of t.t.totlcs

higherthan thc funilarlcntal rnodc is ncgligiblc in prirr'tit'r', ttltlcss ll'tc rlt(itls olnatural frcqucncics in lhc highcr rnodcs to lhc lirntlirlttcttlltl l'n't;trr'ttt'y tttc clost'

v

+1,

i ,!

!

i

bbO

t< =...=.=.??q?=.?=:.*::ta)lnOOo-@-@-€.tdN--dNN__N_

.i oi

,:t=bcc==L=:o:d.--hON€-OnO$@rrNN--NNN-J-:

)-+rt..=-b=cc.=tcco=-,rrOrd-Ai€n@€O___N<_N'-;

:=a ->- -t- - * ^ ^5- ^ - ^ ^--.::: '.::':::-::. N-nNO€-Qq-rc)d *_*__N:

:-d:.r"9ql-:enqE-:,:qq

:rr-ornr€@r@c)r-

*. o-vlq

'1'.. - f:q "-' -: q ql1 =t - o or,n€a>n+o€.or; roi 6iON

aqnrt- -'-.'--SN6O--O€ OOr+r;od6ddri6.i+6G;

^:'-qqEr,:qqEqeoo' j -- -€hc>^i --cj -i ++

-----N-

rro TTol @ ++r' 7 r :o :i- r. - o o * o * -i -i

EE-OO€

qHEt'J

st34t

b{

qqNI

-qen\.1r-s)-oc>n-o---no-^i -joi 6i +i^i

c.r qo$nNcr€.FNOOrcF-n&Orc-+$6++^i^i^i-;d+doi

9r:oh

E. 't i- ^rio n * o nF- o ir o oonn€on^i ^i ++,+ri ri ci

dr;@-:t"leon€n-b>6.o_-a,c-ncn.rd6-; r) d+

r€i.;

ro':oq-5.)-bnon*\-booO,O.O^i Os-i .+sssO>+nognr*r-cEqEtgoQinonnoo6onnesqidoodrodri

\!o€n€odN-oooooc)€oboro€o-,$€rocjr+Adqjd<t

\nnro,':e=CLr,cc=bcc.=-oooonooi -io.o^i -j-: r-<rN* N_N+___

rro f f-: 6 $+{?-9:ta,-o.o'+m+.i-i

" 9?d Ftr--ooLp - o

-i q O<<

!

tro

aa.;+t!

!

xE

a--

a9.E

9poa-Y-ONll o

o

Qc

6.: O

- fo---= --o!-tr .9e

ail{a

il

s4!c

il

ss!

I

tl{s€No

!

o-c;OE.2A>q

oEsfr

r:

-o.'o$ f 6 6N qNN

&>e&

348 LIUILDINGS: WIND LOAI)S, ri I lllJ( ) I t,i l^l lll til,ONt-itr, AND t)t. StGN Ol [(X)t tN(]

1.0

I

0.01 |

nb/U(h)

FIGURE 9.2.2. Shapes of f2 curve in open and"Across-Wind Response of Buildings," J. Struct.

0.1

urban terrains. After A. Kareem,Dlv., ASCE, 18 (1982), 869-887,

to unity, that is, substantially lower than those occurring in typical high risebuildings.

For a square building model with height to width ratios h/b : 8.33 locatedin urban terrain, it was found in19-26] that the across-wind response decreaserfrom the maximum value that corresponds to wind normal to a building face,to about 5O% of that value when the angle between the mean wind directionand the normal to a building face is about l5o. The peak across-wind responseand the peak along-wind response induced by wind parallel to a diagonal ul'the cross section have approximately the same value; that is, they are appn)x-imately equal to 0.8 times the peak along-wind response induced by windnormal to a building face 19-261.

Numerical Example The building considered in thc trtrnrcricll cxamplc ofSect.9.l.2 is again assumcd to bc actcd upun by wirrtl colrrrslxrncling to ufastest-mile spccd al l0 nr abovc gnluncl in opcn lerririrt l//(10) , 7ti rnplt,

n(;t tor;li wtNt) lil tipoN$E 349

l lrc tttcarr lrourly wirrtl sperccl at thc top ol' thrr lrrriltling is thcn [t(h) : 2.5u*lrl(/,/it)), or u(lt) -. 2.5 x 2.98 x ln(2(X)/1.(x)) .tt).4 rn/s. Thc fbllowingf.rrrlrs irrcObraincd: n,blulltl:0.155: h/h ' l'15.7: i' = 0.075 (Tableg.2.l,riulrrul)iu.r tcrrain), o, : 0.23 m (Eqs. 9.2.7 or 9.2.5); oi, : 0.2g m/s2 (Eq.tj .'.2). Assuming that the peak t'actors arc gv = 3.-5,.g0 - 4.0, it follows thitllrr' pcirk across-wind response and accclcration arc L,"* : 0.805 m, l.u^ =I I I rrr/s2. These values are larger than thc corresponding values of the along-rvrrrrl rcsponse calculated previously, that is X-o^ = 0.452 m andX_* = 0.21grrr/s'.

lrigurc 9.2.3 shows the mean and rms along-wind response and the rms*t ross-wind response of a 1/400 model of a 64-story building in urban teq4in.'I'lrt' characteristics of the model were the following: h : 0.658 ^, Ji :(l 154 ttt (where,4 is the floor area), n, : 8.3 Hz, n, : 8.49H2, and l" : l,0.01 (where n and I denote frequencies and damping ratios, respectively)I'l l0l. Results of wind tunnel tests for the model of a 53-story building arerlrrrwrr in Fig. 9.2.4 for open and urban terrain t9-13].

101

I I 1oohl ltl" l!'nl#;l 5

:'I

or/h I2

10 1

5 235J1015U(1.8 h.ln,,1f A

l'l(,illlll'l 9.2.3. Mcln itkrttg witttl tttttl rrxrl rlrrrrr s(liliuc ol irkrng-win<l lntl lcnrss-rr'trrrl tlt'llt'cliotts ttl it (r4-sloly ltrrilrlirrg rrrork'l wrtlr tr t'ilt,rrlirr slrapc irr plln l9-.101.

350 BUILDINGS: WIND t.OAlJS. Slllti(;lt,llAl lll t;l'ONSt ANIJ DFSIGN (]f Il(X]I ING

/Along-wind *------2

-A>----- .r'/'--r-./Across-wind

I = 0.01

Urban exposure

Wind speed ar 47O m above ground (m/s)(a)

Arons-win{--9 -- -26----g-t' ../'/' r = o'01

--/'-r-'licross-wind

oPen exPosure

30 40Wind speed at 300 m above qround (m/se)

FIGURE 9.2.4. Ratios of peak along-wind and peak across-wind response to meanalong-wind response for a 53-story building model with a square shape in plan in urbanand open tenain [9-13].

Figure 9.2.5 19-751 shows the across-wind response in smooth flow, flowover suburban terrain and flow over urban terrain, for prismatic buildings withseveral depth-to-width and damping ratios. The model scale was estimated tobe about 1/600, and for all models the height H, the sectional area BD, aruJ

the density were 0.5 m, 0.0025 m2 and l2Okglm3, respectively. In Fig. 9.2.5,fs, (J, and h*, denote, respectively, natural frequency of vibration, wind speedat building top and root mean square of across-wind response at building top'respectively.

9.3 TORSIONAL RESPONSE

Severe distorti<lns duc to the combincd cllccts ol'ilcl'()rir' wirrtl krittls and lor-sional momcnts occurro(l rluring tho 1926 Floriclt ltttt'rit'tttlc itt lwrl Miarni high

50

ry ll',,,i',/;r5 10 t5

- f], fl ,l o o.oo17' I I ;l:l]ll

20

o 0-00147.0.0180v0.0510

o 0.00234.0.0133v 0.0476

5 10 15*F't

!, it l()lilil()NAl ilt lil'()Ntit 351

'll)l()

o 0.00.0.014v0.04 I

o0.001 4

.f) l) lr'2

kv0. 055

5

o0.001 92.0.0167v0.0626

5 10 t5U/f^B

o0. 001 64 u

.o.o1iz J/vo. o47t f

5 t0 t5o 0.001 90

.0.01 78

25

u/f0B

v0.0534

i'1

l- ,to , to , to , '9 "/to0 5 t0 15 20 ?5

t/ f a,/BD

urban arealil(;t.lRE 9.2.5. Across-winrl rcsponsc of prismatic buildings (circles and triangle in-tlit'lrtc clamping ratios). Fnrln I'1. Krwli. "Vortcx Induccd Vibration of Tall Buildings,".l . Wirul l,)ng. Ind. Affrxl.,4l-U ( I9r)2). Il'7 128.

l)"0Il) 510152025

u/ f o./BD

smoot h

0r0203040+0 5 10 15 ?O 25

u / f ^,/BDU

open terrai n

40

5

o 0.001 96.0.01 56v0.0450

5

o0.00250o0.01 49

w.0423

51015202530oO.00229r 0.01 60

352 llt,ill)lN(iii WtNl) l()nl )li, lilllll(.1(lltnl lll l,l '{rNlil nNl) l)l :;l('N t)l ll{)( )l lN(i

rise structurcs, thc l-5-story llt:alty llrriltling, rltttl lltc l7-s(ory Mtrycl Kiser

Building t9-3 11. Both buildings hacl utrusuillly nitrK)w shapcs in plarr (thc tlimensiois in plan of the Meyer-Kiser Builcling wcrc about 14 x 42 rrr)''l'he:ir'Structural Systems consisted of steel frames. The two transvcrse cnd l'rantcs ol'

the Meyer-kiser Building experienced horizontal deflections of about 0'60 rrr

and -0.20 m, resPectivelY.Following these incidents engineers became concerned with wind-inducctl

torsional edects, as shown by subsequent developments in the literature, irr

cluding a 1939 ASCE report that dealt with such effects in some detail [9-32'9-331 .\everrheless, wind-induced torsion of tall buildings is not mentioned irr

rhc 196l ASCE srate-of-the-art repoft 19-341, or in any U.S. building codc orstarrtlartl tlcvclopccl hclirrc the ASCE 7-95 Standard [9-5]. This deficiency muy

cxpl:rin wltll itppcll-s l<l havc bcen the absence of provisions against wincl'inclucctl torsion in f hc original clesign of the John Hancock Building in Boston'which by virluc ol'its shapc is particularly sensitive to both across-wind antl

torsional cll'ccts.Torsional cffects are due to the fact that in any individual building the centcr

of mass and/or the elastic center do not coincide with the instantaneous pointof application of the aerodynamic loads. Ad hoc tests simulating these effects

have-been conducted for a number of years on individual building models'However, until recently, relatively little work has been performed toward thc

development of design information and analytical procedures for use by struc

tural d-esigners. A first attempt at studying analytically torsion induced ort

buildings -by fluctuating wind loads was reported by Patrickson and Friedmarr

[9-35]. More recently, Safak and Foutch have presented potentially usefirl

methods for estimating the along-wind, across-wind, and torsional response ol'

rectangular buildings 19-36, 9-371. However, owing to the absence of sufficie nl

infonriation on aerodynamic loads, the methods are not presently usable lirtdesign purposes.

WinA tunnet and full-scale research studies of torsional response were firslreported in [9-26] and [9-38]. Reference [9-26] includes information on wintlinduced torsional moments in an isolated square building model having a heiglttto width ratio h/b : 8.33 in flow that simulates urban conditions. Accordirrgto the results of [9-26], torsional moments are largest when the mean wirrtlvelocity is normal to a building face. As the angle cv between the mean wirrtlvelocity and the normal to the building face increases from 0o to 45', the

torsional moments decrease from their maximum value corresponding to o '0o to about 25% of that value for a : 45". Assuming that the mechanicirlproperties of the model are similar to those of typical high rise structurc-s' il

was estimated in [9-26] that, for a : 0", the peak torsion-induced respottst'

of a corner column is approximately 65% of the peak along-wind rcspotrsc

corresponding to a : 0o. For ty:45", the peak torsion-induccd rcsptlnsc ol

u .o-"1. column is ab6ut 15% <tf the pcak along-wind rcsp()nsc corrcsptltttliltgtoa:0o.

Systcmatic wirrtl trrrrrrr:l slrrtlirrs corrtlrtc(etl rrl lhc Illrivt'r'sily ol'Wcslet'rt ()lt

tr I I r 'l l',lr )l lAl i il ',1'r rf l',1

l;rtl() wel-(' sttlrst'tqttcrtlly tt'Potlt'tl irr l() l()l (o l() .f .tl.' 'llr,'r-,,' :,lrrtlrcs lutvt' lt'rll(' llr(' li)ll()wirrg crrrpilit'rrl t'llrtiott lol cslirrr;rlirrl' {lrc Pt';rk lr;r:rt' lortlrrt' /i,,,,1 {/(/r)lrrrrlttccrl by wincls witlr spc:ctl U(lt) al tllc l()l) ol lltt' lruilrllrrl,,:

T',,,,,,1U(h)l : {rl'i' ltttttll I,q/'/,,,,,,1 t t(h)lti (9.3. r)

rvlrcrc ry' is a reduction coefficicnt thut is briclly cliscusscd subscqucntly, gr. =I l"i is a torsional peak factor, and thc Iincar antl nrrs [-rase torque, T[U(hl ana1 ,,,,,1U(h)l are given by the expressions

7 wr.tt)l : o.o38pL4 hn?ru?

I,.-l U(h)l = 0.00167 ! ptonr'rU','rSr

u(h\If

nrL

, Jlrl dsAt2

(9.3.2)

(9.3.3)

(e.3.4)

(e.3.5)

ln Eqs. 9.3.2to9.3.5, p is the air density (p = I .25 kglm3), ft is the height()l lhe buildiflg, trr and f7 are the natural frequency and the damping ratio inrlrr: fundamental torsional mode of vibration, ds is the elemental length of theI'rrilding perimeter, lrl is the torque arm of the element ds (i.e., the distancelrctween the elastic center and the normal to the building boundary at the centerol the element ds; see Fig. 9.3.1), and z4 is the cross-sectional area of thel,trilding. Equation 9.3.2 and 9.3.3 are based upon the experimental resultsslrown in Figs. 9.3.2 and 9.3.3, in which the ordinates are the reduced mean;urtl rms base torque, f, : Tlbfahn?) and o, : Tn,"(t/zl(pL4hn?i, respectively.

l,'l( il ll{l,l 9.-}.1. Notrtliorts

r'lltt'ttsttlls ol lltt'st'slttrlics rvtrc Lrr'll\ I'r"\rl,rl t,r tlr, ,rrrtlr,'r., lrY l)l \ l:;yttrrrov

354 Fltlll l)lN(ifl: wlNl) loAllli, $llit,(llllltAl ltl tit)oN$t , ANr) l)trit(iN ot lt(x)t tN{i

86

2

10- 1

I6

4

2

10'I6

4

2

10-s

1o-1 o uioo ' o u1o, ' o ufo,ur - REDUCED VELOCTTY

FIGURE 9.3.2. Mean base torque for tall buildings with various shapes in plan (cour-tesy Dr. N. Isyumov, Boundary-Layer Wind Tunnel Laboratory, University of WesternOntario).

The torques 7 and f-. are each induced by wind with reduced speed U, anclwith the respective most unfavorable direction. In general, the most unfavorabledirections forTand 7-, do not coincide. In addition, in most cases neitherol'these directions will coincide with the direction of the extreme winds expectedto occur at the site. For these reasons, the coefficient ry' in Eq. 9.3.1 is lessthan unity. It is estimated in [9-39] that 0.75 < t! = I in most cases.

The peak torsional-induced horizontal accelerations at the top of the buildingat a distance u from the elastic center can be written as

FHHHs-\'P*

\)t+

EHOE|-;:;::1"

100

ullaccoFzt!

ouJofotlJc

I

F

^ 2g7T^.uAU =- p6bdhri, (e.3.6)

where 0 is the peak angular accclcration and r,,, is thc rirrlirrs rll'gynrlion. Fora rectangular shapo with unilorrn bulk mass pcr unil volurrrc

ulla(roFa(EouJoloUJtr

I

b

1o 'uo4

100864

10'

2

10 'I

lr ir t oilr;toN^t ilt r;t 'r )Ni it 355

EFHHs\PS

RN

::::l

<dTHOE53 2.7I

;:;i'.r'

64

10 -3864

2

10 -4 2 4 AA 2 4 6a 2 4 6Ato-l 1 Oo 10 1 1O2

UT - REDUCED VELOCITY

lrl(;URE 9.3.3. Root mean square of base torque for tall buildings with various shapesrrr plan (courtesy Dr. N. Isyumov, Boundary-Layer Wind Tunnel Laboratory, Univer-r.rly ol' Westem Ontario).

b2+d2 (e.3.7)

lirl. 9.3.6 was obtained in [9-39] assuming a linearfundamental modal shape;rrrtl ncgligible contributions by higher torsional modes of vibration.

Numerical Example For the building considered in the numerical examplesrrl Sccts. 9.1.2 and9.2, h : 2OO m, b : d : 35 m, U(h) : 39.4 mls, p6 :.l(X) kg/m3. It is assumed that the natural frequency and the damping ratio intlrc lirndamental torsional modc of vibration arenr:0.3H2 and f7: 0.01,rt'slrcctivcly, and that thc air clonsity is p : 1.25 kglm3.

lirrrrrr Eq.9.3.5. t,: tl(ltl2l{ol4)lb :35 m. Then U,:39.41(0.3 x 35).1.75 (tlq. 9.3.4),-rpv.+1 l.11 x 107 Nm (cq. 9.3.2),7,,,,. 139.41 : 1.95

. l0/ Nrrr (lic1. 9.3.3), 7,,,,,, r).2 r 107 Nrn (liq. 9.3.1 in which it is assutttctl

& 12

rG- '/.Ilalo

""AIA^Q

*'/sc

"llaa\I

//I/

il356 tll,lt t)tN(i:i wtNt) l()nt):;. t;ililt(:l,ilnt ltt :,t '()N:it , nNI) t)t :;t(iN ()t tt(x)t tN(i

rl, = 1,8't :3.lt).'l'hc llcrrk l()lsi()rr rtttltttr'tl ltolizorr(lrl rrt't'clcllrlion ill tlte t()l)corner (e,' : 35 x ,1212 - 24.1 rrr; is /,t,,,.,,1, -. 0..17 nr/s'. N.)tt. lltirl llrisexceeds the peak along-wind accclcralions but is substarrtially lcss lhirn tlrcpeak across-wind accelerations calculated in thc previous numcrical oxanlplcs.

According to wind tunnel tests reported in [9-82], fluctuating torsion tlcpcntlsstrongly on building cross section, being largest by far tbr triangular builclings,intermediate for rectangular buildings, and lowest for D-shapecl and diamontlshaped buildings. Such dependence is not apparent from Fig. 9.3.3.

The peak combined effect of the along-wind, across-wind, and torsionlrlloads can bc obtained by summing up vectorially the individual peak effects ol'thcsc krads and rnultiplying the result by a reduction factor (e.g., equal to 0.g)which uccounts lirr thc fact that, in general, individual peaks do not occur.sitttultttttgrttsly. Il'llrc cornhincd elTect so calculated is less than an individr,urlcll'cct, it is tlrc lltlcr tlrat should bc considered in design.

Morc rcccntly, based on acnrdynamic data reported inIg-21, [9-66] presentctla rigor<tus dynarnic analysis of torsional moments which takes into account thcel1'ect of the distance between the elastic center and the center of mass of tht:structure; see also 19-831 .

9.4 TUNED DAMPERS AND VISCOELASTIC DAMPING DEVICES

Two main types of device have been employed for the reduction of tall buildingvibrations in translation and/or torsion: tuned mass dampers, and viscoelasticdampers. Devices intended to reduce torsional vibrations must consist of atleast two units located at sufficient distances from the elastic center of thcstructure.

Tuned dampers consist of a mass, usually of the order of 05% to l% <tl'the total mass of the structure, that is added to and interacts dynamically witlrthe structure. Inherent in or attached to that mass is a system that dissipatcsenergy during the relative mass-structure motion. Active controls may be usotlto improve performance 19-76]. Several types of tuned dampers have becrrproposed which by various means ensure dynamic interaction (e.g., springs orpendular devices) and energy dissipation, and which may have more or lcsselaborate control systems 19-161. In tuned liquid ilampers (TLDs) most of tht:mass is due to liquid contained in a tank, and liquid motion provides or contributes to energy dissipation [9-71 , 9-78]. The latter may be increased hyplacing obstacles in the liquid's motion, for example, cruciform poles [9-791or floating elements t9-801. In Sect. 9.4.1 we discuss in some detail the dcviccknown as tuned mass damper (TMD), which illustrates the basic principlecommon to all types of tuncd dampers. Section 9.4.2 discusses viscoelastit'dampers.

For a recent rcvicw of'clarnping dcviccs firr thc t'orrlrrl ol wirrtl-intlucctlvibrations, scc ll3 1041. ltcl'clcncc l9 ll3l rliscrrsst's t'orr{rol ol vilrr.:rti6rrs 6ltall builclings crlrrsisl irrg ol lrrcglrslnrelrrrcs tlcsil'.lrt'rl t,r rvtllr:,1:urrl wirrrl lrrrrl

lllll l, Irl\Mt'l ll:i nNl) Vl:,(.()l ll\l;ll( li/\Ml'lllri I,l \/lr l . ,|at7

,'.rr{lrtlrltkc lolrtls lrrrtl l() i,nl)lx)rl lttotlttllrt sttlrslltttltttt':' llr.rl lrior,irL'lltt'tl:,:tlrlt'I'urltlrng spirce.

1l-4-1 Tuned Mass Dampersllrt' 'l'MD cclnsists o1'a rclativcly srrurll vilrrrtoly sysl('nl (nlrss, splirrg, rrrrtl

,l.rslr1xlt.) attachcd to a structurc whosc viblrrtiotts it is tlcsigrrctl to rrritigatc. Itrr':rs invcntcd in 1909 by Frahm itntl hlts trntil rcccntly bccn usod prirnarily inrrrt't'hanical engineering systcms. ln thc last dccadc TMDs have increasinglylx ('n cmployed in wind-sensitivc structures, including the Centerpoint Tower,lirrlrrcy, Australia 19-441, the CN Tower, Toronto 19-451, the John Hancock! .rryg1, Boston (equipped with dual TMDs designed to control both torsional,rrrtl lateral motions) 19-46, 9-471, and the Citicorp Center, New York Cityl') 113. 9-49, 9-501. Generally, the purpose of the TMDs is to reduce buildingrrrot ions insofar as they affect occupant comfort, and the effect of the TMD is!r()r (aken into account in strength calculations [9-46,9-41].

A schematic view of a TMD operating on the top floor of the Citicorp Centerr:, slrt)wn in Fig. 9.4.1. The mass of the TMD consists in this case of a400-r.n q1tn.r"," block bearing on a thin oil film. The TMD structural stiffness isI'r,rviclcd by pneumatic springs which can be tuned to the actual frequency oftlr,' building as determined experimentally in the field. The TMD damping is1'rovided by hydraulic shock absorbers. The system includes fail-safe devicesr. l)roVert excessive travel of the concrete block [9-49]. Additional information,'n l'MD equipment and control systems is given in [9-46].

'l'hc theory of the tuned mass damper was developed by Den Hartog [9-51]lrrr tlre system shown in Fig. 9.4.2, with Cr : 0 and a harmonic load F(l).( lrr thc basis of results given in t9-5 11, the theory was subsequently extendedrrr l() -521 to include the case where C1 * 0 and F(r) is a random load with, rrrrstant (white noise) spectral density. In Fig. 9.4.2, Mt, Cr, Kr, and M2, C2,A , rrlc the mass, damping, and spring constant of the structure and of the TMD,r, slrcctively.

I llect of TMD upon Deflections and Accelerations of Structure. The, llt't't ol'the TMD can be viewed as being equivalent to changing the damping,,rrr,, ()l'the original system (not provided with a TMD) from the value f1 :t ,l:J K tM I to a larger value f" Thus the deflections and accelerations of mass,'1/, irr thc system of Fig. 9.4.2 can be obtained by calculating the deflections.urrl rrccclerations of miss M1 in the system with dampinE Cn: ZJX,tt4y",i'wrr in Fig. 9.4.3. Using results fiom [9-49] and [9-52]it can be shown that

I rv1((troj - cYy) - a1la]\,, (9.4.1a)

rr l rt't t'

2 rv11(rv.,rv 1 rvl) I cvl(pj * 2cvr') * rrr

(().1 Ilr)

OEodEoo6dg

:oo=off

>Er6

.:o

.:Ioa

c

oIz

o

3tU9

>|{)ts

o>'UJIo

Boz0)

oOgooU

oq

ii -,i.o-t'F-a ?t)

EF3gtE>s.juiEo\ [lfrl A&a?nE!

Dg,P

€.:Ee!6z

g3*8_Ecoc9ggE8Bo

co

co

9es6!og-

Eo8;a! o53E-

e:r4

oE3e6

egrgcqaE

sbog

358

Vol. 4, No. 5, with pcr-

I

Mt

I

9,4 IIJNI I) DAMPERS AND VISCOELASTIC DAMPING DEVICES 359

FIGURE 9.4.2. Schematic of system equipped with a tuned mass damper.*

@1

Q2

A3

DPr

2f ((tf + b)t+fze+p)+4f(&z2ft + 2fzf (l + p)

2lzf

(9.4.1c)

(9.4.ld)

(9.4.1e)

(e.4.rf)

Itt lJqs. 9.4.Ia through 9.1.4f,

FIGURE 9.4.3. Notations.

+f iigrrrcs 9.4.2 t<t 9.4.6 arc rcprotluccd l'r<tn Engineering Structure.s,tirissiolr ol' thc publishcr, lluttcrwoflh Scicntilic t_td.

360 BUILDINGS: WlNt) lOADri, tilllt,(llllllAl IIfSPONSF, ANI) lll til(iN (ll llO()l lN(i

p : MzlMl

f : a2la1

',: JK1M, (i : 1,2)

| : C/(2M;a) (i : t,2)

(9.4.19)

(e.4. th)

(e.4. li)(e.4. r.i)

(9.4.2)

(9.4.3)

Forexample,if p:0.01,/:0.98, f1 :0.01, and f2:0.0515, then f,, =0.03226. The dependence of f" upon f2 is shown in Fig. 9.4.4 for fr : 0.01,/: 0.98, and various values of p. [9-49]. Note that for each pr there exisrs anoptimal (maximum) value of f" (denoted by l3p') which can be sought byrcprcscnting 8q.9.4.1a graphically. Alternatively, it is shown in [9-53] thatwith ncgligible errors, the following approximate relations can be used tbrpreliminary design purposes to obtain l?Pt and the corresponding value of f2(dcnoted by fin'):

J;fln'=o+0.8fr >fl

,rO, - &12 2

Forexample, tf p :0.01 and f1 : 0.01, then flpt= 0.033 and fipt= 0.05,

alJl

12345678910111213(2 ("/.)

FIGURE 9.4.4. Dcpcndcncc of t. upon t2 and p. Allcr I{. .l . Mr'Nurrrirra, "'funerlMass Dampers in lluiltlings," .1. Strrct. Div., ASC|i. l0.l (l()77). l7t{5 179u.

col lArili(r DnMl|rN(i t)t vt(;t ti 361

Displacements of TMD Mass. lrr dcsignirrg rr 'l'Ml) systln, alkrwancc lnustlrt' lrtadc firr thc clisplitcotttcnts (travcl) ol'the 'l'Ml) nrirss. 'l'ltcsc clisplaccments;rrc in practicc rclativcly largc. Frlr cxarrrplc, irr lher clsc ol'citicorp Center,IMI) displacements induced by a stonrr with l l0-yoar rcturn period wererslirrrated from model tests to bc on thc orclcr ol' 1.00 m.

Lct the displacement of the TMD lnass with rcspect to mass M, be denotedIry r' (Fig. 9.4.2). The displaccrrcnt duc to resonant amplification effects onlyol nrass Mlinthe original system (shown in Fig. 9.4.5) is denoted by"r1,s. (Itis t'rrrphasized that.rl,e does not include contributions due to mean or quasistaticlrxrtlirrg.) Using results from [9-49] and [9-52), it can be shown that the ratiorrl llrc mean square values of x2 and -r1,s, denoted by xlno^, is given by thert'llrl ion

zSPtt(9.4.4)a1(a2c"3-cr)-uso.!

lrrrl cxample, if p1 :0.01, /: 0.98, fi : 0.01, and f2 : 0.0515, thenr1,,,,,,,, : 13.7. The dependence of x22no^tt2 upon f2 is shown in Fig. 9.4.6 for(r ' 0.01, "f : 0.98, and various values of p.19-491.

Ocsrgn of TMDs for Actual Structures. Because buildings are multi-rlcp,rcc of freedom systems, the model shown in Fig. 9.4.2 is not a rigorousrt'prcsentation of a building with a TMD. The error inherent in the assumptionthirl the building equipped with a TMD can be represented by the system of

;-:z'.1nom xi.o

l,'l(il llll,l 9.4.5. Notations.

il362 ll(lll I)lN(;1; WtNl ) t()nt)li, l;ltt(,(.illilnt ilt :;t ,oNt;t nNI) t)t :;t(iN ()t il(){)t tN(i I r rW I Il',1 lll Jll I )ll'l( i:: 363

( )vcrlrll tl:ttttpttt;r. rt'tltrirt'tl :rs lr Iutrt(iorr ol llrt' r,1x't llrctl lt('ln tccurfonccirrlr:t'virl ol'tlrc winrl lorrtlirrg (c.ll., lO or l(X) y(',us).

Irnvinrrrrrrcrrl clrirrlrc'(cris(ics itl (lilnllx't l(x':rll()ls (r..g., trir tcntpcrature).S;lrrco availablc lirr (liulpcr tlispl:rt't'rrrt'rrt, :rrrtl rctprisik: damper stiffness.lir.cqucncics <ll' vibration ol' brriltlirrg (tlrursl:rtiorral and torsional).

llrt' tlatnpcr design includcs llrc sclcction ol'the material properties (shear lossrrr'xlrrlus, loss tangent, and thcir toulpcraturc dependence), and the size andrrrrrrrlrcrof dampcrs; see l9-67,968,9-1O,9-7 1,9-121 fordetails. Buildings, rluippcd with viscoclastic dampers include the World Trade Center, New\ olk. and the Columbia Center Building, Seattle.

(I,5 LOW.RISE BUILDINGS

Itriltlings with relatively low heights are, as a rule, rigid and do not exhibit'.ry'nilicant dynamic amplification effects.

As was shown in Sects. 4.6,4.'7,1.3, and7.4, wind loads on any given.,, trrlrl building or model depend upon several factors, including the character-r'.tr('s of the oncoming flow, model scale, area affected by the wind load, andr.rlro ol'openings to gross area of the building envelope. Recent work on the, llt'cl of these and other factors was reviewed for low-rise buildings by Stath-{,lx)ulos (see [9-84, 9-85] and references quoted therein).

ASCE 7-95 Standard [9-5] windloading provisions for low-rise buildings.,r,' bused to a considerable extent on results obtained in wind tunnel tests attlrt'(lniversity of westem ontario and Concordia University. Despite the small.,;rlc at which the tests have been conducted (usually l:200 to 1:2000), it hasl,t't'n the consensus of code writers in the United States and Canada that theyI'r,,r,itle a reasonable basis for codification, with occasional adjustments reflect-,,r)' l'osults of full-scale tests or the desire to calibrate new provisions against, rrsling practice. The tests have confirmed that the fluctuating part of the load, .rrr in many instances be significantly larger than the mean loid and that, for,ury givcn storm, peak pressures and the ratio between mean pressures andlirrt lrurtirlg pressures decrease as the terrain roughness increases.

lir:sults on the influence of geometric parameters have been used to simplifyt;rrrtlrrrcl provisions. It was found, for example, that for buildings with small

lr, rlht-to-width ratios and length-to-width ratios of 1.0 to 3.0, the loads do notrL pi'nd significantly on length; wind loads increase with building height buttlr,' tk:pcnclcnce of pressure coeliicients on height is reduced if they are refer-, rr, r'tl wi{h rcspect to the velocity pressure at the mean roof height; roof slope'. rrrr irrrl'rotlitnt paramctcr l9 tt6l.

Wt' rtotc llutl stunclutrl lirrrrrlrls lrrc bcing devclopcd that wrtr.rlcl all<lw thc use,'l tl;tlrt bltscs oltlrtittcrl lirrtr winrl lrrrncl lcsls, as opposccl lo thc usc ol'rlata'',ttttttt:tt-it's, wlrit'lr is lypit':rl ()l ( ur('n{ sllrttllutls. Mlrny ol'(lrc silrrplilit'lrtiolrsir:,rttlt'rl lo itt t'tult'n( s(:lrrl:rrrl:. \r'orrlll 11,,',.'1i,11.n,, lorr1,,t.l i,,.n,.,..lr"rl, rrtslr';rtl

3 4 s 6 t'uf' e 10 11 12 13

FIGURE 9.4.6. Dependence of ",*',, upon f, and p. After R. J. McNamara."Tuned Mass Dampers in Buildings," J. struct. Div., ASCE, r03 (1977), 1785-l79tJ.

Fig' 9.4-2 (where Mr, Kr, and c' are equal, respectively, to the generalizccrmass, the stiffness, and the damping in the fundamental mode of the buildingnot equipped with a TMD) was estimated for a particurar structure in t9-53 i.According to the approximate estimate of t9-531, the simplified model of Fig.9 .4.2 led in that particular case to an overestimation of the equivalent dampingratio of the structure by a factor of about 1.2.

It is noted that results reported in [9-54] on the dynamic response of lighrequipment attached to structures are applicable to thl study of the errors inherent in the model of Fig. 9.4.2. These errors are generaily negrigible forstructures with ratios of frequency in the second mode to fiequency in thcfundamental mode of the order of two or larger.

9-4.2 Viscoelastic Dampersviscoelastic dampers are passive devices that have the advantage of not rcquiring constant operational monitoring and of not depending on eiectnc powcr..Like tuned mass dampers, viscoelastic dampers are used for acceleration rc-duction only. The buiiaing damping they achieve can attain 4% or more, antrfor very large buildings their construction costs were estimated to be about0.5% of total construction costs l9-lo, g-jll.

The fbllowing l)ctors ncccl to bc considcrccl in thc tlcsigrr 6l-viscgclastit.dampers*:

'l'l)crsorr:tl (rrrrtttrrrrir':rliorr lry l)r lr M;tlrrrrrrrli, lM (.otilP;rrr\,. ljl l,.rrrl NlNt 1,Ii.,

364 lll ,tt t)tN(iti wtNt )l()nl r:,,:.ltillr tunt\t lt :;t'()N:,1 .nNl Il)l :;t(il,l ()t l(x)t tN{;

of using conscrvativo otlvcl()l)cs ()l l)r('ssur('rllrl:r, tlc:sigrrcl's woul(l lt:soll lo (ltr.more economical or risk-consistcllt ol)ti()ll ol'usirrg thc <lrigirral tlu(a corn.sponding to the set of gcometric paranrcte rs ol'intcrcst. 'I'his issuc is tliscLrsscrlin Chapter 17.

Tests have also been performed to obtain information on the cfl'cct of'buil.|ing orroof configuration on the loads. Forexample, it was found that nega(ivr,pressures are lower on hipped roofs (four-slope roofs) than on gable roofs (twOslope roofs) t9-871. The ASCE 7-95 Standard incorporares results f'rom [9-tt7l,as well as results on two-level flat roofs [9-88], sawtooth roofs [9-89], arrrlmulti-span gable roofs [9-90] (see Chaprer l7).

The influence of tributary area on loads can be ascertained by summing rrPthe sirnultaneous pressures (or pressures multiplied by appropriate influencccocfficients) at a sufficient number of pressure taps over the area of concerrr,using thc pncurnatic avcraging technique t9-911. Recent progress in the devclopment of dcviccs capable o{' rneasuring local pressures and performig spatilrlintegration ol' pressures is rcported in [9-92], which describes a device witlrlength 55 mm, width 35 mm, and depth 25 mm, equipped with 32 pressurrmeasuring ports whose frequency range is 0 to about 2OO Hz.

Architectural features such as parapets [9-93] and roof overhangs, both olwhich are accounted for in the ASCE 7-95 Standard [9-5] (see Chapter l7), aswell as eave details (i.e., whether roof and wall meet at a sharp angle or an'connected by a curved transition surface) [9-94,9-95], were fbund to influencclocal pressures, in some cases significantly.

For a study of wind effects on mobile homes, see [9-l l2].

9.6 DESIGN OF CLADDING AND ROOFING FOR WIND LOADS

The main purpose of this section is to present a risk-consistent procedure lorthe design of glass cladding subjected to wind loads. The procedure is applicable to buildings with specified orientation and requires the availability olsufficient (l) wind speed data characterizing the extreme wind climate in tht,region of interest and (2) aerodynamic pressure data obtained in the wind tunnclfor various zones of the building facades.

The procedure presented here differs from conventional design practicc irrtwo respects. First, in conventional practice the design of each cladding panclis based on the requirement that the nominal wind load corresponding to irspecified mean recuffence interval N lusually iv: so years) may not cxccr,(la load capacity corresponding to a specified probability of failure p7 (usurrllyP/: 0.008). Second, in conventional design practice wind clircctionality is trottaken into account. As shown in scct. 8.1 .2, this cun lc:lrrl to signilicant tliscrepancies betwccn thc norninal loacls uscd in clcsigrr rrrrtl tlrt':rclrurl kratls. 'l'lrcsafety level ol'(hc cllrtltlitrg clrn llrcrclirrc: hc slnrrrl'ly norrrrrrrlirru :rrrroltg llrt.vari<lus ztlncs ol'llrt'lrtriltlirrg llrclrtlr:s trrril t1rr9rr.11 ltlr.rrlii;rl lrlilrlirrl,s lt:rvirrli

l) Itr",t ill(.lrrt tWlilt ) ll)/\l r:; 3(i5

rlrllt'tr'ltl oticttl;rliolrs I llrt' prl;xrst' ol llrr. rLk t orr:,r:,lt.rrl tlt'si1',lt PtrrcctlLtrclrtr'st'ttlctl irt tlris tllrlllt'l rs l() r'litrritt:rlc or rr'rllrtr'srrtlr lrorrrrrrilirrtrtitics.'l'ltt: t'oltvcltliottitl ltrrrl (lrc lisk t'orrsislt'rrl tlt':r1',rr prrrt't'tlures havc a number.l r.t)illnlon s1c:1)s. 'l'lrcsc rrr.t: r.t:vit.wctl irr sr.t.t. ().(r.1, which also includes a{l(:;('lil)lion ol'tltc slcps tlrirl tlislirrguislr llrc lwo plrcctlurcs. Section 9.6.2 sum-rtt:ttizcs rcsults <ll'clcsign upplicirliorrs tlurt illustr-lrte the economic and safety.rlv;ur(agcs inhcrcnt in (lrc risk consisrcnr pnlccdure. Section 9.6.3 lists a fewl',r:;it' rcf'crcnces on wind cllbcts ol'rrxrfing.

!1"6-1 Conventional and Risk-Consistent Procedures for Designing(:ladding GlassI'rrt'ctlures fbr conventional and risk-consistent design of cladding glass entailrlrt' lirllowing common steps:

Obtaining information on the extreme wind climate.Converting basic wind speeds (e.g., fastest-mile wind speeds at l0 mabove ground in open terrain) into wind speeds used for aerodynamicreference purposes (usually, mean hourly wind speeds at the top of thebuilding).Obtaining fiom wind tunnel tests information on the time-dependent aero-dynamic pressures acting at various points of the building facades.Converting the information on time-dependent aerodynamic pressuresinto equivalent wind loads with standardized time history, that is, loadswhose effect upon the cladding panels is equivalent to that of the actualtime-dependent loads.Estimating design wind loads using information on the wind climate andon the equivalent standardized wind loads.Obtaining information on the load capacity of the cladding panels.Adopting a design criterion relating the design wind loads to the loadcapacity of the panels.

tt. Designing the cladding glass.

lior. additional details, see [9-96].

lxtreme wind climate. The conventional design procedure uses information()n oxtreme wind speeds regardless of direction. To apply the risk-consistentrlt'sign procedure, the information needed to characterize the extreme wind,lirrralc in regions not subjected to hurricane winds consists of directional larg-('sl yolrly wincl speeds. Such information may be extracted l'rom rnonthly Local

Altlt'rtsttrt'olllrccllrtltlings:rli'lylcvt'llol;lzorrt lorlrrriltlirrll)isgivt.nbyllrcr:tliolr,/rr,,lrr'lrrlt.rrllt('('xl)('('l('(l ttttlttltt'r ttl lt:tttr'ls llrrrl lrrrl rlrrrrrrl' tlrr' lilr'lirrrr'r)l llr(.slnr(lul.rrrrtl llrr. lpl;rl rrrlrrlpr,rl ;r:trrt'ls lor llr:rl zolrr.(ol lruiltlrlrll)

l.l

t.

(r.

1

366 rrt,'r)rN(i:i wrNr) r()nr ):;. rit*r( rrrnl rrr :,r,()N:ir . nNr) r)r l;r(iN ()r ,()()l rN(i

Climatological l)a(a sttttttttlttics issrrt'rl lr.y rlrt' Nirtiorlrl ()ccrirnit. rrntl Alrrr,spheric Administration (sce Scct. .1.4;.'lllre,sc clatl arc usually rcc.rtlctl .vcropen terrain (airports) and should be rccluccd to a comrnon cllvati.n (usrurlly10 m above ground).In hurricane-prone regions directional information on hurricane wind spcctrscan be obtained by Monte Carlo simulation (see Sect. 3.3.2) or tiom crat,stored in [8-9] (see also [3-71]).

conversion of Basic spe^eds to Aerodynamic Reterence speeds. Givcnthe basic wind speed uf(r},0) (i.e., theiastest-mile wind from direction g arl0 m above ground in open terrain), the corresponding hourly mean speed,u(h,0), at elevation h over the building site can be estimated by using Eqs.9'l'6' 9-I.tl, and 9.r.9 and the micro"meteorological paramete'rs of rabrcs9.1- l' 9- 1.2. ancl 9- r .r. p','r cxampre if u/(10, 0) : ls mph and the buildinghas hcight h : 2oo rn and is lrcatcd in a iown with roughness length upwintrof thc building z0 = 1.00 nr, U(200,0) = 39.4 -1, frJ" "^"-pi", in S".tr.9.1.2 and 9.2). For hurricane wincls, sce Secr. 2.43 and fq_S, p. iSSt.Aerodynamic Pressures on Buflding Facade.s. Information on aerody-namic pressures is obtained fiom wind tinnel tests. It r, pr"ro,t"o in terms or.aerodynamic pressure coefficients defined as

P(M,. 6r1C,(Mi.0^7 : -- i-louzth. ors

; t) il( )( )t [J( , l ot t wll{l) t ( )/\t ): ; 367

,l.Pt'tttlt'ltl tlt't'tt';tsr'ol lltr' y'l,r::. -lrt'r11'(11 ,\(/) l() \til l lrc I;rrlrrn'lotrtl is ol-rtaincclli,rrt lllL: ctttttliliott llt:r( l:rrlrrrt' ('('( lrs wlrt'rr llre tt'lrsiolr stl('ss (r(/), which is inr', n('r'lll lr norrlincltr-lrrrrt lrorr ol llrr. lo;rtl7r(/), is t.r;rurl lo llrc strcngth S(f), whichr'.;r lirrrctiolr ol (lrc rtllitrl strt.rrl'tlr ,\10;:rtrtl ol llrc klrtl p(l) (Fig. 9.6. l). This.rp;rlrriqgll cntails Mortlt' (':rr lo sirrrrrlrr(iorrs ol tlre initial strength from probability,lr:;trrlrutions obtainctl cxpt'r'irrrcnlrrlly, us wcll as thc calculation by numericalrrrr'llttrtls o1'the nonlirrcitr rclirlion bctwccn the loads p(t) and the normal stresses,'tlllr.6r, r), tbrasuflicicrrt nurnbcrol'points M1anddirectionsd/of thestresses.llrt'lpproach is applicd to panels subjected to (1) loads with the time history

/'t /):rrrd (2) constant loads with a 60-s duratiorr, poo, which are commonly usedrrr N.flh American design charts. Probability distributions of the load capacity,'l rlrc panels are obtained for the loads p(r) (indexed by their mean value/'(/)) and for the 60-s load pon. Let these distributions be denoted ay Polpal.rrrtl P2,,,(p6,e), respectively. The 60-s load pfifl equivalent to the load p(r) islrvt'rr by the relation

p"u?, : P;J{P,lp(/)l} (e.6.2)

l lrrrs, fbr any point M1 and wind direction 0p, an equivalent aerodynamic coef-irt rt'nt can be defined as

(e.6. t)

I

t50

I

?fl)

l"l(Jtlltl,l 9.(r.1. livolrrliorr ol tt rr:,r,rrr ,,tr,'l,tr't' rtl 1tl:rss 1ll:rlt'. lr:rilrrrr' ()( ( ltr:. ;tt tr!!rt I

(9.6.3)

-------- s(r)

2s0 300 350 a{t0 450 5{10

/( .)

.ilrii .,tr( n,'tlr r,villr lllrtt' :rl :r lxrirrl orr lltr'11 )(, '. |

(, ')ri I

pZlru,. otC},n(Mi.01,1 : - "

\pU'(h.0*)

w.here p(Mt, 0*) is the pressure at point M1 of the facade, induced by windblowing from direction 91 with u ,''"un hourly speed at the top of the building,u(h,00; p is the airdensiry;,ay.d cr(\, d1) is the pressure.i.m.i"n, ar poinrM; corresponding to lhe_wind direciion'd^. pressure coefticienrs c,,(Mj. d1 ) arcrecorded as functions of time for various wind directi"", B- "i v-Jrious pointsof the building facades' including points near corners and eaves. Measurementsare usually made for angles 0r: k x 15. (k : 1,2, .. ,)q althoughoccasionally the increments may be smaller than 15. to alow detection ordirectional maxima.

Equivalent 60-s Wind l_9ads. Wind pressures p(.M1, 0r) andthe corresponcl_ing pressure coefficients CrMi,01) are randomtynuJiu#ng run"tion., of tirrrcthat depend upon the position M1 and the mean wind direction 01 (e.g., sccFig. 4.7.2.).. . The load capacity of glass cladding panels depends upon the entirc tirrrehistory of the load. This, dependerce cln^in principre be tui.cn inkr acc.unt rryusing basic fracture mcchanics relations to clescribc thc cl'lccl ol'lirtig'c c;*rsctlby the fluctuating loacl, thal is, thc tirnc-clcpcnrlcrr( gnrwth irr l5c sizc: .l. llrrwspresent on thc srrrlirccs luxl crlgcs ol'lhc;.r:rnr.ls, :rrrrl llrr.r..orrst.tlut.rr( lilnt,

l5

MPa

l0

"(t)i'r'?, ih.r, 1{l,r,,{ 'q,o'

50

t368 llt.,ll l)tN(it; wtNl) l()nt )l;, :iillt t{.ililt^t ilt:;t ,{)Nt;t nNt) I)t :it(iN ()t il()()t tN(i

The approach just dcscribcd lrirs so liu'lrt'crr rrscrl orrly irr cxpl1;r.irl6ry ilvcstigations t9-58]. Currently a sirnplcr appnr:rch is usctl lirr rlcsign psrl)()scs, 1rwhich it is assumed that the actual fluctuating loacl causing lailJro is oquivale,rrlto a constant load with asmall duration, tpr, and a magnitudc equal t. thc pcirkfluctuating load averaged over the time ipt , ppr. It is iommonry assumecr thlrrtrp = I s.*

The l-s constant loadprp must in tum be converted into an equivalent 60-sload p[[. It can be shown from basic fracture mechanics relations that rhcstresses o6e and onr induced by the 60-s loadpifi and the 1_s load pr1,, respcctivcly, are equivalent from the point of view of their effect on glass if

o[,, x 60 : oirx I (9.6.4)

pzl,(Mi, 0i : ipcil,,(Mj, 0iu2(h, 0k) (9.6.1)

l)l l:l{,l.J ()l (,1 nl rl)lN(i nNl) il()()l lNt(, t{rt t Wll.ilr l{)/\t r,, :tfi!}

n'lrit'lt ltirs tltc slrntt' lolnr trs lttl. t{. Llt. 'l'lrc rlt.sll'11 lvlntl lt,;rrl:; r.;rrr tlrt.rr.lorr.lrt' t'slirrrirlctl lrs slrowrr in Sccl. lJ. 1.2, irr wlritlr yri,ilt /'tr. t)t.l:rrr.l {',,,1,{ Ilt. 0Al'.lrorrltl bc substitutcd lirr p(rrl) arrtl ('(//)(,,({/). rt'spt.t.trvr.ly

'l'lrc cstirnation <ll'clcsigrt wirrtl loruls tlillt'r's irt'tollurlr. lo 14,11,'11,,., tlrt' r'orrrt'ttlional orthc risk-c<lnsistcn( tlcsigrr pntcctlrrlr'is rrst'tl. lrr corrvt.lrllorlrl tlcsilirrIrltclicc equivalent 60-s l<lacls witlr :r 50 yr.:trl rnc:ut tccut'r't'rrct. intglv:rl,1",,,)\ n1(M) are estimated withtlul cortsitlcring llrc cll'ccts ol wirrtl tlircc(iorrality.l'.r'this reason the actual mcan rccurrcncc intcrval, N.,,r.tlt'the clcsign loadl,,,ij ,,, varies fiom panel to pancl. ln thc case of panels for which the direction,rl (hc most severe extreme winds coincides with the direction of the largest,rt'nxlynamic coefficient, N"., ir indeed 50 years. However, for most otherl);illcls Nacr exceeds 50 years, in some cases by one or even two orders ofrrr:rgrritude (see Sect. 8.1.2).

A second consequence of not accounting for wind directionality is that anytrvo buildings that are identical in all respects but have different orientationsr'ill cxperience different numbers of panel failures during their lifetime. Indeed,',ilrcc conventional practice does not account for wind directionality, it willv rt'ltl exactly the same cladding design for the two buildings even though, owingt. thcir different orientations with respect to the direction of the most severer'\trcme winds, the two buildings will exhibit different degrees of sensitivityto wind effects.

lior the risk-consistent design procedure it is necessary to estimate the mean,rrrrl lhe coefficient of variation of the equivalent 60-s largest lifetime load. Thisr:, tlone as shown in Sect. 8.2.2 (Eqs. 8.2.7 to 8.2.14), in which pffi, and| ,,,,,, should be substituted forQ, ancl Vn,, respectively.

Load capacity of cladding Panels. Information on the load capacity of, l;rtlcling panels can be obtained from manufacturers' charts [9-55, 9-56]. These,lr:rrts include estimates of the standard deviation and of the 0.8 percentageP'irrt of the load capacity of panels with different sizes for annealed, heat-,lrt'rrgthened, and tempered glass.x The charts of [9-55] and [9-56] exhibitrrrrrlual inconsistencies, and apparent internal inconsistencies have been notedrrr l() -551 (see [9-57, 9-58, 9-59], which report research aimed at improvingrlrr'sc chafts).

()wing to fatigue effects, the load capacity of glass panels depends upon thetrrrrc history of the applied load [9-57,9-58,9-59]. The load capacities givenrrr l()--5-51 and [9-561 have a standardized time history; that is, they are expressedrir l('f'rlrs of constant loads with a 60-s duration, denoted by puu.

' Ilrr'o li lx'r(cll(rtgc point ol lhc kr:rtl t lrp:rt ity is thc load to which lhcrc r:orrcs;rontls ;r prolr:rlrility,'ll;rrlrrrtoll.lJritrrclsouloll,(XX)(sct.Sct.t Al5).Irrlirnrurlionorrkr:rrlst.orrt,s;xrrrrlirrl,1oo1llq.1I'r,,lr;tlrililit s ol l:rilrrrc is :rv;rillrlrlt. rn l1) 'r(rl

whcrc rt is thc cxponcnt in thc phenomenological relation describing subcriticalcrack gnrwth. For soda limc glass it may be assumed for practic-al purposcsthat n : 16 [9-59]. From Eq. 9.6.4 and the simplifying assumption that thcload-stress relationship is linear, it fcrllows that

P'fi = o.78n,, (9.6.s)

(9.6.6)

(e.g.' see t9-601). Thus, in this simplified approach, the equivalent 60-s aenrdynamic coeffrcient for any point M1 and wind direction g; ias the expression

c;1,(Mi. or1 = YP'illtJr)lpu)th. o*t

For additional details, see [9-97].

Estimation of Design wind Loads. From Eq. 9.6.3 if follows thar the equivalent 60-s loads p[[ are given by the relation

*The peal valueprl depcnds upon the record length (orstom duration) z. commonly it is assurrrcrlT = 20 min to r hr (full-scale), to which therc corresponds a laborabry record length 7,,,T(D,,,/ D)l(u,,,lu ), whcre D,,,lD .and, u,,lu are the mode I gcomctric and verocity scare, respccrivt.ryFor structural rcliability calculations it is dcsirable to estimate thc nrcan xnd standrrd dcvirrti'rrof thc peak pressurt.Tr,,,. siner.. lirr lny givr,n v:rlrrt./rU). 7r,,r v1rr.it.s 1,,,,,, ra.,,,,1 ,,, ,.....,,,..;. ,,,,,can be donc fntnl sovcr:tl lccortls wilh lcrrgth 71,,, or hy trsirrp k'r.lrrrir;rrt.s h:rsul .rr *rrrlrrrrprocesscs thcory (sct: lirl. A2.:1.1, l4 751, l9 (r.11. irnrl cspr,t.i:rlly l1)(rll, llri,.lr r.lrrt:rirrs rrst.lllpractical rcsults).

370 tltlt t)lN(it; wtNl) l()nt)ll, 1;iliU(.ililtnt ill :;t'()Niit nNt) Ilt lil(iN ()t il()()l lN(i

Design Criteria. Tho convcntionirl tlt'sigrr lllrcr:tlrrrc uscs lhc litllowing rlesigrrcriterion:

p66(0.008) > p"f,,o(Mi) (9.6.13)

wherep[fl.r0(M) is the equivalent 60-s wind load estimated without considcrirr;iwind directionality effects (see discussion above), and p6s(0.008) is the 60-sload capacity of the panel corresponding to a cumulative failure probability ol8 panels out of 1000.

'fhe risk-consistent design procedure is based on the requirement that thcprobability of failure of each panel during the lifetime of the building be lcsslhan a spccilicd valuc Pr. It is now shown that this requirement leads to a designcritcrion cxprcsscd in tcrnts ofequivalent 60-s loads and of60-s load capacitics.

Considcr thc sal'cty indcx p dclined by Eq. A3.29.It is possible to writc

trt .triil rrt i:tnDl)lfl(, /\lll) llt)()lll.lr, Ir111 t7y11.11) l()/\l):, :lIl

:;::iU*" 9.6.2. Division of a high-rise building face into zones of equal glass thick-

rrt'ss of the glass panels may change as a function of elevation, as in the case,,1 the John Hancock Building in Boston. In other cases the same glass thicknessrr used over an entire building face or even over the entire building. ForLrt'1s[iens where wind-borne missiles, including roof gravel, may be expectedtrr lrit the cladding, special zones are suggested in [9-62].

Wc denote a zone in which the glass type and thickness is uniform by D, (il, ...n).If the conventional design method is used, Eq.9.6.8 must be

,;rr(isfied at all points M1 within D,:

[p6o(0.008)]; > max [p"d5o(Mj)]Di

(9.6.11)

It is possible to estimate the expected number of failures inherent in the designl';rscd on Eq. 9.6. 11 as follows: Each zone D; is divided into subzones Aii(j

l, 2, n4) over which it may be assumed that the wind loads do not vary',11'rrificantly.* Using Eqs. 9.6.9 and ,{3.37, it is possible to calculate, forany1'rvc:rr orientation of the building o7, the safety index 0ii1 and the lifetime prob-,rlrility of failure P|; of the panels within A,,.

Lct the number of panels within A,, be denoted by nnii. For the building,lt'signcd by the conventionltl rncthod (Eq. 9.6. 11), the expected number of;':rrrr:l lhilures in subzonc,4;i rlru'ing lhc lifctimc of the structure with orientation

rlrr pnttlitc. ,4,, luc tltc ltilrttlrrry :rrt';r:. ol llrr' prcs:rrrrt lirl)s ()r lllc wirrtl lrrnrrt'l rrrodcl <lf thei'rrlrlirtg (or'. il rt ltilrutltt'y iue:r ('tl('lrrli, lrr'lrrrrrl llrc trrrrlrrt'r ol /),, llrt.porlion ol lltirl tributary,rr,';r tottlltint'tl itt lltc zottc /),).

(9.0.tr1

where p[fl,, and V r"r, are the mean and the coefficient of variation of the largcslequivalent 60-sec wind load during the lifetime of the building, the subscript/, represents the lifetime of the structure in years, andf6s and Vp^, are the mcitlland coefficient of variation of the load capacity. From Eq. 9.6.9 it follows thirt

B60 should satisfy the relation

Pon > P'8,(.Mi;exptP(V'?";,,( Mj) + v?,uJt''l (9.6. r0)

where B is the value of the safety index corresponding to the failure probabililyPJ (see Eq. 4.3.37). Equation 9.6.10 is the design criterion used for riskconsistent design.

The question of the selection of the safety index 0 or, equivalently, of tlrefailure probability Py, is discussed next.

Design. For ease of construction it is necessary to divide the building facackrsinto zones, each characterized by a single type of glass panel. Thus, for altygiven architectural pattern defined by the location and by the height and witltlrof the panels, the design of the cladding consists of (l) dividing the buililirtlqfacades into such zones and (2) selecting the type of glass (i.e., whethcr rrrt

nealed, heat-strengthened, or tempered) and the panel thickness for each zottt'.An example of division into zones of equal glass thickness, suggcslctl irr

t8-111, is shown in Fig.9.6.2. This division rnakcs il possiblc to provirlcstronger panels al uncl ncar thc cclgcs and cavcs, wlrct'e lretrxlyrrrtttic prcssutl'sare usually Iargcsl . Ilowcvcr-, <ltlrcr possibilitics r:xisl . liot r'xlrttplc, thc thick

. ln(Dn lP'A)L)--' tv",t + v;d,1"'

i

ii

i

rill

i

1

i

*A

372 BtJtil)tN(iti: wtNt) t()nt )1i, :;ilt{,(;l('lrnl lll i;l'()Niil . nNl) l)l :;l(iN ()l ll()()l lN(i ?cv1 isx

nt1i1 : n1,iiPI1,1

The expected total number of panel failures per lifetimefor the entire building are, respectively,

-t s-/nn:4ntu-t s-1ny: L ntr

I

lixpcricrrcc appcars to indicate that the cladding in any subzone ,4, designctlin tccortlarrcc with thc conventional method (Eq.9.6.8) is acceptable from a

sal'cty point ol' vicw if thc aerodynamic and climatological data upon whichthc <lcsign was based are adequate. It might then be argued that the probabilityP7 corresponcling to the saf'ety index B used in Eq. 9.6.10 may have the value:

Pf : nllaf {PLJti} (9.6. rs)

(9.6. I2)

for thc zonc 1); itttrl

(9.6. r3)

(9.6.t4t

(9.6. r6)

l)l :,1(ir! ()t ct nt )tilN(i nNt) il(x)t til(, t()t I VVllJt, trr/\t ): , :ll],

irily onc lrirrrt:l tlrrrirrg llrc lilt.tirrrt.ol llre lrrriltlirrli rs lj tt,lrt,,. wlt.r,,. rr,, rr rlrt,Iol:tl lruttttrcr rll llttltt'ls ol'llrc lrrrilrlirrg. 'l'lrc s;rlcty iirtlt.r 1i is tlrcrr t,rrlr.rrlrrlt.tlrrsirrg IJq. 43.17' itrttl lltc clatltlirtg lirr clrch r,ont l), is tlcsrlirrt'tl lry tlrr.rrsk( ()nsistcnt pntcorluro in accorclarrco willr lit;. ().(r. lO:

lpooli > maxT;, {7ri,il,,( M)cx1.tlf}(V),,,,1M; t V,:,,,,,,)t,,1 (9.6. l7)

A c:rmputer program fbr thc crcsign .r'cradding by Eq. g.6.11 in c.nluncti.nwith Eqs. 9.6.16 and 43.37 is rcfbrcnccd in t9-611. Illustrative results obtainedlry using that program are prescnted in Sect. 9.6.2.

9-6.2 Economic and safety Advantages of Risk-consistent DesignProcedureIir illustrate the potential advantages of the risk-consistent design procedure,rt'sults of computations taken^from [9-61] are presented for a 200-ir tail uuitaingrcpresented in plan in Fig. 9.6.3. rt was assumed that the building is locatedin lerrain with uniform roughness in all directions (zo : 1.00 m; anO that therei'c no neighboring structures influencing the building aerodynamics. Aerody_rurrnic pressure coefficients obtained in the wind tunnel were extracted from

I

i

l

li

l

where mnx;.1,, tph| is the largest of the values P!;.However, such a choice of Py for use in risk-consistent design might bc

imprudent. The authors believe that it is reasonable to adopt as a design ob'jective an expected number of panel failures per lifetime for the entire building

n7 : max nl

Indeed, the conventional design procedure, ignoring as it does wind direction-ality effects, can be viewed as providing sufficient safety levels for all buildings,regardless of their orientation. This can be interpreted as meaning (l) that thcexpected number of failures rej inherent in the conventional design procedurcris icceptable even for buildings with the most unfavorable orientation cv1 antl(2) that if the conventional procedure is used, building with more favorableorientations are overdesigned.

If Eq. 9.6.16 is adopted as a design objective, the probability of failure <tl'

*The failure condition for each panel of a zone is defined by the event Pa - P"d < 0. Notc thirl

these events are not in all cases statistically independent, since the loads induced on vatittttr'panels, and in some cases the load capacities ofvarious panels, may be correlated. Howcvet,Eq. 9.6.12 holds regardless of whether the failure events are independcnt or not. This ctn lrt'shown by considering thc simple cxample of n,, coins. Let lailurc dcnotc the occurroncc ol"heads." The expectation o1'thc numbcr of failurcs that would occttt il lltc rt,, coins wt:rc losst'tlonce is 1, : ll2nt,.This is truc rcgarcllcss ol'whcthcr thc luiltrt' ('v('trls ilr(' irrtlt:pcrttlcnl (lts ittthe case ofcoins liaving c:rt'lt lrn intlcpcn<lcnl rttotion) or grt'r'li'clly toltt'lrtlt'tl (lrs itl lltc tltsc ttl il

set ol'n, coins, lixcrl onkr :r wciglrllcss holrrtl wilh :rll lltt' "ltt';rls" ott llt( :i:tlll(' sitle, so llt:tlllilurc 11l grrt: coilt worrltl r'rrllril lirilrrn'ol:rll llrc rr,, toitts) Nolt llr;tl rrlrrlr'lltt t'r1x'tlit(iotts olIr, Wottkl ltt.lltC Slrtrrt'irr llrt lwo r'lrsts. lltL'sl;tttrl:ttrl (l('vl:llllrll'. ttlttlrl ttol

\eo.dr

\

lil(;lJltl,l l).(r..1- | )rrri.rr,,rrrrr'. ol lrrrrlrllrrl, rrr pl:rrr

;374 llt,ll l)lN(il; wlNl) l()nl ):; 1;lltu(.ililtnt ttt t,t,()Nl;l nNl) l)l 1;l(iN ()l tt{)()l lN(i

19-601. 'l'hc wincl clinratc was assunrc(l kr lx'tlt'lirrctl by thc tl:rlrr ol 'l':rlllt'tJ.I.].forwhich summary statistics arc givcn in Itig. -1.4.1. 'l'ho lucadc:s wcrc tlivitlerlinto zones of uniform glass thickncss in accorclancc with liig. 9.6.2. lt w:rsassumed that the cladding consisted of anncaled glass pancls with dirncnsions1.8 x 1.8 m. The information on the load capacity of thc pancls was lakcrrfrom [9-56]. Approximate typical prices per unit area of panels with variousthicknesses were obtained from glass distributors. These were used as a basisfor performing estimates of the nominal cost of cladding glass inhercnt in :rnygiven design.

From an inspection of Fig. 3.4.1 it is apparent that the wind effects arc notcqually severe for the parallel faces AD and BC (or AB and DC) of the building.shown in Fig. 9.6.3. Nevertheless, as noted earlier, the conventional designmcthod would result in this case in identical designs for those faces. It is alsoclcar that the severity of the wind effects on the various faces depends uponthe orientation oi of the building. Again, this is not reflected in the conventionrrlmethod, which results in identical cladding designs regardless of the buildirrgorientation a;.

The cladding of the building shown in Fig. 9.6.3 was first designed inaccordance with the conventional method. The nominal cost of the cladding sodesigned was estimated to be $361,000 for the entire building. Using the pnrcedure described in the preceding section, the expected number of panel failurcsper lifetime inherent in the conventional design was estimated for various buikling orientations cy;. The results of the estimates are shown in column 2 of Tablc9.6.r.

Also shown in Table 9.6.1 (columns 4 and 5) are nominal costs of claddirrg,designed by the risk-consistent procedure on the basis of the following desigrrobjectives: the expected total number of failures per lifetime is equal (l) to tlrcvahe n! of column 2 (see column 4) anil (2) to the value n]: 12.0, whiclrcorresponds approximately to the most unfavorable orientation of the buildirrp,(see column 5).

TABLE 9.6.1. Nominal Costs of Cladding for Various Designs

Conventional Practice

Risk*Consistent Procedurc

I ll I I I ll l.l(.l 37s

('ottsitlt't lltr' tlt's11'111. lr:r:.r'rl orr llrt' lusl ol llrt'st' lwo olrlt.r'lives- lt is scclttlr.rl irt (ltis t':tst'lll('('(()n.int,s lrtltit'vr'tl :rlt'ol lltt'oltlt'r'ol'5'/,, kl l0'/,,.ll,lv1'v1'1', llrc lrrt'l llr:rl llrt' r'orrvt'rrliorr;rl rlr'sip,rr is :rcce:lltlrhlc to building in-.;,,'t (iolt trullrorilit's, lt'1'.:rrrllt"ss ol llrt' lrrrilrling or.icrrtlrliorr, irnplics [hat in ther rr'\\, ()l ll'tcsc uutlrol'ilit's srrtlr :r rlt'sigrr is srrllit'icntly sal'c cvcn in those casesrrlrr'n'lho builcling olit'rrltr(iorr is rrrrllrvol'rrblc; as notcd in Sect.9.6.1, this,'lr:,t'r'vlrtion lcacls to tlrc rrtkrpl iorr irs ir tlcsign objective of an expected number, 'l l;tilttt'cs pcr lil'ctintc apptrrxinra(cly cqual to the largest of the estimated values,,', \l 1,2, . .. , 8). A cornparison between columns 3 and 5 of Table 9.6.1.lr.ws 1[i11, for buildings with favorable orientations, the use of the risk-con-.r'.tt'rr( design procedure can then result in significant savings (in the case, i:rnrirrcd here, almost 25%).

As stated earlier, the results of Table 9.6.1 were obtained for building fa-,.r,k's divided into zones in accordance with Fig. 9.6.2. As indicated inl') (rll. similar conclusions hold for designs in which the glass thickness is, "ilslilnt over an entire building face.

'r (;.3 Effects of Wind Loads on Roofingl.'t t crrl material on the perfbrmance of roofing in strong winds shows that wind,.ur (lruse (l) high suctions, which may induce peeling failures, panel failures,.rrgrlxrrling member failures, or system failures, and (2) scour of roof gravell') ')li l. A procedure for the selection of gravel size and parapet height to avoid,,r:rvt'l scour (displacement) and, more important, gravel blow-offfrom the roof,. I'rolroscd in [9-99] (see also 14-63, 4-eD.

lrr rrrcas of the roof where calculations indicate gravel blow-offwould occur,rlrr' usL: of concrete slabs instead of gravel is recommended. Alternatively, the,,r.rvt'l should be fully embedded using a double-surfacing technique [9-100].It,r:,t'tl on obseruations of roof behavior during hurricane Hugo, it has been,i ( ()n)nrended that for buildings less (more) than 13.7-m high, parapet heightsi', :rr lcast 0.3 m (0.6 m) [9-101 ,9-102].

l;or wind-related information on mechanically attached single-ply systems,,,, lt) l03l; metal edge flashings, see [9-109] and [9-ll4]; asphalt shingles.'rr, I tlrcir attachment, see [9-104, 9-105]; roof fasteners, see [9-ll0]; loose-lrr,l rtrol'insulation systems, see [4-83, 4-84,9-lll]. Studies of wind effects,'rr trlt'nxrf.s, including pressure distributions on gable roofs and around indi-' ',irr;rl [ilcs, are summarized and referenced in [9-106]; see also [9-107]. It wasI, 'rrntl that aluminum shingles behaved poorly in hurricane Andrew, while spray-r 1 r1 rl 11'11 polyurethane fbam roofs performed outstandingly [9- I 08].

III FERENCES

') | lt. 1.. Wirlllrrw, "lttlt'tlt'rt'ttr r' ;rrrrl l'roxirrrily lillccls," in llitrtl l,.rcitttl Viltnrtiutt,\ ()f ,\!ntt lutt',\." Il lior l., l (r rl 1, $1y1i111','r'Vt'rllrg. 11..* y,r1l,, lr)rtl

BuildingOrientation

(l)

Number ofPanel _

Failures, nf(2)

NominalCost ($)

(3)

Nominal Cost.Design Basedon Value Difrom Col. 2

($) (4)

Nominal Cosl .

Design Blsctlon Valuc ri

: 12.0 (:f) (\)0"

45"90"

t3.5"

2.5I 1.9Ltl

l.) I

36 r .00016l .(xx).l(r I .(XX).l(, I .(XX)

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Building Model oJ Square Cro:;s-St<'tittrt otr tt Singlc Mtxltl, itr tlrr Wtrkr rtl rtSimilar Model, andinthe Wake oJ'a Rcctangular Mtxlcl,l{cport VI,l l1 7() ll,Department of Engineering Science and Mechanics, Virginia Polytcchrrit. l1stitute, Blacksubrg, VA, March 1979.

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() I A. Ci. I)avcnport, "'l'hc Application oi'statistical Concepts to the Wind Lo:uling to Structurcs." Prrx'. Insr. Civ. Eng.,19 (1961), 449-472.

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9-14 E. Simiu and D. W. Lozier, "The Buffeting of Structures by Strong WintlsWindload Program," NTIS Accession No. Ptr_294157 lAS, Computer Progrrrrrfor Estimating Along-Wind Response, National Technical Information Servir'c,Springfield, VA, 1979.

9-15 C. Soize, "Dynamique stochastique des structures dlancdes soumiscs irrrrcharges du vent," Revue Frangaise de Mdcanique,60 (1976),57 65.

9-16 G. Solari, DAWROS: A Computer Program .for Calculating Aktng-Witttl llt'sponse oJ' Structures, Pubblicazione dell'Istituto di Scienza dcllc Cilstnrziprrr.Serie IV, No. l, University of Genova, Genova, Italy, lgtt l.

9-17 E,. Simiu, "Revised Procedure firrEstimating Along-Wind llcsponsc," .l . Strttt tDlv., ASCE, 106, No. STI (Jan. l9tl0). I 10.

9-18 G. Solari, "Along-Wintl llcsponsc Ijstiuurlion: ('krsc:tl Iiolrrr Solrrtion."./Strucr. I)iv.. nS('lr. lOtl, No. S'l'I (.lrrrr, l()lil).22.5 1.1,1 .

9-19 A. Kiu'ccrn (llcrsorr:rl torrrrnrrrrictrliorr, l()()(r),

ilt IIiltt.t( I :IT I

')'O l'. A. Itosltlr, .lrt l'.rIt'titrrr'tttttl,\lrtrlt t,l tlt. lit\l't't,rr'1tf tt,\'tlutu'('l'ti.tttt toWittrl ltnttl.lil,W'l'll (rti. lilrtrrlly ol (it:rrlrr:rlt Slrrtlrt':;. Urrrvt:r'sity ol'Wcstcrn()rttlrrio, l,orrtlolr. ()rrlirrio, (':rrr:rtl;r. l(l(rH

') 'f W. C. Iltrrty atttl M. ll . l.ltrlrirr,.;lt irr. l)trt,tttrir'.s t,l :;tnt('tut'(,t. Prcntice-Hall,lirrglow<xrrl ('lill.s, N.l. l(Xrl

ri r.) .f . A. BlLrlno, N. M. Nt'wrrr:rrk. :rrrtl l, ll. ('ornirtg, Design of MultistoryIlcinfttrcr:d Crttt'n'lt' Ifitiltlittg.r lttr I'.rtrtlttltrttkr Motions, Portland Cement As-sot irlion. ('hit'irgo. |(t(r I .

G. T. Taoka, M. llogrrrr, Ir. I(lrrrrr, unrl R. H. Scanlan, "Ambient ResponseolSome Tall Structurcs," .l . Srruct. /)iv., ASCE, l0l, No. ST1, Proc. PaperI l05l (Jan. 197-5), 49 .55.

'l C. S. Kwok and W. H. Melbourne, "Wind-Induced Lock-In Excitation of TallStructures," J. Srruct. Dlv., ASCE, 107, No. ST1 (Jan. l98l),57 72.

.)\ B. J. Vickery, "Notes on Wind Forces on Tall Buildings," Annex to AustralianStandard 1170, Part 2-1973, SAA Loading Code Part 2-Wind Forces, Stan-dards Association of Australia, Sydney, 1973.

'(r T. A. Reinhold and P. R. Sparks, "The Influence of Wind Direction on theResponse of a Square-Section Tall Building," Proceedings Fifih InternationalConference on Wind Engineering, Fort Collins, CO, July 1979, PergamonPress, Elmsford, NY, 1980.

.' J J. W. Saunders, Wind Excitation of Tall Buildings with Particular Referencesb the Cross-Wind Motion of Tall Buildings of Constant Rectangular Cross-Section, Doctoral Thesis, Dept. of Mechnical Engineering, Monash University,Victoria, Australia, 1975.

'11 A. Kareem, "Across-Wind Response of Buildings," J. Struct. Diy., ASCE,108 (April 1982), 869-887.

)() A. Kareem, "Wind-Excited Response of Buildings in Higher Modes," J. Struct.Div., ASCE, 107, No. ST4 (April 1981), 701 706.

l0 A. G. Davenport, "The Response of Six Building Shapes to Turbulent Winds,"Phil. Trans. Roy. Soc. London, A269 (1971),385-394.

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!)

')

l,

')

t,

()

i)

'f

r)

r)

(,

;378 tlttll l)lN(ii' wlNl) l()nl ): ;. :;lllll(.lllll/\l lll :'l '()l'll;l . nNl I l)l :il(;N ()l I

g-39 G. L. Grcig, 'llnttnl rttt l,..ttitrrtttt t,l ll'trttl lrttlttcul I)\'tttttrtit' llttrltrt'tttr'lltllBuildings, Mastcr's'l'hcsis, l)cpl11 . ol lingirrccr-ing, IJrtivr:lsily ol Wt'slt'tttOntario, London, Ontario, Scpt. l9tto.

9-40 G. Lythe and D. Surry, "Wind-Induced Torsional Lrtads on'l'all lltriltlittgs."J. Wind Eng. Ind. Aerod.,36 (1990), 225 234-

g-41 N. Isyumov, "The Aeroelastic Modeling of Tall Buildings," Pnx'cctling.s ttlInternational Workshop on Wind Tunnel Modeling .for Civil Engin.ccring A1t

plication, Cambridge Univ. Press, Cambridge, 1982'g-42 N. Isyumov and M. Poole, "Wind-Induced Torque on Squarc and Rectangulilr

Buil<ling Shapes," Proceedings Sixth International Conference on Wind I'trginrtring, Illscvier, Amsterdam, 1984.

9 4.1 .l N Yang antl Y. K. Lin, "Along-Wind Motion of Multistory Building"'./.I'ttg. Mrclt. /)ir,., ASCti, 107 (198 l),295-307.

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9-45 "Lcad Hula-Hoops Stahilizc Antcnna," Eng,. News Record, 197,4 (July 22.1976), 10.

9-46 K. B. Wiesner, "Tuned Mass Danrpcrs to Reduce Building Wind Motion""Preprint 3510, ASCE Convention and Exposition, Boston, April 2-6' 1979.

9-41 "Hancock Tower Now to Get Dampers," Eng. News Record, October 30'1975, p. 11.

9-48 N. Isyumov, J. Holmes, and A. G. Davenport, "A Study of Wind Effects lirr'the First National City Corporation Project-New York, U.S'A.," Universit.t'of Western Ontario Research Report BLWT-551-75, London, Ontario, Canaclir,1975.

9-49 R. J. McNamara, "Tuned Mass Dampers for Buildings," /. Struct. Dir'.,ASCE, 103 (Sept. 1977),1785 1198.

9-50 "Tuned Mass Dampers Sway Skyscrapers in Wind," Eng. News Record, Attg.18, t977, pp. 28-29.

9-5 I J. P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1956.

9-52 S. H. Crandall and W. D. Mark, Random Vibrations in Mechanical Sys/arr,r.Acadcnric Press, New York, 1963.

9-53 R. W. Lutt, "Optimal Tuned Mass Dampers for Buildings," J- Struct. Div .

ASCE, 105, (Dec. 1979), 27 66-2'772.9 54 A. Dcr Kiureghian, J. L. Sackman, and B. Nour-Omid, Dynamic Responsa ttl

Lig,ht Equipment in Structures, Report UCB/EERC-81i05, Earthquake Engineering Research Center, College of Engineering, University of Californirr.Berkeley, April 1981.

9-55 PPG Glass Thickness Recommendations to Meet Architccts Specified l-Mintttt'Wind Load, PPG Industries, Pittsburg, 1981.

9-56 LOF Technical Information-Strength of Glass Under Wirul Loarl.s, PublicatiorrATS-109, Libbey-Owens-Ford Company, Tolcdo, OH' 19u0.

9-57 W. L. Beason, and J. R. Morgan, "Glass Failurc Prediction Motlcl," .1. Stntt t.

Eng., ll0 (Fcb. 1984). 191-212.9-58 D. A. Rccrl irnrl li. Sirniu, "Wincl l-oarling anrl tlrc Slrcrrgth ol (ilrrss."./.

SIrutr . l')r,q.. I l0 (April 19t34). 715 129.

lll ll lll ll' I

')'r() li. Silrritr lurtl l) A litt.tl, "liinli orr l(rrr;1 li.:,1::rrrrl tlrr.l\lrrrlrlrrr;,ol (,1.111111111,(ilrrss Stlr'lrg,tlr lry tlrt' Wcilrrrll l)islrilrulrrrr. " I't,,,.r.,!rrt.r:t lU t lA! .\\tttlt,,trttrrttttt l'rrilxtltilistir-Mt'tlttxl.s itt tltt'lllt't'ltrtttir'.s rtf ,\'rtlttlt ttttl ,\'ttnt /r//, t.,!il.r( Llrplsr..lurtc l9 21, l9tl,1, N (' l,irrrl :rrrtl s. l11'1'rvr.rrz (r.rl:, ). Syrrrrrl,t'r Vr'rl:r1,. Nt.ryYork. l9lJ-5.

') (() J. A. Pclcrka antl .l . lj. ('erttt:tk, ll'irnl Iirtrtrr'l ,\trtrll,rtf ,ltltttrrtt ()l.litt'lltti!tlirt,q.I)cpartl)lcnt ol'(livil l')rrgilrccrrng, ('olor';uLr Sl:rlt'Ilnivclsity, Iioll ('olli1s, N9v.I 978.

t) (rl E. Sirniu ancl A. Iiilotti, "Wirrtkrw Class Facadcs as Structural Systems: Anlmprovcd Rcliability-Bascd Dcsign procedure," proceedings Internationalconference on strut'tural saJbty and Reliabitity, May 27-29,1995, I. Konishiand M. Shinozuka (eds.), Kobe, Japan.

') (rl J. E. Minor, w. L. Beason, and p. L. Harris, "Designing for windborneMissiles in Urban Areas," J. Strucr. Diy., ASCE, 104 (1979), 1149_1760.

') (r I N. J. cook and J. R. Mayne, "A Refined working Approach to the Assessmentof wind Loads for Equivalent Static Design," J. winrt Eng. Ind. Aerodyn.,6fl980), t25_t31.

') (rl J' A. Peterka, "Selection of Local Peak Pressure Coefficients for Wind TunnelStudies of Buildings," J. Wind Eng. Ind. Aerodyn., f3 (19g3), 477_499.

() (r.5 E. Simiu, "Modern Developments in Wind Engineering, parl 4,,, Eng. Struc.s (1983), 273_28r.

') (r(r A. Tallin and B. Ellingwood, "Analysis of rorsional Moments on Tall Build-ings," "/. Wind Eng. Ind. Aerodyn 18 (Aprit 1985), t9l_195.

't (t/ P. Mahmoodi, "Design and Analysis of viscoelastic Vibration Dampers forStructures," in Proceedings, world Innovation week Conference INovA-73,Eyrolles, Paris, 1974.

rr (rlJ P. Mahmoodi, "Structural Dampers," J. Struct. Div., ASCE,95 (Aug. 1969),t66t-1612.

') (t) A. G. Davenport and P. Hill-carroll, "Damping in Tall Buildings: Its vari-ability and rreatment in Design," Building Motion in wind, N. Isyumov andT. Tchanz (eds.), American Society of Civil Engineers, New york, 19g6.

') /() D. F. Sinclair, "Damping Systems to Limit the Motion of Buildings," BuiLdingMotion in wind, N. Isyumov and r. Tchanz (eds.), American Society of civilE,ngineers, New York, 1986.

't ll c. J. Keel and P. Mahmoodi, "Design of viscoelastic Dampers forcolumbiacenter Building," Building Motion in wind, N. Isyumov and r. Tchanz (eds.),American Society of Civil Engineers, New york, 1986.

't l .' P. Mahmoodi and c. J. Kccl, "performance of Viscoelastic Structural Dampers{or the columbia ccnlcr l}rrilrling," Builtling Motion in wirul, N. Isyumov and'l'. ]'chanz (cds.), Arrrc'rt'irrr S.t'icly .l'civil E'ginccrs, New york, 19g6.

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tlltJll l)lN(il; wlNl) l()nl )l ;, :;lllll(.il,il/\t ilt :;t '()N:;l . nNt ) t)t :;t(;N ()l tt()()t lN(i

9-75 H. Kawai, "Vortex Inclucctl Vilrtirliorr ol 'l'trll litriltlirrgs,"' .l . llirrtl I,,,rt,g. lrrrl.Aerod.,4l-44 (1992), ll7 128.

9-76 S. Yamazaki, N. Nagata, and H. Abim, "'l-uncd Activc l):rrnpcrs Instlllctl irrthe Minato Mirai (MM) 2l Landmark Tower in Yokohama." .l . Wirul I')t,g.Ind. Aerodyn., 4l-44 (1992), 1937,1948.

9-17 L. M. Sun, Y. Fujino, B. M. Pacheco, and P. Chaiseri, "Modclling ol thcTuned Liquid Damper (TLD)," J. Wind Eng. Ind. Aerodyn.,4l-44 (19()2),l 883- 1 894.

9-'78 T. Wakahara, T. Ohyama, and K. Fujii, "Suppression of Wind-Induced Vibration o1'a Tall Building Using Tuned Liquid Damper," J. Wind Eng. Irul.Atnxl.vn.. 4l-44 (1992). 1895 1906.

919 'l'. []ctla. ll.. Nlgakaki, and K. Koshida, "Suppression of Wind-Induced Vilrration by I)yltarnic l)anrpcrs in Towerlike Structures," J. Wind Eng. Inrl.Acnxl.yrr.. 4l-44 (1992), 1907 1918.

9-ll0 Y. 'l'aurura, R. Kousaka, antl V. J. Modi, "Practical Application of NutationDarnper fbr Supprcssing Wind,lnduced Vibrations of Airport Towers," J. WilulEng. Ind. Aerodyn.,4l-44 (1992), l9l9-1930.

9-81 K. Miyashita et al., "Wind-Induced Response of High-Rise Buildings-Effccrsof Corner Cuts or Openings in Square Buildings," J. Wind Eng. Ind. Aerodyn..s0 (1993), 3t9-328.

9-82 D. I. Beneke and K. C. S. Kwok, "Aerodynamic Effect of Wind InduccrlTorsion on Tall Buildings, J. Wind Eng. Ind. Aerodyn.,50 (1993), 271-280.

9-83 M. Islam, B. Ellingwood, and R. Corotis, "Dynamic Response of Tall Buildings to Stochastic Wind Load," J. Struct. Eng.,116 (1990), 2982-3002.

9-84 T. Stathopoulos, "Low Buildings," in Wind Loading and Wind-lnduced Structural Response , State,of'-the-Art Report, Committee on Wind Effects, AmericanSociety of Civil Engineers, New York, 1987.

9-85 T. Stathopoulos, "Evaluation of Wind Loads on Low Buildings: A Brief Historical Review," in A State of the Art in Wind Engineering, Ninth InternationalConference in Wind Engineering 1995, Wiley Eastem Limited, New Delhi,I 995.

9-86 A. G. Davenport, D. Surry and T. Stathopoulos, "Wind Loads on Low-RiscBuildings: Final Report opf Phases I and II, Parts I and 2, BLWT-SS8-1977.Boundary Layer Wind Tunnel Laboratory, University of Western Ontario, London, Ontario, Canada, 1977.

9-87 D. Meecham, "The Improved Performance of Hip Roofs in Extreme WindsA Case Study," J. Wind Eng. Ind. Aerodyn.,4l-44 (1992),1717,1726.

9-88 T. Stathopoulos and H. Luchian, "Wind Pressures on Buildings with SteppcrlRoofs," Canadian J. Civil Eng., 17 (1990), 569-571.

9-89 T. Stathopoulos and P. Saathoff, "Codification of Wind Pressure Coellicicntsfor Sawtooth Roof.s," J. Wind Eng. Ind. Aerodyn.,4l-44 (1992), l1Z7-fl't\.

9-90 T. Stathopoukrs and P. Saathoff, "wind Pressures on Rool.s of Vari6us Gc6rrretries," J. Wind Eng. Ind. Acruxlyn., 38 (1991), 213 284.

9-91 D. Surry and'l'. Stathopoukrs, "An ltxpcrirncnltrl Applr:rt lr lo llre licorrorrriclrlMcasurcrrrcrrl ol'Sllirlilrlly Avcnrgctl Winrl Lo:rtls," .l . Irt,l .lt,trnl .2 (lgjli,l385 197.

lt rrlt ltrI', :lBl

') ().) ('. Sollit'c ltrrtl .l . M:try, "Stlttttll;tttt'otts N4r':r:,rr, rrr,'rrt., oi I,lrrr lrr:rlrrrl, I'rt.ssltresUsrrtg l'iczort'sislivr' Mrrllit'lrirnrrcl 'lrirnstllr( (.r:, ;r:, AgrPlll',1 to Atlrroslllrr'l.it. Wirrtl'f'unltcl 'l'csts," .l . Wirttl l,,tt,q. lrttl. ..lr,rrnl\tt . $(r ( l(r()\), / I X(r.

't t)I It. J. Kincl, "Worst Sucliotts Nt'rrr lill't':; ol lil:r{ liool lirps with l)arapcts,".1. Wind Eng. Ind. Acnxl.\,rt., .ll ( l()l.lli). .)51 .l(rl

tr t)'l A. P. Robertson, "Ell-ccl ol li:tvt's l)r'lrril orr Wirrtl l)rcssurcs ovcran IndustrialBuilding," J. Wind lhg. ltrtl. Aotrlt,rr., -llt ( 199 l), 325 333.

it t)5 'f . Stathopoulos ancl H. l.rrt'lriiur, "Wind-lnduccd Pressures on Eaves of LowBuildings," J. Wirul l,.ng. Irtrl. Acrtxlyn., 29 (1988), 49-58.

't rxr E. simiu and E. M. Hcnclrickson, "Design criteria for Glass cladding Sub-jected to Wind Loacls," J. Struct. Eng.,ll3 (19g7),501_518.

tt t)J E. Simiu and E. M. Hendrickson, "Wind Tunnel Tests and Equivalent l-MinLoads for the Design of Cladding Glass," J. Wind Eng. Ind. Aerodyn.,29(1988),49-58.

'tt)l.i J. E. Minor, "Perlormance of Roofing Systems in Wind Stoms," Proceedingsof the Symposium on Roofng Technology, National Bureau of Standards andNational Roofing Contractors Association, 1977, pp. 124-133.

') (x) R. J. Kind and R. L. Wardlaw, "The Development of a Procedure fbr theDesign of Rooftops against Gravel and Scour in High Winds," Proceedings ofthe Symposium on Roofing Technobgy, National Bureau of Standards and Na-tional Roofing Contractors Association, 1977, pp. ll2 123.

') 100 T. L. Smith and J. R. McDonald, "Roof Wind Damage Mitigation: Lessonsfrom Hugo," Professional Roofing, Nov. 1990, 30-33.

') l0[ T. Smith, R. J. Kind, and J. R. McDonald, "Hurricane Hugo Tests the Per-formance of Aggregate Ballasted Single-Ply Systems," ProJessional Roofing,Aug. 1992,29-34.

't 102 T. Smith, R. J. Kind, and J. R. McDonald, "Hurricane Hugo II: Testing thePerformance of Aggregate Ballasted Single-Ply Systems," ProJ'essional Roof-ing, Sept. 1992,32 38.

') 103 T. Smith, "Mechanically Attached Single-Ply Systems," Profbssional Roofing,Mar. 1992,14.

') 104 D. E. Shaw, "Better Uplift for Asphalt Shingles?" Professional RooJing, Mar.1993,30 32.

() 105 T. Smith, "Asphalt Shingles: The Importance of Corect Attachment," Pro-fessional Rrxfing, Dec. 1992, 54.

') 1 06 C. Kramer and H. J. Gerhardt, " What are the Effects of Wind on Tile Roofs? "Professional Roofing, June 1993, pp. R7-R10.

') 107 c- Kramer and J. H. Gerhardt, "wind Loading in permeable Roofing Sys-tems," J. Wind Eng. Ind. Aerodyn., ff (1983), 34j-358.

') l0l] r. Smith, "Hurricane Andrew: A Preliminary Assessment," ProJessional Roof-ing, Oct. 1992, 58.

') l(X) .1. R. McDonald, P. P. Sarkar, andH. Gupta, "Wind-lnduccd Loadson MctalIrdgc F'lashings," Wirul I)r,qitrecring, Prccecding.s, Ninth Inttnuttiotutl Crn.li'rcrrt'c tn Witul I,.rt.rlirtr't.r'irt,q. Vol. l, pp.69 ltO, Wilcy litrstcr-n l,t(1., NcwI )t:llri.

') Il0 li. A. llirsk;rr';ttt:ttttl () Ilull I r':rltt:rliort ol'llool liitsl('n('ls trnrlt'r I)vrr:rrrrit'

382 UUILDINGS: WIND LOADS, Slllt,(; lt,llAl lll I]PONSF, AND DESICN ()l lttX]l lN(i

Wind Loading," Wind Enginccrittg, l'nx'ttdirt14s, Ninth lntanuttiotuti (Iur.li,r-ence on Wind Engineering, Y<tl. 3, pp. l2O7*1217, Wilcy B,astcrn Ltrl., NcrwDelhi.

9-11I Y. Sun and B. Bienkewicz, "Wind Loading and Resistance of Lr>osc-Laid llrxrl'Paver Systems," Wind Engineering, Proceedings, Ninth International Ctrtli,r-ence on Wind Engineering, Yol.3, pp. 1255-1266, Wiley Eastem Ltd., NowDelhi.

9-ll2 R. D. Marshall, Wind lnad Provisions of the Manufactured Home Constru*tion and Safety Standards-A Review and Recommendation for Improvemenl,NISTIR 5189, National Institute of Standards and Technology, Gaithersburg,MD, 1993.

9-l13 R. Zhang and Q. M. Feng, "Vibration of Tall Buildings under TurbulcntWind," Proceedings, Third International Conference on Stochastic StructurulDynumics, San Juan, Puerto Rico, Jan. 15-18, 1995.

9-114 T. Smith, "Hurricane Hugo's Effects of Metal Edge Flashings," Int. J. RoolingTech.,2 (1990), 65-10.

CHAPTER 1O

SLENDER TOWERS AND STACKSWITH CIRCULAR CROSS SECTION

slcrrtlcr towers and stacks are designed to withstand the effects of both along-wrntl and across-wind loads. The along-wind response can be estimated byusrrg, the computer program of [9-14]. Simplified methods may be used ifirlrlrnrximate estimates of the peak along-wind response in the fundamentalrrrrxlc are sought. Since the gust response factor G depends only weakly uponllre lirndamental modal shape (see Eq.9.1.10), it can be calculated by using'l'irblc 9.1.5. The peak along-wind response is then obtained from the relation,\, ,,,,,^(Z) : G7{z), where.rl(z) is calculated using Eq. 5.3.4 (with t : 1).

'l'hcse procedures must be used with appropriate values for the averagervrrrtlward and leeward drag coefficients C, and C1. For slender towers andrt;rcks with a circular shape in plan, it may be assumed in all cases that c, :( l. so that the total drag coeffici ent Cp : C, . Information on the magnitude of( j, irrrd its dependence upon Reynolds number, surface roughness, and aspectrrrtio is provided in Sect. 10.2.2.

sc:vcral procedures for estimating across-wind response are currently avail-*lrlc. Among these, the procedure developed by Rumman [10-l] has beenrvrrlt:ly applied to the design of reinforced concrete chimneys. The basis of thisluoccrlure is largely intuitive. Nevertheless, it appears that the results of its{rl)l)lication have been satislactory in practice.

ll is generally agreed that thc kriuling ancl response models inherent in Rum-lll;tlls' procedure are not cntirerly crtttsistcrr( with aclvances made over the lastlrvo tlccaclcs in the ficlds tll' ttticttrtttt'lcotrrlogy, ircrrxlynamics, and acnrclastic-tty. According to llG2l tltis corrkl irr t't'rlirirr silrlrtiotrs lcacl to thc unclcrcsti-itutlitttt tll'thc acr<lss-wintl tcslxrttsc, lrrrrlit'rrlirrly in llrc scconcl nurtlc rll'vil'rr-ir.ll(ttt. l)nrcctlurcs in wlticlt tllolt'rrrlvnrrt't'rl rrplrrrrirt'lrcs irrc tttilizcrl wcrc

t'

383

384 :;t f Nt)t tt l()wl ltl; nNl) i;ln(.hl; wl lll{illl{:tll nlt (;ll()l;l; l;l (;ll()N

developcd in [0-31, arrd by Vickerly, l]rrsrr. :rrrtl ('lalk in [10-21 arrtl ll0 ztl ttrtlO-el.

The ESDU procedure [0-31 is bascd on a rrodified vcrsiort ol' lhc lrxrtlc:lconsisting of Eqs. 6.1.4 and 6.1.5. It considers two response rcgions, trnc ittwhich the forces associated with the motion are ignored and another in whiclrthe effect of these forces is taken into account. The response of the structtlltis estimated for each of these two regions, and the structure is designed fbr thc:

higher of the two responses. It is suggested in [10-4] that a drawback of thc:

ESDU procedure is the lack of a natural transition between the two responseregimes, which introduces an element of arbitrariness in the application of thcproccdure.

Thc proccdurcs dcveloped by Vickery and coworkers imply in effect ther

lirllowing appnrach. A nominal response is calculated which coresponds trlthc assurnption that acroclastic effects do not occur. The actual response is thcnobtained through rnultiplication of the nominal response by an aeroelastic cor-rection factor which varics continuously over the entire range of possible aenlelastic effects. The derivation of that factor is explained in some detail in Sect.6. 1.2. Because of uncertainties inherent in them, these procedures should bcr

used with caution.Structures that are light in weight and have low structural damping (e.g.,

ceftain steel stacks) could experience unacceptably severe aeroelastic effeclsunless provided with aerodynamic or mechanical devices for the alleviation ol'across-wind motions. Some of these devices have proven to be quite effectiveand are routinely incorporated in the design of steel stacks.

This chapter describes Rumman's procedure (Sect. 10.1) and the procedurcsdeveloped by Vickery and coworkers (Sect. 10.2). These procedures are ap-plicable to isolated structures.* Also presented in this chapter is informatiotton aerodynamic and aeroelastic devices for the alleviation of the across-wintlresponse (Sect. 10.3).

10.1 RUMMAN'S PROCEDURE

In this procedure it is assumed that towers or stacks with a circular cross sectirlltare subjected to a sinusoidal force per unit length with amplitude

Fo(z) : iCrpLI2(2",)D(z) (l0. r. r)

*If several stacks are grouped in a row, buffcting fbrccs associatcd with vo(cx shctlclirrg tuttcause the response of stacks located downwind of thc first structurc in thc row lo bc irs higlr ir:fourtimes the responsc ol'an iclcntical but isolalcd stack. l-irrri(txl tlulrr ort lltc rcsponso ol grottlx'rlstacksareavailablcinllO(rl,ll024l,ll0-25l.liorallxrnrtrghlcvit'wrtl ittlirt'ttutlionutttl lilct:tlttrcon interference antl proxilnlly cllccls on cylintlrictrl s(lll(lllr('s, scc l() II

r() r ilr rMMAt.t :, t't rr'r.t trt rttt 381=r

\vlr(flc t) is ail tlerrsity 1p 1.J.5 kg/rrr'), l/(,,,',) is llrt'r'ttllt:tl wrtttl sllcctl itl,.lt"vrrli9rr r.,,,. ('t is lltc lill coollicicrrl, irntl /)(l') is lltt'tli:trrtt:let ol sttttclttrc itlt.lt.virli6n e.'l'hc litr-cc /'ir(:) is assruttcrl to l'rc pt'rtr't'll-y toln'littctl splrrtwisc.lit'lcrcrrcc Il0-IIsuggcsts 2,,, : lt' wltcrrt: /r is tlrt'lrciglr( ol strttctttrc' ()thcrr.li'rcrrccs suggcst 2.,., : 2l3h to -5l(r/r I l0- l0l, or t:,,, 213h ll0- I ll- 1'hc wind,,1x.ctls U(2,.,) producc at elcvatioll 2,,, votlcx shotldirtg with licquencies equalIr) lhc natural f'requencies of thc structtlrc, so

I

I

t,

ir

i

Iil

t,

*, : Io,

m(z)Y?Q) dz

":.i)

i

IU(2",) : - niD(2",) (t : 1,2,...) (10. r.2)

(10.1.4)

(10.1.6)

'l'hc Reynolds number at elevation zn, is calculated as follows:

G." : 61,000 U(2",)D(2",) (10.1.3)

tU(:.,.,) inm/s andD(zn) inm). ForGe ) 3 x 106orsoitisusuallyassumed:', 0.220 to 0.25. From Eqs. 5.2.8, 5.2.10,5.2.14, and 5.2.16 it followstlr;rt lhe peak deflection for the structure excited in the ith mode may be written

rvlrt'rc y;(Z) is the ith normal mode of vibration, f, is the damping in ith mode,

(10.1.5)

r:, llrc generalized mass, and m(z) is the mass of the structure per unit height.I .r rcinforced concrete chimneys ratios C1lfi = 13-16 have in many instancesIrr't'rr assumed for design purposes [0-U.

According to [10-11] it was determined from observations that tall reinforced( ()ncrcte chimneys with constant or nearly constant diameter do not appear to, \l)cricnce unacceptably large motions if their Scruton number c;, defined as

Y,(zt : #? D2e",) \',o3ffi4 r,u,

2Mi n-/ f, . PD'(Z,,\\ ,itz.l ,tz.Jrr

r'; lrrlgor than filur.'I'lrc pcak m()tncnt l( trrry clt'vlrlron.'ts rlotttitt:tlt'tl lty crlnlribtttiorrs tltttr ltr

rrrt.llitrl litrccs.'l'hcrrclirrt', rlt'rtolttt!: llrr'rt':rl':tr'tt'lt'tltliotl lt( clt'vitlioll.l lry),(.t), tlrC pclk tntttttcttl:tssrtt't:rlt'rl rvtllr llrt'tllt tttotlt'ol vilrr':tlitttr is

&.i

386 :jLLNt)ftr towilri ANt ) litA(:Ki; wt ilt oiltoUt An (in()tili :it (iil()N

trL,(z) = trt(2,,)Yi(2.)(z.t * z.) dz.,J.

phJ(Li({ = 12rn1)2 \ *(r,)yik)kr - z) dz1

o1

The shear force at elevation Z1 ma! similarly be written as

( 10. t.7)

( l0. r .n)

Si(21; = (2rn;)2 (10. r.9)

Numerical Example Consider a chimney with constant circular cross sectigrrfor which D : 17.63 m, h : 193.6 m, and n1 : 0.364 Hz [10-9]. It isassumed CLlh : 15, y{zlh) : 17lh)t67, m(z) :58,000 kg/m for z < hlL,m(z) :41,000 kg/m for z ) hl2, and S : 0.22. From Eq. 10.1.5, M, =1.87 x 106 kg. The critical velocity in the first mode of vibration is u(zu,) ..29.15 m/s (Eq. 10.1.2). The corresponding Reynolds numberis G" : 3.4 x107 6q. 10.1.3). The peak."rpon*i at elevation zis y,(7) : O.s\(zlhllbT nt(Eq. 10.1.4), and the peak moment at the base is Jll(O) = l.l7 x 106 kNrn(Eq. 10.1.8). If it is assumed fr : 0.02, then the Scruton number cr :4.3(Eq. 10.1.6).

10.2 PROCEDURES DEVELOPED BY VICKERY AND COWORKERS

It was mentioned earlier that these procedures may be viewed as, in effcct,estimating the across-wind response in two phases. First, a nominal responscris calculated by assuming that the structure is acted upon by the across-winrlaerodynamic loads it would experience if it were at rest. The nominal responsetherefore does not reflect any aeroelastic effects, since the latter involve loadsassociated with the motion of the structure. The actual response is obtaincrlthrough multiplication of the nominal response by a correction factor that accounts for the aeroelastic effects. The approach used to estimate the nominllresponse and the aerodynamic correction factor is described in Sect. l0.z.l.Information on the requisite aerodynamic and aeroelastic parameters is proviclctlin Sect. 10.2.2. Approximate expressions for the across-wind responsc arrgiven in Sect. 10.2.3.

It is emphasized that, although the procedures presentccl in this section 1reconceptually advanced, they yield results that may bc rrnccr(1in t9 within irlleast 3O%. This is the case in part because the structurirl tllrrrrgling is in lroslcases poorly known. In addition, much of thc availlblc irrlirrrrrirtion conccrnirrg,the aerodynamic ancl acr<lclastic parillnctcrs (soc Set't. l0.l.l) is orrly lcrrltrliver.

Io, *r',r',r,,, or,

t():' l,l t(x;l l)t,lll 1; l)l vl lol,l l) ltY vl(;hl lty ANI) (;0w(lilt(t nl; 387

Atterrrllts to ohlilin srrc'lr irrliu'rrraliorr lhlrtt wintl trrrtrtcl t('sls (c.9,.. irt ll0 l2l),ilrI llr:norillly rrrrstrcct:sslirl, owirrg (o scvclt scirlc cllc'r'is. l,or llris rcilsott i( ltaslrt'crr pointcd out that wincl tunncl sintullliotts ol'lltt: irctrrss-wintl ltchavior ol'.,lt'ntlcr structurcs with circular cK)ss sccliort tttttlcr wittrl klads cannot bc uscdlor tlcsign purposcs unless caretully intcrprc(orl irr tlrc light ol'acroclastic thcory;rrrrl ol'clata obtained f-rom full-scalc tcsts ll0-lil.

10.2.1 Basic Approach to Estimation of the Across-Wind ResponseLt'l o',1)""(z) denote the rms value of the nominal across-wind response at ele-v;rtiorr z in the ith mode of vibration. The rms value of the actual across-windr('slx)nsc at elevation z in the ith mode is denoted by ou;(z). The followingrrlrrtion holds:

(10.2.1)

ivlrt'r'c fr is the structural damping ratio, far is the aerodynamic damping ratio,;rrrrl l(,/(i I (",)lt'' is the aerodynamic correction factor in the ith mode.

Estimation of Nominal Across-Wind Response. The nominal across-windr('slx)nse is obtained by subjecting the structure to the across-wind aerodynamicloirtls it would experience if it were at rest. No aeroelastic effects are takenrrr(o account, and the only damping that affects the motion is the structuralrl;rrrrping.

ln a turbulent flow the structure at rest would experience a superposition oftw() ilcross-wind loads. The first of these two loads, due to vortex shedding intlrr' wake of the structure at rest, is denoted by L{2, r). The second load, duelo tlro lateral turbulence in the oncoming flow, is denoted by L2(2, r). The loadL1(;-, t) can be written in the form

o,i(z) : (r, . ,,--r;" o]? (z)

LrQ, t) : )pC{2, t)D(z)U2(z)

,,o that its spectral density is

(10.2.2)

Sy,(2, n) : l*pDQ)Ut(z)l2Sc,Q, n) (10.2.3)

Atcrrrding to [10-2], measurements indicate that the spectral density Sgr(2, n), :rn lrc represented by thc bcll-shaped function

u,ltctc rt tlcnolcs lhc lir:t;trcrrt'y, r,i l.i lltc votlcx slrcrllling l'rcqucncy givcn byllrc tclrrliort

nt'{},' "' J n|,,,,,"^n[ I

l-:-vull] (to 2'4)

= 10.'l 4 nr

= 0.1 83

388 sr t NDFn towlnri ANI) s |ACK!; wt il l (:il tcul Ar1 cno$ri st c iloN

1.0

ttS, (n)

cl

0.1

0.0110 2- 10-1 1

ttDU

FIGURE 10.2.1. Power spectral density of lift force coemcient c. measured on Ham-burg television tower. From H. Ruscheweyh, "wind Loadings on the Television Tower,Hamburg, Germany," J. Ind. Aerodyn., I (1916),315-333.

s is the Strouhal number, and B is an empirical parameter that determines thespread (bandwidth) of the spectral curve. This model is compatible with resultsof full-scale measurements (Fig. lO.2.l).

The cross-spectral density ofthe load L{2, r) can be expressed as [10-41:

SU(z)" D(z)

Sr,(zr, zr, n1 : Sl,2Q1, n)St/,2(22, n)Rs(21, 22, n)

Ro(zr, zz) : cos(2ar)exp(-arz)

(10.2.5)

(10.2.6)

(r0.2.7a)

(r0.2.1b).rl- _ - I-l4l 1-21r: D(a) + D(72)

The parameter a in F,q. 10.2.7 a is a measure of the decay of the cross-spectralfunction Sr,(4, zz, n) with the distance lz, - zrl. Associated with the parametera is a correlation length z which is a measure of the spanwise length beyonclwhich the force fluctuations are no longer correlated.

The lift force Lr(t) is the projection on the across-wind direction of the dragforce induced by the resultant of the mean velocity U(2.) and of the lateralturbulent velocity u(2,, t).In large-scale turbulence this lorcc hus an anglc ol'attack with respect to the along-wind direction equal kt t,l L/, irrrtl its pr<r.jccti<lnon that direction is

t0? Ilt(x;l l)lllll r; l)tvl loPIt) llY vloKl ltY ANI) oowolrKElr$

r'(i, /)I I(2.\

I, )(t) tp()1Jl'!17.1

389

(r0.2.8)

Spcctral and cross-spcctrul irrlonrurtion on thc lateral velocity fluctuations a(r)rs givcn in Scct.2.3.-5. lnlilrrrration on the aerodynamic parameters B,3,t'i"t, o, and Cp is givcn in Scct. 1O.2.2.

'l'he mean square valuc of the nominal response induced by each of the loads/,1(/) and lo(t) can be estimated as in Sects. 5.2.7 or 5.3.2.The mean squarevirluc of the total nominal response is equal to the sum of the mean squarevrrlrrcs of the responses due to the loads L(t) and Z2(r). However, becausetlrcsc loads are uncorrelated, the peak value of the total nominal response isIt'ss than the sum of the individual peak responses due to Z,(r) and LzQ).*

Estimation of Aerodynamic Correction Factorl!,lftt + e"ill't'. One of thetlrlliculties that arises in the estimation of the aerodynamic correction factor isllur( relatively little reliable information is available on the structural dampingrrrtios f;. Ranges of values fi suggested in [10-7] are listed in Table 10.2.1.

TABLE 10.2.1. Suggested Structural Damping RatiosType of Structure Structural Damping Ratio

Unlined steel stacks and similar structuresLined steel stacksReinforced concrete chimneys and towers

r'lt is of interest to estimate the extent to which the effect of the load Lz(r) is significant from a

1rr;rtlical point of view. Using the information of Sect. 2.3.5, it can be verified that the lateralvr'krt ity fluctuations differ from the longitudinal velocity fluctuations as follows: (1) the ordinatesrrl rlrc spectral density at high frequencies are largerby 33% forthe lateral than forthe longitudinalllrrr'trrations, (2) the area under the spectral curve is lower by 50% for the lateral than for thel,rrr1'.itudinal fluctuations, and (3) the exponential decay coefficients are lower by about 33% fortlrr' l:rtcral than for the longitudinal fluctuations. Calculations then show that the peak nominal,rr rrrss-wind response due to l4(r) is of the order of 5O% of the peak fluctuating part of the along-rvrrrl rcsponse or, roughly, abottt 25% of the peak total along-wind response.

It lirllows that if the ratio between the along-wind response and the nominal across-windrr'rlx)nso estimated without accounting for IaQ) is small, then taking L2Q) into consideration willlrirvc rr negligible effect on the magnitude of the nominal response, particularly in view of thel;rcl rrrcnlioned earlier that L'(t) and Lr(t) are uncorrelated and that their peak values are thereforerrot rrtltlitive. On the other hand, il'thc ratio of along-wind to nominal across-wind response islrrplr, lhc design will be govcrncrl by thc along-wind response regardless of whether Zr(l) is,t((r)untcd fbr or not. Finally, il'lltc lirlio rrrxlcr considcration is close to unity, accounting for/ ,(l) wrruld incrcase thc pcak notttitt:rl irt't-oss wintl rcsponso hy ahout 25% if Llt) and Lr(r) wcrci ()r('lltcd; howcvcr, sincc this is rrol llrr' trrst', lhc irrcrcrrsc will only bc of thc ordcr ol' l0 tolr',?,. lirrthcsc roasons,1oa lilsl nl)l)roxrrilll()n, llrc lirt'c inrlucctl by latoral turbulcncc lluctuirtrrlns nlity bc ncglcclcrl, ttttlcss lltc clilnuulr'(l ;x'rk rrkrn;i wintl:rtttl irt'toss-wirttl tespottse lt:rvr'trl)l)r1)xinritloly lhc sirrrrc virlrrc, itt wlticlt r'irrf lltr ir|ilrrs wirul rcs;xrrrst' slr()rrltl lrt'itrr1;rrrcrrlcrl lryrorrpllrly l0%.

0.002-0.0100.004-0.0160.004-0.020

390 SLENuEn towfti.ri AND sIACKsi wt ilt oill(:r,t An cn()ri$ sfcTtoN

The approach to the estimation ol' thc ar:nrdynarnic darnping ralio 1,,; is rlc=scribed in some detail in Sect. 6.1.2. Inlbrmation on the acroclastic parartrcterK,6 needed to estimate l,; (see Eqs. 6.1.36 to 6.1.38) is summarizcd in Soct.t0.2.2.

10.2.2 Aerodynamic and Aeroelastic ParametersThe purpose of this section is to provide information on the drag coefficicrrtCp, the Strouhal number S, the rms lift coefficient Czytt2, the bandwidth pa-rameter B, the parameters describing the spanwise correlation of the across-wind load, and the aeroelastic parameter K"s used to calculate the aerodynamicdamping ratio f,,,.

The aerodynamic and aeroelastic parameters depend upon the Reynolds num.bcr

G.. : 6t,OOO U(dD(z) (10.2.9)

where u(e) is the wind speed at elevation z in m/s and D(z) is the outsidediameter in meters; upon the turbulence in the oncoming flow, upon the aspectratio h/D(h) , where h is the height of the structure and D(h) is the diameter atthe tip and upon the relative surface roughness klD of the structure, where ftis the height ofthe roughness elements. For steel stacks and reinforced concretcchimneys and towers l0 3 < kld < 10-5 [10-7]. It is assumed herein thatklD vaies only within this range.

Drag Coefficient Ce. The dependence of Cp upon Reynolds number anrJsurface roughness is represented in Fig. 4.5.5c for cylinders with aspect ratiorhlD(h) > 20. Forstructures with aspect ratios l0 < hlD(h) < 20it may beassumed that up to the elevation h - D(h) the drag coefficient has the valuc

(10.2. t0)

where Cf, is the value of the drag coefficient taken from Fig. 4.5.5c. Fronrelevation h - D(h) to the top of the structure the drag coefficient may bcassumed to have the value Co : 1.4 Ci for all structures regardless of aspcctratio (see [l0-l3l). The main effect of turbulence in the oncoming flow is todecrease the Reynolds number corresponding to the onset of the critical regiondefined in Fig. 4.5.2.

strouhal Number. The following values of the Strouhal number are suggestcdin [10-13] (see, however, [4-86] and Fig. 4.4.4):

S:0.20 G"<2x105 (10.2.11il)

0.22 <,S < 0.4.5 2 x 105 ? (11,, :- t v l0(' (l0.2.llh)

Co: c'o[t - o.otr (, - hl

t0 ? I'll()(il l)lllll li l)t vn ol)fl) nY vloKl ltY nNl) c()woltKt lti 391

* =,.[,,.r., ,(x)71 ',,r,,(r)

_,rl] (n,.= 2 x r0(' (r0.2.nc)

lirrr2 X lOs < 61" < 2 x 106 thc vortcx shcclcling is random, and the Strouhalrrrurrbcr given by Eq. 10.2. I I b corrcsponds to the predominant frequencies oftlrt' lkrw in the wake. In Eq. 10.2. llc the coefficient c depends upon aspectrirlio as follows:

(10.2.12a)

(to.2.rzb)

wlrrrro h is the height of the structure and D(h) is the diameter at the tip (seello t3l).

C7''". The following values of the rms lift coefficientpurposes (see [10-13]):

G,.<2x105 (t}.2.l3a)

2x10s1Ge72xtO6 (10.2.13b)

I / ft\ 12)+0.03515 + log,o(=)l t G" > 2 x 106I "',"\D/l)(10.2.13c)

lrr liq. 10.2.13c the coefficient d has the expression

= 12 (10.2.14a)

d-

RMS of lift (irrc suggested

h< olny < tz (lo'2'14b)

I lrt' lilt coefficient also appcars to depend significantly upon turbulence inten-rrty. However, little inlirrrrrirlion on this dependence is available to date.

Bandwidth Parameter B. l{cli.r'r'rrt'tr l0-4 suggcsts that

)tt'll 00t( , ,,, ( r0.2. r5)

392 .LENDER towflis AND !.itAoK:i Wt llt (;i,(;(,t An (i'oss tiE(;iloN

where u'is the mean squarc valuc ol' krrrgitudinal turbulcncc lluctuations 'rrtluis the mean wind speed. According tu ito-e1, for practicar purposes it rrruybe assumed B = 0.18 for all flows.

spanwise correlation Parameters. For Reynords numbers G" > 2 x rotit may be assumed that in Eq. 10.2.7 the coefficient a : l/3 and that to thisvalue-there corresponds a correlation length L = D t10-41. For G" < 2 xl0'. I = 2.5 [lO-14]. Then, using the noration g : LID,"

(10.2.16u)

(10.2. t6b)

Aeroelastic Parameter Kro. on the basis of tentative information from[10-4] and [10-13], the foilowing approximate expressions may be used toobtain Koo:

^ (z.s G."<zxtos" : [ ,.0 G" > 2 x ro5

", (r t t -,.nt) o.8s < {,.,.0o.55a, t.o = {,< t.t

",(rr,-,t) ,,=[,<r3",(oou -,rtt),, = {,< r 84

o 1.84=t

*", (+,) =

Uu- < o'85 (t0.2.t7a)

(10.2.r7b)

(10.2.17c')

(10.2.17d)

(10.2.17e)

(to.2.t7tl

(10.2. tn)

(10.2. 19n)

(r0.2.19h)(10.2. t9c)

where

A1 : Q(L2A3Q4

( t.O G" ( lOaI

1 1.8 toa < G" < los

[r o ros < G"

at:

L,'lr.u U(10 rrr) I 12

*At:\ l-

Ito u(lonr)= 12r

at:oe+o2|'"r,,,(f) *rllro J-rt.t (ro.Z.z*a)J D(h)oo:\ / n \ r| 1.0 - 0.(X | 12.5 - * I == < 12.5 (to.2.22b)\. \ D(h\/ D(h)

lir;rrirtions 10.2.20 reflect the fact that if the wind speed at 10 m above groundir, rclatively low the atmospheric turbulence may be weak. This can lead to at onsiderable enhancement of the aeroelastic effects (see Fig. 6.1.10).

t0.2.3 Approximate Expressions for the Across-Wind Responsel'lrc across-wind response in the ith mode of vibration may be estimated as

ori(z) : t?t''y,(:r)Y,(z) : gyioyi(z)

Byi : r2ln(36oon;)1,,, * ,ffi;,g, 0o.z.2s)

I s- ll/2C2t/2 _ C2 r/2 | li

Isr snom'r L(r, + r",liph

Si(z) = 12rn;\2 ), mk)Y/z) dz1

ph

SlLik) = 12rni\2 ), ^Q)r,(r,)kt - d dzt

r() I I'ir( xri ilunt li Dfvf I Ol'il) try vr(;Kr ny ANt) (;()w()llK[ nti 393

(tO.2.ZOa)

(10.2.20b)

(t0.2.21)

(r0.2.23)

(10.2.24)

(t0.2.26)

(r0.2.27)

(10.2.28)

rvlrcrc oni(Z) is the rms of the deflection at elevation z in the ith mode ofvrlrlrrtion, t?''' it the rms of the corresponding generalized coordinate, li(z) inllrc ith rnodal shape, )i(z) is thc pcak deflection in the ith mode of vibration,,r;,., is thc peak factor, a1 is llte rtirtural I'rcqucncy in the ith mode in Hz,j- ,i,,,,.,"' is thc rms norrrinirl ge rrcrirlizr'tl crxlrtlinalc in thc ith mode (whichtollr:sponds to thc rcsprrtsc t'sl itttrrlt'tl lry itssrrrnirrg tha( no acroolaslic cfl'ccts

394 SLENDEII lowEllsi ANI) silAcKli wl lll (illl(;trl All (;liosis SFCTI()N

occur and that the motion is all'cctcd Only by structural damping), li/(f, -t

l")\t,, is the aeroelastic correction f'actor, f, is the structural danrping in thcith mode, f,, is the aerodynamic damping in the ith mode, si(z) and sT[;(z) arc.respectively, the shear force and the bending moment at elevation z due to thcacrbss-wind response in the rth mode, and m(z) is the mass of the structure pcrunit length.

To estimate the across-wind response, expressions a19-ne^ eded for the rmsof the nominal generalized coordinate in the ith mode, tiu.,i"'. and the aent-

dynamic damping in the ith mode, f,,. These expressions are given belowr"purut"ly for (1) structures with constant cross section and (2) tapered struc-tuies. In both cases the expressions are valid only for relatively small ratiosor;(h)lD(h), firr e xample 3% or less (to which there would correspond negligiblovalucs <rl'thc sccont.l tcrm within the bracket of Eq. 6.1 .22).It is noted that,in practicc, thc tlcsign of a structure will be acceptable only if the ratios or(h)/D(h) inhcrcnt in that dcsign arc indced small.

Structures with Constant Cross Section. The following approximatcexpressions based on the approach described in Sect. 10.2.1 were proposed in

[10-e]:

Soi =

(r0.2.29)

(10.2.30)

where p is the air density (p - 1.25 kglm3), Mi is the generalized mass in the

ith mode (Eq. 10.1.5), and D is the outside diameter. The critical wind specd

corresponding to the ith mode of vibration has the expression

(10.2.3 r)

Information on the structural damping ratios f; is given in Table 10.2.1. ln-formation on the parameters, 3, Ctt'', S, and Krs is given in Sect. 10'2'2'Note that in Eq. 10.2.2Oathe speed U(10 m) corresponding to the ith modc irr

ffi ,z = LY#t *El, \oo r?u, o,]'''

lnt l0/zn)u(lo m) tnr(y6ieru"''

-# *'""' \oo'?u> o'

fl,DIt - _Lvcr,r S

(10.2.32',)

where h is the height of the structure in meters ancl z1y is lhc: rottghncss lcnglltin meters for the tcrrain that detcrmines thc wind pnrlilc ttvcr' lltc uppcr halfof the chimncy (scc 'l'iltlc 2.2.1 ancl Sccl. 2.4.1).

t0,, I'n(x:t l)unl ri t)[vLLopEu By vtcKERy AND cowoRKERS 395

Ntmerical Examplo ('onsidcr the chimney described in the numerical ex-irrrr;rlc ol'Scct. 10.I (h : 193.6 m, D : 17.63 rrr, n1 : O.364Hz,y(zlh) :(.'lltltl'7, m(2.) :51i,000 kg/m for z < hl2, m(z) :41,000 kg/m for z > hl). Mr : 1.87 x tO6 tg;. It is assumed fr : 0.02, klD: 10-s, and zs :ll0-5 rn. We seek the response in the first mode.

Assuming tentatively that 3 : 0.22, the critical wind speed at elevationthl6: 161.3 m is u...r :0.364 x 11.6310.22 :29.16 m/s (Eq. 10.2.31),to which there corresponds a Reynolds number G..:3.4 x 107 > 2 x l0('tlit1. 10.2.9). The aspect ratio is hlD = ll. It fbllows that

3 = 0.178

S:1.04t'' = 0'143

lt,I vitzt dz : 44.7 m.lt r

ffi''': 0.115 m

mu(10) > 12 -SK'o(l) = 0.465

f"r : -0'0043EtD - 0.130 m

8lr:3'94

(Eqs. 10.2. 1 lc, lO.2.l2b)(Eq. 10.2.16b)(Eqs. 10.2.13c and IO.2.l4b)

(Eq. t0.2.29)

(Eq. 10.2.31 and, 70.2.32)

(Eqs. 10.2.17c, 10.2.18, 10.2.19c,

lO.2.2Ob, 10.2.21, and 10.2.22b)

(Eq. 10.2.30)

(Eq. t0.2.26)

(Eq. 10.2.25)/ \ l-67l7o,rk):or:o(re:..1 ) ^ .Eq.to'2.23)

/ . \167Ytz):o'st(u:.01 ) ^ (Eq'to'2'24)

1I(,(0) : 1150 x 106 Nm (Eq. 10.2.28)

Notc that the results of the calculations depend strongly upon, in particular,llrt' rrssumed value of the structural damping ratio fr . Had the value f1 : 0.01lrt'err appropriate, the rcsults oblainccl would havc been larger than those ob-r;rrrrctl in this example by a lirclor ol'l(0.02 - 0.(n43)/(0.01 - 0.0043)lt'' =I ()().

fapered Structures. 'l'lrc lirlhrwirrp. rrlrlrlrxirrrirlc cxprrssions hirsotl orr lhcirlrprlach dcscribccl itt Sert't . 10.,'.1 wr'rc ;ttulxrsr'rl irt ll0 ()l:

396 st LNDEn towl HS AND fitAcKli wint (;lt(;t,t An cno$li sfcloN

t?o^.,(2",)t'' =O.o I 6C i.t

t ) JJt /2 p Da (2..,) y,(2.,,,)

r)/42M,Ptt2(2",)( r0.2.33)

(t0.2.34)

(10.2.36)

(10.2.37)

. 0.lD(2",) dD(dllrlz,,1 - A - d, 1,,,

t.i(2,,) : -# J, ", (ffi)["#]' v?(z) az (r0 2 3.5)

where the notations of Eq. 10.2.29 are used, Do : outside diameter at basc,2,., is the elevation corresponding to the critical velocity

lJrr(Z",) : ryd,u(z; 2",) : ln(zlzo)Urr(Zn,) ln(zn,lzo)

and e6 is the roughness length for the terrain that determines the wind profileover the upper half of the chimney (see Table 2.2.1 and Sect. 2.4.1).

Since, as in Eq. 10.2.26,

t',{r1''': *;/t"' (- - L )'"\fi * hik"')/it follows that the maximum response in the ith mode corresponds to the max.imum value taken on by the function

F,(2",) : Da(2",)yi(2",)(10.2.38)

{P(z")Ki r (,,(2" )f\t/z

To determine that value, it is in practice necessary to calculate F(2",), and, inparticular, loi(Zr,), for a sufficiently larger number of elevations O I 2", < h.

As pointed out in [10-8], if the structure is very lightly tapered (i.e., ifdD(a)ldzl,:,,.. and therefore p(2",) is small-see Eq. 10.2.34), rhen the approx-imations on which F;q. 10.2.33 is based are no longer valid and Eq. 10.2.33ceases to be applicable. In that case the chimney is assumed to behave as if ithad a constant outside diameter D equal to the average diameter of its top third[0-9], and Eqs. 10.2.29 to lO.2.3I are applied with the same values of rheparameters E, C'rt'', and S as those used in Eq. 10.2.33. In practice, it istherefore necessary to calculate both the value of the rcsponsc yiclde<t by Eqs,10.2.33 and 10.2.35 and the value yielded by Eqs. 10.2.2q ancl 10.2.31. ltfollows from [0-21 that the response to be assunrccl lor sinrctrrrirl rlcsign pur-poses is the smal.l,er <ll'thcsc tw<l valucs.

l0r I'tt()t:f t'{[lt ii t)t vltoPf t) tlY vl(]KillY ANt) cowonKElls 397

Numerical Exampla ('orrsitlcr ir chirnncy with hcightll : 365.8 m, outsiderlirrrrrotcr at thc baser /)o .17.8 rrr, outsidc diarneter at the tip D(h) : 72.6 m,t'trrrstant tapcr (i.c. , dl)(r,)ldr. - lDo - D(h)llh), fundamental frequency n1

0.252 Hz ll0-91. ll is assurncd that the fundamental modal shape y(zlh) :t.'lhf .the mass pcr unit lcngth m(z) :180,000(l - O.9zlh) kg/m, the struc-Irrrul damping ratio in thc first mode f1 :0.01, the relative surface roughnessrrl lhe structure klD - lO-t, and the roughness of the terrain z0 : 0.008 m.We: seek the response of the chimney in the first mode of vibration.

Assuming tentatively S = 0.2, the critical speed U". > 0.252 x 12.610.215.9 m/s (Eq. 10.1136), to which there corresponds G" > 67,000 x 15.9

". 12.6 = 1.3 x 107 (Eq. 10.2.9). The aspect rario is hlD(h) = 29.0. kkrllows that

q+ : 1.0 (Eq. 10.2.22a)

l'lrc coefficient a2 (Eqs. 10.2.20) depends upon the wind speed U(10 2",).As mentioned earlier, the function 4(2",) must be calculated for a sufficiently

lrrrgc number of elevations zei to obtain the value that maximizes the response.Wr: show here calculations for z"r:365.8 m and 2",: 182.9 m'

lior 2", :365.8 m, U".(365.8) : 13.70 m/s (Eq. 10.2.36) and U(10; 365.8)9. I m/s < 12 m/s (Eq. 10.2.37). Therefore az : 2.0 (Eq. 10.2.20a). It

tirrr be verified that f"1(365.8) = -0.0065 (Eqs. 10.2.17 and 10.2.35), and/,1(365.8) = 1.6 x 106 ma (Eqs. 10.2.38 and 10.2.34).

ltor zn, : 182.9 (n, U,, : 27 .61 m/s (Eq. 10.2.36), U(IO; 182.9) : 19.16ttr/s ) 12 mls (Eq. 10.2.37), az : 1.0 (Eq. l$.z.2$b), f"tQ82.9) : -O.OO42tlitls. 10.2.17 and 10.2.35), F(182.9) = 4.6 x 106 ma 1Eqs. 10.2.38 and10.2.34).It can be verified that the largest value of Fr(2",), and therefore thelrighcst response in the first mode occurs for 2", = 182.9 m. It follows that

S = 0.23

4', : o.l5S:1.0

Mr : 3.3 x 106 kg

4r : 1.0

az : 0'9

{i,,,,,. r(182.9)'/2 : o.Q6 m

t',08Lg1t'': 0'079 r'

(Eq. 10.2.11c)(Eqs. 10.2.13c and lO.2.l4a)

(Eq. 10.2.16b)(Eq. 10.1.5)

(Eq. 10.2.19c)

(Eq. t0.2.21)

(Eqs. 10.2.33 and 10.2.34)

(Eq. 10.2.26)

ln (Eq. 10.2.23): o.ole( ' \'\.l(r5.tl /o,tk)

sltNuEtl lOwillli ANI) :i r n(;Kli wr il l oil rct,l Al l 0lrolili ril o l l()N

Ytk) :0.304( == ) n.r (Eqs. 10.2.24 dnd t0.2.25)

\ 36s.8 /Sltr(O) : l29O x 106 Nm (Eq. 10.2.28)

The response will now be estimated by using Eqs. 10.2.29-10.2.31. Thc av'erage outside diameter of the top third of the chimney is D : 16.8 m. llfollows that

22-tr2_ 0.035 x0.15 x 1.0 1.25 x 16.8t / .. ^365.8\r'2I i,,n,. r' - = *, ra ,. q2l - j-n.* ( 16.8 5 /:0.0625 m

mUu: l8'4-S

mu(ro) > t2-S

(8q.10.2.29)

(Eq. 10.2.31)

(Eq.10.2.32)

1.25 x 16.82 365.8t^,: --(0.9 x 0.55)-3.3 x 10" 5

10.2.22a)/ - 12

orlk) :0.0?e(#J) - (Eqs. 10.2.23, t0-2-26)

t r2Y(z) : o'm+(fr) ' (Eqs' t0'2'24,10'2'2s)

fltr(o) : l28o x lo6 Nm (Eq. 10.2.28)

It is seen that in this case the response yielded by Eqs. 10.2.29-10.2.31 ixapproximately the same as that obtained by Eqs. 10.2.33-10.2.35.

= -0.0039 (Eqs. 10.2.30, 10.2. l7 c, 10.2.18,lO.2.l9c, 10.2.20b, 10.2.21,

10.3 ALLEVIATION OF VORTEX-INDUCED OSCILLATIONS

Aerodynamic DevicesA common method of alleviating vortex-induced oscillations is the provisiorrof "spoiler" devices that destroy or reduce the cohcrcrrcc ol'thc shcd vrlrtice:lr[10-26, 10-27]. Of the various types of such dcviccs, ort('ol lhc rrrost oflcctiveis the helical strakc syslcm lirst clcscribctl in ll0:1.51.

ilt :t At il vtAiloN ot vollll x tNt)t,clrtj ofi(il t.AiloNS

'l'lrtr ltclicitl sltitkc syslerrr consisls ol'throc thin rcctangular strakes with aprtt'h ol'onc rcvoluliorr itr .5 ilialrrctcrs and a strake (radial) height of 0.10rllurrclcr (to 0. l.l (liilnlertr:r lor vcry light or lightly damped structures) appliedrrvcr lhc t<tp 3301, b 40%, ol'thc stack height. The effectiveness of the systemrs rrot impaired by a gap of 0.005D between the strake and the cylinder surfacellt) 16l. Referencc ll0-l7j reports the remarkable results obtained by usingllris system (with 5-mm thick strakes, 0.6-m strake height, and 30-m pitch) inllrt.case of a 145-m tall and 6-m diameter steel stack (Fig. 10.3.1).

lr<rr Reynolds numbers 6le 12 x l}s or so, in flow with about 15%Irulrrrlcnce intensity, helical strakes were found to reduce the peak of the across-rvrrrtl resonant oscillations by a factor of about two, as opposed to a factor ofllrorrt 100 in the case of smooth flow [10-23]. It appears that the performanceol s(rakes can be unsatisfactory in the case of stacks grouped in a row [10-28,l(l 291. Also wind-tunnel and full-scale tests indicate that for large vibration'ilnl)litudes (e.g.,3% to 5% of the diameter), the vortex street reestablishesilscll', and the aerodynamic devices become ineffective t10-301. It is noted thatllrc strakes increase drag, as shown in Fig. 10.3.2 t10-181 .

Shrouds can also be effective in reducing the coherence of shed vortices. Ar,thcrnatic view of a shroud fitted to a stack is shown in Fig. 10.3.3. Results

l, l(;llltll 10.3.1. Stccl clrirrrncy witlr hclictl strakes. From G. Hirsch and H. Rus-,lrr'wt'ylr, "lrLrll-Scalc Mclsrrrcrrrcrrls orr Slccl ('hinrncy Stacks," ,l . [ru|. Aerodyn.,lt I t, /(r ). l4l 147 .

400 SLENDEII tOwr..n..j ANt) StA(;KS Wt lil C[l(]tjtAU CROSS SECION

105 106 107Reynolds numberl)ttl

FIGURE, 10.3.2. Effect of strakes on drag coemcient. From L. R. Wooton and C,Scruton, "Aerodynamic Stability," \n The Modern Design of Wind-Sensitive Stru*tures, Construction Industry Research and Information Association, London, U.K,,1971, pp.65-81. By permission of the Director of the National Physical Laboratorl,U.K., and the Director of the Construction Industry Research and Information Asso-ciation, U.K.

Lo.o,

FIGURE 10.3.3. View of shroud fitted to a stack tl0-161.From D. E. Walshe and L. R. Wooton, "Preventing Wind-Induced Oscillations of Structures of Circular Section."Proc. Inst. Civ. Eng.,47 (1970), l-24.

- 1.5ooE

,9Eo!.Ea r.oco!

!a()E 0.5.o.9

oOoGO

T-,f

I'

T/D = O.12

T/D = O.06

Plain cylinder

J,ilr il ilr Nor s 401

ol wirxl tunncl cxpcrinrenls rclx)ilo(l in ll0-l(rl slrowctl thirt oscillutions wcrc.,rrlrstirrrlially rcducod witlr only thc top 25'I, ol'lltc tnorlcl hc'i8,ht shroudcd. Therrrrrsl cll'cctive shrouds wcrc l()und to bc lhrtsc with l gap width w = 0.12D;rrrtl an open-area ratio between 2O%' antl .l(r%, (with lcngth of square s :0 0.52D to 0.070D).

Mechanical DevicesSrrch devices include hydraulic dampers and tuned mass dampers (TMDs).

'l'hc use of hydraulic dampers to reduce vortex-induced oscillations is dis-lrrssod in [0-19]. An example of such an application is given in [0-17],wlrich mentions the use of three hydraulic automotive shock absorbers installed;rl 120" angles in a plane view between a 47-m high stack and a separatetr'ucture at the 18-m level.

'l'hc tuned mass damper (TMD) consists of a secondary vibratory systemiruirched to the structure and located near its top (see Sect. 9.4). If excited bylrulnonic (or quasiharmonic) oscillations of the structure, the TMD will vibraterrr opposition to these motions and thereby reduce the amplitude of the structuralrrslx)nse. The basic theory of the TMD is discussed in [10-20]. One of theIrrs( tuned mass dampers used in a large structure was designed for the Center-poirrt Tower in Sydney, Australia. The mass for the damper was in this casepnrvided by the water tank of the tower t10-2U. Further applications of TMDstrr rcduce tower oscillations are discussed in 19-791, |0-221, and [13-91].

REFERENCES

l() I W. S. Rumman, "Basic Structural Design of Concrete Chimneys," J. PowerDiv., ASCE, 96 (June 1970), 309*318.

lo I B. J. Vickery and A. W. Clark, "Lift or Across-Wind Response of TaperedStacks," J. Struct. Div., ASCE, 98, No. ST1 (Jan. 1972), l-20.

I ( ) .l ESDU, Across-Wind Vibrations of Structures of Circular Cross-Section in Windor Gas Flows,ltem 78006, Engineering Science Data Unit, London, 1978.

l{),1 B. J. Vickery and R. I. Basu, "Across-Wind Vibrations of Structures of Cir-cular Cross-Section, Part 1, Development of a Two-Dimensional Model forTwo-Dimensional Conditions," J. Wind Eng. Ind. Aerodyn., 12 (1983),49-'73.

lo 5 R. L Basu and B. J. Vickery, "Across-Wind Vibrations of Structures of Cir-cular Cross-Section, Parr. 2, Development of a Mathematical Model for FullScalc Application," ./. Wirul Eng. Ind. Aerodyn.,12 (1983),75-97.

lo (r I). .1. Vickery, "Across-Wintl Buft'cting in a Group of Four In-Linc Modcl(lhinrncys," l. Wirul lit,q. ltul. Atnxlyn.,8 (198 l), 171-19?.

lo / R. l. Ilasu und l]. .l . Vickcry, "A ('orrrparison ol'Modcl arrtl Irrrll-Scrrlt: lltrlrrvirrr irr Wintl ol 'lirwe ls:rtttl ('lrirrtrtcys," I'rrtt't,t,tlitrgs Witul 'littrttcl Mtutt'ftff:-'-''

402 SLENDEn tOWEltS AND STACKI] Wl lll clllOULAll (il|OSS SECIION

for Civil Engineering Applir:utiotrs, (iaithcrsburg, MD, April l9tl2, ClrrthritlgeUniv. Press, Cambridg, 1982.

l0-8 B. J. Vickery, "The Aeroelastic Modeling of Chimneys and Towcrs," Prrr-ceedings Wind Tunnel Modeling for Civil Engineering Applications, Gaithcrs-burg, MD, April 1982, Cambridge Univ. Press, Cambridge, 1982.

l0-9 B. J. Vickery and R. I. Basu, "Simplified Approaches to the Evaluation ol'thcAcross-Wind Response of Chimneys," Proceedings 6th International Conl'cr-ence on Wind Engineering, March 1983, Gold Coast, Australia, in J. Wi.nel

Eng. Ind. Aerodyn., f4 (1983), 153-166.l0-10 L. C. Maugh and W. S. Rumman, "Dynamic Design of Reinforced Concrcte

Chimneys," Journal Am. Concrete 1nst., Sept. 1967.l0-ll G. M. Pinfbld, Reinforced Concrete Chimneys and Towers, Viewpoint Publi-

cations, Scholium International, Inc., Flushing, NY, 1975.10-12 K. C. S. Kwok and W. H. Melboume, "Wind-Induced Lock-in Excitation ol

Tall Structurcs," J. Struct. Div., ASCE, f07 (1981), 57-72.10-13 R. I. Basu, Across-Wind Response of Slender Structures of Circular Crost

Section to Atmospheric Turbulence, Vol. I, Research Report BLWT-2-1983,University of Western Ontario, Faculty of Engineering Science, London, On-tario, Canada, 1983.

l0-14 A. G. Davenport and M. Novak, "Vibration of Structures Induced by Wind,"Chapter 29-II in Shock and Vibration Handbook,2d ed., C. M. Harris andC. E. Crede (eds.), McGraw-Hill, New York, 1976.

10-15 C. Scruton, Note on a Device for the Suppression of the Vortex-Excited Oscil-lations of Flexible Structures of Circular or Near Circular Section, with SpeciulReference to lts Application to Tall Stacks, NPL Aero Report No. 1012, Nu-tional Physical Laboratory, Teddington, U.K., 1963.

10-16 D. E. Walsh and L. R. Wooton, "Preventing Wind-Induced Oscillations ofStructures of Circular Section," Proc. Inst. Civ. Eng., 47 (1970), l-24.

l0-17 G. Hirsch and H. Ruscheweyh, "Full-Scale Measurements on Steel ChimncyStacks," J. Ind. Aerodyn., l, 4 (Aug. 1976), 341-347.

10-18 L. R. Wooton and C. Scruton, "Aerodynamic Stability," in Modern Designof Wind-Sensitive Structures, Construction Research and Information Associu=tion, London, 1970.

10-19 A. Brunner, "Amortisseur d'oscillations hydraulique pour chemindes," Jtttrr'nles de I'Hydraulique, 8, Part III, Lille, France Q9e).

lO-20 J. P. Den Hartog, Mechanical Vibrations,4th ed., McGraw-Hill, New York,1956.

10-21 "Tower's Cables Handle Wind, Water Tank Damps lt," Eng. News Reunl,187,24 (Dec. l97l),23.

10-22 R. H. Scanlan and R. L. Wardlaw, "Reduction of Flow-Induced Vibrations."in Isolation of Mechanical Vibration Impact and Noise, AMD, Vol. l, Soctittrt2, ASME, New York, 1913,35-63.

10-23 I. S. Gartshore, J. Khanna, and S. Laccinole, "Thc lill'cctivcncss ol'VtttlexSpoilers on a Circular Cylinder in Smooth and'l'rrrbrrlcttl likrw," in |liudEnginee ring, Procccdings o1' thc Filih Intcrnationll ('ottlcrclrcrr, lirtrt Collitut,CO, July 1979,.1 .li. Ccrnrak (ccl.), Porg,trlrrttlt lltcss, ()rlirnl, l()110.

llr il lil NCt $ 403

l0 14 W. lllnerrkllnl) iul(l W. lllurlrrcr., ..'l'r.iutsvcrse Vihrirliorr llclrirviorol.(.vlirrtlcr.sin [.inc,".1. Wittl l'.,ttg. ltul. Arnxl.yrt.,7 (l9l{l)..]7 5.1.

lo .15 H' Ruschowcyh, "l)rrrblcrns with ln-l,irrc Stlcks: lixpcricncc with lrull-Sc:alcObjccts," Eng. Srrut'r., 6 (l9tt4), 340 143.

l() 2() M. Zdravkovich, "Review ancl Classilicltion ol' Various Acrotlvnarnic andHydrodynamic Means for supprcssing V.rtcx Shctlding," J. winct Eng. Intl.Aerodyn., 7 (1981), 145-189.

lll l7 M. Zdravkovich, "Reduction ol'Eft'cctiveness of Means for suppressing wind-lnduced Oscillation," Eng. Struu., 6 (19g4), 344_349.

l0llt H. Ruscheweyh, "straked In-Line steel stacks with Low Mass Damping," -/.Wind. Eng. Intl. Aerodyn, 8 (1981), 2O3-21O.l{l l9 H. Ruscheweyh, "Dynamische windwirkung an Bauwerken," Bauverlag,

Wiesbaden, 1982.l0 |0 H. Ruscheweyh, "Vortex Excited vibrations," tn wincl-excited vibrations oJ'

Structures, H. Sockel (ed.), Springer-Verlag, New york, 1994, 5l_g4.

CHAPTER 1 1

HYPERBOLIC COOLING TOWERS

Much research into the wind loading of hyperbolic cooling towers has beenconducted following the wind-induced collapse in 1965 of three out of a groupof eight cooling towers at the Ferrybridge Power Station in England tll-ll,Principal areas of investigation have been (1) the spatial distribution and thevariation with time of the wind loading on the tower surface and (2) the responseof the tower to wind loads, including the dynamic effects induced by fluctuatingwind loads. This chapter summarizes and references the principal results cur'rently available in these two areas. * These results are presented in Sects. I I ' Iand ll.2 for towers that are not significantly affected aerodynamically by thepresence of neighboring structures. Information on groups of cooling towers iB

presented in Sect. 11.3.

11.1 DESCRIPTION OF WIND LOADING

Wind-induced pressures acting on a tower are determined by the characteristicttof the oncoming flow, the tower geometry, and the features of the tower sur-face. In addition the pressures depend upon the Reynolds number of the flow,which is in most cases of the order of 107 to 108 for the prototype, and hyabout two orders of magnitude smaller in the wind tunnel. On account of thildependence it has been necessary to complement wind tunnel test by full-scalemeasurements.

*The authors would likc lo acknowlcclgc thc valuablc cottlribttliotts to lltis t ltitDtt:t lty l)tolcstrttl'D. A. Rccd.

404

il.r l)l ti(;llll,ll()N ()l wlNl) lo^t)tN(l 405

As usual, il is eonvcrricrrl lo tlcscrifu (lrc l)lcssurr:s in tcrrrrrs ol'lhc surn ofir nlcAn and a lluclttltittg prrtl .

11.1.1 Mean Pressures'l'hc rlcan pressurc at a point clolinccl by thc hcight above ground z and theirrrgular coordinate d (Fig. I l.l.l) can be expressed as:

pk.0l : jplCne. ilu'(d + CpiU2(H)l (11.1. 1)

wlrorc p is the airdensity (p = 1.25 kg/m3), U(z) is the mean wind speed att'lcvation z in the undisturbed oncoming flow, Co(2,0) is the mean externalf rcrisure coefficient, F1 is the height of the tower, and Coi is the internal pressurecocllicient. Based on results of full-scale measurements, [11-2] suggests Cpr =0..1.t'The following tentative relations, based on wind tunnel and full-scalerrrilsurements, have been proposed for the external pressure coefficient Co(2,0r I I l-31:

Cr(2, 0) = |coQ,o) - I -

0 : oo (ll.l.2a)

0<0<0b (11.1.2b)

0 > 0u (l l.l.2c)

(11.1.2d)

B sinc (r fr)Cok,0) : CoQ,06)

B=1I AC,

FIGURIi I l.l.l. llyltcrltolic ctxrlittg towcr-noliltions.

'tWirxl lttrrrre l rncilsurr:nrcllls <;trolcrl itt l l l .t l sttltpcst lltrrl sornt'wllrl lrigltt:r irtlt'rtritl l)l('sslrr('srrt (r' irr rrrrvcnlc:rl lowcrs; lltrrl is, ( ),, () t ut r'vr'tt (l (r

04 (y)'" +

406 HYPEIIUOLTC C(X)UN(i tOWl ilri

oo

4 6 8ro-2 2 4 6 810-t 2

ROUGHNESS COEFFICIENT k/aFIGURE ll.l.2. Approximate pressure difference ACo as a function of roughnesacoefficient kla for towers with 36 to 144 ribs. After H. J. Niemann, "Wind Effects onCooling-Tower Shells," J. Struct. Div., ASCE, f06 (1980), 643-661.

lnBC- (11. l.2c)ln[sin 90(0"/01)]

where ll : height of tower, ACo is a function of the ratio kla of the rib height,k, to the distance between ribs, a, represented in Fig. 11.1.2, a is an exponentcharacterizing the mean wind profile (Table 2.2.2), and the angle 0 is expresscdin degrees. The angles 0o,0r, and06 are represented in the schematic pressuredistribution diagram of Fig. 11.1.3. Values forthese angles are given in Fig,ll .l .4a (based on full scale measurements on the Schmehausen tower) and inFigs. ll.l.4b and 11.1.4c (based on wind tunnel measurements) [ll-20].Numerical Example Assume that, as in the case of the Martin's Creek towcr,H : 127 m, kla = 0.02, and cv = 0.17. We seek the values of Cnk,0) lbrz : 95.4 m and 0 : 35',70", and 97".

From Fig. ll.l.2, LCo - 0.65. For z:95.4 m, Fig. ll.l.4q yields 0,,

= 35",0r ='loo,and06 = 97".Itfollows thatB :2.1 , C:2.14 (Eqs.1l.l.2d and ll.1.2e). From Eq. 11.1.2b, Cp(95.4 m,35") = 0, Cr,(95.4 rrr,70') - -1.1, and Cp(95.4 m,97") = -0.38.

values of the external pressure coefficient c,, at thc towcr throat obtainotlfrom full-scale measurcmcnt by a numbcr of invcstigirl()rs ltro sh<lwn in lrig.

r r I t)t li(;t ilt ' I t( )N ()t wtND I c)Al)tNCi 407

maxCo

l,'l(iURE 11.1.3. Distribution of pressure coellicient Cr,. After H. propper and J.Wt'lsch, "Wind Pressures on Cooling Tower Shells," in Wind Engineering, proceed-rtt,tl,s rf' the Fifth International Conference, Fort Collins, CO, July 1979, J. E. Cermak(rtl.), Pergamon Press, Elmsford, NY, 1980.

I 1.1.5. Note that the values obtained for the tower of [11-6] differ appreciablyIrorn the other sets of values. This is due to the absence of ribs on the externalsrrrlace of that tower. Note also the agreement to within about 15% betweentlre values obtained in the numerical example and the values measured on theMrrrlin's Creek tower at the throat elevation z : 95.4 m [1]-2]. Figure 11.1.5llso shows an example of differences between values of e obtained from art't of wind tunnel tests on the one hand and full scale measurements on theollrcr.

11.1.2 Fluctuating Pressuressllcsscs induced by fluctuating pressures are usually comparable in value to\llcsses induced by the mean loads. The purpose of this section is to presentr lcscriptions of fluctuating pressures for use in the estimation of tower response.Atltlitional information on fluctuating pressures is presented in [1]-5] andlr 22l.

RMS of Fluctuating wind Pressures. The rms of the fluctuating wind pres-',rrlt's. o,,(2. 0). may bc wrillcrr irs

ot,Q,,0l \1t{'i,8.,0\ll;(:.1 (ll.l.3y

tvltt:ro p is lhc air dcnsity, l/(.:) is llrc rrrt'rrrr wirrtl slrerertl trl clcvirtiorr;, irrrtlt'i,l:.,0) is an cttrpilit:itl llttclttttti!lg prt'Hriur(r r'ocllit'icrrl. Alltrrrrltls lo rclirlc

N

o()

lr

408 HYI)FRBO| tc coot tNG rowFil$

(a)

0oot0o

Aco = 9.6gd =0.13-O.17

(b)

7500

1000 1250

1.0

* o.u

0

Aco = 9.7td = o.18

0o

(c)

01 0b

[\25o 5Oo 75o jOOo 1250

0FIGURE ll.l.4. Angles 0o, 0,, and 0u (after tll_201).

!q\r, 0) to the turbulence intensity of the oncomingIl-3] and [11-8]. According to itt_:1,

Ci,tz, ol = 1.8 o'' u(zt

flow have been reported in

(11.1.4)

where o, is the rms of the longitudinal velocity fluctuati.ns. Thc variation 'r.Ci'Q,0) with d at the elevation of the throat is sh<lw' ftrr.tlrccr scts of mcir-surements in Fig' ' . 1 .6 tr.r ?l) According k) r r r -3 r 'r*r I r r r r i, rhis vari-ation depends upon thc ratio krir, whcrc t is rhc hciglrr or. lrrt. r.irrs irnrr /) is

lr I l)l !i(;llil'ilt)N ()t wlNt) t()nt )tN(i 409

Numbrlrol rlba

o MaominA Weisw€iler 52 8.bx 1O -3

v Martin's Croek 84 2.2x1O-2oscfrnehausen 144 2.3x1O-2- - Wind tunnel

600 900 1200 1 500

Ro5.4x 1O '6.5x 1O 7

1x1O84-6x1O7'| .6x 1O 5

1 800

Ret1 1-61 1-41 1-2-11-4

11-7

tiIIII

lrl(;uRE 11.1.5. Mean pressure coefficient around the throat section of hyperbolicr'rxrling tower for four full-scale data sets and one wind tunnel set. After Tien-fun Sun;rrrtl Liang-mao zhou, "wind Pressure Distribution around a Ribless Hyperbolic Cool-irrg'fower," J. Wind Eng. Ind. Aerodyn., f4 (1933), l8l_192.

0.4

0.3

C;

0.2

0.1

00 30 60 90 t20 150 180

Degrees

lrl(;IIRli ll.l.6. F'lLrctrrlting l)ltssllr c'rrcfiicicnt arouncl thc throat ol'a hypcrbolict'rxrling towor.

-.-.- Full-scale [1]_8]Full-scate [1 1-9]Model [1 1-10]

,'n'l' ,,r \

\.-a7/ \

l.-1.-:-

410 HYPERBOLIC COOLING IOWI-I1!i

k/D = 5.4x1O-a

nro=r.r-to-lFl/--- --l' k/D=5x10 "

o 3Oo 600 goo .l 2oo 1500

0

FIGURE 11.1.7. Ratios Ci(2, 0)lci,k,0) for towers with various roughness param-

eterc klD at elevation z - 0.7 F1. From H. Propper and J' Welsch, "Wind Pressures

on Cooling Tower Shells, " in Wirut Engineering, Proceedings of the Fifih InternationalConferenit, F.rt Collins, CO, July l97g' J. E. Cermak (ed'), Pergamon Press' Elms-ford, NY, 1980.

the diameter of the tower at the throat. Note that the coefncients Ci are lowerfor the rougher than for the smoother towers in the region 60' < 0 < 120'(Fig. 11.1.7).

Spectra of Fluctuating Pressures.of fluctuating pressures was proposed

The following expression for the spectrain [11-3]:

1.0

OloNIN ^-9lv U-Col oolo

1 800

nS,(2, 0. n)o?Q,0)

(11.1.s)

(1 1.1.6)

(11.1.7)

(11.1.tt)

where

| - a'@\d(0): ,'YpQ):[m]"'"'

| ., / D\''''ltuttqt n,xog) : lb;'"(o) (;/ | ,^

where n is the frequency, the parameters a(0), bs(0), and Bo(0) are given in

Fig. 11.1.8, a is the power law exponent (Table 2'2'2), D is the diameter at

thJ throat, and Ii is the integral scale of turbulence (Sect' 2'3'2)'

Cross-spectra ol Ftuctuating Pressures- According to lll-l2l' quadraturcspectra are negligible; that is, the cross-spectra arc ltlr pntclicitl prlrposcs cquillto the co-speitra. Thc lirll<lwing rclations wcro pf()porictl irr lll-l2l lirr the

cross-spcctra <tl' lho pn:ssurc lluctuations:

,/-)) k/D:ok/D =4"5x lO

k/D=2"Ox1O-3 -\

il t t)t licltll'li()N(l, WtNt )l()Al rlN(i 4ll

?.o

1.5

LO

o.5

o

4.O

3.O

2.O

1.O

o

lfr

ir,.t

Ii

o 300 600 9000

1 200 1 800r 50.

a = O.27

1--- -\L0 = O.18

o 3oo 600 90o 12Oo .tsoo 18Oo0

l,'l(;URE 11.1.8. Parameters a,(0), bo(0), and B,,(0). From H. Prcipper and J. Welsch,"Wind Pressures on Cooling Tower Shells," in Wind Engineering, Proceedings of theliilih International Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.),I't:rgamon Press, Elmsford, NY, 1980.

l. Windward region (0 < 100',9' < 100'),

So(2,0, Z',0', n) : R,,(2, {, n)Ry(O,0', n)S}/z(2,0, nlsto/212,,0,, n)

(11.1.e)

2 Leeward region (0 = 100', 0' = 100'),

SoQ, 0, z',0', n) : R,(2, z', n)R,(O,0', nySltz(2,0, n1stotz1z,,0,, n)

(1 1.1.10)

At'cording to [l1-12], cross-spectra of pressures on the windward region, onllrc one hand, and pressures on the leeward region, on the other, are negligible.'l'his is a simplifying assumption that is not entirely consistent with resultsre lxrrted in [ 1-3].

ln Eqs. ll.l.9 and 11.1.10,

R,,(2., z.' . z) : exp( -p,i)R,(0, 0', rt) cxp(- Az.fz)

R1(),0'. tt\ (',(0,0'\R(10 * 0'1, n)

(il.l.lt)(lr.r.r2)(ILI.1.1)

h/l) =4 br ll)k/l): o

412 HYPERBOLIC COOLING TOWEI|ti

c2(0,0')

0 =12Oo

9= 18Oo

FIGURE ll.l.9. correlation coemcients c2(0,0'\. From H. pnipper and J. welsch,''wind Pressures on cooling Tower Shells, " in wind Engineering, proceedings of thaFifih International conference, Fort collins, co, July 1919, J. E. Cermak (ed.),Pergamon Press, Elmsford, NY, 1980.

R(lo - o'1, n1 :

i,

where P1 = 7, 0z = ll, 0t =the gradient height 6 listed inll 1-31, is given in Fig. I I . I .9.

it:exp(- tszf|)nlz - z'l

u(6)

l0 - 0'ltnD' '360'u(6)

25 U1-121, U(6)Table 2.2.2, and

(11.1.t6)

is the mean wind speed utCz(0, 0'), as obtained in

(1 1. l. l4)

(11.1.15)

11.2 ESTIMATION OF TOWER RESPONSE

Several approachcs ttl thc cstirnation ol'towcr rcspolrse lurvt. bcrrlr prlpgsctl,For towcrs that cxhibit no sigttilicanl rcsor)iull irrrrplilir'rrtiorr cllircls, lll-71

- 1.0c2 ( 0, 0',

600

c2(0,0',

60.

il 1' t silrMAlt()N ot lr)Wl tt ttt tji,(lNlit 413

t'rrtPltlys cxprcssirttts lol tlrc ntcritlional irnrl cirr,'rrrrrlt rcrrlirrl t'otrt-lrrtions ol llrcllrtt'itraling prcssttros to obtain tho variarrecs ol'llrc rrrt.r'itliorrrrl, cilt'iiruli'lcrrliirl.;rrtl n<lnnal displaccmcnts ol'tho towcr shr:ll.'l'his approach was superscdcd hy lll l2l, whiclr cnrpl,ys u spcclrirl irp=lrloitch in which models of spcctra artrl c:nrss-sprrctrit ()l'ptt:ssiu'c lluctuir(ions(st'c Scct. ll.l) are used to obtain thc spoclnrl rlcnsilics <ll'thc rcsponso byrrrclh<rds fundamentally similar [o lhosc ol'Sccts. 5.2.7 anrJ -5.3. 'l'hc spcctralirppnrach is applicable to towcrs ltlr which resonant amplification effects arerrlinificant, as well as to towers which-as is most commonly the case-arer.rrlliciently stiff that resonant amplification effects are negligible. In both casestlrc calculations can be carried out by using a computer program similar to thatlrslcrl in [9-14], but modified to account for differences in geometry and in therrrotlcling of pressures, as well as for the fact that a typical response of theItrwcr, rather than having the form of Eq. 5.2.1, is written as

y(s, 0, t) I [q.,,(r).os m0 + qi,,,(t)sin mTly^.,(s) (11.1.7)

wlrcrc s is the distance along the meridian, 0 is the angular coordinate, r is thelntrc' qm.i and q'.,, are the time-dependent symmetric and antisymmetric gen-rlirlized coordinates for mode m, i, respectively, and j^.i3) is the vertical modalslrrrpc, An attempt to use a spectral approach to estimate the response was alsott'lxrrted in [1]-13].

ln [l1-14] and Il-15] finite element methods of analysis are used in con-[rilction with step-by-step integrations in the time domain. one advantage ofsrrclr an approach is that it can accommodate nonlinearities and changes of the|hysical properties of the structure during the loading process. Time histories,l lluctuating pressures used in this approach can consist of measured data, asrrr lll-l4lx and Il-15], or can be simulated by Monte carlo methods fromrlrcctral and cross-spectral information. More recently, ARIMA (Auto Regres-rrvtr Integrated Moving Average) methods have been used for representingllrrc(uating loads in the time domain tll-161. Time-domain solutions, thoughgrotcntially useful for research purposes, are costly and may be impractical forrorrtine design.

spectral methods, as developed in [1]-12], were applied in [1]-4] to studyllre rcsponse of typical reinforced concrete towers with ratio HID :2.0 (D :rlrrrnrctcr at throat). The results obtained indicated that the resonant amplifi-.irlion cffects contributed less than 57o to the total response. A typical diagramol thc ratio N11lq,at the stagnation point is shown in Fig. ll.2.l for U(e,n.ou,)

45.4 mls (N' I is the mcridional stress, q, : (ll2)pUr(z,n^,o,), p is the air(l('tlsily, and U(2,1.,,,,,) is thc rncan wind speed at the clevation of the towcrllrroirl). It is secn that in this clsc tlrc pcak total responsc dillbrs insignilicantly

rlk'r':tttsr: tttcirsutul tl:rlrr wclr. irvtriltthh. ully lor lll(' tllroill sccliotr, il wirs itsstrrlt.tl rrr lll l4ltlr:rt llrr' verlit'irl tlisllilrrrliolr ol llrt. hrrrrlr rr rrrllorlr.

_\-m

414 HYPERBOLIC COOLING TOWERS

FIGURE ll.2.l. Ratios of meridional stress,N1 1, at stagnation point to dynamic pressure, 4r,at elevation of tower throat. After H.-J. Nie-mann, "Wind Effects on Cooling-TowerShells," J. Struct. Div., ASCE, 106 (1980),&3-66t.

*")o",," '

mean

180

160

't 40

't20oo6 1OOE*Bo

soo 1000Nrr/gt (meters)

from the peak quasi-static response (obtained by neglecting resonant amplifl'cation effects). The latter is approximately twice as large as the mean response,

It is shown in [11-4] that for the type of towers studied therein, the designmay be based on an equivalent static pressure

Pk. 0) : Cok.O)qokl (11.2.l)

where, in open country,

qp(z) = lipu2(to)l$ (rt.2.21

where 6 = 280 m, p is the air density (p = I.25 kg/m2), U(10) is the hourlymean wind speed at 10 m above ground, and @ is a factor accounting forresonant amplification effects (1 < d < 1.1).

An equivalent static pressure approach is also included in [11-17], in whichthe expression for the equivalent pressures is consistent with the format usedfor dynamic pressures in the American National Standard A58.1-1972 tll-181,Reference tl1-171 recommends the use in this expression of aerodynamic coef'ficients obtained from wind tunnel or full-scale tests, and of a gust loadingfactor to be determined by a dynamic analysis.

The use of a single gust loading factor implies that the stress amplificationdue to wind gustiness may be considered for practical purposcs to be the samgat all points of the towerand forall types of stress. As shrtwu in Ill-231, thitiassumption is not necessarily correct in all cascs.

7-10.23

'(;)

'llil ? FStiMAiloN ot towt tt nt fit()Nst 41S

Itl(;uRE 11.3.1. Tower locations at Ferrybridge c Generating station. From J. Ar-rrritt, "Wind Loading on Cooling Towers," J. Struct. Div., ASCE, f06 (1990),623_()41.

o

ftr416 ilyt't rtll()t t(: (.()()l lN(i l()wt ttl'

11.3 GROUPS OF HYPERBOLIC COOLING TOWERS

Wind-induced stresses in the tower shclls can bc considcrably trrorc scvclc irt

the case of groups of towers than for isolated structures. This was httrttt: ttttlby the behavior during the November l, 1965, stormx of thc cight lowcls ttlthe Ferrybridge C Generating Station (Fig. 11.3.1), three of which collapst:tlwhile five survived. The inquiry of Il-l] indicated that failure was duc lolarge tensions in the windward face of the towers. On the basis of wind tunne I

tests and of infbrmation on the design of the towers, it was estimatcd irrtll-l9l that the mean hourly wind speeds at l0 m above ground, U(10). rtl

which failurc of thc towers could be expected to occur had the values showtt

'l'Alll,l,l I l.-1.1. listinratcd Wind Speeds Corresponding to Tower Failures (m/s)il l-1.)l

l,l(JURE 11.3.3. Ratios of stresses amplified by interf'erence effects to corresponding'.rresscs on isolated tower (d, is the diameter at throat: d3 is the diameter at base). Afterll L Niemann, "Reliability of Current Design Methods for Wind-Induced Strcsses"rn Nutural Drafi Cooling Towers, P. L. Gould, W. B. Krdtzig, I. Mungan, and U.wirtck, (eds.), Springer-Verlag, Berlin, 1984.

ru'l'able ll.3.l. The wind speeds U(10) during the storm were reported to risellrrn about l8 m/s to about 20 m/s. The reported sequence of tower failuresrv;rs fbund to be consistent with the results of Table tl.3.l [ll-19].

It is noted in [1-19] that higher mean and fluctuating loads ofien occurt'lrcn the wind blows through a gap between upstream towers. Details on,listributions of mean fluctuating pressures on the surf'ace of towers placed intlrc wake of other buildings or in groups are given for specific configurationsrrr lll-l9l on the basis of both wind tunnel and full-scale measurements, andrrr I I l-81 on the basis of full-scale tests.

Stress amplifications due to interlerence eflbcts can also occur in the case,rl lrairs of cooling towars (Fig. I1.3.2). Laboratory data on such amplifications.rre shown in Fig. 11.3.3 for various wind directions and distances betweenllr('towers in a pair (max n11 is the maximum hoop, max n22 is the maximumrnt'r-iclional tension, min n22is the maximum meridional compression, max ln12lr:; thc maximum shear force). It is seen that in some instances the amplifications,rri'considerable (over 3O%).k is noted in Il-23] that cooling towers can alsoI't'uclvcrsely affected by the presence of adjacent buildings within a powerpl;urt.

REFERENCES

ilt IIllt iliI

"'iil ;"' 2 ee q =rn,iil;,7 177 dBloA ,---\ I(o)'

Tin""/ Y/ li";^"

4l I

l{},)m; r\;tl5

(( '"))\i:yi/

, .,.'. '(( ))r\.....-, / |

.lt)t, ,lr.I .1/ (trl

/ ..\ I(( ))t\z/l*, nn

|"u,*"/

Tuwcru(10)

IAt9. t

IBt9.l

2A19.1

2823.4

3A21.5

3B23.8

44, 4l]21.6 21..1

FIGURE 11.3.2. Cooling towers, Lin-rerick Gencrating Sllliotr, l.irrrcrick, Pcnrrsylvania. Courtesy of Philadclphia Elcctric Cornpany, l,irrrclick ( ie rrt'r'rrlirrg Strttiott.

tThc approxitultlc nlciur witttl tlittt'lion is sltowrt irr lrrli II I I

ilt

il.1

Ilcport of'thc Cotttrtritt,'t'rl lrrtluiry intrt thc Cttllrtlt.sc t2l (ixtlitrg'lltvt.'rrs trll"trr.l,britlgt otr Mttrtlttv, I Not't'ttrlrcr /9fi.5, ('cntrul l')lcctl-icily (icnerirlirrgIllolrtl, Il.M. Slitliottrrrv ( )llicr'. l.otttlott, l9(r(r.N. .l , Sollcnlrcrgt'r. l{ ll S,:rrrl:rt. :rtttl l). l). llillirtglorr. "Wirrtl l,o:trlittl' :ttttlllt's;xrrtst'rrl ('txrlttrl, lrtrltr:,. ' .l ,\trtttl.1)lr'., AS('lt, l(Xr (l()li0). (rOl (r'l

418 HYPEntsoLtc cooltNc town*i

ll-3 H. Pnipper and J. Wclsch, "Wirrtl l)russurcs on C)rxrling'lirwcr Shclls." irrWind Engineering, Prcceedings ol'thc Filih lntcrnational Conl'croncc, liort (lol-lins, CO, July 1979, J. E. Cermak (ed.), Pergamon Press, h,lnrslord, Ny,1980.

Il-4 H.-J. Niemann, "Wind Effects on Cooling-Tower Shells," J. Struct. Div.,ASCE, 106 (1980), 643-66t.

l1-5 J. F. Sageau, In Situ Measurement of the Mean and Fluctuating Pressure Fie ldsaround a 122 Meters Smooth, Isolated Cooling Tower, Electricitd de Francc,Direction des Etudes et Recherches, 6 quai Watier, Chatou, France, Sept. 1979.

1l-6 T. F. Sun and L. M. Zhou, "Wind Pressure Distributions on a Ribless Hy-perbolic Cooling Tower," Proceedings 6th International Conference on WirulEngineering, Gold Coast, Australia, inJ. Wind Eng. Ind. Aerodyn., f4 (1933),18r-r92,

ll-l S. H. Abu-Sitta and M. G. Hashish, "Dynamic Wind Stresses in HyperbolicCooling Towers," J. Struct. Diy., ASCE, 99 (Sept. 1973), 1823-1935.

1l-8 J. F. Sageau, Caract€risation des champs de pression moyens et fluctuants dla surface des grands airorefrigdrctnrs, Electricit6 de France, Direction desEtudes et Recherches, 6 quai Watier, Chatou, France, July 1979.

1l-9 H. Ruscheweyh, "Wind Loadings on Hyperbolic Natural Draught CoolingTowers," J. Ind. Aerodyn., I (1976),335-340.

I l-10 A. G. Davenport and N. Isyumov, The Dynamic and Static Action of Wind onHyperbolic Cooling Towers, Vol. 1, Research Report No. BLWTI-66, Univ.of Western Ontario, London, Ontario, Canada, 1966.

l1-ll M. Pimer, 'Wind Pressure Fluctuations on a Cooling Tower, J. Wind Eng,Ind. Aerodyn, 10 (1982), 343-360.

ll-12 M. G. Hashish and S. H. Abu-Sitta, "Response of Hyperbolic Cooling Towersto Turbulent Wind," J. Struct. Div., ASCE, f00 (1974), 1037-1051.

l1-13 M. P. Singh and A. K. Gupta, "Gust Factors for Hyperbolic Cooling Towers,"J. Struct. Div., ASCE, 102 (1978),371-386.

1l-14 P. K. Basu and P. L. Gould, "Cooling Towers Using Measured Wind Dara,"J. Struct. Diy., ASCE, f06 (1980), 579-600.

1l-15 R. L. Steinmetz, D. P. Billington, and J. F. Abel, "Hyperbolic Cooling TowerDynamic Response to Wind," J. Struct. Diy., ASCE, f04 (1978), 35-53.

ll-16 D. A. Reed and R. H. Scanlan, "Cooling Tower Wind Loading," in pro-ceedings of the 4th U.S. National Conference on Wind Engineering Research,Department of Civil Engineering, University of Washington, Seattle, July 26-29, 1981, Vol. 1, pp.254-261.

ll-17 Reinforced Concrete Cooling Tower Shells-Practice and Commentary, ACI334, lR-71 , American Concrete Institute, Derroit, Michigan, 1977.

I l-18 American National standard Building code Requirements for Minimum DesignLoads in Buildings and Other Structures, A58.1, American National StandardsInstitute, New York, 1972.

ll-19 J. Armitt, "Wind Loading on Cooling Towers," J. Struct. Dly., ASCE, 106(1980), 623-64t.

ll-20 J. Welsch, Der Einfluss des Windprc{ils aufdia stu!ix'lt,rt Witullnttttspruthuttgen von rotationshypcrfutlischcn Ktihllurmschulcrr, l,clrlsltrlrl I, lnslitut liir korr.struktivcn Ingcnicrrrhau, Ruhr-Univcrsitiit llochutrr. l t.lrnrruy l t)ll{.

trt l tit N(l ll 419

ll2l I). A. l{ecrl irrrrl li. Sirtriu, "Wintl l,oarls rur ('rxrlinp'lttwt't's," l)rrrlt Slrrlc oltho Art l{cpol't on Wintl lill'ccts ort Sintcltttcs, ('ontntillr't'ott Witttl lilli'c'ls,Arncrican Strcicty ol' (livil linginccrs, l9ll4.

ll 22 Y. Kawarabata, S. Nakac, and M. Haracla, "Srttuc Aspccts ol'thc Wilrtl l)csigrrol'Cooling Towcrs," J. Wind ling. lrul. Aanxl.yrt. l4 (l9tl3), 167 lti0.

ll73 H.-J. Niemann, "Reliability ol'C,'urrcnt l)csign Motlrods lir Wind-lnclucotlStresses," in Natural Drrsught Cutling'l'owers, Procccdings tl'thc 2nd Intcr-national Symposium, Ruhr-Bochurn, Gcrmany, P. L. Gould, W. B. Kr:itzig, l.Mungan, and U. Wittek, cds., Springer-Verlag, Berlin, 1984.

xCHAPTER 12

TRUSSED FRAMEWORKS ANDPLATE GIRDERS

Trussed frameworks subjected to wind loads have routinely been used in struc_tural engineering applications for more than a century. Nevertheless, the statcof knowledge conceming the effects of wind on thii type of structure is stillimperlect and provisions concerning such effects included in various standards,codes, and design guides are in some cases mutually inconsistent and in clis-agreement with experimental data ll2_ll.

For any given wind speed, the principal factors that determine the wind loaclacting on a trussed framework are:

o The aspect ratio \, that is, the ratio of the length of the framework to itswidth. If end plates or abutments are provided,lh" flo* around the frame_work is essentially two-dimensional, so that for aerodynamic purposes thclength of a framework may be considered to b" infinit".

o The solidity ratio @, that is, the ratio of the effective to the gross area of'the framework-x For any solidity ratio s the wind load is-for practicarpurposes independent of the truss configuration, that is, of whether adiagonal truss, a K-truss, and so forth, is involved.

o The shielding of portions of the framework by other portions locateclupwind. The degree to which shielding occurs depends on the configu-ration of the spatial framework. If the framework consists of parailcr

*The effective areas of a plane lruss is the area of the shadow pnrjccrctl by ils rrrcrrrhcrs .. :rplane parallcl to the truss, thc pnricction bcirrg norrnal kl thl{ plrurc. 'l lrt' 1,1i;.r :rrr.;r .l :r pl:r'ctruss is thc arca ctlntlinctl within thc oLrtsitlc conlorrr ol' (lurl tlrrss. Ilrt. ,lll,.r.tir,,. lrrt.:r lrrrtl llrt.gross area of-a spalial It:ttttcwrtrk itte rlclirrctl, rcsyrr't'livcly, ;rs llrt. r'llr.r trrr. rrrr.;r .rrrrl tlrt. g.rssarca ol'ils upwintl Ilrcc.

420

l:, 42 I

ltltsri('s (or'1r,i1111'1s1, (lrc slrit'lrlirr11 tlt'pt'rrtl:; orr llrt' rttttttlrt't lrlrtl slttrciltg tll'tlrc tt'ttsscs 1ol ginlcrs).

. 'l'lrc slr:r;lc ol llrc rrrr-:rrrbcrs, tlurl is, wlrt'llrcr tlrc tttctttbcr are rounded orlurvc sltrrp ctlgcs. Iiorccs orr lorrrrtlt:tl rrctttbcts dcpcnd on Reynolds num-bcr 61" ancl on lhc rougltttcss ol lltc tttotttbcr surface (see Fig. 4.5.5). Fortrusscs with sharp-ctlgctl rttcrttbcrs the elI'ect of the Reynolds number and<ll' the shapc ancl surlacc rtlughness of the member is, in practice, negli-gible.

. The turbulence in the oncoming flow. As noted in Sect. 4.5, the effect ofturbulence on the drag force acting on frameworks with sharp-edged mem-bers is relatively small in most cases of practical interest ll2-2, l2-5,12-141. A similar conclusion appears to be valid for frameworks composedof members with circular cross section in flows with subcritical Reynoldsmembers. For this reason, and owing to scaling difficulties, in most caseswind tunnel tests for trussed frameworks are to this day conducted insmooth flow [l2-l , l2-5, 12-6].

o The orientation of the framework with respect to the mean wind direction.

l'his chapter reviews the aerodynamic behavior of trussed frameworks and plateliirders, including single trusses and girders, systems consisting of two or moreyrlrrallel trusses or girders, and square and triangular towers. Test results areoltcn presented from several sources with a view to allowing an assessment ofllrc errors that may be expected in typical wind tunnel measurements. Through-rrrr( this chapter the aerodynamic coelicients are referred to, and should berrscd in conjunction with, the effective area of the framework, A1 .

Wind forces on ancillary parts (e.g., ladders, antenna dishes) must be takeninto account in design in addition to the wind forces on the trussed frameworksllrcmselves Il2-1 , l2-l7l. Drag and interference effects on microwave dishirntcnnas and their supporting towers were studied inU2-271. Drag coefficientslirr an unshrouded isolated microwave dish with depth-to-diameter ratio 0.24wcre found to be largest for angles of 0 to 30 degrees between wind directionrrncl the normal to the dish surface, and are almost independent of the flowItrrbulence (Co = 1.4). For a single dish the ratiofo between the incrementaltotal drag on the tower due to the addition of a single dish and the drag for theisolated dish depends on wind direction, and it is higher than unity (as high as1.3) for the most unfavorable directions. This is due to flow accelerationsrrrtluced by the dish. As more dishes are added at the same level of a tower,irrlcrf'erence factors are still greater than unity, but tend to decrease as thenrrrnber of dishes increascs. According to |2-271 an empirical formula for theinlcrl'crcnce factorgivcn in ll2 2l'll is loo krw by a factorof more than two fort't'rlirin wind directions; utr ltlte I'rutlivt' lirrrrrttllr is proptlscd in l12-271.

Akrng-wincl cl'l'ccts on (()w('r'r.i rrr;ry lrt't'slinr:rtt'tl lry rrsirrg procctlurcs suclt:ts wcrc rliscussccl in Sccl. 9.2.1 'l'lrt' tlt'r,t'lopttrt'rttlrl ('()lnl)ulcr bltsc:cl vt:rsittttol'tlrc AS('li 7 ()5 Stlntliu'tl ;lnrvisrorrs u:,('r,:,u( lr rr pror't'tltrt'lirr lltrxiblt'l()w('r's(ll7 5l stc tliskt'lttr:tppt'rtrlt'tl lo lltrs lrool.)

t422 ilttit;l;t t) tnnMt w()t il\ii nNt) t,l /\il (,ilil)t rr:;

Ref-erence U2-291 rop()ns lirll-scirlc nlcirsurcnrL'nts ircconlirrg (o wlriclr ircrosswind effects on square towers with anglc rrrcrrrbcrs arc conlllaritblc (o rrlorrgwind effects. It proposes a semiempirical proccdurc lirr cstintating lolsiorurleffects, which are due largely to the presence of eccentrically locatod antonnirdishes.

For studies on wind effects on cranes and guyed towers, see [12-l-5 1,

|2-161, and [12-17] to 112-261, respecrively.

12.1 SINGLE TRUSSES AND GIRDERS

Figure 12. I . I summarizes measurements of the drag coellicient C$) tor a singlctruss with infinite aspect ratio normal to the wind. The data of Fig. l2.i.lwere obtained in the 1930s in Gottingen for trusses with sharp-edged membcrsl2-2, 12-31,* and in the late 1970s at the National Maritime Institute, U.K.(NMI), both for trusses with sharp-edged members and trusses with membcrsof circular cross section.i It is seen that differences between the Gottingen antlthe NMI results for frameworks with sharp-edged members do not exceed l5/,,or so. For single trusses normal to the wind and composed of sharp-edgctlmembers, ratios Cg)(},)/Cg)(\ : o) of the drag coefficients corresponding roan aspect ratio \, on the one hand, and to an infinite aspect ratio, on the othcr,are shown in Fig. 12.1.2 tl2-31.

Drag coeflicients C!j) reported in [12-5] for trusses normal to the wind.composed of sharp-edged members, and having aspect ratios l/6 < \ < 6,are listed in Table l2.l.l. Also listed in Table l2.l.l are values C$)(X : *lobtained from the drag coe{ficients of [12-5] through multiplication by thcrappropriate correction factor taken from Fig. 12.1.2. It can be seen that dil'ferences between the values C!'(\ : oo) based on U2-51 and the correspondingFlachsbart U2-2, 12-31and NMI values of Fig. 12.1.1 do not exceed 20%.

Figure 12.1.3 [12-7] summarizes results of tests on trusses with membcrsof circular cross section (x : -) conducted in the subsonic wind tunnel ltPorz-wahn, Germany [12-8] and in the compressed air tunnel of the NationarPhysical Laboratory, U.K. tl2-101.+ Note that for Reynolds numbers G" <lOs the drag coefficients in Fig. 12.1.3 differ by about 5% or less from rhcrcorresponding results of Fig. l2.l.l.

A framework whose solidity ratio is 6 : I is a solid plate (or a girdcr).

*References tl2-21, 112-31, and [12-41 are available in English as Building Research Estahlislrment Library Translation No. IT2202, Building Research Station, Garston, watford, U.K.lThe NMI measurements for trusses with members of circular cross section ret'erred to in tlrischapter were carried out at Reynolds numbers 104 < 61" < 7 x 101 [12-61.+Figures 12.1 .3 and 12.4.5 to 12.4.8 are reproduccd with pcrmission oI CIDEC'I'(Corniti Irrtcrnational pour le D6veloppement et I'Etude clc la Construction Tulruluirc) |ront llirul [iprct,.s rtrtUncladTubular Structurcs, H. B. Walkcr (cd.), Constrado Publit:rtiorr l/75. ('onstructiorrirl SlcclResearch and Develttptttcnl Orgirnization, ('nryrlon, ti.K., 1975. Ilrr.y ;rrt. lr:rst'tl irr plll orrrcscarch work carrictl oul by ('ll)li("1'irrrtl n:ylrrlr:tl irr ll2 ltl ln(l ll.'(rt

l;,t :;lN(,ll

--.._._--. o

.+ =- t-l Angle-sectionmembers

_- ))

I norno section members

lllllllllllllll ---* [ nectaneutarmembers

0 0.1 0.2 0.3 0.4 0.5 0.6 01 0 8

Solidity ratio ro

lrl(;URE l2.l.l. Drag coefficient C$) for single truss, \ : o, wind normal to truss.linrm R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experiencedlry Arrays of Lattice Frames," in Wind Engineering, Proceedings of the Fifih Inter'rttttional Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.), Vol' 1, Per-y'rrrrron Press, Elmsford, NY, 1980, pp. 405-420.

l'lrc drag coefficient coffesponding to wind norrnal to the plate can be obtainedlrrrrn Figs. l2.l.l and 12.1.2. Additional information on the aerodynamiclrrrhavior of rectangular plittt:s is givc:n in Sccts. 4.5 and 4.6.

ll was shown in Scct.4.(r llr;r( llrc lrt'rorlynlrtttic lirrcc norlnal t<l a rcctangularplirtc with aspect ratirl \ - 5lrt lO is lrrlllt'l wltt'lt llte yitw:tttglc'r'is rv -'1O"tlurl il'lhc winrl is nortttirl lo llrt'pl:rlt'1l'r1i '1.(r. l). llowt'vt't, lirr lrrrsses willr

,,l,lfl:]/,,*;rrrglt'is lltr'lrolizorrl;rl,rrr1'1, l',1\\{(rt lll rrr,.rr rrtttrl rlttr'rlloll.ttrrl llr ttotttt;tl lrt lltt'

-Qi tad '.OEooo 1.0b0oo

0.8

0.6

0.5

1/^

FIGURE 12.1.2. Ratios C!j,)(\)/Ctj)(\ : o), wincl nonnal to truss [12-3].

TABLE l2.l.l. Drag Coefficients for Simple Trusses

o.t4 o.29 0.4'7 o.77

l.l8 + 5% 1.28 I r*

r:':,r'nllr:;()l lltl ,:;1 ,l :,ntlll{)l l'l All (;llll)lll:: 4?:i

I | ' I'l I llll I I I I t l' | | ll ll ' I3 1 s 6 7 E 9 105 2 a 4 s 6 7 8 9106 Rg 2

I, l(;URE 12.1,3. Drag coeflrcient C!i)forsingle truss with members of circularcross',r'tlion, X : o, wind normal to truss [12-7] (courtesy Comitd Intemational pour lel)('vcloppement et I'Etude de la Construction Tubulaire, and Constructional Steel Re-'.,'rlch and Development Organisation).

cB): clj)(v, * vrr) (t2.2.r)

liigure 12.2.1 shows values of Vr and Vyy, repofted in [12-41 for three types.l truss, all with sharp-edged members and infinite aspect ratio, as functions,,1 thc solidity ratio d, and of the ratio between the truss spacing in the along-rvirrd direction, e, and the truss width, d. Values of Vy and Vn, also reportedtn ll2-4], for four types truss of truss with sharp-edged members and aspectr:rtio \ : 9.5 are shown in Fig. 12.2.2. On the basis of the data of Figs.l.l.2.l and 12.2.2, [12-4] suggested the use for design purposes of the con-',('lvative values C(3)lCr:) given, for eld > 1.0, in Fig. 12.2.3.

l{ocent measurements conducted at the National Maritime Institute, U.K.,rNMl) on trusses with infinite aspect ratio are summarized in Fig. 12.2.4.l(r'lercnce l2-6 suggests thc lbllowing approximate expressions based on ther,'srrlts ol' Fig. 12.2.4:

Illr..,{isttJ lltnMl w()l tKli nNl) I't n il (,ilil)t il:;

,i''

03

1.0

(r) c8, (: . ^(2) Clj)(X : o.)

. u)- 1.40 + 5% 1.54 + 5%

- 1.45 - 1.65

1.21 + 5%

- t.45 - 1.35 *2.l0*Reference [l2-51.

solidity ratios { < 0.4 or so the maximum drag occurs when the wind isnormal to the truss Il2-21.

12.2 PAIRS OF TRUSSES AND OF PLATE GIRDERS

We consider a pair of identical, parallel trusses, and denote the drag coefTicicntcoresponding to the total aerodynamic force normal to the trusses byC9@), where a is the yaw angle. Forbrevity, the notation C|Q): C!'zr)itused. The cases where the wind is normal to the truss (a : 0) and wherc rv* 0 are considered in Sects. 12.2.1 and 12.2.2, respectively.

12.2.1 Trusses Normal to the WindTwo parallel trusses normal to the wind affect each olhc:r lrc:nrtlynirrrrically, srrthat the drag on thc upwincl and on the clownwincl trrrss will lurvt'rlrirg cocrllicients VrCtj)and V,,Cff), rcspcctivcly, whcn: f ilj) is tlrt'rlr;r1, 1.1y,.11;,.1t'rrt lirr.;rsingle truss n<tnnlrl lo lhrr wirrtl trntl, in gr:ncrirl \lrr / ,1,, / I ll lollows tlurl

,r," ''(:;)''' '

lrrt (rttsst's willr slrlrlll t'rlgt'rl tttt rrrlr,'t:'. ;ttttl

lirr0 < (, < 0..5 ( l :.1.1)

TRUSSED FFAMEWORKS AND PLA'TE OIRDERS

til+l FtLr-*i

1.0

0.8

0.6

o.4

0.2

0

-o.22.0

e/d(a)

2.0 3.0e/d(b)

Factors V' and VilIl2-41.

3.01.0 4.0 5.0

:

1.0

0.8

0.6

0.4

o.2

0

-o.2

'P=O 234

4rt q=O.404

'P = 0.234-a---- --,

. tr.. .^ = n LnA,-z ae a_ _ - _b _ _ _ - -- T ;,r- ;;;; - :

y,z--x''- -^-__---Lu--------b' --x-' $NN4a'

-.r-t'

o = 0.545

4Em---":91:: l"t/nt'arP:0

lou I{:,' f *,

L--zt.tzazzO'

'{----y'"t ,

,.a"=9.545

FIGURE 12.2.1a,b.

PAIRS OF TFUSSEB AND OF FiI ATF (tiItDFItIt

o.2

4.0 6.0

(r2.2.3)

fbr trusses composed of members with circular cross section. The nominalsolidity ratio $" in Eq. 12.2.3 is related to the actual solidity ratio as shownin Fig. 12.2.5.

Figure 12.2.6 shows ratios CB)|C$) for trusses with sharp-edged membersand aspect ratio X : 8 t12-11.

1.0 2.0 .l 0 4 0 lr,0 6,0 7.0 8.0rl

FIGURE n.2.2. Factors V, nnrl V,, lirr lirur selr ol'two purullol trussos with sharp-odgccl menrbcrs, \ : 9.5, wind nortttal tr tntsl:,s ll2 41,

1.0

0.8

0.4

==270!2t-",-ffir'o{\f./t'--)'

. .^ = o 427 ,p = 0.430 ---;>o!t-*;;iJ=;>-+--a---:;---r' -_arv-'-v-llostt

1.0 2.0 3.0e/d(c)

FIGURE 12.2.1c. Factors V, and Vnfl2-4).

c'B' , or/ ,\r'-o ot

c*:z-Qe \t/

-!!1'-"t9-r-YT!9:':-'-'-"-,- "\- ------- -o-t"oer@ c= 0 411a---

--- :-----------?o./ 1-.-.s:j-i-*- - - - - --'-r',*r@ 7=; o!{-l - - -- + - - --+

,2":irl_ _ " _ _) - - o- - - - -'e- - - - - 4- - - -'/ito''- - rvlodetc),r=0627v

428 TRUSSED FRAMEWORKS ANI) PI AII (IIIII)I N!]

2.2

2.L

2.0

1.9

1.8

CB)

cg)

r.7

1.6

1.5

r.4

1.3

r.2

1.1

0.6 0.7

FIGURE 12.2.3. Approximate ratios c$) lC$) proposed for design purposes by Flachs-bart |2-41.

Examples:

1. Consider a truss with sharp-edged members, solidity ratio @ : 0.1g,spacing ratio eld: 1.0, and aspect ratio \ : oo. From Fig. l2.l.l,cS) = 1.70 according to both Flachsbart and the NMI tests. From bothFlachsbart's and the NMI t.e^lts, C$ttC$\: Vr * Vrr = 1.5 (Figs,l2.2.la and 12.2.4a), so C$) = l.7O x 1.55 = 2.65. Note that ac-cording to Fig. 12.2.3-p.loposed by Flachsbart as a deliberately con-servative design chart_C(]tlC')t = 1.83, which exceeds the value basedon Figs. 72.2.1a and l2.2.4aby about2O%.

2. Consider a truss with sharp-edged members, solidity ratio @ : 0.46,spacing ratio eld: 1,.Q, and aspect ratio X = 9.0. Approximate valuesof drag coefficients C$), ratios.CSrtCg, : Vr * V,,, anO correspondingcalculated drag coefficients C$), based on the Grittingen tl2-41, NMI[12-6] and western ontario [12-5] information, are listed in Table 12.2.1.It is seen that while the difference between the values c$) based on|2-41 and [2-5] is abott l2%, the corresponding values Cfi) are vir-tually identical in this case. Note also that thc clilibrcncc between thcvalues Cf;) based on [12-61, on the one hancl, arrtl orr l12-41 or l12--51,on the other, is about 25%.

0.3 0.4 0.5

a

l? ! PAltls of Ilil,Jssl ti ANt) ot I'l Alt oillt)t nfi 429

*t,,lttt

0-0 0.1 02 0.3 0.4 05 0.6 07 0.8q

(o)

lflGURE 12.2.4. Factors Vr and Vtr for two parallel trusses with (a) sharp-edgedrrrcmbers and (D) members of circular cross section, \ : o, wind normal to trusses.lirom R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experiencedlry Arrays of Lattice Frames," in Wind Engineering, Proceedings of the Firth Inter-rttttional Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.). Vol. l. Per-gamon Press, Elmsford, NY, 1980. pp. 405-420.

12.2.2 Trusses Skewed with Respect to Wind DirectionWc now consider the case in which the yaw angle is cv * 0. For certain valucsol' a the effectiveness of the shielding decreases, and thc drag cocfliciontflll'}(cv) characterizing thc total l<rrcc normal to thc trusscs is largcr ilrln lhcvaluc C!]). (Rccall that, by dclinition, Cll\fq : C'i|'.)

Ilati<rs rnax lC(,ltkyl )/('If ) rcporrccl in ll2-51 lirr lnrsscs wilh sharp ctlgtrl

I*,, 1

_^[,(Ut{llel

430 TRUSSED FRAMEWORKS AND PLATE OIRDEFE

tl,ltt

FIGURB 12.2.4. (Continued)

0.2 0.3 0.4 0.5 0.6

e(b)

members and aspect ratio X : 8 are shown in Fig. 12.2.7.. For example, foreld: l.o, O :0.286, and X : 8, the ratio max {C?'tol}lc8't = 1.77,versus CptC$) : 1.59 (Fig. 12.2.6).

12.2.3 Pairs of Solid Plates and Girders

Figure 12.2.8 shows the dependence of the factors Vl and V11(see F;q.12.2.1)upon the spacing ratio eld for a solid disk and for three girders normal to thewind [12-4, l2-lll. For certain values of the horizontal angle cv between thewind direction and the normal to the plates the ratio C\\@ltCllt rnay be largerthan unity. For example, for a ptate with aspect-ratio \ = 4 und spucing ratioeld:0.5, if 40' z .r < 65", then Cl3)(a)/C\3' = 1,20 Il2"ll.

0.35

0.30

o.25

0.20

0.15

0.10

0.05

0.3 0.4 o.7 0.8

FIGURE 12.2.5. Equivalent solidity ratio {. for trusses with members of circularcross-section and solidity ratio d. From R. E. Whitbread, "The Influence of Shieldingon the Wind Forces Experienced by Arrays of Lattice Frames,' ' in Wind Engineering ,

Proceedings of the Fifth International Conference, Fort Collins, CO, July 1979, J. E.Cermak (ed.), Vol. l, Pergamon Press, Elmsford, NY, 1980, pp. 405-420'

ri)

--'--a=0286T "t"'---i/tt\ /!iviri l,*-*--*- --'*---* --- -----+--- e= 0'464

ifl4t

rc\**--o____.o\\\c= 0.773

FIGURE 12.2.6. Ratios C!l)/C!i) ftrr tnrsses with sharp-edged members, \ : 8, windnormal to trusses. From P. N. Gcorgiou rrnd B. J. Vickery, "Wind Loads on Buildingljrames," in Wind Engin,eering, Pnxvcding,r ol' the Fifth Intcrnatbnal Crryfercnce,Fort Collins, co, July 1979, J. Il, ('ennpk (ctl,), Vol, l, Pcrgamon Prcss, Elmslirrd.NY, 1980, pp. 421-433.

431

452 TRUSSED FRAMEWoFKS AND PLATE GIFDERS

TABLE 12.2.1. Drag Coefficients Based on the Giittingen, NMI, and WesternOntario Studies

Flachsbart NMI Westem OntarioReferences t2-4 t2-6 t2-5cg)cgtcg'CB'

1.5x0.95=7.43o0.23+O.92=1.15bl.l5xt.43:r.64

1.7x0.95=1.62"1.29',t2.08

12.7'1.30r1.65

"Figs. 12.1.1 and l2.l .2.brig. 12.2.2.'Figs. l2.l.l and 12.1.2.'|Eq. 12.2.3a or Fig. 12.2.4a..Table 12. l. l.trig. t2.2.6.

\-o= o;;i:.------i \ \ \

o oj 1.0

FIGURE 12.2.7. Ratios max {C|@)llCg) for trusses with sharp-edged membeni, )\: 8. From P. N. Georgiou and B. J. Vickery, "Wind Loads on Building Frames,"Wind Engineering, Proceedings of the Fifth International Conference, Fort Collins,CO, July 1979, J. E. Cermak (ed.), Vol, 1, Pergamon Press, Elmsford, NY, 1980,pp.42r-433.

-A-- _.':\l'--t' -^--'(t::lo

. \rQ\'\ \l. ---- -^--a\.. -o----:1.5 -..\\.r\- \ \.\- tt^io.'a--: - --_"\lr..\o-i)..t*..-- : -=

1.0'\.----.=i;'\ \\\oi\\ -:t\t' \+ ---- +\ =0.50

0.6

-0.2

-0.4

e/d

FIGURE 12.2.8. Factors V, and V,, for two parallel solid plates (girders) [12-4,tz-rtl.

Data concerning the effect of bridge decks on the aerodynamic forces actingon pairs of plate girders are available in [12-12].

12.3 MULTIPLE.FRAME ARRAYS

'l'he first attempts to measure aerodynamic forces on multiple frame arrays wereroported in [2-l] and [2-6].

For frames normal to the wind, thc drag coefficients for the first, second... n-th frame may be writtcn ns VrClj), Vrcltt V,,Ct|, where c[j)islhc drag coefficient fora single l'ruttte nonnul to thc wind. Thc clrag cocllicicntlilr the array of frames nomral lo tlre winel is tlrur

a----l ---r

A --+-xtffitta=2.0 -t-"dt4 Ud=L3.6 +"BtW, l/d= 9.5--c--

cl'j' = clj'tvr * v2 r

I2,3 MULTIPLE.FFAME ARRAYS 433

til+l [lrt#ei

iiltltl!lill

8.0

t1.0

rttt2.0 3.0 4.0 5.0 6.0 7.0

I V,) ( 12.3. l)

434 TRUSSED FRAMEWoFKS AND PI ATF oInDET:IFI

Factors \fj (,1 : 1,2, . . . , n) for arrays tlflthrce, four, and five parallel trusseswith sharp-edged members and infinite aspect ratio are given in Figs. l2.3.laand l2.3.lb for spacing ratios eld: 0.5 and eld: l, respectively [12-6J.Drag coefficients C$) for the same arrays are shown in Figs. 12.3.2a and12.3.2b tl2-61. Also shown in Figs. 12.3.2 are measurements of c(p tor trusseswith infinite aspect ratio and members with circular cross section 112-61.

,ltl,Qtt It

s+\

tltn

OC)@ilnl

o

A

o

+

te

e/d = 0.5

1.0

2.0

3.0

4.O

(a)

FIGURE 12-3.1. Factors'rj (i : 1,2, . . . , n) for arrays of r parallel trusses (n =3,4, and 5) with sharp-edged members, X : oo, wind normal to trusses. (a) Spacingratio eld: 0.5. (b) Spacing ratio eld: 1.0. From R. E. Whitbread, ..The Influenccof Shielding on the wind Forces Experienced by Arrays of Latticc Frames," windEngineering, Proceedings of the Fijlh International ConJbrence, F<lr1 collins, co, July1979, J. E. Cermak (ed.), Vol. l, Pergamon Press, Ehnslirnl, Ny, l9tt0, pp. 4051420.

1.0

0.9

05

0.3

02

01 0.10.50.4 u.b0.3o"2

I2,4 SQUARE AND IIIIANGULAH TOWERS 435

lstFrame -{

ttitrirtl12 3 45

0.3 0.4 0.5 0.6 0.7

(b)

0.2

Symbol

o .......-.=

-----+ro

-

Frameconfiguration

lTIL-J

FIGURE 12.3.1. (Continued)

Ratios cfltc$) measured in [r2-rl for trusses with sharp-edged membersand aspect ratio x : 8 ar9 shown in Fig. 12.3.3. As pointed out in Sect. 12.2,the drag force normal to the trusses doei not reach a maximum when the trussesare normal to the wind, but for some yaw angle cy ;e 0. Ratios maxlc?@>l tc\) measured for the trusses just iescribed are shown in Fig. 12.3.4lt2-tl.

12.4 SQUARE AND TRIANGULAR TOWERS

As pointed out earlier, thc aenxlynarnic crrcflicicnts givcn in this chaptcr aroin all cases referred to, ancr shourcr be usotl in conjurition with, thc clJ.ectivcirrca of thc fiamework, A.s. For nqunre rrnrl triunguliir lowcnt, zll is rhc cll.ective

436 illt,lilit l) lltnMl w()llKli nNl) I't n ll (,llll rl ll:i

x\--x--x-n: I

OM0 0 1 0.2 0.3 0.4 0.5 0.6 0.7 0,ua(b)

FIGURE 12.3.2. Drag coefficients C$)forarrays of n parallel trusses, 1 : oo' wintlnormal to trusses. (a) Spacing ratio eld:0.5. (b) Spacing ratio eld: 1.0. FrorrrR. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienced byArrays of Lattice Frames," Wind Engineering,, Proceedings of the Fiilh InternationulConference, FortCollins,CO,July 1919,1. E.Cermak(ed.),Vol. l,PergamonPress.Elmsford, NY, 1980, pp. 405-420.

area of one of the identical f'aces of the tower. The influence of wind gustinesson the tower loading and response can be determined by using the methods firr'estimating along-wind response discussed in Chapter 9'x

For information on guyed tower response and design, see [4-10], 14-111,and I l2- l1l ro 12-261.

12.4.1 Aerodynamic Data for Square and Triangular Towers'l'ho rcsults of wincl tbrce measurements on square towers can be expresscd irr

tcrrns of the aeroclynamic coefficients C,v{cv) and C7(a) associated, respectively,with the wind force components N and Z (N = 7) normal to the faces of tlrt:

*The width of the structure used as an input in these methods should be equal to thc actual witlllrof the framework. This ensures that the lateral coherence of the load lluctuations is lakctt ittloaccount. On the other hand, the depth (along wind dimcnsion) ol'thc ll-atttcwork slurtrltl lrr'assumed to be equal to zero in order not to ovcrcstinlatc thc lavorahlc cllcct ol tlte itlortg wittrlcross-correlations ol thc fluctuating loatls (scc Scct. 4.7.4). lrirrlrlly, thc itrc:t ol lltt' li:ttttt'wotlper unit height at any givt:rr clcvlrliorr. uso(l to cslirnirlt: llttr ttterttt ittttl llrt' llur'ltlltliltli tlr:rg lirt t s.

shoulcl bc cqual to lhc clli'r'livr':rr('it l)cr rrlrit lrt'ig,hl :tl lltltl t'ltv:tliort

l:,4 :i(.)llnlil nNI) ilInil(ilil/\t I towt il,. 4:ll

,6

III

A, - ___. _ ----. -- -L' = 0.286

a{\,I

e,d

If IGURE 12.3.3. Ratios c!j)/c!j) for arrays of five trusses with sharp-edged members,\ - 8, wind normal to trusses. From P. N. Georgiou and B. J. Vickery, '.Wind Loadstrn Building Frames," wind Engineering, Proceedings of the Fifih Intemational con-li,rcnce, Fort Collins, CO, July 1979, J. E. Cermak (ed.), Vol. l, pergamon press,lilmsford, NY, 1980, pp. 421-433.

tower (Fig. 12.4.I) and in terms of the aerodynamic coefficient Cp(a) asso-ciated with the total wind force Facting at a yaw angle cv : tan-r (Z/N). Notethat Cp(cv) : tcfo") + C2r1a11t/2, since, as indicated earlier, all aerodynamiceoemcients are referenced to the effective area of one face of the frame-work, Al .

For a triangular tower (which has in practice and is therefore assumed hereIo have equal sides in plan), the results of the measurements can be expressedirr terms of the aerodynamic coemcients Cp(cv) (Fig. 12.4.2). The aerodynamict:ocfficients Cr(0") and CF(60') correspond, respectively, to wind forces actingin a direction nornal to a side and along the direction of a median (Figs.12.4.2e and 12.4.2b).

Measurements of loads on a tapcrcd square tower model with sharp-edgedrttcmbcrs, aspect ratio \ = oo. lrrrrl soliclity ratio averagcd over the height clflhc k)wcr d = 0. 19 (rangilrg lnlrr y'r O. 13 irt (hc hasc to @ : 0.47 at tltc(ip) wcrc rcportccl in lhc I().]Os lry lr.:rlzrrrryl rrrrtl Sirilz ll2-131. Ilrrtil rcrcc:ltlly(ltcsc: Ittoasurclncnls ltltvc lrt't'rt llrt' l)ur( rl);rl s()ur'((' ol'rllrlrr ()n s(luiu1' l()w('t'ri.'l'ltc: crlcllicic:nls (i(rv), (',(rv). ;rttrl {', 1rv) olrl;unt'rl in llJ l.\l ltl' listt.tl lilv:rriorrs irrrgltrs rv in 'l'lrblt' I J .l I

Ittt-rv 45" tltt' v:tlttt's ol (''1,(,r ) .rnrl ( r(,r) :,lrottltl lrt' r't1rr:rl, lr:l Porrrlt'rl

-\_lr:2

-*--*---)5*" tt-l

-r---I4.4

4.0

3.6

-1 Ir-@)-D

1.6

2.0

L6

t.?

4.8

4.4

4.0

n:5n:4

oAngle ser:tiorrmernbers

x Crrcular-sectronmembers

3.6cg)

J.Z

2.8

2.4

2.O

16

r208

-ra_ --\!--x-n:2

0.8

0.4

00 0.1 0.2 0 3 0.4 0.5 0.6 0 7 0.8

a(a)

438 TRUSSED FRAMEWORKS AND PLATE OIRDERS

-L']l^t^"flti Eo;lNIEI

$t.^\;'...\'6.\. \^

u-\*i..:--o -i3:r o-{-_ ___A-V- -' J-d-=g zi- -. - - - --{'ts\'i-*'x.-_;atji.-_-::1

;;l;-:l=Tr- -----:

1.0 0 0.5 1.0

FIGURE l2.3.4.Ratios max {Cf,'1ultClt, a.t"."r, of five trusses with sharp-edgedmembers, \ : 8. From P. N. Georgiou and B. J. Vickery, "Wind Loads on BuildingFrames," Wind Engineering, Proceedings of the Fifth International Conference, FofiCollins, CO, July 1979,1. E. Cermak (ed.), Vol. 1, Pergamon Press, Elmsford, NY'1980, pp. 421433.

FIGURE 12.4.1. Notations.

qfI?.4 SOUARE AND II]IANGIJIAII iOWrIIH 439

(b)

FIGURE 12.4.2. Notations.

out in [12-131, the 4% difference between these values in Table 12.4.1 is dueItr measurement errors. Note that the value C^(0") :2.54 is close to the valuesinf'erredfrom [12-5] and [12-6], whichare, respectively, C,,(0") : Ct) = I.5x 1.73 :2.60 (as obtained by linear interpolation for 6 : O.I9 and eld :1.0 from Table 12.1.1 and Fig. 12.2.6), and C"(0") : Cg) = t.7(0.93 +0.58) : 2.57 (Eq. 12.2.1, and Figs. 12.l.l and 12.2.4a). Note also that while(hc largest tension (compression) in the tower columns is caused by windsrrcting in the direction a : 45" , the largest stresses in the bracing memberstrccurfora:27".

Measurements of forces on square towers with sharp-edged members (\ :oo) were more recently conducted at the National Maritime Institute, u.K.(NMD ll2-141. coefficients cr(0") and ratios cp(u)rcp(O") based on thesenrcasurements are shown in Figs. 12.4.3 and 12.4.4, respectively. Note, forcxample, that for 6 = 0.19, Cr(0.) = 2.60 (Fig. 12.4.3), versus Cr(O") :2.54, as obtained in [12-13] (Table 12.4.1). The agreement is less good forlhc ratio cF(45")/cF(O"), which is about 1.12 according to Fig. 12.4.4, andrrbout 1.40 according to the data of rable 12.4.1. As shown subsequently inllris section, data on square towers composed of members with circular crossscction suggest that the NMI results are more reliable than those of [12-13].

'I'ABLE 12.4.1. Aerodynamic Coefficients: C"(o), C.(a), and Co(o) for aSt;uare Tower with f = 0.19 and ), = o [12-13]

18" 270 36" 45"

(c)(a)

9"0"(t(,v(tr)(', (c)('1"Qx)

2.54

2.54

)110.192.76

2.970,7{)J,05

3.0I1,36J..r0

2.842,053.50

2.602.493.60

o=30'

f,r

o Angle members-smooth flow.o Angle members-turbulent flow.

+ Square shaped members-smooth flow

FIGURE L2.4.3. Drag coelficients Cp(O") for square tower with sharp-edged membcrrmeasured at National Maritime Institute, U.K. From A. R. Flint and B. W. Smith."The Development of the British Draft Code of Practice for the Loading of LatticcTowers," Wind Engineering, Proceedings of the Fifih Intemational Conference, FctrlCollins, CO, July 1979, J. E. Cermak (ed.), Vol. 2, Pergamon Press, Elmsford, NY,1980, pp. 1293-t304.

cr@)cr(0.)

I oo: 15' 30' 45'

FIGURE 12.4.4. Ratios Co(a)/Co(O') for square tower with sharp-edged mcrnlrt'rrmeasured at National Maritime Institute, U.K. From A. R. Irlint and B. W. Srrritlr."The Development of the British Draft Code of'Practicc lirr tlrc Loltling ol'Lirtlit'cTowers," Wind Enginct'ring, Pnx'rcdings rl tha [,'i.lilt ltttt'rttttti,ttrrtl ('rtr.li,rt,rrcr,, ltrrtCollins, CO,.luly l9l9,.l .1,1. Ccrrrrlrk (ctl.), Vol. 2, ll'r1'.:rrrron I'n'ss. lllrrrslirrtl, N\',1980, pp. 1293 I304.

440

l;'4 l;()unl rl nl.Jl) ilttnil{irt t\il t1)wt tt,. 441

Lr(0")

5 1 5 6 7 8 9 105 2 3 4 s 6 7 8 9106 4z 2

FIGURE 12.4.5. Drag coefficients Co(0') for square tower with members of circularr'mss section [12-7] (courtesy Comitd International pour le Ddveloppement et I'Etudetlc la Construction Tubulaire, and Constructional Steel Research and Development( )rganisation).

Sguare Towers Composed of Members with Circular Cross Sec-fion. Figures 12.4.5 and 12.4.6 [12-7] represent, respectively, proposed aero-(l_vnamic coelicients Cr(O') and Cp(45') as functions of Reynolds number Gelirr towers with aspect ratio }, : oo, based on recent wind tunnel test resultsrcported in [2-8] and [2-9]. The values CF(45') of Fig. 12.4.6 may beregarded as conservative envclopes that account for the loadings in the mostrrnlavorable directions. Rcsul(s ol'tcsls conclucted at NMI in both smooth andtrrlbulent flow at Rcynokls rrtrnrlrt't.s (11,. : 2 x 101 li)r solidity ratios @ :O ll , O : 0.23, ancl <,f - 0. I I (^ rx') rrurtt'lt thc r.rrrvcs ol'ljig,. 12.4.-5 urrcl11.4.6 to within ab<lul 5%, or lt':i:; ll.' l.ll

N<rtc lhat lirr 0 < ,h . 1', \ llrt' r;rtrr' (; (,1i")/('/ (O") is r'plsltlcl.lrfly t'lost.r'to LI thitlr lrt lltc virltrc l.'l rrlrt'tt'trl rrr'lrrlrlt' I-).l I 'l'lris worrlrl lr.rrrl lo,ottlirnr lltc hrrxrrl vlrlirlity ol llrt NNll rr',,rrll . on .,(luju(. 1()\v(.1\ rvrllr slr:rrPt'rlgt'rl lttclttlrt'r's rlisr'rrsscrl t';ultt'r ln llu', '.( ( lt'n

080.2

- - n 1?,

e = 0.535

*442 Inul;l;l l) I nn Mt w( )t rh:, n l\| ) t,t n l ( ,l l,t tr.

c n(45)

3 1 5 6 r s 9 1S 2 3 4 s 6 r I 9 10. !/te ?

FIGURE 12.4.6. Drag coeflicients Cr.(45') fbr square tower with members of circulalcross section [12-7] (courtesy Comit6 Intemational pour le Ddveloppement et I'Etuilcde la Construction Tubulaire, and Constructional Steel Research'and Developmcnt0rganisation).

Triangular Towers composed of Members with circular cross sec-tion- Figurcs 12.4.1 and 12.4.8 [12-7] represent proposed aeroclynamic cocr'-ficients c/.(0") = cr(60') and cp(30") as functions of Reynolds number 61"fttr t<twcrs with aspect ratio X : oo, based on measurements reported intl2-81, It2-91, and [12-101.

FIGURE 12.4.8. Drag coefficients C,.(30") for triangular t()wcr with rncrrrbcrs olcircular cross section ll2-71 (courlcsy c.miti rrrtcrn.ti..:rr prrrr lt' r)t:vt'r.ppcrr*rrr t.rI'Etude de la Constnrctitln'ftrbulairc, lrnrl ('onslrucliorr:rl Slt.r'l ltt.st':lt.lr rrrrtl l)t:vt,lopmcnt Organistrl iorr ).

l;'.1 :;(Jl ,nt il nt..ll | iltlnt.t(,t,t nll l()Wl tt:; 4A.-l

5 1 5 6 7 8 9 10s 7 r 4 s 6 r I 9106 q, 2

TIGURE 12.4.7. Drag coe{ricients c1.(0") and co(60') for triangular tower with mem-bcrs of circular cross section [12-7] (courtesy comitd Intemation"al pour le D6veloppe_rncnt et I'Etude de la Construction Tubulaiie, and Constructional Steel Research andI)evelopment Organisation).

I i,,(30')

'1.0

t444 ilttjlitit t) |tnMt w()nKt; nNt) r'r n r {;lu)l rr:i

REFERENCES

l2-l P. N. Georgiou and B. J. Vickery, "Wind Loads on Building lilrrrrcs." l4lirrrlEngineering, Proceedings of the Fifth International ConfcrcnL'c, Forl Collins,CO, July 1979, J. E. Cermak (ed.), Vol. l, Pergamon Press, Elrnslirrd. NY.1980, pp. 421-433.

l2-2 O. Flachsbart, "Modellversuche iiber die Belastung von Gitterlachwcrkcn durclrWindkriifte. l. Teil: Einzelne ebene Gittertniger," Der Stahlbau, T (April 1934).65-69.

l2-3 O. Flachsbart, "Modellversuche iiber die Belastung von Gittedachwerken durchWindkrdfte. l. Teil: Einzelne ebene Gittertreger," Der Stahlbau, T (May 1934)13-79.

l2-4 O. l.'lachsbarl, and H. Winter, "Modellversuche iiber die Belastung von Gittcrlachwcrkcn tlurch Windkr:itte. 2. Teil; Rdumliche Gitterfachwerke." DrrSrultllnu, tt (April 1935), 57-63.

l2-5 P. N. Gcorgiou, B. J. Vickcry, and R. Church, "Wind Loading on OpcrrFramed Structures," Proceedings Third Canadian Workshop on Wind Engineering, Vancouver, April 198 l.

12-6 R. E. Whitbread, "The Influence of Shielding on the Wind Forces Experienccdby Arrays of Lattice Frames," Wind Engineering, Proceedings of the FililtInternatktnal Conference, Fort Collins, CO, July 1979, J. E. Cermak (ed.),Vol. I, Pergamon Press, Elmsford, NY, 1980, pp. 405-420.

l2-7 Wind Forces on Unclad Tubular Structures, H. B. Walker (ed.), ConstradrrPublication l/75, Constructional Steel Research and Development Organization, Croydon, U.K., 1975.

l2-8 G. Schulz, The Drag of ktttice Structures Constructed from Cylindrical Metnbers (Tubes) and its Calculation, CIDECT Report No. 69/21, Drisseldorf, WcstGermany, 1969 (in German).

l2-9 G. Schulz, International Comparison of Standards on the Wind Loading ol'Structures, CIDECT Report No. 69/29, Drisseldorf, West Germany, 1969 (irrGerman).

12-lO R. W. F. Gould and W. G. Raymer, Measurements over a Wide Range tlReynolds Numbers of the Wind Forces on Models of lnttice Frameworks, Nlttional Physical Laboratory Sc. Rep. No.5-72, Teddington, U.K., May 1972.

l2-ll G. Eiffel, ln Rlsistance de I'Air et l'Aviation, H. Dunod & E. Pinat, Paris.l9l 1.

12-12 J. M. Biggs, S. Namyet, and J. Adachi, "Wind Loads on Girder Bridges."Transactions, ASCE, f2f (1956), l0l-113.

12-13 D. Katzmayr and H. Seitz, "Winddruck auf FachwerktLirme von quadratischcrrrQuerschnitt," Der Bauingenieur, 2lI22 (1934),218-221 .

12-14 A. R. Flint and B. W. Smith, "The Development of the British Draft Coclc olPractice for the Loading of Lattice Towers," Wind Engintt'ring, Prot'cctlittglof the FiJih International Conferenr:e , Fort Collins, ('O..luly l()79, J. U. ('crmak (ed.), Vol.2, Pergamon Press, l9fl0, pp. 129.1 llt),I

12-15 J. F. Eden, A. lny, and A. .1. Bullcr, "('rlrncrs itt Slorrn Wirrrls." l'.'rr,q. ,\trttt't..3 (r981). r75 rtto.

nt I r lt N(.t :; 445

l.) l(r .l . li. lltlcrr. A..l . lhrllt'r. rttttl .l . I'rtlit'ttl . "Wirrrl lurrnt'l 'lcsls orr Mtxlcl (-'mrrcSlrttt'ltrcs." l'.rr,q. ,\trrtr't ., 5 (l()8 ]), .ll"i() -)()S

I ) l1 (i- A. Savitskii, (lrlt'rtltttittrt.t litr' ,,ltttt'ttrrtr ltt.tttrllrttiotr.s, 'l'cchnical Translation'l"l'79-52040, prrblishcrl lirl llrt' Nrrliortirl Sticnec Iioundation by Amerind Pub-lishing Cb., Ncw l)cllri, lgllf , :rv:ril;rlrlc liurr National Technical InfbrmationScrvicc, Springlicltl. VA 22 l(, I .

Il llt V. Kolouick, M. Pirncr, (). Iiischcr, and J. Niiprstek, I{ind Effects on CivilEnginecring St rudur(s, Elscvicr, Amsterdam, 1984.

l -l- 19 R. J. McCaffrcy and A. J. Hartmann, "Dynamics of Guyed Towers,', J. Struct.Dlv., ASCE, 98 (1912), 1309-1323.

12-20 J. W. Vellozzi, "Tall Guyed Tower Response to Wind Loading," proceedingsFourth International Conference on Wind Effects on Buildings and Structures,Heathrow, September 1975, Cambridge Univ. Press, Cambridge, 1976.

l22l R. A. Williamson, "Stability Study of Guyed Tower under Ice Loads," -/.Srruct. Div., ASCE, 99 (1973),2391-2408.

12-22 J. E. Goldberg and J. T. Gaunt, "Stability of Guyed Towers,,' J. Struct. Div.,ASCE, 99 (1973),'741-1 56.

l) 23 R. A. Williamson and M. N. Margolin, "Shear Effects in Design of GuyedTowers," J. Struct. Diy., ASCE, 92 (1966),213-260.

l)-24 F. Rosenthal and R. A. Skop, "Method for the Analysis of Guyed Towers,',J. Struct. Div., ASCE, f08 (1982), 543,558.

l)-25 D. M. Brown and J. W. Melin, Guyed Tower Program Listings and (Jser'sManual, Technical Report sponsored by United States Coast Guard, U.S. De-paftment of Transportation (Contiact DOT-CG-52604-A), J. W. Mellin andAssoc., Urbana, lL, 1975.

1226 A. G. Davenport and B. F. Sparling, "Dynamic Gust Response Factors forGuyed Towers," J. Wind Eng. Ind. Aerod., 4l-44 (1992),2237-2248.

l) 27 J. D. Holmes, R. W. Banks, and G. Roberts, "Drag and Aerodynamic Inter-f'erence on Microwave Dish Antennas and Their Supporting Towers,', J. WindEng. Ind. Aerod.,50 (1993), 263-2'70.

ll28 rnttice structures: Pan 2-Mean Fluid Forces on Tower-like space Frames,Engineering Science Data Unit, ESDU Data Item 81028, 1988 (rev. ed.).

l)29 K. Hiramatsu and K. Akagi, "The Response of Latticed steel rowers to theAction of Wind," J. Wind Eng. Ind. Aerod.,30 (1988), 7-16.

CHAPTER 13

SUSPENDED.SPAN BRIDGES,TENSION STRUCTURES, ANDPOWER LINES

Structures that consist of or depend for their integrity on cables or membranesmay exhibit an increased susceptibility to wind effects. Notorious examples arcthe Brighton Chain Pier and the original Tacoma Narrows Bridge (Figs. l3Aand l38). The purpose of this chapter is to present information and referencesconceming such structures, including suspension and cable-stayed bridges, ca-ble roofs, fabric structures (air-supported or otherwise subjected to tension)'and power lines.

13.1 SUSPENDED-SPAN BRIDGES

Suspenclcd-span (i.e., suspension and cable-stayed) bridges must be designedto withstand the drag fbrces induced by the mean wind. In addition such bridgcsare susccptiblc to aeroelastic effects, which include torsional divergence (orlatcral buckling), vortex-incluced oscillation, flutter, galloping, and buffetingin thc prcscncc of silf--excitecl fbrces. The study of these effects is possiblconly on the basis of infbrmation provided by wind tunnel tests. Various typcsof such tcsts arc briefly clescribed in Sect. 13. 1. I . Procedures for analyzing thc:

susceptibility of suspended-span bridge decks to aeroelastic effects and pertincntdesign considerations are presented in Sects. 13.1.2 through 13.1'5'

It is noted that the action of wind must be taken into account not only lirl'the completed bridge, but for the bridge in the construclion stagc as wcll. Irr

general, the same methods of testing and analysis ap;rly ilr (ltt' two cltscs' 'l'tr

decrease the vulnt3rability of thc parfially cornplotr:(l bt.irlp,c lo wirrrl. t(:lltp()rlll'yties ancl damping rlcviccs arc uscrl. Also. lo tttittilttizt'lltt'rt:;l' ol sl trrttg wittrl

446

b,it'-otsEp

;i=o cJ u-qs >

!!^

> :::zv=-

^ t --

-Ez-JV'34)io€.5ar.o -6}EOcn--"!36-|:.. s

-a@t'=o\.'! ! cOr5CsN ^ U_L A - /ros.=U91|,=c9!Y!,XgHe^a)zAv*

P e.? 3-o=Yaoa.-9V

,r! = ! ^-!d*,@H+aJ r Coo

H p.i v.:5bt*-a(.)e3_.F 3;3:,-E e -b05 h."'tr@.=tm,=A-i!!;or3-o(a" -a i.A'beedO!d;

!FASU c.)V li-9 g'^Firn4=-. H 9'-,<-a2a4=&R

.Ugdl.-Ea)i,-titu .1\ il

btItN.

{!

&q{\It*

$N

$

dw

w:,#,

ri

It!ttttd(

o

Rt',s .l)*\sI\9)t

[,I{t\-J\jt

l,i:i

E,

t'It;

i*,1,}

'od.,

ai* r

#,

*444 t;U:;ltt Nt)t t)r;t,nN tit rilxit i; il N:,t()r..r :,rtr,(.l,nr :; nNr) r,()wr tr lNt :

Flutter of the Tacoma Narrows Bridge, November 10, 1940 [13-1,

loading, construction usually takes place in seasons with low probabilities oloccurrence of severe storms.

Aeroelastic phenomena may affect, in addition to the deck, bridge toy_or'hangers. and cables. Problems relatecl to the design of these. or similar. clcments are dealt with in Sect. 13.1.6.

13.1.1 Types of Suspended-Span Bridge Wind Tunnel TestsThe following three types of wind tunnel tests are currently being used to obtaininformation on ihe aerodynamic behavior of suspended-span bridges.

l. Tests on models of the full bridge. In addition to being geometricallysimilar to the full bridge, such models must satisfy similarity requiremc-ntspertaining to mass distribution, reduced frequency, mechanical damping, arrrlshapes of vibration modes (see Chapter 7). The construction of full-britlgcmodels is thus elaborate and their cost relatively high. The usual scale of srrclrmodels is of the orderof 1/300, although scales of l/100 have been usecl irr rr

few cases [3-l] to [13-6]. A view of a full-bridge model in a wind tunncl issholvn in Fig. 13.1.1.

2. Three-dimenskna.l. partial-hridgc mtxlcls. ln rrrtxkrls ol'tlris typt: thc rrririrrspan (or occasionally lrull'ol'il) is rrrrxlclcrl in rrrr t't'orurrrrir':rl ;rlrpnrxirrurliorr.

lflGURE 13.1.1. Model of Akashi Strait suspension bridge (courtesy of T. Miyata,Yokohama University, and M. Kitagawa, Honshu-Shikoku Bridge Authority, Tokyo).

l'ypically a support structure consisting of taut wires or tubes, or of a fine-wiret'ltcn&rl, supports the geometrically simulated deck structural form. Usuallylirndamental vertical and torsion modes are simulated. The model is envelopedlry i1 three-dimensional simulated boundary-layer flow in the wind tunnel.

'3. Tests on section models. Section models consist of representative span-wisc sections of the deck constructed to scale, spring-supported at the ends to:rlkrw both vertical ancl torsional motion, and, usually, enclosed between endplrrtcs to reduce aeroclynrrtrtic c:ntl cllbcts (Fig. 13.1.2). Section models aret'lrrtivcly inexpensivc. 'l'lrt'y t':rrr lrt' r'onslnrctctl to scales of the order of l/50to ll25 s<l that thc tliscrcp:rrt'ics lrt'lwt'r'rr lirll-scllc and modcl Rcynolds num-lrt'r''r'irrc srrrallcr lluru irr (lrc t'rr:rc ol lrrll lrlrrllit'lcsls. Scclion trttlclcl.s are quite

ll,ol lr tlistrrssir)il ()l lt('ylt()l(ls ililrillr'r ',rrrrtl.rttl\ ri ilIrtt ilr{ [l',. sct ('lt:t1tlt'ts'1 itrrrl 7.

r:r r :;U:;l 'l Nlll t):;l'nN lllillxit li 449

YW!,,,.,,.,.qsaiid:,,,'i;,' ll''1,1. ltti*.'..';l

.iirg$rr,:-rrrip ..,,r.;{. ., id

&",!q4$.,i.,

**lit,lil,l Nl)l l)til'nN lililtxit :;, il Nl,t()t'J :;lltl,(iil,1il:i. ANt ) t,()wt n ilNl i

FIGURE 13.1.2. Section model of the Halifax Narrows Bridge (courtesy Boundary-Layer Wind Tunnel Laboratory, University of Western Ontario).

useful for making initial assessments, based on simple tests, of the extent ttrwhich a bridge deck shape is aeroelastically stable. Finally, section modelshave the important advantage of allowing the measurement of the fundamentalaerodynamic characteristics of the bridge deck on the basis of which comprc-hensive analytical studies can then be carried out. These characteristics includc:

a. The steady-state drag, lift, and moment coefficients, defined as:

FIGURE 13.r.3. Drag, lift, and aerodynamic moment coelficients for replacement

Tacoma Narrows Bridge [13-l].

Tacoma Narrows Bridge [13-1] and in Fig. 13.1.4 for a proposed stream-lined box section of the New Burrard Inlet Crossing t13-81'

b. The motional aerodynamic coefficients. These coefficients characteizethe self-excited forcls acting on the oscillating bridge and are discussed

in Sect. 6.5.2. Examples oimotional aerodynamic coe{ficients F1,f , .4f(t : 1, 2, 3, 4)for various types of bridge decks are given in Fig' 6'5'3'questions pertaining to the laboratory ditermination of H,f , A! are te-viewed in [13-9] and [13-88].*

c. The Strouhal number S (see Sect. 4.4)'

1g.1.2 Torsional Divergence or Lateral Buckling

I-ateral buckling of a bridge deck may be viewed as that condition wherein,given a slight deck twist, the drag load and the self--excited aerodynamic mo-

r-ncnt will precipitate a torsional divergence instability- Thc ttlrsional divcrgcnccphcn<lmcnon has been analyzccl in Sect. 6.4 in thc casc ol-a lwtl-tlittlcttsiottltl

r,l' rrr.rt, rct.t:.1 slrrtlias t.rx.llit.it'rrls // ] ;rrrtl ,1 f' hlrvt: bccn irrclrrtlgl; :tlso, tlr:t1' r't'lltlt'tl t rx'llit icttl:

t,',!' li l, 2, l, 4) lltvc lrt't'tl iltltrxlrtr ''rl (:;t't' Ii; l I l 4l :rrrtl I I l ltt1l)'

l:| I i,l,l;l'l tllrl lr l;l'nll llllll)(il ', 451

(lt t

o.lr

04

0.30.2

,v (rlr'r1)

0.8

0.6

0.4

o.2C,,0

-o.2

-0.4-0.6-0.8

0

-0.8CM

- l.t)

-2.41a

-30 -20 -10 0 10 20 30a (deg)

vD.o - wrBC, : , L," lpu'n?M,-, _

i,urB,

(l3. r. r )

(13.1 .2)

(13. r.3)

where D, ,L, and M are the mean drag, lift, and moment per unit span,respectively, p is the air density, B is the deck width, ancl U is thc nrcarrwind speed in the oncoming flow at the deck clcvalion. 'l'lrcsc cocllicicntsare usually plotted as functions of thc anglc rv lrt'lwt't'rr tlte lrorizrlrrltrlplane and thc planc of thc briclgc clcck. ('ocflicicrrts ('t,. (', . irrrtl ('p, rrrl'shown in Fig. 13.1.3 lirr lhc opcrt lntss britlgt' tlt'r'l' ol llrt' rt'lrlirt't'rrrcrrl

+nlF--Q--n

B

-30-20-10 0 10 20 30a (deg)

452 st,sPFNI)H)t;t)AN Slillxit li, llNlit()N tilnuott,nl t;, ANt) t'()wt n ilNt I t:l I lit,lit'l Nt)t t) l;t'nN Bnil)(it l; 453

(:lcnrcnls ()l lhc nritltrx ('1 :rrc tlt:rrolc:tl by r';; arxl rcprcsotr( thc anglc ol'twist(yi irl .r' - .r'i irttlrtccrl lry lr rrrril lolsional nr()nlcnt acting at x : xi.

Lct lrr) rcprcsurl tlrr colurun rrratrix of the angles of twist a;. In matrixnolll ion

{cv} : Cr{M} (r3.1.4)

where {M} represents the column matrix of the torsional moments M1 appliedat x : x;. These moments can be written as

u, : )pu2B2tt,Cr1ai) (13.1.s)

where Al, is the span length associated with point xi. The problem is nowsusceptible of solution by iteration on Eqs. 13.1.4 and 13.1.5. First it is as-sumed o; : 0 for all j and M1 are calculated from Eq. 13.1.5. Inserting thesercsults in Eq. 13.1.4 yields a column of values cy;; reinserting these into Eq.13.1.5 develops new moments, and so on. The process will converge for anychosen velocity less than the critical divergence velocity that conceptually isapproached in an asymptotic manner by the iterative method suggested.

The process is simplified, however, in the case where Cr,(o) can be ap-proximated by a linear function

Cy(u) = cro (13.1.6)

where CMs : Cu(O). Using the notation

\J*rl-ttlr

tttt

J--/<-.-".-"-/

rl

1.20

CD

0.0

-.t0

.20

CL

0.0

_.40

*.80

-1 .20-.30- 10.0 0.0

a0.0d

A Handrails - guardrails

Q No handrails - no guardrails

CM

tttrtt

lrlt

0.0

-.10

-.20 lnd assuming A/, : A/ for all i yields

{o\ : c,

I- oU'B"A,L,2'

* arr]

I ac^, Ilr, - -n c.l {o) : Cr{Cuo}

1:p

dC,-a-Fda

| (dcM,Ld" "

(13. 1.7)

(r3.1.8)-.30-10.0 0.0

0

FIGURE 13.1.4. Drag, lift, and aerodynamic moment coeltrcients for proposed deckof New Burrard Inlet Crossing [13-8]. Courtesy of the National Aeronautical Estab-lishment, National Research Council of Canada.

structure. In this section the analysis of Sect. 6.4 is extended to the case of atull bridge.

The data needed for the analysis are the experimcntully rrrr::rsrrrcrl rngrncnlcoefficient C7a@) and the torsional flexibility matrix (',.oIthc tlt't'k. l.ct.r', arrclx1 G, i : 1,2,. . . , N) ilcn<ttc valttcs tll'tho c<xlnlirrirlr' r irkrrrlq tlrt'splr1. 'l'lrc

(13.1.9)

licluation 13.1.9 will havc inlirritc (lolsiorrllly divergent) solutions when thetlctorrninant

10.0

tttt

tt

tttt

ttlt

1," ",','n''''l ' (r3.r.r0)

454 lltjljl'l Nl)l l)l;t'AN nnllxil :;, ltNt;t()N tiilil,{:t{,ilt :;, ANt) t,()wl ti ilNt :

Equation 13. l.l0 yiclds a sct tll' c:hllrrctclislic valucs 7r ol' whiclr llrr: lrrr.gr:sl 1r: pc corresponds to the lowcst vclocily IJ - IJ,. lilr torsional divcrgr:ncc:

f l ltt2U-:l----:--l' lp,pB'LL) (r3.1.il)

In general it is found that only torsionally weak bridges incur the actualdanger of torsional divergence/lateral buckling at wind speeds attainable irrpractice. It should also be noted that for many bridge decks the moment inducctlby the horizontal wind is negative (i.e., it twists the bridge deck so as to creatca negative angle of attack, the wind then approaching the upper side of thcrdeck). Such decks are not highly susceptible to torsional divLigence at wintlspeeds in the usual range; however, if the slope of the curve dCTalda vs. o ispositive, a thcorctical torsional divergence is still possible.

13.1.3 Locked-in Vortex-lnduced Responseopen truss sections generally "shred" the oncoming flow to such an extenlthat large, concerted vortices cannot occur and vortex-induced oscillations ol'the deck are weak. However, in the case of bluff deck sections of the box-oropen box-type, instances of severe vortex-induced response are known to havcoccurred.

one such instance is cited in [13-10]. To reduce the oscillations, fairingswere added to the section as shown in Fig. 13.1.5, which includes results ol'wind tunnel measurements. It is noted that in this case the water surface isclose to the underside of the projected prototype and could thus be expectcrlto affect significantly the flow around the deck. For this reason the water surfacewas also modeled in the laboratory.

Additional examples of streamlined bridge deck forms are shown in Fig.13.1.6.

Analytical Procedures for Estimating the vertical vortexJnduced Re-sponse. Under the action of the mean flow and of the shed vortices, the moclclsection will be subjected to a self-excited and to a vortex-induced lift. wirhnotations used in Sect. 6.5 and assuming that the vertical and torsional modcsarc uncoupled aerodynamically, the equation of motion of the section will htr

mfi + z(1,a]t + aihl : (t3.1.t2)

where o is the voftex-shedding circular frequency and 11,f and c1.v are cocllicients to be determined. If the model is given some initial vcrtical cle{irrrnali91.its response will have the form

;l,l;l 'l Nll,l t):it 'nN ilhil){:t:

l,'l( Jl lltl,l l.l, l.(r. St rr.,rrulrrrr.,l I rr rrly,1.,;,., l, lor rrr:.,

455

Velocity (m/s)

FIGURE 13.1.5. Vertical amplitudes of vortex-induced deflections for various bridgedeck sections of the proposed Long creek's Bridge [13-10]. Courtesy of the NationalAeronautical Establishment, National Research council of canada.

) ou'n ["rt," L, * ,,,,in ,,1

1) 1.8 rr trrin,r,

h : (ho + h,e l'')sin(at I 4t\ (t.l.I.n)

456 suspENDED-SPAN BRtDGES, rr Ntit()N riiltr,cluHEs, AND powEn ilNtti

where fte is the steady-state amplitudc, rf is a phasc anglc, ancl 7 antl fi, arcconstants identifiable from the experimental observations. It can then casily bcshown that

47maf,h6Lrt/ -

-

PU:B(13.1.r4)

(13.1.16)

(13. r.19)

where U, : n6Al3, fl.p: apl2r,,4 is the net area of bridge deck projected ona vertical plane normal to the mean wind (per unit span), S is the Strouhalnumber for the bridge deck, and that

(13.1. ls)

At lock-in 0) = 0)h.The dimensionless quantities Cry and HI are applied to the prototype bridge

in the following manner. If fo is the assumed mechanical damping ratio of theprototype, the total (aerodynamic plus mechanical) damping in the prototypecase can be written as

Ht:,+[n3-"']

: )pu!nc,,[J'a,r"r axl sin1,,r + 4;

To:lr-*'fThe prototype being assumed to respond in an early bending mode ft1(x) ac-cording to the relation

h(x, t) : hr(x)q(t)

q{t) is governed by the following equation:

(13.1. l7)

Mi| I2Tpoflr + ,lqrl

(13.1.18)

In Eq. 13.1.18, c,r1 is the circular frequency of the chosen mode and M, is thcgeneralized mass of that mode:

*, : I: hllxym61 ax

where m(x) is the mass per unit span and L is the span ol'tlre pnrtotypc bridgc.The maximum amplitudc at vortex-induccd rcsonancc is llren givcrr hy

llr(x) 1,,,,,, :

fit. I t;t,t;t,l Nl)l l) til ,AN lll ilt)ol s

hlx)pU .?n{:, r ll; 1,,(.t) ,/.t+U,rli,,

For example , if h(x) is a half sinc wavc ovcr the span of a bridge with auniformly distributed mass, the del'lection at the span center is

lr(:)l _eu(ne_rvI \2 / 1."^ mrofif, (t3.t.2r)

The accuracy of the above procedure is acceptable only if the differencebetween the mechanical damping ratios of the model and of the prototype issmall. If this difference is large, the procedure may become inapplicable onaccount of strong nonlinear effects.

An altemate, nonlinear model (see Sect. 6.1.1) may also be employed. Ifthe description of section activity as given by Eq. I3.I.l2 is modified to thefollowing (Van der Pol) form:

m[li + 2l6aph + ..lh] : ou'nxnf (t

then I1f and e become the aerodynamic parameters. These are presumed to beevaluated from section model tests in a manner similar to that described inChapter 6.

The steady-state amplitude of a bridge deck section model is then given as

ho _ "1ur - +s,,lt''B-'l ,HT I

where S., is the Scruton number defined as

h2\ h-e -l;. (13.1.22)B'/ U

(13.1.20)

(t3.r.23)

(t3.1.24)

The coefficient F1f may be viewed as the value obtained at low oscillationamplitudes by any one of the several identification schemes employed to obtainl'lutter derivatives. If the steady-state (vortex-induced) amplitude he is alsorneasured in a section model test, then e is given by

(mc -_Lru - OB2

(13.1.2.s)

Altcrnatcly, if Hf is not obtaincd lrcm a low-amplitudc rrrotlcl lcst, lrrrlirrstcacl thc m<ldcl is alkrwcrl to oscillllc cl<lwn lnrnr an inititl lalgcrittttplittttlc

. Hr - 45,,e:+&nrBfHf

*,458 :;t,tit,t Nt)l t):;t 'nN trl lt)(it :;. l N:;t()N 1;ulr{:tlu i;. nNt) t'()wt n ltNt :;

Auto a stcady, lockcd-in stal.o ol'nrclrsuretl anrl)lilu(lc: 11;, tlrc vlrlrrc ol'//'f' rrurybe determined fiom

I j I :;1,:;l't lll )l I ) : il'n fl I ]t ilt x;l i

lr( r)/l'9'( rylj.(/) tJi,,r1,l; tcr;,lrlr

' t, pI/ltlKl/i' x lt r.r'(r){r(/)l{(r).p'r(r) f (x)th (13.1.33)

in which /(x) is a l'unction atltlitionally inscrtcd to account for spanwise lossol'coherence in thc vortcx-rclatcd forces.

If integration of the left-hand side is extended to the full bridge, integratingthe right-hand side of Eq. 13.1.33 spanwise results in

IE +zfuri + r?,t]:]pun3mufrc,- eczt2lt 113.1.34)

where 1 is the generalized full-bridge inertia of the mode in question and

KHT

where

K :2trSS is the Strouhal number and a is given by

t t)mlfio:-__10-+2pB'l B' l6rrs ]

( 13. r .26)

(13.1.21)

(13.1.32\

( 13. I .28)

R,, being dcfined as the response amplitude ratio of first to nth cycles of am-plitude decay (Eq. 6.1.14).

The information given in Eq. 13.1.23 is applicable to the section modelonly. To extrapolate it to a full bridge, the oscillatory structural mode involved(usually a simple, low-frequency one) must be considered as well as the prob-able nature of the spanwise correlation of the lock-in forces. Referenccll3-941has considered these parts of the problem.

The sectional equation of motion is Eq. 13.1.22:

mtti + 2tp6h + @iht: )ou,nrcu, (' - ,U;)L o3 r.2er

where it is further assumed that

h : h(x, t) : ,p(x)BtG) (13.1.30)

<p(x) being the single dimensionless mode of frequency co1 responding to locked-in vortex shedding and {(t) the corresponding generalized coordinate. This isassumed to undergo the purely sinusoidal oscillation

t@ : {o cos <,l/

at the Strouhal frequency, that is, where

(13.1.3 r)

The strength of vortex-induced forces is dependent upon the local oscillationamplitude of the structure; there is also a loss in their coherence with spanwiseseparation. For example, Fig. 6. 1.2 depicts the correlations between locallateral pressures separated spanwise along cylinders displaced vertically sinu-soidally with different relative amplitudes.x A general review in [3-94] sug-gests that under such conditions an appropriate correlation loss function can beapproximated by selecting/(x) to be the mode shape <p(x) itself, normalized tounit value at its highest point. For example, with a mode representing a half-sinusoid over a span L, f (x) may be estimated as

7fXf(x):sin7

4sB2.l,cl,-nlnil(x--lllr

-r

nh; I At-h6,)c,: [ 'P2(x)f(x\(tx' Jrpon L

c, : I e4$)I(x)d-r- Jrpon L

(13.1.35a)

(13.1.3sb)

(13.1.37)

(13.r.36)

At steady-state amplitude, as noted earlier, the damping energy balance percycle of oscillation will be zero, a condition that defines the vortex-inducedamplitude

If /z from Eq. 13.1.30 is inserted into Eq. 13.l.Zg unrl tlrc n'srrlt rrrrrltil-rlictl hyBp(x), the action of thc scction r/x <11'lhc slnrclrrrt: :rssot'i;rlctl witlr sp:rrrwisr:point x is sccn to hc rlcsc:rihccl by thc: clrlutrliorr

,, -)lcr'r *otol"'q{r 'l rCaHf ]

whcre thc Scruton numbcr is rlcrlirtul rrs

tllcsttlls tlrrirlitlrlivcly sintillu kr llrrrsc ol l;r1' lr L) lr:rvr"lrt't'rr rt'lxrrlt.rl lirrI I l ()61.

2trSUu--uL

B

\(luill(' l)r t:]nr, ilr

460 SUSPENDED-SPAN BRIDGES, TENSION I;IIIIJCIURES, AND POWIF LINES

s,r:h (13.r.38)

(13.1.39a)

(13. r.39b)

For the case of a sinusoidal mode the values of C2 and C4, respectively, are

cz:

cq:

fL "rxdxJnsin'TT:0.4244

I, ,*' Tf :o 33es

An example [13-941 will be drawn from the historic Tacoma Narrows caseof 1940. This bridge underwent considerable vortex-induced disturbance priorto its demise by torsional flutter [13-93]. Pertinent data, forexample, relativeto an 8-noded vertical mode of this bridge are

nl : natural frequency : 0.66H2

B:39ftslussp :0.002378 -^;-' tt-

^:ffi:88.sypL

l - | ^n"'dxJo

For <p : sin rxlL, I : mfiLlz slug ft2,

I : 0'0025

Interpreting data from Ref. [13-94]

BaK : : 3.1343 at lock-inU

frU : 35.2 mph : 51.6 -Hf : l.l8l at lock-in

e : 4l'l I at lock-in

Then

I3,I SUSPENDED.SPAN BRIDGES 461

, l. (m (0.0025)(88.5)5,, : (trrB. , : ntl't,: ,E: 4O,,'A..u;: 0.030586

Cz : 0.4244, C4 : 0.3395 ("standard" values)

Hence by Eq. 13.1.37,

I c"n! - 4s--lt'2su -L eC+Hf l^lto.+zul x 1.r87 -4(0.03059)lr/2:21' ' r :0.03013I (417r)(0.33es)(1.187) _l

so that the predicted peak-to-peak amplitude is 2lsB : 2.35 ft. From visualobservations at the site reported in [13-1], for modes of this type the doubleamplitude "could hardly have exceeded 3 ft."

13.1.4 Flutter and Buffeting of a Full-Span BridgeTheory. The flutter phenomenon was studied in some detail in Sect. 6.5 underthe assumption that two-dimensional geometrical conditions hold. In the caseof a fuIl-span bridge, the deformations of the deck are functions of positionalong the span so that this assumption is no longer valid. A generalization ofthe results of Sect. 6.5 to the case of the full-span bridge is presented herein.An example is included.

Let h(x, t), p(x, r), and u(x, t) represent, respectively, the vertical, sway,and twist deflections of a reference spanwise point .r of the deck of a fullbridge:

N

h(x,t): Z tt,14ng,1t1j: I

N

a(x, t): I a;(x)t(r)i: I

N

p(x, t) : 4rn,{;4ffi,{r)

(13.1.40a)

(13. r.40b)

(13.1.40c)

where h,(x) , p i@) , ui@) are respectively the values of the ith modal deformationfbrm at point x of the deck and {,(t) is the generalized coordinate of the ithmode.

If 1r is the generalized iner1ia ol' thc lull bridge in mode l, the equation ol'motion for that mode is

IlEt t 2l'ie,L, | *it,) - et (r3.r.4r)

;*462 sust,t Nl)t t)til,nN ilnlxit i;, il Nr;t()N liiltt,oil,nt t;, ANI) t,()wl tt ltNl :l

where f, is the damping ratio tll'tho ilh rturrlc, o; is its mdian nalrrrlrl lictprerrcy,and Qi is its generalized force, delincd by

I

Q, : I f(Lo" + L)hiB + (Do" + Dr)pB * (Mon * M)ail tkJ deck

(t3.t.42t

In the expression for Q, the following definitions of forces per unit span atsection x hold:

Aeroelastic (self:excitation) forces under sinusoidal motion:

(13.1.43a)

(13.1.43b)

(13. 1.43c)

Buffeting forces:

:,t ,:;t ,t Nt)l t):;t 'nN tlt ltxit :i 463

tlrrtttititt, ttr itplltrrpt t;r|,,' ttrltrtilltttrr'r' ltrrrt liorrs l(r ()71 (lrlrt lypiclrlly tlcpict a clirrr-irtutiott witlt ittcrcrrsrrrg llet;rrt'nt'.y ol llrc lirlt'r: lcvcl liirrrr lltut ol'thc stcady-slatc lilrcc. Itr wlritl is tlist'ussctl subset;uurlly lhc ccluivalcnt ol'unit aerody-rtatnic adtni(llnce: is lrrcilly irssrrrrrr:rl. 'l'lrcsc ussunrptions arc usually conserv-Irtive or do n<lt inlnrclucc iurporlanl crnrrs, as in an cxample to follow. Further,sclf:developed local, or signuturc, turbulence efl'ects are also not representedcxplicitly in the buffeting fbrce expressions. In the example to be presentedbelow, these do not happen to be important forces, though they could be incertain specific cases.

In whgl"f.qllews only a.si11gtg;ggde approximation to the total response willbe postulated. This ttinO'of isdffiption is justifiable from observation of thelact that typically just one prominent mode will become unstable and dominatethe flutter response of a three-dimensional bridge model in the wind tunnel.Clearly multi-mode response can also occur. This somewhat more complicatedproblem has also been treated in the literature [13-86] t13-9U. On the otherhand, the mode-coupling forces of the wind are usually not strong comparedto those of damping. This problem will not be pursued here.

Following the 5-i5rgle-mode folm of analysis, any mode i may be consideredin Eqs. 13.l .41 . The corresponding modal forms are then introduced into Eqs.13.1.40 to 13.1.44. This results, when all but those flutter derivatives shownure ignored as of lesser imponance. in

(r3.1.4s)

in which

lq, : h,, Pl or a;f (t3.1.46)

Because of the linear nature of the resulting equation of motion, it may be seenthat under this formulation the conditions of system stability are independentol the buffeting lorces.

The system equation

t:i + Z^yioioti -f <,sioti (13.1.47)

rnay bc rcwrittcn with a rrcw ll't:tlrrcncy oig, a new damping ratio "y,, anrl irhuli'cting firrcc Q;7, dcfinctl. r'cspcctivt'ly, hy

Gqq : f ,..*n?olf

r*:)ou'alr*of Lr* ratB] + K2H{a + K,r|.|,)

u*:)ou"'l*f Lu* xe;ff + x'eto + K'U*]

o*:)ou'alxrf er+ Kpf B+ + Kzpla + K,fp'f

+ (49! +\ d.t

+ P'f Gpp, + Atc,,.,ftiI " f u(x-tlLo : : pU-B l2C, -" 2' | - U

\ w(x- r) l

")il (r3'r'44.)* j*.0 [LbhiB -t D6p;B * M6u;l dx

M, : : puzBz ?rr,oT - (*) t#l

+ x'.e{c,,;,)t . - (KBQ, : ,pu'B''t ffi wf cr,o,

)ou'alzc,u@, t)fUI

(13.t.44b)

(13.1.44c)Do:

Note;' In the force expressions above, it is assumed that there is no interactionbetween the aeroelastic and the buffeting forces. This circumstance is partiallycompensated by measuring the aeroelastic forces under conditions of turbulcncc[3-89].x Furlher the sectional buffeting forces are written in a form that cxpresses their dependence both upon time-independent gust components antlsteady-state force components, this again being partially accounted for by assessing the "static" force coefficients at their mean values under turbulcnlflow. Modifications to these expressions introducing indicial lift-growth lirnctions can be made t6-971. These lead, in the frequency or power spcctr.rrl

*Turbulence was fbund ttl havc a stntng llvorablc cll'cct on (hc llrrttt r vckrt ily lurtl lhc rcslxrrrsr.to vortex shcdding of a sccliort trurdcl ol'{hc Quincy l}ritlgc. :rntl orr tlrt llrrrlt'r vckx'ity lirr lr lrrllmodel ofthcl,ion's(iirlcllrirlgr:rrcrrsstlrcllrrlr':rnl irrlt'l (Vlrntorrvtr)llt$\l:st.r':tlsolll lO,ll

_ Qio(t)Ii

.r;;' ,,' "',!,," ,','II(;, ., ( l.l.1.,1r{)

484 suspENDED-spAN BRIDGES, TFNritoN ,.ilnuctuRES, ANt) powrFt uNr s

27iaio : 2(iai - ff 4nyor,h, + pf Gp,t,, + Af G,,,,",1 (13.1.49)

lrQiilt) : )ou'n\ ).,,n[Luh,n * D6piB -t M6a;l dx (13.1.50)

For instability it is then ne-cessgry thqt "yr < 0; this leads finally to the sinsle-*g9S-,,,*11"1instabilitv crirerion ""

Hf Go,n, + pf Gp,p, + Atco,o, -- #,1, . + oto,,,,)t, (t3.l.sl)

in which only the important flutter derivatives F1f, Pf , At,,4f have beenretained. An a.ssumpJion inherent in this criterion is that the flutter derivativesretain full coheidiiCe among spanwise sections. The effect of reduied ioiierencecan be seen qualitatively as analogous to a reduction in the values of theGq,q,'

In practice the flutter derivatives I1f and Pf are most often negative invalue,* while,4f may take on positive values for advanced values of reducedvelocity

(13.1.52)2n

The effect of the flutter derivative A{ (an "aerodynamic stiffness" effect) is,in many practical cases, almost negligible. This reflects the relative magnitudesof the larger structural, versus the aerodynamic, stiffness for typical bridges.

For buffeting analysis, the generalized force may be rewritten as

/l\ ouzB2tl ax\i )O,rt,t

: ,,, Jo*. [.rr * DP, + Ma;|7 (13'1.s3)

where

L : 2c, 4-'') + (c! w(r' t\- U -*Cr)t (13.1.54a)

*Pf may be obtained by equating the expressions for the drag

/r\, : \;) p(u - n)BCD

/ l\ ou2Bxp,i,,: \r/ ,-Ncglecting thc tcrm in /2, thc rcsult is Pl = *-2ClK.

I) :2Cr'r'f

M:2cu+!+c;Y9't),.'U"'UDefining two new functions p(x), rl,@),

13 I utrftPt Nt)t t) llt,AN tthilxll ti

p@):2[Cyh;(x) * Cpp;(x) + Claai@)]

*(x): (CL+ Cp)h;(x) + C'1aai@)

( 13. r .s4b)

(13.1.54c)

(13.1.55a)

(13.1.ssb)

the integrand of Eq. 13.1.53 becomes

Lh1 r Dpi t Ma; : er;\4f + {(x)W; (13.1.561

The method of solution adopted here will be to seek the power spectraldensity of the bridge deck deflection. This is partly motivated by the fact thatthe power spectral densities of the wind components u(t) and w(t) are knownor can be reasonably estimated from the results of research.

Defining the Fourier transform of {, as

(13.1.s7)

with; : JJ, tne Fourier transform of the response equation for {, becomes

lr',0 - <,sz + 2i,pioc,rlfi : ry J.".* [rt"l Wf + t@)-*f]+

(13.1.s8)

Multiplying Eq. 13.1.58 by its complex conjugate and by 2lT, we obtain, ingoing to the limit T --+ @, the result

fTEi(<at : l* J, tiu)e-i"tdt

y?t(r?o - r')' + (2tio:ioa)2lElf

_ ( eurnrty' [ [,,,. ..,dx"dxr,: \ ,L ) l.,l " tx,,' xb'') r t

whcrc

(13.1.5e)

SUSPENDED-SPAN BRIDGES, TENEION STRUCTUHES. AND POWEF LINES

TI (r", xo, @): j* '7fit A,>u(xo, o) -t rlt(x,)w(x", a)l

x lg@)u*(xb, a) + rl,@)w*(xa, a)l (13.1.60)

Since the power spectral density of {; is defined as

)Sg,g,(c,r) : l* ;#f (13.1.6r)

we find that

,*[(' - (fr)')' * (,, *e)'],,,,,r,rlolt2g2tlz ll I: I "' I J J Zt le@')e@6ts"(xo' x6' <'t)

deck

+ t!t(x.)rlt(x)5,,@o, xt, a)l++ 03.1.62)'' I I

in which the cross spectra Su, and S,, have been neglected. (While limiteddata presently suggest that this is a conservative assumption, knowledge ofthese quantities in applications can improve accuracy.)

From this point on, the distributed cross power spectral densities of z andw will be assumed to take the real forms (neglecting their imaginary compo-nents)

su(xo. x6. c'r) = sr(co) I cl*" - xnl I"*ot--l {13'l'63a)

where C is a constant (see Chapter 2) satisfying

5nl 2onl_<c<_U_-_ U

s*(xo. x6, c,r) = s,(co) I cl*' - x'l I"*PL- / I

qt + 6OrzlU)P

z being the frequency of mode i.According to chapter 2, power spectral densities of z and rv in the atmo-

sphere may be approximated by the expressions (Eqs. 2.3.21 and 2.3.33),

2OOzuzx

(13.1.63b)

(13.1.64)

S,(n) : (13.1.65a)"i,:l#)'[^,[ t#*6u2*

(r3.r,7r)

13,1 gUSPENDED.SPAN BRIDGES

336zulS-(n) - UU + l0(nz/lJ)5t''l(13.1.6sb)

where z* is the friction velocity defined by Eq. 2.2.5, ru is frequency, and z isdeck height.

In calculating.lg,4, as in Eq. 13.7.62, it can be observed that the followingtypes of integrals require evaluation:

so that, finally,

R, : j f e@)e@"-ctxa-xut/t++ (13.r.66a)

*, : J f ,r,<*.1,r,<rol"-ctx'-x6ttt ++ (13.1.66b)

[1 - (<,r/c,r;s)2]2 + f2y,(alc,:,o)12

ptttzt,x?o{ReS,+ nrs,}h 03.1.67)st*,(') :

The variance of {; is

"?, : J; s4,4,(n)dn

S,(n)dn : 6u2*

S*(n)dn = l.7uzx

(r3.1.68)

(13.1.70a)

(13.1.70b)

which, for example, can be approximated with the aid of the formula (see Eq.5.3.39),

[- s(,r)an _ l* g1r1a, * rno_s@o) (13.1.69)Jo tl - (nlnsy2lz + [271;nlns)]2 : Jo o,t)u" ' 4,y

Referring to Eqs. 3.I.65, we find that

J;

J;so that a buffeting calculation f<lrmula is obtained for the variance of {;:

. orl*Y*tr,,.lJ#

*+468 liL,st,t NL)LD:;t'nN tlnt{xit l;, n Nl;t()N liilt(,(:tt,t u li. nNt) t,()wUt ilNt l;

Then from Eqs. 13.1.40 thc standanl rlcviatiorrs arc ohtairrctl

on,@) : h;(x)Bog,

oo,(x) : p,(x)8o4,

o,,(x) : a;(x)og,

l.o0. ti

0. (i

0.,'1

i;t,t;t 'l t{t)t II:,l'nt.t |ililtr(it ,, 4ti9

\,

ArA;

Example In this example parameters of the Golden Gate Bridge are employed.A I :50 scale section model was used to obtain flutter derivatives H,t and A!(t : l, . . ., 4). A set of these derivatives forzero-degree wind angle of attackin smtxrth flow is prcscnted in Figs. 13.1.7 and 13.1.8 [13-97].'l'hc vibration nxrdcs and ficquencies of the bridge, together with their modalin(cgrals (),,,,,, warc obtaincd for the first eight modes with the results given inTablc l3.l.l.

Modal lirrtns arc suggested by the notations S : symmetric, ,4S : antisym-metric, L : lateral, V : vertical, and Z: torsion. Values of the modalintegrals G,/;,,,, suggest the importance of the mode: in Table 13.1.3 the largestin each category (i.e., vertical, lateral, torsion) is underlined. The most pro-nounced modes are mode 6 (vertical), mode 1 (lateral), and mode 7 (antisym-metric torsion).

Flutter. The torsional aerodynamic damping coefficient ,4f exhibits a pro-nounced change of sign with increasing velocity, indicating the possibility ol'single-degree torsional flutter (Fig. 13.1.8). Mode 7 was selected as the mosrvulnerable to flutter instability (Fig. 13.1.9). It is the torsional mode with boththe lowest frequency and greatest Gn,r, value. Experience has shown (e.g., in

0 z 4 6 B 10{ IrlnB

I'IGURE 13.f.8. Aerodynamic coefficients A! (i : 1,2,3,4), Golden Gate Bridge(courtesy of Dr. J. D. Raggett, West Wind Laboratory, Carmel, CA).

the original Tacoma Narrows case) that the lowest antisymmetric torsion modeis typically the most flutter-prone in long-span bridges. In the Golden GateBridge case this mode is practically a complete sine wave along the main span,with a node at center span and practically zero amplitude on the two side spans.

The pertinent parameters in this case are, in the units* kip, ft, s:

fz : 0.005 (arbitrary choice)

1z : 8.5 x loe lb ft s2

TABLE 13.1.1. Frequencies, Types of Modal Forms, and Modal Integrals for(lolden Gate Bridge

Frequency Type Gn,n, G,, Gon',

(13.1.72u)

(13.1.126)

(13.1 .72c)

3

l0

( iolrlt'lr ( iirlt' liritlgt.('A)

0.2:0.0

l

-+./..

- 0.2

-0.4

"'Irl.o l

p : 2.38 x l0 6 kip ft-a s2 : 0.002378 Ib ft-a s2

: 0.002378 slugs/ft3FIl+'

7.0

0.0

-7.0I I7,r r lil

FI(;URIt 13.1.7. Acnxlyrlurric cocllicicrrts //1" (i l. I, t,,l)(cttutlcsy ol'l)r. .l . l). l{lrggell. Wt:sl Wirrrl Lirlxrr':rlory. (':rlrrt.l,

I

234.5

()

1

8

0.0490.087o.lt20.1290.1400.t640.t92o.t91

LASVI

LSV,

n s'l'rS'l',

3.33E-01'7.398-t53.09E-017.828-t45.58E- l43.87E- I 3

3.32F.-022.11tt.-Ot

8.03E-05t.tlE-15l.24E 021. 16E- 142.438-141.25F.-14l.29lr I (X)

2 5511 0l

2.62E-163.25E-011.728-141.90E-01I .91E-013.44!,-0_L6.611;.-12.) 50ti t2

l .l.lli Nr'ulrrrr'.. r'rl kip lO(X) lb lirrcc; I lb lirrtt'Iil.

t.' .) ltls () lil rrr/s . I lt o tl),ll,i

{)

o9Fqtroo

o.63

O-oo

()

o8iotrd>E!t=oeg.xhou!r+6)A(){:5rtrxoqfE9

0)

'roC!r\6\OV:ErqEIF

-.r&e,\JE-4Fr6

o60ao(9 0l At(soqouD luourocqdslp leed ol lead

boort

470

I3.I SUSPENDED.SPAN BHIDOES 471

B=90fiI : 6451 ft

')_9troro, - L.z

The flutter criterion in this case reduces to

u * * 4hltt1 1 <-- pB"Goro.,l

or

At - o-tzt

From the graph forel'6ig. 13.1.8) the corresponding reduced velocity value(with n - nt : 0.192 Hz) is

U1: 4'32

which corresponds to a critical laminar-flow flutter velocity of

ftmikmU,,: (4.32)(0.192X90) :74.65 - : 50.9 * : 81.9 *Bufteting. Four modes, mainly active over the main span (see Table 13.1.1),were examined, as listed in the following table:

TABLE 13.1.2. Generalized Inertia of Full Bridge for Four ModesMode i Frequency (Hz) l0 e I, 0b ft s2)

ASVrSV'

ASTrST'

These additional data were used

lus : 4144 ft : main span length

zo : 0.02 ft

z : 220 fl = dcck hoight

U' 2.5 ln(z/r,')

2478

0.08700.12850.19t6o.1972

15 .716.158.508.59

472 SUSPENDED-SPAN BRIDoES, TENSIoN STnUoTUHES, AND PoWER LINES

Co : 0'34

Ct : 0.215

ft:,,,Cu:0

dC,,d" : -0.lll

The modes involved were assumed to have the forms of simple sinusoids:

/ ?r"r \hsr, : h6 sin( ,lrsl/2nx\

hnsv, : &6 sin( a I" \lrt /' /nx\otsvt : as sin[ - I" \/rrl. /2rx\

QASVT : "O

.rn\ ,^ /

Using these approximate expressions, we obtain the following results:

For a vertical mode

For a tonional mode,

Ra: 4c,htRc)Rv : (CL * Co)z n?AtC>

nr: +C2naf,ng\

n, : c,ja{n1cy

(t3.t.73a)(13. r.73b)

(13.t.74a)

(13.1.74b)

where, for symmetric modes,

Rrcl: f f TXo Trxn

J"J,t" ffi sinffi e-cv"-',,,, ++ (r3 r.75a)

R(c) : (T)' or-' ( r3. r .75b)

13,1 SUSPENDED.SPAN BRIDoES 47s

with

(*tn''- K '2r2{l+e-K\{ i"" = P +7 + 1V;;y (r3'r'76a)

I r: c+ (r3.r.76b)\tand, for antisymmetric modes,

R(c) : ll ,i,2f:,i,zfr:,-Ctx'-xut ++ e3.t.77a)

R(c) : (T)' O,,' (t3.1.77b)

with

fw':p:vp.W.# (13 178a)

I *: c+ (r3.r.78b)\tWe now have the two following forms of Eq. 13.1.71:

For a purely vertical mode,

4 :l#l'l<",'r\t# + 6".)

+ (cL t c)z "?,lt# * t.r",.))V 03.r.7s)

For a purely torsional mode,

4 :lutill'f,rr*,,,'lY# + ou'*)

+ (c'vai)2lt#? * t.r,'.))V (13.1.80)

Using the data, the results of Fig, 13. 1.9 ure culculated. In each casc ni6 is thenatural trequency of the mocle in qucntion.

*474 t;t,til,t Nt)t t) lit'nN nnlxit :i. n Ni,t( )t.l i.llrt(;llill :; nNl ) t,()wl l ltNl :

The calculations just prcsontc(l arc inten(lc(l lo hc illuslralive . l)ct;rils rrrrtlapproximations may difl'er sorncwhat accorclirrg t.o thc tlcsigncr's .jutlgrrrcrrt.

Dependence of Aeroelastic Stability upon Bridge Characteristics. 'l'hc:

aeroelastic stability of a bridge is controlled by several factors:

l. Geometry of the bridge deck. Unstable shapes include solid girdcr or"H-section" types of deck form; open-truss deck sections with closed,unslotted or unvented roadways; and certain very bluff cross sections.On the other hand, stability is enhanced by streamlined forms and byopen-truss sections that contain vents or grills through the roadway sur-flce.I'-rcquencits o.l'vibrution rf the bridge. High torsional frequencies tendto cnhancc stability. Examples of torsionally stiff shapes are closed tor-sion box scctions, rlr dccp trusses closed by roadway and wind bracingto constitute a latticed tube. On the other hand "H-sections" are tor-sionally weak. Stability is also enhanced if the torsion-to-bending fre-quency is high.Mechanical damping of the bridge. Aeroelastic stability is clearly en-hanced if the mechanical damping ratios of the bridge are high. We alsomention the possibility of enhancing the aeroelastic stability of a bridgcby vibration reduction devices. Such a device, consisting of tuned massdampers (TMDs; see Sect. 9.4.1) provided with disk brakes and notrequiring any power source, was installed on the 1939 Bronx WhitestoncBridge t13-1081.Deck inertia. Heavier systems increase the flutter threshold.

2.

13.1.5 GallopingThe susceptibility of a bridge deck to galloping can be determined by inspectingthe plots of the lift and drag coelficients C7 and Cp versus a (e.9., Figs. 13. 1.3and 13.1.4). The condition for incipient galloping instability is (see Sect. 6.2):

dCtda +cD<0 (13.1.8 r)

Cases of large-amplitude across-wind galloping of suspended-span bridges havc:not bcen reported to date.

It follows from Eq. 13.1.81 that avoidance of deck shapes with regions olstrongly negative lift curve slopes is conducive to stability.

13.1.6 Structural MembersTowers and bridge members of circular, squarc, I or H scclion lnay bc susceptible to wind-induced vibrations, particularly urrrlcl llrt':reliorr ol'shcrl vor.tices.

Iil I :;lll,l'l lllrl Ir'.1'l\lI Illlllr:l 475

ll srlst't'pt ilrilrly (() v()tl('\ tntlttt't'rl vibt;rlrorr r,.; lr prnlrlt'rrr. out'ol llrrr't'ly1x.sol solrttittlt ctttt, itt gcnt't;rl, lrt'rrst:rl. l;ilsl, llrr. slrlllrr.:.s ol llrr- rrrt'rrlx't t:rlr lrt'ittcrcasccl so lhal llrc o.ilicirl witttl vcloci(.y t'xr't't'tls llrt' vt'lrx rllt's llr:rt rrril'lrl lrt.t:xltoctccl to occur clurirtg the: lili'ol llrt'strrrtlrrrr'. l'o t'rrlt'rrl:rtt.llrt't'lilicrrlvclocity U,,, thc lirllowing rr.:llrlion is rrsrrl:

tt,l)u,, : s- (13.1.82)

4.

whcre n I is the fundamental frequency of vibration in the across-wind direction,/) is the across-wind dimension, and S is the Strouhal number of the member.Ilccause the dimension D of an individual member is small compared to theintegral scale of the atmospheric turbulence, it may be assumed that the memberbchaves aerodynamically as if the flow were smooth so that the Strouhal numbercan be taken from Table 4.4.1.

Second, devices may be used that spoil the coherence of the shred vortices.llelical strakes and shrouds of the same design and with the same proportionsrrs indicated in Sect. 10.3 may be employed on circular members. Figure 13.1.10shows a spoiler device consisting of staggered fins that was successfully used{o suppress the oscillations of a pipeline suspension bridge [10-23,13-48]. Thistlcvice would not be effective if the member were exposed to winds blowinglrom any direction (as would be the case if the member were vertical) ratherthan from just the direction parallel to the plane of the fins. Figure 13.1.11shows perforations in the web of an l-section member that reduce vortex-induced response and galloping if the wind direction is normal to the web butwcre found to aggravate the galloping problem if the wind direction is parallello the web.

Finally, in certain cases tuned mass damper (TMD) devices may be em-lrloyed. The principle of these devices was discussed in Sect. 9.4. An exampleol' a TMD used to control the oscillations of bridge I-beam members is de-scribed in detail in [l3-49]. The device consists of a cantilevered rubber-shank

;#-

Itl(illltl,l l-1.1.10. Sl:rg,picrt'tl lrrr.,,rrr;r lrrlrr'lrrrt':,rr:,1x'rrsiorr lrlitllit'II t.lli, lO.ltl

476 :il,t;t 'l Nl)t t):;t,nN tnil)ct :i, llN:il()tt :iililr{jil,1il li. nNt) t'()wt il ilNt :

FIGURE 13.l.ll. Perforated web of l-section member.

pendulum weighted at the lower end. The weight employed may be of the order<tf O.l5% or more of the weight of the structural member.

To reduce vortex-induced oscillations of individual cables such as those incable-stayed bridges or the deck hangers of suspension bridges, cable-to-cableties, friction or hydraulic dampers, or TMD devices may be employed. In casesin which the oscillations cannot be prevented, fatigue-free cable terminationsrnay have to be used to avert damage at the supports.

Mitigation measures may also be necessary to reduce large-amplitude vibra-tions (0.6 to 2 m double amplitude) that were observed in cables of cablc-stuyccl bridges under the combined action of rain and wind. Wind tunnel studiescstublislrcd that the vibrations are due to mechanisms that include the formationol't[:strrhilizing rivulets at the upper part of the cable [3-105]. The use olt':rbles with protubcrances was found to be effective in suppressing the vibra-l iorrs .'llrt' rrriligirlion ol' bridgc tower oscillations by means of TMD and tunctllrrlrritl rLrrrrpcr ('l'l,l)) dcviccs (see Scct. 9.4) is described in [9-79].

13.2 TENSION STRUCTURES, POWER LINES, AND POLES

13.2.1 Cable RoofsVilrr':rlions ol' crrblc-srrl.rpotlctl nxll,s arc causctl prirrcipirlly by btrll'clirtg lirrccstlrrt: lo irrcitlcrrt trrrtl slrrrcttrr-c inrlrrcctl lurbrrlc:rtcc. ll is lil.t'ly llr:rl llrrttr:r'(srrll.

rr:' ll tJ:,1()N:illll,{:ll,l tl :,, l'0Wt n ilUt '. /\l.lt) l,{rl l.. 477

cxcilctl rlst'illitltolt) ol r';tlrlt' trrols is t'irt't'. sirrtt' ruoi{ tool :ilnt( lul(':. (l() u()lpcrtrrit erttrlttglt tlerllcclion lo itttlttt:c signilit;rrrt tlr;rrr1',t'r. rrr llrr' ;rt'r,rrlVnilnl('lilrccs.'l'hc rturgrritrrclc: ol tlrc htrllctirrg lirrct's tlrrr lrt'rrrvt'slrp,;rlt'rl rrr wrrrtl lrrrurt'ltcsting ol'acroclastic or rigitl nurtlt:ls. lrr llrt' lrrllt'r t'lrst' lrxrtlrrrll Irrnt'tiorrs lo lrt.used in dynamic studics can bc: tlt'vt'lopt'tl l.nrrrr llrt'n'r'otlt'rl llrnt'tlt'grt.lrrltrrlpressures.

Unwanted vibrations will rtol occrrr il lhc cirblc nxrl'is sullicicntly still'.Stiffness is achicvcd by thc pnrvisiorr ol'sullicicnt wcight, fbr examplc. in theform of precast concrctc r<xrl'pancls, by prctcnsioning of cables; and/or by theprovision of a stifl'ening systcm of tensioned cables with curvature opposite tothat of the main, load-bearing system. In double-curvature roofs, the load-bearing and stiffening cables form a network-in most cases, orthogonal. Un-less carefully designed, such roofs may exhibit serious vibration problems thathave, in the past, necessitated the provision ofadditional ties and the lubricationofcable intersections to reduce noise caused by cable-to-cable friction. In singlecurvature roofs, stiffening cables may be provided at some distance underneaththe load-bearing system, as in the case of the well-known Utica, New Yorkauditorium. The two layers of cables and the vertical members joining themfbrm elements with considerable stiffness that prevent the occurrence of anysignificant wind-induced oscillations.

Recent studies on wind effects on cable roofs are reported in [3-50] toI l 3-s3].

13.2.2 Air-Supported and Tensioned Fabric StructuresLong-span fabric structures, especially of the air-supported type, are a relativelynew architectural and engineering development Il3-541 to [13-58]. In manyinstances their design has been based on rudimentary representations of thewind loading tl3-581. Attempts to develop more realistic and elaborate windloading criteria or wind tunnel modeling procedures are reported in [13-59] toI l3-631.

13.2.3 Power Lines and Poles'l'he design of power lines requires the estimation of drag forces and the pre-tliction and/or mitigation of wind-induced vibrations.

A comparison between drag coefficients on standard aluminium conductorswith a steel core and trapezoidal wire conductors is reported in [13-98]. Windtunnel tests showed that for wind speeds higher than 85 km/hr, drag coefficientsIirr trapczoidal wire concluclrlrs urc srnirllcr than for standard conductors. Ref-Lrrcnce ll3-991 reports thut wintl lrnrrcl rrrrrl lit'ltl lcsts rlrr 3.(16-rrr long slanrlanlt'rtttcluctors yicldcd sirrtilur tllrrp, t'ot'llit'icrrls. liir'ltl nr('lrsurcnlcnls ol' swing lrrrglcs antl insulaltlr lirtccs sltowt'tl llr:rl rvrrrrl lortt's t':rlt'rrl:rlr'rl lry lrssrrrrrirrl', rrrrilirrlrr wirrtl lrllrtls hitsctl ()tt rn(';r:rur('{l tlr;r11 trx'llt( l('nl:. ()v('r('sltnritl(' lltt' lrtlu:rllirtcc:s ll.l-99, l3 l(X)l A lrkt'ly ctlrl;111;s111111 ltr'r,.;rl lt'rr:,1 rrr l|:rt, rr tlrt'rrrr1lt'r'li'r't sPirliirl coltcrcrtt't' ol llrr' ;r,'r,r,lyniunr( l,r;rrl:, I I | / l I

U

U

U

U

478 lit,r;t 't Nl l t) r;t'nN nt lilxit :, l f..t:,tot.t r,ll,{ nll :;. nNt) t,()wt n nNt :;

'l'hr: rrtaill viltl'atiolr ;ttrrlrltrtns ol lorrp, sp;rrr t'irlllt:s:rntl Jllwt:r lirrt.s:rle irssociated with voncx-shodding, lirll slllrn glrlkrpiltg, rrrrtl subsp:rrr wirker intlucctlgalloping. These problems arc briclly discussocl bclow. I,or aclcli(iorurl irrlirrmation and studies on wind effects on power lines, sec ll3-641 kr ll3-7 ll, rrnrlll3-1011 to [13-103].

VortexJnduced Oscillafions. Vortex-induced or Aeolian oscillations in longspan cables are generally caused by winds with speeds of the order of 2 to l0rn/s. The oscillations generate packets of narrow-band random waves arrivingirl lho cirblc supports. Since the cable is not perfectly flexible, the waves causr:oscill:rlory honcling strcsses near the supports that result in fatigue damagc,rlrlt'ss lrrrr(c:ctivo rrrcasurcs are taken [13-72, 13-73, l3-],4]. In the casc ol'slr':rrrtk'tl wiri.:s lirtigrrcr tlirrnagc can be produced by shear-induced friction, which:rlli't'ts rrr:rirrly tlre irrrrt.r wircs.

APPtrtrtcltcs Ltsctl to prcvcnt fatigue failure include the provision of speciulcttsltiortccl suppotls that allcviate the bending stresses and applications of thcrIr-urctl rrrass clarnpcr ('l'MD) concept such as the classical Stockbridge dampcrll3-7-5, 13-16, 13-771. The Stockbridge damper (Fig. 13.2.1) consists ot'irroactivc countcr-vibrating mass with a fairly wide band of frequency possibil-itics. 'l'hc cllbct of the mass is to suppress to a large extent the last half-wavc(ncarcst thc support) generated by the cable oscillation. Like all TMD deviccs(scc Scct. 9.4.1), the Stockbridge damper is not an energy dissipation device:to any apprcciable extent; it is, instead, an anti-resonant spring-mass devicc.stockbridgc dampers or similar devices can be readily purchased for a widenrngc ol' spccific applications.

Full-span Galloping. Full-span power line galloping occurs most characru.istically when ice forms on conductors and creates a new surface contour tlr:rtis pnrne to galloping tl3-781 to [13-80].

\,

ilt II nt N(;t :; 479

Mclttts ol ltllt'vt:rlrttl' llrt'l';rllopirrg ol'powcl lirrt.s lrirvr: ilrclutle:tl rrrcltirrg 6l'ict: by cilt-ryirrg lril'111'1 ('un('nls lr'rupoliu'ily irrrtl llrus hcrrling lhc clblcs, installlrtion ol'g:rlltlllirrg w:tttrirtll scns()rs localr:rl al sul)p()rt towcis in rcgigns whcrecablo icing is knowrr lo tlrkc pl:rco, urrti-galklping acroclynamic {evices designedkl spoil thc l<lcal llow, and tuncd rlass clampers at the center of cable apun..

subspan Galloping. Subspan galloping, or wake-induced lateral galloping(see Sect. 6.3), has occurred repeatedly in grouped or bundled conductorsI l3-8ll to [13-84]; for a recent review see [9-ll. countermeasures have in-cluded (l) detuning the various cables in a bundle from each other by meansof special spacers and (2) increasing damping by providing energy-diisipatingspacers or by lowering the cable tensions, a measure that results in an increaseof the inherent self-damping of the cable. None of these solutions has beenf'ully effective. In particular, some highly complex and costly spacers witharticulated and spring loaded arms have been found to be unsaiisfactory.

conceptually simpler-although again costly-solutions have included a largeincrease in the number of spacers used between supports so that subspan lengthsare cut down and the corresponding frequencies are raised, and a continuoustwist of the conductor bundle from support to support, which breaks the span-wise coherence of the vortices shed in the wake of the windward conductor.The continuous twist solution has been judged, so far, to be impractical forapplication in the field.

Poles with Partial lce coating. Experiments on circular cylinders with ap-proximately uniform coverage by ice or snow over about a third of the circum-f-erence (Reynolds numbers based on diameter 50,000 to 500,000) indicatedthat such coverage can create strong susceptibility to galloping motion, pafiic-ularly for ice or snow thicknesses of about 3% to 6% or tfie rytinaer diameterll3-1061. The research of t13-1061 was motivated by massiu" loss", of poleswith partial snow coverage in an Aleutian island wind storm [13-107].

REFERENCES

l3-l F. B. Farquharson (ed.), Aerodynamic stability of Suspension Britlges, partsI-v, Bulletin No. ll6, university of washington Engineering ExperimentStation, Seattle, 1949-1954.

l.\ 2 C. Scruton, "Sevem Bridge Wind Tunnel Tests," Sur-veyor, 107, No. 2959(Ocr. 1948;, 555.ll -l C. Scruton, "Expcrirtcnt:rl Investigation of Aerodynamic Stability ol suspcl-

sion Bridges with spccill llct'crence to proposed sevcm Briggc.,, l,rpr. Irr.st.Cir.,. l'.ng., l, Pull l. No. 2 (M:rr. l9-52), lg9_222.

ll4 A. Hirai, I. ()klrrrclri,:trrrl l. Miyrtl:r, "On thc Bchlviolol'srrspt'rrsrorr llritlpr.srrrrrlcr wirrtl Acri,r," irr l't., t ttlirt,g.t rtl rlta lnttrtutri.rttt! ,\\,rrtltrt.tit,tt rtrt ,\'rt.sJ)t'tt\i.)tt llritl,4'.t, l.:tlrorrlotio N;r( i()nitl tlc lirrgcrrlrllilr ('ivll, l,islr11rr. l()(r(r,pp. 24() 2.56.

lr'l(;tilllt l.].2. l. Stoc'kbritlge tlrrrrr;x.r

48O l;tJlil't Nl)l I)l;l'nN llllll)(il :i, llN:;l()N :;lllll(:l(,lll i;' nNl) l'()wl ll llNl '

I 3-5 A. G. Davcnpo(, N. Isyutttov, I)..1 . lilttltrr-, ittttl ('. li. l'. llowert' y' '\trttlt'ttlWintl Action on u Suspcnsion Ilritlgt'luring lircctitttr ttnd (ltnltltlitnt,llc.lxttlNo. BLWT-3-69 with Appendix BLWT-4-70, F-aculty of Engincorirrg Scicttce:.university of western ontario, London, Canada, May 1969 and March 1970. 1.1,2 I

W. H. Melbourne, West Gate Bridge Wind TunneL Tcsl.r, lntcrnal Rcporl,Department of Mechanical Engineering, Monash University, Clayttln, Victoria, Austrialia.

13-6

l3-7 A. G. Davenport, 'The Use of Taut Strip Models in the Prediction of thcResponse of Long-Span Bridges to Turbulent Wind Flow-Induced StructuralVibrations," in Proceedings of the IUTAM-IAHR Symposium on Flow-lnducecl Structural Vibratktns, Karlsruhe, West Germany, 1972, Springer-Ver-lag, Berlin, 1974, PP. 373-382.

13,t3 R. L. Wardlaw, Static Force Measurements of Six Deck Sections for thcI'n4xxr:d Ncw Burrard Inlct Crossing, Report No. LTR-LA-53' NAE, Na-{ir>nal Rcscarch Council, Ottawa, Canada, 1970.

l3-9 R. H. Scanlan, Rct:cnt Methotls in the Application of Test Results to the WirulDcsign o.f' Long, Suspended-Span Bridges, Report No. FHWA-RD-75-11-5'Fecleral Highway Administration, Office of Research and Development, Wash-ington, DC, 1975.

13-10 R. L. Wardlaw and L. L. Goettler, AWind Tunnel Study of Modifications tttImprove the Aerodynamic Stability of the I'ong's Creek Bridge, Report No'LTR-LA-8, NAE, National Research Council, Ottawa, Canada, 1968.

13-11 Y. K. Lin, "stochastic Analysis of Bridge Motion in Large-Scale TurbulentWinds,' ' in Wind Engineering , Proceedings of the Fifth Intemational Confer-ence, J. E. Cermak (ed.), Pergamon Press, Elmsford, NY' 1980'

t3-12 W. H. Lin, "Forced and Self-Excited Responses of a Bluff Structure in a

Turbulent Wind," Doctoral Dissertation, Department of Civil Engineering,Princeton University, 1977.

13-13 R. H. Scanlan and R. H. Gade, "Motion of Suspended Bridge Spans undct'Gusty Wind," J. Struct. Dlv., ASCE, 103 (1971)' 1867-1883'

13-14 D. B. Steinman, "Aerodynamic Theory of Bridge oscillations," Proc. AscF),75, 8 (Oct. 1949), ll47-1184.

l3-15 D. B. Steinman, "Aerodynamic Theory of Bridge oscillations," Proc. ASCF),76, I (Jan. 1950), 154-158.

l3 16 R. H. Scanlan and A. Sabzevari, "Experimental Aerodynamic Coelficients irr

thc Analytical Study of Suspension Bridge Flutter," J. Mech' Eng' Sci',lnstituti<rn ol'Mcchanical Engineers, London, ll, 3 (June 1969)' 234-242'

13 l7 R. H. Scanlan and J. J. Tomko, "Airfoil and Bridge Deck Flutter Deriviltives," J. Eng. Mech. Div., ASCE,97, No. EM6 (Dec. 1911), l7l1-l'737

13-18 R. H. Scanlan, J. G. Bdliveau, and K. s. Budlong, "Indicial Aerodynartrit'Functions fbr Bridge Decks," J. Eng. Mech. Div', ASCE' 100, No' EM4(Aug. 1974), 651-672.

l3-19 R. H. Gade, H. R. Bosch, andw. Podolny, Jr., "licccnt Acnxlynarnic sltttlies of Long-Span Bridges," J. strucr. Div., AS('li. 102, N0 S'l'7 (.lrrly 197(r),1299-13',t5.

1320 T. A. Rcinhokl, H. W.'l'iclcttt:trt, itrttl li..l . Mitltt't. Iltr' l,'r.tiurrttl Ilr'sltrtrt.st'pl tt Stt,sltt,tt,titur lltitl,qr Stilli'rtin,q'l'nt.s.t Mrnltl lrt ltrtltttltrtrt ttrtt! lo rtrr lllt

lll lllll N(:l :l 481

.\'tt.'unt ()lt.\tttr'lr', llr'1xrr1 No. Vl'l li /'l .lli. l)r'p:ttltttctll ol lingincclilrg Sci-clcc 11tt Meclrirrrics, Virgirriir l'olylt't lrtric lrtsliltrlc alrtl Statc Univcrsity,llllcksburg, 1974.'f. A. Rcinholcl, H. W.'l'iclctrrrrtr, rrtrtl li..l. Mahcr' Dccay of the Wake of aBtsx (iinlcr, Roporl No. Vl'}l l1'7.5 lt3, l)cpartment of Engineering Scienceand Mcchanics, Virginia l)olylcchnic lnstitute and State University, Blacks-burg, 1975.

13-22 T. A. Reinhold, H. W. Ticleman, and F. J. Maher, Wake Study of a Suspen'sion Britlge stillbning Tnzss, Report No. VPI-E-75-17, Department of Engi-neering Science and Mechanics, Virginia Polytechnic lnstitute and State Uni-versity, Blacksburg, 1975.

13-23 G. Roberts, "The Sevem Bridge. A New Principle of Design," Proceedingsofthe International Symposium on Suspension Bridges, Laboratorio Nacionalde Engenharia Civil, Lisbon, 1966, pp. 629-639.

ll-24 C. Scruton, Experimental Investigation oJ'the Aerodynamic Stability of Sus-pension Bridges, Report No. 165, National Physical Laboratory, Teddington,u.K. 1948.

ll-25 C. Scruton, Experiments on the Aerodynamic Stctbility of Suspension Bridg,es:Results of Tests of a Fult Model in Horizontal winds , Report No. I 85, NationalPhysical Laboratory, Teddington, U'K., 1950.

l.\-26 D. E. Walshe , The Aerodynamic Investigation for a Box-Type Roadway Deckfor a suspension Bridge Proposed for the Humber Crossing, Special ReportNo. 012, National Physical Laboratory, Teddington, U.K., 1968'

l.\-21 D. E. Walshe, The Aerodynamic Investigation for the Suspended Structure ofthe Proposed Bosporus Bridge, Special Report No. 020, National PhysicalLaboratory, Teddington, U.K', 1969.

1.1,28 A. Hirai and T. Okubo, "On the Design Criteria Against Wind Effects forProposed Honshu-Shikoku Bridges, ' ' in Proceedings of the International sym-posium on suspension Bridges, Laboratorio Nacional de Engenharia civil,Lisbon, 1966, pP. 265-272.

l \ 29 M. Ito, "On the Wind-Resistant Design of Truss-Stiffened SuspensionBridges," in Proc. second (lsA-Japan Research seminar on wind Effects onStructures, Kyoto, 1974, University of Tokyo Press, 1976, pp' 285-296'

I ] l0 N. Shiraishi, "on the Aerodynamic Responses of Truss-Stiffened Bridge Sec-tions in Fluctuating Wind Flows," in Proceedings of the IUTAM-IAHR Sym-posium on Flow-lnduced structural vibrations, Karlsruhe, west Germany,I 972, Springer-Verlag, Berlin, 197 4, pp. 40 I -405.

I l .ll I. Konishi, N. Shiraishi, and M. Matsumoto, "Aerodynamic Response Char-acteristics of Bridge Structures," in Proceedings of the Fourth InternationalConference on Wind Effects on Buildings and Structures, London, 1975, Cam-bridge Univ. Press, Cambridge, 1976, pp. 199-208-

I t 12 T. Okubo, N. Narita, and K. Yokoyama, "Some Approaches for ImprovingWincl Stability f or Cablc-stayed Girder Bridges," in Proceedings of the FourthIntcrnutitnal (\rqli'rrtrt'c ott Wirul I')ffctts on Buildings and Structures, Lon-don, 1975, ('rrrrrbritll',c (lttiv. I'ttss, ('rrrrrbritlgc' 1976' pp' 241 249.

II iI Y. Nlkulrrtrlr tlul l. \'oslrrrrrr:r, "llitt;tty lilttllct ol'sttspcnsion Bridge DeckSt'cliorrs, .l . l'.tr11. NIr'tlr l)tv n S('lr. 102' No lrM'1 (Atrg' 1976), 6U5 700'

482 1;t,t;t't Nt)t I)l;l'nN ltllll)(il l;, llN:;l()1..1 1;llll,(.lt,lll :;, nNl) l'()wl ll llNl

l3 34 'l'. Miyata, Y. Kuho, irrrtl M. llo, "Arurlysis ol Acrocluslit' ()st'illrlitttrs olLong-Span Structurcs by Nonlincar Mr.rlti l)irncnsional l)nrcctlurcs," irr /)zrceedings of the Founh Inlcrnationul Confcrcru:c tn Wirul Ellcct.s ort liltiltlirt11.tand Stucture,r, London, 1975, Cambridge Univ. Press, Carttbt'idgc, 1976,pp.215-225.

13-35 M. Ito and Y. Nakamura, "Aerodynamic Stability of Structures in Wincl,"IABSE Surveys 5-20182, International Association for Bridge and StructtinrlEngineering, ETH-Hcinggerberg, Ziirich, 1982.

13-36

t3-31

R. L. Wardlaw, A Preliminary Wind Tunnel Study of the Aerodynamic Stabilit,-ofFour Bridge Sections for the Proposed New Burrard Inlet Crossing, RcponNo. LTR-LA-3 l, NAE, National Research Council, Ottawa, Canada, 1969.R. L. Warcllaw, "Sornc Approaches for Improving the Aerodynamic Stabilityol'f]ritlgc lLoatl l)ccks," in Pnx:ecdings of the Third International Conferenctort Winrl lifli't't:; ott Builtlings urul Slructures, Tokyo, 1971, Saikon, Tokyo,l()12, pp.93 I 9.+0.

l3-3tt ll. 1.. Wardlaw, A Wirul Tunncl Study of the Aerodynamic Stability of thcPn4toscd PasLrt-Kenncwick Intercity Bridge, Report No. LTR-LA-163, NAIlNational Rcscarch Council, Ottawa, Canada, 1974.

13-39 H. P. A. H. Irwin, Wind Tunnel and Analytical Investigation of the Responstof'the Lions' Gate Bridge to a Turbulent Wind, Report No. LTR-LA-2 10,NAE, National Research Council, Ottawa, Canada, 1978.

13-40 C. Ostenfeld, G. Haas, and A. G. Frandsen, "Motorway Bridge AcrossLillebaelt. Model Tests for the Superstructure of the Suspension Bridge," inProceedings of the International Symposium on Suspension Bridges, Labora-torio Nacional de Engenharia Civil, Lisbon, 1966, pp. 587-608.

13-41 K. Kloeppel and G. Weber, "Teilmodellversuche zur Beurteilung des aen)-dynamischen Verhaltens von Bruecken ,' ' Der Stahlbau, 32, 3 (1963) , 65 19 .

13-42 K. Kloeppel and F. Thiele, "Modellversuche im Windkanal zur Bemessungvon Bruecken gegen die Gefahr winderregter Schwingungen ," Der Stahlbau,32, t2 (1961), 353-365.

13-43 F. Leonhardt, "Zv Entwicklung aerodynamisch stabiler Haengebruecken,"Bautech, 45, l0-11 (1968), 1-21.

13-M F. Leonhardt, "Latest Developments of Cable-Stayed Bridges for LongSpans," Bygningsstakiske Medd., 45, 4 (1974), 89-143.

l3-45 H. Loiseau and E. Szechenyi, Etude du comportement airo€lastique du tablicrtl'un pont d haubans, T.P. 1975-75, Office National d'Etudes et de Rcchcrchcs Airospatiales, Chatillon, France.

13-46 J. Roche, "Les Mdthodes d'etude a6rodynamique des ponts i haubans," irrProceedings oJ the Conference on Cable-Stayed Bridges, Paris, 1974, pp.7s-86.

13-41 D. Olivari and F. Thiry, "Wind Tunnel Tests of the Acroclastic Stability olthe Heer-Agimont Bridge," Tech. Note 113 von K:irmiin lnslitutc lirr FluitlDynamics, Rhode St. Genbse, Belgium, 197-5.

l3-48 R. C. Baird, "Wind-Induced Vibration ol'a Pipc l.irtc Srrspcrrsiorr llritlgc lntlits Cure," Trans. ASME,77 (Aug. 1955), 797 t't(tl .

13-49 H. P. A. H. Irwin, K. R. (itxrpcr, irnrl ll. 1.. W;rlllrrw, .llt!tlittttirtrt rtl Viltntlion Alt,vtrlttr':; to (lntlntl Witul ltrtltttt'tl Viltntli,,rr rtl I llt'ttrrr Iirt.tt lllt'trtlu'r',s

nt t t llt Nct l; 483

pl'tltt ('(),nnrt,(lt,t1, Iiltrrt'Ilritl,qt,l,rtlrtr':tloty lt'cltrricrrl ltclxrrl No. l,'l'R-l,n '

194, NAl,, Nrrliott:rl ltt:st:itrclt ('ottttcil, ()(tawir, ('anada, 1976.

l-l--50 'l'. Mutsurrrolo, "An lnvcstigation on thc Rosponsc 9f Pretensigned One-Way-'l'ypc Suspcnsion Roof,s to Wind Action," J- Wind Eng. Ind. Aerodyn.' 13(198.1,). lltl 394.

l3--5 I E. Kimoto and S. Kawamura, "Aerodynamic Behavior of One-Way TypeHanging Roofs," J. Wind. Eng. Ind. Aerodyn., 13 (1983), 395-406'

13-52 I. Elashkar and M. Novak, "wind Tunnel studies of cable Roofs," J. windEng. Ind. Aerodyn., f3 (1983), 4O1-42O-

13-53 H. Yamaguchi and L. Jayawardena, "Analytical estimation of structuraldamping in cable structures," J. Wind Eng. Ind. Aerodyn',41-44 (1992)'t96l-19'72.

13-54 T. Herzog, Pneumatic Structures, Oxford University Press, New York, 1976'

l3-55 Practical Applications for Air-Supported Structures, International Conferenceheld at Las vegas, oct. 1974, Canvas Products Association International, St.Paul, Minnesota, 1974.

l3-56 "Technics: Fabric Structttes," Progressive Architecture, 5l (1980), 110-t20.

l3-57 A. Morrison, "The Fabric Roof," Civ. Eng., 50 (1980)' 60-65.13-58 Air-supported Structures, State of the Art Report, American Society of Civil

Engineers, New York, 1979.

13-59 M. Horcic, Windbelastung und Berechnung des Spannungs- und Vetfor'mungszustandes im zylindrischen Tell von Traglujlhallen mit besonderer Be-riicksichtigung des Konstruktionsmaterials (H. von Gunten, Referent, H. H.Thomann, Korreferent), Eidgencissische Technische Hochschule, Zldlich,19'74.

13_60 B. V. Tryggvason and N. Isyumov, A Study of the wind-lnduced Responseof the Air Supponed Roof for the Dalhousie University Sports Complex, BLWT-SS7-1977, Boundary-Layer Wind Tunnel Laboratory' University of WestemOntario, London, Ont., Canada, 1917.

l3-6t B. V. Tryggvason, "Aeroelastic Modeling of Pneumatic and Tensioned Struc-tures,,, in wind Engineering, Proceedings of the Fifth International confer-ence, Fort Collins, CO, July 1979, I. E. Cermak (ed.), Pergamon Press,Elmsford, NY, 1980.

13-62 H. P. A. H. Irwin and R. L. wardlaw, "A wind Tunnel Investigation of a

Retractable Fabric Roof for the Montreal olympic stadium," in wind Engi-neering, Proceedings of the Fifth International Conference, Fort Collins CO,July 1979, J. E. Cermak (ed.),'Pergamon Press' Elmsford, NY, 1980.

l3-63 R. J. Kind, "Aeroelastic Modeling of Membrane Structures," in Wind TunnelModeling for Civit Enginccring Applications, T. A. Reinhold (ed.), Cam-bridge Univ. Press, Crttttbt-idgc, 1982'

l3-64 J. C. R. Hunt and l). .l . W. ltich:rnls. "()vcrhcad-Linc Oscillations and theE{1cctof Aero<Iynarnic l):trttPt'ts," I'r'tn.Irr,st. Iiltt" ling.' Il6(1969), 18691814.

l.l 65 .l . ('. R. Hun{ ancl M. l). l{owllrllorr, "Mr'lt'oroloy'11';rl ('otttlilituts Associittctlwi(lr thc Ilull-Sp:rtr (itrlloPing ( )st rll;rltort:, ol ( )r,'r'rltr';ttl 'l'l:tttstttissiotl l.irlcs,"I'ttt<'. ltr.st. l'.lt't . l'.ltr,g., 120 ( l() / 1). ti /'l li /('

.t484 t;ut;t,t Nt)l t)lil'nN ulltxit 1;, IIN:;t{}N :;ntt,(:l,nl i;. nNt) t'()wt il ilNt I

l3-66 V..1. lJrzozowski and It. ll. llrrwks. "Wrrkc Intluc:ctl liull SPan Instrl)ilily ()lBundle Conductor Transrnission l,incs," AIAA J., 14 (lgj6), 179 Iu4.

13-67 A. N. Hoover and R. J. Hawks, "Role of Turbulcncc in Wakc-lrrtlucctl (irrlloping of Transmission Lines, " AIAA J . , 15 (1917) , 66-70.

13-68 A.S. Richardson, Jr., Dynamic Load Model study for overhead T'ransmit-siotrLines, HCPIT-2O6312, U.S. Department of Energy, Division of Elecrric En-ergy Systems, Washington , DC, 1977 .

13-69 A. H. Peyrot and A. M. Goulois, "Analysis of Flexible Transmission Lines,"J. Struct. Div., ASCE, f04 (1978), 763-769.

13-70 "Device Reins in Galloping Power Lines," Eng. News-Record, Nov. 30,t978, p. 17.

l3-7 I A. (i. l)avcnporl, "Gust Rcsponse Factors for Transmission Line Loading,"i^ lilitul l')tgitrtrrittl|, Pnrccdings of the Fffih International Conference, Forl('ollins, ('O, l()79, Pcrganron Prcss, Elmsford, Ny, 1980.

ll1) .l . S. ('irrnrll, "l,ahoralory Studies of Conductor Vibration,,, Trans. AIEE,55, -5 (Mly 193(r), -543 547.

l3-73 lr. ll. Irarquhamon and R. E. McHugh, Jr., "wind runnel Investigation ol'Cl.ntluckrr Vibration with Use of Rigid Models," Trans. AIEE,75, part 3(Oct. 1956), 871-878.

13-14 J. s- Tompkins, L. L. Merrill, and B. L. Jones, "euantitative Relationshipsin Conductor Vibration Damping," Trans. AIEE,75 (Oct. 1956), 879-896.

13-15 G. H. Stockbridge, "Overcoming Vibration in Transmission Cables,,, Electr.World 86,26 (Dec. 1925), 1304-1305.

13-76 R. G. Sturm, "Vibration of Cables and Dampers-I, il," Electr. Eng., SS( r936), 455-46s, 673-688.

13-77 R. A. Komenda and R. L. swart, "Interpretation of Field vibration Data,"Trans. IEEE, PAS-87, a (April 1968), 1066-1073.

13-78 A. S. Richardson, J. R. Martuccelli, and W. S. price, ,,Research Study onGalloping of Electric Power Transmission Lines, " in proceedings of the Sym-posium on wind Effects on Buildings and structures, vol. 2, National physicalLaboratory, Teddington, U.K., 1963, pp. 6ll*686.

13 19 A. s. Richardson, "Design and Performance of an Aeroelastic Anti-Gallopingl)cvice," IEEE Summer Power Meeting, Chicago, Conference paper No. 68,('t)-670-PWR. 1968.M. l). Rowbr)tk)m and R. R. Aldham^Hughes, Sub-Span Oscillation: A Rc-vit'n, ..1 l,)ri.stittlg Knowledge, Report No. 22-71(SC)-02, Central Electricityl(cseirlt lr l,ulronrtorics, Lcathcrhead, Surrey, U.K., 1971.

It i'i I () Nrgol :rrtl (i. J. cllarkc, "conductor Galloping and control Based orr'lir'si.rurl Mc:chanisnr," IEEE, Power Engineering Meeting, New york, con"li'rt'rrt'e I):rpcl No. C74 016-2, 1974.K. ll. ('txrpcr anrl R. L. Wardlaw, "Aeroelastic Instabilitics in Wakcs," inI'nnttrlirtgs tl tha'l'hinl Interncttional Confereru:c on Wintl lifli,tt.;.n Builtlitr,q.s rttul ,Structun's,'lirkyo, 1971, Saikon, Tokyo, l()72, pp. (r.17 (r5.5.

l{. 1,. Wardltrw, K. R. C<xrpcr, ancl R. H. Scurrllrn. "()lrsr.r.vrrli.ltri on tlt(.l)rrrlrlcrrr ol' Srrbsparr ()scilla(ion ol' lluntllcrl l)owt,r ('orrtlrrt.r'rs." I)MI,:/NAt,.'Q. llttll. l()7.1( l). Nirliollrl Ilcrscirrch ('orurt.il. ( )lt:rw:r. (';rrr;rrl:r

lll lllll ll( | '; 485

l.l tt4 lt. ll. St.;rrrlurt,,4 W'ittt!'litttttrl lrtIt',tti!:tttt!,t'rtrlr, lltr' lt'trtlIrtrrtrrrr '\trtltrltlvrtl Btttttllttl I'tntt'r I'irrt'('t'tt(ltt(l('r'\ lttr Ilttlr" ()ttr'1"'' ' l';rrl Vl l{t'1rotl No

l.'l'R-l,n l2l NAli, Nlrtiorlrl l{t'scirrt'lr ('ottttt tl. ( )llirw:r. ('itruttl;t, l()/'lll tt5 R. 1,. Wilnllilw, "liltltt(:r ittttl 'lot'siottitl lrrs(irlrrlily''' rt ll'rtrtl l"ttrtt'rl I'rltrtt

tiotts 0l slructurcs,lf . Sockel (etl.1, Splrrrllcl Vt'r.llrg, Ncw Yolk. l()()1.

l3-86 R. H. Scanlan, "On Iiluttcririltl lhrlli'lirrg Mccllitttistrts irt l.rtng Sprrrr llritlgcs"'Probabilistic l"nginttritr,q Mtt lttttrit"r, -l ( l9ttl{)' 22 21 '

l3-87 A. Jain. N. P. Joncs, antl l{. tl. Scanlan, "F'ully Couplcd Bul1'cting Analysisof Long-Span Briclgcs," in wind Engineering, Proceedings, Ninth Interna-tional Confercn('c, pp.962-91 1, Wiley Eastern, New Delhi' 1995'

13-88 L. Singh, N. P. Jones, R. H. Scanlan, and o. Lorendeaux, "SimultaneousIdentification of 3-DOF Aeroelastic Parameters' in Wind Engineering' Pro-ceedings, Ninth International Conference, pp.972-981, Wiley Eastern, NewDelhi, 1995.

13-89 R. H. Scanlan and W. H. Lin, "Effects of Turbulence on Bridge FlutterDerivatives," J. Eng. Mech. Div., ASCE, 14 (1918), 713-133'

13-90 P. P. Sarkar, N. P. Jones, and R. H. Scanlan, "Identification of AeroelasticParameters of Flexible Bridges,'' J. Eng, Mech.,l20 (|994), |718_1742.

13-91 A. Jain, N. P. Jones, and R' H. Scanlan, ..Coupled Flutter and BuffetingAnalysis of Long-Span Bridges," J' Struct' Eng' (forthcoming)'

13-92 R. H. Scanlan and N. P. Jones, "Aeroelastic Analysis of Cable-StayedBridges," J. Struct. Eng.,116 (1990)' 219-297'

13-93 K. Y. Billah and R. H. Scanlan, "Resonance, Tacoma Bridge Failure, and

Undergraduate Physics Textbooks," Amer. J. Physics,59 (1991), ll8-124.l3-g4 F. Ehsan, "The Vortex-Induced Response ofLong, Suspended-Span Bridges,"

Doctoral Dissertation, Depaftment of civil Engineering, Johns Hopkins Uni-versity, Baltimore, MD, 1988.

13-95 M. Novak and H. Tanaka, "Pressure correlations on vertical cylinder,"Proceedings, Fourth International conference on wind EJfects on structures,Heathrow,U.K',pp'221-232,CambridgeUniv'Press,Cambridge'1972'

13-96 R. H. Wilkinson, "Fluctuating Pressures on an Oscillating Square Prism,"Aero. Quarterly,32, Part 2 (1981), I: 97-110; II: 111-125'

l3-g1 J. D. Raggett , section Model wind Tunnel studies, Golden Gate Bridge, westWind Laboratory Report' Carmel, CA' 1995'

l3_98 M. Tabatabai' S. G' Krishnasamy, J. Meals, and K. R. Cooper, ..Response

of smooth Body, Trapezoidal wire overhead (compact) conductors to windLoading," J. Winrl Eng. Ind. Aerodyn', 4l-44 (1992),825-834'

13,99 L. Shan. L. M. Jenke, and D. D. cannon, Jr., "Field Determination ofConductor Drag Coelficients," "/' Wind Eng' Ind' Aerodyn" 4l-44 (1992)'835,846.

l3-100 N. G. Ball, c. B. Rawlins, and J. D. Renowden, "wind Tunnel Errors inMcasurcments ol'P<lwcr C<tnductors,'' J. Wind Eng. Ind. Aerodyn., 4l-4(1992),847 t151.

l.l l0I y. Iruiino, M. llo. irrrtl Il. Y:rrr;rgrrchi. "'l'ltrcc Diltlcnsional Bchavitlrof Gal-Ieping irr'l'clccornrrrrrrrir':rliorr (':rlrlt's ol liigrrn: ll Scc(ioll," .l . Wind ling- Irul-A(r(,(l\ttt., -l(l ( 191{t'l). I / .)('

I r ri.)

l] til

.,r.il! ilillxil ti, lFNtil(lN rillil,t;il,n] l;. ANt) l,OWl il llNl li tf.l-102 'l'nttt.sttri.s:;iort Litrt, lIqli.rnrt.r, liltrA ll,trrtl lrtt!ttr.ttl (.otttlttt.tot.

DL- 100-4, ljlcctr.ic:al l)owcr ltcsrrirrclr lrrstitutc, .]4 l2 HillvicwAlto, CA 94304. tgigl3-103 M' E' criswe, and M. D. Vanderbirt, Reriabirity-based Dasil4rt ..r..r,n,t.srnission Line structures, E.RI EL-4793, Electricj po*", n",iurch rnstirurc,3412 Hillview Avenue, palo Alto, CA 94304, l9g7.l3-104 G' H. Hirsch, "Damping Measures to control wind-Induced vibrations,,, iny,:(.ti;{;o vibrations of structures, H. sockel ("d ), a;;;;Vertag, Newl3-10-5 M. Matsumoto, N. Shiraishi, and H. Shirato, .,Rain-wind Induced vibrations.l'Cables of Cable_Stayed Bridges,,, J. lVind Eng. Ind. Aerodyn., 4l_44(te92) 2011_2022.I I l0(r 1"' H' Durgin' D' A. palmer, and R. w. white, ..The Galloping Instabilityo| Ice Coated poles," J. wind Eng. Ind. Aerodyn., 4l-4 (rgg2), 675-6g6.l.l 107 Ir. H. Durgin (personal communication, 1995).l'l loti 'Harmonizing with the wind," Eng. News Record., oct.2, 1944, pp.l'l 109 P. p. Sarkar, New lcrentification Methods Appried to the Response oJ.FlexibreRridges ro wind,.D^epartment of civil Engineering, J.h;. H;;;il-- university,Ilaltimore, MD. 1992.

CHAPTER 14

OFFSHORE STRUCTURES

wind loads affect offshore structures during construction, towing, and in ser-vice. They are a significant structural design factor, especially ii the case oflarge compliant platforms, such as guyed towers and tension leg platforms.

wind can also affect the flight of helicopters near offshore platiorm landingdecks il4-1, 14-2, 14-31, as potentially dangerous conditions may be createdby flow separation (see Sect. 4.3) atthe edges of the platform. Let the horizontaldistance between the upstream edge of the platform and the upstream edge ofthe helideck be denoted by d, and let the depth of the upstream surface-pro-ducing the separated flow be denoted by l. on the basis bt *ind tunnel tests,it has been suggested that the elevation ft of the helideck with respect to theupstream platform edge should vary from at least h = 0.2t if d = 0 to at leasth = O.5tif d = tIl4-21.

This chapter includes information on wind loads on offshore structures ofvarious types (Sect. 14.1), and on the treatment of dynamic wind effects in thecase of compliant structures (Sect. 14.2).

14.1 WIND LOADTNG ON OFFSHORE STRUCTURES

Methods for calculating winrl loacls on offshore platforms are recommended inll4-4lto tl4-8]. Howcvcr, lrrlrorirtory irnrl lirll-scalc mcasurements indicatc thatthcsc mcthods may, itt sttlttt' ittsl:rnt't's, lurvc scriorrs lirni(utions, particularlyinstllar as thcy d<l nol it('('()lrrtl tor llrt'l)r('scn('c ol'lili lilrcrrs, rrrrtl irccolllirlsullicicntly--rlr ttol :tl itll lirr slrit'ltlirrli rrrrrl rrrrrlrr;rl irrtgr.li.r.t.rrcc t'llr'cls. I;6r.cxiuttplc, accrlrling lo wirrrl lrrnrrr'l (t.:,1 rt.:;rrlls olrl:rirrt.tl lirr lr jltt.k rr;r (st.ll.clcvlrlirrg) 1ll:rllirrlrr Il4 ()1, llrt'rttt'llrorl:, ol Il.l .ll:rrrtl Il.l 5lovt.n'sl ilrurlt.wintl

M(,lit,,t. li,ltl<lAvcrtuc, Itrrlo

447

*488 ()t I liil( )nt 1; il tt,o t(,nt s;

kriuls on.jack-up units by at lcast .)-5,/n. 1q,;1;"r,rtcs bascd on lirll,scalc tluta lirrIttl atrchorcd scmisubmersible platlirrnr ll4-l0l suggest that thc rtrolhotl 6l'f l4-5f ovcrprcdicts wind loads by as much as loo%. It has therclirrc bcconrcc()llllll()ll practice to obtain design information on wind forces on platlirrntslirrrrr laboratory tests. Most tests provide data on mean, as opposed to gustingkrirtls. ln using such data the effect of gustiness should be accounted fbr by:rrurlyticirl ffleans (see for example, Sect. 14.2). Possible Reynolcls numbcre llccts shoulcl also be assessed with care.'l'lris sr:ction briefly reviews a number of wind tunnel tests conducted forst'rrrisrrbnrc:rsihlc units and for a large guyed tower platform. Wind tunnel testirrlirrrrrirtiorr on.jack-up units, on jacket structures in the towing mode, and ontw() tyl)cs ol'concrete platform is available in [14-9], t14-lll to [14-14],ll,t \51, ll4-191, and [14-40].

14.1.1 Wind Loading on Semisubmersible UnitsA st'lre tttltl ic view of the model of a semisubmersible unit used for tests reportedin l14 l-51 is shown in Fig. 14.1.1.*

'l'hc sirlc lbrce and heeling moment coefficients are defined as

CY: , -YtpU'(5O)A,

CK:, "K>pU'(50\A.H,

(14. l. l)

(14.1.2)

whcrc Y is the side force, K is the heeling moment, p is the air density, U(50)is thc mcan wind speed at 50 m above sea level, .4" is the projected side area,:urtl H, is the elevation of the center of gravity of ,4". Coefficients CY and CKlu'c obtaincd separately for the overwater and for the underwater part of thcrrnit. 'l'hc overwater coefficients reflect the action of wind and should be ob-t:rirrtrtl irr a llow simulating the atmospheric boundary layer. The underwatert'rx:llit'icrrls account for hydrodynamic effects and should therefore be measureclirr rrnilirlru ll<lw.

liigrrlcs l4.l .2 'tntl 14. 1.3 show values of CY and CK measured in [14-l-51Ior tlrt' t'rrst' ol'trn upright dralil 7r,,. : 10.85 m (corresponding, for the unitlrt'rrr1, 11111fr'.;t'tl. (o:r tlisplirccrncnl'of lJ,729 tons). As noted in [14-151, thcl!urlx)sr'ol tltr'lt'sls lirr lhc undctwater part is to determine the elevation of thcr't'rrtr'r ol rt'lrt'liorr (i.r:., thc point of application of the resultant of the undcr-w:rtt'r lort'cs) lirl thc l'r'cc-lkrating unit. For an anchored or dynamically posi-

rlrlittrt's l'l.l.l tlrlorrglr 14.1.(r arc cxccrptcd lrom E. Bjerrcgaar{ :rrrtl S. Vt:lsglu1r, ..Wirrtl()v(rlrrrrrirrP, lllli'(l()rl:tSctnisttbtttcrsiblc,"PapcrOTC3063, Pnx'raliu.rit,()llshort'l't:c6nokrgy('orrl('11'tr((', lLtttslott,'l'X, May l()71'1. ('opyright 197t3 Ofl,shrlrt:'li.t.lrrrolo;,y ('orrlt'rt.rrt.t'.Ilrcrr|li1',lrttlr;tli71.11 rstlrctlcplhrtlirtutrclsionolthcrrnitirrllrt.t.vt.rrlrr'r'lrlrllitrprr(c.g..li1.

:ttt:tlt1,,lt'ol lrcr'l y'r O").'Ilrc tlispl;ttt'tttt'tt1 is lltt'vrtlttntr'ol w:rlcltlisllllrct'rl by tlrc illrnrr,r:t'rl p;nt rrl llrr'rr1t

ti

ir

I

i

$oo

0).oao-o

E4)

oo

o()

V)

t

Dtu

489

*490 ()l l.(;l t()nt llilt(,(;tt,nt I

1

0.5

0

- 0.5

-1

0.5

0

0,5

1

ft'f(illf{ltl 11.1.2. Values CY and CrK as functions of wind direction rlt at different:rrr11li's ol lrt'cl y'r lirr the overwater part [14-15].

tionerl rrrrit llrc ccnter of reaction is determined by the anchoring forces or bylltc tltnrslcrs I l4- l5l.

liigrrrc 14. 1.4 shows estimated values of the heeling forces induced by 100-krrol lrcrrrn winds* for various values of the upright draft T1as" and of the angleol' lrccl <f . The elevations of the center of action of the overwater (wind) forcelrrtl ol' thc ccnter of reaction on the underwater part are shown in Fig. l4.l.5.It is sccn that as the angle of heel increases the elevation of the center of actionol'(ltc wind force decreases. This decrease is due to lift forces arising at nonzeroirrtglcs ol'hcel @.

,l

0,s

0

0,5

1

Udeg

\-90\di360 deg

Itl('lliltl'l 14.1.-1. Valucs C)'and C'Kas functions of direction ry' at dill-crcnt anglcs ol'lrrt'l y'r lirl lltc rrrrt[:rwllcr pan ll4-l-51.

rWitttls blowirtp', rtkrttg lltt' :rxis Y (lrig. 14. l. l). Wind blowing rrkrrrl', tlrr. :rrrs \ :rn' 1.li'r1'tl (g;rs lrt':ttl (or lxrw) wirrtls. Wirrtls wlrost.tlirt:t.liorrs ltisccl lltc:uty',lr.s lrr.tu't.r.lr,rrr'., .y:rrrtl y:rr.r.tt'lr'trr'tl lo :rs rlululrrirrlt, wirrtls.

t4 r wtNl) t()nl)tN(i ()N ()t tlit t()t il 1iilil,(:tUnt li 49'l

FIGURE 14.1.4. Wind heeling forces corre-sponding to 100-knot beam winds tl4-151.

The heeling lever is defined as the ratio of the overturning moment to thedisplacement of the vessel. Values of the heeling lever for 100-knot beamwinds, obtained from the wind tunnel tests of [4-15], on the one hand, andby using the American Bureau of Shipping method U4-41, on the other, areshown in Fig. 14.1.6. (The displacements listed in tl4-l5l for the 6.43 m,9.00 m, and 15.25 m drafts are 12,740 tons, 16,963 tons, and 19,495 tons,respectively.) It is seen that for large angles of heel the differences betweenthe two sets of values are considerable. This is largely due to the failure of[4-4] to account for the effects of lift. It is noted in [14-16] that the largestovertuming moments are commonly induced by quartering winds.

In the tests of [14-15] and [4-16] the water surface was modeled by therigid horizontal surface of the wind tunnel floor. Following the method de-

Tggo=$,1, I6" 9.00 m" 10.85m" 15.25 m

-'-e"\..-'y''--'-

lDistonce obove2SJVoterline (m)

\-."- -$,1

,

,,1

\l

lr'l(ltJRll, 14.1.-5. Illcvlliort ol'ccrrlcl ol :rt'liorrol wirxl lorccs lrtxl corrcslxrrxlirrl', ccrrle r ol rt'lrt'(iorr orr llrc rrrrtlcrwrrlcr lllrrl ll.1 l5l.

WIND

TM0o:6.4 3 mu 9.00m" 10.8 5m" 15.25m

-db*'w't"

*492 I4 I WINI) IONI)IN(i ON OI I SHOIII] SIHUCTUBES

2 DRILLIN. RlGs BENorNc sHoE

2oo ToN (1a1 Ms)CLUMP.WEIGHT

I

l

i

I

493

€ffiffil. '3o' I

ModolTesls A B S{ts-- ---- 6.43m- {- - - --- 9.00m--)(-- ---10.85m----o-

-.15.25m /

utu -tt-

HEELiNG LIMIT

0' 5o 10o 15' 20' Angle of heel

frl(;lJltl,l 14.1.6. Wind heeling levers obtained from wind tunnel tests and from Amer-icirn llrrrcuu ol'Shipping (ABS) method [14-15].

scrilrr:rl in ll4-171, tests reported in [14-18] were also conducted by placingllrc rnoclcl in a tank filled with viscoelastic material up to the level of the windIturrrcl flxrr. This facilitates the testing of models of partially submerged units.Itclbrcncc [4-18] also includes results of tests conducted in the presence ol'rigirl ohstructions aimed at representing water waves. The results revealed thatwuvcs could increase the overturning moments substantially. This suggests thcncccl lirr improving the simulation of the sea surface in laboratory tests.

'l'hc acrodynamic testing of the Ocean Ranger semisubmersiblex is reportcdin ll4-3t)1. The problem of combining hydrodynamic and wind loads was;rtltlrc:sscrl by conducting I : 100 scale aerodynamic models in turbulent flowovcl rr lltxrr with rigid waves, and using lightweight lines to apply the measurcclrrrcirrr rrrrl lluctuating wind forces and moments to a l:40 hydrodynamic rnodclIt'stt'rl irr contlilions sirnulating those experienced during the storm. Additionalwintl lrrrrrrt'l tt:sts ol'scrrrisuhmcrsiblc units are reported in[14-14] and [14-l9lIo I l.l ))1. I l/t 401. lnrl I l4-411.

14-1.2 Wind Loads on a Guyed Tower Platforml{t'lr'r't'rrt't' ll,X 2ll prcscnts rcsults of wind tunnel measurements on a l:120st';tlt' rrotk'l ol'llrc ovcrwittcr part of a structure similar to Exxon's l-cna guyctl

r'lltt'()tt'itrt lllrrrgcl llrtl citltsizctl olt licbntatry 15, 1982 in Hitrcurirr l'rcltl. Illr krrr sorrllrt:irst olSl .hrlur's Nt'wlirtlxllittttl. itt it slonrt with l7 rrr lo 20 rn wavc heiglrtr, lrrrrl l.)o hrrr/lr to IlO krrr/Itt wrttrl s1x't'rls. ll w:ts lltc l:rr11t'sl sttbtttt'tsibk: oll,slrore tllillirrl', plirlllrrrr rrr llrr'rror|r1.,1(r rrr lrililtlrottt Lrt'l lrt ogrt'ltlirttts tlt't k rttttl witlr l2O rtt ktttp. pottlrxrrrs. All ol llr, li l , rlu' rrrcrnlx rs wt.r(.losl rtt lltt' :rtr'trlr'nl

FIGURE 14.1.7. Schematic view of Lena guyed tower platform. From M. S.Glasscock and L. D. Finn, "Design of a Guyed Tower for 1000 ft of Water in theGulf of Mexico," J. Struct. Eng., ll0 (1984), 1083-1098.

tower platform. (A schematic view of the platform, installed in over 300 m ofwater in the Gulf of Mexico, is shown in Fig. 14.1.7 |4-241; see also Fig.14.1.8.) The mean wind profile created in the laboratory matched closely boththe power law:

( 14. l .3)

and the expression for sustained winds (i.e., winds averaged over at least oneminute) recommended by the United States Geological Survey [14-7] for usewithin the Gulf of Mexico:

22" (ra2shm) DtA.PEBIMETER PILES

64" (1372mm) OlA. MAIN PILES12OO' (oAOm) ANCHORCABLE - 6" (l27mm)OIA. COATED

f2" <182snm,DIA. ANCHOR PILE

/ - \l'12u(z') : urrol (101)

(14.1 .4)

where z7 : 2.2 m and z is the elevation above thc still watcr lcvcl in rlclcrs.The airlwater boundary was modeled by the rigicl horizonlirl surlircr: ol' llrcwind tunnel floor. Forcc irnrl rrxrrncnt cocliicicnts wcrc tlclirrctl llv n'llrliorrs olthc typc

J"('t llrtl'1 11,;,'1,'

/ - - \01128rJkt: uoo) (:r-:,)

(l,l I 1)

494 ()t I lill()l11 1;lllt,(;l{,lll li

+ 5B.O'-

l.l I wlNl) l()nt)lN(i ()N

YCD, CMD

x -tI

u(r6t

FIGURE 14.1.9. Notations. From P. J. Pike and B. J. Vickery, "A Wind TunnelInvestigation of Loads and Pressure on a Typical Guyed Tower OlTshore Platform,"Paper OTC 4288, Proceedings, Offshore Technology Conference, Houston, TX, May1982. Copyright 1982 Offshore Technology Conference.

()t t:iu()nt t;nl,clt,lil l; 495

Drilling Derrick (2 t

Derrick Structure (2)

- + 163.5'

SubslructureWell Conduclors

+224.9'

Flore Boom

@ Decx Slruc'iure(enclosed )

@ Dritting Pockcges ( 2 )@ P- ronts@ Crews Quorters (2)

Elv. O.O (14.1.6)

Skid Bcse67'x32'x8'

lrodction of

BoomI eo'

Nole: Helidecks Resl onTop of CrewsQuortcrs

where F and M are the mean force and the moment of interest, p is the airdensity, U(16) is the mean wind speed at 16 ft above the water surface, andthe reference area AR and length l,a were chosen as 1 ft2 and I ft, respectively.The force and moments obtained in ll4-231are represented in Fig. 14.I.9,which also shows the notations for the respective aerodynamic coelficients. Themoments characterized by the coemcients CMD and CMT were taken withrespect to a distance of 6.2 in. (62 ft full-scale) below the still water level.The measured values of the aerodynamic coefficients are represented in Fig.14. l. l0 for several platform configurations. The configuration for the base casewas the same as in Fig. 14. 1.8, except that the deck structure was not enclosed.Additional results in Il4-231 show that the effect of enclosing the deck isncgligible for practical purposes, as is the effect of the well conductors. Re-rnoving the flare boom results in torsional moment reductions, but has negli-gible effects otherwise. It is shown in|4-231that drag forces and drag momentsbased on wind tunnel measurements are smaller by about 30% and 2O%, re-spectively, than the calculated values based on tl4-71.

To check the extent to which the results depend upon the laboratory facilitybcing used, the same structurc was subscqucntly tcsted indcpendently in adifferent wind tunnel ll4-2-5 1 ltt tnosl c:rscrs ol' signilicrrncc lnrrn ir rlcsignvicwpoint the results obltrirrt:tl irr ll,1 2.5 1 wt'n' liugr'r llurrr tlrosc ol'll4 211 hyanlounts that did not cxcct'rl lO to \Oi)l, I

rl)illr'rerrtt'slrt'lwt'ettttsttllsol;r,'ro,l\tt.rtttt, nr.l.nrrrrrl.,rrrrrlrril'1lrr'Il'r'n'l{lll\ utrltll(t.11lirtilitrr's;ttt':tlso ttrtlt'rl itt Srr't I (r

Deckr56'x 156'x 57

rJ-

Flore Boomto'.-J

IIIt+2'

(b)

CT. CMT

Itl(llJlll,l 14. l.tt. (ittycrl lowcl-Platlirrlrr: (a) siilc t'lt'v:tltott, (/') l)l;trr ll4 25 1

F.

U

uO

r:

N

I'IIND DIBECTION

(b)l,'l(Jlllll,) 14. 1.10. Wintl tLrnncl tcst rcsults. From P..1. l)ikc rrntl ll..l . Vickcry, "AWirrrl 'l'trrrrrcl Irrvcstigalion ol'l-uacls antl Prcssurc on a'l'ypic:rl ( irryt'rl lirwcr-Oll,shon'l'l:rllilnt,"l'irPt:r'()'l'('42llll, I'nxcttling.s,O.ll.shrtrt'll'r'ltttolrt,t:\'('t,nl('tt'ntt'.llorrslorr.l'\. Miry l()1{2. ('opyriglrl l()tl2 ( )ll.slrorc 'l'ccltrtology ( 'ortlt'tt'rrt r'

496

1.1 ;, IrYNnMt(. Wtt.tt) | lllclli ()N (;()Ml'l lnNl ()l l:;l l()l l :;llill(:l([il :; 497

(c)

FIGURE f4.1.f0. (Continued)

14.2 DYNAMIC WIND EFFECTS ON COMPLIANT OFFSHORESTRUCTURES

Compliant offshore platforms are designed to experience significant motionsunder load. An important advantage of compliance is that the forces of inertiaIrssociated with these motions contribute to counteracting the external loads.

ln the case of large offshore structures installed in deep water, compliancehas the additional advantage of making it possible to design platforms withvcry low natural frequencies in the surge, sway, and yaw degrees of freedom*( c. g. , I /30 Hz to I / 150 Hz, depending upon type of platform and water depth) .

Wavc motions have narrow spectra centered about relatively high frequencies(c.g., trom lll5 Hz for extreme events to about I Hz for service conditions).'l'lrus, aside from possiblc second-order effects, compliant platforms generallytkr not cxhibit any dynarnic amplilication of the wave-induced response.

t.lnlikc wave motions, winrl spcccl lluctuations in the atmospheric boundarylrrycr arc charactcrizccl l'ry brorrtl bltrttl spcc(ra (scc Fig. 2.3.4). For this rcason

ll,ilrt'irt tttolions ilr lltc lorrliitrrrlur;tl. tr;rr:vrt:r', rrttrl vr'tlir':rl tlirct'liolt ltlt' lt'lt'rtt'tl l() its .rrrrli(,llt\',;rtttl ,/tcilr,, tcspcclivcly. Arrl',rl;u rlolroli rr .r lr;rrtv('rs(', lurrlr,iltrtlirr;tl, :ttttl ltotizottlltl Pl;tttr'rrtt ttlt'rtul lo lts toll. pilt lr, rttr,l \/it\\. lr".l! ' lr\' lY

b)

zU

IIua

UEa

rIJt

:)

-. () BASI CAS[L w/0 fasl otRRlcK

w0 B0rll oEnnlcl(s' w/0 DBtU-t]{G EOUIPiIEM

- O zDECl(COilFlGURATloll

rLua

UOOLCoFzUIa=

l,,llND DIRECTI0N

0 SIASE CASE() WO EAST OBREKI WO BOIH DERRrcT\S- W0 oRlUJilG EoUlPillEl{Io 2 DECK GoilHGURATtoil

/ ,**-tt-/P

./ *F*t^:

- ' , :.--t","--- \:\V''iTM,T{SVEBSE MOMEIIIT (CMT)

I'IIND DIRECTION

rti&.

498 ()t Il;l t()nt liutt,(itt,lu I

il ltrrs ltccrt sllr(ctl irr lltc litcr-ulurc llrrrt wintl incluccrl rlynanric iurrlllilir'ir(ioncllccts orr cotttpliilnt structurcs arc signiliclril ll4-23,ll4-261. A rnorc gt-liu:(lc(lirssossrfronl ol'thc cl]'ects of wind gustincss was prcscntetlinl14-271as pafl rll'ilrr cvaluation of thc response to environmental loads of the North Sca Huttontcnsion lcg platform (Fig. 14.2.1, see also tl4-281). According to ll4-271:"Wirrd gusts are typically broad-banded and may contain energy which couldcxcitc surgc motions at the natural period. These would be controlled by surgctlrrrrrping. 'l-hcorctical and experimental research is required to clarify the im-l)()r'liulcc ol' this mattcr. "

l'f (Jllfll'l 14.2-l- Sclrcttntic vicw ol'thc Hutkrn tcnsion lcg plrtlirrrrr. Irnrrn N. llllis..l ll.'l'clkrw. li. Atttlctsott, urttl A. L. Wrxxlhcatl. "lltrllorr'l'l ,l'Vcsscl Slnrclrulrl('rrttlillttr':tlrrrrt rtlttl l)trsig,lt liclrturcs," l'}apcr ()'l'(' 442J, l'tttt t,t'rlirr.r;.t. O llillrrc 'l'ct.lrllrlol'y ('ottli'rt'trt't', l lottslolt, 'l'X, Miry l()t12. ('opyl'iglrl l()l.i-! ( )llslrort' 'll't lrrrology('orrlt'tt'lrt r'.

l;l l' l)YNnMl(: Wl[]]) I lll(;11; ()N O()Ml'l lnNl ()l l:;l l()l tl l;llll,(;ll,ltl l; 499

Irrve:sligirliorrs irrlo llrt'bchirvior ol'(otlsi()tl lcg pl:rtlirrrrrs utrclor wind loadsrcJrortotl in ll4 291 trrrtl ll4-.101 wcro basccl ott lhc assulnption that the responseto wind is dcscribccl by a systcrn with proportional damping, the damping ratiobcing ol'the rtrclcr of 5%. Howcver, it was shown in [14-31] that for structurescornparablc to the Hutton platform, the effective hydrodynamic damping isconsiderably stronger, and that the wind induced dynamic amplification forkrw-f'requency motions are for this reason negligible. Section 14.2.1 describesthc approach used in t14-311 to estimate the response of a tension leg platformto wind in the presence of current and waves, and a simple method for esti-lnating the order of magnitude of the damping inherent in the hydrodynamicloads.

14.2.1 Turbulent Wind Effects on Tension Leg Platform Surge

Under the assumption that the extemal loads are parallel to one of the sides ofthe platform shown in Fig. 14.2.1 , the equation of surge motion can be writtenAS

Mt : F,(t) (14.2.1a)

where

F,(i) = F,(t) + Fh(t) + R(t) (14.2.tb)

In Eq. 14.2.1b, F,(t), FhG), and R(/) denote the wind force, the hydro-dynamic force, and the restoring force, respectively. Not included in Eq.14.2.lb is the damping force due to internal friction within the structure, whichcorresponds to a damping ratio of the order of l% and is negligible comparedto the damping forces associated with hydrodynamic effects.

Wind Loads. Like the hydrodynamic loads, wind loads consist of a componentclue to flow separation, and an inertial component associated with the relativefluid-body accelerations. However, it can be verified that the inertial componentis about two orders of magnitude smaller than the component due to flowscparation, and can therefore be neglected in practical applications.

To estimate the wind-induced drag force it is assumed that the elementalrlrag fbrce per unit of area projected on a plane P normal to the mean windspccd can be written as

p(y.:. 11 - !p,,C,,(t, z)Iu(y, z, t) - i?)f (14.2.2)

whc:rc /r,, is thc air tlcrtsily. { ),( r'. l) is thc pressurc cogfiicient at clcvatiolr l:

Irrrtl lxrrizolr[:tl crxrtrlitut(t' t'ttt lltc plrtttc P, / is thc titttc,,r is thc surgc tlisplirt't.r1etr(, lhc tl<lt tlt'trolt:, rlrll, r, rrlr:rliorr witlr tcspcct trl litrrc, ltlttl tr(.y, :, /)is lltt' wirrtl spccrl rrpwirrrl ol llr,' :,lnr(lur(' irr tlrc tlirct'liott ol-lltc tttclttt witttl.

t500 otIr;il()ril riilr(,(;ilililli

'f'lrc spccrl u(.y, ?., l), can bc cxprcssr:tl lts a stttrr ol'lhctlrc llrrctuirting spccd u'(y, 2., t)'.

u(y, z, t) : U(z) * u'(y, z, t)

'l'hc totul wind-induced drag force is

(14.2.4)

wlrt'rt' ..1,, is thc projection of above-water part of the platform on a plane normalIo tlrt' rrrcrrn wlnd speed.

'f 'lrt' nrerrn speeds can be modeled by the logarithmic law (Sect. 2.2.3). Thcsl)r'( llr ol thc longitudinal velocity fluctuations can be modeled by Eqs. 2.3.25.*'l'lrt' t'r'oss spcctra of the longitudinal velocity fluctuations are modeled by Eq.l.l.lO. 'l'hc cllcct of longitudinal separation should also in principle be takenirrto lrecotrnt . However, it follows from information presented in [2-89] thattlris cllct't is ncgligible as far as fluctuating aerodynamic loads on offshoreslru('tlrrcs irrc concerned.

lt cirrr hc verified that the mean square values of u' and i and the meanvrrlrrc ol'thc product u'* are small compared to the square of U. It then followsl'nrrrr llr1s. 14.2.2, 14.2.3, and 14.2.4 that the mean drag load can be writtenlts

I

F,(r) : J^,,rrr.z. 1dy dz

F,(i) = ip"C/"U'(2")

whcrc lhc overall aerodynamic drag coefficient is

'' : #Tu\^"'"'' z)u2(z) dY dz

I'i,.,(r) - ,,,\n"Cn', z)u(z)u'(2, t) dy dz

l4:' l)YNnMl(i wlNl) I lll(;l:; ()N (i()Ml'l IANI ()l l:ill()l ll :;llltl('ll'lll :; 501

wlrcrc l[c sgbst'r'i1tt l rcl'crs to llre llrct tlrir( tlrt'pl:rllirlrrr is irl lr's( . 'l'ltt'liottltt'ttranslilnrr ol' (hc auloc<lvariattcc litttc(irtlt ol' /"i ,(l) yie ltls

(t4.2.9)

(14.2.10)

The spectrum Sn,(n) can be estimated numerically by assuming Cr(li, z) :c,(i :1,2). An equivalent wind fluctuation spectrum can then be defined as

S" (n)s,.*o(n) : lrsr^"r;rt

From Su,"o(n) it is possible to generate by Monte Carlo simulation realizationsof the process

uLqU) : ? "'"rt cos(Zrn1 -t $,)

In Eq. 14.2.10 the phase angle S; is generated by random sampling from a

uniform distribution in the interval 0 < fi < 2tr.Let the spectrum of the force F',c,,,(t), defined as

ntcan spc:ccl {/(l) unrl

(14.2.3t

(r4.2.5)

s7,;,.,(n) : ,i,\^, Jr,, ,r,,.u,' ;1)( ),( v.,' ::.,){/(::')l/(;:,)

(l.r.l.ri)x SL",(.y,, lz, 2.1, z) dy1 dv; tl?.1 tl:.;

(14.2.6)

:rrrtl l;,, is thcr clcvation of the aerodynamic center of the above-water pan of thelrlrrtlirrrrr. linrrn lic;s. 14.2.2 to 14.2.5 it follows that the fluctuating part of thewirrrl rlrrrlq lorrtl (hut would act on the platform at rest (i.e., with i : 0) is

F 1c,,,(t) : p oC oAoU(2")uLq(t)

be denoted by Sp"o,,(n). ClearlY

s"*,,I,) = sr,.,(n) (t4.2.12)

Thus aio(l) can be viewed as an equivalent wind speed fluctuation that isperfectly'coherent over the area Ao and whose effect upon the structure at restis the same as that of the actual fluctuating wind field.

The total wind load acting on the platform can thus be expressed as

F,(t) : ip,CA,lU(2,) + u'.r(t) - *(t)12 (14.2.t3)

Numerical calculations have shown that if the difference between the ele-vation of the helideck (or the top of the crew quarters) and the underside ofthe lower deck in a typical drilling and production platform is less than abouttwo-thirds to three-quartors ol'tlre-witlth ol'thc rnain clcck, thc tcrm C](z' -z2)2 of p,q.2.3.30 can bc ncglct'tctl wlrt'rr ev:rlrurling tlrc ilt(c:grlrl in lit1. 14.2.|t.Thisisaconsequenccol'the llrt'ttlr:rt('issrrritllel. lltlrrr(',by.lt litcttlIol'ltlXrtt(1.5. 'l'hc appnxinruligrr illtt.tt'ttl rrt ttclr,lct lirrp, ( "(.11 l:,)' is sliP,lrlly t'6trscrvltivc l-nlln a slntt'trrllrl cnllln('('r lrll lt()lttl ol vir'w ltltorrlllr trrsll'.ltillt';rlrtIV

(r4.2.1t)

(14.2.1)

rNolt' lh:rl lor lhc licrlucncy r - 0 Bqs. 2.3.25 yicld a spectral onlirurlc ,S(0) prrrgrrlional to tlrcirrlt'1ir:rl lrrrllrlcncc scllc /,), in accordancc with lundarncntal princilllcs (st'c Iitq. 2.1 l9). On llrt^ollrt'r lr:rrrtl. li1. 2.-1.2.1 (c.9,., quotc(l in ll4-231) yiclds S(0) O, urrtl i1 tlrcn'lon'rrult'rcstirrrirtt'slltt'spt'tlrrl otrlirtirlt's itt llrc nrttgc ol lypicirl nalttral lictlttt'tttit's lot torrrplrurl slrrrclrrrcs (liig..t. l.rl).

()l l:;l t()ilt liiltt,(;tUtil I

lil(lllltl,l 14.2.2. Integration domain andI r lrrrslirrrrrrl iorr ol' variables.

s()). N()linlt, thcn, that for any arbitrary function O,

(14.2.14)

(l;ig. 14.2.2), and assuming Co(li, zi) = Co, U(zi) : U(2"), and Su,(n) =,t,(:,,, rr1, (i - I ,2), it follows after some algebra from Eqs. 14.2.8,2.3.30,rrrxl 14.2.9 that

S,."q(ru) = Su(zo, n)J(n) (r4.2.15)

wlrcro 5',,(2,, , n) is the spectrum of longitudinal velocity fluctuations at elevation:,, , and J(n) is a reduction factor accounting for the imperfect coherence amongllrc llucl.uating wind pressures at different points of the platform, given by thecxprcssion

nr'; : c.b -' u(2") (t4.2.17)

lrr lirl. 14.2.1'7, b is thc width of main deck. Equation 14.2.15 can be used inIrt'rr trl lirls. 14.2.213 antl 14.2.9 forthe Monte Carlo simulation of the equiv-rrlcnt vclot'ity lluctuations aiu(t) (see F,q. 14.2.10) needed in the expression oftlrt'lolrrl wintl kxrcl acting on thc platform, Fu(t).

Hydrodynamic Loads. The total hydrodynamic load F1, can be written in thclirrrrr

[rr,: F,, * F,,- Ax - I].r (l4.2.lti)

wlrcrc /'), is thc (ollrl hytlr<ltlytltttric viscotts lirlcc, /,,. is llrt'lotrrl w:rvc irrtlrrccrlr'xt'itirrg lirrcc, y' is llrt'srrrgc:trltlt:tl rrrirss, lrrrtl /J is llrt'srrr1't.wlrvr'r'lrrli:rliorr

A/llJl) IIrI(;t:i ()N (;()Mt't tnNt ()t t:il l()nt tiillt,(:tt,llt ti 503

tllrrrrpirrg cocllicicrrl . ll wlrs rrssrrrrctl lirl converrit'rrct' irr ll4 .lll tlrirt lltt' wirvt'rrrotiorr is rrronoclrronlrtic', lrt:rrcc lhc:rbscttcc rtl sct'otttl ortlcl tllilt lirtccs irtl;.q. 14.2.ltt ll4-311. l( was lssunrcd in lrtklilion lhlr( /J O, sincc llrc I'lrrlilr(iontlarrrping at low licqucncics is ncgligiblc ll4-121.

'l'he total wave-induced exciting lirrcc arrtl tltc sut'gc-irtltlctl tnrrss cutt bccstimated numerically on the basis of p<ltcntial thcory. Altcrn:rlive:ly, thcsc lwoLcrms may be assumed to be given by the incrLia c()rllp()ncnl ol'lltc Morisottequation:

ll4-34, p. 3l], where p, is the water density, vu is the elemental volume ofthe submerged structure, C-u is the surge inertia coefficient corresponding tov,r, X is the horizontal distance from some arbitrary origin to the center of V;along the direction parallel to surge motion, ui and u;1 are the current velocityand horizontal particle velocity due to wave motion, respectively, at center ofv,,. Equations 14.2.19 and 14.2.20 may be employed if for the componentbeing considered the ratio of diameter to wave length, DIL < 0.2 ll4-34, p.2831. Since forT* = 15 sec, L: gTz*l2T = 350 mll4-34, p.2831, where7", is the wave period and g is the acceleration of gravity, it follows that formembers of typical tension leg platform structures, for which D < 2O m orso, the use of Eqs. 14 .2.19 and 14.2.20 is acceptable if three-dimensional floweffects are not taken into account. The wave motion can be described by deepwater linear theory, so

(14.2.21)

where 11 is the wave height and k,, is the wave number given by

(t4.2.22)

ll4-34, p. 1571. The total hydrodynamic viscous load may be described by theviscous component of Morison's equation

F,, : 0'5P,,' * ttii *l[ui + u,, - *l (14.2.23)

whcrc ,4r,,, is arca ol'clcrut'n(rrl volrrrut' v,, grnricclcrl <ln lr planc rtolrturl (o (lrcr!ir-ccliorr ol'lho ctrrrcrrl , irrul ( ,7,, rs llrc tlr;r1', tocllit'it'rtl r'olrt'sporttlirtp. lo ,'1,,,,.

ll tltc lclirlivc rnoliorr,rl 1111' lrrxly wrllr rr's|t't'l to tlrc lltritl is lr:rt'tttorur'. lltt'

A= p.Xlv4(c-u- t) (14.2.19)t.l

F" = p.llr,,r. l* . tDi t u11 - rt*) (14.2.20)

rH r.- /. 2z-r\uii:Te ^"'cos\k.Xt -T).r(n) : -3 [-*0,-E) + (r - ;) texp(-r) - r]] (14 2 t6)

I /2r\2o" :; (r"i

I,, J, *,, Y, * Y,l) dY, (tY2: I' *u,,, * t) dt

L L {'1,A,,,1.t',

*504 ()t Iriilolit r;ilrt,(;t(,til l;

(lnrg an(l incrliu crrcllicicnts itt Mrtrisort's c:(lulriion can bc clctcrntirtrrl ort lltcbrrsis ol'cxpcrirrrcntal results as lunctions ol'local oscillaklry Rcynolrls rrrrrrrbcr.(11, , 2TD2l(il.t), Keulegan-Calpenter number, K : VTt lD, an<l rclalivr: b<xlysrrrlircc nrughness, where D is the diameter of the body, a is tho kincnraticviscosity, V and T1 are the amplitude and period of the relative fluid-brxlyvclocity. Howcver, actual relative fluid-body motions are not harmonic. I'hisintrirtlrrccs rrnccftaintics in the determination of the drag and inertia coeflicientst'vcrr il t'xlrclirnontal information for harmonic relative motions were availablcrr t('nns ol (11,, rrncl K. Unfortunately, such information is not available tbr thc:rrrrrll A rrrrrrrbcrs (of the order of 2) and large Reynolds numbers (of the ordcrol l0(') ol inlcrcst in tension leg platform design. Forthis reason calculations:lrrrrrltl lrt't':rllicd out for various sets of values Ca, C., and investigations:.lrorrltl lrt't'orrtlucted into the sensitivity of the results to changes in these values.

Restoring Force. The surge-restoring force in a tension leg platform is sup-plit'rl lry (hc horizontal projection of the total tension force in the tethers. Mostol lhis lirr-cc is the result of pretensioning, which is achieved by ballasting thelloirting platlirrrn, tying it by means of the tethers to the foundations at the sealkxrr'. llrcrr dcballasting it. The tension forces in the tethers should exceed thet'onrprcssion fbrces induced by pitching and rolling moments due to extremelrxrtls.

l.lrrtlcr thc assumption that the tethers are straight at all times, the restoringlirn't: c:rr.r bc written as

FIGURE 14.2.3. Notations.

to the hydrodynamic viscous load F,, (Eqs. 14.2.1, 14.2.18, and 14.2.23). Forthis reason it is appropriate to solve Eq. 14.2.1 in the time domain.

The nominal natural period in surge is

Tn:2r (14.2.26)

where M"6 is the coefficient of the term in -t and ft the coefficient of the terminx in Eq. 14.2.1. From Eqs. 14.2.1, 14.2.18, and 14.2.24, it follows that

505

(ry)"R(/)__(r+srth+Lh

wlrcrc 7'is the initial pretensioning force, AZis the incrementalstrrgc rrrotion, /, is the nominal length of the tethers at r :incrcrrrcnlal lcngth, and

(14.2.24)

tension due to0, A/, is the

(r4.2.2s)

T.- :2trltu + 'qlt^f""lTl

wlrt'n' (',11 is thcr tkrwnclraw coefficient, equal to the weight of water displaced;r; tlrt'rlr':rlt is incrcusctl by a unit length [4-32] (Fig. 14.2.3).

lrr n'lrlity. lryrllotlynumic and inertia forces cause the tethers to deform trans-vt'rst'ly 'l'lrc rrrrglc bctwccn the horizontal and the tangent to the tether axis attlrt'plrrtlirrrrr hccl can thcrcfore differ significantly from the values correspond-irrp to lhc crrsr: ol'a straight tether. Nevertheless, owing to the relatively smallrolt'ol llrc rcs(oring lirrcc in thc dynamics of typical tension lcg platfbrms, thct'lli'cl ol srrclr tlill'crcrrccs on thc motion of thc platfirrrns appL)irrs to hc ncgligiblclirr' prircticll purp()scs ll4-36, l4-37, l4-381.

Surge Response. 'l'hc strrgc rcsl)onsc is olrtiritrctl by solvrrrl, lrt1. 1.1 .J.1. 'l'ltisr't;tlrliorr is norrlirrt'ru', llrt' sl nrngt'st corrllilrrrliort lo llrc rrotrlrrrt';uity lrf i111' 1111q'

(14.2.26a)

A calculated time history of the surge response is represented in Fig. 14.2.4as a function of time for a platform with the geometrical configuration of Fig.14.2.5, under the following assumptions: platform mass M : 34.3 x 106 kg;total initial tension in legs T: 1.56 x 105 kN; Morison equation coefficientsC-i;: 1.8,x Ca,,: 0.6, wave height and period H :25 m and 7. : 15 s,respectively; current speed varying from 1.4 m/sec at the mean water level to0.15 m/s at 550 m depth; aerodynamic parameter C,1": 4320 m2 elevationof aerodynamic center 2,, : 5O m; atmospheric boundary-layer flow parametersr:0.002, P :6.0, Li,: 180 m,J, :0.01,f,:0.2, Ct: 16 (scc Eqs.2.2.23,2.3.2,2.3.4,2.3-25, lLl,tl 14.2.17); and mean wincl spccd [./(:,,) - 45m/s. lt is shown in ll4-3 ll llurt thc contrihutions ol'(hc rucrrtt wintl itttrl ol (ltt'wincl lluctuations lo thc llelrk r'('sl)onsc ol'l"ig. 14.2.4 ut-c rtltottt 1ll')/,' lnd l)'/,'.

'1'l16r'llre pllrtlirrtrt gl lriA, l,l .t \. rl lollows lrolrr llrcsc irsstuttltliotts itlrl ltottt lirlr l.l .t l(),rrrrll,'l.,l.l(rr llt:rl lltt' tuttttittttl tt;tlttl;tl ltlrlttr'ttr 1' tr /,, l(Xl s

rtlr T / a

l, 1 N,,: ,,,* C'*[l - tl I - (xll)'l

506 ()t I1iil()ilt tiiltu(;l(,1il r

0 500 1000 1500 2000

TIME [s)

FIGURE 14.2.4, Calculated time history of surge response [14-31].

rr:spcctively. It can be verified that this conclusion is equivalent to stating thatwincl-induced resonant amplification effects are negligible in the cases inves-rigarcd in u4-311. Sensitivity studies showed that the results were affectedinsignificantly by uncertainties with respect to the actual values of the atmo-sphcric boundary-layer fl ow parameters.

Nominal Damping Ratio of Pseudo-Linear System Representing the Re-sponse to Wind. It was indicated previously that the estimation of the effect

0rprh; - 600r

Itl(llllll,l 14.2.5. (it'otnt'lly ol ;tl:tllotnr

I'l :' llYll/\Ml{ Wllllrllll(.1:,()N(.{)Ml'l lAlJl {}l l:'ll()l ll "lllll{lll'lll ': ltll7

0l'trtrltttlClrl wirul orr sUt!t,(' wits:tltt'tttlltt'rl irr llrt' li(i'l:rllll(' (tll lltt' lr:r:ltr-' t'l lltt'Irsstrrrrllli6rr llrlrt tltc ctlrrirliort ol'strlgt'rturliott l('l)l('s('lll:j:l ltttt':tt :;vsl('llr wllll:lviscotr,s tlirrlrping (crltt cllltrltclcrizul lry it ttotttitl;tl tlirtrrpilrl' llrlto' ( Ilrt't'llt't (

.l'this tcrrn is p6stulatccl to bc cclrrivllcrtt lo tlrc tl:rrrrpitrl', t'llt't'l ol lltc lrytlrrr

tlynamic viscous lirrcc.Such an approach is acccptahlc il tlrc or.tlcl ol rrutg.triltttlt' ol lltt' ttortttlt:tl

damping ratiit is consistent wi(h thc hyclnrtlynalrtic'bclt:tvior.ol tltt'syslcllr.Calculations are now presentccl that illustratc lrow such lttltttittitl tlltttt;'riltg rltlioscan be estimated.

It is assumed that aio(r)(see Eq. 14-2.10) is given by the harmonic function

2ruLqQ) : uLot cos ,t

50

E40U(5E.=(_/) 35

where I is the natural period in surge, and the system under considerationis linear with mass M + A, natural period 7,, arid damping ratio f' The ampli-tude of the contribution to the surge response of a harmonic forcep,C.A,U(2")uJq r cos Ztrnt is denoted bY r,*u*, and is given by the relation

+ 12ynT,)2\t/2

(r4.2.27)

(14.2.28)

(14.2.29)

i

I

J! !max

The nominal damping ratio f is estimated from the condition that x,*,* be equalto the contribution to the surge response of the force poCoAoU(Z)u'", l cos(2tlTn)t, as obtained by solving- Eqs. 14-2.1' By substituting l/f for n in Eq'14.2.28, it follows that

. jp"C,AuIJ{z)uio 1;:6

Calculations were carried out in 114-311 for the platform shown in Fig'14.2.5 (with tether lengths In = 600 m) and for a similar platform with tetherlengths l, = l5O -, .,iittg the mechanical, hydrodynamic' aerodynamic' and

arm"osph;ric boundary-lay"i flo* parameters listed previously. Equation 14.2.29yieldei I : 0.5 andf - O'2 for the platforms with /, = 600 m and ln = l5Orn, respectively.

The- values C,1,, : o 6 and Cmj: 1'8 on which these results were based

rnay not be realisiic fbr members with large diameters, such as those depictedin nig. 14.2.5 |4-34|. For lhis rcason calculations were carried out in [14-3lllilr a-numbcr of possihlc scts ol' vlrltrc (,1, . C,,,,. Thc calculated nominal clartlp-

irrg rati.1s f wcic in trll clrst.s lirruttl tri l-rc sullicicntly largc t<t prcvclt( tltt',raa,,rr"n"" gl- s(nlng wirrtl irrtlut t'rl tlyrutrrtit' :rrrrplilicalirln cllccls. Ilrtwt'vt't '

litr sorttc vlrlttr:s tll' (',1,,, ("ll('tllilli()lls irr wlrit'lr {ltt' :tssttttlctl ('tll'rcllls wottltl lrt'l.wt:r ll''t 1111;5;1:61'lt,i f f l r'orrl,l lr':,rrll rn rcrlut't'tl ttolltitt:tl tl:rlltpirrll l;tlios lolt't'rlrtilt clirrrit(olo1t,it'ltl t'otttllltott:' ll('i 'ltr"('rvttltl w;tvt'lt"sls viol;tlt'llrt'ltt'ytroltl:;

p,CuAoUko\u',, 1

tnt + ,cnnlT)) 11l - tnf lrl:

()t I r;il()1il :illlt,(ilt,lrl l;

lln(l Kculcguu-Carpcntcr nuntbcrs, tltcy crtttltot pnrvitlc it rrsclirl ilrtlit'rrtitltt ol'tlrc c:ll'cctivc darnping lirr thc pn)totypc. 'l'his, in addition to thc ahscrrcc ol'lcliablc tlrag data fbr large Reynolds numbcrs and small Keulegan-Clilrpclllol'Irrunbcrs, is a continuing cause of uncertainty in the assessment of dynarlrict.llr:cts incluccd by wind acting alone or, in the case of a nonlinear analysis, irr

c()n.ir.ulc(ion with wave-induced slow drift.

REFERENCES

ll I M. lr. l)avics. L. R. Cole, and P. G. G. O'Neill, The Nature of Air Fktwsrtvtr oll.rlutrc Pluforms, NMI Rl4 (OT-R-7726), National Maritime Institutc,lit'lllrirrrr, U.K., June 1977.

l.l .l M . lr. l)avies , Wind Tunnel Modelling of the Local Atmospheric Environmentttl oll.slntre Plaforms, NMI R58 (OT-R-7935), National Maritime Institute,lrclllrarrt, U.K., May 1979.

1.1 ] 1.,. ll. Lirtleburg, "wind Tunnel Testing Techniques for offshore Gas/oilI'r.otluction Platforms," Paper OTC 4125 Proceedings, offshore Technology('tnt.lt'rcncc, Houston, TX, 1981.

ltl 4 llult's .for Buitding and classing Mobile offshore Drilling unirs, Americanllurcau of Shipping, New York, 1980.

l,,l .5 I?ulcs for the Consffuction and ClassiJication of Mobile Offshore Units, Detnorskc Veritas, Oslo, 1981.

14 6 Rulcs for the Design, Constuction, and Inspection of Offshore Structures,Appendix B, Loads, Det norske Veritas, oslo, 1977 (Reprint with correctionst919).

l4-1 Rcquirements for Verifuing the Structural Integrity of OCS Plaforms, Appen-rliccs, United States Geological Survey, OCS Platform Verification Division,l{cston, VA, 1979.

1,.:l lt AIrl Rccommended Practice for Planning, Desig,ning, and Constructing Fixed()l.l.sltorc Platforms, API RP 24, American Petroleum Institute, Washington,t)(

" 1981 .

l.l () I) .l . Norton and C. V. WollT, "Mobile Offshore Platform Wind Loads,"l';rlx'r'()'l'C 4123, Proceedings, Offshore Technology Conference, Houston,'l x. l()t31.

l.l lo ll. lltxrrrsllrr, "Analysis of Full-Scale Wind Forces on a Semisubmersible Plat-lrrnrr (lsirtlt ()pct':tlor's Data," J. Petroleum Technol. (1980), 171-776-

I I I I l' .l . l,rrrrslirrtl. Mutsurt,rncnts of the Wind Forces and Measurements of an OilI'r,t,lrtr'tittrr .lttt'ktt Structura in Tow-Out Mode, NMI R30 (OT-R-7801)' Na-lrorr:rl Mrrli(ittlc Ittstittltc, Fcltham, U.K., Jan. 1978.

l.l l.) ('. li. ('owtlrcy, 'l'itrrt-Avtrul4cd Aerodynamic Forces and Moments on a Mtilclol tt Ilrrrt' I.tg,gctl Crnt'rctc Production Plaform, NMI R35 (OT-R-7808)'Nrrliorlrl Mrrritirnc Institutc, Fcltham, U.K., June 1982.

I,l I \ ('. li. ('owtlrcy.'l'irtrr'Atrrugul At'ruxl.ynutnit'Flrtcs ttrul Mttnrnls rtn tt Mtxlcltll tt'l'ltrrr'l,t',r4gul ('tntt'rt'tt' Itnxlutliott l'lttl.lrtrnr, NMI It.l(r (O'l'It-7ltol3),N:rliotutl Mltritirrrt' ltrslilttlc. licl{llltltt' I J. K.' .lrrlle l ()112

lt l llt N(it :; 509

l':l l4 l]. L. Millcr lrrul M. ll. l)rrvics, lilirtrl ltxttlirr,r4 ttrt ()llsltort'Slru<'lurcs-ASttttrttrttry ol Wittrl 'litttrrtl ,\rrttlir,s, NMI l{l-16 (()'l'R-tJ22-5), National Maritimelnslittrtc, licltlrarrr, l..J .K., Scpt. l9tt2.

l4-15 lr. l).jcrrcgaarcl ancl S. Volschou, "Wind Ovcrturning Effect on a Semisubmer-siblc," Papcr OTC 3063, Pntcccdings, Offshore Technology Conference,Houston, TX, May 1978.

14-16 E. Bjcrrcgaard and E. Sorenscn, "Wind Overtuming Effects Obtained fromWind Tunnel Tests with Various Submersible Models," OTC Paper 4124,Proceedings, Offshore Technology Conference, Houston, TX, May 1981.

14-11 J. H. Ribbe and J. C. Brusse, "Simulation of the AiriWater Interface for WindTunnel Testing of Floating Structures," Proceedings Fourth U.S. NationalConference, Wind Engineering Research,B. J.Hartz (ed.), Department of CivilEngineering, University of Washington, Seattle, July 1981.

l4-18 J. M. Macha and D. F. Reid, Semisubmersible Wind I'oads and Wind Effects,Paper no. 3, Annual Meeting, New York, Nov. 1984. The Society of NavalArchitects and Marine Engineers, New York, 1984.

14-19 R. F. W. Gould, The Prediction of the Wind Loading on a Wind Tunnel Modelof an Offshore Drilling Unit Based on the SEDCO 700 Design, NMI Rl8 (OT-R-1729), National Maritime Institute, Feltham, U.K., July 1979.

14-20 C. F. Cowdrey and R. F. W. Gould, Time-Averaged Aerodynamic Forces andMoments on a National Model of a Semisubmersible Offshore Rig, NMI R26(OT-R-7748), National Maritime Institute, Feltham, U.K., Sept. 1982.

14-21 P. J. Ponsford, Wind-Tunnel Measurements of Aerodynamic Forces and Mo-ments on a Model of a Semisubmersible Olfshore Rig, NMI R34 (OT-R-7807),National Maritime Institute, Feltham, U.K., June 1982.

14-22 A. W. Troesch, R. W. Van Gunst, and S. Lee, "Wind Loads on a 1:115Model of a Semisubmersible," Marine Technol., 20 (July 1983),283-289.

14-23 P. J. Pike and B. J. Vickery, "A Wind Tunnel Investigation of Loads andPressure on a Typical Guyed Tower Offshore Platform," Paper OTC 4288,Proceedings, Offshore Technology Conference, Houston, TX, May 1982.

14-24 M. S. Glasscock and L. D. Finn, "Design of a Guyed Tower for 1000 ft ofWater in the Gulf of Mexico," J. Struct. Eng. ll0 (1984), 1083-1098.

14-25 T. A. Morreale, P. Gergely, and M. Grigoriu, Wind Tunnel Study of WindLoading on a Compliant Olfshore Plaform, NBS-GCR-84-465, National Bu-reau of Standards, Washington, DC, December 1983.

14-26 J. R. Smith and R. S. Taylor, "The Development of Articulated BuoyantCofumn Systems as an Aid to Economic Offshore Production," Proceedings,European Offshore Petroleum Conference Exhibition, London, Oct. 1980, pp.545 551.

14-27 J. A. Mercier, S..l . LcvL:rc(lc, antl A. [,. Bliault, "Evaluation o1'Hutton T[,PResponse to Envirorrrrcntirl l,()r(1s." I'irpcr'O1-C 4429, Procccrlirgs, Ofl,shorcTcchnology Ctttrli'tctrtr', Ilottslott.'l X. Mlry l()t32.

14-28 N. tillis, J. H.'l't'llow. l; Aruk'r.';orr. lntl A. 1.. Wtxxllrt'ltl, "lltrltolr'l'l,l'Vcsscl Strlrclrrrrl ('otrlrl'ur;rltorr :rrtrl I )r':,i1'tt lir':tltttt's," lt:rlx'r ( ) l'(' 'l'l.l'1. I'r,tt't'ulitt.qs, Ollslron' 'lt'r'lrttnlol't, ('rltrlr'tt'rt( r'. lltttlsl()tt. 'l \. Mlry lt)li,l .

142() A. Klrrcerrr lrnrl (' l);rll,rrr. lllrr:rrrrt l,llr'rlr, ol Wltrl otr'li'lt:'iott l.r'1'l'llrl

510 ()t t:;il()nt :;tlrt,(;l(,lll 1;

lirrrrrs," I'apcr O'l'C 422(). I'nn't'r'tlirt.q.s, ()ll.slutn"l?clrtutltt,q.v ('rtrr.li'tr'rrr't',Houslon, 'l'X, May 1982.

l4 l0 lJ. J. Vickery, "Wind Loads on Compliant Offshore Structurcs," l'nrtulitt,g.s'Ocean Structural Dynamic Symposium, Oregon State University, DcpitrtnlcttlI'Civil Engineering, Corvallis, Oregon, Sept. 1982, pp. 632-648.

l4-3 I E. Simiu and S. D. Leigh, "Turbulent Wind and Tension Leg Plattbrm Surgc"'J. Struct. Eng., ll0 (1984), 785-802.

l4-32 N. Salvesen et al., "Computations of NonlinearSurge Motions of Tension LcgPlatfbms," Paper OTC 4394, Proceedings, Offihore Technology ConJbrenct.Houston, TX, May 1982.

1.1 .ll J. A. Pinkster and G. Van Oortmerssen, "Computation of First- and Second-Order Forces On Oscillating Bodies in Regular Waves," Proceeding,s, Seutrullnternational Conference on Ship Hydrodynamics, Univ. of California, Berkc-ley, 1917.

l,J .14 T. Sarpkaya and M. Isaacson, Mechanics of Wave Forces on Offshore Struc-tures, Yan Nostrand Reinhold, New York, 1981.

14-35 A. G. Davenport and E. C. Hambly, "Turbulent Wind Loading and DynamicResponse ofJackup Platform," Paper OTC 4824, Proceedings, Offshore Tech-rutlogy Conference, Houston, TX, May 1984.

14-36 E. R. Jefferys and M. H. Patel, "On the Dynamics of Taut Mooring Systems,"Eng. Strucr., 4 (1982),37-43.

14-3"7 E,. Simiu, A. Carasso, and C. E. Smith, "Tether Deformation and TensionLeg Platform Surge," J. Struct. Eng., ll0 (1984), 1419-1422.

l4 38 E. Simiu and A. Carasso, "Interdependence between Dynamic Surge Motionsof Platform and Tethers for a Deep Water TLP," Proceedings, Fourth Inter'national Conference on Behavior ofOffshore Structures (BOSS), pp.557-562,I 5 July 1985, Delft, The Netherlands.

14-39 R. L. Wardlaw, P. H. Laurich, and G. R. Mogridge, "Modelling of DynamicLoads in Wave Basin Tests of the Semisubmersible Drilling Platform OceanRanger," Proceedings, Interncttional Conference on Flow-lnduced Vibrations,Bowness-on-Windermere, England, May 12-14, 1987 .

1.1 40 .l . M. Macha, "Modeling Wind Loads on Mobile Offshore Structures-A Sum-rrrrry of Wind Tunnel Results," in Wind Elfects on Compliant Offihore Plat-.litrns, C. E. Smith and E. Simiu (eds.), American Society of Civil Engineers,Ncw York, 1986.

l,l ,ll lr. Iljerrcgaarcl and S. Hansen, "Wind Effects on Semisubmersibles and Otherliloirling ()lllshorc Structures," in Wind Effects on Compliant Offshore Platlitrrns, (. li. Srnith and E. Simiu (eds.), American Society of Civil Engineers,New Yor.k. 19t36.

CHAPTER 15

WIND.INDUCED DISCOMFORT INAND AROUND BUILDINGS

It is required that structures subjected to wind loads be sufficiently strong toperform adequately from a structural safety viewpoint. Recent experience hasshown that in the case of tall, flexible buildings, the designer must also takeinto account wind-related serviceability requirements. The latter may be for-mulated, in general tems, as follows: structures should be so designed thattheir wind-induced motions will not cause unacceptable discomfort to the build-ing occupants.

Wind-induced discomfort is also of concern in the altogether different con-text of the serviceability of outdoor areas within a built environment. Certainbuilding and open space configurations may give rise to relatively intense localwind flows. It is the designer's task to ascertain in the planning stage thepossible existence of zones in which such flows would cause unacceptablediscomfort to users of the outdoor areas of concern. Appropriate design deci-sions must be made to eliminate such zones if they exist.

The notion of unacceptable discomfort, which is seen to play a central rolein the statement of serviceability requirements, may be defined as follows. Inany given design situation various degrees of wind-induced discomfort may beexpected to occur with certain frequencies that depend upon the degree ofdiscomfort, the featurcs ol'thc clcsign and the wind climate at the location inqucstion. The discomlirrt is uttircccptlrhlc il'any of these frcquencies is .iuclgcdto bc too high. Statcrrrt:rrls s;x't'ilyirrg nurxinrtrrn acccptahlc tncan l't'ct;ucnt'ics<ll'<lccurrcncc firr virriorrs rlt'1'.rt't"s ol rlist'orrrlirl ruo ktrowtr:ts cotttlirll ttilr'r'i:r.

ln crlrnlilrt critcritr tlt'vt'loP1'11 lor rrsr' irr rlt'sigrr il is irrtPlrrctit';rl lo Itli'r'cxplicitly to rlcgrccs ol tlisr'orrrl,rrl ;r: r;rrt lr ltlrllrt'r', ttlt'tt'rtt'r' is ttt:trk' lo ;t

suilirltlt: l)itf:illc:(L:r, vluiotr- v;rlrr':,,r1 s,lrt, lt:ttt':tssor'irtlt'rl willt v:rrl()tts (l('l't('('sol'tlist'orrrlirrl. ln llrc t'lr:u'ol \\'irrrl rrrlu(,'il 111111111111' ltlrliotts, lltls Pltl;tlttt'lt't ti

5l I

512 wtNt) tNt)t,ot t) t)[;(;()Ml ()t tt tN nfil] nt t( )t tNt) ttltll l)lN(iti

tlrr: lruiltlirUl ilccclcralion- ln critct-ia pr:tlrrinirtg to thc scrviccability ol'pc:tlcslrirtrrrrrcirs, (hc paranrctcr cnrployed is an appnrpriatc mcasurc rtl'thc wintl s;lccrlnoar lhc gnrund at the location of concern. Clearly, to develop comfirrt c:ritcria,i{ is rcquircd that parameter values be established that correspond to varioustlcgrccs <ll'human discomfort. Furthermore it is necessary that to various clc-grccs ol'discomtirft-or, equivalently, to the parameter values that correspondIo thcnr thcrc bc assigned maximum acceptable probabilities of occurrencc.

Vcrilying thc compliance of a design with requirements set forth in a givenst't ol cornlil( critcria involves two steps. First, an estimate must be obtaincdol tlrc wirrrl vckrciLics under the action of which the parameter of concem willr'xct'ctl tlrt' vrrlrrcs specified by the comfort criteria (these values may be referredIo irs t'r'ilir'irl). Socond, the frequencies of occurrence of these velocities mustlrt't'stirrur(cd on the basis of appropriate wind climatological information. lfllrt' ln't1rrt'rrcics t.hus estimated are lower than maximum acceptable frequenciesspccilictl by thc comfort criteria, then the design is regarded as adequate fromir st'r'v it:cirbil ity viewpoint.

l{clcvrurt c:ornputational fluid dynamics methods are discussed in [4-89].l{c:rsorrirhlc qualitative results have been obtained in some instances, but not[:lirritivc validations appear to be available. The development of comfort cri-rclia lirr thc dcsign of tall buildings and questions related to their practical useirrc tliscusscd in Sect. 15.1. Comfort criteria forthe design of pedestrian areasrrntl rclatcd design information are dealt with in Sects. 15.2 through 15.4.

15.1 SERVICEABILITY OF TALL BUILDINGS UNDER THE ACTIONOF WIND

15.1.1 Human Response to Wind-lnduced VibrationsStuilics ol'human response to mechanical vibrations have been conducted withinthc: lust two decades mainly by the aerospace industry. Because the frequenciesol'vibrir(iorr of interest in aerospace applications are relatively high (usually Illz to .15 Hz), the usefulness of these studies to the structural engineer is11t'nclrrlly lirrritccl. Ncvcrtheless, results obtained for high frequencies have beenr'xtrrrpol:rltril irr ll5 ll to frequencies lower than 1 Hz, with the followingt'on1'slxrnrlt:rrcc bcirrg prlposed between various degrees of user discomfort andllrt' ;rt't't'lt't'rrliorts cltttsing them:

I)cgrcc ofl)iscomfbrt

Acceleration (inpercentages of the

acceleration of gravity g)

lrnpcrccptiblcPcrccptiblcAnnoyingVcry AnnoyingIrrlolt:t':rblt:

||) I :il llvl(.1 nllll llv {)l lnl I llt,ll I)lN(ill llNl)l ll llll n(.ll()N ()l wlt'll) 513

llcsul(s 6l cxprl.irrrcrrls ltirrrctl :rt r:stlrblislrirtg pt't't't'grliott llttt'slrolrlr lol pt'

rirxlic rrrrtlions ol' 0.(Xr7 llz to 0.2 Flz Irtvr: ltt'r.'tt t't'1rot1t'tl ilr l l5 Jl. 'lltr'cxpcrirncnts, carricd out ort Il2 sLrb.ic:cts itt tttol iott sitttttlltlots l('l)r( \('llljrll\/('of an ofiicc environmcnt, wcrc clcsigrrctl to litkt' ittto lteeottttl lltt' tltllttt'ttt t' ttlrottperception thresholds of bocly oricrrla(iott, llrtly tturvr:lttt'rtl, lrotly lxtslttlc, ;ttttlthe extent to which the motion is anticipatctl by (lrt: srtbict'ts ol lltc t'xpt'lttttt'ttlsThe perception thresholds as reportcd 1'>y 50')(, tll'lltc: sttll.ic('ls w('le lirtttttl trr

be approximately 1 % g,0.9% g, and O.6% g lirr frcqucncics ttl' v ibrat ion ol 0.(Xr7Hz, 0. I Hz, and 0.2 Hz, respectively. It is noted that within this l'rcquoncyrange-the perception thresholds decrease as the frequencies increase. Addi-tional experimental results are used in [15-2] as a basis for a tentative relationbetween the horizontal acceleration of a floor and the percentage of the indi-viduals on that floor for whom the acceleration will be perceptible'

Studies of human response to vibrations of a motion simulator have alsobeen reported in [15-3] and [15-4] for frequencies in the range 0.1 Hz to 1 Hz.Average perception thresholds were found to vary from about o.6%s for fre-quencies of 0.1 Hz to about 03%S for frequencies of 0.25 Hz. Motions weredistinctly perceptible and the subjects were annoyed while working at theirdesks ilthe accelerations exceeded 1.2%g. Beyond accelerations of 4%g, theperceptions were described as strong and the subjects experienced difficultiesin *ulking. The motions were described as extremely annoying or intolerablebeyond accelerations of the order of 5%g to 6%9. Similar results have beenreported in [15-5].

A study presented in [15-6] and [15-7] is based on observations of humanresponse to actual rather than simulated wind-induced accelerations. The in-vestigation covered the behavior during a storn of two buildings and of theiroccupants. Estimates of the rms value of the top floor accelerations during thestorms were based on response measurements for one of the buildings, and onwind speed measurements and wind tunnel testing for the second. These esti-mates represented averages (1) in time over the periods of highest storm inten-sity (20 min to 30 min)* and (2) in space over the entire area of the floor-thespace averaging being performed to account for wind-induced torsional mo-tions. (For the effect of torsional motions see also t15-321 .) The rms valuesthus obtained were 0.2%S for the first, and 0.5%g for the second of the twobuildings. Interviews with building occupants then revealed that about 35% ofthe persons on the higher floors in the first building experienced motion sicknesssymptoms during the storm. For the second building the reported percentagewas about 45%. It is noted in [15-7] that creaking noises that occur during thebuilding motion may increase significantly the feeling of discomfort and shouldtherefbre be minimized by pnrper structural detailing.

Rcsults of surveys contlrrc{ctl among occupants of tall buildings in .Iapan arcrcportcd in ll5-81.

'r'l inrt'lrvcr':rl'.t's wt're ttlso cllt'tlt'rl .r't't lottltt'l 1x'rirxls I l5 (rl.

<)ns)nga!'t,gll%,s 5'/,,,c,5%,g l5'n,7i) l5'x,,q

*514 wlNl) tNt )1,(.t t) t)[i(,()Mt ()l il tt,t At.]t r /\l()t,Nr) nl ,l t)tN(i:i

15-1-2 Comfort Criteria('oltrlittl critcria should in principlc bc basctl on an cxtcnsivc ktr<lwlctlgc ol'thellLrgrcc ttt which building users are prepared to accept discomlort ass<lciirlcrlwi(lr wirrd-incluccd accelerations. However, at present such knowlcdgc is sculcc.

A sirnplc comfort criterion has been proposed in [5-9], bclicvcd by itsrrrr(hors to hcjustificd by the results of [5-2]. This criterion, which lin-rits thc:irvc:rirgc nrrrnhcr ol'occurrences of 1%g accelerations at the top occupied lkxrrItr :rt rnost 12 pcr ycar, has been applied to the design of the World Tradc('t'tt(t't'11.5-91. ln ll5-61 an attempt is presented to develop comfort criteriu orrtlrc brrsis ( I ) ol'r-ccorclccl objcctions by building users to the recurrence of wind-rrrrlrrt't'rl lrrriltling vihrations and (2) of estimates by owners or developers ol'llrt' possibk' t't'ottoruic repercussions of user dissatisfaction with the buildingpt'r'lrrrr;urt't'. Iinrrrr interviews with building occupants who had experiencc(lnr()tr(fns wilh lur rrns value of the top floor accelerations of about 0.5%g, Irv:rs cslirrurlctl lhat about 2% of the people in the top one-third of a buildingworrltl olr jer'l l() rnore than one occunence of such motions in six years. Inter-vit'ws witlr brrilding owners and developers suggested, on the other hand, thatrcrrlrrl orsrrlcsof o{Iicespacewouldnotbeaffectedsignificantly if atmost2%ol' thc occupants in the top one-third of the building found the sway objection-lblc. On thc basis of those findings, it is suggested in [5-6] that the followingtlcsigrr critcrion appears to be reasonable: "The retum periods, for stormscrrrrsirrg iln rrns horizontal acceleration at the building top which exceeds 0.5%5,shall rxrt be less than six years. The rms shall represent an average over the20-rrrirr period of highest storm intensity and be spatially averaged over thebuilcling floor." This criterion is presented in [15-6] as tentative and in possiblencccl of ad.justment as additional information becomes available.

rrrt r,t ilvl(.1 nlill llYol lnl I lll lll l)lN(i:itllllrl ll llll nr llrrl l{rl wllll r fil.'i

FIGUR.E l5.l.l. Wind speeds inducing critical building accelerations.

15.1.4 Frequencies of Occurrence of Winds lnducing CriticalAccelerationsThe second step in verifying the adequacy of a design from a serviceabilityviewpoint is to estimate the frequency of occurrence of accelerations o higherthan the critical value o* specified by the comfort criteria. As shown in[5-6], it is reasonable to define this frequency as the mean number per yearNs(o > o*) of storns causing accelerations o > o*.It is acceptable, inpractice, to approximate Ns(o > o*) by the number of days per year Np(o >o*) during which the maximum wind speeds exceed the values correspondingto the curve of Fig. 15.1.1. It may be argued that, for office buildings, highspeeds occurring at night should not be counted in estimating the mean fre-qucncy Np. However, in view of the many uncertainties inherent in the designfirr building serviceability, such refinements do not appear to be warranted evenlhough they might reduce Np by a factor of the order of two.

'l'hc number of clays pcr ycar Nzr(o > o*) during which wind vclocitiesc:xccctl ccrlain spccifictl virlrrt's (that is, the valucs dcfinccl hy thc curvc <ll'Fig.l-5.1"l) cirn hc ohtirirtctl n';rrlily l.nrrrr l.ocal Climatological I)ttrr (1.('l)) she:cts

15.1.3 Relation betweenAccelerations

Wind Velocities and Building

A lirst stc:p in vcrifying the compliance of a design with requirements set forthirr t'orrrlirrl crilcria consists in the estimation, for each possible direction, of thewintl spt't'rls lhirt would induce the acceleration levels of interest. Wind tunnelIr'sl rt'srrlls rrrry bc uscd to obtain plots of speed versus direction for the windvt'kx ilit's llurl ilttlucc critical building accelerations (that is, accelerations equalt. tlursr'sPt't'ilitrtl by thc comfirrt criteria). An example of such a plot is shownrrr l;i1' 15.1 I. (Notc that the methods of Chapter 8 can be applied in thist'orrlt'xl.) Spr:ctls corrcsponding to points outside the curve of Fig. 15.l.l willirrrlrrt'c irccc:lcnrtions such that-if a criterion of the type proposed in [15-6] isrrst'tl o ) o {', whcrc o is the spatially averaged rms value of the top floor;rt't'clt'lrlions ancl o,' is thc critical value of o specificd by thc cornf<lrt critcria(c.g., irr f l-5-(rl, lt" : O.5ol,g). For cstirnatcs ol'huiklirrg irct'clcrltions, sccrrlso ('lrirplcrr ().

516 wtNt) tNt)(,(;t t) t)ll;c()Ml ()nl lN nul l nll()llNl) llt,ll l)lN(ili

lilr thc wcathcr slatitln closcst trl lltc klt'irliorl itt clucstitltr (scc Sccts. -1' l' ]'4'ancl 8.3.1). 'Ihe LCD contain daily rccords ol'thc fastcst-ruilc or pcrtk gtrsl

spocds ancl of the corresponding wincl directions. To usc thc inlilrrrr:rtiotr ob-tiiincd 1'rom the LCD in conjunction with Fig. 15. 1 . l, proper adjustmcnts ttrust

bc rnacle to account for anemometer elevation, roughness of terrain, and av-oraging of the wind speed with respect to time, as shown in Sect. 3.1.

ihe estimated mean yearly frequency Np(o > o*) must be compared withthe maximum acceptable annual frequency of occurrence of accelerations o >ox specified by the comfort criteria. Let this frequency be denoted by N,a(o >a*) ie.g., the value of N7(o > d*) proposed in [15-6] is 1/6 peryear)' If N2< Nr,1he design is regarded as adequate from a serviceability viewpoint.

15.2 COMFORT CRITERIA FOR PEDESTRIAN AREAS WITHIN ABUILT ENVIRONMENT

The problem of wind-induced discomfort in pedestrian areas is _not new (see

Fig.15.2.1andp'188).However,inrecentyearsnewtypesofbuildingandop""n rpu"" configurations have evolved. These may exhibit under certain un-favorable conditions zones of intense surface winds causing unacceptable dis-

,l:ull,lr,h,i\ Lll'l.l, ll"'lrr. J' l! \

F.I(;URFI 15.2. l. Thc Gust. Lithograph by Marlct, collccliort ol llrc llibliothi:c1trc tlcla Villo tlc l'itt'is (pltoto llogcr Viollct, I'irris)'

t" (.()Ml ()l ll (,lil llllln l{'l I l'l lrl :,llllnNnl ll n:;Wl llllNAlll ,ll llNVlll(}NMI Nl 5lI

t'orrrlirrl lo rrst'rs ol ;x'rlt'sl r;ur iurits.'l'ypit':rllyr'suclr t'ottligttt':tliotts ittvolvctall ltuiltlirrgs lisirrg wcll :rlxrvt'llrt'srrrnrrrrttlittg lrrrill eltvilorttttcttt atttl atli:rccntlo opcn riplccr; sut'lr rrs l)lirzirsi ()r'rrrrrlls. As irtrlicrrtctl prcviously, t<t dclinc thcnolion ol'urracccptablc tliseorrrlirrl (luan(i(rrtivcly it is rcquircd (l) that a cttr-rcspondcncc bc cstablishctl bc(wccn vlrrious dcgrccs of'pcdestrian disc<lnrlirfland thc wind spccds causing thcrn ancl (2) that maximum acceptable frequenciesol'occurrence be specified fbr thesc wind speeds. The present section is devotedto a brief discussion of these two requirements.

15.2.1 Wind Speeds and Pedestrian DiscomfortLet V denote the mean wind speed measured at approximately 2 m aboveground and averaged over 10 min to t hr. Observations of wind effects onpeople and calculations involving the rate of working against the wind suggestthat the following degrees of discomfort are induced by various speeds Zll5-10, t5-111:

V : 5 mls onset of discomforttr/ : 10 m/s definitely unpleasantV : 2O mls dangerous

A more detailed description of effects of winds of various intensities (as definedby the classical Beaufort scale) is presented in Table 15.2.1t15-101. Tentativeinformation on comfort of strolling pedestrians under various sun exposure,ambient temperature, clothing, and wind speed conditions is provided int1s- l 21.

Experiments reported in [15-13] and [15-14] suggest that pedestrian comfortis a function not only of the mean speed /, but of wind gustiness as well. Itis therefore reasonable, in principle, to study wind effects on people in termsof an effective wind speed Z" defined as follows:

I d2t/2 |V':Vll+k-lI v I(15.2.t)

where V is the mean speed, ,121/2 i" the rms of longitudinal velocity fluctua-tions, and k is a constant reflecting the degree to which the effects of thefluctuations are significant. According to the results of[15-13] and [15-14], anappropriate value for this constant is ft = 3.0. However, other investigatorsuse the value k : I 5 ll.5-l-51 or k : 1.0 tl5-161. According to [15-14] windtunncl cxperiments anrl obscrvrtlirlns of pcdestrian performance suggest thelirllowing crlrrcsp<lnclcttt't'betwt't:rt spcotls Z'' (with k : 3.0) and various de-grccs ol' discorllil11.

rllrrl rtol cxr'lrrsivr'ly; st'r' I l'r l(rl

518

'l'Alll ,l,l

llcrrrr lirltN tt t ttlrt:t

wlNt)tNl)(,ol t) t)t1ic()Ml ()l ll lN nNl r nll(,llNl) llt,ll l)lN(;l;

15.2.1. SumrnarY ol'Wintl lilli'cls ll5-l([ _Dcscription ol'

Wind Spced (m/s) Description ol' Wintl lrllccts

ll,:' (l()Ml {)lll (:lll llllln l()l I I'l l)l :;tltlnN nt ll Ati wt lillt'l A ilUil I til\/iltrrt.tMt ilt 5lq

whe:rcr /r is tlrt' wirrtl spt:t'tl trvcttrgctl ovt'r' 1 s ll5 I /1. A:, rrolt.rl rrr I l'r I /1.tltcsc crilcrirt ittc cc;uivlrlLrnt Io rtr rturrgitlrlly trlrt't',sr'v('r('llr;rrr llto:;r'rrlIr.5-141.

Thc ability ol'pcdcslriarts to ittlittsl l() slrl)nll wirrtls is lrllt't'letl lrtlvt'r'st'ly ilthe exposure to such wincls is lclirlivcly stttklt'n, lrs is llrc t'lrsc in zolrr's willrflows that are highly nonunilonn irr s1.xrcc. lt is thcrclirrc rxrtctl in ll5 l.rll tlrirtif the mean speed varies t>y 7O%, ovor a distaucc ol' lcss thirrr 2 rrr or so, lltceffects of wind on people are more severe than suggcstod abovc.

Measurements of wind drag on people are reported in [5-291.

15.2.2 Comfort CriteriaComfort criteria were previously defined as statements specifying maximumacceptable frequencies of occurrence for various degrees of discomfort. Thefollowing simple criterion based on extensive experience with the study ofground level wind effects in built environments is suggested in [15-11]. Com-plaints about wind conditions are not likely to arise if, in pedestrian areas,winds with mean speeds V > 5 m/s are estimated to occur less than 1O% ofthe time. Complaints might arise if such speeds are estimated to occur betweenlO% and2O% of the time. Estimated frequencies higher than20% coffespondbroadly to situations where in existing shopping centers remedial action had tobe taken to reduce wind speeds.

More detailed comfort criteria reflecting individual opinions on acceptablefrequencies of occurrence of various wind speeds have been proposed in[5-151, t15-18], and [15-19]. An example of such criteria is given in Tablers.2.2l15-181.

The first criterion in Table 15.2.2 is roughly equivalent to the criterionpreviously quoted of [5-11]. The limiting gust speed of 25 m/s correspondsto winds that could knock a frail person to the ground 115-191. Otherwise, asindicated in [15-18], the values of Table 15.2.2 are subjective and have beenarrived at in the absence of reliable data.

TABLE 15.2.2. Comfort Criteria for Various Pedestrian AreasLimiting Wind

Speed

oI

Jt

CalnrLight airsl.ight brccze(lcntlc breeze

Mrxlct':tlc brccze

lire slr btccze

Strong breeze

Moderate gale

Fresh gale

Strong gale

Less than 0.40.4-1.51.6-3.33.4,5.4

5.5-7.9

8.0-10.7

10.8-13.8

13.9-11 .r

17.2-20.7

20.8-24.4

No noticeablc windNo noticeable windWind felt on faceWind extends light flagHair is disturbedClothing flapsWind raises dust, dry soil,

and loose paperHair disarrangedForce of wind felt on bodYDrifting snow becomes

airbomeLimit of agreeable wind on

landUmbrellas used with

difficultyHair blown straightDifficulty to walk steadilyWind noise on ears

unpleasantWindbome snow above head

height (blizzard)Inconvenience felt when

walkingGenerally impedes progressGreat difficulty with balance

in gustsPeople blown over bY gusts

V":V":V"-V', :

6 m/s9 m/s15 m/s20 m/s

onset of discomfortperformance affectedcontrol of walking affecteddangerous

AreaCriterion Description

Frequency ofOccurrence

Srrltsetlrrcttl obscrvations of pedestrian perfolrnance in a large wind tunnelIrntl :rt tlic birsc ol' a high-rise building, conducted in Japan on over 2000pctlcstriirns, lravc lctl to the dcvelopment of the following proposed criteria:

/, < -5 rrr/s pcrfbrmance not all'cctctl5 rrr/s ( /1 < l0 rn/s pcr'lirnnant:t: rrll'cctcxll() rrr/s < V| < 1.5 Irr/s pcrlilrlnanc:o scrirrtrsly lrlli't.lctll5 rrr/s .- /r porlilrtttltttcc vct'y st'tiottsly lrllct'tcrl

Plazas and Parks

Walkways and other areassubject to pedestrianaccess

All of abovc

All ol-abovcr

Occasional gusts toabout 6 m/s

Occasional gusts toabout 12 m/s

Occasional gusts l()ahout 20 rn/s

( )t'r'irsiortal gusls l():rlxrrrl lJ5 rrr/s

l0% of the time or about1000 h/yr

I or 2 times per month orabout 50 h/yr

Alxrrrl .5 lr/yr

Less llrirrr I lr/yr

52O wtNl) tNI)t,ot l) l)llio()Ml ()lll lN nNl ) /\ll()t,Nl) ltl,ll l)lN(il;

As slurwl in Scct. l-5.4, thc culcrrllrlccl licrlucncy ol'()ccttrrL:llcc: ol witttlsltcccls in pcdcstrian arcas dcpends vcry stK)ngly up<ln thc cstilnation pnrcctlttrcbcingusctl. ltisnotedthatthecomfbrtcritcriaof [5-lll-andsirnilarcrilct'ilsrrggcstccl by other authors-are applicable only if the wind speed I'rcqucne icsiuc cslilnatccl by the simplified procedure of Sect. 15.4. These critcria arc rro

krngcr applicablc if the detailed procedure of Sect. 15.4 is used.ln thc abscncc ol established criteria, decisions regarding the acceptability

ol'corrrlirr1 conclitions in a pedestrian area are left, in practice, to the judgmcnlol'llrr: sitr: ()wncrs ll-5-201.

15.3 ZONES OF HIGH SURFACE WINDS WITHIN A BUILTENVIRONMENT

15.3.1 Wind Flow near Tall BuildingsAs rlrtctl irr ll-5-l ll, high wind speeds occurring at pedestrian level around tallllriltlings arc in gcneral associated with the following types of flow:

l. Vortcx flows that develop nearthe ground, as shown in Fig' l5'3'1'2. I)csccnding air flows passing around windward corners, as shown in Fig.

15.3.2.3. Air flows through ground floor openings connecting the windward to the

lccward side of a building (Fig. 1 5. 3.2) or cross-flows from the windwardsidc of one building to the leeward side of a neighboring building.

'l'hc flow visualization in Figs. 15.3.1 and 15.3.2 was obtained by injectingsrrurkc in the airstream. It is seen that the flow pattems in the immediate vicinityol' thc windward face are consistent with the pressure distributions shown ontlrc winclward face in Fig. 4.6.7b (i.e., the air flows from zones of high tozoncs ol' krw pressures). Part of the air deflected downward by the buildingliy'rrrs ir voncx (Fig. 15.3.1) and thus sweeps the ground in a reverse flow (area..1 , rrlrrkerl "vorlcx flow" in Fig. 15.3.3). Another part is accelerated aroundtlrt. lruiklilg c()ntcrs (Fig. 15.3.2) and forms jets that sweep the ground neartlrt. lrrrrIirr1i sitlt:s (irrcas B, marked "corner streams" in Fig. 15.3.3). If an()lx.nlnll r'orrrcctirrg tlrc winclward to the leeward side is present at or near the

1ir,,rrrr,l lcvcl, put-l ol'fhc dcsccnding air will be sucked from the zone of rela-tively higlr prcssurcs <ln thc windward side into the zone of relatively low1r,"rir,.", (suctions) on the leeward side (Fig. 15.3.2). A through-flow will thusswccp the area C shown in Fig. 15.3.3. Through-flows of this type have causedserious discomfort to users of the MIT Earth Sciences Building in Cambridgc,Massachusetts, a structure about 20 stories high [15-21.|. Cross-flows bctwccnpairs of buildings are caused by similar pressure differcnccs, its slrown in Fig.15.3.4.

The pattern of thc surfircc wincl flow within 11 5llg 1ls:l.tctttls itt tt t'olttltlt:x wlty

r,, r .'()Nr :; ()t ilt(ilt :i(,1il A(;t wtNt )lt wt ililN A Iil.Jil I t NViltoNMLNt 521

l'I(;URE 15.3.1. Wind flow in frontright.). *

of a tall building (wind blowing from left to

lfl(;URE 15.3.2. Wind lkrw rte:rr lltc wintlward facc of a tall building (wind blowinglirrrr lcli to right).

rlrillrrr.es 1.5..1. I lhn)ullll 15. t I'1.llriltlirrg llcst'irrt lt Iislirhlislutrcnlliorrcry ()llitc.

l.r l -),1.;rrrrl l\ l.)\torrlrilrrrletl lrylx'r'rrrissiorrol llrt'l)rrr.tlrrrllh ('olr\'rlllrt, ('orrlrollcl ol llcl llriltrrtrrit M:rjcsly's St;r

*522 wtNt)tNt)tt(;l l) l)lt;(i()Ml ()l il lN Aul) nl r{,{,Nl } lrt,ll l)lN(i:;

Vortexf low

lil(llJltl,l 15.-].-1. Regions of high surface wind speeds around a tall building (afterlr.5 r rl).

ll'lt /'()Nl 1,1,1 lll(ill :itllil n(;l WlNl):; Wl llllN n ltlJll I lNVlll()NMl Nl 523

rrgxrn tlrc rclaiiv(: loclr(itlrr, lltc: clitttclnsirttts, tltt: sltitl)t:s, ittttl t't:rlltilt rll tltc ltt'chiir:c(rrnrl lbl(uros (c.g., glorrrrtl lkxrr opr:rtittgs) ol' tlte: lttriltlirrgs ittvolvctl,upon thc K)ughncss and thc lopoglirlllriclrl lclrtttl'cs ol'tltt: lcrrltitt:rtrrtttttl lltt:sitc, and upon the possiblc prcscnco ncul lltr: silc ol'ottt: rlr scvtritl tlrll lrttiltlirrgs.'fo study the surface wind llow in any givcrr brrilt cttvinrrrrtcttl , it is thctclirrt'necessary, in general, to conduct wintl turrncl tcsls. Ncvcrtltclcss, its irrtlit'itlt:tlin [5-ll], experience has shown that inlirnlation hirsctl ott ircrotlyttitlttic sltttl-ies of the basic reference case represented in Fig. 15.3.3 is uscl'ul lor thcprediction of surface winds in a wide range of practical situations. Such infbr-mation is presented in [15-11] and will be summarized below. Its range ofapplicability includes built environments that retain a basic similarity with theconfiguration shown in Fig. 15.3.3 and in which the height of the buildingsdoes not exceed 100 m or so. Detailed information on the wind environmentaround single buildings and around groups ofbuildings is presented in [15-30].

15.3.2 Wind Speeds at Pedestrian Level in a Basic Reference Case[1s-11]Surface winds around models of the tall building shown in Fig. 15.3.3 weremeasured in wind tunnel tests conducted at a 1/120 scale. The roughness con-ditions simulated in the tests were typical of a suburban environment, the meanwind profile being given, approximately, by a power law with exponent cv :0.28. The surface winds depend upon the dimensions H, W, L, and h definedin Fig. 15.3.3 and are expressed in terms of ratios VlVs, where V an:d V, aremean speeds at pedestrian level and at elevation Il, respectively. In certainapplications it is useful to estimate the ratio VlVs, where Iz0 is the mean speedat l0 m above ground in open terrain. The ratios VlVs can be obtained asfbllows:

(rs.3.l)

Approximate ratios V1/V1, corresponding to the experimental conditions re-ported in [15-11] are given in Table 15.3.1 for various heights I1.

In the material that follows, the wind direction is assumed to be normal tothe building face (angle 0 : 0') unless otherwise stated.

Speeds in Vortex Ftow. Vo and V11 denote the maximum mean wind speedat pedestrian level in zone A of Fig. 15.3.3 and the mean wind speed at

'l'Alll,E 15.3.1. Approximatc Ratios VrrlV,) [5-lllII

(nr) 20 30 ,10 50 60 70 (x)

tlt

streams

htr

Main wind directrlV -VVHvo va vo

co.Fo

*!cto

_9 1{( ) t(x)

lt,tV,,vu

Through-

Itl(,llll{l,l 15.-1.4. ('lrss lkrw lrt'lwt't'rr lwo llrll lrtrrlrlirrl'. (.rltlr ll5 I ll)0.7.r 0.1{2 I O,l I Oli

524

o.7

0.6

0.5

.r 0.4

>< 0.3

o.2

0.1

0

WIND.INDUCED DISCOMFORI tN AND AnOUNt) DUil t)tNGS

L/H = O.258> H/h> 2

o.1

0.6

0.5

L/H = 1.08> H/h> 2

\<o.4

0.3

o.2

0.1

01.50.5 1.0

w/Ho.7

0.6

o.2

0.1

0

0.5

o.4

s 03S= 0.2

a

L/H = 2.O4 > H/h> 2

05

$ 04s 0.3 0.1

01.0

w/H

1.0 1.5 2.Ow/H

FIGURE 15.3.5. Ratios VnlVo [15-11].

elevation Il, respectively. Approximate ratios VAIYH are given in Fig. 15.3.5as functions of WlH for various ratios LIH and for the ranges of values Hlhshown. The height ft corresponded in all the model tests to typical heights ofsuburban buildings (7 m to 16 m). It is noted that as the building becomesmore slender (as the ratio WIH becomes lower) the ratio VAIVH decreases.

Typical examples of the variation of Vn with individual variables are shownin Fig. 15.3.6. If the distance Z between the low-rise and high-rise building issmall, the vortex cannot penetrate effectively between the buildings and Z7 issmall (Fig. 15.3.6h).If I is very large or if h is very small, the vortex thatIonrrs upwind ol the tall building will be poorly organized and weak; Vn willthcrclbre be relatively low (Figs. 15.3.6b and 15.3.6d).It h approaches thevalue of H, the taller building will in effect be sheltered and the speed Z,a willthus be low.

It is noted that the ratio VllVn is of the order of 0.5 for a range of practicalsituations.

Speeds in Corner Streams. Figure 15.3.7 shows the approximate depen-dence upon Hlh of the ratio VBIVH, where VB and Vn denote the maximummean speed at pedestrian level in the zones swept by thc corncr strcams andthe mean speed at elevation //, respectively. A typical cxatttplc ol'thc vlriirtion

0_5L/H = O.5

8> H/h> 2

15.3 loNl li ()l lll(lll frulltAct, wtNt)ti wt ililN A iltilt I I NVItoNMfNr 525

H (m)h)

H=O.4mL=O.4mh=0.1 m0 =0"

0 0.1 O.2 0.3 O.4 0.s 0W (m)

(c)

FIGLIRE 15.3.6. Examples of the variation of vn with individual parameters tl5-111.

4lt67il/ h,

Ir'l(llfllfrl 1.5..1.7. lldtios V,/V,, I l5-l I l.

ooF1 o

1

0.1 0.2 0.3 0.4 0.5h (m)

u)

s 0.6\Q- 0.5

L=O.4mW=O.4mh=0.1 m0=o"

0 0.1 0.2 0.3 0.40.50.60.7 0.8

lt=O.4m|l= O.4 mh=O.1 m0 =0"

Il-O.4mW=0.4mL=O.4m0 =O"

w/H > o.5L/H4*

il'I llr il :'( )Nl !: ( )l ilt( ;il : iUnt Aot wtNt )ii wl il llN A ilt It I t NVil t( )NMt N t 527526 wtNl) lNl)(,()l D t)ll;coMl ()lll lN ANI ) nl l()ttNt) lll'll l)lN(lt;

L =O.4mW=0.4mh =0.1 m0 =0'

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

o.2 0.3 0.4w (m)

ti5

34

r*,1

0

6

5

E453

,t

0

i:I'., 4l

',:ll,I

0

65

1i^E3,_e 2

1

0

o.2 0.3 0.4

h (m)

0.5 Wind direcJt_)

t ron

0.1 0.5

L (m)

FIGURE 15.3.8. Examples of the variation of vB with individual parameters [15-11]'

of /3 with the variables H, L, W, and ir is given in Fig' 15'3'8' The speed Z6

issecntodependweaklyupontheangle0betweenthemeanwinddirectionand thc normal to ttre uuilding face. However, the orientation of the corner

strcarns and, hence, the positioi of the point of maximum speed ZB may depend

signilicarrtly upon the clirection 0 of the mean wind'lnlirrrrration on thc wind speed field around the corner of a wide building

rrrrxlcl (// - 0.4 n, W :0.4 m, L : O'3 m) is given in Fig' 15'3'9' The

wintl spcccl clccrcases rather slowly within a distance from the building corner

cqual, apprcrximately, to H. The iatio Yl(Dl2), where Y is defined as in Fig'

f j.:.q ana D is the building depth, provides an approximate measure of the

position of the comer streani. Mlasuied values of this ratio for various values

of 11and of wl(Dl\) are shown in Fig. 15.3.10. It is seen that the points ofFig. 15.3.10 are fit reasonably well by a curve of the form y: constant xrt'. p* example, if W :4j m and D : 15 m' then Wl(Dlz\ : 6' Yl(Dl2)

= o.a 6ig. 15.^3.10), and the maximum speed on thc ccn(crlirrc: ol'tho builcling

would occur at Y = 0.8 x Dl2 : 6 m'Itisn<rlocl thattlrc r.lioV1,lV1,is<ll'lhctlrclcrtll'0'()5 lirr illlllll''('ol pritctic:ltl

sil r url irttts.

o.2

0.3 m

FIGURE 15.3.9. Surface wind speed field in a corner stream [5-ll].

Speeds in a Through-Flow. Let Vg and Vo denote the maximum mean windspced through a ground floor passageway connecting the windward to the lee-ward side of a building and the mean wind speed at elevation F1, respectively.f rigure 15.3.11 shows the approximate dependence of the ratio V7IVH upon thelrrrameter Hlh as determined in [5-11] by semiempirical formulas and windIunnel measurements. Examples of the variation of Vrwith H, W, L, h, and 0

f = constant X1

W2

wr\Itl(lllRl,l l5..l.lll. l;rrrlririt:rl r'rrlvc l/X vc:rsus l,//X ll-5-lll.

r'oO.8 mI .n^ -nl ro.4 mI oo.a *\.0.2.

H=0.4mW=0.4mh =0m

-90 -45 0 45Wind angle (0')

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

528 WIND.INDUCED DISCOMFORT IN AND AROUND BUILDINGS

1.4

1.3

1.2

1.1

w/H > o.5O.1<I/H<-

1.0

0.9

* 0.8

!t 0.7

0.6

0.5

0.4

0.3

o.2

0.1

0 456H/h

FIGURE 15.3.f 1. Ratios V.IV, [15-l l].

are given in Fig. 15.3.12. It is seen in Fig. I5.3.12b that for WIH < 0.5 theratios VglVn are lowerthan in Fig. 15.3.11. Figure 15.3.12e shows forvariousvalues of d the range of variation of Z6' with opening width.

Thc graphs of Figs. 15 .3 . 1 1 and 15 .3 .12 are based on measurements in andneur passageways with sharp-edged entrances. If the edges of the entrance arenrunded to form a bellmouth shape, the speeds Vg can be reduced with respectto thosc ol'Figs. 15.3.11 and 15.3.12 by as much as25% or so [15-11].

It is noted that the rat\o Vs lV H is of the order of I .2 for a range of practicalsituutions.

15.3.3 Wlnd Tunnel and Full-Scale Measurements of Surface Winds:Case Studies*Case 1. Oftlce Building (H - 31 m) Spanning a Shopping Center115-111. A 31-m tall building for which Hlh : 4.4, WIH : 1.6, and LIH =*Thc sourcc ol thc material is indicatcd by reference numbers in euch cuse, For ndditional cascstudics, sec ll5-311,

I6,3 ZONES OF HIOH EURFACE WINDS WITHIN A EUILT ENVIRONMENT t2c

1".0.3m14 -O.4m/r -0.1 m0 -0"

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8H (m)

(a)

Hwh0

=0.4m=0.4m=O.1 m=0o

o.2 0.3t7 (m)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

I7

6Fi

€4>Q3

2

I0

8

7

6

5Gi4>o3

2

1

o

5

4

3

2

1

E

.c)

7

6

5

QqEle

2,l

0

E

\o3

0.1

FIGURE 15.3.12.I rs-1 11.

I=0.3mH=O.4mw=o.4m0=O"

o.2 0.3

n (m)

u)Examples

t (m)(c)

-90 -45 0 45 90Wind angle (0')

(e)

oi /,. with individual parameters

0.1

I =0.3H =O.4,r = 0.10 =0'

mmm

0f thc vari&tion

wtNll lNIlr,lcLlJ Dlscc)Ml olll lN ANI) nll(xlND BtllLDlN(ls

---+->N

aI5.

FIGUR-E 15.3.13. Plan view, case study 1'

0.85 is shown in plan in Fig. 15.3.13. Full-scale measurements of ground levelspeeds (;; at locations i : 1,2, ... ,9 (see Fig. 15.3.13) and of the speeds236 measured at location l0 at a 36-m elevation above ground were made ontcn occasions. The results of the ten SetS of measurements are expressed inTable 15.3.2 in terms of ratios V11y'\6. Also shown in Table 15.3.2 are av-crages of the measured ratios V1iy'V36 for west winds (measurement sets athrough h) and for east winds [measurement sets j and k]. These averages weremultiplied by the factor (36/31)0 28 - 1.04 to yield approximate ratios V()IVH,whcrc V,,is the mean speed at elevation H : 3l m.

It is notccl that the measured ratios V1i1lV36 vary in certain cases considerablyl'nrrrr rncasurcmcnt to measurement (e.g., V6y'V36: 1.33 and 0.56 for mea-surcnrcltt scts c and f, respectively). No explanation is offered for these vari-irlirrns. ljor purposcs ol'comparison, Table 15.3.2 also includes predicted ratiosVAlVy, V4lVrr, anrl V1.lVs based on Figs. 15.3.5,15.3.7, and 15.3.11, re-spoctivcly. Thc agrccment with the average measured values is seen to be fairlygrxrd.

Case 2. Model of a Building in Utrecht, Netherlands [15-11]- A proposed80-m tall building with width W : 50 m, depth D : 22 m, and for whichHlh : 8.0, WIH : 0.63, and LIH : 0.5, is shown in plan in Fig. 15.3.14-Contours of ratios VlVs, shown in Fig. 15.3.14 for south and lor north winds'were obtaincd in [-5-lll using wind tunncl data rcporlocl in ll5-221. Moasurcdrali<ts VnlV11 and VlVllurc about 0.65 (at lhc ccnlor li111r ol'llrc lrttiltlirrg) itntl

o0

q)

0)

aq?

m

t\rnrn

rcFJ

Fr

q

FC)

>

ctrOi.9\- O

q

\q

a)oo

a)

t^Sl-v'-: I.-l ^

I

€!o

\- ()H

\':\

c)bodro

bo

C)

o

.o

cd

o(!?J

@

a- 6l lr) c.)O\ OOOO

o\ -ot\oO\ O\co- ca

r) <'O\ Oq \\q n999U4

cr,5.{!\oEo\ Srot^;6:ii

UF6l Cq C.t \o\ \\q

\lr)c.i

9

nF-

9oo

vl

\co

trJl3 -.t.Fq 9 .-r !9Ern E O\ C Cr ',tr;O:L--=/Ec o-cUF

r)F-\O -O\- r)cr.F-$Nca .a.aO ooOOU9UUU-

ca\olr)ncln\o6lonnnOOOc.l c.l c.)

@ O\ c.)nac'l

OOt"- V-rco@ r]|.)F-\nc..! cj n09 qq\9UUU_UC-r)\O \OOr€ C\F-oOco*- :v1h r)\o0o,- ^'-: .- .- -J :ueu uJlJ aOO*F-O NciO r)\Ocn\oicq (\cI\oo oooo--i,^^' ,^.^^' -:;99V999!'9-*\n\O cir)ca O\-ra)v?clq clqq \qq999UUaa-i\Oca\O n-$ OO+oon1- -:oqv? -q\qUUU aaO OOO

o rrrca co

nn\OOclnOO

\oooc! coo6. Qca!n-cl -:01 'q tr:\999999U)a-

Ni\O cq:a3, CO\a.id a.l c..l c.l O t'- O cq Ouuu (-::-= -.O-*N- ^t\OO. .JC€n':- -:!n nn\99999999U

tn-* o\cai *\o-qaa -nn 1199VV99!rJUU

O\ cnioo

Ocvo

u96^Q9.;avtsl.rOt'J='- rl r.r =l- 'ar ll:l

53t

532 wtNt ) tNl )1,( :l l) l)l:;(;( )Ml ( )l ll lll

v/vH = o.15

n I t( )t tNt ) lil,ll l)lN( i:;

'NtI

l', r ,'ot It , ot ilii.i! ',t,ltl n(.1 Wtt'lt): , Wt ililIl /\ iil,[ I til\/!ltol.JMt Nt 533

O.(X). tcspt't'ltvt'ly I'rt'tltr'lt'rl nrlios l',l l'1, ;rntl 1,,/1,, lr;r:;t'tl orr Iiigs. 15.-1.-5irtl(l l-5..1.'/ lue :rl)()u( O.(rO lrrrrl l-(X). n'sllr't lrvt'ly. Ilrc :rgrccrrrcnt l)clwccuprctliclctl iut(l ln(:asur'(:tl vrrlrrcs is sr't'rr lo lrt' rt'ltsorurlrly grxxl. ll is no(cd,Itowcvcr, thal thc vorlcx llow is irsynun('lrt;rl rrrrrl t'orrlirins rcgions in whichllrc rati<rs VlV11 arc as high its 0.1{

Case 3. Models of Place Desjardins, Montreal 115-231. Figure 15.3.15slrows a model (l/400 scalc) ol'onc arnong several designs considered for atlcvelopment in Place Dcs.jardins, Montreal. The predominant wind direction,tlctcmined from measurements at the top of a tall building near the site, isslxrwn in Fig. 15.3.16. Wind tunnel tests were conducted for that directiononly. Surface flow patterns were observed by using thread tufts taped to the

l,'l(,llllll,l l-5.-l.l-5. l)llrtt'l)r':;;;ttrlttt:, tnotlt'l ((()url(:iV ol lltt'N:rliott:rl At'nlurrrticlrll:rlrlisluttr'ttl. Nlrlioturl ltt'sr';rrt lr ('orrrrr rl ol (':rrlrrl;r)

Wind direction

)..

\50403020l0

Scale in meters

-

0.8 \ \/o.a5\\----==-!--====-a- \ \_===-=_= \ \ .r_)

Wind direction

FIGURE 15.3.14. Plan vicw. clst: 2lis

?.4.1I L t",1,

(Llil,)Qz) ,rta t"r,'t

4.i5

534 wtNt)tNt)t,ct t) t)t:ic()MI ()t il tN nNIr nt t()t,Nt) tt(,ilt)tN{i:

1.6646.gyo

{1 .1 5)(46.57o)

FIGURE 15.3.16. Wind speeds and turbulence intensities, place Desjardins [15-23](courtcsy of the National Aeronautical Establishment, National Research Council ofCanarla).

nrotlcl surllccs, a w(x)l tuft c;n thc cnd of a hand-held rod, and a liquid mixtureol'kcnrscrro-chalk (china clay) sprayed over the horizontal surfaces of the model.As thc wind blows over thc model, the mixture is swept away from high speedzones and accumulates in zones of stagnating flow. After the evaporation ofthe kerosene, the white acc:umulations of chalk indicate zones of low speedswhile areas that are dark represent zones where surface winds are high. windspeed measurements were made in these latter zones. The numbers givcn inFig. 15.3.16 represent ratios of mean wind speeds at the locations sh<lwn t<rthe mean speed at 1.8 m above ground at thc norlhwcs( c()n)cr ol'tlrc: tlcvcl-opment. The percentagcs of Fig. l-5.3.16 rcprcscnt tullrult'rrt't' irrlcnsilir:s, :yttlthe arrows show lltc: tlirccliorr ol'lhc wirrrl c()nrlx)n('nl llurl w:rs rrrt':rsrrn'tl by

r ill( ,l tl:uttl n(.1 wlNt): ;wililllt/\nUilI tl]vilt()l.lMl Nt 535

llre llnllrt','l'ltt't1rr:rrrlilit's llrirl lue rrol lrt'lwt.t'rr lr;rt.rrllrt.:;t.s t'or.r.csl-lottcl to lncltstlrclllLlllts Iltlttlc ilt tlte: itbscltct: ol rr plrjt't lr'rl 'rO slory l()w('r'nclu-lhc s<luthwcsl('()nlcr ol'thc tlcvclopluclrt. 'l'o irrvcslil',:tlr. tlrt't.llt.t.( ol tlrc tower upon lltcstrrlhcc winds, tncasulctllcnts w('rr lrlso rnlrtlt' willr (lrc rrrodcl of the towcr irrlrllrcc. Results of thcso nrcirsllr.t:rn('nls :rn' sltown bctwccn parentheses in liig.r-5.3. t6.

case 4. commerce court Plaza, Toronto lls-241. A l/400 scale morlc:l;rnd a plan view of t-he commerce court project in Toronto are shown in Figs.l-5.3.17 and 15.3.18, respectively. surface flow patterns obtained by smokcvisualization are shown fbr two wind directions in Figs. 15.3.19 and 15.3.2ol15-251 . Ratios VlVr, where V and V11 are mean wind speeds at2.7 m an<l240 m above ground, were obtained from measurements in the wind tunneland, after the completion of the structures, on the actual site. The results ofllrc measurements are shown in Fig. 15.3.21 as functions of wind direction fbrIocations 1 through 7 (see Fig. 15.3.18). The agreement between wind tunneland full-scale values is seen to be generally acceptable, although differencestrl'the order of 3O%,50%, and even more can be noted in certain cases.

Case 5. Model of the DMA Tower, Paris 115-261. Models of the l2}-mlall DMA tower and of adjacent projected structures are photographed in Fig.

Itl(l(lltl'l l-5.-1.17. ('otttttttttt ('rrttrl [\lrrkl ll'r '.1; (to11111'5y l]orrrul:uy l.:rycl-Wintllttttltt'l l.:tllotlrloly, 'l'lrt' lJrrrr,, r'.rlt ol \!r..,tr'rri ( tll.rttot

2.66

'.oo @5O.3o/" t- t

traL'3.11

41.4./.(s.38)

134.O%t

k-@

(3.82)(36.'t%l

>,.q

.ut5%

2.5631.O%(2.48133.1%l

T-(0

iii! toweriii s+ stories

536 wtNI) tNt)t ,(:t t) t)l;(;( )Mt ( )t lt tN n t,|l r /\ltoLNt ) iltil t)tN( i

FIGURE 15.3.18. Plan view, Commerce Court. After N. Isyumov and A. G. Dav-enport, "comparison of Full-scale and wind runnel wind speed Measurements inthe Commerce Court Plaza," J. Ind. Aerodyn., I (1975),201-212.

l5.3.22 against the background of the actual site. Let v" and v"H denote speedsdcfined as in Eq. 15.2.1 with k : I and measured at2 m and l2O m aboveground, respectively. Ratios v"lv"H obtained in wind tunnel tests for the south-wcsl wind direction are shown in Fig. 15.3.23. It is noted that for this direction(lrr: highcst winds occur between the two curved buildings located northwesttrl llrc lrrwcr (circled value VnlV"11 : 1.08 in Fig. 15.3.23) rather than in theinrrrrt'tliirtc vicinity of the tower itself. The increase of the wind speeds by thet lr;urrrclirrg ol'thc flow between buildings forming an angle in plan is sometimesrt'lt'rctl t() irs it Vcnluri el1-ect [15-16|.

15.3-4 lmprovement of Surface Wind Conditionsll rrt r'r:r'tlrirr klcutions suface winds are judged to be too high and thus to causerrrr;rt't'ePtrrblc rliscornfirrl to pedestrians, ways must be sought to imprclve cn-vinrttttrcttlitl wincl conditions or otherwise protect pedestrians from unplcasantwitrtl clll'cts. lrr cctlltin cxtrcmc cases it may bc ncccssltry lo tlcsign builclirrgsol lowcl' hciglrl or ol-tlill'crcnl configuratitlns than wrrrr.. origirurlly irrtcnrlctl. Il'possilrle, ()l)clt ltleirs sltottltl ltc so rlcsigrrctl :rs lo pn'vt'rrl 1x.rk'sll'iirrr lllrllic

l|):t .'()t.ll , {)t ilt(,il :il ,1il n(.1 wlNlt):; wt illlll n llUlt I lt{vilt()NMl NI 537

\lltr,tlt

|*:-:: Wind

-//I"IGURE 15.3.19. Surface wind flow patem, commerce courr (easr wind) [15-251

through high wind zones. Also, as suggested in [15-12], handrails should beprovided in potentially dangerous areas. In certain extreme cases it may benccessary to enclose windy areas frequently used for pedestrian traffic.

Local improvements of surface wind conditions can be achieved by provid-ing (l) roofs over pedestrian areas and/or (2) solid or porous screens at suitablelocations. studies of sheltering effects due to screens are reported in [15-271rrnd [15-28]. However, no general design rules exist to date on the basis ofwhich sheltering effects could be predicted reliably within a built environment.Also, as noted in tl5-121, solid screens merely deflect the wind from onekrcation to another so that the consequences of their use must be investigatedcarcfully.

A f'ew case studies illustrating rcmcdial measures aimed at reducing pedes-triitn level wind speccls ilr(: l)rui(l)tc:cl bclow.*

Case 1. Shopping Center, Croydon, Engtand 11S-ttl. Figurc t5.3.24 isIt vicw l'rot'tt thc wcsl ol :rtt ttllir'r'lruiltlirryi,7.5 rrr lirll,70 rrr witlc, irrrtl Ill rrrtlccp:ttlitlittittg tt slrrt;lpinl'. (('rt( r /'r rrr lorrli A plrss:rll('wity l2 r1 Irilglr:r1tl \. /

r'lltcstttttr't'olllrctrr:rlcri:rlirrrrrlr,,ri,rll,q 1,l,rrrr,i rrrrrrrlr.r.,trrt.;rrlrr.:rst.

wtNt)tNl)l ,(;l l) l)l:;(;()Ml ()l ll lN nNI) nl l()llNl) ll(lll l)lN(i:;

FIGURE 15.3.20. Suriace wind flow pattern, Commerce Court (southwest wind)t l 5-251.

m high connects the shopping center on the west side of the building to thestrect on the east side (Fig. 15.3.25). The shopping center was designed andhuilt without the curved roof over the shopping mall that can be seen in Fig.15.3.24. Alicr the completion of the building complex, it became apparent thatrcnrcrliul rncasurcs wcre necessary to reduce wind speeds in the passagewayirrrtl in llrc shopping mall. The ground level wind flow was investigated in thewirrtl tunncl, Iirst for thc complex as initially built (i.e., with the mall notcovcrcd) and thcn with various arrangements of roofs over the mall and ofscreens within the passageway. Ratios VlVo measrtred in the wind tunnel (lzand V, are the mean speeds at 1.8 m and15 m above ground, respectively)are shown in Fig. 15.3.25 in three cases. For the complex as first built, thehighest values of the ratio VlVswere 0.68 in the vortex flow zone and l.0l inthe through-flow zone. The provision of a full roof over the mall but ol noscreens within the passageway reduced considerably pedestrian level spcctlscaused by west winds. However, with east winds, thc lklw wlts lritppctl ttntlcrthe roof ancl the wind spccds within thc rlall wcrc. lir lltis rcrtsott, higlr; rrs

o5lLF-

occo

t-C=;;t.I

o,9c!ts6-

-t!a).o-Za

91 iruE:-*<tTs 9Rht 5 -:Fhtsr-^Ci-:.; JqJ>

^ t--q 9€ -6 E E€oL^Q tr0)IJ:6ru\)ao-1.s!.:-\

tri-a'9F(JvNa t=

@ >95! aY\J; d' oH H"H- rPE9 Ftr

'i: .) Xtr >-Yal\ -..-c

a{v trn) -rn*EF! oJY

,^!2="')OaaYtsO: F =s6lr>,<ooJ

a\

1

t\

o\\Olo

l'lb.o

oc.9ooJ

,a>\>1 .*r7

c.9

oJ

539

*540 wtNt)tNl)U(;t t) l)t:i(i()Ml (,1 il ll\1 nl']lr nl r()l lNl) ltl lll lrlN(,:;

I,'IGURE 15.3.22. DMA Tower (courtesy Centre Scientifique et Technique du BAti-mcnt, Etablissement de Nantes).

shown in Fig. 15.3.25, the speeds were also high at the east entrance of thepassageway. A solid roof close to the tall building followed by a partial roofover the rest of the mall, and a screen obstructing 75% of the passageway arearcsultcd in a significant reduction of surface winds, as shown in Fig. 15.3.25.It is notcd that to protect the mall from strong vortex flows caused by westwincls, thc solid roof had to extend for at least 18 m from the building face.

'l'hc solu(ion actually applied consisted of providing (l) a full roof over thecntirc rrurll (l;ig" l-5.3.24) and (2) screens with75% blockage in the passage-wiry.'l'his solulion proved elTective in ensuring a comfortable wind environ-IilL:nt.

Case 2. Models of Place Desiardins, Montreal115-231. It is seen in Fig.15.3. l6 that the ground level winds in the Place Desjardins mall (Fig. 15.3.15)are relatively high: with the 5O-story tower southeast of the development notinstalled, V$/V(o: 3.11 and Vosy'Vo, : 2.96: with the tower in place, V6,lVtrl : 3.38 and V11s:)lV(\ : 2.48. Wind tunnel measurements of pedestrianlevel wind speeds are also reported in [15-23] for the casc in which thc mallwas covered. With the -5O-story tower in placc, thc cll'cc( ol'r'ovt'rittg thc rrrallwas to rcducc thc rncan wincl spcccls by ir luctor ol'livc:tl ltx':tliott tl rrrrtl l.ry a

l', L'illll'. r)t iltriil ,t,ilt n(I WtNl)t; Wt ililN n tJlilt I tNVilt()NMl Nl 541

. 0.62

FIGURE f5.3.23. Surface wind speeds near the DMA Tower 115-261.

lirct<rr of about I .61 at rocation 10. However, with the tower not installed,while the mean speeds were reduced by a factor of almost three at location g,tlrc rcduction at location l0 was insignificant.

case 3. commerce court Plaza, Toronto lls-ls\. After the completion,l thc building corrrplc:x slrowrr in Fig. 15.3.18, conditions were found to bePrrrtictrlarly annoying ott wintly tlrrys lor pcdcstrians walking from the relativelyPlrrlcctcrl zonc n<lt'(h ol'tlrt' ll st()ly lowcr into thc flow funneled through thel)ilssllllcwlly 2.1. Wirttl lttttltt'l (t'sls irrrlit'rrtcrl llrlr( thc pnrvision of screens attllt'gtrrtllttl lcvt'l lts sltowtr rrr lir1l. l.5.l.l(xr woulrl n.sull :rl l<lcirtigrrs 2, -5, and

0.35o 0.34 o

o:r o:'o.44o 0.89 0.87 o 7j

0.96r o. -l0.74. ffi,..x o.zs

0.43 o 0.63.

D.M.A.Tower

.0.61

542 wlNl) lNl)l l( )l l) l)ll;( l( )4,41 ( )ll I lN n l.ll ) n I 11 )l ,Nl) llllll l)lN( ;:;

FIGURE 15.3.24. Tall building and shopping center, Croydon [5-11].

(r in rcdLrctions of undesirable mean speeds of the order of 40%. However,whilc cll'cctivc acrodynarnically this solution was rejected for architectural rea-sons. lr.rstcad, pottcd cvergreens about 3 m high were placed as shown in Fig.15.3.26b.'l'his rcduced the mean winds by about2O% at location 2, l0% atIocation 5, and 33% at location 6.

15.4 FREQUENCIES OF OCCURRENCE OF UNPLEASANT WINDSWITHIN A BUILT ENVIRONMENT

15.4.1 Detailed Estimation ProcedureLetVn(V,0) denotc thc wintl spccrls at l0 rn 11111;vs grtrrrrul irr o;x'tt lt'n:ritt tlutlinduce petlcslrilrn lcvt'l wintl spct'rls /rrl :r g.ivcrt lot';rtiort itt ;r Itttrll ('nvit()nnr('nl,

l',.1 llll rll ll llr ll or ililr 'r r /\:;nNl wtNt): i wt ililt.J n nl,il | I Nvlli()NMt Nt 543

As first built 053 057With f ull roof and no screen o 4s o 24With partial roof and '15y" screen 023 o1j

As f irst built 026With full roof and no screen 048With partial roof and 75o/o screen 021

065 068 065 049 036 012019 020 025 021 02a 032019 028 023 023 019 040

007 017 044 052 056 078 10'l045 052 061 067 063 071 088o17 011 023 043 047 053 059

FIGURE f5.3.25. Model test results, Croydon [15-l ll.

and let the angle 0 define the direction of the velocity vector with speed Zu.'l'he frequency of occurrence at the location concemed of wind speeds largerlhan V, denoted by f', can be written approximately as

f':n2t?

in which fv,o arc the frequencies of occurrence in openspccds larger than Vo(V, 0i) and the directions 0; - rlnrrngle d; being defined as

2ri0i

II(i 1.2....,n)

(ls.4.l)

terrain of winds with<0<0,+r/n,the

(ts.4.2)

Irt ltractical applicatiorts lr l(r lxrirrl ('()nrl)irss ts t'onunonly rrsctl so that in Eqs.I5.4.1 irnrl 15"4.2. tt l(t

'l'rr rrltlitirt.lI it is lr('('('ss;ry, Iirsl, lo t':,lrrrr:rlt'llrt'v;rlrrt's rl l/,,(l/, //,). Iinrrrrwilttl t'littlt(okrgit'lrl rllrt;r. i( is llrt'tr l)osstlrlr'lo t':,lrrrr;rlt'llrt'lrt't;ut'rrt'i.'s /li'.

N

544 wtND tNI)tJCt t) t)t$(:()Mt ()ilt lN ANtr Alt(lt tNlt null l)tN(ili:;tr\rt

o

a ooa

(b)

l''l(;uRIt 15.3.26. Rcrnccriar nlc:lsures at Commerce Court: (a) screens; (b) trees. AftcrN. Isyumov and A. G. Davenport, "The Ground Level wind Environment in Buirt,fp.|1eas," in Proceedings of the Fourth International conference on wna E;ffects .ttBuildings and structures, London, 1975, cambridge Univ. press, camlridge, r97(r,pp.403-422.

Trees---> -"o

It, I lill {lilFll{'l|!r fil ltl ll,t tA!i^til Wlf.ll[; Wiililti A tilllt I tNVilt()NMt:Nt

f lrc r1x'ctl J't(]', ll, ) t;rrr lrt. wnllcn :ts

545

)'tlv,0i) I Vo\o,) ..vtvtt(q') w0)' ( 15.4.3)

(ts.4.4)

llrt' r;rlios Vtl|i)lvil(?i) characterize the site from a micrometeorological stand-P,rrr(. llrr standard nrughness conditions in open terrain, these ratios dependrrurrr (lrr: clcvation Hand upon the roughness conditions upwind of the site, as',lrrrwrr irr Sccts. 2.2 and 3.1. The ratios VIV{0;) at a given location are an;rr'rrxlyrrarnic property of the wind environment and are estimated on the basisol wirrrl tunncl tests, as seen in Sect. 15.3 (e.g., Fig. 15.3.21).

A rrsclirl basis for the estimation of frequencies flo is provided by weather"tirliorr rccords of wind speeds and directions, observed at three-hour intervalsirrrrl puhlished in monthly Local Climatological Data sheets (see Sect. 3.1).( '()rrrii(lcr, fbr example, all the three-hour interval observations in a year (8 obs/rlry X 365 days : 292O obs), and assume that 58 out of these observationsrr'plcscnt NNW winds with speeds in excess of 6 m/s. The frequency of oc-r'un('nco of such winds can then be estimated as follows:*

f?: #= 2%

It is desirable, in practice, to base frequency estimates on several years ofrlrtir. 'fhis is the case fortwo reasons. First, one yearof data might not reflecttlrt'wind climate in a representative way. Second, the observations taken attlrn'e:-hour intervals are instantaneous values, which are sometimes lower,:;rrilctimes higher than the mean speeds. The estimation error associated with:,rrt'h differences is small if the sample size is large.

ln certain applications it may be of interest to estimate frequencies for in-rlrvitlual seasons, or for a grouping of seasons (e.g., spring, summer, and fall).lrr such cases the only data used to estimate wind frequencies are those thatt rvcr the season (or seasons) of interest. It is also noted that winds occurring,s:r,y, fiom 1l p.m. to 5 a.m. are, in many cases, of little concern from ther.lrrrrclpoint of pedestrian comfort. In estimating wind frequencies, midnight andI rr.rn. observations can then be eliminated from the data set.

lrrfbrmation on frequencies of wind speeds at a weather station fvio may beprrscnted either diagrammatically or in the form illustrated by Table 15.4.1.

An example is now prcscntcd of the calculation of frequencies /2. Thetrrlculations are carried out lirr krcation 4 of Fig. 15.3.18 for which the plotl/1V,, is given in Fig. 1.5.3.21. lt is irssurncd that the ratio VolV, = 1.5 andtlurl thc wind climatc is tlcrsclibcrl by 'l'rblt: 1.5.4. l. The frequency.fv is soughl

r'l'he sttpcrscripl in lht'nolitliott /l'tr'prcrrrrl: tlrt's1x'r'rl l',, (r rtr/s, whilc llrc srrbst.r'ipl cortr.s;xrtttlslttlltcvitlttci Iiltit l(r;rrtttl (orrll;riirrrrvlrrrlrllrt irrrp,lr'//isrrrt'trsrrrt'tl totrttlt'rtlotlwistsl:rr'(irrg lirrnr thc NNW rlircclrorr (rn' l,r1 ll 'l I I

o<g

xH9z

E]2zz+ lr.1az

r!2zIJ.]

6.1 .,r

troq,)o0

q)

dliliqJ30)

a

qJ

CB

aoq)

a

q()

o)

Ltri

tf,lai

f-lrla1E

tIl-, u)gl

9H/aO

r!O\ 6Da

@u)

naa

olB

z

\oF(h

VI U)

.+>

/zzooc)

o

546

r)orono\o c.l

t-- a.l $ r

c.;-id I

Ott-61 ,

cq-o I

-0OO.l ,

co-Ol

-Oc.l 'c.: ol o I

\OOc7) ,

(.{*O I

.??n ro.t-Ol

..lci$ r

c.{io I

?.lc'l rc.{io I

0909.1 r

-ivln rcoio I

q\vl-:\Oc.lOO

C!F-61 ,

*-Ol

nnq-F-c..lOO

nnn-\O.TOO

':nnat ^lOO

09 09 cl -:CiiOO

6q a a'>\\\!tr tr troco\ooo-nnnn\a \o \o \o

a

l.n

ll

!

crt

dQt\tra

rc

F

s.l2zz+ tr.l-z

2z

IJJ2zr!cn '.r

n-a r!ot4ra

ElO\ (,(n

6A

*erA

-F(/)

F.h u)

sFB

c^, I

NTz

z

.EooxH

d; o

cdo

\oO

odd

oi oioq-:coO

c.; o

+oNrr

c.;o

n'1\OO

9-:caO

c.i o

naa.t o

BSc\lV

t-\rci*Ltr* lo-ri. lt* l-( -^.5l\\ \u: ' I

la i \ \

+

9eru*

t::

F.i t

lr),1 llll ljl 'l

I'll :ll !'; lrl llNl'l IAUANT WlNl)S Wl tlllN A llt,ltl t NVlll(lNMl Nl 547

lirr' ;rcrkrstriiut lcvcl wirrtls with spccds V > 5 m/sec. Equation 15.4.3 can thenbc wli(lcrn ls

7.sVo6.0i) : WVA;)

'f'hc calculations are given in Table 15.4.2.

(ls.4.s)

(15.4.6a)

(1s.4.6b)

15.4.2 Simplified Estimation ProcedureA simplified version of the procedure just presented is suggested in [15-11] forbuilt environments similar in configuration to the basic reference case (Fig.15.3.3) dealt with in Sect. 15.3. In this version the aerodynamic informationused, rather than being a function of wind direction (as, e'g., in Fig. 15.3.21),is limited to the results given in Figs. 15.3.5, 15.3.7, and 15.3. 1 1. The ratiosVnlV1lof mean wind at elevation F/in the built environment to mean wind atl0 m above ground in open terrain may be taken from Table 15.3.1. As far as

the climatological information is concemed, the data needed are the frequenciesof occurrence of all winds with speeds in excess of various values Zs, regardlessof direction (in the example of Table 15.4.1, these data are given in the lastcolumn). It is noted in [15-11] that this simplified procedure, even though not"exact," provides generally reliable indications on the serviceability of pe-destrian areas in a built environment of the type represented in Fig. 15.3.3. Itis emphasized, however, that the procedure can only be regarded as useful ifapplied in conjunction with the comfort criteria proposed in [15-11] (see Sect.t5.2.2).

To illustrate the procedure proposed in [15-l l], consider the case of a build-ing complex for which H : 70 rrr, W : 50 m, ,L : 35 m, and h : 10 m.*From Figs. 15.3.5 and 15.3'7, VAIVH = 0'6 and VBIVH = o'95, whete V1

^nd VB are the highest mean speeds in the vortex and in the corner flow,rcspectively. For 11 : 70 m, VHIVyOO) = l.O4 (Table 15'3'1), so

Yt = o.ozvo

h=t.mvo

'l'hc frequencies of wincls /2 ) 5 m/s and VB ) 5 m/s are now sought,assuming that the wincl clittlttc is clcscribed by Table 15.4.1. It follows fromliq. 15.4.6a that, in onlcr lltitl V1 > 5 ,nls, Vo > 510.63 = 8 m/s. From'l'ahlc 15.4.1, the I'rcqtrrrttt'y ol srrclr witttls is 5o/o. However, to speeds Z6 )

tlirr lhcsc notalions, see lrig, l5 t I

548 wtNt ) tNl )ll(;l l) l)li;(l()Ml ()l ll lN nl ll ) Alt()tll

51r/s lltcrc corrcsl.rotttls sllculs (, - 5/l 5 rrr/s, wlticlt ltt't: sct:tl irt'l':rlrlt'15.4.1 trt ()ccttr itb()ut 30%, o|(hc tittlc.

'l'lrr: cornlirrl critcrion proposccl in [15-lll and prcscntod in Scc{. 15.2'2s(:rtt:s tllrt lrcas in which wind speeds in excess of 5 m/scc occur rngrc lhan)ll'/t, ol'rhc tirnc are generally unsatisfactory from a pedestrian comlirrt pointol' vicw. 'l'hcrclirrc, according to this criterion, the wind conditions of thclirrcg<ling cxittnplc arc unacceptable.

REFERENCES

l\ I li. K. ('hang, "Human Response to Motions in Talt Buildings," .1. Struct./)ir',, ASCIE, 98, No. 5T6 (June 19733), 1259-12'12-

l\ I l'. W, ('hcn and L. E. Robertson, "Human Perception Thresholds of HorizontalMrrriort." J. Struct. Div., ASCE, 97, No. ST8 (Aug' 1972),1681-1695'

l.\ I M. Yurnada and T. Goto, Criteria for Motions in Tall Buildings, College oflirrginccring, Hosei University , Koganei, Tokyo, Japan, 1975'

l5 4 ',l'. (ioto, "Human Perception and Tolerance of Motion," Monograph of Coun'cil on Tall Builctings and Urban Habitat, Vol' PC (1981)' 817-849'

l5 -5 lr. R. Khan and R. A. Parmelee, "Service Criteria for Tall Buildings for WindLoading, in Proceedings ofthe Third International conference on wind Effectson Buiidings ancJ Structures, Tokyo, 1971, Saikon' Tokyo, 1972' pp' 401-401.

l5-6 R. J. Hansen, J. W. Reed, and E. H. Vanmarcke, "Human Response to wind-Induced Motion," J. Sffuct. Div., ASCE, 98, No' ST7 (July 1973), 1589-1605.

15 1 J. W. Reed, WinrJ-lnduced Motion and Human Discomfort in Tall Buildings'Research Report No. R7l-42, Department of civil Engineering, MIT, Cam-bridge, 1971.

l5-ti T. Goto, "studies of wind-Induced Motion of Tall Buildings Based on oc-cupants Reaction," J. Wind Eng. Ind. Aerodyn', 13 (1983)' 241-252'

l5 g L. Ircld, ..superstructure for 1350 ft. world Trade center," Civ. Eng., ASCE,41, (r (Junc l97l),66-70.

15 l0 A. I). l)crrwunlcn, "Acccptable wind Speeds in Towns," Build. sci.,8,3(Scl)l . l()7.1), 259-261 .

15 ll A l). l)crrwrrnlcn rrntl A. F. E. Wise, Wind Environment around Buildings'lirrikling l{cscarclr Establishmcnt Report, Department of the Environment.llrriltling ltcscarch tjstablishment, Her Majesty's Stationery Olfice, London,t915.

15-12 T. V. Lawson and A. D. Penwarden, "The Effects of wind on People in theVicinity of Buildings," in Proceedings of the Fourth International ConJerenceon winrJ Effects on Buildings and Structures, London, 1975, Cambridge Univ.Press, Cambridge, 1976, PP. 605-622.

l5-13 E. C. Poulton, J. c. R. Hunt, J. C. Mumford, and J. Poulton, "Thc McchanicalDisturbance Pro<Iuced by Steady and Gusty Winds ol'Mtxlc:rrttc Strcrrgth: SkillctlperiormanceandScmanticAsscsstncnts," I'.rgrtttttrtit.t,ltt,6(l()75),65 I 673

|, 54q

l5 l,l .l . ('. lt. llrrrrt. li (' llrultorr.;rrrrl .l (' Mtttttlorrl, 'llr, l'.llr'rl" nl Wtrttl ottItr.olllt'. Nt.w ('rrlt'rur lllrst'tl orr Wnttl l'ntrrr'l l't1x'tttttr'ttlr," llrtrl,l Ittttt,'ttI I, t l()/{r;. I Jl{

l-5-15 N. lsytrtrrov:lrrtl A (i l):tvt'ttlxrrl. " llrc (itottltrl It'vll wttrrl l''ltt'tlrtlltttr'ttl ttiIltrilt.trp At-clts," it'r I'tt'tt'ttlirt,qs r'l lltt l\trttllt ltttt'tttrtltttttrtl < t'ttlt'tI ttt t ttttWind Iillcct,s ott Ihtiltlirr.q,t tltl ,\trrtttrttr'.r, LottrLrtt. l()ll. ('lttttlrtttl;',t'llrtrvPrcss, Cltttbritlgt:, l()7(r, pp. '10.| "12J.

l5-16 J. Candctlcr, "Wind linvir'onrrrerrt Anrurttl lltriklings: Atrtrxlyttitlttit'('ottcepts," in Pnx'culirtl4s ttl'tltc ["ourlh Inttnrutitnul (1nl|rrttct' tnt Witul l',llt't t,ton Buildings und Slru(:ture.r, London, 1975, Cambridge Univ. Prcss, Catn-bridge, pp.423-432.

15-17 S. Murakami and K. Deguchi, "New Criteria for Wind Effects on Pedestrians,"J. Wind Eng. Ind. Aerodyn., 7 (1981), 289-309.

15-18 L. W. Apperley and B. J. Vickery, "The Prediction and Evaluation of theGround Level Wind Environment," in Proceedings of the Fiilh AustralasianConference on Hydraulics and Fluid Mechanics, University of Canterbury,Christchurch, New Zcaland, 1974.

l5-19 W. H. Melbourne and P. N. Joubert, "Problems of Wind Flow at the Base ofTall Buildings ," in Proceedings of the Third Intemational Conference on WindEffects on Building and Structures, Tokyo, 1971, Saikon, Tokyo, 1972' pp.105-l 14.

15-20 E. Arens and D. Ballanti, "Outdoor Comfort of Pedestrians in Cities," inProceedings of the Conference on the Urban Physical Environmenl, 1975, U.S'Forest Service, American Meteorological Society, and Syracuse University,Syracuse, NY 1975.

15-21 M. O'Hare, "Designing with Wind Tunnels," Arch. Forum (April 1968),60-64.

l5-22 R. Poestkoke, Windtunnelmetingen aan een model van het Transitorium II vande Rijksuniversiteit, Lltrecht, Report No. TR72l10L, National Aerospace Lab-oratory NLR, The Netherlands, 1972.

15-23 N. M. Standen, A Wind Tunnel Study of Wind Condition.t on Scale Models ofPlace Desjardins, Montreal, Laboratory Technical Report No. LTR-LA-101,National Research Council of Canada, National Aeronautical Establishment,Ottawa, 1972.

15-24 N. Isyumov and A. G. Davenport, "Comparison of Full-Scale and Wind Tun-nel Wind Speed Measurements in the Commerce Court Plaza," J. Ind. Aero-dyn., 1,2 (Oct. 1975),201-212.

15-25 A. G. Davenport, C. F. P. Bowen, and N. Isyumov, A Study of Wind Effectson the Commcrct ()turt Pntict:t, Pan II, Wind Environment at PedestrianLevel, Enginecring Scient'c llcscarch Rcport No. BLWT-3-70, University ofWcstern On{ario, lrrtt'trlly ol lirrginccring Sc:icncc, London, Canada, 1970.

15 26 J. Ganilcrncr, li)trt,l,' ,!, ltr tt,ut l) AI 1., I'rtrtit 2, I)tttt'rrttitttttiotr rlrt <'lrttrttlt tlt't,ilt'.t.st'tttt tuti,sitttt.tlt',!tt,',trt1,l,'t,'l,,tti ,!,'ltt l.,ttt-l).M.A., liN n l)YM 75'l('.('cntor Scicntilirltrt' t't 'li't ltrttrlttr' rltt ll:tlttttt'ttl, N:tttlt's, lilirlrcc, l()75.

l5 27 M. ()'lllrrt. untl lt l, krrrrr;rrri r. "l'( r(,' l)t sillrt:r lo l(t't'I Witul llrtttr llt'ttt;' ;t

Nttislttlt't'." ,4t, ltit Ii,', ( lrrlr l(f{r'}} l i l l ilr

l

550 wrNurNr)rJCr.r) DtscoMfont tN ANI) Anot,Nt) RtJilDtNGrl

15-28 V. K. Shrirrin, "Wind Comlirrt and Wind Shcltcr," in I'nx'ctdings r2l'thaSymposium on External Fktws, University o1 Bristol, 1972.

15-29 A. D. Penwarden, P. F. Grigg, and R. Rayment, "Measurements ol' WindDrag on People Standing in a Wind Tunnel," Build. Environ., 13 (1978),75-84.

15-30 W. J. Beranek, "Wind Environment around Single Buildings of RectangularShape, and Wind Environment around Building Configurations," Heron, 29(1984), 1-70.

l5-31 F. H. Durgin and A. W. Chock, "Pedestrian Level Winds: A Brief Review,"J. Struct. Div., ASCE, f08 (1982), 175l-1767.

15-32 A. Tallin and B. Ellingwood, "serviceability Limit States: Wind Induced Vi-brations," J. Strucr. Eng., ll0, (Oct. 1984), 2424-2437.

CHAPTER 16

TORNADO EFFECTS

'fornadoes are storns containing the most powerful of all winds (see Sect. 1.3).However, their probabilities of occunence at any one location are low com-pared to those of other extreme winds (see Sect. 3.5). It has therefore beengenerally considered that the cost of designing structures to withstand tornadocffects is significantly higher than the expected loss associated with the risk ofa tornado strike (the expected loss being defined as the product of the magnitudeol the loss by its probability of occurrence). For this reason tornado-resistantdesign requirements are not included in current building codes or standards,Ior example, the Uniform Building Code [6-1], the Southern Building Code116-21, or the ASCE 7-95 Standard [17-l].

However, in designing facilities for which the consequences of failure wouldbc exceptionally grave, the effects of a tornado strike must be explicitly takeninto account. Such facilities include nuclear power plants, for which it is re-quired that "structures, systems and components important to safety . . . betlcsigned to withstand the effects of natural phenomena such as . . . tornadoes. . . without loss of capability to perform their safety functions" [16-3]. In thetJnited States, construction permits or operating licenses for nuclear powerplants are issued or continued only if this requirement is satisfied in a mannerconsistent with Regulatory Guides* issued by the U.S. Nuclear RegulatoryCbmmission (e.g., [6-31 and I l6-41) or otherwise acceptable to the Regulatorystafl'of that agency. lt is (hc purposc ol'this chapter to describe studies undcr-lakcn, as well as dcsigrr clilcliir irrrtl llrlcctlurcs developed, with a vicw tocnsuring an adcqualc rcsislirttee rtl'ttttclcirr l)()wcr plants t<l tornackr cllbcts.

+'l'hc licgulatirry (ittitlcs rtte n'vir'wrrl l*'rirxlilirllv, irr nt'crlt'tl. l() itr'('onrrKXlitlc corrrrttt'rtls rrttrllo t'trllcc( nrrw ittlitnttitlirrr or r'rpi:ri.:ttr r. I ifr ,ll

551

il552 t()nNnt)() Il tot:;

Tornado cllbcts uray bc diviclctl itt(o tltrcc gn)ups:

1. Wind pressures, caused by the direct action upon the structurc ol'thc airflow.

2. Pressures associated with the variation of the atmospheric pressure fieldas the tomado moves over the structure (atmospheric pressure changceffects).

3. Impactive forces caused by tornado-borne missiles.

To estimate these effects, it is necessary to assume a model of the tornadowind flow. A model currently accepted fbr use in engineering calculationsconsists of a vortex characterized by the following parameters: (l) maximumrotational wind spccd V^r,* (2) translational speed of the tornado voftex 2,.,(3) radius <lf'maximum rotational wind speed R^, (4) pressure drop po, and(5) rate ol' prcssurc drop dp"ldt. (Values of these parameters proposed for thedesign of nuclcar power plants in the United States are listed in Sect. 3.5.).The tomado voftex flow model must then be complemented by assumptions onthe detailed features of the wind flow. Such features are discussed as neededin the subsequent sections herein. For a survey of recent developments inengineering practice related to tornado effects, see [6-5].

16.1 WIND PRESSURES

A procedure for calculating wind pressures is now described; it is taken from[16-6], and assumes the following:

1. The wind velocities and, therefore, the wind pressures, do not vary withheight above ground.

2. The tangential wind velocity component is given by the expressions

rV, : ^ V^ (0<r=R.)

K_

wltt'l'c K rs it (()ns(iurl ol lttrrlxlltotItlrly Ilrl; ,'rprr':,:.rorr, wlrrr lr l. rrolrigonrttsly ('()r l('('l, is r'ottvt'ttir'rtl in t ;rlr'ttllrl torr,

'l'hc wintl prosstlrc yr,,, ttsctl irt tlt'sip,rrirrg, s(tttt ttttt's ()r l)iuls :rttl lxrrlrorrstlrcrcol'rnay bc writtcn irs

lr, t1,(',, I t1y(',,i 1l(r.1.211

whcrc Q, is thc cxtcrnal prcssurc crrcllicicnt, C,i is thc internal pressure coef:licient, qp is the basic extcrnal pressure,* q, is the basic internal pressure.Values for the pressure coefficients C, and Cpi zre given, for example, inll6-31. The quantities qp and eu may be calculated as follows:

Qr: CIP^^

Qv : CYP^ *

(16.1.s)

(16.1.6)

whcre V,, is the maximum tangential wind velocity and Ru is the radiusof maximum rotational wind speed.

3. The total horizontal wind speed is

V:KV, ( r6. l .3)

+The rotational wind spcccl is dcfinccl as thc rcsultanl ol'lhc l:rngcrrtiirl ;rrrtl r:uli:rl wirrtl vt'kxilycomponcnts ll6 l3l.

where

P^o : *PV'-^ (16.1.7)

ln Eq. 11.16.7, p is the airdensity and Z-u^ is the maximum horizontal windspeed (see Sect. 3.5.1). If Z-o* is expressed in mph andp-"" inlblft?, |p :0.00256 lblftzl(mph)z. The quantities Cf and C! are reduction (or size) coef-licients that account for the nonuniformity in space of the tornado wind field.'l'he size coefficient Clmay be determined from Fig. 16.l.l as a function oflhc ratio LlR., where L is the horizontal dimension, perpendicular to the windtlirection, of the tributary area of the structural element concerned (if the windIoad is distributed among several structural elements, e.g., by a horizontalrliaphragm, L is the horizontal dimension, perpendicular to the wind direction,ol'the total area tributary to those elements). The size coefficient Cf may berlctermined as follows. If the size and distribution of the openings are relativelytrnifbrm around the periphery of the structure, C! is determined in the sameway as Cf using a value of t equal to the horizontal dimension of the structurepcrpendicular to the wind direction. If the sizes and distribution of the openingsrrrc not uniform, the following weighted averaging procedure is used:

l. Determine quantity 11lR,,, such that

t't R,,,

1i,,, t't I L(r6. r.r3)

rllt't:tttse tto tlislirtt'liorr is ttt;trlr' rtt lltr'. lt,r, , ,lttrr' lrlu'r'r'rt lr;tsir' Ptt'sstltt's tlscrl rtt lltr' rlt stl'.tr olslnt('1ilt('s, rttt (ltt'0nt'lIttttl. ;ilt,1 "l l',ilt, ;ilr,1 l','llr,'r', r'il llr( ollr('t lr;il11, llrr'n()lllll{)il rl, u\('(l rrI l{r }l lor l)r('ssll('s ()n p:ul!;tttrl lxrtltltt, r', ri}l t rrl'lr'\'r'rl ltr'tr'tl

V, : R'n

r V,n (R-<r(o)

(16.1.1)

(16.1.2)

554 rotlN^rx) tlIloll-;

1.0

0.9

Fcs

o.7

0.6

0.5

0.4 1.0 1.2 1.4 1 .6 1 .8 2.O

FIGUR.E 16.1.1. Size coefficient Cf 116-61.

2. Locate plan of structure drawn at appropriate scale within the nondimen-sionalized pressure profile of Fig. 16.1.2, with the left end of the struc-ture at the coordinate rtlRm.

3. Determine factor Cn from Fig. 16.1.2 for each exposed opening.4. Determine Cf; from Eq. 16.1.9

Lft-

-, _ Dl Asicq,Ls - -FX rLl no,(16.1.9)

1.0

0.9

0.8

o.7

0.6

0.5

0.3

o.2

C,I

rR-

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.O 2.2 2.4 2.6 2.8 3.O

0.1

FIGURE 16.1.2. Cocfticicnt C,/ ll() 61. li'l(illltl,l 16.1..1. $r'lrr.rrrrrtit vit.w ol lrrriltlirrpq

ilr r wlNt) t,llt lil;(,nl l; 555

whcrc,rl,, is thc arca ol'opcning itl locirliorr i. (i,, is thcr lirctor (',r ull<tcatiorr ri, ancl N is tho nutttbor 9l'gpcrri1gs. ('l'lrr: cgcllir:ic:rrt (1, in liig.16. 1.2 rcprcscnts nonditncnsionalizcrl winrl prcssures lnrl was calculatcclusing Eqs. l6.l.l, 16.1.2, 16.1.3, anrl l(r.1.7.'lir obtain trig. t6.l.l,the nondimensionalized prcssurcs ol' tJig. 16.l .2 wcrc intcgratcd bctwccnthe limits 11 and 11 * L, whcrc 11 is givcn by Eq. 16. 1.8, ancl the rcsultsof the integration wcrc thcn nonnalizcd; the coefficient Cl is thus anapproximate measure of the average pressure over the interval L t16-61).

Numerical Example The building of Fig. 16.1.3 is assumed ro be in regionI. The sizes and distribution of the openings (not represented in Fig. 16.1.3)are assumed to be uniform around the periphery of the structure. The ratiobetween area of openings and total wall area is AolA*: O.25.It is assumedV^u :360 mph (161 m/s), R. : 150 ft (46 m) (see Table 3.5.1). Thepressures on the 100-ft (30.5-m) side walls induced by the wind blowing inthe direction shown in Fig. 16.1.3 are calculated as follows:

pmu* :0.00256 x 3602

For basic external pressures,

(Eq. 16.1.7): 330 # (rr,roo {)

L:20Oft (61 m)

L 200 : 1.33R_ 150

Cf : 0.50 (Fig.

ae : 0.56 x 330 : (r*uo {) (Eq. 16 r.5)

r6.1.1)lb

185 -ft'

4l.'9(.'"/n{'

;556 t()trNntx) Il t(;tli

Iiol basic inlcrnal prcssLltts,

L : 200 ft (61 m)

L_ : 1.33R*

cf : o.so

4u : o.56 x 330 : 185 qft2

I ior' Pr.t'ssrrrc cocfficients.

Co:

Lt'i -

lirlr wind pressure,

p*, : -0.7 x 185 - 0.3 x 185 :

(Eq. 16.1.a)

16.2 ATMOSPHERIC PRESSURE CHANGE LOADING

('onsitlcr thc cyclostrophic wind equation (Sect. 1.3) written as

dpo v?dr: P; (16.2.1)

wlrt'n'r/,rr,,/r/r'is tlrc ulrnosphcric pressure gradient at radius r from the centerrrl tlrr'torrrirrlo vorlcx. 'lir obtain thc pressure drop po,Eq. 16.2.1 is integratedlrorrr irrlinily lo r'. ll'thc cxprcssionfor V, given by Eqs. 16. l.l and 16.1.2 isrrst'tl I l(r (rl:

r8s g (*ruo I\ft' \ m'l

1,,,1n -,t;' (, #i) (o s r - R,,,)

l'lll :,',l ,l tl ( ll^tl( il I ( )n I !ll.1r i 551

lrt tlrc ctrsc ol slrrrclrrrcs with rto opt.rrings (tttryt'ttlt,rl \.!ntt.tut(,\ ), llrr' rrr{t.rrr;rll)rcssuro tttn:ritts ctlttirl lo (lrr: trlntos;rltt.r'it'pn.ssrrrt. lrt.lort'llrt- lrlr.;:;lr1't'ol llrt.tornado-'l'hcrclirrc during (lto p:tssirgt'tlte rlillt'rt'rrtt'lrt'lwccrr llrt'lrrtt.r'rr:rl prt.ssurc and the atnrosphcric prcssutc is ctlturl lo 2,,. ll lollows lrorrr llt1s. l(r.l Jand 16.2.3 that thc maxirtrurrr vtrlrrc ol'yr,,, wlriclr (x'(.ut.s ltt /. O, ls

1l',t,tt'^ : PVl, ( l (r.2..11)

lf the structures are completely open, the intemal and external pressures arecqualized, for practical purposes, instantaneously, so the loading due to at-mospheric pressure changes approaches zero. In structures with openings (ventedstructures), the intemal pressures change during the tornado passage by anamountp,(r). Denoting the external atmospheric pressure change by p,(t), theatmospheric differential pressure that acts on the extemal walls is p"(t) - piG).

A useful model for p"(t) can be obtained by assuming, in Eqs. 16.2.2 and16.2.3, r : Vot, where 2,, is the translation speed and r is the time. A simplermodel in which the variation of p.(t) with time is given by the graph of Fig.16.2.1 may also be used [16-6]. The time-varying internal pressures p,(t) maybe estimated by iteration as follows. Assume that the building consists of anumber n of compartments. The air mass in compartment N (where N < n) attime f +1 is denoted by Wy(\+1) and may be wrirten as

Wy(t1+t) : Wp(\) + [GN(i")(t) - G,v,.",,(4)J Ar (16.2.4)

where G1r,,", snd Grq,,",y denote the mass of air flowing into and out of compart-ment N per unit of time, respectively, and Ar is the time increment. The airmass flow rates G7y can be calculated as functions of the pressures outside andwithin the compartment N and of relevant geometrical parameters, including

(tt.o I) (Eq 16 r 61

-0.7 tl6-31

+0.3 ("r*< 0.3, see lro-:l)

Vi,, Ri,,- p ;-7

(t6.2.2)

(t6.2.3\

A,

p,,(r) (R,,,sr(oo) lll(;lJltl,l 16.2.1. ltleirlizt'rl :tlttto:ltltltt, lrr".',uri'r lr;rrr1',' v{'r:,n:, lrn(' lurrt'liorr Il{r {rl

558 TORNADO EFFECTS

opening sizes, as shown subsequently. The internal pressure in compartmentN at time ti+r,Piu(ti+1), is then written as

(16.2.s)

where k : 1.4 is the ratio of specific heat of air at constant pressure to specificheat of air at constant volume.

A computer program for calculating loading on vented structures due toatmospheric pressure changes is briefly described in [16-6]. The program in-corporates the following type of model [16-7] for the air mass flow rate:

G : 0.6C,Azf2l,,(pr - p)lt'' (t6.2.6)

p i.(tj +, : lry#lo o,r r,,t

ft [1k - tl- It | - (A2tAt)2 -lJ"t

where. / \2lk*: [(,1)

- (p2lpr1{t'- rtr*

- Pzlqr | - (AzlA)2(prlpr)''o

(16.2.7)

FIGURS 16.2.2. Illustration of pressure distribution and flow pattern during buildingdepressurization [6-6].

Air flowpattern

N

(?)(oll

\o\o

0.)

o

oN

a)Htrod)

e

t:or

raal\o

r{/tf[i

N

oO)(til

N

@ro(')il

NOEE(f, @o)NOOdc;I lt

NO

N

Nlt

sf(fJ

ll

N

so)Nil

N

(o(oll+

zIIZZ=V

o

oo6N.ci$Err(J>

N

NstI

oooo;toFrO

Elo.(J>

oo@

sf LOo)ciNErro.o>

ooNrodNc! c')Erro.(J>

N

(o(oil

oo@..roErrE>

6t9

r560 r()t tNnt x) I ttt(;t:;

ancl zl 1 is thc arca (rln tho sitlc ol'c()lnl)iu-lrncnl l) ol'tlrc wull bctwctrtt cottt-partmcnts I and 2,,42 is thc arca cotltloctirrg cornpartttlcnts I antl 2" (', is ir

nondimensional comprcssibility cocfficient, k: 1.4, p1 is thc prcssurc in cottt-partment l, p2is the pressure in compartment2 (p2 < Pr), and "y1 is thc massper unit volume of air in compartment 1. If, in compartments provided with ablowout panel, the differential pressure exceeds the design pressure for a panel,a statement in the program transforms the blowout panel area into a wallopening. In view of the presence of three-dimensional effects not accountedfor by Eq. 16.2.6, the atmospheric differential pressures on extemal wallsobtained by the procedure just described are multiplied by a factor of 1.2t l 6-61.

An illustration of the pressure distribution and of the flow pattern in a build-ing during clcprcssurization is given in Fig. 16.2.2. An illustration of a structuredcprcssuriz.ation model with values of geometric parameters required as inputin thc conrputer program, and an example of a corresponding differential pres-surc-tirnc history calculated using the program, are shown in Figs. 16.2.3 and16.2.4, rcspcctively.

Between compartment 3 and outsideatmosphereBetween compartments 1 and 3Note: lnput time history per Fig. 11.2.1

using3R-,//rr=9secand po = 432 lb/tt2Structure depressurization modelshown in Figure 11.2.3

Time (s)

llbltt2 = 48 N/m2

Note: This example is for illustration lxrrp()sr::i ()nly

I l,r, 5ti I

16.3 TORNADO-BORNE MISSILE SPIEDS

'lir cs(itrtltlc sPcctls irlliritrt:rl lry lrn olrjt't'l nt()vltl, rrrrrlt.r llrr. ;rt.ltorr ol ;rcrorlyrtitltric lorccs intlrrccrl lty lot'llrtkr wilrtls.;r sr.l ol lt:i:illntl)l t()nr, l:. rt'rlurrt.tl

o On thc acnxlynalrric t'lurr':rt'lt'r'islit's ol llrc olr;t.t'l .

o On thc dctailctl lctrtrrrcs ol llrt' wilrtl lLrw lickl.o On the initial positiotr ol tltc olr.jcc( wi(h rcspcct lo llrc grourrtl untl (o lhc

tornado centcr, ancl its irritial vcl<rcity.

objects commonly considered as potential missiles in the design of nuclearpower plants are bluff bodies such as wooden planks, steel rods, steel pipes,utility poles, and automobiles.

The purpose of this section is to review approaches to the tornado-bornernissile problem based on (l) deterministic modeling, (2) probabilistic modelinginvolving numerical simulations, and (3) modeling of missile transport as aMarkov diffusion process.

16.3.1 Deterministic Modeling of Missile MotionsEquations of Motion and Aerodynamic Modeting. The motion of an objectmay be described in general by solving a system of three equations of balanceof momenta and three equations of balance of moments of momenta. In thecase of a bluff body, one major difficulty in writing these six equations is thatthe aerodynamic forcing functions are not known.

It is possible to measure in the wind tunnel aerodynamic forces and momentsacting on a bluff body under static conditions for a sufficient number of positionsof the body with respect to the mean direction of the flow. on the basis ofsuch measurements, the dependence of the forces and moments on position andcorresponding aerodynamic coefficients can be obtained. Aerodynamic forcesand moments can then be calculated following the well-known pattern used inairfbil theory; for example, if an airfoil has a time-dependent vertical motionh(t) in a uniform flow with velocity V, and if the angle of attack is .' : const,the lift coefficient is [16-81

dc, /Ct: ::lar/cv \ (r6.3.1)

'l'his procedure fbr calcullrling aerodynamic lilr-ccs lrrrtl rrulnrcrrrs rnrry lrr: irsstttned b be valid il'thc Itrotions tll'thc horly corrccrnt'tl :lt. slrr:rll. Ilowr'vt'r',irr thc casc ol'unctlnstrltirrt'tl bltrll'borlics rrxrvirrg irr rr wintl llow. llrt' v;rlirlityol'such ir pr<lcctlurc rclrurilrs lo bc tlcrrrotrsll-irlt.tl.

Itt lhcr lrhscncc ol'tr s;tlrsl;tt loty rrrrxlt'l lirr llrt' rrt'trtlyn;urrrt rlt.:rr'llrl rorr olllrc rrlissilc lrs lr riliirl (six rlt';'11'.',' ol lttt'rlrrr) llrtly. t( is r'u:;lorrr;rrv lo r(.:,()rt

40

nr20loa0.9cb20:o

l dft\* rrtr)

FI(;tjRFl l(t.2.4. Difl'cn:ntiul prcsstttc-litttc hislory lirr'('()nrl)iulnr('nls I :rrrrl I ll(r (rl.

562 l()llNnl)() I lll(:1.'

(. tlrc.llcrrrtr(ivc 11l tlcsclibiug tlrc rrrissilt: rrs u tttatcriitl lxlirtl ilctL:(l tllx)ll l)y ilclrag lirrcc

D : )pCoAlV, * Vrl(V' - Yu) (16.3.2)

( 16.3.3)

where p is the air density, V. is the wind velocitY,Yu is the missile vclocity'.4 is a suitably chosen aiea, and C7, is the corresponding drag coefficient. Thismodel is reaionable if, during its motion, the missile either (l) maintains a

constant or almost constant attitude with respect to the relative velocity vectorV' - V', or (3) has a tumbling motion Such that, with no significant eITorS,

,o*. -.un value of the quantity CpA can be used in the expression for the

drag D. The assumption of a constant body attitude with respect to the flow*oild b" creclible if the aerodynamic force were applied at all times exactly at

the center of mass of the body-which is highly unlikely in the case of a bluffbody in a tornaclo flow-or if the body rotation induced by a nonzero aerody-namic moment with respect to the center of mass were inhibited by aerodynamicforces intrinsic in the UoOy-nula system. The question thus arises as to whethersuch forces are present. This question has not been studied exhaustively in theliterature. However, simple experiments suggest that in the case of bluff bodiesthe aerodynamic damping forces have a destabilizing effect. Wind tunnel tests

reported in 116-91 tend io confirm this view. The assumption that potential

tornado-bome missiles will tumble during their motion appears therefore to be

a reasonable one.Assuming then that Eq. 16.3.2 is valid and that the average lift force van-

ishes underhmbling "onditionr, the motion of the missile viewed as a three-

degree-of-freedom system is govemed by the relation

dY, I CDA ,-,; : ;, =; lv' - v'l(v' - v,rz) - gk

ll, I l( )lltln l)r I l1{,lllll [Il:.].ll I ',1'l I lr', lrli.l

('1,r1 r'( (),,.'l I I ('/,,"1 ' I ( 1,, 11) tl(' 1'l)

whcro (ir,,4,( i l" 2, -l) ,,n'Prrrlrrtls ol llrt'Ito;t'tlr'rl ;rrt';r:' tottt':,Potttltttl'lothc cascs in which ilre Prirrcilrrl rrrcs rl llrt' lxrtly ;rtt' r;tt;tllt'l lrt lltt' vr'r l.t V,,

- V, by tho rcsJrcclivc sl:rlit'tlt:r1r, tot'lltt'tt'ttls, ;utrl r' ir. :t t ot'lltt tt'ttl :t:;:'ttttrt'tlto be 0.50 lirr planks, nrtls, pipcs. lurrl ptrlt's rrrrrl O. l I lot rtttlotrtolrilt's llr llrt'case of circular cylintllicltl botlit's (lirtls, pipt's, lxrlt's), lltt'ltsstttttpli,'tt ,'0.50 is clearly cottse t'v:tlive .

Computations and Numerical Results. A computer program fbr calculatingand plotting trajectories and velocities of tomado-borne missiles is describedin [6-12]. The program includes specialized subroutines incorporating theassumed model for the tomado wind field and the assumed drag coefficients(which may vary as functions of Reynolds number). Input statements includevalues of relevant parameters and the initial conditions of the missile motion.

In Eq. 16.3.3 both Vr and V, are referred to an absolute frame. The velocityV, is usually specified as a sum of two parts. The first part represents the windvelocity of a stationary tornado vortex and is referred to a cylindrical systemofcoordinates. The second part represents the translation velocity ofthe tornadovortex with respect to an absolute frame of reference. Transformations requiredto represent V,in an absolute frame are derived in [16-12] and incorporatedin the computer program.

For tornadoes with parameters given in Tables 3.5.1 and 3.5.2 for regionsI, II, and III, and referred to as Type I, Type II and Type III tornadoes,respectively, calculated values of the maximum horizontal missile speedsVfio* are given in Fig. 16.3.1 as functions of the parameter CrAlm- Thesevalues were obtained on the basis of the following assumptions:

o The tangential velocity of the tornado vortex Z, is described by Eqs.16.1.1and 16.1.2.

r The radial velocity component V, and the vertical velocity component Zare given by the expressions* t16-l3l

V,: o.5oV'

v.:0.67vt( 16.3.5)

( r 6.3.6)

The radial componcnt is tlircclccl toward the center of the vortex (Fig.16.3.2); the vcrtical ('()rrllx)rt('tll is tlircctctl trpwarcl.

o The translation vclot'ily ol'llrr' lottrrrltt vrtt'lt'x /,, is rlir.t'c(ctl lrlortg lhc.rrrxis (Fig. l()..1.1).

rlirr :rllentirlivr' ttttxlt'ls. s( (' I I 'l('l

where g is the acceleration of gravity, k is the unit vector along the verticalaxis, and ,'? is the mass of missile. It follows from Eq. 16.3.3 that for a givenllow liclcl ancl given initial conditions the motion depends only upon the valueol'tlrc pirratrtcrir C1,Alm. For a tumbling body this value can, in principle' bc

rlctcrnrinccl cxporimcntally. Unlbrtunately, little information on this topic ap-

l)crrrs to hc prcscntly available. Ref'erence tl6-10] contains information on

turnbling .,rti,rnli undcr flow conditions corresponding to Mach numbers 0'5to 3.5. th" dotu of tl6-l0l were extrapolated in t16-111 to lower subsonicspeeds; according to this extrapolation, for a randomly tumbling cube the quan-ttty CpA equals, approximateiy, the average of the products of the proiectcd

ur"u, "or."rponding to "all positions statistically possible" by the rcspcctivc

static drag coefficients t16-11, pp. 13-17, and 14-161. In thc abscncc ol'tnorcexperimeital information, it appears reasonable to assul.llo ([at thc cllbctivcproduct CpA is given by the expression

a^ olai G^4?r-E; EEEE&':.rrEd!ga; nixis,Sssnlf € #€#gf €#€# Efie frEEUEgEEeB#€ f€fi9#€f€ff€Ss EoEoEoEaSs.

oooc-.to+t--+n6!?!tY!t!d

o.t.iO+<+O1 n\Od6Nj

\qcl\\o6too

6i 6i 6i 6i 6i

F-Oola-r-o6i-;oo

_--9p =* fvFEo ^- -= - -F r ^- C;oEvE-us i5::g:hE: E

FN ^hv - -.,OXo_d ^-;^X6u^-^n^ e-\ .,-xE<-: x Hx9Ex^E d..-jeaSej^aF€=e::;€= JXxE*:;= E^g:K9i Eesh*3s,iFJ-:l.;-;=:-i ;^i O

or<.SEo.!?>S:o9.8'aE?;;,E++<t E E tEh#iSnt<6{ ci + rr) \o

tr-qi

cl

ci

6i

6l

al

- ar ^"2 clnlcl +o gq $;d=: 3Ee q$e^i - "aK ;E+ T "bSai- -6 9oi:--r)^:X s i;u ssni+ = =E.i n*$Jo+tsr'\o-€J-!i\-

dHEs.o>,

YH8.ExFF:-Ec.!>-vriF

!a\

su+{u5.<-o

\J

AdotU

aCd\2&

E:.s0 =O3>e

o

o

aq)

oqzc)

o)q)ra

oq)6)

(h

N

xz

aIqq)

Q

(n\ot't

Frc6

FIGURE 16.3"2. Horizontal components of tornado wind velttcity.

TORNADO EFFECTS

1o*4 1.5 3 b 1o-3 2 3 5 1o-2 1.5 2.5

S ,^',on,

FIGURE f6.3.1. Variation of maximum horizontal missile speed as a function ofCpAlm for various types of tornadoes.

o The initial conditions (at time r : 0) are r(0) : R., y(0) : 0, z(0) :40 m, Vy,(O) : Vu"(O) : Vu,(O) : 0, where x, !, z are the coordinatesof the center of mass of the missile and V74,, Vur, Vu, are the missilevelocity components along the x, !, z axes. Also att :0 the center ofthe tornado vortex coincides with the origin 0 of the coordinate axes.

Table 16.3.1 lists assumed characteristics of selected missiles and the cor-responding horizontal speeds Yfiu* as obtained from Fig. 16.3.1. A computer

100

60

:50F-sqo

\

Ce A/m = 1 m2 /kg <---> Cp A/w = 4.902 It2 Ab1 m/s = 3.28 ftls

Tornado type ITornado type IITornado type III

t

z-qo

Nn l)( ) lll )l lt.ll Ml:,1 ;ll I :;l'l I l):i 567

lrlrrt rrl lltc ltttrizottl;rl ptrrjt't'liorr ol tlrc lr;tit'tlory ol it nlissil(.wrlh ('11Alttr =.0. I irr ir'l'y1lc I (ot'rurtlo is slrowrr irr liig. l(r. I l,

Sensitivity study for the Maximum Horizontal Missite speeds. In viewol'the unccrtaintios irtvolvctl in llrt' rrrrxlcl ol lltc rnotion, it is of interest tostudy the scnsitivity ol'tlto rrr:rxirrrrn lrorizorrtal nrissile speed Z|]u* to variouschanges in thc assuntptions.just tlcscribccl. In cach of the cases examined below,all the assumptions ollror tharr thosc under study as noted-are the same asused to obtain thc tornado 'fypc I curve of Fig. 16.3.1.

l. Initial conditbns x(O) and y(0). Results obtained for CpAlm: 0.001and CpAlm: 0.01 are shown in Table 16.3.2 forthree sets of initial conditionsx(0), y(0). In Table 16.3.2 the arrows represent the directions of the tangentialand translation wind velocities. It is noted that the initial position correspondingto the largest calculated value of z[u^ depends upon cpAlm [position (c) forCpAlm: 0.001; position (b) for CpAlm: 0.011.

2. Initial elevation z(0). calculations show that if the parameter CpAlmcorresponds to the middle branch of the S-shaped curves in Fig. 16.3.1, thenthe values of vffo* decrease as the initial elevation decreases. However, if themissiles are relatively light so that the parameter CpAlm corresponds to theupper branch of the S-shaped curves, then Vffo* is independent of z(0).

3. Initial missile velocity. If the missile is injected in the flow, for example,by an explosion, the assumption that its initial velocity is zero no longer holds.All other conditions being equal, a nonzero initial velocity does not necessarilyrcsult in values of vfio'higher than those corresponding to zero initial velocity.This is illustrated in Table 16.3.3 in which the conditions Z1a,(0) : 0,Vt,(O) : l0 m/s, and V7a,(0) : 2O mls; Vu,Q) : 0, and Vr(O) : O wereassumed.

4. Translation velocity 2,,. Depending upon the initial conditions x(0), y(0),lhe speed v'fi^ may increase or decrease as the translation velocity 2,, of thet<rrnado vortex decreases. For example, if Vt : 0, for .r(0) : 46 m, y(0) :0, and CpAlm: 0.001, Vfro : 25 mls, rather than 1 mls, as in Table 16.3.2(in which it was assumedVn:31 m/s). However, forx(0) : 0, y(0) : -46rn and CpAlm: 0.001, Vfru : 25 mls rather than 5l m/s, as in Table 16.3.2.

'f'ABLE 16-3.2. Maximum Missile speeds vfi"* (m/s) for various rnitial(londitions x(0), J(0)

,r(0)(nr)

, l(r

,)l

0

v(0)(rrr)

o

o

' lr,

C,,A/m : 0.001 C,,A/m: 0.01ooo t\

o+

o

{€

I

xtIxtIxtIxlIxtIxl

xt

oa)E

o

'd

>'

a).?

o

E

Odo

a_)'aa

N

(n(t)

r{&FlrI

Ell

O

Ic

I

+

o

oooo++++ctsN@6icFaJou:-NNrl{: m U, tstlll

oooo++++F(no9nor@NOuooo@OF9Q

(t2

I'l( )

566

(b1

\l (t I

I ORNADO tI I EC I tJ

TABLE 16.3.3. Maximum Horizontal Missile Speeds V)',1"* (m/s) Corrcspontlingto Various Initial Velocities

CoA/m: 0.001 CoA/m: 0.01vr,(o) vr,(o)

tI()

(a)

(b)

r(0) y(0)(m) (m) 20102010

0

*23

20 62 58 53

35 63 59 5945

8

35

5. Modcl ofthe wtrtexflow. A vortex flow model proposed in [16-14] andt16-l5l dlllbrs from the model previously described-in which the radius R,is constant-essentially in that it assumes a significant linear increase of R.with height above ground. It is shown in [16-12] that if this model is used, thecorresponding calculated missile speeds are in most cases higher than those ofFig. 16.3.1.

Some meteorologists have expressed the view that the actual radial velocitiesZ, are considerably lower over most of the tomado wind field than indicatedby Eq. 16.3.5 t16-161. The radial drag forces available to maintain the trajec-tory within the region of high winds-where the missile gather momentum ata high rate-would then be comparatively small, and the missile speeds wouldbe considerably lower than those of Fig. 16.3.1. It is also believed that theactual vertical wind speeds are lower than indicated by Eq. 16.3.6, so missilestend to hit the ground sooner than calculated on the basis of this equation, witha consequent reduction of missle speeds Vn* [16-121.

For additional deterministic studies of tornado-borne missile speeds, thereader is referred to [16-17] through U6-231.

Missile Velocities Specified in ANSI/ANS-2.3-1983 Standard. Table16.3.4lists maximum horizontal missile velocities specified in [3-48] for eachtornado wind speed corresponding to probabilities of l0-5, 10-6, and 10-7 peryear (see Table 3.5.3). The velocities were obtained by methods of the typepnrposcd in [6-12l for two standard missiles "thought to cover the entirerangc of characteristics associated with all potential objects that can be pro-pcllcd by a tornado" t3-481. Also included in Table 16.3.4 are estimates listedin [3-48] of maximum horizontal ranges of the missile paths.

Reference [3-48] also provides estimates of the maximum altitude reachedby the missiles (not reproduced in Table 16.3.4) and a simple procedure forestimating vertical missile velocities.

16.3.2 Probabilistic Modeling lnvolving Numerical SimulationsA probabilistic approach to the tomado-bornc missilc pnrblcrrr was prcscntcclin |6-241, which is applicablc in situations whcrc tltc ttttntbct'rrtttl gcrrntclt'y

a0)o!

I

aq,)

O

C)

o0ozo0oq)

L€(atrn\of*]rq4,

o,E8

o,

O$

o.

\

a

oo

a

ON

aOra)c{

a

\oa.l

ac.l

oo

>

=Gil:*oxFA

(lJ

t,.9thdoO

zooE

o

o

-oo.C)

Fo

€o\clN(az(hzE

U)

o2zdOo

trIt()E

tr..l

3

-tr9++ts'tr b'tsRgooo.aEa E.rr)Or)O!+ oo an oO

aata)OciO

c.l

oo.E,E H,tsNO\NO\or)+oo\t

a.oF,f ts.!r)OOOF-Or)O\O C.l

ao.tarOOOA-RAoo+

Ea EuOO\ OO \n F-.+ioo +

ao.r)oofi

.9tr-

OOOr}=f

a-)ca6aJ- ]J2 2"- b€ Sxd

H-,t-Jn,::l ,* 1 Hd: ia

569

570 l()liNnl)() I lll(illi

ol'thc p()tcntiill targcts is spccilictl, lrlttl tllcir locittitltr is cilllcr sPt:cilictl or

cleterminecl by Monte Carlo sir'ulati''. 'l'hc t.axit.ltuttt spcccls .l't1c lrrissilcs

hitting these targets depend upon the fbllowing f-actors:

l. Maximum tomado wind sPeeds'

2. Structure of tomado wind field.3. Tornado Path.4. Type, number, and initial positions of missiles'

5. Aerodynamic characteristics of missiles'6. Rcstraining force, that is, force that must be overcome by the aerody-

narnic firrcc in order for the missile to be injected in the flow field.

Wc clcn<rtc tlro cliscrctc events associated with these six factors by Ar,', Ar,l'. ., Ati,,.'l'hc probabilitics p(,4;r,) are such that for each l (l : l' 2' "'' 6)'

)) p\A1,t : I

These probabilities are obtained from available information or on the basis ofjudgment.

Acomputerprogram|16-25ldevelopedinconjunctionwiththisapproachyields the'rp""d, oT the missiles that hit each target, given any ser of events'A, (i : 1,2, ... .6). The program automatically performs calculations cov-Ll"g "' the combinations of the postulated events ,44..^Assume, for example,

thatiits by missiles with speeds lirger than some specified value occur for two

combinations of events, denoted by Atr, Azi, ' , A6,nand A*' Azq' "' 'A6,.Theprobability of occurrence of those hits in PgPzr "'Po- * PtpPzq "'Por'

Amoreelaborateapproachisemployedil116-26]'whichcountsamongltsfeatures the use of aeiodynamic force coefficients that depend upon missileorientation. The latter is assumed to vary randomly with time and is obtained

at each time increment by Monte carlo simulation. For the purpose of modelingmissilc injcction, ob.iecti are clivided into "minimally restrained missiles" and..scqucntiatly rcstrainecl missiles." The first category includes objects re-

strainctl ,,nty try their weight and by friction, objects lying on the top layer ol'

a stack 1o, uncie-cath the top layer if the latter is easily removed by wind)'and components of buildings ihat would fail under the action of tomado winds'The second category, which includes atrl other missiles, is included in thc

computations toithe sake of "completeness in the probabilistic formalism"

116-261. The restraining forces are sampled by random simulation from prob-

uUitity-OirttiUutions selected on a subjective basis. Tornado characteristics arc

atso sampteO by random simulation' However, the numbcr of potcntial missilcs

on the site is specified deterministically. This is clono otr lhc bilsis ol'sttrvcysof nuclear power plant sitcs rcportccl in [16-261'

(i,:1,2,"',ni) (16.3.7)

{ r}: , 57 I

Irr lrsst'ssirrl', slrut'lrrlrrl irrtp:rt'(, Il{r l{rl ir(((}ul{:, lot 'lt;t,',,,',,t'si ol ttttssilt'willr tcsPcct (o tlrt' t:rtgt'l stttlrrt't' ;rl lltc {rttr' ol rrrP;rt l. rrl:.:.ll(' :.rzt' tt'l;rlrvr' ltrltrrgct tlirrrcrrsions, strsct'ptilrility ol t'ortt'rr'lt' lo :lt;rlrlrrrl', tl;rrtt;r1',t', str:,t t'plilrrlrlyol'concrclc lrnrl slccl lo ltct'lirllrliorr. rrrrtl rit'ot'lrt'l rrro( iorr ol rrrir.silr's llr:il thrrrot pcrlilralc thc ltugc:l . All tlrt'Morrlc (':ulo srrrrul:rliorrs rrtt'pr'rlotrrrt'rl lrylssuming thal only rlrrc rrrissilt'is lrvrrilirlrlt'orr llrt'sitc.'l'lrr'pnrbrrlrilily oloccurrencc ol'an cvctt( ,4 1 irlli't'lirrg rr s;x:t ilit'rl lrrr'1ir.'l tlrrl'irr1'. tlrc itlr sll'ikc by ;r

trlrnado with intcnsity 1, is tlc:notctl by /'(,4'l/, 1,.'l'hc corrcspontling prob:rbility<rf event At if N missilcs arc availahlc on sito is thcn assuntctl [o bc

P@)(A,lril : l - tl - P(Atlr)jlN (16.3.8)

Il'the missile population is subdivided into subpopulations M; (each character-ized by the fact that the missiles originate in region R; and have a specified setol properties zr;),

P(Alr)j : z p@,lrt, M)jP(Mi) (16.3.9)

where P(,41|11t, M)i is the probability of event-41 due to the action of a singlerrrissile belonging to the subpopulation Mi, P(Mi) : nilN, and n; is the numberol'missiles in subpopulation Mt.

16.3.3 Probabilistic Modeling of Missile Motion as a MarkovDiffusion Process 116-271ljor various components of nuclear power plant installations the question arisesin practice whether protection against tornado-borne missiles is required.

lf the probability that a target will be hit by a missile is smaller than somevalue acceptable to the Nuclear Regulatory Commission (e.g., lO-7 lyear), thensuch protection is not needed. The probability that a target will be damagedby tornado missiles in any one year may be written as

P7 : P7PsPp (16.3.10)

whcre Pp is the probability of occurrence of a tornado at the nuclear powerplant site, Ps is the conditional probability of hitting a target given a tornado()crcurrence, and Pp is thc probability of damage given a hit. For unprotectedlirrgcts it may be assunrcrl cotrscrvativcly Pp : 1.

'f'hc probability Py trlry lrt' wt'illt:tt rrs

I',,,t,,'1,)',,,,]y'|i\\(l;\{,(;, l;1 il6..1.1 1)

wltclc lt, is tltc lttttttlx'r ol pol( nlr.rl trtt','.t1,':' l)('r unll :u(':r rl llrt' srlc lrtlrl'toltsitlt'tr:rl, ,4 is llrt" irt('ir ()l llrr l.rr1'r l. ,/,(/') r:, llrt' tt'l;tlrvt' ltt't1ttt'ltty ol lot

572 l()l lNnlx) lllloll;

tradocs witlr intcnsity 1"<ln thc IlLriitlr scltlc itr thc n:gitln bcing cttrrsitlcrctl' 4(l'')is the probability that a missilc will bc injcctccl in thc wind llow givon (hc

occurr;nce of a tornado with intensity F, and {,Q, F1 is the probability that a

unit area whose center is located at elevation z will be hit by an airbornc missilcouring a tornado with intensity F, the site swept by the tornado being assumed

to hai" a potential missile density of one missile per unit of horizontal area'

For the sake of simplicity, ii is assumed in U6-271 that the density ofpotential missiles on the site is uniform. It is further assumed that (1) the wind

speedisconstantandequaltothemaximumtornadowindspeedthroughoutthc tornado path and 1Z; ttre angle g between the vertical and the drag force

incluccd by tirc maximum tornado wind speed is uniformly distributed between

tlrc valuci 0 : -rl2 antl 0 : rl2.Both these assumptions are conservative'(For cxaurplc, whcn 0 : o the direction of the maximum tornado wind is

assuilrocl tri bc vcrtical. This overestimates the probability of injection of the

rnissilcs.) I{cstraint forces are specified in a manner similar to [16-26].To cstimate the function ,lr(2, F) it is postulated that the motion of the tomado

missile can be represented as a Markov chain.* This postulate is justified by

the assumption alio used in 116-261that the missile undergoes purely random

tumbling, so the aerodynami- force it will experience at any one point depends

on the iandom position that the missile has at that point rather than on itsprevious geometric attitudes.

Once the Markov chain model for the missile motion is postulated, a prob-

ability density function, (ro, Yo, /0, r - r0, v - vo, / - to, F' 1) is delined

such that

| ', 57.1

lly irpplyirrg lltr'lr.olrttollotrrv ('llrpttutlr ('(lu;t(ror (o lltc lttnr'ltor (i:urrl rrrlr'gr-atirrg ittul itvt'titgitrg lltt' t'srrlls lo olrl:titt lltr' ptolrtlrlltltr':; (.)) ,ur(l ( l) llr'.1dolinctl, ll6 21 | tlr-:rivcs closetl lirttn tt'lrrliotts lot (ltt' lttttr'lrrrtt ,l't.. t;l

16.4 COMBINED TORNADO EFFECTS

Let the total tornackl cll'ccts consiclcrocl in dcsign bcr dcnotql by W,.'l'hc lirllowing expressions lor W, arc spccificd in ll6-61:

W,: W,n

W,: W,,

W,: W,,,

W,: W,n + 0.5Wh

W,: W,n * W,,,

W,: W,o + O.sWr, * W,,,,

P* : P* i*',

, (A ('/i (O r- li,,,l (l(r.,1())

(16.4.1)

(16.4.2)

(16.4.3)

(t6.4.4)

(16.4.s)

(16.4.6)

(16.4.8)

dP : G dx dy dz du* du, du, (t6.3.12)

where W,o is the maximum wind pressure effect, W,o is the maximum atmo-spheric pressure change effect, and W,,,, is the maximum missile impact effect.Equations 16.4.1 through 16.4.6 are justified in [16-6] as follows.

16.4.1 Wind Pressures Plus Atmospheric Pressure Change EffectsIf the structure is unvented, the total pressure p,o due to the direct action ofwind and to the atmospheric pressure change can be written as

Pro: P* * Po

where p, can be represented in the form

(16.4.7)

where x, !, Z ?te the coordinates in space, u,, uv,,uzare the missile velocity,o*pon"ntr, and. dP is the probability that, given the occurrence of a tornado

wittr intensity F and characteristics .y, a missile that becomes airborne at mo-

ment /0 hits the volume dx dy dz around point r during a unit time interval at

the insiant r with a velocity between v and v t dv (r : xi t yj + zk; v :zr,i + auj * zr-.k, where i j, k are unit orthogonal vectors)' The function G

is rcl'crrccl to as the originai (fundamental) Green's function of the problem'Mrxliliccl Grccn's functiins can be derived by integrations and/or averaging ofthc original Grccn's function. The modified functions correspond to (l) the

pr.bability that given the occurrence of a tornado with intensity F and a set ofcharacteristics 7, the missile will hit the volume dx dy dz around point r duringa unit time interval at the instant r with any velocity, (2) a similar probabilityfor the case where the hit occurs over a unit area with orientation O, and (3)

a similar probability averaged over all possible tornadoes having intensity F'

*A Markov chain is a process in which the probability of transitirtn l.r(rtlt ottc lx)inl l() llnotltcl

depcnds only on the cclordinates ofthcsc points and on lhc slittc ol lltt'syslcttr rrl llrt'iltililtl poirtl;

that is, thc probability is indcpcntlcnt tll lhc prcviotts hisloty ol lltt'syslt'ttt

and Cis a constant (scc Eqs. l6.l.l through 16.1.7). Using Eqs. 16.2.2 and16.2.3, the expressitln lirr lltc tolirl plcssttrc 7r,,,, bcc{)mcs

Pn,: l),,i,,1 , Irl l)

t ,i,, rt,i,,

1,,,,, tt : ' tl I A ( )

',I(/i { l() .1 lo)

574 t()t tNnlx) llll(;l:l

F'rom Eq. 16.4.9 it firllOws thal, lilr Kr('< l,7r,,,,, is lt Ittaxirttttltt itl r' 0'where it is equal to the maximum atlntlsplrcric prcssurc chaltgc cllcct (liq'16.2.3); tor t?C ) l, Pn, is a maximum at /" : R,,, whcrc it is cqual to tllomaximum wind pressure effect plus one-half the maximum atmosphcric prcs-

sure effect. Equation 16.4.10 rho*, that regardless of the value of K2C, pn,,

is maximum ui , : R., where its value is again equal to the maximum windpressure effect plus one-half the maximum atmospheric pressure effect. Thescconsiderations justify Eqs. 16.4.2 and 16.4.4. For completely open structuresthe atmospherit pressure change effects approach zero so that the maximumloading is given bY Eq. 16.4.1.

16.4.2 Load Combinations lncluding Missile Effects

It is assurrtotl in [16-61 that the maximum speed is attained by the missile at a

clistancc r l'nrnr thc ccnter of the tornado vortex nearly equal to R.. It followsrhcn fkrnr Eqs. 16.4.9 and 16.2.3 that Eq. 16.4.6 will hold. The case of openstructurcs coiresponds to Eq. 16.4.5. Finally, if after attaining its maximumspeed near r: R^ the missile is ejected without-or with little-loss of mo-mentum, then the wind pressures and the atmospheric pressure change effects

could be negligible at the time of the missile strike. This situation correspondsto Eq. 16.4.3.

REFERENCES

16.1 unform Buitcling Code, Intemational Conference of Building O{ficials, Los

Angeles, CA, 1975.16-2 Southern Building Code, Birmingham, AL, 1965'

16-3 Cod.e of Fecleral Regulatbns, Title 10, Part0, Appendix A, criterion 2 (DesignBases for Protection Against Natural Phenomena), Office of the Federal Reg-ister. General Services Administration, Washington, DC, 1976'

16-4 De.sign Basis Tornado for Nuclear Power Plants, Regulatory Guide 1.76, Di-rectorate of Regulatory standards, U.s. Atomic Energy Commission, 1974.

16-5 J. D. Stcvcnson and Y. zhao, "Modern Design of Nuclear and other Poten-tially Hazardous Facilities," Nuclear Sap4' (in press)'

l6 6 J. V. I{0t2, G. C. K. Yeh, and W. Bertwell, Tomado and Extreme Wind DesignCrircria.fitr Nut:Lcar Power Plants, Topical Report No. BC-TOP-3A Revision3, Bcchtel Power Corporation, San Francisco, CA, 1914'

16-1 R. c. Bincler, Fluid Mechanics, 2d ed., Prentice-Hall, Englewood cliffs, NJ'1949.

16-8 Y. C. Fung, An Introduction to the Theory of Aeroelasticiry, Dover, New York'1969.

l6-9 R. H. Scanlan, "An Examination of Aerodynamic Responsc Thcorics ancl MoclclTesting Relative to Suspension Bridges," in Prot'ccdings rtl the'l'ltinl lttttrnational Confe rence on Wirul EfJccts tn Buildings turrl Slntclttrr.s, 'lirkyo' 197 I 'Saikon, Tokyo. 1912, 1'tp.94 I 95 I'

t( l :; 575

l(r lo (; li. ll:tltst ltt' :rtttl .l S l{inr'lurtl, "Arr l }r:rll orr ( 'rrlx's rr( Mlrt.lr Nrurrllcrs 0.5lrr .1..5." J. ,'lrtttrtttrtl. ,\i i , l9 (l;(.1) l()5-l). li t l.l.l

I6 ll S. l;. IIot:lrtr't-, l"lttitl l)\trtuttir' /)rir,r3 lPrrlrlislrt'tl lry llrc irrrllror, l9-5tl).l6 12 l'.. Sirrrirr rrrrtl M. ('ollt's. lltrttnltt rhrtrrt, furi,s.tilt, s1x't,tls, NBSIR 76-1050,

National lJurcau ol st:rlrrlrrnls, Wirslrirrglorr, lXl, 1976.l6-13 J. R. Mc[)on:rltl, K. ('. Melrrrr,:rrrrl .l . lr. Minor, "Tornado-Resistant Design

<rf Nuclcar P.wr:r l)li'rt St*rclrrrcs," Nuct. SaJ., 15,4 (July_Aug. 1974),432'439.

16-14 w. H. Hocckcr, "wind Spccd and Air Flow in the Dallas Tornado of April2, 1975," Mon. Wtarhcr Rev.,88,5 (1960), 167_190.

l6-15 F. c. Bates and A. E. Swanson, "Tornado considerations for Nuclear powerPlants," Trans. Am. Nucl. Soc., l0 (Nov. 1967),712_713.

l6-16 J. R. Eagleman, v. U. Muihead, ancl N. williams, Thunderstorms, Tornatloesand Building Damage, Lexington Books, Lexington, MA, 1975.

16-ll D. F. Paddleford, characteristics oJ'Tornado Generated Missiles, Report No.WCAP-7897, Westinghouse Electric Corp., pittsburgh, pa., 1969.

16-18 A. J. H. Lee, Design Parameters for Tornaclo Generated Missiles, TopicalReport No. GAI-TR-102, Gilbert Associares, Inc., Reading, pa., 1975.

l6-19 The Generation of Missiles by Tomadoes, Reporl No. TVA-TR74- l, TennesseeValley Authority, Knoxville, 1974.

16-20 R. c. Lotti, velocities of rornado-Generated Missiles, Report No. ETR-1003,Ebasco Services, Inc., New york, 197-5.

16-21 D. R. Beeth and S. H. Hobbs, Jr., Analysis of rornado Generatetl Missiles,Report No. 88 R-001, Brown and Root, Inc., Houston, TX, lgi.5.

16-22 B. L. Meyers and w. M. Morrow, Tornado Missire Risk Mottel, Report No.BC-TOP-10, Bechtel power Corp., San Francisco, CA, 1975.

16-23 A. K. Battacharya, R. C. Boritz, and p. K. Niyogi, Characteristics of rornarloGenerated Missiles, Report No. VEC-TR-oo2-0, united Engineers and con-structors, Inc., Philadelphia, 1975.

16-24 E. Simiu and M. Cordes, Probabilistic Assessment oJ Tornaclo-Borne MissileSpeeds, NBSIR 80-2117 , National Bureau of Standards, Washington, DC, Sept.1980.

16-25 computer Program for Probabilistic Assessment of Tornado-Borne Missilespeeds, National Bureau of Standards, computer Tape No. pBgl-12g423, Na-tional Technical Information Service, Springfield, VA2216l ,lgg1.16-26 L. A. Twisdale and w. L. Dunn, Tornado Missile simulation ancl DesignMethodobgy, EPRI NP-2005, Electrical power Research Institute, palo Alto,CA, Aug. 1981.

16-27 J. Goodman antl .1. li. K.ch, "'l'hc pnrbability of a Tornado Missile Hitting aTarget," Nuclattr l,)t,\. l)r,,t., 7-5 ( l9ti2). l2-5 l-5,5.

CHAPTER 17

STANDARD PROVISIONS FOR WINDLOADING

As recently as two decades ago most building code provisions on wind loads

were not more rhan 3 or 4 pages long. The ASCE 7-95 Standard [17-11 pro-

visions and commentary on wind loads are more than 50 pages long' reflectingthe vast increase in the amount of information available on wind loads, as wellas the need to provide more differentiated provisions on various types of build-ings and structures.

However,noteventheASCET_g5Standardcontainsenoughinformationto yield economical and risk-consistent designs (see sect. 17. 1) . For this reason

standard formats are being developed that would allow the use of data bases

obtained from wind tunnel tests, as opposed to the use of data summaries'

which is typical of current standards. Many of the simplifications resofied to

in current itandards would therefore no longer be needed; instead of using

envelopes of pressure data, designers would resort to the more economical or

risk-consisteni option of using the original data corresponding to the parameters

frrr thc spccific design situation at hand ll7-2, l'l-3)'Scction l7.l briefly summarizes some conclusions of a critical study on

sta'clarcl pnrvisi.ns dtvclopcd in fbur countries for wind loads on low-risesrftrcturcs ll7-41. Bascd on l17-51, Section 17.2 summarizes the main provi-

sions of thc ASCE T-95 Standard. Section 17.3 contains a detailed numerical

example illustrating their use. Since many of those provisions are exceedinglyintricate, it was found necessary to develop an interactive program representing

a computer-based version of ihe Standard. The software for the program is

included in a diskette [17-5] appended to this book. Besides being useful lirrgaining familiarity with the way standard provisions are developed in a corn-

iu,"'. ior-u,, using the diskette- in conjunction with the Standarcl will hclp thc:

reader to connect many of the issues dealt with in this btxrk wi(h tltc rcrrlity ol

design practice.

576

Wltllr I{ )nl ) l'l l( )Vll,l( )Nll; i )l lltl /\'.r | /'lr', ,t/\tll}/\Illi lr77

17.'I STANDARD PROVISIONS ON WIND TOADS ANDHISK-CONSISTENCY

ln this scc(iott wc stttttttru-iz,c s()nr('('()lrr'lu:;lon:; rtl lr t rtltt:rl :;ltttlv ott s{lrtttl:utlpnrvisions clcvclopctl irr lirrrr t'orrrrlrit's lor wnrrl lo;rrls orr low rrst'slrrrr'lrrrr'slll-41. Acconling (o llre sltuly, nS('11 Sllrrrrl:rnl llrl)vlsr()ns lor wirul loirrls orrtypical low-risc builtlirrgs (l()tlll vu'siorr) lerrrl lo ovu'cslirrur(e perrk wirrrl lrxrrlson structural systonrs, ('irrratlilrrr pnrvisiotts 1t:rttl to rtrttlcrcslirttatc tltcrrt, atttlAustralian and Swiss pnrvisions tcncl to undcrcstirrratc pcak wind loads firrlower wind speeds and structurcs with light root's, and overcstimate them tbrhigher wind speeds and structures with heavy roofs. In addition, for all fourcountries, standard provisions fare poorly with respect to risk-consistency. Theshortcomings of the standard provisions are attributed to the fact that the spec-ified wind load induces stresses governing the design that may differ from thoseinduced by the actual load, which depend, among other factors, on wind speed,gravity load, and type of structural system (e.g., on whether a frame hascolumns hinged or fixed at the base; see, for example, [17-6] and [A3-9]. Infact the very location of the governing stresses induced by the standard loads,on the one hand, and the actual loads, on the other, may be altogether different.While the critique of lll -41 needs to be carefully scrutinized, it raises the validpoint that provisions on wind loading should be based on both aerodynamicand structural considerations, which may be strongly intertwined (e.g., seelll-7D. Although this principle has been applied to some extent in the devel-opment of the ASCE and Canadian provisions, much work remains to be donetoward the development of judicious and transparent procedures that accountfbr both aerodynamic and structural behavior.

As was mentioned earlier, lack of risk-consistency also results from the needto compress large amounts of information in simple envelope curves coveringa wide variety of design situations. Risk-consistency can be expected to beimproved significantly with the future development of a new generation ofstandards equipped with appropriate knowledge-base and procedural systems.

17.2 MAIN WIND LOAD PROVISIONS OF THE ASCE 7-95STANDARD

Wind loads depend on thc wincl fkrw ancl on the aenrdynamic and structuralproperties of the structrrrc bcirrg tlt:sigrrt:tl. 'l'lrcy lrlso tlcpcnrl on thc sll'cty lcvclinhercnt in the choicc ol'tk'si1',1v t'r'ilcli:r. Wc recrrpitrrllrtc brielly tlte llclols lhrt(intlucncc thc wintl ll<tw :tsstrttt,'tl lor tlr'si1'11 l)trt lx)s,('s'

l. 'l'hc hltsic wirll s1x't'tl {',t t' ('lt:rlrlr'r l}2.'l'lrr:illclilttr'('ut]('n(('ntlr'rr';rl,rlllr, \\'llrl.lrr'r'{lllt.rltttrltt,r".llt, ttlltltt,rlr'

loirtls jtrtlgt'tl lo l)('irppropllirl{ lrrr llr( {l{',t}!rl ril llrt lypt rtl lrttllrltttl'rtt

578 :;lnNl)nl tl) I'll()vllll()N:; l()l I wllllr I ( )nl tlN('

slructuro unclcr consi(lcnrliou (sr:c ('ltltplcl' -3, Apltrrtlix Al, ltrrtl A1l

pcndix A3).3. The characteristics of the terrain surrounding the building or slructuro

(see ChaPter 2).4. The characteristic height above ground for the point or system being

considered (see ChaPter 2).5. Directional properties of the wind climate* (see Chapter 8).

The effect of the wind flow on the structural system or structural or nonstructuralcomponent being considered depends upon the following:

6. The aerodynamics of the building or structure (see Chapter 4).7. The position(s) of the area(s) acted on by the wind flow (see chapter

4).8. The magnitude(s) of the area(s) of interest (see Chapter 4).9. The porosity of the building envelope (see Chapter 4).

10. The selection of the probability that the peak fluctuating wind loadacting on a system or element will be exceeded during the wind stormconsidered in design (see Appendix A2) '

11. The structural properties of the system under consideration, includingthe effect of gravity or other loads and the susceptibility to dynamiceffects induced by the wind load fluctuations (see Chapters 5, 6, 9, and10).

'17.2.1 Calculation of Wind Loads: Basic Steps

The Standard recognizes two methods for estimating wind loads: the analyticalprocedure and the wind tunnel procedure. We briefly discuss in this chapterihe analytical procedure, which proceeds in two steps. The first step is intendedto reflect the properties of the wind flow (items I through 5 above). The secondstep reflects the aerodynamic properties of the structure and dynamic propertiesassociatcd with effects of the longitudinal (along-wind) wind turbulence (items6thnrugh ll above).

Step 1: Determination of Velocity Pressure. The velocity pressure at ele-vation z is

q,: 0.oo256K,K,tVzI

q,: o.6|34K,K,,V2I

(17 .2.1a)

(t].2.tb)

Al ll) l, /9

lrr tlrt'p;rrlicul:rt r':rst'rrr wlrit'lr 17 is t'vlrltt:rlt'rl :rl t'lcv:rltott /r, tl rs rlt'rto(r'rl lr-y

r77,.'l'lte Stltntlltnl ltlso ttst's (lrr'1it'rrt'n( n()(rlr()n {/ lr) (l('ur)l('vt'lotily l)r'('ssttll'irrrtl irrrlic:rtt's wltc(ltt:t irt ir 1t:rt1it'ttl;rt insl;uttt' tl tl tt .l tlt,.

'f'hc /zl.rlr'vt.,itttt sltt't'tl /t:rlttt'sltortrls rtotrrirr:rlly lo;r 5O y(';u nr(':rr r('('ulr('n('r'intcrval and is pnrvitlc:tl lirt rttosl pt'ogt:rplrit'rrl lot'lrliorts lry Irip. (r. l ol'llreStandard. ll rcpr-cscrr(s llrr: spr't:tl lirrrrr irrry rlin't'liorr ir( irrr t'lt'vrrtiort ol lO rrr

above ground in lla( ol)cn c()llnlry (l')xlxrsrrn: (', sct: llclow).The vektt'ity pr(ssur( (.\l)(),tur( txJ.licit'ttt K.. rcllcc:ls llrc rlcpcnrlcrrcc ol'thc

velocity pressurc on lcrnrin nrughnoss (i.c., cxposurc catcgory) and heightabove ground. lt is givcn in'l'able 6-3 of the Standard. Section 6.5.3.1 of theStandard defines in detail four exposure categories: A, B, C, and D. [n sum-mary terms, exposure A corresponds to terrain in large city centers, exposureB to urban and suburban areas, exposure C to flat open country, and exposureD to terrain close to a shoreline and with wind coming from over the water.

The importance factor 1 is provided in Table 6-2 of the Standard as a functionof the building and structure classification given in Table l-1. For Category IIbuildings or structures, that is, for most ordinary cases, 1 : 1. For CategoryI buildings or structures, that is, buildings or structures representing a lowhazard in the event of failure (e.g., agricultural facilities), 1 : 0.87. ForCategory III and Category IV buildings or structures, that is, buildings orstructures representing a substantial hazard to human life in the event of failure(e.g., buildings where more than 300 people congregate in one area, and es-sential facilities such as fire stations),I : 1.15. The importance factor maybe interpreted as defining the mean reculrence interval of the effective basicvelocity Itt2V. For I > I (1 < 1), that interval is larger (smaller) than 50years.

The topographic factor

K,,: (l + KIK2K)2 (11.2.2)

reflects the speedup effect over hills and escarpments. The multipliers K1, K2,and K3 are given in Fig. 6-2 of the Standard.

Step 2: Estimation of Aerodynamic Pressures or Forces. Aerodynamicpressures are specified in the Standard for buildings other than open buildings.*'l'hc aerodynamic force on an area acted upon by a uniform pressure is equalto the pressure times thc arca. However, the Standard has specific provisionslilr aerodynamic forccs (lrs opposctl lo prcssures) acting on monoslope roofs()vcr open buildings; clrirrrrrt'ys, llrtrks, iuttl sirrrilar slructurcs; solid signs; opcnsigns and latticc I'rarrrt'wor.ks, ;urtl lrrsst'rl l()wt'ls. 'l'lrc crrlcullrlion ol'lrcnrtlyniurric prcssuros iurtl lirtr'r's ;r:, :;pt't tlrt'tl irt llrc Slrrrrrlrrnl is tlcscr-ibctl irr Secls.IJ.2.2 irntl 11.2.3.

r();x'rr lrrrilrlirrl',s lrt'lturl,lttt1l, lr;rrttt; :rll ",rll, ,rl l':r',1 li(l';. r'l'r'n ('icrl (r.'ol llrc Sl;rrrrl.rrrlt

(n. * *q, r'in -nr')/m\(a. in Pa, v in )

*Thesepropcrtics arc not accountcd lirrclcarly antl cxplicitly irr llre AS('li 7t)5 Sl;rrrtlrrll

*580 :;tnNl)nl ll) l'll()vliil()N:i l()l lwlNl) l(rAl )lN('

17.2.2 Calculation of Aerodynamic Pressures

The Stanclard specifies aerodynamic pressurcs used lbr the design ol'builclings']'his is clone separately for main wind force resisting systems and fbr conrpo-nents and cladding.

Main Wind Force Resisting Systems. Pressures calculated for the design ofmain wind force resisting systems are specified in the Standard for nonflexiblebuildings,x low-rise buitdingst (which are a special class of nonflexible build-ings), flexible buildings, and arched roofs'

Nonflexibte Buitdings. Pressures are given by the equation (Table 6-1 of stan-dard):

p:eGCp-qo(GCpt) (r7 .2.3)

The first and second terms in Eq. 17.2.3 represent the external and the intemalpressure, respectively. In Eq. 17.2-3,4 : Qzfor windward ya-ll, q : qoforieeward wall and side walls, and /z denotes the mean roof height- G is a gust

effect factor that accounts for the fact that the wind load is not perfectly coherentthroughout the surface over which it acts (G : 0.8 for exposures A and B and

G :b.gs for exposures c and D-section 6.6.1 of the Standard). cn is the

pressure coefficient from Fig. 6-3 of the Standard, and GCri is a factor pro-portional to the intemal preisure, which depends on the degree to which thefuilding is open (see Table 6-4 and Sect. 6.2 of the Standard). Figure 6-3 ofthe Standard was developed from tests on flat roofs and gable roofs. To the

writers' knowledge, there is no published evidence that it applies to buildingswith other types of roofs.*

Low-Rise Buitdings. Pressures are given by the equation

p: anlGCpr- (GCpi)) (17.2.4)

'Ihc crrcllicicnt GC,,S is proportional to the extemal pressure and is obtainedl'r<rrrr Fig. 6-4 of thc Standanl . GC,,i is defined as for Eq. 11 .2.3. Note that the

Starrtllri givcs thc clcsigncr thc option of using for low-rise buildings eitherlrig. 6 4 or Fig. 6-3.

*Flcxibility is defined in Table 6 I of the Standard by at least one of two properties: ratio height/

least horizontal dimension > 4, fundamental frequency of vibration/ < l Hz'iLow-rise buildings are defined in the Standard as buildings that are not open and have mcan r<xrl'

height not exceeding 60 ft and the building's least horizontal dimension.+However, according to Note 4 of Fig. 6-3 of the Standard, coeflicicnts g, ol tlut ligurc ntay

also be used for the estimation of piessures for the calculati.n .l'tttairt wirttl lirret .csislitrp'

systems of monosloPe rool.s'

l/';' [,lnll] wlNl) l()nl I I'l l()vl:;l()Nll; (ll llll A"r I

Floxible lSuiltlitttls. I'r't'sstttcs irll'l'.rv('rl lry tlrt't'tltt:tllolt

,,,t'. .;tAiltr^l ilI arlll

1t 11( i, ( ',, tl/ -' "t

Whilc thc gusl clloct lirt'lor'(i (st'c lir; l-l..) l)ir(('()unl:; ottly lot lltt'ttttlrt'tlrt'1spatial c<lhcrcncc ol'(lrc wirrtl krtrtls, (i1 ir('r'()unts itt:ttkliltrtn lol lltt'rlytuttttttarnplification ol'thc trlong-witttl tr,spottst tlttt'lo llrt'vrrri;r(iott ol lltt'wirttl lorrtling with timc. Thc Stanclar-cl itscll'tlocs rrol spccily tttc(ltotls lirr.cstirrurting (i,and limits itself to indicating that C7 is obtaincd by ratior.ral analysis. A rtrcthodthat may be used to estimate G7 is provided in Sect. C6.6 of the Commentaryto the Standard.

Arched Aoofs. Pressures are given by Eq. 17.2.3, where the coefficient Qis obtained from Table 6-5 of the Standard.

Effect of Parapets. The effect of parapets on the main wind force resistingsystems is merely to add to the wind load acting on the walls up to the eavelcvel the load acting on the external surface of the parapets.

Roof Overhangs. Roof overhangs must be designed for a positive pressureon the bottom surface of windward roof overhangs corresponding to q, : 0.8in combination with pressures indicated in Fig. 6-3 or 6-4 (Sect. 6.7 .2.1 ofthc Standard).

Torsional Effects. For buildings with mean roof height h > 60 ft (18.29 m),torsional effects are to be accounted for in accordance with Section 6.8 of theStandard.

Components and Cladding. The Standard provides procedures for the cal-culation of pressures on building components and cladding for two types ofbuilding: buildings with mean roof height h < 60 ft (18.29 m) and buildingswith height h > 60 ft (18.29 m). Pressures depend on the location (pressurezone) covered by the component/cladding. If a component/cladding coversrrurre than one zone, the pressures are automatically calculated in the programon the basis of the weighted mean method described in [17-8].

Buitdings with Height h '< 60 ft (18.29 m). Pressures are given by the equa-l irln

1t t1,,1( i( ,, ( (;( ;,, ) l (11.2.6\

wltclc r77, is calctrllrtctl llslll| ('\lx):'rtrt'('lt'1';rttllt':;:; ol lt'tt:tttt. lltt' l;rtlol (), is

1',ivcn itt lrigs. (r--5, (r (r.;rrul (r / ol llrr'l-il,rttrl;rtrl. ;tttrl (,'(,,, r'.,1t'llrrt'rl;t:; ttt Iltll1'2.\ I'or-ltuilrlirrlls :ilt'rl tvtlltltt ,'rp,r',tttt ll. lltt' ptr':,:,tttt'', l'l\r('rr lry l'.t1

582 litnNt)nl t{) t,il()vll;t()Ni; t()t twtNt) tr)nr)tN(:

11 .2.6 must bc multiplicd by tltc Iirctor'0.13-5. 'l'his is u lirrrrr ol crrlilrlrtisltagainst past practice, since in thc abscncc ol'lhc lactor 0.t15. it was notctl tlrirlcalculated loads were high compared to loads spccifiecl in prcvious standards.That the same factor is not applied to buildings sited within cxposurcs othcrthan exposure B introduces a clear inconsistency in the Standard. This wastolerated presumably because low-rise buildings are built predominantly firrexposure B, and once the 0.85 factor was accepted for this exposure, there waslittle interest in pressing for its use with other exposures. Note that the Standanlcontains provisions on pressures on wall components or cladding only forbuildings with gable roofs (Fig. 6-54 of Standard); no such provisions areavailable in the Standard for buildings with hipped roofs (Fig. 6-58), steppeclrool.s (Fig. 6--5c), multi-span gable roofs (Fig. 6-6), and monoslope roofs orsawlrxrth r<xlf.s with two or more spans (Fig. 6-7).

Buildings with Height h > 60 ft (18.29 m). pressures are given by the equa-fion

p:eIGCp-(GCri)) (17 .2.7)

where q : q, for positive pressures and q : q1, for negative pressures, thefaactor GC, is given in Fig. 6-8 of the Standard, and GCoi is defined as in Eq.17.2.3.If 60 ft (18.29 m) < h < 90 ft (27.43 m), Tabte 6-l of the Standardstipulates: ' ' GCo values of Figs. 6-5 , 6-6 , and 6-7 shall not be used unless theheight-to-width ratio does not exceed l, q is taken as e1,, an:d Exposure C isused for all terrain."

Effects of Parapets. For flat, gabled, hipped, and stepped roofs of buildingswith lz < 60 fr (18.29 m), if a parapet with height of 3 ft (0.91 m) or higheris provided around the perimeter of the roof, then the roof corner zones aretreated as the roof edge zones; that is, pressures at the roof comers are sub-stantially reduced.

Roof overhangs. Roof overhangs must be designed for pressures determinedI'rom prcssure coellicients given in Fig. 6-5B (Sect. 6.7.2.1 of Standard).*

17.2.3 Galculation of Aerodynamic Forces'l'hc Standard contains the requisite design information on aerodynamic forceslor main wind force resisting systems. For structures that are not flexible theaerodynamic forces are calculated in accordance with the formula (Table 6- Iof Standard):

*Ifthe tributary area for a component includes (1) areas outside thc ovcrhang :rntl (2) an 6vcr6angarea, depending upon the application the designer rnight nccd lo crrlcrrlirlc tlrt. lrvclrgc pr.osslr.coverareas (l) (i.e., excluding thc overhang arca), antl llrc irvcrirgt.l)ressurl.ovt.r :r1':rs (l) rrrrtl(2) (i.e., including the ovcrhang arca)- Tlrc colllpulcl l)r'oltr':un clli.t ts lxrtlr t.;rk rrlrrtrorrs.

l/ I

l'' t1.(i(',,1,

{ l\lr llll\llr)ll I -/\Nll'lt lrllll

(l/ .' lir

whcrc (i is:r gtrsl cll'ct:l litt'lor llt:rl ;tt't'otttrl:; lor llrt' l:rt l llr:rl llrt' rvrrrrl lo;rrl r:;

rtot pcrl'cctly coltctcttl (lttrrttgltottl lltt' stttl:rt t' ovt'r wlttt lr r( :rt l:; ((,' (l fi lorcxposurcs A and t] arrtl (i O.l{5 lor cxl)()slr('s (' :rrrtl l), Sr'tl (r (r I ol (lrt'Standard), Ci is a lirrcc cocllit'icrrl p,ivt'rr rrr'litlrlt's (r (r lo (r lO, .'1, rs:ur irt(';rtlefined in Tablc 6 I .

For flexible structuros thc lillkrwiug lirlnrullr is spccilicrl:

I; - q,G1C1A.1 (11.2.9)

that is, the gust effect factor G7 must be used in lieu of the factor G.For comments on the gust effect factor GJ, see text following Eq. 17.2.5.

The method for estimating G7 provided in Sect. C6.6 of the Commentary tothe Standard is applicable to rectangular buildings whose horizontal cross sec-lion is uniform with height. However, that method can also be used to obtainrough estimates of GJfor structures such as chimneys, tanks, or trussed towers.If the chimney or trussed tower is tapered, then reasonable approximate resultscan be obtained by assuming that throughout the structure's height the cross-sectional dimensions are equal to those at two-thirds of the height of the struc-t.ure. However, the fundamental frequency used as input in the program shouldhe calculated on the basis of the actual dimensions of the structure.*

17.3 CALCULATION EXAMPLE

'Ihe calculations presented here are intended to help the readers to familiarizethemselves with the use of the ASCE Standard 7-95 provisions on wind loads.We recommend as an exercise the reproduction of these calculations and theparallel use of the computer program [7-5] appended to this book. Also in-cluded in this section are typical outputs of the program. Occasional smallclifferences between hand calculation and computer program results (typicallyon the order of 0.1%) are due to roundoff errors.

17.3.1 Building (Mean Roof Height h < 60 ft (18.29 m))

Wind and Building Characteristics: V : 100 mph (44.70 m/s), exposureC, building classification: catcgory II; building not situated on or near hills oroscarpments; building nol opcn; builcling dimension normal to ridge line: 40

r'ln irrrplcnrcnting this itpprirrrt lr rrr llr r'.rrtnrl('r l)r()1',rinrr ol' ll7 51, llrt' lirllowing tc:tsott:tblt':rssrutrptions havc bccn rtr;ttlt' l,rt llr, lttrt1u,:,, o1 r'slirtlrlittg lllc gusl t'lli'tt llrttor (,',. l;or tirtullrrsltlctulcs, dcpth O, wirltlr rlr:rrnclr'r. lor lrcr:r1'orrrl ;urrl rx l:rl1rrrl slnr( lln1's. (l('l)llr O,

rvitlllr tli:utrr:lt:r ol t itt rrnrsr'rrlrtrl t rr, lr lrrr ',rprr.rrt rilnr( llll's, (l('l)llr rvtrlllt sttk' rtl r,tltrur'(wirrtl lroln;rl lositlc),rrrrtl rlt'1rtlr {l.rrrrltlr (.')' -,;r,1.'(rvirrtl ;rlorrlrli:r11otr;rl;,lor tn;ur1'ttl.rtlrtrvcts, rlt'pllt O, witlllr '.trlr ll lt t;tttl'1,'

*584 lltnNt)ntlt) I'lt()vll;l()Nli l()ll wlNlt t()nl)lN(i

li (12.19 rn); ditncnsion parallol to ridgc lirrc: (t0 li (18.29 rtr); crtvo hciglrt: tlft(2.44 m); gablecl roof with slope 0 : 2O", no openings in walls (i.c-' builclingis not partially enclosed; see Sect. 6.2 of Standard); building not in a hurricanc-prone region; no parapets; no overhangs; fundamental frequency larger than I

Hz. Use Fig. 6-3 for main wind force-resisting system.Importance factor: 1.00; mean roof height: h : 8 + (4014)tan(2o") : I 1.64

ft (3.548 m); Kh 0.85 (Table 6-3 of Standard); Kzt l; Qn

(0.00256X0.85)(1.00X100)2(1.00) : 2t.76 psf (1041.87 N) (see Eq. t7.2.1);internal pressure : (21.76X+0.18) : +3.92 psf (187'69 Pa) (see Table6-4 of Standard).

Main Wind Force Resisting SYstem

WAt-t-S (scc Eq. 11 .2.3 of this chapter, and Table 6-1 and Fig. 6-3 of Stan-dard)

Wind Normat to Ridge LIB : 40160 : 0.67 (see Fig. 6-3 of Standard; l, :horizontal dimension of building parallel to wind direction; B : horizontaldimension of building normal to wind direction)

Windward wall: Cz : 0.8Leeward wall: Cp : -0.5Side wall: Cp : -O.l

G : 0.85

Windward Wall

Extemal pressure: (21.76X0.85X0.8) : 14.80 psf (708.63 Pa).

Netpressure: 14.80 - (-3.92) : 18.12 psf (896'32 Pa) (negative internalPressure).

14.80 - (3.92) : 10.88 psf (520-94 Pa) (positive internalpressure).

Leeward Wall

Extcrnal prcssure: (21.76X0.85X-0.5) : -9.25 psf (442.89 Pa).

Net prcssure: 9.25 - (-3.92) : -5.33 psf (-255.2 Pa) (negativeintemal pressure)

-9.25 - (3.92) : -13.17 psf (-630.58 Pa) (positivcinternal pressure).

Side Wall

External pressure: (21.76X0.85X-0.7) : - 12.95 psf ( -620.0-5 Pa)-

Net pressure -12-95 - (-3.92) : -9'03 psl' (-423'3(r l)a) (rrcgalivcin(crnal prossurc).

tlr rAt (rit AltnlIxnMt,r t 5ll5

12.95 1\ ()2) l(r.ti/ |sl ( /()l ,lO l':r) (lxrsrlivr'inlr:rrurl llressrr le ).

Wind Parallel to Ridge L/l] .,10/(10 l5

Windward Wall Satnc us :rtxlvc (up to rirlgc lervc:l ).

Leeward Wall

External pressure: (21.76X0.85X-0.4) : -7 .40 psf (-354.31 Pa).Net pressure: -1.40 - (-3.92): -3.48 psf (-166.6 Pa) (negative

internal pressure).-1.4O - (3.92) : - 11.32 psf (-542.0 Pa) (positive in-

temal pressure).

Side Wall Same as above.

ROOF See Eq. 17 .3 of this chapter, and Table 6-1 and Fig. 6-3 of Standard.

Wind Normalto Ridge

h t1.64 : 0.291L40Windward Slope

Cp: -0'3 +(-0.1x0.291 - o.2s) : -0.3160.5 - 0.25

(0.2)(0.5 - o.zer)C,:

Leeward Slope

0.5 - 0.25: 0.161

Cr,: *0.6

Windward Slope

Ilxtcrnal pressurc: (21.7(r)((l l{5)( O. } l{r) .5.ti-l psl ( 27().(r2 l'rr).Not prcssurc: .5.tt-l ( I ().)) I r)l psl ( ()l () P:r) (rrt'1';rtivt'irr

lcttlrl ptr':.r.trt')5.l,l,l ( I r).') r) /(r;r:,1 I .l(tl I l':t) (lrr,:,tttvt.rnlt,llr;rll)t1's:iull')

lixlrlrrlrl l)rl'ssulr': (.tl /t,lt(l li'rttl) lrrlt 1 {)t) P:,1 (l.l/ t)i l';r)

586 stnNDnnD t,trovtstoNs t()n wtNt) l()nt )tN(i

Net press: 3.09 - (-3.92) : 1.Ol psl' (335.64 Pa) (ncgativcpressure).

3.09 - (3.92) : -0.83 psf (-39.74 Pa) (positivcpressure).

f l l I nl I lll n lll ,ll liAMl'l I '-ttll

ltor h 2lt,

Ncl pross. : 9'25 ( l()2) 5 I I psl { .t1\ ,}O l';r)(ncrglrl ivc irrlent;r I l)t'(.r+ilu r )

: __9.25 1.1.e2) 1.1. l7 psl ( (r-l0.-5tl l'rr)(positivr: intcrrtal pre:ssurc)

For >2h,

Net press. : -5.55 - (-3.92) : -1.63 psf (-78.04 Pa)

(negative intemal pressure)

-5.55 - (3.32) : -9.47 psf (-453.43 Pa)

(positive internal pressure)

Components and Cladding

WALL COMPONENT In the calculations that follow the component coverstwo pressure zones. The areas of the component in each zone and the total areaof the component are

Zone 4: A : 40 sq ft (3.72 m2)Zone 5: A : lO sq ft (0.93 m2)Total area: 50 sq ft (4.65 m2)

As was pointed out in Sect. 17.2.2, if a component is located in more thanone pressure zone, the pressure on the component is calculated by the weightingrnean method, which is used below.

ExternalPressure Coefficients: Based on total 4.65 m2 (50-sq ft area)

Zone 4: GCp : 0.877 and -0.978*Zone 5: GCo :0.877 and -1.152Zone 4:

Neg. ext. press. : (21.16)(-0.978): -21.28 psf (-1018.89 Pa)Net press. : 21.28 - (-3.92) : -11.36 psl'(-tt31.20 Pa)

(ncgativc internal prcssurc)

r'f'ltc virlrrc ('t, - 0.81'l is obl;rirrcrl lry int('tl)i)l:tti()tr its lirllows. lirr',4 l0 li', (,, I O. lor.4 -5(X) llr, (;, 0.7 (lrip.(r 5 ol llrt Sl;rnrlrrlrl). lirrrn llrt'st'lw() lx)inls il is lrrssilrk.lo ol)l;lnlltt'v:tlttt'srr:urtl/rttllltt'littcrln(.11 | /, (r, llrclt'srrllsrrrr'rr { o t/llrr(5(X)t lr{l{})ll.lt I | {{) I lln(lO)l/lrrt5(}t. l, lot I \(} lt'. ( i, O.1{//

iutcrrrll

intcrnal

Leeward Slope

External pressure: (21.76X0.85X-0.6) : -11.10 psf (-531 .47 Pa).Net pressure: - 11.1 - (-3.92) : -7 .18 psf (-343.8 Pa) (negative

internal pressure).-ll.l - (3.92) : -15.o2 psf (-719.16 Pa) (positive

intcrnal prcssure).

Wind Parallelto Ridge

h -ll'&:0.194L60Horizontal Distance from Windward Edge

For 0-hl2 (0-5.82 ft; O-1.77 m), Cp : -0.9,Ext. press. : (21.16)(0.85X-0.9) : -16.65 psf (-791.73 Pa)

For hl2-h (5.82 - 71.64 ft:' 1.71-3.55 m), Cp: -0.9,Ext. press. : - 16.65 psf (-797 .73 Pa)

For h-2h (n.e - 23.28 ft; 3.55 - 7.10 m) Q : 0.5,

Ext. press. : (21.76)(0.85X-0.5) : -9.25 psf (*443.18 Pa)

For >2h (>23.28 ft:' >7.10 m),

Ext. prcss. : (21.76)(0.85)(-0.3) : -5.55 psf (-265.73 Pa)

For 0-hl2

Net press. : -16.65 - (-3.92) : -12.73 psf (-609.52 Pa)

(negative internal pressure)

: -16.65 - (3.92): -20.51 psf (-948.90 Pa)

(positive internal pressurc)

For hl2-h, santc as abovc

&*

Pos. ext. press.Net press.

SIANDARI.) PROVISIONS FON WINI) IONI)IN(i

*21.28 - (3.92) : -25.20 psl'(* 1206.-5tt Pa)(positive intemal pressure)

(2t.76;)(0.877) : 19.08 psf (913.56 Pa)19.08 - (-3.92) : 23.0 psf (1101.25 Pa)

(negative internal pressure)19.08 - (3.92) :15.16 psf (725.86Pa)

(positive intemal pressure)

(2r.16)(-r.tsz) : *25.07 psf-25.07 - (-3.92) : -21.15 psf

(negative internal pressure)-25.0t - (3.92) : -28.99 psf

(positive internal pressure)(21.16)(0.877) : 19.08 psf (same as for zone 4)19.08 - (-3.92) :23.0 psf (1101.25 Pa)

Zone 5:

Neg. ext. press.Net press.

Pos. ext. press.Net press.

(negative internal pressure): 19.08 - (3.92) : 15.16 psf (725.86 Pa)(positive internal pressure)

Average Negative Pressure

{(40)(-2s.2) + (10)(-28.99)}/s0 : -25.96 psf (-1242-97 Pa)

TABLE 17.3.1. Building and Wind CharacteristicsNotat i on

St anda rd Program Value

BasicwindspeedExposure categoryCl ass i f icat ion categorylmportance factorDimension parallel to

ridgeDimensron normal to ridgeEave he i ghtAngleof roof planePa rapet he i ghtOve rhangsMean roof heightGust effect factorVe I oc i ty p ressu re exposu re

coefficientVe I oc i ty pressu re

I

;

;G

exposu reclassificationimportance-factorridnoreave_he i ghtthetaoa rao ho-dimh

100 mphcI

1

60 f t40 f t

B f t20"0 f t0 f t

11 6 f t0. 85

0852l-76 rssl

Kh khqh qh

tir I ul N(;r s 589

'l'Alll,lt 17.3.2. lltxtl'ltrt'sstlt'es: Mulrt Wirrrl-lrirrct. llrsislhlg S.yslcrn lirr (labled RoofWi tttJ Norrrra I to llrrlpio

Net Pressu res

External lnternalp ressu res p ressu res

(psf) (psf)

Wi th Pos i t i ve Wi th Negat i velnterna I lnterna I

Pressu re Pressu re(psf) (psf)W r ndwa rd

I cewa rd

3.1-5.8

-11.1

-0.8-9 .8

-15.0

7.O-1 .9

-7 .2

3.9-3 .93.9

-3.9WindParallel toRidge

llor i zonta l Di stance Externa lI rom W i ndwa rd Edge p ressu res(ft) (pst)

Net Pressu resWith Positive With Negativelnternal lnternal lnternal

pressu res Pressu re pressu re(psf) (psf) (psf)

0-5.85.8-1 1 .6

tI .6-23.2

>23.2

-16.6-16.6-9.2-5.5

3.9-3.93.9

-3 .93.9-3.93.9

-3 .9

-20.6-20.6-L3 .2

-9.5

*L2 .7

-12 .7

-5.3-1.6

N')lc lmportance factor. / : l.

Average Positive Pressure

{(40)(23.0) + (10)(23.0)}/s0 : 23.00 psf (1101.25 pa).

'f irbles l7 .3.l and l7 .3.2 show typical input listing and outputs (based on Fig.6-3 of the ASCE 7-95 Standard) of the compurer program [17-S].x

Ilir print a record of a scssiott or pltl lltcrt:ol, :rlicr starting the session the user should click''Slop," "File," "New," "l)illop.," "(.)ucstiorrs/Answcrs and Advice," "Consult," ..Cgnlinuc('onsultation." Bcfrlrc cntl ol st'ssiol tlitk "slop," "l;ilc," "Print." To print a filc (c.g., {ilc('(' WAt,t,S.TXT), wltcrrr irskctl "S;rvt.'l';rlrk.s ilt lilt.('(' WAl.l.S.'fX1'1" itrrswcr Ycs. click''Slop" and "Opcn," cnk:r ('(' WAI lS'l \'l' irr hililrlrlilrtctl spircc rrrllcnrcllh "l;iL' Nrrrrrr.,";rrrtl click "OK," thcn "liik"' rtttrl "l'ttttl " l'or rrrurt'rlt'lirils scc instrrrclions lirr llrc rrst.r irr llrt'/rlxxrl lltc I)isk scctiolr arxl llrc l{r'lrl rrrr, ltk.ol rlrcl.r'ilr.

*titnNt)Al il) I'lr()vll;l()Nl; t()l I wlNl) l()nl )lN(i

REFERENCES

17-1 ASCE 7-95 Standard, American Society of Civil Engineers, New York, 1995.

l1-2 E. Simiu, J. H. Garrett, and K. A. Reed, "Development of computer-BasctlModels of Standards and Attendant Knowledge-Base and Procedural Systems,"in Structural Engineering in Natural Hazards Mitigation, Structural Congress'93, April 19-21, 1993,Irvine, CA, pp. 841-846.

1'7-3 T. Stathopoulos and H. Wu, "Knowledge-Based Wind Loading for EnvelopcDesign," t. Wind Eng. Ind. Aerod.,53 (1994), 177-188.

17-4 M. Kasperski, "Aerodynamics of Low-Rise Buildings and Codification," J.Wind Eng. Ind. Aerod., 50 (1993), 253-262.

l7 5 M. M. Schcchter, E. Schechter, and E. Simiu, Developmental Computer-basetlvcrsion ot' ASCE 7,95 Standard Provisions for wind Loads, NIST TechnicalNotc 1415. National Institute of Standards and Technology, Gaithersburg, MD,I 995.

17-6 E. Simiu, "Wind Climate and Failure Risks," J. Struct. Div'' ASCE, 102(1976), 1103-1707.

l7-7 M. Kasperski, H. H. Koss, and J. D. Holmes, "Limit State Design of Low-Rise Portals Using Wind-Tunnel Tests," Wind Engineerireg, Proceedings, NinthInternational Conference on Wind Engineering, Vol. 3, pp. 1243-1254, WileyEastem Ltd., New Delhi.

ll-g K. C. Mehta, R. D. Marshall, and D. C. Perry, Guide to the Use of the WindLoad Provisions of ASCE 7-88 (formerly ANSI A58. 1), American Society ol'Civil Engineers, New York, 1991.

APPENDIX A1

ELEMENTS OF PROBABILITYTHEORY AND APPLICATIONS

A1-1 INTRODUCTION

Definition and Purpose of Probability Theoryl,ollowing Cramer tAl-ll, probability theory will be defined as a mathematicalrrrodel for the description and interpretation of phenomena showing statisticalrrgularity. In the field of wind engineering, such phenomena include the windrntcnsity at a given location, the turbulent wind speed fluctuations at a point,lhc pressure fluctuations on the surface of a building, or the fluctuating response,rl'a flexible structure subjected to wind loads.

Belative Frequency and Probability of an Event: Randomness('onsider an experiment that can be repeated an indefinite number of times andwhose outcome can be the occurrence or nonoccurrence of an event,4. If, forlrrrgc values of n, the ratio mln, called the relative frequency of event A, differslittlc from some unique limiting value P(,4), the number P(A) is defined as thel,nhability of occurrence of event A. For example, if a fair coin is tossed, therrr(io of the number of heads observed in a very large recorded sequence ofll's (hcads) and T's (tails) woulcl hc close to j so that, in any one toss, thepnrbability of occurrcncc ol' l lrrrirtl worrkl hc l. Consider, however, thc rc-( orllccl scqucnce

lt'l'il't il t il't'il't'il't't'orrsis(irtg ol'tltcnllttinf, Il':;:rrrrl I ". ll. rrr llrr:r :rctlrrt'rrct', llrt'obst'r-vt'rl ()ul(()nr('rrl :r loss is it ltt:irtl, llrt'lrrolr;tlrrltll ol :r lrt';rrl rrr llrr'rrr'xl loss will olrviorrsly rrolt'r. 11nt 2;

t592 I n Mt Nt:; ()t t,n()nnilil ily ilil ()t ty ANt ) nt't't t(;n l()Ni;

Indeed, fbr thc dcfinition ol'probability.just advanccd kr br: nrcrninglirl, itis required that the sequence (S) previously relbrred to satisly thc conclition olrandomness. This condition states that the relative fiequcncy ol-cvcnl ,4 nruslhave the same limiting value in the sequence (S) as in any paftial sequenccsthat might be selected from it in any arbitrary way, the number of tems ineach partial sequence being sufficiently large, and the selection being made inthe absence of any information on the outcomes of the experiment [A1-31. Thchypothesis that limiting values of the relative frequencies exist is confirmed fbra wide variety of random phenomena by a large body of empirical evidence.

A1.2 FUNDAMENTAL RELATIONS

Addition ol ProbabilitiesConsider two events A1 and,42 associated with an experiment. Assume thatthese events are mutually exclusive (i.e., cannot occur at the same time; anexample of mutually exclusive events is, in the case of one die, the throwingof a "five" and of a "six"). The event that either A1 or A2 will occur isdenoted by ,41 U A2. The probability of this event is

P(AtUA):P(A)+P(A2) (Al.l)

The empirical basis of the addition rule, Eq. A1.1, lies in the fact that, if therelative frequency of event Alis mlln and that of event A2is m2ln, the frequencyof either A1 or A2 is (ln1 * m)ln. Equation A1. 1 then follows from the relationbetween frequencies and probabilities, and can obviously be extended to anynumber of mutually exclusive events Ar, Az, . . . , A,.

Exampte For a fair die the probability of throwing a "five" is I and theprobability of throwing a "six" is i. The probability of throwing either a "five"ora"six"isthen*+*:l

Lct the nonoccurrence of event,4 be denoted by A. E,vents A andA arcmutually cxclusivc. Also the event that ,4 either occurs or does not occur isccrtain; that is, its probability is unity:

P(AUA):t (Al.2a)

Equation Al.2a follows immediately from the addition rule (Eq. Al.1) applicclto the events A and A, the probabilities of which are the limiting values of thcrelative frequencies mln and (n * m)/n, respectively. The probability that /does not occur can be written as

P(A)-t-P(A) (A 1 .2b1

n I :' tt,ut)nMt rJtnt ilt In il()Nti 593

Compound and Conditional Probabilities: Tlre Multiplication RuleCttrrsiclcr two cvcnls A;ttttl Il lltirl rrury occru ll llrc s:rrrrt: lirrrc. 'l'lrc pnrbabilily<rl'thc cvcnt that,4 and 1l will occrrr.sirrrrrl(lrrrr:ously is cullcrl lltc ctttrtltttttttdpnbahility of cvcnts A antl Il rntl is tlcnort:dby t'(A o 1i).'l'hc pnrbability ol'cvcnt,4 given that cvcnt B has allcacly occunctl is clcnotcd by p(Alll) unrl isknown as the conditional prolxthilit.y ctl'cvcnt ,4 unclcr thc condition that ovcn(B has occurred. Formally, P(AIB) is dctincd as fbllows [Al-l];

P(AIB) : P@nB)P(B) (Al.3a)

ln Eq. Al.3a it is assumed that P(B) is different from zero. Similarly, whenP(,4) is different from zero.

P(BIA) : P(A a B)P(A )

(A1.3b)

Example In a certain region, records kept for a long time show that in anaverage year 60 days are windy, 200 days are cold, and 50 days are both windyand cold. Let the probability rhat a day will be windy and the probability thara day will be cold be denoted by P(A) and P(B), respectively. If ir is knownthat condition B (i.e., cold weather) prevails, the probability that a day iswindy, or P(AIB), is 50/200 : (50/365)/(200t365).

From Eqs. Al.3a and A1.3b, it follows that

P@nD:P(B)P(A|B): P(A)P(B:A) (A1.4)

lrquation Al.4 is referred to as the multiplication rule of probability theory.

Total Probabilitiesll'the events Br, Bz, ... , B, are mutually exclusive and p(B,) + p(B) +' ' ' + P(8,) : l, the probability of event ,4 is

p(A): p(AlB)p(BJ + p(AlB2)p(B) +... + p(AlB,)p(B,) (Al.s)

l')c;uation Al.5 is rclcrrorl to as the thcorem ol lorul pnhubility.

Example With rcl'crt:rrcc lo (ltc ptcvious cxarnplc:, wc n()w rlcrrolt' llrt. pnrlrrrllility ol'()ccurrcltcc ol wirrrls:rs /'(21 ), thc pnlbtrbili(y ol (x'('un'(.n(.('ol wirrtl:;lqivc:rr (hat:r tliry is t'oltl lrrrrl liivt'rr llrrl it is rrol colrl irs t'(.1lBt) :rrrrl /'{.ll/1 ,;,rcspr:clivrrly, lurtl lltt'Prirlr;rlrlllly llr:rl:r tl:ry is coltl lrrrrl llr:rl il ls rrol t.olrl rr:;/'(/lr) :rrrtl /'(/1,), n'slx'r'trvr'ly l,to11s;,,r1 Al.5 it lirllows llt;tl /'(..1 I 1i( )/.l(X))("1(X)/l(r5) | (lo/l(r\t (t(r',/ t(ri) (il)/t(r5Tw<rcvcnlsAtutdA lilrwhiclrIx; Al.?blrolrlslrn's;ritl ktl\\'((,tnl,l('tttl'nt(u.\'

*594 t:lIMFNIli ()l I'll()llAllll llY llll ()l lY nNl ) nl 'l'l l(ln ll()Nli

Bayes's Rule

If Bt, 82, . . . , B, are /? mutually exclusive events, the conditional pnlbabilityof occurrence of B; given that the event,4 has occurred is

PGIBi)P(Bi) (A1.6)P(BIA) : P(AIB)P(B]) +''' + P(AIB")P(B,)

Equation A1.6 follows immediately from Eqs' A1'3b and A1'4 (in which B is,"pla,edbyBl)andEq.Al.5.Equation,{l'6allowsthecalculationofthepirtnrio, probabilities f1n,1,l1in terms of the prior probabilities P(B), P(B),'.

. . , P!i,,)and the coniiti.,nal probabilities P(A|B), P(A\B), " ' , P(A\B,)'

Exampte on the basis of experience with destructive effects of previous tor-nadoe.s, it was estimated subjectively that the maximum wind speeds in a

rornado were 50 to 70 m/s. It was further estimated, also subjectively, that the

likelihood of the speeds being about 50 m/sec, 60 m/sec, and 70 m/s is P(50): 0.3, P(60) : 0.5, and P(70) : 0.2.* These values are prior probabilities.Subsequently a detailed failure investigation was conducted, according to whichthe speed was 50 m/s. However, associated with the investigation were un-

certainties that were estimated subjectively in terms of conditional probabilities

P(fi|4^.), that is, of probabilities that the speed estimated on the- basis of the

investigaiion is 50 m/i given that the actual speed of the tornado was Z,-"'

The estimated values of P(5012,*") were

P(frlso) : 0.6

P(fi160) : 0.3

P(fi170) : o.l

It follows from Eq. A1.6 that the posterior probabilities, that is, the probabil-ities calculated by taking into account the information due to the failure inves-tigation, are

P(solfr) : P(filso)P(50)P(50lso)P(s0) + P(fi160)P(60) + r(SblzolPfzol

: 0.51

P(6olo :0.43P(7olo : 0.06

*Equating such a sub.jectivcly estimatcd likclihtxxl t() a prrrbilbilily h:rs Plrikrsoltltititl irttltlit:rlioltsthat arc bcyoncl lhc scopc ol'this lcxl.

n I ] llli.il,/\Ml till\t ilt tn ilr ril,, 5!llr

li is sct'rr llurl wltt'r't':rs llrt'llrior'Prrrlrlrlrililit's l;rvrilt'rl tlrr';r:,:,rlrrlrlrolr llr;rl llrr.spcctl wits (r0 rrr/s, trr't'otrlin11 lo llrt't'lrlt'rrl:rlcrl Po..lt.rror Irolr:rlrllrltt.:,. r( l:, ntorr.likcly that thc spcctl wits ottly 50 rrr/s. Il ,.;ltotrlrl lrc rrotr'tl. ol toursr'. llr;rt llrr:,rcsult is usclirl tlnly lo tltc exlcnl llrirl llrr'vrrriorrs srrlr;t'tlrvc r':;lrrrurlr':. lrssrrrrrt.rlin thc calculalions arc c()nccl .

lndependenceIn the example fbllowing Eq. A l.3b thc occurrcncc of winds ancl thc occurrcnccof low temperatures are not independent phenomena. lndeed, in the region inquestion, if the weather is cold, the probability of windiness increases.

Assume now that event ,4 consists of the occurrence of a rainy day inPensacola, Florida, and that event B consists of an increase in the world marketprice of gold. It is reasonable to state that the probability of rain in pensacolais in no way dependent upon whether such a price increase has occurred ornot, Consequently in this case it is natural to assume that

P(AIB) : P(A) (A1.7)

Two events A and B for which Eq. Al.7 holds are called stochastically* in-dependent. By virtue of Eqs. A1.3a and A1.7, an altemative definition ofindependence is

P(AAB):P(A)P(B) (A1.8)

Example The probability that one part of a mechanism will be defective is0.01; for another part, independent of the first, this probability is 0.02. Invirtue of Eq. A1.8, the probability that both parts will be defective is 0.0002.

Three events A, B, and Care said to be (stochastically) independent if andonly if, in addition to Eq. A1.8, the following relations hold:

P@nc):P(A)P(C)P(Bnc):P(B)P(C)

P@nBac):P(A)P(B)P(C) (Al.e)

In general, n events ur-c sirirl to bc inclcpendent if relations similarto Eqs. A1.9lxllcl lor all combinatiorrs ol lwo or nrorc cvcnts.

r'f ltt'wtttrl .tltxltrt.tlit lll(';rlr5 "(oiln({lr'(l $'rllr Lrrrtkrll exllt'rirrrt'rrls lrrrtl Plrlrtrlrilily' lAl ll,;rrrrlil is rltlivt'tl liirtrr llrt'(ilcr'k rrrrr\rri,,/r,rr. ilr(,[ilil], '10 irirrr:rl, stt'k :rllr.l, 1,ilr'sr, r;Urrrri:,r.

*596 Ll I Ml Nls ()l I'll()llnllll llY llll ()llY nNl) nl'l'l l(;n ll()N1;

AI.3RANDoMVARIABLESANDPRoBABILITYDISTRIBUTIoNS

Random Variables: Definition

Let a numerical value be assigned to each of the events that may occur as a

result of an experiment. The resulting set of possible numbers is defined as a

random variable.

Examples

l.Acoinistossed,Thenumberszeroandoneareassignedtotheoutcomehcacls and to thc outcome tails, respectively' The set of numbers zero

ancl one constitutcs a random variable'2..lilcachmcasurementofaquantity,anumberisassignedequaltothe

result ol that measurement. The set of all possible results of the mea-

surements constitutes a random variable'

Random variables are called discrete or continuou^s according as they may

take on values restricted to a finite set of numbers (as in example 1) or any

value on a segment of the real axis (as in example 2). It is customary to denote

random varia-bles by capital Roman letters, for example, X, Y, Z' Specific

values that may be takenon by these random variables are then denoted by the

corresponding lower case letters x, y, z'

Histograms, Probability Density Functions, CumulativeDistribution FunctionsLet the range of continuous random variable X associated with an experiment

be divided into equal intervals AX. Assume that if the experiment is carried

out n times, the number of times that X has assumed values in the given intervals

X, - Xo, Xz - Xr, " ' ,Xi - X,-r, ' ' ' is n1' flz' ' '' ' fli' " ' ' respectively'a'grupti in *hich the nu-t"r, (or frequencies) n; are plotted as in Fig. A1.l

xo x1 x2 x3 x4 xs x6 x1 x8 xs

FI(;tlRlt A l.l. Il istogt'rtrtt

ilANt )( )M Vn lltn ilt I :; ANt) I't t( )ilAnil I I y I rti;il til tt il t( )Nl; 597

is t'irllctl t lti.tltt.4ntttr. lSirrrilrrr p,r'rrplrs rrur.y lrt' plollcrl irr llrc crrsc ol'cliscrctcr-lrntlorrr vrrriirblcs. )

Lct thc onlinltcs ol'thc histogrrrrr in lrig. Al.l bc tliviclcd by nAX. Thercsulting diagram is known ts l.lh'r1ttt'ttt'.t,tlcttsity distributkm IAl-4|. Therelativc frequcncy ol'thc cvcnt X, r ( x < X; is thcn equal to the product ofthe ordinate of the I'rcqucncy clcnsity tlistribution, n;l(nL,X) by the interval AX.Since the arca undcr thc lristograrn is (n, * n2 *... + n,..)LX: nLX,the total area undcr thc l'rcqucncy density distribution diagram is unity.

As AX becomes very small so that it can be written LX : clx, and. as nbecomes very large, the ordinates of the frequency density distribution approachin the limit values denoted by f (x), where x denotes a value that may be takenon by the random variable X. The function /(x) is known as the probabilitydensity function of the random variable X (Fig. Al.2a).It follows from thisdefinition of f(x) that the probability of the event x I X < x + dr is equalto f (x) dx and that

f(x)dx:l

l,l(illlll,l A1.2. (rr) I'lrlrrlrilrly rlt.rrsityIrrrrr lrorr (1,) ('rrrrrrrl:rlivt. rlislrilrtrlron lrrrrtItorr

L

F(x)

-l.l--

598 EL.EMENIs oF PllotlAlllLlrY I lltclllY n Nl) nl'l'l lon Il()Ns;

In the experiment reflected in Fig. A I . I , thc nutnbcr ol' tinrcs that X has

assumed values smaller than X; is equal to the sum n1 I n2 I .. ' * n;'Similarly the probability that X < x, known as the cumulative distrihution.function of the random variable X and denoted by F(x), can be written as

(Al.l0)

that is, the ordinate at X : "r in Fig. Al.2b is equal to the shaded area of Fig.Al.2a.

It follows from Eq. Al.l0 that

F(x) : \' _ro, *

dFtxtf{xt: *:Changes of VariableWe consider here only the change of variable y : (x - a)lb, where a and bare constants. We assume that the cumulative distribution function Flg@) is

known, and we seek the cumulative distribution function Fy(y) and the prob-ability density function"fy(y). We can write

Fx(x) : P(X < x) (A1.12a)

lx-a x-af:pl",'<- -::l tAl.l2b)lb b I

: Fy(y) (Al.12c)

Since Eq. Al.l2c implies that dFy(x) : dFt(y), or fv(x) dx : fvj) dv, itfollows that

Ifxk) : ufvO)

Joint Probability DistributionsLet X and Ybe two continuous random variables, and let/(x, y) dx dy be thcprobability thatx ( X < x -f dx and y < Y < y * dy. Thequantity f(x, y)is called the joint probability density function of the random variables X and Y

(Fig. A1.3). The probability that X < x andY < y is called theioint cumulativcprobability distribution of Xand yand is denoted by F(x, y). From the definitionof f (x, y) dx dy, it follows that

(A1.1l)

(A1.13)

F(x, y) : J" - I'-.r',", v) d.tr: ttv (A r. r4)

l(t

Al :l llnNl)()M v^lllABt ftj nNt) t,n()ilAHil ily l)tslRtBUT|ONS

FIGURE A1.3. Probability density function f (x, y).

and

It follows from Eq.

I-J-f'-,v)dxdv:1A1.14 that

o2FG. v\J \x. y)

dx dy

.fx',) : f* *ro, r, o,

(A1.ls)

If f(x,y) is known, the probability that x < X < x I dr, denoted by.fr@)r/x, is obtained by applying the addition rule to the probabilities /("r, y) dx dyover lhe entire I/ domain:

(A1.16)

ol'X.Finally, thc probahility tlrirl r, - l' - r, I r/y unrlcr thc contli(iorr lhut.r .,-

X < x * r/-r is tlcrlllctl lrv /(rlr) r/r,. 'l'lu' lrrnr.tiorr /'(.vl.r) is krrown irs tlrctrtnrlilitnttl pntlnbilitv tlrtt,t'it\' lrtttttitttt ol l'1iivt.n thrrt X ,r. ll'lir;. Al. Iris uscrl, it lollows tlrirljr

's

&i

*600 ELEMINIs ol I'll()tlAtlll llY llll ()l tY nNl) nl'l'l loAll()Nl;

f(x, v) : f*(x)fr(Y) (Al.l8)

Similar definitions hold for any number of discrete or continuous random vari-ables.

Application: The Basic Problem of Structural Safety

Clonsiclcr a sirnplc structurc subjected to an external load L. Let S denote thestrcngth ol'thc structure, that is, the value of the external load at which failure,clcfincd as thc attainlnent of some speci{ied limit state' occurs. The basic prob-lem of structural safety consists in determining the probability of failure of thestructure if the joint probability distribution F(/' s) : Prob(L ' /, ,S < s) ofthe load Z and strength S is known.

It follows from the definition of the strength s that failure occurs if theinequalityL > Sholds. Theprobabilitythat/ ( L < l-l dland s < S < s

+ is will be denoted by f (t, s) dl ds. The probability of failure will then beobtained by summing up the elemental probabilities f (1, s) dl ds over the entiredomain L, S over which L > S (shaded area in Fig. A1.4):

Al4 l)l :icltll'l()l (; ()t ltnllr{tMvnilt^l|t I ilt ilnvt()t i 60l

Il is rt:itsottitlrlc to ltsstttttc tlurt /, rrrrtl ,! llt' urtlr'lrr'rrtlt.rrl. I'lrt.rr, l)y villu(.()ltrq. A l. lt3.

(A I .20)

(A1.21)

A1.4 DESCRIPTORS OF RANDOM VARIABLE BEHAVIOR

Mean Value, Median, Mode, Variance, Standard Deviation, andCorrelation Coeff icientThe complete description of the behavior of a random variable is provided byits probability distribution (in the case of several variables, by the joint prob-ability distribution). useful if less detailed information is provided by suchwell-known descriptors as the mean value, the median, the mode, the standarddeviation, and, in the case of two variables, the correlation coefficient.

The mean value, also known as the expected value, or the expectation ofthe discrete random variable X, is defined as

/ (-r', v)J (ylx) :

Jrt*t

If X and Y are independent, f(ylr) : /r(y) and

(Al.l7)

(A1.19)

Itl.:;t lr(ltl:(.rtSubstituting Eq. A 1.20 into lr1. A I . lt)

rrrlr rl

", - J,, /i(/) J,, .l',t';t tt: ,il

f-: Jn Irr,tFs() (tt

"r: f at \'of Q, il a,

Figure A1.4 and Eq. A1.19 reflect the assumption that L > 0 and S > 0'

where m is the number of values taken on by x. The counterpart in terms ofrelative frequencies of the quantity E(X) is

ln

E(x) : Z *,f,i: I

(4t.22)

If the random variablecomplete analogy with

rmmL; t Xitli \- ni: /t x - (Al -235n i:t''nX is continuous, the expected value of X is written inEq. A1.22 as

t,;(x) J-" nu, ^ (A't.24)

'l-hc mediun ol'lt cottlittttotts t:rttrlrrrr vlrlilrlrlc X is lhtrl valuc ol'(hc viu'ilrblcwltich crlrrospontls lo lltr'vrtltrt'Iol llrc r'urrrrrl:rlivc rlislrilrrrtiorr lrrnt.liorr.'l'lrcrtttxlc ttl'it cttttlittttotls tiutrIrn viur;rl)l(' .{ r:; tlr;rl vlrlrrc ol'llrr' v;rlilrlrlc llt:rlt'rlt'tt'sltotltls ltr ll1,' t,, ,atnlunt l)olrl ll llr Irolr:rlrrlrly (l(.nsity ltttrt lrort ll rsl,'l(;tlltt,l Al.rl. l)orrririrr ol irrlegrrliott lirt-r':tlt'trlrtliott ol ;ttrrlr;llrility ol llrilrrn

602 rltMt Ntl; ()t t'll()llnllll llY llll ()llYnNl)Al'l'l l(.All()N:;

recalled that Prob(x < X < x + dx) : .l $) ri-r; tho ttttttlc tttity (lrtrs lrt:interpreted as the value of the variable that has the largcst probability ol'<rc-currence in any given trial. The mean value, the median, and thc modc arcreferred to as measures of location.

The expected value of the quantity [x - E(X)]2 is defined as the vuriunccof the random variable X. By virtue of the definition of the expected value (Eq.Al .24), the variance can be written as

(Al.2s)

Thc quantity SD(X) : [Var(X)lr/2 is known as the standard deviation ofthe random variable X. The ratio SD(X)/E(X) is referred to as the cofficientof variation of X. The variance, the standard deviation, and the coefficient ofvariation are useful measures of the scatter (or dispersion) of the random vari-able about its mean.

The correlation cofficier?/ of two continuous random variables X and Y isdefined by the relation

Con(X, Y) : fi- J:- tx - E(x)l f.y - E(Y)lf @, y) dr dy (Ar.26)sD(x)sD(r)

The correlation coefficient is similarly defined if the variables are discrete. Itcan be shown that

-1 < Con(X, Y) < | (At.21)

lt can be easily shown that if the two random variables are linearly related:

Y:a-fbX (A1.28)

then

Corr(X, Y) : +t (At.2e)

The sign in the right member of Eq. A1.29 is the same as that of the coefficicntb in Eq. A1.28. It can be proved that, conversely, Eq. A1.29 implies Et1.A1.28. The correlation coefficient may thus be viewed as an index of the extctrtto which two variables are linearly related.

It is noted that if X and Y are independent, then Corr(X, Y) : 0. 'l'hisfollows immediately from Eqs. A1.26, Al.l8, and A1.24. Howcvcr. thc rclation Corr(X, Y) : 0 does not nccessarily imply thc inclcpcnclcncc ol'X arttlY lAt-41.

n t,, (;t riMt ilttc, l'()t:;: ;()N. N()t tN/nt nNl I l()(,tJ{rt tN,l/\l trt:,ililtiUil()N:l 603

41.5 GEOMETRIC, POISSON, NORMAL, ANDLOGNORMAL DISTRIBUTIONS

The Geometric DistributionConsideran cxpcrimcnt ()l tltc lylrc krrowil irs lfurtttttrlli triuls in which (l) thconly possible outcotllcs rttc tltc ()('crrrrcncc :rntl thc n()noccurrence of an eventA, (2) the probability p .r' r:vcrrt .,1 in any onc trial is constant, and (3) theoutcomes of thc trials arc irrdopcndcnt of each other.

Let the random variablc N be equal to the number of the trial in which theevent ,4 occurs for the first time. The probability p(n) that event .4 will firstoccur on the nth trial is equal to the probability that event A will not occur oneach of the first n - I trials and wiil occur on the nth trial. Since the probabilityof nonoccurrence of event,4 in one trial is 1 - r, (Eq. A1.2) and since the ntrials are independent, it follows from the multipiicatibn rule (Eq. A1.g)

p(n) : (1 - p)'-'t, (n : I,2, 3, . . .) (A1.30)

This probability distribution is known as the geometric distribution with pa_rameter p.

The probability P(n) that event Awill occurat least once in n trials can befound in the following manner. The probability that event A wlll not occur inn trials is (l - p)'. The probability that it will occur at least once is therefore

P(n):1-(l-p)" (Al.3l)The expected value of N is, by virtue of Eq. Ar.z2, in which Eq. A1.30 is

used,@

N: I ntt - p)n 'pn: I

The sum of this series can be shown to be

N:rt, (A 1.33)

Var(X):E{Ix-E(X)l'\r-- J _ lx - E(Xtl2f(xtdr

{'ottsitkrr lt clitss ol t'vt.rrl:,. r.;rr lr ol rr.lrr, lr rrr;r1ollters :rtttl willt t'r1rr;tl ltlur.lrlt,,,rrl ;rl iriry l!1r,. r,l

(A1.32)

{rr r ur lt(l{'li(.rtrlt-lrlly ol llrt.irulltlr'tvitlo t / A

The quantity N is referred to as the return period, or the meen recurrenceintervel, of event ,4.

Example Foradic, thc P*rbrrhility thar ir "lirur" occurs ir r, : *. If thc t<ltalnunrber of trials is larl-ic, it rrur.y lrr. t'xpeclctl llurl. irr lhc: krng, niir, u ..lirrp"will appear on thc av(:tirll('orrr't' irr /V 11,1, (r tr.irrls.

The Poisson Distributiorr

604 I tt Mt Ntl; ()l I'll()llnllll llY lllt ()l lY nNl ) nl'l 'l l(,n ll()N:;

random variable is definccl, which consists ol'thc nuttthcr N ol'cvctt(s tlrlrt willoccur during an arbitrary time interval t : lz - lr Qr > 0, /r < l, '< 7'). l'ctp(n, r) denote the probability that rz events will occurduring thc intcrval z. Il'it is assumed that p(n, z) is not influenced by the occulrence of any nutnbol'of events at times outside this interval, it can be proved [Al-4] that

rt i , (i(li,

41.6 EXTREME VALUE DISTRIBUTIONS

('llrssiclrl cxll-cll)('v;rlrtt: lltt'ot'y is lrrrsctl on llrt';rn;rlysl, ol tl:rl:r r.onirslrrrl, 'lthc lalgcst vitluc in c:trclt ol lr nrrrrrlrt'r ol lr:rsrt t.orrrP;rr:rlrlt.st.(s t.;rllr.tl t.ltt,(.1t,\(c.9., a scl consistirrg ol lr yt'tl'ol t.t.orrl. ot ol :r s;ltrplr.ol tl;rl;r ol 1,,rvt.rr srzt.;in wind cngillccring, it lttts lrt'crr t'trslorruuy lo tlt'lrrrt'r'lxrt'lrs lr.y t.:rlt.rrtlirr.ycil.s).For indepcntlcnt, itle:nlit'irlly tlistrilrrrtctl vrrrilrlcs witlr crurrLrl:rlivc tlislr.ilrrrliorrl'unction F, tho clistribtrlirlrr ol llrr: lrrlgcst ol'lr scl ol'rr valucs is sirnply ,iq'".with properch<licc ol'lhc conslun(s rr,, and /.r,,, and firr rcasonable functions F,F"(an * b,,x) convcrgcs to a lirniting distribution, known as the asymptoticdistribution. A notablc result of the theory is that there exist only three typesof asymptotic distributions of the largest values known, in order of d".r"uringtail length, as the Fr6cher (or Fisher-Tippett Type II), Gumbel (Type I), andreverse Weibull (Type III) distributions. Of these distributions only the reverseWeibull has limited upper tail tAt-25, At-261.

In contrast to classical theory, more recent theory makes it possible to ana-lyze all data exceeding a specified threshold, regardless of whether they arethe largest in the respective sets. An asymptotic distribution-the generalizedPareto distribution (GPD)-has been developed using the fact that exceedancesof a su{ficiently high threshold are rare events to which the Poisson distributionapplies. The expression for the GPD is

G(y):Prob[}Z<]l:l-

(A1.36a1

llquation Al.36a can be used to represent the conditional cumulative distri-bution of the excess Y : x - u of the variate X over the threshold u, givenX > ufora sufficiently large. The tail length parameters c > 0, c : 0, andr' ( 0 correspond, respectively, to Fr6chet, Gumbel, and reverse weibulltlomains of attraction. For c : 0 the expression between braces is understoodin a limiting sense as the exponential exp(-yla) [Al-25, p. Zl5].

The threshold approach can increase the size of the sample being analyzed,.('onsider, for example, two successive years in which the respective largestwind speeds were 30 m/s and 45 m/s, and assume that in the second yearwinds with speeds of 3l m/s, 37 rn/s, 4r m/s, and 44 m/s also rccordcd.Iitrrtherassume all thosc sptrctls wcrr rccotrlcrl irl tlatcs scparalcd by suflicicntlylong intervals (c.g., lorr1lt.r'lllrrr lr wt'ck);rrrrl t':rrr llrcn'lil.c lrc virrwcrl lrs irrtlcpcndcnt. Ftlr thc l)lttl)()s("s ol llrrr'slrolrl llrt'ory (lrt. lw() y(';us worrltl srr;rplysix cla(a ptlinls. 'l'ltc t'l;tssit:rl {ltt'oty ivorrlrl nr;rk(' us(' ol orrly lrvo rllrllr l16irrts.lrl lirct il rtrlry ltc lrtl',rrt'tl llr;rt. lrl i lroo:,1r1,. :r ,,orlr.\\.1!:rl I,ru,,., 11,,,..',olrl. llrt.rtttlttbct ttl'tllttlt;xrittls tr:,t'tl lo r'.,lrnr:rlr.ll!(' li;rt:t!lr'lr-r,, ol llrr,(il'l) r.oultl lrr.t0ltgllli'1',1r;t I:il1',t'r.llutrr r'r\ rr 'rrr r r,rrrr;rt, llrrrrr \, r. rl llrl llrrr':,lrrlrl r:, lorr

(Xz)" \ -p(n.'rl : i, "' (n : 0, 1,2,3, . . .) (,{1.34)

If Eqs. A1.24 and Al .25 areused, it is found that the expected value and thcvariance of n are both equal to Xr. Since Xr is the expected number of eventsoccurring during timc r, the parameter X is called the average rate of arrivulol' thc proccss and rcprcsents thc cxpected number of events per unit of time.

Thc appticability of Poisson's distribution may be illustrated in connectionwith the question of the incidence ol'telephone calls in a telephone exchangctAl-51 . Consider an interval of, say, a quarter of an hour during which thcaverage rate of arrival of calls is constant. During any subinterval the incidenceof a number n of calls is as likely as during any other equal subinterval. Inaddition it may be assumed that individual calls are independent of each other.Therefore Eq. 41.34 applies to any time interval lying within the quarter ol'an hour.

Normal and Lognormal DistributionsConsider a random variable X which consists of a sum of small, independentcontributions X,, Xz, . . . , Xn. It can be proved tAl-ll that under very generalconditions, if n is large, the probability density function of X is

/ crta)0,\t*;[[' . (?)]-"'],

)'o(Al.3s)

where trr" : E(X) and o] : Var(X) are the mean value and the variance of X,respectively. This statement is known as the central limit theorerr. The distri-bution rcprcsented by Eq. A1.35 is called normal or Gaussian.It can be shownthat thc probability distribution of a linear function of a normally distributctlvariable is normal. Also the sum of two or more independent normally distrib-uted variables is normally distributed.

Normal distributions are used in a wide variety of physical and engineeringapplications, for example, the description of errors in measurements. At tlrcsame time, it should be carefully noted that many phenomena may nol bcnormally distributed, for example, the extreme wind speeds occurring at atrygiven geographical location.

If the distribution of the variable Z : log X is norrrutl, (lrc tlistl'ibution olthe variable X is said ro bc ltsgrutrmul.

p1 : -!- "*p(-t* - P')2\

:/2ro^ \ -t{ )

606 ELEMENIS of l)ltolJAllltllY llll ()llY nNl) nl'l'llon ll()Nli

low, the consequent increase in the clata satrtplc will not itrtpnlvo thc clturlily

oftheestimates:asignificantbiaswillsetinowingtothestrongvitllaLionol'ir," ur..r_ption that the threshold is asymptotically high.

Giventhe*"u,,1'(r)undstandarddeuiatio''s(Y)ofthevariateY|^|-21|.

nt{, lxllilMl v^l Ut trt,.illilililto| t., lilll

In Eq. A1.39, p itttcl rr tttc t'trli:t-tctl lo:ts lltt'lot'rrtiorr rrntl llrt'st:rlt.plulurrt.lr,r,respectivcly.* ll can bc slrowrr, rrsing llt;s. n l.14 luul Al.15. tlurt (lre nrclrrrvalue and thc standarcl rlcviltion ol'X arc

@

@

@

':;E(r)['. [#i'],::[ '-Lffil']

Fx(x) : [Fy(x)]'

(Al.36b)

(A1.36c)

(A1.38)

E(X):p+0.5772o7fSD(X) - - o

v6

(A1.40)

(Al.4l)Type I and Type ll Distributions of the Largest Values' Mean

Recurrence lntervals

Let thc variablc X be the maximum of n independent random variables Y" Yz'

...,Y,,tA1-61. Sincetheinequality X < x implies Yi 3 xforalli(i:1'2'

. . . , n), it follows that

F(X=x): Prob(Yt3 x,Y23 x' "' 'Y'=x) (A1'37a)

: Fv,(x)Fvr(x) ' ' 'Fv,,(x) (A1'37b)

where, to obtain L,q. A1.37b from Eq' Al'37a' the generalized form of Eq'

Al.8wasused.Thep-uuuiti.i",Fy,(y)arereferredtoastheunderlying(orthe initial) distributions of the variabies Ii. The latter are said to constitute the

;-;;;;;;whtion from which the largest values X have been extracted' In the

particular case in wtrictr att the variables Y1 have the same probability distri-

bution Fv(Y), Eq. A1.36 becomes

The cumulative distribution function for the Type II distribution of the lctrgestvalues (also referred to as the Type II Extreme Value distibution, or the gen-eralized Frichet distribution) is

Frr(x) (41.42)

7 )0where ;.r, o, and 7 are the location, the scale, and the shape (or tail length)parameter, respectively IAl-81. In the particular case p : 0, Eq. A1.42 isreferred to as the Fr6chet (as opposed to generalized Fr6chet) distribution.

Equations A1.39 and A1.42 may be inverted to yield the percent pointfunction, that is, the value r of the random variable that corresponds to anygiven value of the cumulative distribution function. In the case of the Type Idistribution,

x(F,;: tt - o ln(-ln17,)

whereas for the Type II distribution

(A1.43)

x(Frr) : p -t o(-ln Fi)-'', (At.44)

It is convenient to dcnotc thc curnulativc distribution function value F, or F,'by p and x(Fs) orx(Fl1) by (i1(1t).'l'hcn. lirr thc Type I distribution,

(it(p) ;r rr 11( l' 7,l) (A I 45:r)

'r'As sllttwtt irr li;s. A l..lO ;rrrrl A I 'l I . tlrr",r' pru,trrr'lr'r., ;ut rrrrl llrr. (.{lx.( l:tliol lurrl llrt. r,l;rrrrl;rrrlrk'vi:rliort ol .Y

[n the case in which they are unlimited to the right, the initial variables Y are

said to have distributionloithe exponential rype if their cumulative distribution

iun",ion, converge (with increasing y) toward unity at least as^fast as an ex-

p""""ii"f functio"n; the initial variables 1, are said to be of the Cauchy rype if

lim [l - F(Y)]Yr : A (A > O:k > 0))-@

As the number n becomes very large, the distributions F1,'@) of the largest

;; approach limits known us tfte Type I and the Type II distributions ac-

;il;g;: the initial oi*triuution* ur" of rhe exponential and of the cauchv

type, respectivelY [A1-4, A1-7]'The cumulative distribution iunction for the Type I distributfun o.l'lhc lttrgc'tt

values(also referred to as the Type I Extremc vat.uc distribution, or thc (itrrrrht:l

distribution) is

608 ELEMENIS oI l'lloBntllLl lY llll ()l tY nNl) nl'l'l l(in ll()N1;

and for the TYPe Il distribution,

Gx@) : pt * o(-ln P)-'''

(Al.46b)

Relations Between Type I and Type ll Extreme Value Distributions

Let the Type II distribution be written as

4r0) : exp{-[(Y - prJlorr]-7] (A1.47)

(In the present context it is convenient to denote the location and scale param-

"t",. of itr" Type II distribution by px and on, respectively) . If the transformation

! - ltil: exp-t

is applied to Eq. Al.4'1, the expression obtained is a Type I distribution withparameters

1t" : ln og (A1.48)

(A1.4e)

It is now shown tA1-121 that as .y approaches infinity, a Type II distributionapproaches a TYPe I distribution.

Consider the distribution of the standardized variate

n r {; I xilu Mr vnr ur t)r:iiliiltuil()Nl; G{l{l

wltcrc krc (X)rrrrtl scrrlc (X) ilro lllci-rsurcs ol. ltlclrlion urrd scalc, reslx't'lrvt'ly.ol'tlrc tlistribttliort ol'X. [Jxarnplcs of tneasurcs ol'locaticln ola runtlonr virrrirlrlt.X arc its cxpcctcd value E(X) and its median Gx(0 5) Examplcs ol'rnt'lsurt.sof scale of a random variable X are its standard deviation SD(X), its intc:rrlrrrr rl rledifference 65s : Gx(0.75) - GxQ.25), and its95% difference 6e-s : (ix(0.t)'/.s)- Gx(0.025).

The percent point function Gz(D is given by

Gz@) : Gx(p) - loc (X) (0<p<l) (Al.sl)scale (X)

With no loss of generality, a reduced variate with pr : 0 and o : I may beused in the demonstration. Substituting Eq. Al.45a with p : O and o : I intoEq. 4.1.51 and choosing, for simplicity, loc (X) : Gx(0.5) and scale (X) :6nr,

Gz@) = t-ln(p)l-r/'v - [-ln(0.5)] "" (0<pcl) (A1.s2)[-ln(0.975)f '', - [-ln(0.025;1-'/r

As 7 - oo, this expression becomes indeterminate. However, application ofL'Hospital's rule yields, after simplification,

Gz@) : -ln[-ln(p)] - { -ln[-ln(O.s)]] (0<p<1) (Al.s3)-ln[-ln(0.97s)] - { -ln[-ln(0.02s)]]

As can be seen from Eqs. Al .45a, the terms in the numerator and denominatorof Eq. A1.53 are, respectively, the percent point function, the median, and the95% difference of the reduced variate for the Type I distribution. It has thusbeen demonstrated that, as 1 approaches infinity, a standardized Type II variateapproaches a standardized Type I variate and, hence, a Type II distributionasymptotically approaches a Type I distribution.

Type I Distributions: Mode of the Largest Value from a Sample ofSize n as an Approximation of the Percent Point FunctionG"[1/(1 - n)]LctZbe the largest ola sct clf n values of a random variable X, each of whichhas a Type I Extrcrnc Valuc clistribution (Eq. A1.39). The cumulative distri-bution function ol'this ltrlgcst vlrlrrc is

FromthedeflnitionofpandEq.Al'2,itfollowsthatProb(X>x):l-p i,tthe random variabll X represent the extreme annual wind speed at some

!i*n to.ution. Each year maf then be viewed as a trial in which the event

inut ,rr" wind speed X will exceed some value x has the probability of occurrence

|-p.ByvirtueofEq,Al.33,themeanrecurTenceintervalofthiseventis

(A I .45b)

(A1.46a)

Thus the wincl spced x corresponcling to a mean reculrence interval N is equal

to the value of the percent point function of X corresponding to

IAI -

-

l-p

1P:1-N

I^l

X - loc (X)7-scalc (X)

(A I .50) Ii,,l;l l/',(.:)1" cxpl -rr cxp( n)l (Ar.s4)

*}610 t LLMt N|l:; ()t I't t()ilAtril ily ilil ()ny nNr) At,r,l t(]AiloNti

where

-oo<z<@z- po

0 <o(mThe corresponding probability density function is

If,kl::nexpf-ne-* -rlo

Thc root ol'thc cquation

(A l .5-5)

(Al.s6)

(Ar.s7)

(A1.60)

llil Ml v^t t,l I rl,, t ltlllU I l( 'f'l:; 6 I I

Ilt liq. Al.(tl, (ir(I l/rr) is lltc vlrlrrc ol .\ totrt':rlxurtlilrp, lo llrt.rrrt'irrrlccurrcn(:L: irttcrvaI lt.

Il can bc vcrilicrl llurl lirr rr srrllrt'icrrlly l;rl1',r'- siry. l lO,

r,, l t,,( r ')l ,,'( r ) (Ar (,r)| \ il/l \,,/(For examplc, lirr rr -. 20, tlrc right and lcli nrcrrrbcrs ol' Bc;. Al .62 arc cqualto -2.970 and -2.996, rcspcctivcly. For n : 40, thcy are cqual to -3.616and -3.689, respcctively.) It follows therefore that

is, by definition, the mode* of the largest of the set of n values considered.From Eq. A1.57 it follows immediately

(A1.58)

or if Eq. A1.55 is used,

mode(Z):p_ oln! (A1.s9)

Consider now the initial random variable X. Since X has a Type I distribu-tion, its percent point function is

Gx(D: tt" - o ln(-lnp) (A1.43a)

or making use of Eq. A1.45a in which N is the mean recurrence interval,

'lJ,,Q) I

; - 1nexpl-ne-'" - wlfne'- ll :0

",.(' -*) : p-or"[-r''('-+)]ln the parricular case in which N : n

o, (' -;) : tr - or" [-r' (' -;)l

_x, Ie :-n

(A1.63)

Equation A1.63 shows that if X is a random variable with a Type I distribution,the mode of the largest value in a sample of n values of X is very nearly equalto the value of the random variable corresponding to the mean recurrenceinterval n lAl-91.

An interesting experimental verification of this statement is provided by thedata of [Al-ll], which cover a period of 37 years. For example, for the firstfive sets of [Al-11], the values of the largest of the maximum yearly windspeeds recorded in 37 years a-u*, and the values of the estimated 37-year wind,u37, zr? (in mph)

Cairo, Alpena,TatooshIsland Williston, Richmond,

IL MI WA ND VAu^u^ 51 50 84 50 48uzt 52 5l 81 52 50

The probability that the largest of a set of n values of the random variableX with a Type I distribution is contained in a given interval can be easilycalculated using Eq. A1.54. For example, from a 37 -year record of the largestannual wind speeds at Richmond, Virginia, the values of p and o were estimatedto be 36.8 mph and 3.78 mph, respectively [A1-11]. Using these values, theprobability that the largest wind speed Z : V-u* in a set of n : 37 largestannual speeds is contained say, in the interval h,(l + 0.24) : 50 * 12 canbe estimated as follows:

P(38 < Z < 62) - \ ltr(;t,l;. I;,,162) /,'1./(3ll) . 0.9.5 (Al.(r3l).l ru "

Joint Extreme Value Dislribrrtiorre'l'ltc.ioinl'l'ypc I Iixlt'ctrtr'V;rltrt'It,rlr;rlrrlrly rlr:,lrl)lrlron ol lwo t'orrt'llrlcrl v;rrrltblcs X, l'ltits lltc e xl)r('sston

/r\rGrl | - - | = t" - oln : : mode(Z)\ n/ n

(Al.6r)

*It is recalled that the mode of a variablc X is the valuc ol' that vrrriirhlc rrrosl likcly to ott rrr irrany given trial (Secr. n 1.4).

Fxy(r,yr - *p[-L*r( -^7)* ".0(--=,*')1""J

and the correlation coe{icient pxy 2 0 [Al-23]. It can be verified that forprobabilities of interest in structural reliability calculations (e.9., Fxv@, y) )0.99) and for values pxy 3 0.7, saY,

Fxv(x, y) = F.y(x)Fy(y) (Al.64c)

where Fx(x) and Fy(y) are the Type I Extreme Value distributions of X and Y,

respectively, that is, it may be assumed that Xand Yare statistically indepen-dent.

The Reverse Weibull DistributionThe expression for the reverse Weibull distribution is

( | ,, - x l')Frrr(x) :.^pL l';lJ x<trt

*612 t tt Mt Nlli ol l'll()llnllll llY Illl ()llY nNl) nl'l'l lcn ll()N:i

where

m:(l_ p*r)-t''

(A1.65a)

The relations between distribution parameters and the expected value E(X) andstandard deviation s(X) are

s(x) (A1.6sb)

(A1.65c)

o:- tr0 + 2li - Ir(1 + tly)l'\

LL : E(X)

where I is the gamma function tAl-81. Forexample, forE(X) : 50' s(X) :6.25, and^y :2, o : 13.49, and p : 61.96- The tail length parametery is

related to the parameter c in the GPD distributions as follows [Al-28]:

nlli lXllll[ill vAltll I'l'.llllllllll(lr]:; lil:l

or, il wc tlenole (lrt'pt'tt't'rrl lxrurl ltrrrt'liort lry (,',(/r);rtt(l 1"1,1 lry 1,.

(i,l1t'1 lr ttl lrrt1,1;r' (Al ('11)

Note that, lirr Tr , I (i.t:., lirr' prirlr:rlrilily ol cxt't't'rl;trtr't' | 1r 01, {i,17r;: lr, so that p rcprcscrtts tlre Itt:rxittttutt lxrssrlrlt'vltlttt'ol llrt'vrttiittt' t.

Mean Recurrence lnterval lor Epochal and Threshold ApproachesWe first consider the case whcrc the variate X represents thc largcst valuc duringa fixed time interval or epoch. The probability I - p that X ) x may beviewed as a probability of "success" during any one epoch (i.e., in any one"trial"). It follows from Eq. A1.33 and the definition of 1 - p that the meanrecurrenceinterval isN: l/(l - p)(Eq.Al.46a). Sincep: l- l/N,thevariate corresponding to a mean recurrence interval N can be obtained imme-diately from the expression for the percent point function (Eq. A1 .45a, Al.45b,and A1.65f for the Gumbel, Fr6chet, and reverse Weibull distribution, respec-tively). For example, winds for which the probability of exceedance is 0.02lyr have, by definition, a ll(O.O2lyr) : 50 yr mean recurrence interval. Notethat in accordance with the geometric distribution, the probabilities that windswith a 5O-year mean reculrence interval will be exceeded at least once in nyears is P(n):1 - (1 - O.O2)'(Eq. Al.3l). For n:50 years, P(50) =0.63.

We now consider the threshold approach and give expressions for estimatingthe value of the variate corresponding to any mean recurrence interval N 1inyears). Let tr denote the average number of exceedances of the threshold peryear. The average number of "trials" in N years is then \N. The probabilityof "success" in any one trial is l /(\F) (success being defined as the occuffenceof the event X > xor, equivalently, Y : X - u > y : x - a). We thereforeset

Prob(I < y) (A1.66a)

Using Eq. Al.36a.

l- (Al.66b)

'l'herefore

(A I .64a)

(A1.64b)

(Ar.6-sd)

1-1 }.N

lnversion of Eq. Al.65a Yields

* "'(' . i)

It- (,

I ,'ul"' Ill+-1 | :l-:I ,rl XN

lAl-291. 'l'ltc vrtltrt' lrt'iltp' 511111'111 ,.'x(firr) : p - ol-ln1F,',))r/1 (A l.(r-5c)

rlI (XN)'l(A l.(r(rt')

614 I tlMl Nl1; ()l l'li()tlnltll llY llll ()l lY nrJl) nl'l'l l(;n ll()Nli

r(N) .Y I r/

wh()rc n is the threshold value.

A1.7 PROBABILITY THEORY AND STATISTICAL DATA

Goodness of Fitl)ata obtained-or that may be obtained-from actual observations may be

vicwed as observed values of random variables. The behavior of the data is

thcn assumed to be described by models governing the behavior of randomvariables, that is, by such mathematical models as are used in probabilitytheory.

In practical applications two important problems must be dealt with. First'from ihe naturs of the phenomenon being investigated (or on the basis ofobservations), an inference must be made on the probability distribution thatwill adequately describe the behavior of the data. Second, the data must be

used for drawing inferences on the parameters of the distribution or on someof its characteristics, for example, the mean or the standard deviation-

In practice, given a set of observed data, ot a data sample, it is hypothesizedthat iis behavior can be modeled by means of some probability distributionbelieved to be appropriate. This hypothesis must then be tested. Tests incor-porate quantitative measures of the degree of agreement, or goodness of fit,tetween the data and the hypothetical distribution or' conversely' of the degreeto which the data deviate from that distribution. If the measure of this deviationis appropriately small, then the hypothesis will be accepted, and vice-versa.Associated with the testing of a hypothesis is a level of signiJicance that rep-resents the probability of rejecting the hypothesis when it is in fact true. Testscommgnly used in applicaiions,-including the well-known X2 test, are dis-cussccl, lirr cxample, in [A1-1] and [A1-4]. Brief mention is made of thepnrbability plot correlation coefficient test tAl-lQl that has been used in thestrrrly <rl'rhc hchavior of extreme winds [Al-11, Al-12]. The probability plotcorrcluliort cocflicicnt is defined as

D (Xi X)lMtQ) - M(D)IfD: B rx - x)t t (tvt,(o) - u(D)'z1

in which X : D Xiln, M(p) : D Mi(D)ln, n is the sample size, and D is thcprobability distribution being tested. The quantities X; are obtained by a rear-

iung"-"nl of the data set: X1 is the smallest, X2 the second smallest, . . . , Xithelth smallest of the observations in the set. The quantities M,(D) arc ob(ainctlas fbllows. Given a random variablc X with probability tlis(ribtrtiorr 1) arrtl

given a sample size n, it is p<-rssiblc fron-r pnrbabilistic consirlct:tliolts lo tlcrive:

n I /' I'll()llnllll llY llll ol lr nlllr ',lAll',llr Al lrl\l/\ {il!r

rrrirlll:rrrirltt'trlly (lrc rlislrilruliolrs ol lltt'stttltllr'sl. :.r'tutttl "ttt;tllr"'l. .tttrl. tu 1', tl

oral, tlrc illr strrirllt'st vlrlut's ol I irr llr:rl srrrttplt' 'l lrt' tltt;tttltlt,', /1I,(/rl .rri llrr'tnodians ol' caclt ol' lltt:st' tlisllilrrrtiorls

Il'thC Clatlr wCrC gCncflttctl lry tlrt'tllr;l ltlrttlt'tt /). llrr'rr, ;tr.trlt' ll'tlt ;t l.t;lll'tland scalo lact<lr, X; will lrt' :rlrprorrrtrrrlt'ly t't1tt;tl lo llrt'lltt'olt'ttt;rl r';rlrrt". 41,(/))fbratl j so that llrc plo( ol .\, vt'rsrrs It,(t)l (rt'lt'rtt'rl lo;ts ptolt;tlitlrly plol ) rvrllbeapproximatcly lincirr'.'l'lris lirrcluil.y will, rrr ltttlt. tt'sttll lll :t ll('ilr tttttly v;tlttt'of rr. Thus thc bct(or li( ol'tlrc rlisllilrrrliorr /) (o tlrc rl:rlrt lltt't'lrtst'l /1; will lx'to unity.

To test whether thc bchavior ol'a givcn sct ol'oxtrcl'ttc clata is bct(cr dcscribcclby a Type I distribution or by a Type lI distribution with some unknown valueol the tail length parameter 7, the probability plot correlation coelficient rp iscomputed for a large number of extreme value distributions, defined by variousvaluis of 7 suitably spaced from "y : I to 1 : oo (it is recalled that 7 : 6o

corresponds to a Type I distribution). The variable in these distributions iswritte; in standardized form so that for any given set of data the coefficientsrp depend solely upon 7, that is, are independent of the location and scaleparameters y, and o on which therefore no prior assumptions need to be made

iar-rtl. The distribution that best fits the data is that which corresponds tothe largest of the calculated values of r2.

Estimation of Distribution Parameters

From the data of a sample it is, in principle, possible to make inferences onthe parameters of the distribution that describes the behavior of the populationfrom which the data are extracted (or on characteristics of the distribution, e.g',the mean). An estimator may be defined as a function A(XI, X2, ' ' ' , &) ofthe sample values such that & is a reasonable approximation to the unknownvalue a of the distribution parameter (or characteristic). The particular numer-ical value assumed by an estimator in a given case is referred to as an estimate'As a function of random variables &(Xr, Xz, . . . , X,) is itself a randomvariable. This is illustrated by the following example.

Consider the observed sequence of 14 outcomes of an experiment consistingof the lossing of a coin:

HTTTHTHHTHHHTH (A1.67a)

The random numbers associatccl with this cxpcrimcnt are the numbers zero andone, which are assignccl to tht: orrlcorttc ht:atls:tntl lo thc ttttlcornc tails' rcspectively. The data s:rrrrplr'r'ot'tt'slxrtulitrg lo (lrt'obscrvt'tl ()tlt('()lll('ri is lllt'rt

0. I. l. l. o. l. o. (). l. o, o. o. l. () (Al (r/lr)

'l'his strrtrplc is ltsstrrru.tl lo lx'r'rlt;ttlr'rl ltottt.tlt tttlttrtlt'pr)l)lll;llr{)lt llr;rt, rrt llttcitsc ()l'ltn itlt:lrlly l:ril t0lrr, tVrll lr;rVr. ;t !!tr';!lt t';rlttr'. r['ttol(.rl IIr lltt" ,;t:,r' lrv

(A I .((xl)

616 I lt Mt Nl; ()t I't r()tlnlltt ly ltt ()t ry nNt) nt't't t(;n tt()N1;

o, equal to j. n rcasonablc cstiurator lilr thc rncan (y is thc sartrplc tttcutt &'t'

l-d:- LXill i:l

n I /' l'l t{)ilAtilt ilY llil{rt t.r l\ill ).,1n it.,ltr At trl\tA lill

('ottsitlt't it sc(ltl('tt('e ol rt ttttilortrtly rli..;trrlrrrlcrl r;rrrrlorrr rrrlrrlrr.r,. () l,I (i l,2, ... . rr) srrt'lr trs:rrt'lislt'tl irr lAl l.llor;r:, rrr;r1 lr,, 1,r.rr.r;rl1rl lry.pft)ccdutostliscttsst'tl irr lAl .ll,lAl I ll. or lAl l.ll llrt':,t'rrrrrrrll'r:,.r!( \,rr.\r'i.(las pnlbabilitics ol ()ccut'rcrrt't'ot llrt'rl:r{:r .\'( }, ) olrl:rirrt'rl lry llrr. lollorvltl,. tr.rtr,,formation (1,)q. A I.4 l):

X( t; ) ;r o lrr( lrr ), ) (n L()())

From the samplc ol'siz.c n, X(Y) (, - I ,2, . .. , rr), it is 1'rossiblo to obtairrestimates of pr and o (i.e., the distribution parameters) and of Gx@) (the percentpoint function corresponding to any given value of p; see Eq. Al.43a). Since,as was previously indicated, the estimates are random variables, the estimateswill differ, in general, from the known parameters and percent point functionof the underlying distribution. The procedure just described can be repeated alarge number M of times. Then M sets of values fu, tr, and d1g(p) and corre-sponding histograms can be obtained. From those sets it is possible to calculatesummary statistics (e.g., the mean, the variance, the standard deviation) forit", 6, and Gx@).

A Monte Carlo study of the behavior of a random variable with a Type Idistribution conducted for the purpose of predicting extreme wind speeds wasfirst reported in [A1-15]. A similar study, subsequently conducted by the writ-ers, is now summarized. The parameter values of Eq. A1.69 used in this studywere p : 36.8 and o : 3.78 (these values in mph represent estimates of TypeI distributions found in [Al-11] to best fit the annual extreme wind speedsrecorded in Richmond, Virginia, between l9l2 and 1948). Two sets of 100samples each were generated, the size of the samples being n :25 forthe firstset and n : 50 for the second. The main results of the study are listed in TableA1.1. For example: Gx(0.98) : 51.57 (calculated from the underlying distri-bution with parameters p : 36.8, o : 3.78); the mean of the 100 estimatesGx(0.98) based on the samples of size n : 25 is Mean[d76(0.98)] : 52.58;the standard deviation of these estimates is s1G*10.98)l :3.46; the largest ofthe estimated Cr(O.qS) is maxlCrlO.OS;] (1 + 16.6/100) x{Mean1G"(0.98)l} : Mean[Gr(O.98)] + 2.5s1Gr10.98)1. A histogram of theestimates Gx(O.999) for the 100 samples of size n : 50 is shown in Fig. A1.5.

The results of Table Al . I were obtained by fitting a Type I Extreme Valuedistribution to the data samples generated from sequences of random numbersby Eq. A1.69. However, it is conceivable that, because of the random characterof the sampling, the behavior of some of the samples would be bcttcr describcdby Type II Extremc Valuc distributions rather than by a 'l'ypc I clistribtrtion.'Io verify whether this is inclcccl the casc, the probabilily pkrt t'orrclirtion t'rx'l'fic:icnt tcst was applictl (o circlt ol'lhc satnplcs. 'l'lrc: rcsrrlts obl:rirrt'rl, wlrit'lr;rrt.inclcpcndcnt ol'lhc pirt'lrtttr'lt't.s;r iuttl o ol'lhc Lrntlcrlyirrg tlislr.ilrrrlior.:ut slrowrrin'l'ablc A1.2.

As sltowtt irr Sr:cl . 1..). 1rt'lt't'nlrr11t's srrc'lr tts lltosc ol 'l'lrlrlt' A l .) i:rrr l,t.t'tttttPtttt'tl ltt silttilltr'pt'tt't'ttl;t1't':; olrl:rilrt'tl l'torrr llrr.';rrr;rlysls,rl lltt':rsrrrt'tl t',,

(A l .6n)

where n is the sample size (number of observations) and Xr are the observecldata. In the case of the sample consisting of all 14 observations in Al.67b,A : i.If the samples consisting of the first seven and of the last seven obser-vations in Al.67b are used, e : + and d : ], respectively.

As a random variable an estimator & will have a certain probability distri-bution with a nonzcro dispersion about the true value a. Thus, given a sampleol' statistical data, it is not possible to calculate the true value cv of the parametersouglrt. Rathcr, t'onfidcnce intervals can be estimated of which it can be stated,with a spccificcl conlidence level q (level of significance I - q), thatthey willcontain thc unknown value cv.

In ordcr that the confidence interval corresponding to a given confidencelevel q be as narrow as possible, it is desirable that the estimator used be

fficient. Of two different possible estimators dq and &2 of the same parametercv, th" estimator d1 is said to be more efficient if E[(dl - cv)2] < EI(dz *oizl.

Details on procedures for estimating distribution parameters can be found,for example, in [A1-1] and [Al-4] (see also tAl-171 and [A1-22]). The questionof parameter estimation for the Type I Extreme Value distribution-which iswidely used in the study of extreme wind speeds-will be examined subse-quently in this appendix. Before proceeding to this topic it is useful to discussfirst the simulation of the behavior of a Type I Extreme Value distribution bymeans of numerical techniques commonly referred to as Monte Carlo methods.

Monte Carlo Methods. Simulation of a Type I Extreme Value Process

As defined in [,{1-13], Monte Carlo methods comprise that branch of experi-mental mathematics that is concerned with experiments on random numbers.The simulation of the phenomenon of interest is achieved by subjecting avail-ablc scqucnccs of random numbers to appropriate transformations. The newscqucnccs thus obtained may be viewed as data, the sample statistics of whicharc rcprcscntativc of the statistical properties of the phenomenon concerned.Examples of engineering applications of Monte Carlo methods can be foundin lAl-41 and [Al-14].

The simulation of the behavior of a random variable with a given distributionis a simple application of Monte Carlo techniques that is now discussed. It isassumed that the distribution is Extreme Value Type I with given parametcrsp. and o (Eq. A1.39).

*Thc synrbol ^ is rrsctl t() (lcnolc cstirl:rlul v:rlrrc.

*i618 fl tMtNtt; ()l t,r.rorr^tl[ily ililony nNt) Apt,t tc^iloNl; n I / t'il()tlnililily ilil(lt ty nr.Jt I r;tAil:,il(jnt

'l'Alll,l,l A1.2. l'r'rt't.nl:tg(. ol' Srurrplrs li'orrr ul)opulaliorr wilh u 'l'.ypc I l)islrilrulioll 'l'llrl ll't.lhl.tcr l,'illcrl b.y 'l'.y;tr I :urrl 'l'ylx. ll llislrilrrrtiorrs

Strrrrplt'Sizt.

rr .1.5 rr 5( )

51 11

^J6o: -,s7f

[,t :7 - 0.57722a

rrnrn {il!l

(A1.70)

(A1.7r)

20-t9 It8 It7 tt6 Its -tc I

g 13 I.l tz too lt IE ro -b elI 8l5 rrz tt

5-9Jll2tlt

III

:IIII

;III

Extrcrrrc ValrrcDistribution

Typc I or'l'ypc ll(y . 13)

IXIX X X

XXX X Xxt x xxxxxxxxxxxxxxxxrxx

xxxxxxxxx xxxxx xxxxrxxx t

Type Il7<7<132-^y<7

l330

t2llxXX XX

xxxxx x xxx x

52.76r3-1

55.37qq 59.9A76-2 -t ? 3 Standard

dwiations

--- r --------- t --------- I ---------l63.6008 67.2t39 7o.az7t 7,r.e{02 mph obtained by replacing the expectation and the mean square value of the randomvariable Xby the corresponding statistics of the sample. In the case of the TypeI distribution, using Eqs. A1.40 and Al.4l,FIGURE A1.5. Histogram of estimated value G*10.999) for 100 samples of size n :

50.

treme wind speed data in an attempt to draw inferences on the applicability ofthe Type I distribution to the modeling of extreme wind behavior in certaintypes of climate. For details on such inferences, see [A1-21].

Estimators for the Type I Extreme Value Distribution:Epochal ApproachA classical method of approaching the problem of estimation is the method oJ'moments. In this method it is assumed that the distribution parameters can be

TABLE A1.1. Monte Carlo Simulation of a Type I Extreme Value Process

p o cx(0.98) cx(0.99) Gx(0.999)

Original (Underlying) Distribution 36.80 3 .78 5t .67 54.24 62.97

where X and s are the sample mean and the sample standard deviation, re-spectively, that is,

X_ (41.72)

(A1.73)

From the estimators (A1.70) and (A1.71) the following esrimator of Gy(p) canbe constructed:

Gx@): X+s(y -0.5772) (A^1.74)

where

y : -ln(-ln p) 1A 1.75)

Undcr thc assutnptiott tltitl (ltt' nutrlottt variablcrs X rrrrtl .r tlt'lilrt'tl lry llr1s. A I /Jand A1.73 rrrc, itsyrrplol it':rlly. rrorrrrirlly tlistributetl, i( cirrr lrt'slro1y11 lAl /.1111. 10, 174, trrrtl 22t{l lh;rl lor l;tr.11t' s:rrnlrlt's ol' size rr

lz*,n

[tr,t -x)').c-

Mcan"

Standard Dcviation"

Maximum Deviation BelowMean" (Percent of Mean)Maximum Deviation AboveMean" (Percent of Mean)Maximum Deviation BelowMean" (Standard Deviations)Maximum Deviation AboveMean" (Standard Deviations)

n : 25 36.90 4.01n : 50 36.80 3.89n - 25 0.86 0.81n : 50 0.65 0.50n : 25 5.90 52.00n : 50 3.80 32.00n : 25 6.60 64.00n : 5O 4.00 32.00n : 25 2.50 2.60n : 5O 2.20 2.5On : 25 2.80 3.2On - 50 2.30 2.5O

52.5851.923.462.14

19.6010.2016.6010.703.002.502.502.60

55.3854.634.002.49

21.3011.0019.0012.OO3.002.502.602.()0

&.6463.605.8s3.61

25.7013.6025.5014.702.802.402.{i02.60

"Estimutcd lnrm IOO sirnrplcs ol size l.

fr620 I il Mt Nt:; ()t t,n()tl^Bil ilY ilil ()r ly nNt) nt,|t tcn il()Nr;

sDIG,@)t :lT+ l.r3e6(y - o.siiz)fr *,.r(y -,.ttzt'l'" {,(A1.76)

A more efficient estimator of Gy(p) has been developed by Lieblein on thcbasis of the method of order statistics [Al-7, A1-16, Al-17]. A method frc-quently used in applications is based on least squares fining of a straight lineto the data on probability paper. This method is used in the computer programof [Al-ll]. A simplified approximate version of this method is presented in[Al-7, pp. 34,227, and 2281. For a discussion of other estimation methodsused firr thc Typc I distribution, for example, the maximum likelihood method,thc rcadcr is rcfbrrcd to [Al-7] and [Al-18].

It can bc shown that the standard deviation of any estimator of a parameteris larger than, or at least equal to, a theoretically specified standard deviationknown asthe Cramtr-Rao lower bound. In the case of the percent point functionof a Type I distribution, the Cram6r-Rao lower bound is

sDcRlcx(p)l : (0.60793y'+ 0.51404y + 1.10866)'/2 T, (At.77)

where y is given by Eq. A1.75 tA1-191. For n : 25 and n : 50, the ratio(1/o)SDgp[i'x@)] is now compared with the ratios (l/o)SDtGr(p)1, whereSDIGx@)l denotes the standard deviation of the percent point function esti-mated by the method of moments, by Lieblein's method of order statistics, andby the method of least squares fitting.

In Table A1.3 the quantities of line l were calculated by Eq.41.76. Thcquantities of line 2 were obtained from [A1-16, p. 131] (through multiplicationof corresponding quantities given for n : tO Uy JtOfZS und Jl0/50 o. nl'quantities given for n : 2obv J2U25 una lfzoro). The quantities of line 3were obtained from Table Al.lx (as shown in [Al-11], these quantities arcindependent of the parameters p and o used in the calculations). Finally, thoquantitics of line^(4) were calculated by Eq. A1.77 .

Assumc that Gy(p) is normally distributed. The approximate statement canrhen bc madc that the intcrval G*Q) X SDIGx@)l will contain the true un-known parameter Gy(p) in about 68% of the cases. This interval (referred toas the 68% confidence interval) is said to correspond to the 68% confiden<'cIevel. For the interval^ Gr@) + 2SDIG1@)1, the percentage rises to 9501,.while for the interval Gx@) X 3SDlGx(Dl, it rises to over 99% (99.7%). Asnoted above, these percentages should be viewed as only approximate; how-

*The standard deviation of Gr1p1 in line 3 is an estimate bascd on a finitc sarrrplc, ln acconltrrct'with the convention adopted herein, the notation.r rathcr than 57) slrorrkl tlrt'r'clirrt: bc rrsctl lirrthe quantities of line 3. This was not donc in 'l'ahlc A 1.3 lirr llrt' s:rkt' ol r'l:rrily.

qx.O&U

qb

x

qb

q

cc&{4FJF.1e4l.i

.:

O

O

O

all

Oa.l

O

c.l

c.ltl

re

oo2o

r!

Id.)

ctt\O

\ocaonnnrn c..l |r) \on n v?a

O\ O0o

(..I NOq q\

r N tt-\o \o \a)-i .{ .-

a€to\'.t 1OO

tooA!

'59o

aa.9 ?aii96,^ 1 Eryb :' avbO6i:

S1.( 'r

cr, tf,L]r'.- at/t

t-t.,1r,

f;2 I

622 I l Ml Nll; ()t I'n()t]nllt ily ilil ()lty nNt) At,t,t t()n l()Nl;

ever, the approximation is satislactrlry lirr rcasonablc sanlplc sizcs such us lrrcused in the analysis of wind speed data.

Reliability of Extreme Wind Speed Predictions for Type I DistributionIt is of interest to examine the effect of the estimation methods upon the rcli-ability of predictions of extreme wind speeds corresponding to mean recurrcnccintervals used in structural engineering calculations.x Consider, for examplc,the case n :25. The 68% confidence interval for the 100-year wind, x1s6 :Gx(O.99), is crlo.oo; x SDtG"1pll. If the most reliable method of estimationof Table Al.3-the order statistics method-is used, then the interval isCr(O.qq) + 0.90o = G*(O.gg) * 0.7s (Eq. 41.70). If, on rhe other han<t,thc lcast rcliablc mcthod of Table Al.3-the method of moments-is used.thcn thc cstirnatcd intcrval is C"(O.gg) * 0.88s.

Numcnlus analyses of wind records show that the ratios slX are of the orderof 0.07 to 0.15 [A1-11, 4'1-15]. Then the 68% confidence intervals obtaineclby the mcthod of order statistics and by the method of moments are (using thcratio slX : O.l2) Gx(0.99)[ + 0.061] and Gy(0.99)[1 + 0.077], respectively.The dillbrence between the respective reliabilities of the estimates of the valucsof X corresponding Io p : 0.99 (or, in virtue of Eq. A1 .46, to a mean rc-currence interval N : 1 00 years) is seen to be quite small, that is, of the ordcrof 2%. Results of similar calculations carried out forp : 0.9, p : 0.99, p :0.999; n : 25, n : 50; and s/X : 0.12, are shown in Table A1.4. Thcdifferences between the reliabilities of the various procedures can be verifiedto be negligible also fors/X: 0.07 and s/X: 0.15.

It is seen from Table Al.4 that any of the methods listed will provide anacceptable estimate of the order of magnitude of the 68% confidence limits.The width of the 95% confidence limits is approximately twice the width olthe 68% limits; for example. for N : 20 and n : 25, the nondimensionalizecl95% confidence limit estimated by the method of moments is 1 * 0.098. Thcdifferences between estimates based on various procedures are seen to remainacceptably small for the 95% confidence limits as well.

It has previously been shown (Eq. A1.63) that if X is a random variablcwith a Typc I distribution, it is possible to view the largest value in a samplcof n valucs of X as an estimator of the value of X corresponding to a meanrecurrcncc intcrval n. While this estimator has the obvious advantage of extremcsimplicity, its reliability is relatively poor. This can be shown by the followingexample. If Eq. A1.76 is used to estimate the 95% confidence interval fbr the3j-year wind speed at Richmond, Virginia (i : 36.8 mph, & : 3.78 mph;see [Al-ll]), the interval obtained is (50 + 5) mph. Using the largest valucin a set of 37 values as an estimator of the 37-year wind, the estimatcd 95%,

*Of two different possible estimators &1 and &2 of thc samc quantity rr, lhc cslirrrrlor rl, is srritlto be more reliable than &2 if (assurning thc cstinlutors to bc rrnbi:rsctl) S/)(ri1) ,!/)(rv,)lA1-161.

Ntl

tr

F.]

&Iti

trOq)

a

0)a

qrio

q

q

L

()

QNa€\o+4Ir.1rl4,F

O

tl

o'1

oto

coO\Odt)

3o.N

o*

dla

€

t .1.'

t-J

||t.' .1 . _,() -+ t.t (.1o- '1 1::u99All 1-t lt+l

':, v.),E 'L'

|!

(..l F- \Ovl <t69U99

+l +t +l

r)aqco ca co99Uv9u+t +t +l

|r)OO *oO@F- O\\O99UU99

+l +t +t +l

t-- * c..t oot-- \O F- \OU9UU999U+l +t +t +l

ao \o|/)t-- \O r)9UU99+l +t +t

O\ It- co$+ +

+t +t +l

aEA(t

- i, v

;LAX4lla).)l:7.) TJ - _1l: . 4t -t

_1 ^)I trr,, ,r:,1 ] ,l 'l',,,'i,rt1,,:,;.lii,-,

rkI

ooo.g

3'oN:it::

t:I

,=!

:

624 EttMtNt:; ()t t,n()t]ntJtly nn()t ry nNl) nl,r'r ton l()N:;

confidence limit interval obtained is (50 + l2) rnph (scc Bq. Al.63a). thai is.an interval more than twice as wide as the interval estimated by thc nrolhod ol'moments.

Estimators for Extreme Value Distributions: Threshold ApproachAs indicated earlier, analyses of largest yearly wind data show that extremcspeeds are better modeled by the Gumbel (Extreme Value Type I) than by thcFr6chet (Extreme Value Type II) distribution. In recent years the developmentof the peaks over threshold approach has made it possible to take a closer lookat extreme wind modeling and, in particular, at the question of the relativcrncrits of thc rcvcrsc We ibull and the Gumbel distributions as models of extremcwincl spccds (lAl-301 to lAl-371).

Thc Gumbcl model has infinite upper tail, whereas it is a physical fact thatextremc wind speeds are bounded. One would expect a good probabilisticmodel to rcllect this fact. To the extent that an extreme value distribution wouldbe a reasonable model of extreme wind behavior, one would therefore expeclextreme winds to be best fitted by the reverse Weibull distribution.

ln addition to the fact that wind speeds are bounded, there is at least oncother indication, albeit indirect, that the Gumbel model might be inappropriate:estimated safety indices for wind-sensitive structures based on the Gumbelmodel imply unrealistically high failure probabilities tA3-91. This is likely due,at least in part, to the use in those estimates of a distribution with an unreal-istically long-infinite-upper tail.

A study of several estimators developed for the peaks over threshold method[,{1-32] concluded that the performance of the de Haan estimators [Al-35] isabout as good or better than that of the Pickands tAl-251 and Cumulative MeanExceedance estimators tAl-291. We now present a summary of the de Haanmethod and of results based on its application.

Let the number of data above the threshold be denoted by ft so that thcthreshold ,, represents the (ft + l)th highest data point(s). We have )t : klnr,,,where nr., denotes the length of the record in years. The highest, second,. . . , A1h, (k + l)th highest variates are denoted by X,.n, Xn t,n, Xn (k+l).a,Xn-k., = rr, respectively. Compute the quantities

rt)

The standard dcviati<ln tll'lhc irsyrrrptolit':rlly uolrrr:tl t'slirrurlor.ol r:rs

nI / 1,il()ilntrlt ltyilll {)t ttArul |:,tnil:,lt{At

s.d.(i) : [],' - (,)2(t - z,,l l4 - 8(l - 2(')U | (l-3.)

trt Irtr

t)n tn (i21'

(Al ril )

(A l.tt2rr)

< 0 (A1.82b)

see [A1-35].The estimate of the tail length parameter d depends on the choice of thresh-

old, and it is common statistical practice to assay the results of the analysisfrom the plot of i as a function of threshold. If, for any given data set, thethreshold is high, the number of exceedances of the threshold is low; hencethe sampling errors are high. Lowering the threshold decreases the samplingerrors, but an excessive decrease ofthe threshold causes a large bias. A plateauwhere d changes relatively little as a function of threshold provides a goodindication of the actual value of c. Examples are shown in Fig. Al.6 tAl-361.For both Albany and Albuquerque d < 0 (e.g., -O.ZO to -0.25). Theseresults, and similar results obtained for a large number of stations, were re-ported in [Al-36]. They were based on data sets extracted from 15- to26-yearlargest daily wind speed records so that mutual dependence among the data be

ALBANY,NY (19 YEARS) ALBUOUEROUE,NM (I9 YEARS)

lil(Jtlltl,l Al.(r. lislirrr;rlr.s ol 1r;rr;rrrn.tlr r .lr(l (f 'r,'i, t orrllrk.nt.t. lroulrrls, Plollr.rl ;r1,;rrrrrtlltn'slroltl (rrr rrrgrlr) llrrl slrrrrlrk' ,,rzr. l,rr lrro rrr.:rtlrt.r :l;rllo1:, rvrllr l(l V(.:1 t(.(!1(l l(.llt,llt

)2e4e)

1l 1i)(3rx

+(5(l ]*o]"' e

ti I

M,;, :o I {togtx, ,,,1 - tog(X, r,)}'

The estimators of c and A are

r:1,2 (A1.18)

e:m(;')+t

^ uMtt)Qt

(A I .7e)

(n I .80)

4f 40ilu Int16 41 65 1S 247 42il finu utr ttfit

2{t - (ul\\t(tttli; ))

626 I il Mt N11; ()t I'tt()l]nl]ll llY llll ()l lY nNI) nl 'l'l lon ll()Nl;

reduced as much as possible. This was accrlrrrplisl.rcd by a clatlr sclocliotr tttc:tltotlaimed at eliminating all but one-the largest-among thc wincl spcctls associated with one individual storm event [Al-33, A1-36]*. The results o1'lA l-3(rl,and similar results for hurricane wind speeds 13-711, have significant consc-quences on the estimation of load factors for wind-sensitive structures, as showttin Section A3.3.

REFERENCES

Al-l H. Cram6r, Thc Elcmcnts of Probahility Theory, Witey, New York, 1955.

Al-2 A. G. Mihrar:n. Simukttion, Academic, New York, 1972'Al-3 R. v<rn Miscs, Probability, Statistics nnd Truth, Allen and Unwin, London,

anrl Mactnillan, New York, 1957.

Al-4 J. R. Bcnjamin and A. C. Cornell, Probability, Statistics and Decision forCivil Engineers, McGraw-Hill, New York, 1970'

A1-5 'l'. C. Fry, Probability and Its Engineering Uses. Van Nostrand, Princeton,r 965.

Al-6 B. E,pstein, "Elements of the Theory of Extreme Values," Technometrics, 2,I (Feb. 1960),27-41.

A1-7 E. J. Gumbel, Statistics of Extremes, Columbia Univ. Press, New York, 1958.

Al-8 N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distri-butions,2d ed., Vol. 2, Wiley, New York, 1994.

Al-9 E. Simiu and B. R. Ellingwood, "Code Calibration of Extreme Wind ReturnPeriods," Technical Note, "/. Struct. Div., ASCE, 103, No. ST3 (Mar. 1977)'125-729.

A1-10 J. J. Filliben, "The Probability Plot Correlation Coelficient Test for Normal-ity," Technometrics, 17, 1 (Feb. 1975), lll-117.

Al,l l E. Simiu and J. J. Filliben, Statistical Analysis of Extreme Winds, TechnicalNote 868, National Bureau of Standards, Washington, D.C., 1975.

A1-12 E. Simiu and J. J. Filliben, "Probability Distributions of Extreme winclSpccds," J. Struct. Dlv., ASCE, 102, No. ST9, Proc. Paper 12381 (Sept.1976). t861 t877.

Al-13 J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methucn,London, and Wiley, New York, 1965.

Al-14 J. H. Mize and J. G. Cox, Essentials of Simulation, Prentice-Hall, EnglewootlCliffs, NJ, 1968.

41-15 P. Duchene-Marullaz, "Etude des vitesses maximales annuelles du vent,"Cahiers du Centre Scientffique et Technique du B1fiment, No. l3l, Cahicr'1118, Paris,1972.

*The data analyzed in [A1-36] and computer programs for thc cstitnation ol'rcvcrsc Wcihtrlldistribution parameteni ancl of wind speeds with various mcln rccttrtctlcc iltlt:rvitls rtlc:tcccssiblein electronic lbrm; scc IAl-371 fbr dctails.

nt I I ilt Nct :: 627

A l l() .1. Licblcin, A Ncn, Mttlrotl tf'Analyzing Extr<'mc-Vttlut' l\tttt, Nlrliorrirl lJrrlr.rruol'Standards Report No. 2190, Washington, DCl, l9-5:1.

n I I7 J. Lieblein, Elficient Methods of Extemc-Vulue Mt'tlrtnhtl(,1,y, ll('lx)r1 N()NBSIR 74-602, National Bureau of Standards, Wrrslrington. l)('. l()7,1.

Al-lll J. Tiago de Oliveira, "statistics hrr Gurnl.rcl lrrtl lirciclrt.t l)istrilrrrtiorrs" rrrStructural Safety and Rcliability, A. lrrcrrrlt'rrtlr:rl (etl.), l,t'r1,,:rnurrr, lrlrrrsl'rrl.NY, 1972, pp. 9l-10.5.

Al-19 F. Downton, "Lincar []stiltt:ttcs ol'l)irtiurrclcrs in tlrc llxtl'rrrc Vlrlrrt. l)islribution," Tec:hrutmcrricr, 8, l (lrcb. 196()). .t 17.

A1-20 G. I. Schueller and H. Panggabcan, "l)nrhahilistic l)ctcrrninatiorr ol l)csignWind Velocity in Germany," in Pntc. Inst. Civ. Eng., 61, part 2 ( l9j6),673_683.

Al-21 E. simiu, J. Bi6try and J. J. Filliben, "sampling Errors in the Estimation ofExtreme Winds," l. Struct. Div., ASCE, 104 (1978), 491-501.

Al-22 I. I. Gringorten, "Envelopes for ordered observations Applied to Meteoro-logical Extremes," -/. Geophys. Res., 68 (1976),815-826.

A'l-23 N. L. Johnson and S. Kotz, Distribution in Statistics: Continuous MultivariateDistributions, Wiley, New York, 1972.

Al-24 A. W. Marshall and I. Olkin, "Domains of Attraction of Multivariate ExtremeValue Distributions," The Annals of probability, lf (19g3), l6g_1j7.

Al-25 E. castillo, Extreme value Theory in Engineering, Academic press, New york,1988.

Al-26 J. Pickands (1915), "Statistical Inference using order statistics," Ann. sra-tist., 3 (1975\, 1 19-131.

Al-27 J. R. M. Hosking, and J. R. wallis, "Parameter and euantile Estimation fbrthe Generalized Pareto Distribution," Technometrics, 29 (1987), 339_349.

Al-28 R. L. Smith, Extreme value Theory, in Handbook of Appticabte Mathematic:.s,Supplement, W. Ledermann, E. Lloyd, S. Vajda, and C. Alexander (eds.),pp. 437472, Wiley, New York, 1989.

Al-29 A. c. Davison, and R. L. smith, "Models of Exceedances over High rhresh-olds," /. Royal Statistical Soc., Series B, 52 (1990), 339 442.

A1-30 G. R. Dargahi-Nougari, "New Methods for predicting Extreme wind Speeds,"J. Eng. Mech., ffs (1989), 859,866.

Al-3 I J. A. Lechner, S. D. Leigh, and E. Simiu, "Recent Approaches to ExtremeValue Estimation with Application to Extreme wind Speeds. part I: The pick-ands Method," J. Wind Eng. Ind. Aerodyn.,4l-44 (l9g}),509-519.

Al-32 J. Gross, A. Heckert, J. Lechner, and E. Simiu, "Novel Extreme Value pro-cedures: Application to Extreme wind Data," in Extreme value Theor.y urulApplication.s, V.l. I. J. (i'lirrnb.s, J. Lechner and E. Sirniu (cds.), Kluwcr,Dordrecht, l9(),1.

Al-33 J. L. Gross, N. n. Ilt't'kt'r'l ..1 . A. l.t'cltrtct-, irnrl E,. Sirniu, "A Slruly ol ()ptirnlrlExtremc Wirxl lisllrrr:rtiott llrot ttlttti's." Wirul 1,,)ttgittu'ritt,q, l'rrnt,t,tlirt,rl,s, Nirrtlrlnlcrrutlittnrtl ('rtt-li'r't'ttr't' r'rr ll'irttl l'.tt.tiirrt't,rirr,4, pp. (r() ll0, Wilt.y 1,,:rslt.rrr l,ltlNcw l)clhi.

Al 34 I). Wlrlslr:rw, "'(;('lllnll tlrr'l\1o:,t ltont \'()ll lixlr.t.tnt.Wirrtl l)trt:t A Slr.;r l,yStcp (irritle." .1. lit't N'tr lrrtt ,\trtrtrl L,,lrttttl, .r.) (l()().1). l()() ,ll.)

*628 FLEMENIs ()l I'll()llnllll llY llll ()llY nNtl nl)l'l l(;n ll()Nli

A1-35 de Haan, L., "Extrcr-nc valuc statistics," in lixtrtnc vulttt'l'lrtrtr,- ttrul A1t

pLictttions,Vol.l,J'Galambos,J'Lechner,andE.Sirrriu(cds,),Klrrwer,Dordrecht, 1994.

4l-36 E. Simiu and N. A. Heckert, Extreme Wind Distribution Tctils: A 'Peaks OvtrThreshold' Approach, NIST Building Science Series 174, National Institute ol'

Standards and Technology, Government Printing Office Stock No. SN003-003-033222-7, Washington, DC 20402, 1995.

41-37 E. Simiu and N. A. Heckert, "Extreme wind Distribution Tails: A 'Peaks

over Threshold' Approach," J. Struct. Eng. (May 1996), 539-547 '

APPENDIX A2

RANDOM PROCESSES

Consider a process the possible outcomes of which form a collection (or anensemble) of functions of time {y(r)}. A member of the ensemble is referredto as a sample function. The process is called a random process if the valuesof the member functions of the ensemble at any particular time constitute arandom variable. A sample function in a random process is referred to as arandom signal.

A random process that is a function of time is called stationary if its statis-tical properties are not dependent upon the choice of the time origin [A2-l],that is, if "whatever started to happen at some time could equally likely havestarted at any other time" tA2-21. A function belonging to an ensemble thatmight be generated by a stationary random process, or a stationary randomsignal, is thus assumed to extend over the entire time domain. Its mean andits mean square value do not vary with time (see Fig. A2.l).

The ensemble average, orthe expectation, of a random process is the averageof the values of the member functions at any particular time. A stationaryrandom process is said to be ergodic if, for that process, time averages equalensemble averages. Ergodicity requires in effect that every sample function betypical of the entire ensemble (Fig. A2.2).

It is convenient in applications to regard a stationary random signal as :rsuperposition of harmonic ost'illirlions ovcr a continuous range of frequcncics.It is the main purposc ol'this ltppctttlix lo prcscnt a description of st:rliotr:rlyrandom signals f.iorrr this lxrirtl ol'vit'w. As rr prcrequisite to the tlcvcloprrrcrrtof such a descriptiott, t't't1;ritt lr:rsit tt'urlls ol'hlrrrnonic analysis irrc rt'vit'wr'rllirst. For a morc rigotrttts lrc;rlnr('nl ol llrc logrics rliscusscd hcrcirr, llre rt'lrtk'ris rclbrrctl, lilr cxurttplc. lo lA.' ll, lA.t .ll" irrrtl lA2-121.'l'lrc uplrcrrtlix t'orrcltttlcs willt srttttc tl()l('s (ttl tto!t ( i;rtt:,:,t;ur l)r()(('ss('s lutrl rtonslittir)nlu'y l)to('(.sh(.:;

ft,!u

RANDOM PROCESSES

@)

FIGLIRE M.l. (a) Stationary signal. Nonstationary signals with: (b) time varyingmean value; (c) time varying mean square value; (d) time varying mean and meansquare value. After J. S. Bendat and A. G. Piersol, Random Data: Analysis and Mea'surement Procedures, Wiley-Interscience, New York, 1971, p' 345.

A2,I FOURIER 8ERIE8 AND FOURIER INTEORALE 631

FIGURE A2.2. Sample functions of an ergodic stationary random process.

42.1 FOURIER SERIES AND FOURIER INTEGRALS

consider the case of a periodic function x(r) with zero mean and with periodZ. It can be easily proved that "r(r) may be written in the form

x(t) : c, * .!, C1, cos(2rknrt - 6r)

where n1 : llT is the fundamental frequency and,

1 l'/2Co:=l x(t)drI J ,Tt)

ct :@1 +E)t'', 8,.Ao: An ' 4,

2 l't2Ar : i J_,,, "(t)"o, 2tknP dt

2 ln2B* = i J_r, ,(r)rin 2rkng dt

(A2.1)

(42.1a)

(A2.lb)

(A2.lc)

(A2.1d)

(A2.le)

Equation 42.l is known an the Foarier,reric's cxpansion of ,r(r) tA2-31.If a function y(l) is aetually nonperkilit, it is still possible to regard it asperiodic with infinite perlod, It can be ahown lA2-3, 4,2-41 rhat if y(r) in

t632 HANDOM t)Roctssl s

piecewise differentiable in every finitc intcrval and if thc intcgral

f-J-- lrtrll ar

exists, the following relation, analogous to Eq. A2'1, holds:

n:, lt l;l'l (:lilnt l)l Nl;ily u,N(;il()N ()t n l;tn il()t.lAtty ltANt,t)M i;iliNnt 633

42.2 PARSEVAL'S EOUALITY

'fhc rnoan squitrc virlur. oi ol tlrt. pr.rirxlit. lirnt'liorr r(/) witlr 1rt.r'itxl 7'1lu;.A2.l) may hr: wriltern ls

, I f"'(,, 7 J ,,; \'(l) ,lt (A'2.71

The substitution of Eq. A2.l inro Eq. A2.7 yieldsy(t) : J]- .,r,.o, f2trnt - 6@)ldn

In Eq. A2.3, known as the Fourier integral of y(0 (in realcontinuously varying frequency and

C(n):le'@)+r2(n)lttz, B(n)efu):tan'7un)

f-A(n\ : J__ -vff)cot 2rnt dt

f-B(n): J*r{tltin2rntdtFrom Eqs. A2.3a through A2.3d and the identities

tan dsing: ^ --1 ^(n' (I + tan- 0)1cosQ: - :------ ^u2' (l * tan'0)

(A2.2)

(A2.3)

form), n is a

(A2.3a)

(A2.3b)

(A2.3c)

(A2.3d)

(42.4a)

(A2.4b)

(A2.s)

(A2.6a)

(A2.6b)

"l: I so (A2.8)

The functions y(r) and c(n), which satisfy the symmetrical relations A2.3 antlA2.5, are referred to as Fourier transform pair-

It is noted that successive differentiation of Eq. A2.3 yields

Jl- l,l"or12trnt - 6@)l dt : c(n)

r-i(I) : - J__ 2rnCft)sinl2rnt - 60ll dn

r-i'(ry - -J " 4r2n1C(n)cosl2rnt <lt(rt\ltln

where Ss : C3 and So: lC? (k : 1,2, . . .).'the quantity S1,is the contributionto the mean square value of x(t) of the harmonic component with frequencyftn1. Equation A2.8 is a form of Parseval's equality IA2-3, A2-4, AZ-51.

consider now the case of a nonperiodic function y(r) for which an integralFourier expression exists. In virtue of Eqs. A2.3 and A2.5,

[- y21r7 dt: J* y{, J- C@) cosf2trnt - g(n)l dn dtJ-- -

: J:- c(r) J* vQ) cosl2trnt - g(n)l ctt dn

('-: I c2@) dnJ--f-: , J, C21n,y dn (AZ.g)

Equation A2.9 is the form taken by Parseval's equality in the case of a non-periodic function [A2-3, A2-4, A2-51.

A.2.3 SPECTRAL DENSITY FUNCTION OF A STATIONARYRANDOM SIGNAL

A relation similar ttl lJc;. A2.li will ttol bc sorrglrl lirr lirlrclirlrrs gcrrt:r'lrtctl bystationary processcs.'l'hc: s1x't'lrirl tk'rrsily will lrt'tlclinctl irs llrt.t'orrrrl(.tl)i11 .

lbr these functions, ol' llrc tlrrlrrtitrt's ,\* .

Consider a staiittttitt'y tittttlttttt sillltrrl .'(l) willr zr'ro rrrt':rrr. lk't';rrrsc rl rhrt'snot satisfy thc ctlntlitiolt A.J..'. tltt' lttttr'liotr . (/) tlrcs rrol lxr:ih('hs ;r l;otrnt'rintcgral. An auxiliary litttr'ltotr t'(/) rvrll llrt'trlort'lrr'rk'lrrrt'tl ;6 lollorv:, (liryA2..1):

there follows immediatelY

rnANtx)M l,lt(x;t li:ll :i

FIGURE A2.3. Definition of function y(r).

/r r\y(t):zG) (-"-<I<;)\ Z L/

Y(t) : 0 elsewhere

The function y(r) thus defined is nonperiodic, satisfies F,q. A2.2,has a Fourier integral. From the definition of y(r) it follows that

lim y(t) : z(r)

By virtue of Eqs. A2.lO and A2.9, the mean square value of y(r) is

The mean square value of the function z(r) is then

,?: )'::"i

n:"t :,t't {:iltnt t)l Nl;ily nNI) Alil()c()vniltnN(il It,Noil()N

2.\ (r) lirrr "_ ('r{rr.yt 'q T

Equation A2.13 bccomcs

Sr(n) : 4nzn2S,1n1

S,(n) : l6ranaS,(n1

S,(n) dn (A2.15)

The function s.(n) is defined as the spectral density function of z(t). To eachfrequency ,?(0 < n < oo) there corresponds an elemental contribution s(n) dnto the mean square value ol; o! rs, of course, equal to the area under thespectral density curve s.(n). Because in Eq. A2.15 the spectrum is defined overthe range o < n 4 o, the quantity s.(rz) is referred to as the one-sirJed spectraldensity function of z(t). It is this definition of the spectrum implicit in Eq.y'.z.r5 that has been used throughout this text. However, a different conventionmay be used whereby the spectrum is defined over the range - @ < n < @and the integration limits in F,q. A2.15 are -@ to o. This convention yieldsthe so-called two-sided spectral density function of ze) IA2-6, p. 751.

From Eqs. 42.6, following the same steps that led from Eq. A2.3 to Eq.A2.14, there result the expression for the spectral density of the first and secondderivative of a random process:

":: J;

63s

(A.2.14)

(42.16a)

(A2.16b)

, .T/2^ lt "oi : i )_r,rt'{t) dr

: ; j-- v2(o dt

(A2.10)

and thus

(A2.l l)

(42.t2)

42.4 SPECTRAL DENSITY AND AUTOCOVARIANCE FUNCTION

From Eqs. A2.3a, A2.3c, and A2.3d there follows

?_crrrt :1vr{nt + Br1n71TT2: 7tA(n) ' A(n) + B(n) . B(n)l

{-, *, )cosztrntl *, I*_*y(r2)cos 2rnt2dt2

2f*: r Jn C'(n) dn

'tlI- 't'l

f, r'(/1)sirr2f*:lim-lr-* T Jo

With the notation

c2(n) cln (A2. r3)

whiclr lrriry hc wl'illt'rr :rs

I.) tttr!, tltl .! ,,, r'U,)sin 2rnt; tlt,

I636 nANt)()M l'ttool rilil f;

t, I-- J--

n:, l, ( lll( t:;1, (.( )Vn I lln N(.l I l,N(;l l( )N 637

Denoting

- I f-R(r) : 7 J__ v(tt\v(tt * r)dt1' (A2.18)

where r : tz - /r, Eg. A2.17 may be written as

2P-,C2{nl: , J -,Q{z)cos

2rnr dr

Equations A2.19, A2.ll, and A2.14 thus yield

S.(n) : J-- ,^.{"). os 2trnr dr

y(11)y(r2)cos 2rn(t2 - t,)dtt dt, (A2.l1t

(A2.le)

(A2.20)

t''/t.( r ) 1,, .t.{r,)r.',,t )nttr tltt

Similarly, by virtuc ol'Eqs. 42.20 ancl A2.22.

R,(l < ol

f-S,tr) :4 Jo R-tr)cos2rnr dr (A2.2.51

Equation A2.25 permits, in principle, the calculation of the spectral densityfunction corresponding to a given signal z(r). Details concerning the practicalcomputation of the spectra can be found in [A2-1], [42-6], and [42-7].

The definition of the autocovariance function (Eq. A2.21) yields

R,(0) : o! (A2'26)

For r ) 0 the products z(t)z(t * r) in Eq. A2.21 will no longer be alwayspositive as in the case in which z : 0, and

(Al.l-lhl

(42.27)where

The function Q(r) is defined as the autocovariance function of a(t) and providesa measure of the interdependence of the variable z at times r and t I r.

From the definition of the autocovariance function (Eq. A2.21) and thestationarity of z(r), there follows

R.(r) : R.(-z)

Since R.(r) is an even function of z,

(1.2.22)

Since the variation of z(t) with time is random, it is to be generally expectedthat for large values of r the products z(t)z(t + r) will be sometimes positive.sometimes negative, and that their mean value will vanish. Thus

lim R.(r) : 0 (42.28)

The nondimensional quantity Rr(llo!, known as the autocorrelation function,is equal to unity for z : 0 and vanishes for r : o.

A2.5 CROSS.COVARIANCE FUNCTION, CO.SPECTRUM,QUADRATURE SPECTRUM, COHERENCE

It is useful in applications to employ tools similar to those developed for theanalysis of a random signal to describe certain properties of two random sta-tionary signals zr(r) and zz(t). Consider two such signals, each with zero mean.The function

| [, il),

R , ,{r) ,',1'l ,. \ ,,,:t(r\z.t1t + r)dt (A2.29\

is clclincd as thc crrr.r,r t'r,r'rtrittnt'r' lrtttt'litttt ol llrc sigrrlrls lr(/) lrnrl t_,(/). linrrrrthis tlc:linition rrnrl lltc sllrliort;rtily ol llrt'lwo sip,rurls. llrcrc lirllows

R.(') : ;* 1, \",,,rt)rQ -t r) dr

J]- ,o.r'lr, n2trnr dr : o

(1.2.2t)

(A'2.23)

It can be seen from a comparison of Eqs. A2.5 and A2.20 that S.(n) and 2R,(r)form a Fourier transform pair. Therefore

f-R,tr) : I I S,(n)cos Zrnr dn (A2.24)" -J -

Since, as follows from Eq. A2.2O, S,(n) is an even function ol'n, liq. A2.24may be written as

"1 I

638 HANDOM I'I]OCLSIJI S

z2(t )

l'l(;URE A2.4. Functions zr(/) and zt(t) -- zt(t - rl.

Rr,rr(') : Rrrr(-t) (A2.30)

However, it is noted that, in general, Rr,rr(r) + Rrrrr(-r)' For example, ifz2Q) = zlt - zs), it can immediately be seen from Fig' A2'4 that

A2.6 PROBABILITY DISTRIBUTION OF THE LARGEST VALUESOF A NORMALLY DISTRIBUTED STATIONARY RANDOM SIGNAL

Consider a normally distributed stationary random signal z(/) with zero mean.Let E(k) denote the expected number of peaks per unit of time that are greaterthan ft times the rms value of z(r). It can be shown that the fbllowing expressionis adequate for use in practical calculations [A2-5, A2-8'l:

n;' ri l.lr )l lMn I I

('oIr..,..,(rr) 1,S1, ,lrr)l' r l,tli r,r)lS.., ( rr ),S, ,( r )

I rANt x )M lit( iNn | 639

Il

(n 2..r6)

(A2.37)

(A2.38)

/ t<2\E(k):zexl(_7/

where

( Jff n's.trl dn-)'''':Lffs,utd, )and S.(n) is the spectral density function of z(t).

R.,.r(zo) : R.,(0)

R.,.r(-zo) : Rr,(2ro)

s?u,@): , I:- R.,.,(z)cos 2trnr dr

and

r-sf;,,rln) : 2 J _ Rz,z2(r)sin 2rnr dr

It follows then from Eq. 42.30 that

s1,,,(n) : scur,(n)

s?,,,(,) : -s9,,@)

The coherence function is a convenient measure of the extentsignals zr(r) and zz|) ate correlated- Its square root is dcnotcdand is defined as

The co-spectrum and quadrature spectrum of the signals zr(r) and z2(r) are

defined, respectivelY, as

Peaks greater than ko, may be regarded as rare events. Their probabilitydistribution may therefore be assumed to be of the Poisson type. Thus theprobability that in the time interval Z there will be no peaks equal to or largerthan ko, may be written as

p(0, T) - e-E(k)r (42.39)

(see Sect. A1.5). p(0, T) may also be viewed as the probability that, given theinterval 7, the ratio K of largest peak to the rms value of z(r) is less than k.Equation 42.39 may then be rewritten as

P(K<klD:e-E(br (A2.4o)

where P(K < klD is the cumulative distribution of the random variable Kgiven the interval 7. Thc probability density function of K, that is, the prob-ability thatk < K < k * r/k, is obtaincd from Eq. A2.40 by differentiation:

/,r (i( l7') k'l't.117. r':&tt (n2.4t)

Thc cxpectcd valuc ol'lltr' lltr'1,,t'st pt';tk rx't'tttrirrg in lltr: ittlcrvirl '/'rrr:ry tlrr'lr lrccalculatcd as showrr irr Sct'1. A 1..1.

(A2.31)

(A2.32)

(A2.33)

(A2.34)

(A2.3s)

(A2.35a)

to which tw<rby Coh,.,,..,(rr)

u : J-_ kPK(klr) dk

k : tz ln vT\t'2 . ;#f"where z is given by Eq. A2.38.

640 rtANrxlMt)tl()ot.s;sl.1i

Prob(b ( z

bc wrilltrrr rrs

ilt ililt Nct l; 64 I

(A2.47a)

(42.47b): J", Bfftx,lx : xr.ly(x) rix

vp : Effl*,t Jo,/*{*l a"

(A2.42\

The integral of Eq. A2.42 was evaluated in [A2-9] and is, approximately,

(A2.43)

where f,, : conditional probability density of X,, given that X : x, ./x :probability density function of the vector X, and Efr t&lx - xl : average ofpositive values of X, given that X : x. If Xn and X are independent,

A2.7 MEAN UPCROSSING AND OUTCROSSING RATES

Considcr thc stationary random process z(r) represented in Fig. A2.la. Weseek the mean rate of occurrence z6 of the event, denotedby A, that the processcrosses the line Z: b in an upward direction (i.e., the mean upcrossing rateof z(r)). The probability of occurence of this event during a time intewal dtin the ncighborhood of the time r is equal to the probability that 2 > o and b< z < b * dz, which can be written as

(A2.48)

A2.8 NON.GAUSSIAN AND NONSTATIONARY PROCESSES

It was pointed out in Sect. 4.7.5 that pressures can have strongly non-Gaussiandistributions, especially at building corners and edges. For techniques ondealing with non-Gaussian processes, including simulation techniques, seelA2-r31.

Hurricanes and thunderstorms have strongly nonstationary wind speed timehistories (Figs. 2.4.7 and 2.4.8). Their effects on flexible structures may beestimated by using wavelet techniques, which account for the variation in timeofthe energy content for any specified frequency band; see lA2-14 to A2-241.

REFERENCES

A2-l A. Papoulis, Probability, Random Variables, and Stochastic Processes,McGraw-Hill, New York, 1975.

A2-2 R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra, Dover,New York, 1959.

A2-3 F. B. Hildebrand, Advanced Calculus for Applications, Prentice-Hall, Engle-wood Cliffs, NJ 1962.

A2-4 D. C. Champeney, Fourier Transforms and Their Applications, Academic,London, 1973.

A2-5 J. D. Robson, Art ltrlrtulut'littt to Rttrulom Vibration, Elscvicr, Anrstcnlant,The Netherlarrtls. l(Xr4.

A2-6 J. S. Bendat antl A. (i. I'icrsol. Iittnhntt l\tttt: Atutlvsi.s tttrtl Mr'rt,tun'nt(,ttPnx:cdurcs, Wilt'y lrrlcrst it'rrt c. Nt'w Yolk, l()7 L

A2-1 L. D. [inocltsott lrtttl l{ h ( )ltrr':,. I't,,r:rrrrtrrtritt,t: trtnl .'hrrtl\',si,y liu l)i!:tt,tl Itrrtt'Scrits I\tltt,'l'lrt'Slr,xL ;rrrrl Vilrrrrliort lttlottnlrlion ('('nl('t, ll S l)('l)illlt('llol' l)clcrrsc. |()(rl'1.

<b+clz,2)r: J:

,, : I; ,r,..(b, z)dz

JJ ''' "o' *')d*'

:dt

f, i@' 2) dz d2

f-)n zf,.,ta. zt az (A2.44)

where fr.r(2, 2; : joint probability density of z and 2. The mean upcrossingrate u6 ii, Uy Oennition, the probability of occurrence of the event,4 per unittime [A2-8]:

(A2.4s)

This result was extended to the case where the random process is a vector x.Let vp denote the mean rate at which the random process (i.e., the tip of a

vector with specified origin, O) crosses in an outward direction the boundaryFrof aregion containing the point O. It can be shown that the mean outcrossingt?le vp has the exPression

,o: \or*(A2.46)

wherein : projection of vector i on the normal to Fp. and l'.i,,(x, -i,; : .ioittlprobability density function of x and i,, lA2-10, A2-lll. lit;trirtiorr A2.4(r crrrt

642 nnNt)()M I't t(x;l l;lil I

A2-8 S. O. Ricc, "Mathcrnatical Anllysis ol'Rirtttlotn Noise " irr ,\r'lr'r'l l\t1x'r,s ttttNoise and Stochastic Processcs, N. Wax (cd.)' Dovcr, Ncw York' l()54.

A2-9 A. G. Davenport, "Note on the Distribution of the Largest Valuc ol a l{attclottrFunction with Application to Gust Loading," J. Inst. Civ. Eng.,24 (lgil).187-196.

42-10 Y. K. Belyaev, "On the Number of Exits across the Boundary of a Region bya Vector Stochastic Process," Teoriia Veroiatn. Primen. ' f3 (1968)' 333-33'l(In Russian).

A2-11 D. Veneziano, M. Grigoriu, and C. A. Comell, "Vector Process Models lirrSystem Reliability," J. Eng. Mech. Div., ASCE, 103 (1977),441-460.

A2-12 E,. Vanmarcke, Random Fields: Analysis and Synthesis' MIT Press, Cam-bridge, 1983.

A2-13 M. Grigoriu, Applications of Non-Gaussian Processes: Examples, Theory'Sirtultttion., Linear Random Vibrations, and Matlab Solutions, Prentice-Hall,linglcwood Ctift.s, NJ, 1995.

A2-14 R. Adhikari and H. Yamaguchi, "A Study on the Nonstationarity in Wind andWind-induced Response of Tall Buildings for Adaptive Active Control"' WindEngineering, Proceedings of Ninth Wind Engineering Conference, Vol. 3, pp.1455-1466, Wiley Eastern Ltd., New Delhi, 1995.

A2-15 K. Gurley and A. Kareem, "On the Analysis and Simulation of RandonrProcesses Utilizing Higher Order Spectra and Wavelet Transforms," Compu-tationaL Stochastic Mechanics, P. Spanos (ed.), pp. 315-324' Bakkema, Rot-terdam, 1995.

A2-16 O. Rioul and M. Vetterli, "Wavelets and Signal Processing," IEEE SignulProc. Mag.,8 (1991), 14-38.

A2-17 C. Meneveau, "Analysis of Turbulence in the Orthonormal Wavelet Reprc-sentation," J. Fluid Mech., 232 (1991), 432-520.

42-18 M. Yamada and K. Okhitani, "An Identification of Energy Cascade in Tur-bulence by Orthonormal Wavelet Analysis," Progr. Theor. Phys., 84 (1991)'799 815.

A2-19 M. Farge, "Wavelet Transforms and Their Applications to Turbulence," Ann'Rev. Fluid Mech., 24 (1992), 395-457.

A2-20 C. Chui, An Introduction to Wavelels, Academic Press, San Diego' 1992.

A2-21 I. Dubuchies, Ten Lectures on Wavelets, Soc. Industr' Appl. Math.' Phila-dclphia, 1992.

A2-222 Y. Mcyer, Wavelets: Algorithms and Applications, Soc. Industr. Appl. Math.'Philadelphia, 1993.

A2-23 G. Strang, "Wavelet transforms vemus Fouriertransforms," Bull. Amer. Math.Soc., 28 (1993), 288-305.

A2-24 G. Strang, "Wavelets," Am. Scient., 82 (1994), 250-255.

APPENDIX A3

ELEMENTS OF STRUCTURALRELIABILITY

The objective of structural reliability is to develop criteria and verificationprocedures aimed at ensuring that structures built according to specificationswill perform acceptably from a safety and serviceability viewpoint. This ob-jective could, in principle, be achieved by meeting the following requirement:failure probabilities (i.e., probabilities that structures or members will fail tosatisfy certain performance criteria) must be equal to or less than some bench-mark values referred to as target failure probabilities.* Such an approach wouldrequire [A3-1. A3-91:

l The probabilistic description of the loads expected to act on the structure.2. The probabilistic description of the physical properties of the structurc

which affect its behavior under loads.3. The physical description of the limit states, that is, the states beyond

which the structure is unserviceable (serviceability limit states) or unsafe(ultimate limit states). Examples of limit states include specified defor-mations (determined from functional considerations); specified acceler-ations (determined from studies of equipment performance, or from er-gonomic studics on user discomfort in structures experiencing dynamicloads); speciliccl lcvcls of nonstructural damage; structural collapsc.

4. Load-structural rc:slxtnsc relationships covcring lhc rrrrrgc ol' r't'sporrsr'sfrom zero up l() tlr(' lirrril statc being consiclc:rc:rl.

'IAn:tltcrnativc slirl(flll('rll ol llttr rcr;urrcrr('nl is llr:rl llrc lcli:rlrrlrlir'r rolr",lrlrtlrll, trr tlrr.\:lrrlr.,litttil st;tlt:s tttttsl lrr' t't1rr:rl lo r)t ('\( r'{.il llrc rt"sPr't livt' (:trJlr.l rt.ll;rlrllrll,.., (rr lr;rlrrlrtl lr rrr1, rlr lrlr.,l:ts llrr' rlillr'rt'lttc lrt'lwt't'rr urrrlr, ;rrrrl llrr. l,rrlrrrr lrolrlrlrillly)

{i4:l

644 illMt Nr1; ()t l;tttt,ol(,ltnl ltl llnltll llY

5. The estimation of thc pnlbabililics ol'cxcccding thc vurirtus lilrri( stirtcs(i.e., of the tailure probabilities), bascd on thc clcmcnts listccl in itctns1 through 4 above and on the use of appropriate probabilistic and stttis-tical tools.

6. The specification of maximum acceptable probabilities of exceeding thcvarious limit states (i.e., of target failure probabilities).

The performance of a structure would be judged acceptable from a safety orserviceability viewpoint if the differences between the target and the failuroprobabilities were either positive (in which case the structure would be over-designed) or equal to zero.

The appmach .just described is seldom applicable in practice. Owing tophysical and probabilistic modeling difficulties and to the absence of sufficientstatistical data it is in general not possible to provide confident probabilisticdescriptions ol'the loads, particularly within the loading range correspondingto ultimatc limit states. Comprehensive probabilistic descriptions of the relevantphysical properties of the structure are also seldom available. In some caseslimit statcs are difficult to define quantitatively. Difficulties may also arise inattempting to describe relationships between loading and structural responscthat involve material and/or geometric nonlinearities or contributions by non-structural elements to the total structural capacity. For certain types of struc-tures-for example, redundant structures subjected to dynamic effects-the es-timation of failure probabilities can be analytically unfeasible or computationallyprohibitive, at least in the present state ofthe art. Finally, there are few agreed-upon values of target failure probabilities, particularly for limit states involvingloss of life, as opposed to mere economic loss.*

The reliability analyst is therefore forced to accept various compromises. Inpractice, in the absence of sufficient data and proved probabilistic and physicalmodels, it may be necessary to use conservative models, or models based atleast in part on subjective belief. In addition definitions of limit states mayhave to be adopted on the basis of computational convenience rather than onphysical grounds. (For example, ultimate limit states are defined in most casesas the collapse of individual members rather than as the collapse of the structurcas a whole;l membercollapse is in certain cases conventionally defined as thcattainment of thc yield stress at the most highly stressed section of the member.even though this does not usually entail physical collapse.) Finally, to simplifythe computations, various approximations may have to be used.

Estimates of nominal (or "notional") failure probabilities are thus obtaincdthat can differ-in certain instances significantly-from the "true" probabili-ties. However, if there are grounds to believe that the ratios of nominal (o"true" probabilities for two given designs do not differ significantly, thc tw<r

*For questions pertaining to safety goals for the operation ol'nuclcar powcr pl:tnls. sct: lA.1 l()1.lIn recent yean a number of studies conccrncd with structur:rl syslcnls hrrvc bct:n lclxrrlcrl. lirtuseful reviews of thcsc dcvcloprncnts, scc l43-ll antl lA3 251.

t.t()Millnt tAil1ilil I'l t()ltnilll illl l; nNt) :;nt I tY tNt)t(:t l; 645

tlt:sigrts ttury lrr't'orrrp:rrt'tl llrrrr tr n:li:rbility vit'wlxrirrt orr lhc llrsis ol'thcrcsltoclivr: ttontinlrl. r':rllrt'r llr:ur "lnrt', " ;rnrb:rbililics.

ll wotrltl, ol r'outse, lrc rlcsilrrblt'(o t:slirlllislr targct (i.c., maximum accept-ablc) norrrirrul lirilure: pnrblrbilitic:s. lrr prirrciplc, this could bedone if the "true"targct lailurc pnrbability antl (hc ratio bctwccn "true" and nominal failureprobability wcrc known. This is not thc case, and attempts are therefore madeto infer target nominal failure probabilities from the reliability analysis of ex-emplary designs, that is, designs that are regarded by professional consensusas acceptably but not overly safe. Such inferences are part of the processreferred to as safety calibration against accepted practice.

While there are instances where such a process can be carried out success-fully, difficulties arise in many practical situations. For example, structuralreliability calculations suggest that current design practice as embodied in theASCE 7-95 Standard [9-5] and other building standards and codes is not risk-consistent. In particular, estimated reliabilities of members designed in accor-dance with current practice are considerably lower for members subjected todead, live, and wind loads then for members subjected only to dead and liveloads [A3-3, A3-4], especially when the effect of wind is large compared tothe effect of dead and live loads.* Whether these differences are real or onlyapparent, that is, due to shortcomings of current reliability analyses, remainsto be established. Thus it is difficult in the present state of the art to determinewhether it is the lower or the higher estimated reliabilities that should beadopted as target values.

Despite serious weaknesses due to both theoretical and practical difficulties,structural reliability tools can in a number of cases be used to advantage indesign and for code development purposes. The objective of this appendix isto present a review of fundamental topics in structural reliability as applied toindividual members, which have found application or are potentially applicableto wind engineering problems. These topics include: the estimation of failureprobabilities; safety indices; and safety (or load and resistance) factorsIA3-241. For a number of recent applications of structural reliability to windengineering, see lA3-27 to ,{3-301.

A3.1 NOMINAL FAILURE PROBABILITIES AND SAFETY INDICES

The estimation of nominal failure probabilitiest is the core of structural reli-ability. Safety indices arc, at lcast in theory, measures of failure probabilities.The use of load and rcsistlrncc lact<lrs in design criteria is intended to ensure

*The carlicst.justilicir(iorr rtl r'tntctrl rlt'rr1'rr ;r:rr'litc willr lcspt'tt to winrl loatling was tlrcctl bythc authtlrs to [rlcrrtinP,'s l()l5 tttrtttol'.rrplt ll rrr,/,\lr, rr,'r lA I .)1, wlrich slitlrs (ctscly: "M:rxirrrrrrrrwind ltlirding ctltttcs st^ltlttttt :tttrl l;r:,1: lrttt .r ',lrorl trrrrr' l lrt' worl,urli sln'ss(.s us(l1 lirr lltis lo:rrlirr1lrttty lltt'tclirtc lrt: ittctt:rst'tl 5l)'.'l';rlrrrvl llro.,r'u.,r'rl lor orrlnr:rrv livr' ;urrl tk':rtl lo:rtls."llitt lrrr'vily, ttotnitt:tl l:tiltttr'lrtolr:rl,rlrtrr", :rrr' lrlrrr r'lorllr rr'ltrcrl lo:,rrrrply rrs l:tilrrrt'prolr;rlriltlrr.:;

646 I il Mt Nlli ()l :;lllt,olt,l{nl lll llnltll llY

that the members to which thc critcria arc appliod havc acccptublc lailrrrcprobabilities within any specified period of interest (c.g., onc ycar, or tltcliletime of a structure).

A3.1.1 Modeling of Loads as Random Processes andRandom Variables

Quantities that vary continuously and randomly with time (e.g., the wind speed,the wind velocity vector, or the wave height at a given location) can be modelcclas random processes. Quantities that are constant in time (e.g., the dead weight),or whose variation in time follows a deterministic law, can be modeled morcsimply as random variables.

ln problems involving combinations of two or more randomly time-depen-dent loads, it is gcnerally necessary to estimate failure probabilities by resortingto models and tcchniques drawn from the theory of random processes. If thesystem being considered depends, in addition to the randomly time-dependentloads, upon the random variables, Xr, Xz, . . . , X^, estimates of failure prob-abilities P(failure lXr, Xr, . . . , X^) are obtained that are conditional upon thevalues Xt, Xz, . . . , X^ taken on by these variables. The probability of failurcof the structure, Py, is then estimated by applying the theorem of total prob-ability as follows:

N( )tvlll.ln I lnll lllll I'll()lJAllll llll :, n Nl) :inl I lY lNl ll( I ', 647

Example l,el ,{(tt Il,, rlerxrtc tlrc: llrrgcst yt:lrlly wirrtl spt.t'tl irt ;r l,,rv(.nItlcittiort- 'l'ltcrr l("' {/ tlcrrotcs thc l:rrge:sl wirrtl s;lcetl ()c('lu nnl'. ;rl llrlrlItlcation rltrrirrg iur ,, yL:rl'pcr-iorl (cqual to thc lil'ctirrrc ol'tlrc sllrrt'lrrrt') lt isassumcd lhat thc largcst ycarly wind spccd U,, has an lixlrcrrrr'V;rlut.'l'ypt. l

distribution:

Fu"(u,,, - "^ol -"-o(-"o. ') (A-1.-]t

(A3.4)

(A3.5)

(1'3.1)

(A3.7a)

(A3.8)

(A3.9)

It can be shown that

fP, : )P(lailurelx,. x2, . . . . x1,)

' f*,.*r.....xr(xr, xz, . . , xr) dxr dxr. . .dxr (A3. r)

F=Uo-0.45ory,,o = 0.78ou,,

Fn:D*0.450yon = O.J8ou

O : U" -l O.78ou,,ln n

ou : ou,,

wherefy,.yr....y,(x1, x2, . xo) : joint probability density function of Xt, X2,. . . , Xr.For treatments of load combination problems based on random processrepresentations, see tA3-51 to [A3-8].

In problems involving only one randomly time-dependent parameter f(t),the question of combining time-dependent random processes no longer arises.It is therefore convenient in applications to use, in lieu of the random processf(r), its largest value during the lifetime of the structure, denoted by X('). Bysubstituting the random variable X(") for the random process f(r), the treatmentof thc rcliability problem is considerably simplified. The largest lifetime valucX(,) can bc characterized probabilistically as

Fyat(x) : [&r'r(x)]" (A3.2)

where X(r) : extreme value of f(r) during a time interval tt : Tln, T :lifetime of structure, n : integer and Fyr,r : cumulative distribution functiottof X(n). The cumulative distribution function of ll), Fyor, is refbrrcd to as thcunderlying distribution of X(n). Equation A3.2 holds if successivc valucs ol'X(t) are identically distributed and statistically indepenclcnt. An applica(ion ol'Eq. A3.2 is presentcd in thc fbllowing cxamplc.

and n : lifetime of structure in years.Finally, we consider the case where the structure is acted upon by two

randomly time-dependent loads with the following properties: (1) their exrremevalues have negligible probability of simultaneous occulrence, (2) their mostunfavorable combination ()ccurs whcn one of the loads reaches its largest life-time value while thc olhcr hlrs lur "onlirurry" (also termed "arbitrary-point-in-time"), rather than all rxtrcnrt' . vtrluc.'t' Sirrcc thc "arbitrary-point-in-tirnc"loading can be morlclcrl lty;rn;rplttrrpriirlcly t'lroscrr 1i11lq: i1111lL:pcntlcrrl pnrlr

*lirt cxtttlplo, il nt:ty lrt' :rr.;rttrtr'rl llr;rl lltr',r prrlx rlrr", r lr;rr;rr lcrrzc llrl rvtrtrl lo;rrl lurl llrr lrvr'kr:rtl :rr'littll rttt rttt'tttlrcts ol lttl,lt tr", lrtttl,ltttl' lt:tttr,',

Fu1u1 :.,,01 -.^o( -'t)l

where U, and ou": sample mean and sample standard deviation of the largestannual wind speed data Uo, recorded at the given location over a sufficientnumberof consecutive years (e.g.,20 years ormore) (Eqs. A1.40 and A1.41).From Eqs. A3.2 through A3.5, it follows that the probability distribution ofthe largest lifetime wind speed U is

(A3.6)

where

648 ELEMENIs; or slnuotUltnl nt ilnllll ily

ability distribution [A3-91, the reliability pnrblcrn can in this casc also br:reduced to one involving only random variables.

A3.1.2 Failure Region, Safe Region, and Failure Boundaryconsider a structure or member subjected to a load Q, and let the value of thcloading that induces a certain limit state in the structure (e.g., the yield stress)be denoted by R. It is assumed that both Q and R are random variables. Thespace defined by these variables is referred to as the load space. By definition,failure occurs for any pair of values, Q, R, satisfying the relation

R-Q<O (43.10)

Equation ,A3. l0 defines the failure region in the load space. The survivalregion, or safe region, is defined by the relation

R-Q>O (A3.1l)

Thefailure boundary, which separates the failure and safe regions, is definedby the equation

R-Q:o (1.3.12)

Relations similar to Eqs. A3.10, A3.11, and ,{3.12 can be written inthe loadffict space, defined by the variables Q", R,, where e" is the effect, or state,induced in the structure by the load Q @.g., a state of stress or deformation),and R" is the corresponding limit state (e.g., the yield stress or a specifie<ldeformation). The equation of the failure boundary in the load effect space is

R"-Q":o (A3. r3)

In general, Q and R are functions of random variables Xr, Xz, . . . , Xn (e.g.,aerodynamic coefficients, terrain roughness, cross-sectional area, modulus ofelasticity, breaking strength: x

ll| il[ :; nNt) :int I ty tNt)t(]t li 649

Sr-rbstituliorr ol lirls. A l l.l rrrrtl A.1 l5 irrkr lrr1s. A.1.10, n-l.ll, ancl A.l.llyiclds thc: Irrttppittl='ol lltr'lrultrrt'r'r:gion, slrlc rcgiorr, and lailurc bounrllrry orr{rrthc spacc ol'lhc vrrliirbles X1 , X.,, , X,, . 'l'lrc cquation o1'thc lailurc bountlirrycan thus bc writtcrr as

(A3. r6)

The well-behaved nature of the structural mechanics relations generally en-sures that Eq. ,A'3.13 is the mapping of Eq. A3.I2 onto the load effect space.Equation A3.16 is thus the mapping onto the space X1 , X2, ..., Xn not onlyof Eq. A3.12, but of Eq. A3.13 as well. Therefore, once it is made clear atthe outset that the problem is formulated in the load, or in the load effect,space, it is common practice to refer generically to Q and Qn as "loads" andto R and R" as "resistances," and to omit the index "e" in Eq. A3.13.

It is useful in various applications to map the failure region, the safe region,and the failure boundary onto the space of the variables Yr, Yz, . . . , Y,, definedby transformations

(A3. r7)

For example, if in Eq. A3.16 Xr : p, and X2: U, where I : air densityand U : wind speed, a variable representing the dynamic pressure may bedefined by the transformation yr : |pU2, and,Eq. A3.16 maybe mapped ontothe space of the variables Y1 , Xt, X+, . . . Xn. Another example is the frequentlyused set of transformations

Ir : lnR

Yz:lnQ(A3.18)

(A3.19)

*These are sometimes referred to as basic variables. We will use here simply thc ten-n .,variables," since what constitutes a basic variable is in many instances a mattcr of convention. lirlexample, the hourly wind speed at 10 m above ground in open terrain, which is rcgardcd in nurslapplications as a basic variable, depends in turn upon various random storm charactcr.istics, sut'lras the difference between atmospheric pressures at the center ancl thc pcriphcry ol'(hc st()r-r)r olthe radius of maximum storm winds.

It follows immediately from Eqs. A3.12, A3.18, and A3.19 that the mappingof the failure boundary onto the space I,, Y2, is

Yt-Yr:g (A3.20)

lrr/l lnp:6; (43.2t)

The failure bountlaty is lt;xrirrl, r ('urv(', rr sru'llrt'c, <lru hypersurface accordingto whcthcr thc pnrblt'rrr :rl lr:rrrrl rs lorrrrrrl:tlt'tl irr :r slttrr't: ol'<lnc, twcl, three, orttxlrc lhan thrcc t'iurthrrn v:u r;rlrl,':,

Q:Q6t,X2,...,X,)R:R(Xr ,X2,...,Xn)

(A3. r4)

(A3. rs)

650 I r Mt Ntl; ()l linlr,(]tUnnl nt ilnt]l uy

43.1.3 General Expression for Estimation of Failure ProbabilityLet the failure region in the space of rcduccd variablcs* x1,, xz,. -r,,,, l)cdenoted by O. The probability of failure Pycan be written as

tPr: 1^ f,,,.,r,.....,,,@t, xz,, ' '' , xn,) drt,dxr, ' ' ' dr', (A3'22)'Jo

where the integrand is the joint probability density function of the reduccclvariables.

In most cases the estimation of failure probabilities by Eq. A3.22 is cont-putationally unwicldy, if not prohibitive, and the use of altemative methods isattempted instcacl. Various such methods, whose applicability depends uponthc charactcristics of the problem at hand, are described subsequently in thisappendix. Howcver, we first discuss in the next section the useful notion ol'safety inclcx.

A3.1.4 Safety lndicesThe safety index is a statistic that under certain conditions, which will bcillustrated subsequently, can provide a simple and convenient means of as-sessing structural reliability.

We consider a failure boundary in the space of a given set of variables, ancldenote by S its mapping in the space of the corresponding reduced variables.The safety index, p, is defined as the shoftest distance in this space betweenthe origin and the boundary S [A3-10].1 The point on the boundary S that isclosest to the origin, as well as its mapping in the space of the original variables,is referred to as the checking point. For any given structural problem, rlzcnumerical value of the safety index depends upon the set of variables in whi<.hthe problem is formulated. The examples that follow illustrate the meaning ol'the safety index and the dependence of its numerical value upon the set ol'variables being used.

Example 7 lt is assumcd that the only random variable of the problem is thckrad (cllcct) p. 'l'ho rosistancc-a dctcrrninistic quantity-is denoted by R. Thc

'r"l hc lotlucctl (.r slantlutlizctl) variahlc -r, currcsponding to a variable X is defined as

t,:"-iOX

where i and oy are the mean and standard deviation of X, respectively.tThis definition is applicable to statistically independent variables. If the variablcs ol'lhc pnrblcrrrare correlated, they can be transformed by a linear operator into a set ol-uncorrclalcrl variitblcstA3-101. Note that an alternative, generalizcd saf-cty indcx was pnrposctl irr lA3 221, wllrscpelformancc is superior in situations whcrc thc lailurc boundarics urc norrlirrcrrr (scc:rlso ('h:rplr.r9 of l43-231)

n:r r N()MtNnt tAil t,ilt t,l t()ilnill il[ I nfll' ';Al I l\ ltll'lr i ', fi5l

ll,

FIGURE A3.1. Indcx /J lol trtcrtrlrct willr t:rtlkltr krrrl ;rrrrl rk'lt'rrrrirrislrt r('srsliur({IA3-241.

mapping of the tailure boundary

Q-R:o (43.23)

onto the space of the reduced variable q, : (Q - Q)loo (i.e., onto the axisOq,; see Fig. ,{3.1) is a point, qf , whose distance from the origin O is B :1n - Qltoo The safety index represents in this case the difference betweenthe values R and Q measured in terms of standard deviations on. It is clearthat the larger the safety index 0 ti.e.. the larger the differenceR - A lor anygiven oq, or the smaller op for any given different R - Q), the smaller theprobability thatQ > R.

Example 2 Consider the failure boundary in the load space (Eq. ,{3.10), andassume that both R and Q are random variables. The mapping of Eq. A3.12onto the space of the reduced variables q,: (Q - Qltonand r,: (R * R)/oa is the line [A3-9]

,, RO,' ,ro

t'0l 1.,,

opQr I Q - oor, - n: O

(Fig. A3.2). The distance between the origin and this line is

(A3.24)

R-OD-p - GTi;Tfn (A3.25)

l,l(il lltl'. A l,l. lrrrL r 1i l.r rrrlrrrlrlr wrllrtililrlont l{rir(l iill(l tilil{1ililt rr':,t:,1;utr r' lA Lt.ll

*652 ELLMENtti ot sttttj(;lt,ltnl lll llnllll llY

Example 3 Instead of operating in thc load spacc R, Q, wc consiclcr lit1.A3.2O, that is, the failure boundary in thc space Yr, Yz, dcfincd by Eqs. 43.lt3and ,A,3.19. Following exactly the same steps as in the preceding examplc, butapplying them to the variables Y1 and Yr, the safety index is in this case

A.r r I'J('Mll.]At tAil l'ltt I'n()nnilil ilil l; ANt] iint I tv tNt)t0t li 653

tlrirt is, Q * 1.1..1 ksi :rrrtl rr,r - 3.21 ksi. 'l'hc cquution ol'thc lirilurc surlaccin thc sp:rctr ol llrc virlirrblt:s, {/, R is

a(J2 * R:0and its mapping in the space of the reduced variables u,, r,is

(A3.33)

vvn rl t2

'- (o2y,+ozyr)t'2

Expansion in a Taylor series yields

(43.26)

(A3.28)

(A3.29)

(A3.3 r)

(A3.32)

(', *:")' : -*(',. *)(1^3.27)

and a similar expression for Y2. It then follows that if R and Q are uncorrelated,

(A3.34)

The valire of the safety index being sought is B : 4.31 (Fig. A3.3). Thecoordinates of the checking point are rf, : -2.51, u! : 3.50, to which therecorrespond in the U, R space the coordinates U* : 100.14 mph, R* :26.76ksi. It can be verified that the values of the safety index corresponding to thevariables Q, R (Eq. A3.25) and ln Q,ln R (Eq. ,{3.29) are p - 4.66 and B= 3.69, respectively.

Note that the mean and standard deviation of the largest lifetime wind speedor of the largest lifetime load, which are needed for the calculation of the safetyindex, cannot be estimated directly from measured data but must be obtainedfrom the probability distribution of the lifetime extreme. This distribution isestimated from the underlying distribution that best fits the measured data.Knowledge of, or an assumption concerning, the underlying probability dis-

rr : lnn + (n - D + - 1<* - &'-^,! +'''R 2' R'

where Zp and V9 denote the coefficient of variation of R and Q, respectively.*If higher-order terms in the numerator of Eq. A3.28 are neglected,

^ ln n - iv'- - (ln O - lvbtt)- 1V2o+V26t'2

^ tntRte)Ll

-' (vh * ,b)""

Example 4 Consider a linearly elastic member whose stresses Q can be writ-ten as

Q: aU2 (A3.30)

where c : deterministic influence coefficient and U : wind speed. We assumcthat a : 0.00267 ksi/(mph)2, the mean and standard deviation of the largestannual wind speed are (lo : 43.73 mph and ou,, : 8.61 mph, the mean anclstandard deviation of the resistance are R : 35.3 ksi and on:3.39 ksi, andthe lif'etime of the member is n : 50 years. From Eqs. A3.8 and A3.9, itfollows that thc mcan and standard deviation of the largest lifetime wind spectlarc U : 70 rnph, ou : 8.61 mph. An expansion of Eq. A3.30 in a Taylorseries yields

and

Q: a02e * v'r)

VQ = 2Vu

*For the definition of thc cocllicicnt ol'variatittn, scc Scct. A L4 l,'l(Jlllll,i A.l,,l. lrrrh'r /i rrr ,,|;rr'r' ol viur;rlrlt.s r,. rr, lA l .),ll

654 t tt MLNlli ()l l;lllt,(;l(lllnl lll llnlrll llY

tribution is required Jbr the estirnutfun t1l'tht su.li'ty irulcx irt ull ctt,st's intutlvitrga random variable that represents a LiJ'etime extreme-

A3.1.5 Safety Indices and Failure Probabilites: The Case ofNormal VariablesConsider the space of the variables Q and R, and assume that both variablcsare normally distributed. Note that the failure boundary (Eq. A3.12) is lincar.Since the variate R - Q is normally distributed, the probability of failure can

be written as

t ', li!!r

'l'ltc t;rtirttlilics ltclwcclt lltl'cttlltcsrs irr lit;:, A I l/ ;rtr'tr'r,r1'tttzr'rl.r', lltt'r'r.rrlanrl lpprttxinlit(c) oxl)r(:sliiott lirr llrt' s:rlt'ly utrlrr 1l ( ()r.".lr)u(lrtrl' lo llrr' 'lr;r,,'ol'tho variablcs lrr /1, lrr Q (lx1s. A.l.l(r ;rrrrl A I .'tt1

A3.1-6 Safety lndices and Failure Probabilities: The Case ofNonnormal VariablesIf the variablcs X1, X2, . .. , X,,, ol litttctiotts (hcr-col , lrrc n()l)n()nlurl, lic;.A3.36 is not applicablc.'l'hc lac( thal Iic1. A.1.-3(r ckros ttot ltolcl rtrcatts (halmembers having the same safbty indcx will, in gcncral, havc ditlbrent failureprobabilities. To illustrate the relationship between safety index and failureprobability, we consider the four members for which the means and standarddeviations of the load and resistance are listed in Table ,A.3.1. Members I andII have the same value of p in the space of the reduced variables lr,, !2,(conesponding to Eqs. A3 . I 8 and A3 . 19) ; their failure boundaries in that spaceare shown in Fig. A3.4. Members II, [I, and IV have the same value of B inthe space of the reduced variables r, e, representing the resistance and theload, respectively. The probabilities of failure of the four members based onthe assumption that all the variables are normal are shown in column 8, TableA,3.1. Those corresponding to the assumption that all the variables are lognor-mally distributed are listed in column 9. Column 10 shows the failure prob-abilities calculated by Eq. A3.22 and based on the assumptions that (1) theload is given by Eq. 43.30, (2) the distribution of the wind speed is ExtremeValue Type I, and (3) the reduced variable representing the resistance has theprobability density function,6(r,) shown in Fig. 43.5.*

It is seen from Table A3.1 that to equal values of the safety index p,calculated by Eq. A3.25 (i.e., based on the assumption that the probabilitydistributions of Q and R are normal), there correspond failure probabilitiesobtained by quadrature (column 10) that can differ from each other by as muchas one order of magnitude (members II, III, and IV). On the other hand,numerical studies reported in [,{3-12] show that the probability of failure Py ofcolumn 10 is uniquely determined, to within an approximation of abott l5%,.by the safety index p calculated by Eq. A3.29, regardless of the relative valucsof Q, Vn, R, and Zp. This interesting conclusion-which, as seen previously,does not hold forthe safety index 0 calculated by Eq. A3.25-is explained bythe approximate similarity between the shapes of the lognormal distribution,on the one hand, and the distributions used in Eq. A3.22, on the other hand.

Note, however, that whilc this conclusion is valid in the particular case.iustexamined, it does n(x rlr:cL:i;riltrily hold in other situations. For example, it ispossible that two rrrcrrrlrt'r's. ont' srrbicctcd to gravity loads and the othcr to

t'Figurc A3.5 corrcspontls :rpprorrrrr;rtr'lv t,r prrlrlislrul tlirllr orr llrc yicltl slrcss ol'A33 stccl, lorwhiclr thc ttotttirurl vrrlrrt'. llx rrrtiur r;rlrrr'..rrrrl llr tot'llit'icrrl ol vrui:rliorr llt'//, - 33 ksi, R -1.07/",,:rrrtl /,, - O.O()(rlAl l.' 1\l ll p .rl/l

Pr:F@-O<o)_ , [u

Jzno* o J-*R -a \

-l

J;tr;4t

')*x"*[j( -(R-O)\ono/one

:1-O (A3.3s)

where O : standardized normal cumulative distribution function, and the quan-tity between parentheses is the safety index 0 corresponding to the space ol'the variables R, O (Eq. A3.25).

More generally, it can similarly be shown that the relationship

Pt: | - O(p) (A3.36)

is valid if all the independent variables, Xr, Xz, . . . , X, are normally distrib-uted, the failure boundary (Eq. A3.16) is a linear function, and B is the saf-ety

index in the space of the reduced variables xr,, xz, x,.. Equation A3.36'of which Eq. 43.35 is a particular case, can be used even if the variables X',X2, ... , X, are not normally distributed and the failure boundary g(Xt, Xt'. . ., Xn) : 0 is nonlinear, provided that a transformation of variables { =

fi(X)(i : 1,2, . . . , n) canbe found such that Y; are normally distributed antlthe failure bounclary in the space Yr, Yz, . . . , Yn is linear. For example, assumcrthat R and Q have lognormal distributions. Then Yr : ln Q and Y2: ln R arcnormally distributed, and the equation of the failure boundary is Y1 - Yz : O

(Eq. A3.20). Applying Eq. 43.35 to the variables Y1 and Y2,

/lnR-lnO\Pr:t -o( --)' \tl oinp I oinp/

-,-o(9,)\vlzi? I vi,/( A 1..17 )

qocc

0)

a

0

o

b0

E

U)o)

Q0Q)

t)-h0)

trqq)

cll-olollileil.t..; I:tFllcat<tFI

C\c.lm1'<=+trl

F-coc.i ^<e+r!

ra)nco^<?9Cir!

o\qcO^a.< D-:t!

rnclcq^aa < io'-:.H

s3 0

rq.? O

-v aa i\'eJii

i\rorS O

Ho€{Jvz

tttlv999ii**

XXXX--rnO\*ic..l\o

ttoooo****XXXX\O C-l N C!**Sr+

ittlvv9u-*EiXXXX.Jnqqi*cico

\ocaco6\O O\ O\ O\I c.i cd cr;

O\ O\ O\ C}\cQ co co e.)cq ca c.) c.)

t-- F- F- t-rcJ cl el cl\ In!n\co co cn co

cao\orncA oO O\ \n

O\O\ooO\o \o c) c{co c.l * co

t'-O\OONF-iF-caalNca

:,oI

oAFo(!

oEgX

rI]

d

old

v?c")

bbtJr

,o

c)

&

oE

o

od

c0

oo-o.a6

v!-d

=s!)!)ddolol

0<q€€

cddOO

ooa9 cdFS

656

''^ry}-

A3,I NOMINAL FAILUFE PROBABILITIES AND SAFETY INDICES 657

FIGURE A3.4. Failure boundaries formembers I and II [A3-24].

wind loads, will have widely different failure probabilities even if their safetyindices calculated by Eq. A3.zg are nearry equal. In this case, or in similircases, a comparative reliability analysis would require the estimation of thefailure probability by Eq. A3.22, or by altemative, approximate methods. Afew such methods are briefly described in the following section.

l-ll-2.732,01

FIGURE A3.5. Probabitity denelry funerlon.4,(r,) lA3,A4l,

658 I ltMt N(; ()t l;tttt,ol(illnl ltl ll^llll llY

A3.1.7 Approximate Methods for Estimating Failure Probabilities

We first describe the method referred to as normaliz.ation at thc <:hccking pttinl.The principle of the method is to transform the variables, X;, into a sct ttlapproximately equivalent normal variables, Xi, having the following propcrty

n:l:' l()nl )nl.ll,lll ',1!.lAlltl lA{ l{rl t!, lilll

ritlltc:r lhan ir (lrypcr) plirnt'. 'l'lre llrilrrlc ;rlolr;rlrrlity th.pctrrl:, rrpotr llrr. ,,;rlt.lyindox corrcsponcling t<l tlrc splcc ol lltt. t.oorrlrrr;rlt.s ,tf' ;tn(l llxrn llrr. r lr;rr;rt.tcristics ol thc quatlric. 'l'hr-: lrtlttl rrrt'(lrlcnnnt('(l lrolrr llrc r'orrrltlrorr llr;rl llrr.quadric appnlxirnatcs lhc nolrlirrt.rrr l:rilrrrc lrorrrrtl;rrv;rr t.lost.lv:r:l yrossrlrlt,;rlthe chccking point.

Alternative appr<litchcs trt tltt't's(irturtiorr ol llilrut'lrrrlr:rlrrlrlics lr;rvt'lrr.t'rrproposed in [A3-l7l antl lA.l ltll.

A3.2 LOAD AND RESISTANCE FACTORS

Consider a structure characterized by a set of variables with means and standarddeviations X; and oy,, &Dd checking points with coordinates X,f and xl 1t : l,2, . . ., n) in the space of the original and the reduced variables, respectively.By definition,

Xf:X, +ot,xlEquation A3.42 can be written as

py,(Xf) : f (xi.)

:Lo<*ftoN:

Pr(Xf ) : F(X'i.): o(xll_)

xi : x: - Q-t[Px,(xI)]"x,

(A3.38)

(A3.39)

where i : l, 2, . . . , m, the asterisk denotes the checking point, pa, and Pxi: probability density function and cumulative distribution function of X;, re-spectively, I'and S : normal and standardized normal probability densityfunction, respectively, F and iD : normal and standardized normal cumulativedistribution function, respectively, xi : reduced variable corresponding toXi, and 04, : standard deviation of Xi, (see Eqs. Al.l2 and ,41.13). FromEqs. ,{3.38 and ,{3.39, it follows that

d(o 'tP(x,r)l) (A3.40)

(A3.41)

OX,: :Px,(Xf )

where

Xf : ^yx,x,

"Y7, :l-fvlxf

(43.42)

(A3.43)

(43.44)

where the bar denotes mean value.t Once Vi and oyy ?te obtained from Eqs.A3.40 and 43.41, the problem can be restated in the space of the reducedvariables xi,. The safety index B is the distance in this space between the originand the failure boundary. A computer program for calculating this distance,based on an algorithm proposed in [A3-14], is listed in [A3-9]. An alternativcrncthocl lor calculating 0 was proposed in [A3-15]. More recently it has beennotcd thut p can hc obtaincd by nonlinear programming methods [A3-lll.lirlkrwing tlro c:alculati<tn of [3, thc probability of failure is estimated by Eq.A3.3(r. (Frlr an altcrnativc normalization procedure, see [,43-26].)

'Ihc procedure just described is approximate because Eq. 43.36 does nolhold if the failure boundary is nonlinear. A method for reducin! the errors ducto the nonlinearity of the failure boundary was proposed in [A3-16]. Followingnormalization at the checking point, relations similar in principle to Eq. A3.36are used that correspond to the case where the failure boundary is a quaclric:,

lNote that the procedurc is inapplicable if Px,6:) - 0, a casc that cun ulisc lirr liril lirnitetldistrihutions.

The quantity 716 is termed the partial safety factor applicable to the mean ofthe variable X;.

In design applications the means, Xi, are seldom used, and nominal designvalues, such as the 50-year wind, the allowable steel stress Fo, or the nominaly_ield stress F, are employed instead. Let these nominal values be denoted byX;. Equation A3.43 can be rewritten as [A3-9]

X! : tx,X, (t: 1,2,...,n) (A3.45)

where

(A3.46)

The factor 74 is thc plrrliirl srrli'ly lirtlor irpplit'irlrlc lo llpr.' ,',r,', ,,irl tlcsilin vlrlrrcof the variablc X,.

In the particular cltsc irr wlrit lr lltt' v:rtt:rlrlt's ol (()n(('rrr rrrt' llrr. le:rrl (,,:rrrtlthc rcsistancc /1, tlrc pttrlilrl s;rlr.lv l;rclor:,;rrr.rt.lt'rrt'rl lo;rr llrr lo;rtl;rrrrl lt.sr:ttancc litcl0r. lior llrc lt'sisl;urtt' l;rt lor llrr' l()t,tlron ,1,,, rtt ,f,y t\ 1,,r.(l lrr lrt.rr rrl\x {il' \p.

xi^Yx - x ^Yx,

660 LLLMI Nll;ol slllt,oltlllnl lll llAlrll llY

From the definition of thc partial sul'cty llctor (lrct. A-1.44) arttl llrc tlclirritionof the checking point in the space of the rcduccd variablos corrcsponrling to I,: ln R and Y2: ln Q, it follows that if higher-order terms (scc Fq. A3.27)are neglected

n:l.l nl)l ()llnl )Y ()l wlNlr l()nl) lAol()t ri; rit,t oil tt I) tN t;tnNt)nt il):; 66 I

A3.3 ADEQUACY OF WIND LOAD FACTORS SPECIFIED INSTANDARDS: ANALYSES BASED ON THE REVERSEWEIBULL DISTRIBUTION

It was seen in the last section of Appendix A I that according to roccnt thrcshoftlanalyses, the reverse Weibull distribution is a crcdiblc rnodcl ol'hoth norrhur.ricane and hurricane extreme wind spccds. lt was rcccntly shown thirt, lirrregions not prone to hurricanes, this modcl yiclt.ls ltxrrl laclors tl.urt :rlc corrsistcrrlwith the valueTq: 1.3 specified in the ASCE 7-9-5 Stanclanl l9--51. whcrc pdenotes the wind load corresponding to a nominal 50-ycar mcan recurrenceinterval. However, for hurricane-prone regions the value Io: 1.3 was shownto be inadequate [A3-31].

As indicated earlier, estimates of quantities indicative of the reliability of astructure or member are strongly dependent upon the probability distributionof the variates. This is true, in particular, of the mean recurrence interval ofthe nominal ultimate wind loads, defined here as the nominal wind loads (e.g.,the 5O-year wind loads) times a load factor 7p.

A3.3.1 Wind Load Factors for Regions not prone to HurricanesAs an example we consider statistical analyses of Denver data [Al-36]. A

probability plot correlation coefficient analysis of Denver largest yearly fastest-mile speeds recorded between l95l and 1977, based on the assumption thatthe best-fitting distribution is Gumbel, yielded a 62.3 mph estimate of the 50-year wind speed at 10 m above ground. 19, la: 1.3, the correspondingnominal ultimate wind speed would be 1.3rl2 x OZ.Z :71.0 mph, to whichthere would correspond, under the Gumbel assumption, a mean recurrenceinterval of about 500 years t3-91. If taken at face value, this would be analarmingly short recurrence interval, since it would entail an unacceptably largeprobability of exceedance of the nominal ultimate wind load during the life ofthe structure.

However, the 500 years mean recurrence interval is based on the Gumbelmodel. Results of statistical analyses support the assumption that the appro-priate model is predominantly reverse weibull distribution. we assume that asinferred from threshold analysis results reported for Denver in [Al-36], the taillength of the best-fitting generalized Pareto distribution is d = -0.2, and thewind speed with a 5O-year mean recurrence interval is x5u = 60 mph. For athreshold of 37 mph-a valuc consistcnt with lhcsc choiccs lA l-361- thc nurrr-ber of exceedances ol'tlrc: llrtcsltoltl is )t ().2llyr'. irrrtl it lirlkrws l'nrrrr lit;s.Al.66c, dthat A = 5.(A lrrplr. 'l'lrc t'sl irnrrlt'tl rrurxirrrrrrrr lxrssilrlt. wirrrl spt't'tlr,,,u* obtained by lcttirtg N , rr, 111 lr11r Al.(r(x. tl is r,,,.,. tt til, \l+ 5.6410.2 - 65.2 lrrlllt.'lltt't':,1 tltt;tlr'rl rn(';ur r('(lnr('l(('rrl('rv;ll ()l llrr'1()lrrinal ultimatc wind spcctl I \l''r.,,, I lr (r( ) (rli .l I rrrlrlr (rjr -l rrplris thcn:lirrc inlinity (i.r'.. srr,lr rr rvrrrrl r,lr('r'rl r', r'.,lrrr;rl(.rl lo nr'\'r.l.rrrrr) 'll1:,

dn = exp(-cxn1vn)

'Ys = exp(aTavs)

aa : cos[tan-'1VnlV*11

ap : sin[tan-t1vntvoll

(A3.47)

(A3.413)

(A3.4e)

(A3.s0)

whcrc (l is thc salbty indcx givcn by Eq. 43.29.Carc rnust bc cxercised in using simplified approximate expressions for par-

tial safbty facl.ors. Consider, for example, the following expression for the loadfactor 7q1, proposed in [A3-20] as an approximation to Eq. A3.48 for memberssubjected to wind loads:

"YA: | + o.sspvo (A3.s l )

where B is given by Eq. A3.29. For members I and II of Table ,{3. 1, it followsfrom Eq. A3.51 that'yar: 1.0 and ^yA,: 1.31, respectively. However, if Ec;.,{3.48 is used, 'YAt:2.32 and ^yp,, : 1.63. It is concluded that Eq. A3'51may result in misleading estimates of the load factor 78.

Equations A3.41 and 43.48 show that load and resistance factors depenclnot only on the safety index B but on the coefficients of variation Vn, Vp aswell. For this reason, to members having the same safety index B there cancorrespond widely different sets of load and resistance factors as calculated byEqs. A3.47 and ,{3.48. This is illustrated in the following example.

Example Members I and II of Table A3.l have the same safety index calculatcd by Eq. A3.29, P : 3.69, as well as approximately the same failurcprobabilitics, sce column l0 of Table A3.1. Although the values of tr/a for thctw<r nrcnrbcrs arc the same, the respective values of Vgdiffer (Table .A3.l). ltcan bc vcrificd that owing to this difference, the load and resistance factors lirr'the two members (given by Eqs. A3.47*A3.50) are @p, = 0.88 versus dR,, =0.83 and, as indicated earlier, 7p,:2.32 versus lA,,: 1.63.

Conversely, the use in the design of various members of the samc sct rtlload and resistance factors does not ensure that those members will havc tlrcsame probabilities of failure. This creates difficulties in the developmcnl olrisk-consistent load and resistance factors for codified design. Thesc diflicr"rltics,and proposed approaches for dealing with them, are discusscd in l,43-91 lntl[A3-211. Problems related to codes of practice arc also tliscttsscrtl irr lA3 221.

lI

662 t l t Mt Ntti ()t tiilt(.,ott,nAt nl ilnlll ily

estimate is of course subjcct to sarnplirtg crn)rs: thc actual rnaxirtturn 1'rossiblcwind speed may be higher than65.2 mph, and the mean rccurrcncc intcrval ol'the nominal ultimate wind speed may be finite, though likely much krngcr than500 years. Despite the uncertainty inherent in our estimates, our result suggcststhat, for nonhurricane winds, a load factor of l.3-specified in the ASCE7-95 Standard on the basis of practical experience-is adequate from a proba-bilistic point of view. This is contrary to what would be concluded if thcanalysis were based on the assumption that the Gumbel distribution holds.

Based on the data of [A1-36], it was shown in [A3-31] that similar resultshold at most stations not subjected to hurricanes.

A3.3.2 Wind Load Factors for Hurricane-Prone RegionsAs indicatcd by thc analyses of [3-7ll and [A3-31], the reverse Weibull dis-tributions that are appropriate for the modeling of hurricane extreme winclspeeds generally have longer tails than is the case for non-hurricane windspeeds. In hurricane-prone regions nominal ultimate wind loads obtained inaccordance with the ASCE 7-95 Standard [9-5] are based on an effective winclload factor equal to 1.3 x (1.05)2, where 1.05 is a factor applied to basic(50-year) hurricane wind speed estimates. Extensive calculations in [3-71] andtA3-31] showed that estimated mean recurence intervals of nominal ultimatchurricane wind loads based on the ASCE 7-95 Standard can be of the order ol'500 years or less. This is substantially less than for loads induced by non-hurricane winds, and is clearly inadequate.

REFERENCES

A3-l P. Thoft-Christensen and M. J. Baker, Structural Reliability Theory and Ap-plications, Springer-Verlag, Berlin, 1982.

A3-2 R. Fleming, Wind Stresses, Engineering News, New York, 1915 (Reprintctlfrom Engineering News, 1915).

A3-3 E. Simiu, J. R. Shaver and J. J. Filliben, "Wind Speed Distributions antlReliability Estimates," J. Struct. Div., ASCE, 107 (May l98l), 1003-1007,and (crrata) Oct. 198 I , 2052.

A3-4 T. V. Galarnbos ct al., "Probability-Based Load Criteria: Assessment of Current Design Practice," J. Struct. Div., ASCE, 108 (May 1982),959-977 -

A3-5 Y. K. Wen, Statistical Combination of Extreme Loads," J. Struct. Div., ASC'Ii,103 (May 1977), 1079-1095.

4,3-6 Y. K. Wen, Clustering Model for Correlated Load Processes," J- Struct. I)iv.,ASCE, 107 (May l98l), 965-983.

A3-7 R. D. Larrabee and C. A. Comell, "Combination of Various Loacl l)nrccsscs."J. Struct. Div., ASCE, 107 (Jan. l98l),223 239.

A3-8 R. D. Larrabee, Approximate Stochastic Anul.y.ti.s tll ('rntrltittul lttulittg,l)c1tl.of Civil Enginccring, MI'f, MIT CU RTU-2u, Onlcl No. (r.l(). ('rrrrrbritlpt',Scpt. l97lt.

ilt II nt N(;t l; 663

n3 9 li. lillrrrliwrxxl ct ll., l)t'ra'lt4trttt'trt ttf tt l'nilttltilit.v Iilt.srtl l,rxttl ('ritt,rirttt.littAtilt'ri( (ttt Ntttitttutl Stttrrtlttnl /-58, Nlls Spcciul l)ublicltion -577, Naliolrtrl llrrrcirrr ol S(:rnclards, Washington, DC, Juno l9tl0.

A3 l0 A. M. Hasol'erand N- C. Lind, "Exact and Invariant Second-Mgmcnt ('txlcIrorrrrat," J. Eng. Mech. Div., ASCE, 100 (Feb. 191.4),829_944.

A3-lI M. Shinozuka, "Basic Analysis of Structural Safety," J. stuct. Eng., llltl(March 1983),721-740.

A3-12 E. Simiu and J. R. Shaver, "wind Loading and Reliability-Based Design,"in Wind Engineering, Proceedings of the Fifth Intemational Conference, F91Collins, CO, July 1979,Yo|.2, Pergamon press, Elmsford, Ny, 19g0.

43-13 W. McGuire, Steel Structures, Prentice Hall, Englewood Cliffs, NJ, 196g.A3-14 R. Rackwitz and B. Fiessler, Nonnormal Distributions in Structural Reliability,

SFB 96, Technical University of Munich, Ber. zur Sicherheitstheorie der Bau-werke, No. 29, 1978, pp. l-22.

A3-15 O. Ditlevsen, "Principleof NormalTailApproximation,', J. Eng. Mech. Div.,ASCE, 107 (Dec. 1981), l19l 1208.

A3-16 B. Fiessler, H.-J. Neumann and R. Rackwitz, "euadratic Limit states inStructural Reliability," J. Eng. Mech. Div., ASCE, 105 (Aug. l9-t9),661_676.

A3-17 M. Grigoriu, "Methods forApproximate Reliability Analysis," struct. safety,t (1982t 1983), 155-16s.,43-18 o. Ditlevsen, "Gaussian Safety Margins," in proceedings Fourth Interna-

tional Conference, Applications of Statistics and Probability in Soil and Struc-tural Engineering, University of Florence, June l3-li, 1983, pitagora Edi-trice, Bologna, 1983.

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Application to offshore structures , NBSIR 84-2921 , Nat. Bureau of Standards,Gaithersburg, MD, 1984.

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ELEMENTS OF STRUCTURAL RELIABILITY

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A3-30 M. Matsumoto and P. Prenninger, "Consideration of Higher Vibration Modesin Reliability Analysis of Bridge Structures," J. Wind Eng. Ind. Aerod..,32(1989), 17r-180.

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iln

APPENDIX A4

PRESSURE COEFFICIENTS FORBUILDINGS AND STRUCTURES

This appendix presents pressure coefficients for various buildings and structures[4-101.x These coefficients were obtained from tests conducted in uniform,smooth flow. (For information and comments on the limitations of results ob-tained in such tests, see Sections 4.5 and 4.6.) In the tables that follow localpressure coefficients are denoted bV Cil".

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INDEX

Accelerations, building, estimation of ,209,336,3M,354,357, 512

and human discomfort, 512Across-wind correlations, 65, 184, 334Across-wind forces, 2, 157, 219, 302Across-wind galloping, 230Across-wind response, 219

analytical models, 219chimneys, towers, and stacks, 224, 383suspended-span bridges, 454tall buildings, 343

Added mass, 502Addition of probabilities, 592Adiabatic lapse rate, 9Adiabatic processes, 8Admittance, aerodynamic, 182

mechanical, 197Advection, turbulent energy, 36Acroclastic instability, 2 l6AcK)clasticity, 2 I 6Aeroclastic stability, infl uence o1' bridge

characteristics on, 414Air infiltration, 188Air mass, specific, 336Air masses, 25Air-supported structures, 477Air viscosity, 138Alleviation of wind-induced response, 356,

398, 475, 477, 478Along-wind bridge response, 259Along-wind forces, 2

676

Along-wind pressure correlations, 67, 186Along-wind tall building response, 328American National Standard, A58.1, 46, 84American National Standard

ANSI/ANS-2.3-1983, 128, 568Antenna dishes, 421Anticyclonic circulations, l6Arctic hurricanes, 24Arrival, rate of, 604Aspect ratio, 390, 420Atmospheric boundary layer, 16, 33Atmospheric circulations, 5, 18Atmospheric hydrodynamics, 11Atmospheric motions, 18Atmospheric pressure, 7, ll, ll2, 127Atmospheric scales, 19Atmospheric thermodynamics, 5Atmospheric turbulence, 51Autocorrelation function, 637Autocovariance function, 636Averaging time, wind speeds, 69

for hurricanes, 78

Background response, 214Balance, frictionless wind, 14Barotropic flows, 35Base pressure, 159Bayes' rule, 594Beam winds, 490Beaufbrt nurnbcrs. 5 I tlBdnard, l4tJ

llelrulrlli's ctlruliorr, l4ollkrckagc, wintl turrncl, 2()l{Illowing ol'nxrl gruvcl, llttJ, .l75Blull:body acnrclynarnics, I ll5Bora winds, 2tlBoundary laycrs, a(nxrsphcric, 16, ll

hurricane, 11, l16intemal, 7llaminar, 145thickness of, 41, 46three-dimensional, 3tlturbulent, 145two-dimensional. 38

Bow winds, 490Bridge decks, buffeting response, 258,461

flutter,246, 461galloping, 474torsional divergence, 243, 451vortex-induced response, 454

Brighton Chain Pier fatltre, 447Buckling, lateral,45lBuffeting, 328

of bridges, 259, 461response, general problem of, 265of rall buildings, 328

Buffon, obsewations on pedestrian-levelwinds, 188

Building codes, 576Building response, 327Buys-Ballot's law, 16

Cable roofs, 476Cables, bundled, 239Cauchy type distributions, 606Center, aerodynamic,2

elastic, 2, 243, 247, 352Central limit theorem, 604Centrifugal force, 14Change of terrain roughness, 71Chaotic dynamics, 81, 236Chimneys, across-wind response of , 224, 383Chinook winds, 28Circular frequency, 196Circulations, atmospheric, 5, 18Cladding design, 364Climatology, 2, 9lClosure relations, 35, 3(r, 37Coefficients, aerodynarnic, I 55

pressure, 157, 665Cocllicicnt ol' variation. (()lCohcrcncc lunclion, (4, (r ll((lrrnlirrl critcril, 5 14, 5 l()(\rrnpli:rrrl oll,slrorc pllrllornn. ,l() /('orrrlxrrnrrl prirhirbililics. 5') I

lNr rt x 6ll('olililill{tlloilirl lltttrl rlyttiIrrtts. l.l(r. 5l]('oilllIrl.'r lrroliilllils, rrlorrli wirrtl rcsl)()ltsr:,

I lt)( irnrlertstrliort. 10. 2l{('()n(lallor!irl prob:rbilitrcs, 59.1( irrrthrt'tion, l0( irnlitlt'rrt'c irrlclvlls, 616, 623

(:xlr'(:nr(: wirxl prcdictions, 97(irnlitloncc lovcl, 6 l6Construction stage, bridges, 446Continuity, equation of, 34Convergence, air, 18Cooling towers, see Hyperbolic cooling

towersCoriolis, 12

effects and wind tunnel testing, 292force, 12, 38parameter, 13parameter values, 14

Corpus Christi, extreme winds in, 104Correlation, see Cross-correlationCorrelation coefficient, 602Co-spectrum, 64, 637Cram6r-Rao lower bound, 99,620Cranes,422Critical divergence velocity, 245, 454Critical flutter velocity, 256, 461Critical region, flow about cylinders, 158Cross-correlation, naffow band, 65

across-wind, 65along-wind, 67effect upon response, 342

Cross-covariance fu nction, 637Cross-spectrum, 64Cumulative distribution function, 596Cumulative mean exceedance estimator.

624Cyclones, 16

extratropical, 25tropical, 2l

Cyclostrophic wind, 30Cylinders, flow past, 148, 159

Dalton's law, l0Damage caused by storms, 24, 30Dampcrs, 356, 398, 476I)unrping, l9(r

rrcnxlyrr:rrrrit, .1.l.lcrilical, l(Xrrreg;rlivc,.l,l)titlio. lt)(rl:rll lrrriltlirrp's, I l.)

rlc Ilruttt ('sllilrirlor!, a),t.1

l)t'tr lltrtIrp', t llr'rrolr, .) l,l

678

Density function, probability, 597spectral, see Spectral density function

Depressurization, during tornado passage, 556Derivatives, material, 137

motional aerodynamic, 251substantial,137

Deviating force, 12Deviatoric stress, 139Difftrsion, turbulent energy, 36Dimensional analysis, 275Directionality, wind, 120, 308Discomfort, wind-induced, 5 I IDissipation, turbulent energy, 36, 56Distribution function, cumulativc. 596Distributions, pnrbability, Cauchy typc, 60

cxponcntial typc, (106Frdchct. (106

Gaussian, 604geomctric, 603Cumbcl, 606joint, 59ti, 6l I

of largcst valucs:Typc I,606Type II, 606Type IIl, srr Reverse Weibull

distributionlognormal, 604mixed, 104, I 19normal,604of peaks in random signals, 639Poisson, I 15, 603reverse Wcibull, see Reverse Weibull

distributionsDivergencc, air, l8

torsional,454Downdrafi, thunderstorms, 28Drag, 2, 155

coellicients, 155, 157surfacc.44timc-dcpcndcnt, 175

Drift, story. 328Drifting, snow, ltJtlDry adiabatic cquation, 9Duration of storm, 58, 335Dynamic pressure, 140Dynamics, structural, 195

Easterly wind, 13Eckert number, 277Ecliptic, plane of, 6Eddies, 53Eddy conduction, l0Eddy viscosity, 35Efficicncy ol cstiltr:rtors. (rl(r

l'lkrrr:rrr laycr, (urbulcrrl, .19Ekrnan spiral, 3tiElastic center, 2,243, 247, 352Energy, turbulent kinetic, 36, 55Energy cascade, 55Energy dissipation, 36, 55Energy production, 36, 55Energy savings, 188Energy spectrum, turbulent, 55Ensemble, 629Epochs, 605Equation of state, perfect gases, 7Equivalent static wind loads, 330Ergodic processes, 629Enors, 96

extreme wind estimation, 96, 98, l16associated with quality of data, 96modeling, 96response of tall buildings, 339sampling, 98, 116

Escarpments, flow over, 73Estimates, statistical, 615Estimation of extreme winds, 91Estimators, efliciency of, 616

reliabllity of,622reverse Weibull distributions, 624

Evaporation, 10Expansion, air, 7Expectation, 602Expected value, 602Exponential decay coelficients, 65, 334,342Exponential type distributions, 605Extratropical cyclones, 25Extreme value distributions, joint, 6l I

relations between Type I and Type IIdistributions, 608

reverse Weibull, see Revene Weibulldistributions

Type I, 605Type II, 605Type III, see Reverse Weibull distributions

Extreme winds. confidence intervals forestimates of, 96, 116

estimation of:numerical example, 98from short-term records, l0O

in hurricane-prone regions, 102in marine environment, 100in well-behaved climates. 95

Eye, hurricane, 22

Failurc, probability ol,.]17. 16(). ((X)Fastcst rnilc winrl, 7olrilling, ltl

l;ilsl 1',rrsl, .111. /()Iiirsl l:rw ol lltclrrrrxlyrr:rrrrrrs" /Irlutlcr. 2.1(r

classicll. l,l(rpancl, 24bsinglc<lcgn'e ol licctkrrn, 246stall,246of suspcnsion span bridges, 247vcbcity, 256

Foehn winds, 27Fourier integrals, 632Fourier series, 631Fourier transform pair, 632Fr6chet distributions, see Type II distribution

of largest valuesFree atmosphere, 16Frequency, circular, 196

natural, 196reduced, 250relative, 591

Friction, effect on air flow, 16Frictionless wind balance, 14Friction velocity, see Shear velocityFront,26,79Froude number, 277

Galloping, 230across-wind, 230of coupled ban, 236incipient, 233power line, 478,479suspension-span bidge, 47 4wake,237

Gaussian distribution, 604General circulation, 20Generalized coordinates, 201Generalized force,2O2Generalized mass, 201Generalized Pareto distribution, 605Geographical effects, atmospheric

circulations, 20Geometric distribution, 603Geostrophic wind, 14, 35Girders, 421,422Glass breakage, 364Glauert-Den Hartog critcrion, 234Goodness of fit. 614Gradicnt height, l7Gradient wind spccd, l6Gravcl, bkrwing ol rrxrl, lt{tl. 175Gust, lirst, 28, 7(.)(lrrst Iront, 79(irrsl lcsporrsc lirt'lor', I ll(irrsl s1x'ctls, (r(), 7()

irr lrrrniclrrrcs, 7l{

INDEX 679

(iuyctl towcrs, 422lirr ofl'.shore platfbrms, 492

Harmonic load, response to, 196Head winds, 490Heat of condensation, l0Helicopter landing decks, offshore platlirrrns,

487Hills, wind flow, over, 73Histograms, 596Horse latitudes, 20Hourly mean speed, 70Human response to vibrations, 512Hurricanes,21

atmospheric pressure in, 22boundary layers of, 77destructive effects of, 24, 30estimation of extreme winds in, 102fosses due 1o,24,30statistics, U.S., 108, lll

world,23, l18structure of, 22

Hydrodynamics, atmospheric, I IHyperbolic cooling towers, 404

groups of, 416

Importance factor, 579Improvement, of surface wind conditions,

536Impulse function, unit, 197Incompressible flow, 139Independence, stochastic, 595Indicial functions. 256Inertial forces in flows, 143Inertial subrange, spectra in, 56,292Infiltration, air, 188Initial distributions. 606Instabilities, aerodynamic, 216

aeroelastic,2l6Insured losses, 30Integral scale, definition, 53

relation to turbulence spectrum, 58reproduction in laboratory, 290variation in atmosphere, 54

Intensity, turbulence, 52, 304Internal boundary layer, 7 IIntcrnrl prcssurcs. 172. 40.5Inviscid lluitls, 140Isobars, l2Isotnrpy, lrral, 5(r

.lack-up plltlirlrrrs,,ll{7

.1cl, wrrll. 2tl

.krirrl prob:rhilily rlistrilrrrtiorrs, 5()llcxlr'('nr(' vrrltrt', (rl I

680 TNDI X

Keulcgan-Carpcntcr nunrbcr, 504Kinematic viscosity, 144Kiissner coefficients. 25 I

Lapse rate, 9Largest values, see Extreme value

distributionsLatent heat of vaporization, l0Lateral buckling, 451Law of wall, 39Least squares fitting, 620Lieblein's method, 620Lift,2, 155Lift, coellicients, 157Linear systems, 195, 200Load tactors, lllt, 659, 661

Location paramctcr, 607Lock-in phenomcnon, 217

in turbulcnt flow.229Logarithmic law, 42Lognormal distribution, 604

Marginal probability density, 599Markov chain, 572Mature stage of storm, 22, 28Mean recurrence interval, 603, 608Mean turbulent field closure, 36Mean value, 601Mean velocity field closure, 35

profilesMedian, 601Mesoscale, 19Meteorology, IMicrometeorology, 2Microscale, l9Missile, tornado-bome, 561Mistral. 2llMixing lcngth, 35Mrxlc, 601, (rll

Nculrll stratilication, 9, JllNcwtonian fluids, 138Nonstationary flows, 328Normal distribution, 604Normal modes, 200No-slip condition, 143

Ocean, winds over, 45, 81Offshore structures, 487Order statistics, method of, 620Orographic winds, 28Orthogonality of normal modes, 201Outcrossing rates, mean, 640Outer layer, atmospheric, 39

Parent population, 606

Peak local wind loads, 187, 366, 581Peaks in random signals, 639, 641Peaks over threshold method, ffis, 613, 624Pedestrian discomfort, 5 17Percent point function, 607

Prandtl number, 277Prediction, weather, 19Pressure, coelficients, 665

drop in tomadoes, 127, 556dynamic, 140fluctuations, 182,366gradient force, I Iinternal, 172,405

Probability density function, 597Probability distributions, see Distributions,

Reverse Weibull distribution, 97, I 15, I 17, rcgirncs, 48605,612,624

Reynolds number, 144

llarttkrrtt sigrrlrls, (r2()

llandonr v:rrirrbk:s. 5()(rRandorn viblittiotts. l()(), lO,l . ,'ll/Rapid distortion lhc()r'y, Ill5

Reattachnrcnl, lLrw, 151Recurrcncc irrtcrvlrl. ilrcril, (r(ll, (r(lN

Reliability, ol cstinr:rlors. (rllstructural, (r29of wind spccrl rlirla, ()2

Resonant responso,2l4Return period, 603

148around flat plate on, 146

effect on wind tunnel tests, 297relation to Strouhal number, 150

Rib height, cooling towers, 406Richardson number, 277Roofs, air-supported, 477

blowing ofgravel, 188, 375cable-supported, 476design of, 375

Rossby number, 276effect on wind tunnel tests, 292

Roughness, cylinder surface, 161Roughness length, 42, 333Roughness regimes, wind speeds in different, 199

47Roughness of tenain,42

Safety, of cladding, 364factors, 659indices, 650

Saffir-Simpson scale, 118Sample data, 614Sampling errors, 98, I 16Saturation pressure, l0Scale parameter, 607

ll il rl r aflt I

liltr';tr rllrr, rlt t')lilrrrl lt'rrrr rr ilrl r'1t','rl r,', rrr,l ' lllll5lrrlrrrl:,, ltl','it1,11,111,,t ltltlrltl{ tt(', I il{

:irllliu tly r ootrlttt;tlr'r. i /llutliutly trl;rlrortr. nrlr r.p!'r'rl lrtrrltlr',,.

.lHll'lwr'r'rr lilt lton vt'lrrt tly ;rrrrl 11r'orlloplttt

wirrtl, ,l I

s1x'clr:r irr irrcrlilrl srrlrr:rrrgr'. 5(riuxl wirxl spccrls irr rlillcrr:rrt lruglrtcss

Similarity requirements, wind tunnel tcsting,274

Carlo, 616Single-degree-of-freedom systems, 195Skew wind, tall building response to, 330Slender towers, 383Snow drifting, 188Solar heating of Earth's surface, 5Southem Building Code, 78Spectral density function, 633

definition, 633of lateral velocity fluctuations, 68of longitudinal velocity fluctuations, 55of multi-degree-of-freedom system

response, 207of one-degree-of-freedom system response,

one-sided, 635two-sided. 635of vertical velocity fluctuations, 68

Spectral gap, 59Spectrum, see Spectral density functionSplitter plates, 155Spoiler devices, 398, 475Squally winds, 28Stability, infl uence of bridge characteristics

on aeroelastic, 474Stable stratification, 9, 50

Ratc ol arrivitl, l'0issorr rlrslrrlrillroilrt, (r(l.l :itllillltr .t[r r= lr=vr=1. arl.l, frlf,

dependence of flow, around cylinders on, Simulation of random processes, MonteLocal Clirnatological Data, 92,95, l2l Parseval's equality, 633

Losses, hurricanes vs. earthquakes, 30 Phenomenological relations, 34Pickands estimator, 624Poise,138Poisson distribution, 115, 603Poisson's equation, 9Poles, iced,477Porosity, l72Power law. 46Power lines, 238, M6,471

Mean velocity profiles, see Wind speed and tornado winds, 129

nor-rnal,2(X) probability Scales, atmospheric, 19 Stacks, 383Molocultr conduction, l0 Probabitity plot correlation coefficient, 614, Scale of turbulence, see Integral scale Stagnation pressure, 159Mon)cnts, mcthod of', 618 6l'7 Scruton number, 223 Standard clcvialion, 602Monin coordinates, 57 Probability theory, 591 Seasonal effects, atmosphcric cinrrl;rlions. 20 Sl:rlioruu'y nrrrtkrrrr signrrl, (r29Monsoons, 20 Production, turbulent energy, 36, 55 Second-order closurc, 37 St:rtistrcs. (rl4Monte Carlo methods, 516 Profiles, mean speed, see Wind speed profilcs Section models, bridgc lcslirrli, ;l,l') Strxllrstrr'. 5()5Morison equation,S03 Scll'-cxcitcd firrccs, 216 .'itoke . l.l,lMulti-degree-of-freedom system, 200 Quadrature spectrum, 64, 637 Scmisubrnorsihlc pl:rllirlrrrs, ll{8 lilrrrrrr rrrfrr'. .r'lMultiplication rule, probabilities, 593 Quartering winds, 490 Scparation, 19, 14.\ lilirl.r ',1",1r'nr'.. I'rr)

Sh:rpc pltr:rutclcr', .rrr"l;ril lr'rt1'.llt |,u:rrttrllr I'lrirlrlr' ;rlrrrrr, ,rlurr'.Ilrr'ur (]

NationalBuildingCodcof'Canada,329,344 Ramps,windllowovcr',71,579 Sh:rrpetlgctl plrtlcs. lltwl)irrl , l.l{} ',lrr'irlrtt';rr . llrrNatural frcqucncy, 196 Rantkrntncss, 5<)2 Sltt':tl, sttll:ttt', .l() ',lt,rrrlr,rl rrrrrrlri r. lill l', ' '/a'Navicr Slokcs ct;ruttions, l3() llitntkxrt plrlt'csscs. (r.)() Slrcltr l;tycr, l.l(r ',trr, trrr;rl ,l\nirnrr,'. l{,i

682 tNt)t x

Stnrctural rcliability, 64:lSuhspan girlloping, powcr lincs, 479Su;rcrcritical rangc, lkrw about cylinder in,

l5tlSurlacc drag cocflicient, 44Surlircc laycr, 39Surlacc roughness, effect on pressures, 159,

29rJ. 390Surlace shear, 39Surlircc winds in built environment, 520Suspcndcd-span bridges, see Bridge decksSynoptic scalc, l9

'lrurrrtrr Nanrrws bridge, 448'l'uil lcngth l)lrilrnctcr:'l'ypc ll tlistlibution, 607

'l'ypc Ill tlislr-ibution, 612'llll buiklings. 327'l'atsurrrakis..lo'l'aut strip rruxlcls, 448'l'aykrr's lrypothcsis, 53, 57'l'cnsioncd lirbric, structures, 477'l'cnsion lcg platlorms, 497Tcnsor, stross, 139'l'crrain roughness, 42'l'csts, statistical, 614'l'lroodorscn's circulation function, 249'l'hcrrnal convcction, effect on wind profile,

49'l'hcrrnally direct circulations, 20'l'horurodynamics, atmospheric, 5

first law of, 7'l'hrcshold,605'l'hundcrsturnrs, 28, 79'l'irrrc-rl:pcnclcnt fbrces, 174'l'Ml) rlt:viccs, 356, 398, 476' lirpogr':rphit' lirctor, -579'lirrrntkr lxrrtt' tttissilc spccds, 561'lirtrluItr's, .111, 122, 55 1

'lotsuurrl rlivt't11'tu t'. 241, 45 1

'lirt:,tottirl t('rilx)ns(', llrrrrlgcr, l,l l. .t'17, .l6ll( ( nltitrl rlist otnlrttl (ltte 1o. -5 14lrrll llrrhlinlls, l5o

lirt;rl ptob:rbiltly. llrcolcttt ol', 593'lirwcrs willr t ilt rrllu closs-scction. 3tl3

(rrrssctl.4.l5'l r:rrrsclilicul llow, abt>ui cylindcrs, 158'l nrrrsrrissiott Iittcs, scc Powcr Iincs'l'rir't:llrrliu' rrrclitlional circulation. 20'l'rrrpit:rl cycloncs, 2I

cxlrenrc wirrtls, 102

slirlislios, LJ.S., l0t{, I I I

world, 23, I ltlstructure of, 22

Trussed frameworks, 420Tuned mass dampers, 356,398, 416Turbulence, 33, 51

decay of, 55effect on aerodynamics, 155, 168, 295effect on bridge stability, 462energy, 36intensity, 52mechanical, 33scale, 53simulation of , 29O, 292, 3O4

Type I distribution of largest values, 96, 606Type II distribution of largest values, 96, 606Type III distribution, see Reverse Weibull

distributionTyphoons, 2l

Underlying distributions, 606Unit impulse function, 197Unstable stratification, 9, 49Upcrossing rate, mean, 640Updrafts,22,28

Van der Pol oscillator, 222Variance, 602Variation, coefficient of, 602Velocity, shear, 39Velocity defect law, 40Velocity fluctuations, 51

coherence,64co-spectra, &cross-spectra, glateral, 68longitudinal, 55quadrature spectra, 64spectra, 55vertical,63

Vclocity pressure, 578Vclocity profiles, see Wind speed profilesVenturi cffect, 536Vibrations, human response to, 512Virtual mass, 177Viscoelastic dampers, 362Viscosity, air, lM

eddy, 35, 145kinematic, 144units, 144water, 144

Viscous eflbcts, 143

vorr K:ilrrriirr corrslturl, 42vorr Kiirrrriirr spcctrunl, 60von Kiinn{n vortcx trail, l4llVortex, constrainecl, I 42

flow. 140formations, two-dimensional llow. I4(rfree. 14l

Vortex-induced responsc, 2 l7alleviation of, 356, 39llbridges,454stacks, 383structural members, 474tall buildings, 343two-dimensional flow, 199

Vortex shedding, 148Vortex trail behind cylinder, 148

Wake buffeting, 257Wake galloping, 237Wakes in two-dimensional flow, 148Wall, law of, 39Wall jet, 28Wave length, 55Wave number, 55Weather prediction, 19Weibull distribution, see Revene Weibull

distributionWell-behaved climates, 95Westerly wind, 12

rl rFi: tirl3

Wirrrl lrrcrrks, lflHWittrl tlitct'lionrrlrly, l,tll, l(tHWiltrl hrrtrh', 1x'irl. lll/, lfifi,5tslWintl lrrcsstrrr.l. llrit luirlrrrll, lH.', lfl(rWirrtl spt'crl tIrlrr, ().), lO.'Wintl sptttl prolrlt.r, l/

itt tlrllt'rr'rrt ttttt;tltttt'r.l rt';',rrrrr.l, ,l /irr lrrrlicrrrrt. llrwr, //krg:rtitltrrrrt' lirw. .llnr'irr clrirrrllc ol rorrglrrrt'ss. 7lovcr hills, 7.1ovor wator, 4-5powcr law, 46stable stratification, 50thunderstorms, 79unstable stratification. 49

Wind tunnels, blockage of, 299dimensional analysis. 275effect of incoming turbulence, 172,295,

304effect of Reynolds number, 297effect of Rossby number, 292low-rise buildings, 295, 303, 363similarity requirements, 274suspended-span bridges, 448tall buildings, 292,3mtypes of, 280

Zero plane displacement, 44

ABOUT THE DISK

DEVELOPMENTAL COMPUTER-BASED VERSION OF ASCE 7.95STANDARD PROVISIONS FOR WINDLOADS

INTRODUCTION

A rcport with the above title [7-5] was issued in the public domain as NIST TechnicalN<xc 1415, National Institute of Standards and Technology, Gaithersburg, November 1995(GPO Stock No. SN003-003-O3377-4, Govemment Printing Office, Washington, DC 20402).'l'hc soliware included on the disk attached to this book is excerpted from [7-51. Thc diskis usclul as an instructional/computational aid allowing wind loading calculations based onthc ASCE 7-95 standard and its commentary to be performed dependably and in a smallli'ircti6n ol'thc timc required for manual calculations. The software from [7-5] was creatc(llirr rlcvckrprncntal purposes, chief among which was the verilication of the correctncss.t.orrrplcttrx'ss irnrl consistency of the ASCE standard provisions. It is not an ASCE documcntor' :rrr ollit iirl vt.rsion ol'the ASCE 7-95 Standard. The provisions on which the disk is basctl:rrr.torrlrrint.tl irr llrc ASCE,7-95 Standard [7-ll, which can be ordered from the AmcricattSot rclv ol ('rvil lirrginccrs, Ncw York, NY 10017.

f'lr.:rsr'rrolr llurl tlrt'l/lintl Itxuls 7-9-5 soltware (35 installed files) are not subject trrr,Iyrr1'lrt llorvt'vr'r. tlrt'irrsl:rllation soliwarc and fbnt file on the disk are protcctcd byropyril'lrl ;rrrtl llrrrs t:rrrrrol hc lrccly tlistributcd without approval from the software manul;rr trrrt-r 'llrt. lollowing 1r:xl cxccrptcd liorn [7-5] pertains only to the 35 installcd progritlll:rrrrl rr':rrlrrrt' lilcs: ^"1'lrc soliwarc is not subjcct to copyright in the United States. Rccipicrttsol tlrt.soliw:rr'(.:tssunlc all rcsponsibility associated with its operation, modification, rrtairrk.nlur(r., untl srrbscrlucnt rc-distribution. Any mention of commercial products is lttr inlolrnlrliorr orrly; it tkrcs not irlply recommcndation or endorscmcnt by thc pnrclttccr ol'lltt'sollwlrre, nor rkros it irnply that the products mentioned arc ncccssarily llrrr bcst lirr tlrcl)u rl)( )s(: . ' '

A t':rlt'rrlirliorr rrxarnple (Scction 17.3) and ailtlititlnal inlirnttlttiotr ltte ittt ltttlt'tl irt ('lrltplt'r17. Sce Il7 5l lirl tttott t'itlcttlitlion cxittttplcs-

684

Alti illl llil r rr;t' ltllii

SYSTEM REOUIREMEN IS

o IllM l'('ol t'orrrP:rlilrk'tultllttr.t rr'rllr lH(r l\.lllz ,l lll,lt, r 1,r,,,,.,,.,,ro Wintltlws .l.l or lrrlilrt.r (Wlrrhrn:.t)l r orrrl:rlrlrk.lo 4 MII lurrrl rlisl sp:rr'r.o 3.-5" lLrppy tlisk tlr rv.'

Notc: Thc dclirrrll lirrrt lol scrt'ctr tlrslliry r:, llrt. WrrrrIrtv:;'l'rl.ltrirurl lurrl (:oltrt. ,,y,,1(.ntr, u:,(.85l4oem). I)o lx)l cllltttA('lltr tlcl;rrtll lottl;rllcr rrrsl;rll:rtiorr unl(.s:i yr)ll lr:rvt.:r prolrlt.nr rvrllrthe screen display. Sotttc ptittlt'rs (c.g. llcwlcll Itrrt kirll) rr'r;rrilt. rr rct.crrl vt.rsiorr pl llrt'Microsoft True Typc lont MS l,incl)raw ('l'ruc'l'ypc). Scc bclow arrtl lcirtlrrrc lilc lirl rrurrcinformation about fbnts firr scrccn display and printing.

HOW TO MAKE A BACKUP DISK

Before you start to use the enclosed disk, we strongly recommend that you make a backupcopy of the original. Making a backup copy of your main disk allows you to have a cleanset of files saved in case you accidentally change or delete a file. Please take the time nowto make a backup copy, using the instructions below:

L Insert the Wind Loads 7-95 disk into the floppy drive A of your computer.2. At the DOS prompt A: ) rype DISKCOPY A: A: and press Enter. you will be

prompted to put in the source disk. Press any key to continue.3. When prompted to insert the target disk, remove the original disk and insert a blank

disk into the floppy drive. Follow the instructions on the screen to complete the copy.4. Remove the original Wind Loads 7 95 disk and store it in a safe place.

INSTALLING THE DISK

The enclosed disk contains files saved in a compressed format. Running the installationprogram will uncompress the files and copy them to your hard drive in the default directoryC:\DWL795.

l. Insert the disk into the floppy drive of your computer.2. Choose File, Run in Program Manager and type A:INSTALL. Click OK and the

opening screen of the installation progftrm will appear. (In Windows 95, choose Runfrom the Start Menu.)Click OK to accept the default destination drive and directory C:\DWL795. If youwish to change the default destination, you may do so now.Thc installation prograrn will urrconrprcss and copy 35 files to your hard drive in thedcfault (C :\DW!,795 ) or rrscr tlt's igrr:rletl tl i rt't'lory.

-5. A Program (irorrp, Wirrrl l,rxrrls 7 (t5, willr tw() l)nr11rrr ltr.rrrs. Wilr-star and Rcad Mr:, will lrt' t tt':rlcrl

RUNNING THE PROGRAM

'l'lrt'irtlr'rlirt't'rll lltt'1ttrr1',t;rtn ltjtf,;t :l;llil;ltrl \\'lr,l,rrr.. rrrr lrr lr;rr ,rt tlrr lo;r rlr'l;t',tittl., lltr.Itl lltt'lrolloltt. ltt lltc rttirlrlll lltr'tl tr jt \\rll,ttt!' jrt!'jl itlr. r, \\ltrl'r\\.. ;rrrrl rlr,tll1, lrorr,, lrll

J.

6BG nil()t,l llll l)llih

bc;rrcscrrtcrl. lt is hclplirl lirr uscrs ol llris prrrgrrrrt lrt know solltc basic Wirttkrws littttliottsancl opcration proccdurcs. Cornmands can bc c()nlr'ollctl thnrugh options l'nrttt the: Wintlowspull-tkrwn mcnu.

l. To launch the program, double click on the Winestar icon. Click OK. OK'2. An opening screen with will appear with introductory information. Click OK kr vicw

the next information page. Click OK again.

3. In order to view and save a log of your consultation session, Open File, New l'rottr

the top menu-bar. Choose the Dialog option and click OK. The default settings inclutlc

Questions/Answers and Advice. Click OK. (You can also add Pictures and Titlc ilyou wish.) A second window Dialog untitled.log will appear in the background whichlogs information as you go through the consultation.

The interactive consultation is directed by dialog windows with lists to pick from orblank spaces to fill in. When you enter information and click the OK button, the next qucstiottwill appear. Some screens provide explanatory information you can access by clicking onthe EXPLAIN button.

The whole knowledge base of the program is incorporated in files with the extension Kl|called knowledge bases and in the picture database file PICTURE.DBA. The current KIJfile appears on the top title bar. Knowledge bases are divided into sections. The currentsection appears on the bottom status line.

If you launch the program from the icon, you will automatically begin a consultation inINTITIAL.KB.

To interrupt a consultation, click the Stop button. To resume the consultation, selcctConsult/Continue Consultation.

If for some reason you decide to intemrpt the consultation without leaving the progrant(Option Stop) and want to begin another consultation, you should do the following:

1. Select File, Qpen . . .

2. Select INITIAL.KB.3. Select Consult/Begin Consultation'

You can also select GENERAL.KB which repeats your consultation with old input data.Caution: Do not resume an interrupted consultation with a knowledge base other than

INTITIAL.KB or GENERAL.KB.Somc options allow restarting consultation with old or new values.Valucs that you introduce can, in general, be changed by using the option Consult,

What if - ptramctcr changes its value as long as you do not leave the current knowledgc

ll];I",,1'tr. irrterrlirl n:ilrrc ol'rhc paramctcr is shown on the status line at the bottom of thc

ARCHIVING

Report ol a Session

Creating and saving a report of a session:

1. Log your session following instruction 3 in the above section Rulltiltg lltt l)rogrartt.2. Bcfbrc closing the dialog window, you will be askcd il yotr wrttrt ltt srtvr' ('lick Yes

and cnlcr a lilc nanrc ((filcnanrc).L(Xi).

S:rvirtg lr st'lt:tliott llirttt llrc rclx)fl Iilc:

n|lrri,I IIil Irt.;t. {ill/

l. St'lcct lilit, ('o1ly A \r,rn,l,,n ( npl lcrl ltolrr rliirkrg lor.lilrlrorrrtl \1rll itlrlrr;u utllt:rll tcxt higlrliglttt'rl llrlllrltl'ltt tlr,. .,.l,.,ttol \ol u,,rrl,l lrl,.trr ropr, ;1;1rl,lr,l, rrrr tlr,(lopy brrttorr

2. Sclccl Iiilc, Ncrv;ttttl r ltoo:r"llrl. r\ I'l;rrL \\'rrll"\r rrll ,r;,;,r.,rr ll;rrlrl trr;rr r.\r,.lu!t,filc, Sc:lct't l;ilt., ( )1x'rr rtrrrl r ltol,,r. tltr ltlr \,rl u,l.,lr lo tr.,r.

3. Sclccl litlit. I'rrstt.

You calt t:rlttlittuc lo;ttltl st'lr.r'lrorrs lronr llrr.rorr.,rrll;rlrorr lo;,lo you! l{.\l lrlr.l,y kt.r.Prn1,thc untitlcd.ltlg witttlow rrpt'tr:rrrrl trrllirrll;rntl |;r:lrrrll lollowirrl, 11,,.,r|r,,u,' utsltu(.llon:;.

Saving TablesAfter a table is displayed in a window, you will bc asked if you want to save the table. Forexample, after viewing a table lor pressures on walls, the next dialog box will ask: ..SaveTablc(s) in file wALLS.TXT? If you click on yes, the liles will automatically be savedunder the default name (e.g. wALLS.TXT or GUST.TXT). NorE: This will overwritefiles from previous consultations. If you DO NOT want to overwrite old files do the followingbefore you begin a new consultation:

l From the main menu-bar choose File, Opcn, select WALLS.TXT.2. The table will be displayed on the screen.3. Select File, Save As and save thc file undcr a new name (c.g. WALLSI.TXT).

lnserting Tables or Report Files lnto Other FilesText and tables can be imported into other filcs such as word processing documents. SelectFile, open and choose the table (e.g. WALLS.TXT) you wish to exporr. Highlight the tableusing your mouse or Edit, Select All. Select Edit, Copy. Start your word processing programand open a new or existing file. Choose the Terminal firnt and adjust the font to a smallSize. Select Edit, Paste to insert the table into your word processing file.

PRINTING FROM THE PROGRAM

Printing the report file while the file is on the screen:

Sclcct File, Print

Printing a selection of the report file:

l. Select Edit, eopy. A window copy text from dialog to clipboard will appear withall text highlighted. Highlight thc sclcclion you would like to copy and click on theCopy button.

2. Sclcct File, Ncw atttl tlrrxrst"l'cxl. A lrl;rrrl' wirrtkrw will ;rp1x';rr.3. Sclcct Edit, Paslc.4. Sclcct Filc. I)rirrt .

I)rirrling tirblc lilcs:

L St'lt't't lrilc. ()1x'n" llr,rr,,,l,,l , lrlrr;rrn, ' I \ I

.1. llrt't:rhlc will lx'tlr:,;rl;rr'(rl .rr tlr, .,, r' ' rr

i Stktl lirlt'- l'|rirrl

688 ABOU] THE DISK

PROBLEMS WITH SCREEN DISPLAY AND PRINTING FONTS

Display

1. If the screen display is unsatisfactory, (e.9. shows character symbols instead of vcr-tical and horizontal lines), you need to change the font and/or font size.l. Select Edit, Font ...2. Select the Terminal, 8514oem, or MS LineDraw (True Type) font.3. If necessary, adjust the font size by entering a number in the Size box.If your display shows screen points instead of letters you need to install a more recentversion of MS LineDraw (True Type). (See instructions below.)

PrintingSome printers will print character symbols instead of the horizontal and vertical lines thatyou see on your screen. You need to change the font before you print by doing the following:

1. Select Edit, Font ...2. Select Terminal or 8514oem font.3' Ifyour printout still does not display horizontal and vertical lines, change the font to

MS LineDraw (True Type). (See instmctions below.)

Note that you can still use the Terminal font for display only.

lnstalling MS LineDraw (True Type) FontIf MS LineDraw (True Type) font does not appear on the font list do the following:

1. From Windows Program Manager, open Control Panel in the Main program group.2. Double click on Fonts icon.3. Select Add.4. Open the c:\windows\systems directory.5. Select MS LineDraw (True Type) if available.6. If MS LineDraw (True Type) is not available you will need to load the fonr from

the Wind Loads 7-95 floppy disk. Put the disk in the floppy drive and choose driveA. Sclcct MS LineDraw (True Type) from the list of fonts and click OK.

Plcasc rcl'cr to thc rcadmc filc on thc disk for more information about solving font problemslirr display and printing.

USER ASSISTANCE AND INFORMATION

If you have questions or problems with the installation of the disk, please call our technicalsupport number (212) 850-6194 between 9am-4pm EST, Monday-Friday.

To place additional orders or to request information about other Wilcy pr<xlucts, plcasccall (800) 879-4539.

2.

ISBN [_q?I- IEI57_h90000

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