1 ISBN0-89464-S03-X IIIIm ~ l r l l l l l BEDNAR e-u m:u cnm -en Glen Zc ::c:u m z< em men oen om ".-r i B! CE 8 PRESSURE VESSEL DESIGN HANDBOOK SECOND EDmON PRESSURE VESSEL DESIGN HANDBOOK 1 PRESSURE VESSEL DESIGN HANDBOOK SecondEdition HenryH.Bednar,P.E. TECHNIP ITALY S.p.A. BIBLIOTECA INVENTARIONQ........................................ . KRIEGER PUBLISHING COMPANY MALABAR, FLORIDA Second Edition 1986 ReprintEdition 1991 Printed and Published by KRIEGER PUBLISHING COMPANY KRmGER DRIVE MALABAR, FLORIDA 32950 Copyright IC> 1986 by Van Nostrand Reinhold Company, Inc. Reprinted by Arrangement Allrightsreserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including information storage and retrieval systems without permission in writing from the publisher. Noliability is assumed withrespect totheuseo/the information contained herein. Printed in the United States of America. Library of Congress Cataloging-in-Publication Data Bednar,Henry H. Pressure vessel design handbook /Henry H.Bednar. p.cm. Reprint.Originally published:2nd. ed. New York: Van Nostrand Reinhold, cl986. Includes bibliographical references and index. ISBN 0-89464-503-X (lib. bdg.: acid-free paper) 1.Pressure vessels--Design and construction--Handbooks, manuals, etc.1.Title. TA660.T34B441990 68I'.76041--dc2090-5043 CIP 10 PrefacetoSecondEdition InrevisingthefIrsteditiontheintenthasbeentoimprovethehandbookasa referencebook byenlargingitsscope.Thestressanalysisof pressurevessels has beengreatlyenhancedinaccuracyby numerical methods. These methods repre-sentagreatadditiontotheanalyticaltechniquesavailabletoastressanalyst. Therefore,chapter12describingthemostimportantnumericalmethodswith illustrativeexampleshasbeenadded.Throughoutthetextnewmaterialand newillustrativeexampleshavealsobeenadded.Thewriterbelievesthatany technicalbookinwhichthetheoryisnotclarifIedbyillustrativeexamples, canbeof littleusetoapracticingdesignerengineer.Alsosometypographical errors have been corrected. Itshouldbekeptinmindthatpracticalengineeringisnotanabsoluteexact science.Therearetoomanyvariablefactorsandunknownquantitiessothat onlyareasonableestimateofforcesandstressescanbemade,particularlyin more involved problems. Almost all problems in engineeringpractice do not have asingle-valueanswer,andusuallytheyrequireacomparisonof alternativesfor solution. Therefore,nodefIniterulescanbegivenfordeciding how to proceed inevery case,andtheruleslaiddowncannotbeappliedinflexibly.Thedesignermust be guided by hisformerexperience andhisbest personal judgment sincehebears the fmal responsibility for the adequacy of the design. Thewriterwouldliketoextendhisgratitudetoallreaderswhoofferedcon-structivecomments, particularly to Dr.A.S.Tooth of University of Strathclyde, Glasgow,Scotland forhis comments onthe stresses inshellsatsaddlesUl'ports. HENRYH.BEDNAR 1 Preface toFirstEdition Thishandbookhasbeenpreparedasapracticalaidforengineerswhoareen-gagedinthedesignof pressurevessels- Designof pressurevessels has to bedone inaccordwithspecificcodeswhichgivetheformulasandrulesforsatisfactory andsafeconstructionof themainvesselcomponents.However,thecodes leave ituptothedesignertochoosewhatmethodshewillusetosolvemanydesign problems;inthisway,heisnotpreventedfromusingthelatestacceptedengi-neering analytical procedures. Efficiencyindesignworkisbasedonmanyfactors,includingscientific train-ing,soundengineering judgment, familiaritywith empiricaldata,knowledgeof designcodesandstandards,experiencegainedovertheyears,andavailable technicalinformation.Muchof thetechnicalinformationcurrentlyusedinthe designofpressurevesselsisscatteredamongmanypublicationsandisnot available in the standard textbooks onthe strength of materials. Thisbookcoversmostoftheproceduresrequiredinpracticalvesseldesign. Solutionstothedesignproblemsarebasedonreferencesgivenhere,andhave beenprovenby long-timeuse; examplesarepresentedastheyareencountered inpractice.Unfortunately, exact analyticalsolutionsfora numberof problems arenot known at thepresent time and practical compromises haveto be made. Mostengineeringofficeshavedevelopedtheirownvesselcalculationpro-cedures,mostofthemcomputerized.However, it ishopedthatthisbookwill providethedesignerwithalternativeeconomicaldesigntechniques,willcon-tributetohisbetterunderstandingof thedesignmethodsinuse,andwillbe convenientwhenhandcomputationsorverificationsof computer-generatedre-sults haveto be made. Noparticularsystemof notationhasbeenadopted.Usuallythes y ~ b o l sas theyappearinparticulartechnicalsourcesareusedanddefinedastheyoccur. Onlythemost important references are givenfor moredetailed study. It isassumedthatthereaderhasaworkingknowledgeof theASMEBoiler and Pressure Vessel Code, Section VIII, PressureVessels, Division1. Thewriterwishestoexpresshisappreciationtothesocietiesandcompanies forpermission to usetheir published material_ Finally,thewriteralsowishestoexpresshisthankstotheeditorialandpro-ductionstaff of thePublisherfortheircontributiontoasuccessfulcompletion of this book. HENRYH.BEDNAR 1 Contents PREFACETOFIRSTEDITIONlv PREFACETOSECONDEDITION I vii 1.DESIGNLOADS/! 1.1.Introduction II 1.2.DesignPressure 12 1.3.DesignTemperature 13 1.4.DeadLoads 14 1.5.WindLoads 15 1.6.EarthquakeLoads I 13 1.7.PipingLoads/21 1.8.Combinationsof theDesignLoads 122 2.STRESSCATEGORIESANDDESIGNLIMITSTRESSES 124 2.1.Introduction 124 2.2.AllowableStressRangeforSelf-LimitingLoads 125 2.3.GeneralDesignCriteria of ASMEPressureVesselCode,Section VIII,Division1/26 2.4.GeneralDesignCriteria of ASMEPressureVesselCode,Section VIII,Division 2/29 2.5.DesignRemarks 138 3.MEMBRANESTRESSANALYSISOFVESSELSHELL COMPONENTS 139 3.1.Introduction 139 3.2.MembraneStressAnalysisof Thin-ShellElements 143 3.3.CylindricalShells 146 3.4.SphericalShells andHemisphericalHeads 155 3.5.SemiellipsoidalHeads 159 3.6.TorisphericalHeads 162 3.7.ConicalHeads/66 3.8.ToroidalShells 171 3.9.Design of ConcentricToriconicalReducersunder Internal Pressure 173 ix 1 xCONTENTS 4.DESIGNOFTALLCYLINDRICALSELF-SUPPORTING PROCESSCOLUMNS/80. 4.1.Introduction 180 4.2.ShellThicknessRequiredfora Combination {)fDesignLoads 181 4.3.SupportSkirts 185 4.4.Anchor Bolts/91 4.5.Wind-InducedDeflections of TallColumns 1103 4.6.Wind-InducedVibrations I 107 4.7.FirstNaturalPeriodof Vibration 1121 4.8.IllustrativeExample 1129 5.SUPPORTSFORSHORTVERTICALVESSELS 1143 5.1.SupportLegs 1143 5.2.SupportLugs 1153 '6.DESIGNOFSADDLESUPPORTSFORLARGEHORIZONTAL CYLINDRICALPRESSUREVESSELS/161 6.1.GeneralConsiderations 1161 6.2.MaximumLongitudinalBendingStressintheShell I 161 6.3.MaximumShear StressesinthePlaneof theSaddle 1165 6.4.CircumferentialStressattheHornof theSaddle 1169 6.5.AdditionalStressesina HeadUsedasa Stiffenerl172 6.6.RingCompressionintheShellover theSaddle 1173 6.7.Designof RingStiffeners I 175 6.8.Designof Saddles 1177 7.LOCALSTRESSESINSHELLSDUETOLOADSON ATTACHMENTS 1186 7.1.Introduction 1186 7.2.Reinforcement of OpeningsforOperatingPressure I 186 7.3.SphericalShellsor HeadswithAttachments I 188 7.4.CylindricalShellswithAttachments I 193 7.5.DesignConsiderations 1208 7.6.LineLoads/210 8.DISCONTINUITYSTRESSES/217 8.1.Introduction 1217 8.2.Procedure forComputingDiscontinuityStresses bytheForce Method/220 8.3.CylindricalShells/221 8.4.HemisphericalHeads/226 8.5.SemiellipsoidalandTorisphericalHeads 1230 8.6.ConicalHeads andConicalReducerswithout Knuckles 1230 9.THERMALSTRESSES 1241 9.1.GeneralConsiderations/241 9.2.BasicThermalStressEquations/241 9.3.ExternalConstraints 1242 9.4.InternalConstraints 1244 CONTENTSxl 9.5.ThermalStressRatchetunder SteadyMechanicalLoad / 254 9.6.DesignConsideration 1256 10.WELDDESIGN/259 10.1.Introduction 1259 10.2.GrooveWelds 1260 10.3.Fillet Welds 1264 10.4.PlugWelds 1277 10.5.DesignAllowableStressesforWelded 10ints/278 10.6.StressConcentration FactorsforWelds I 279 10.7.DefectsandNondestractive Examinationsof Welds 1280 10.8.WeldingProcesses 1281 10.9.WeldSymbols 1283 11.SELECTIONOFCONSTRUCTIONMATERIALS 1284 11.1.GeneralConsiderations 1284 11.2.NoncorrosiveService 1285 11.3.CorrosiveService 1290 11.4.BoltingMaterials 1293 11.5.StainlessSteels/296 11.6.Selectionof SteelsforHydrogenService 1306 11.7.AluminumAlloys 1309 12.NUMERICALMETHODSFORSTRESSANALYSISOF AXISYMMETRICSHELLS 1312 12.1.Introduction/312 12.2.FiniteElementAnalysis,(FEA)DisplacementMethod I 314 12.3.FiniteElementAnalysis,ForceMethod I 376 12.4.Method of StepwiseIntegration 1380 12.5.Method of FiniteDifferences1381 APPENDICES 1385 AI.Wind,Earthquake andLowest One-DayMeanTemperatureMaps/ 387 A2.GeometricandMaterialChartsforCylindricalVessels 1389 A3.Skirt Base Details 1391 A4.SlidingSupportsfor VerticalandHorizontalVesse1s/392 AS.Glossaryof TermsRelatingtotheSelection of Materials 1394 A6.StandardSpecificationsPertaining toMaterials 1404 A7.Flanges 1405 A8.ElementaryMatrixAlgebra 1410 A9.References/416 AlO.AbbreviationsandSymbols/423 INDEX I 429 PRESSURE VESSEL DESIGN HANDBOOK 1 DesignLoads 1.1.INTRODUCTION Theforcesappliedtoavesseloritsstructuralattachmentsarereferredtoas loadsand, asinany mechanicaldesign, the firstrequirement invesseldesignisto determinetheactualvaluesof theloadsandtheconditionstowhichthevessel willbesubjectedinoperation.Thesearedeterminedonthe basisof past experi-ence, designcodes, calculations, or testing. Adesignengineershoulddetermineconditionsandallpertainingdataas thoroughlyandaccuratelyaspossible,andberather conservative. Theprincipal loads to be considered inthe designof pressurevesselsare: designpressure (internalorexternal), dead loads, wind loads, earthquake loads, temperature loads, piping loads, impact or cyclicloads. Manydifferentcombinationsof theaboveloadingsarepossible;thedesigner mustselectthemostprobablecombinationofsimultaneousloadsforimeco nomicalandsafedesign. Generally,failuresofpressurevesselscanbetracedtooneof thefollowing areas: material:improperselectionfortheserviceenvironment;defects,suchas inclusions or laminations; inadequate quality control; design:incorrectdesignconditions;carelesslypreparedengineeringcomputa-tionsandspecifications;oversimplifieddesigncomputationsintheabsenceof availablecorrect analytical solutions; and inadequate shop testing; fabrication:improperorinsufficientfabricationprocedures;inadequatein-spection; careless handling of specialmaterials such as stainless steels; 2PRESSUREVESSEL DESIGNHANDBOOK service:changeofserviceconditionstomoresevereoneswithoutadequate provision;inexperiencedmaintenancepersonnel;inadequateinspectionfor corrosion. 1.2.DESIGNPRESSURE Designpressureisthepressureusedtodeterminetheminimumrequiredthick-nessof eachvesselshellcomponent,anddenotesthedifferencebetweenthe internalandtheexternalpressures(usuallythedesignandthe atmosphericpres sures-seeFig.1.1).Itincludesasuitablemarginabovetheoperatingpressure (10percentofoperatingpressureor10psiminimum)plusanystaticheadof theoperatingliquid.MinimumdesignpressureforaCodenonvacuumvessel is15psi.Forsmaller designpressuresthe Code stamping isnotrequired. Vessels withnegativegaugeoperating pressure aregenerally designedforfullvacuum. Themaximumallowableworking(operating)pressureis. then,bytheCode definition,themaximumgaugepressurepermissibleatthetopofthecom-pl!!tedvesselinitsoperatingpositionatthedesignatedtemperature.It isbased onthenominalvesselthickness, exclusiveof corrosion allowance, andthethick-nessrequiredforotherloadsthanpressure. Inmost casesit willbe equalor very closetothedesignpressure of thevesselcomponents. BytheCodedefinition,therequiredthicknessistheminimumvesselwall thicknessascomputed bythe Codeformulas,not including corrosion allowance; thedesignthicknessistheminimum requiredthickness plusthecorrosion allow-ance;thenominalthicknessistherounded-updesignthicknessasactually used inbuilding the vesselfromcommercially availablematerial. Ifthenominalvesselthicknessminuscorrosionallowanceislargerthanthe requiredthickness,eitherthedesignpressureorthecorrosionallowancecanbe 1standard atm =14.69 psia standard atmospheric ga!ge (pr) negative(psigl (vacuum) 5. local atm o absolutezero pressure (fldlvac'Jum) Fig.1.1. I absolute(psia) DESIGNLOADS3 increased,oranyexcessthicknesscanbeusedasreinforcementof thenozzle openings inthevesselwall. Thevesselshellmust bedesignedto withstand themost severecombination of coincidentpressureandtemperatureunderexpectedoperatingconditions.The nominal stressinanypart of the vesselascomputed fromthe Code and standard engineeringstress formulas,without consideration oflarge instressesat thegrossstructural discontinuities, cannot exceedthe Code allowable stress. 1.3.DESIGNTEMPERATURE Designtemperatureismore adesignenvironmental condition than adesignload, sinceonlyatemperaturechangecombinedwithsomebodyrestraintorcertain temperaturegradientswilloriginatethermal stresses.However, it isan important designconditionthatinfluencestoagreatdegreethesuitabilityof the selected materialforconstruction.Decreaseinmetalstrengthwithrisingtemperature, increasedbrittlenesswithfallingtemperature,andtheaccompanyingdimen-sionalchangesarejustafewof thephenomena tobetaken into accountforthe design. TherequiredCodedesigntemperatureisnotlessthanthemeanmetalvessel walltemperature expectedunderoperating conditions and computed by standard heattransferformulasand,if possible,supplementedbyactualmeasurements. Formost standardvesselsthedesigntemperatureisthemaximumtemperature oftheoperatingfluidplus50Fasasafetymargin,ortheminimumtempera-tureof theoperatingfluid,if thevesselisdesignedforlow-temperatureservice (below -20F). Inlargeprocessvesselssuchasoilrefineryvacuumtowersthetemperatureof theoperatingfluidvariestoalargedegree,andzoneswithdifferentdesign temperatures,basedonexpectedcalculatedoperatingconditions,canbeused forthedesigncomputations of therequiredthicknesses. Thedesignmetaltemperatureforinternallyinsulated vesselsisdetermined by heattransfercomputations,whichshouldprovidesufficientallowancetotake careof theprobablefutureincreaseinconductivity of therefractoryto gas, deterioration,coking,etc.Ataminimum,thedesignershouldassumeacon-ductivityfortheinternalinsulatingmaterial50-100percenthigherthanthat givenbythemanufacturer'sdata,dependingonoperatingconditions.A greater temperaturemarginshouldbeusedwhenexternalinsulationisusedaswell. Thepossibilityof alossof a sizable lining section andtherequiredrupturetime ofthe.shellshouldalsobeconsidered.Extensivetemperatureinstrumentation of thevesselwallisusuallyprovided. Forshut-downconditionsthemaximumdesigntemperatureforuninsulated vesselsandtheconnectingpipingwillbetheequilibriumtemperatureformetal objects,approximately230Fforthetorridzone,190Fforthetemperate zone, and150F forthefrigidzone. , 4PRESSURE VESSELDESIGNHANDBOOK Thelowestdesignmetaltemperatureforpressurestoragepesselsshouldbe takenas15Fabovethelowestone-daymeanambienttemperatureforthe particular location, seeFig. A1.3. Thedesigntemperatureforj7I1ngethroughboltsisusuallylowerthanthe temperatureoftheoperatingfluid,unlessinsulated,anditcanbesafelyas-sumedtobe80percentof thedesignvesseltemperature. However,the external tapandtheinternalboltingshouldhavethesamedesigntemperatureasthe vessel. Whenthedesigncomputationsarebasedonthethicknessof thebaseplate exclusiveof liningorcladdingthickness,themaximumservicemetaltempera-ture of thevesselshould bethat allowedforthebaseplate material. Thedesigntemperatureof vesselinternalsisthemaximum temperature of the opera ting liquid. 1.4.DEADLOADS Deadloadsaretheloadsduetotheweightofthevesselitselfandanypart perm.anentlyconnectedwiththevessel.Dependingontheoverallstate, a vessel canhavethreedifferentweightsimportantenoughtobeconsideredinthe design. I.Erection(empty) deadloadof thevesselisthe weight of the vesselwithout anyexternalinsulation,fireproofing,operatingcontents,oranyexternalstruc-turalattachmentsandpiping.Basically,itistheweightof astrippedvesselas hoistedonthejobsite.Insome small-diameter columns theremovableinternals (trays)areshop-installed,andtheyhavetobeincludedintheerectionweight. Each such casehastobeinvestigated separately. 2.Operatingdeadloadofthevesselistheweightof thein-placecompleted vesselinfulloperation.Itistheweightof thevesselplusinternalorexternal insul!ltion,fireproofing,allinternals(trays,demister,packing, etc.) with operat-ingliquid,sectionsofprocesspipingsupportedbythevessel,allstructural equipmentrequiredforthevesselservicingandinspection(platforms, ladders, permanenttrolleysetc.),andanyotherprocessequipment(heatexchangers) attachedtothevessel. 3.Shoptest deadloadof thevesselconsistsonlyof theweightof thevessel shell, after allwelding isfinished,filledwithtest liquid. Fieldtestdeadloadistheoperatingdeadloadwithonlyexternaland/or internalinsulationremovedforinspectionpurposesandfilledfullywiththe testliqUidinsteadofoperatingliquid.Thisloadisusedasadesignloadonly if thevesselisexpectedtobetested infieldat some futuredate. T h ~iceorsnowloadaswellasanyliveload(weightofthemaintenance personnel with portabletools) isconsideredto benegligible. DESIGNLOADS5 1.5.WINDLOADS Windcanbedescribed asa highly turbulent flowof airsweeping overthe earth's surfacewith avariablevelocity,ingustsratherthaninasteady flow.The wind canalsobeassumedtopossessacertainmeanvelocityonwhichlocalthree-dimensionalturbulentfluctuationsare.superimposed.Thedirectionof theflow isusuallyhorizontal; however, it may possess a vertical component when passing overasurfaceobstacle.ThewindvelocityVisaffectedbytheearthsurface frictionandincreases with the height abovethe ground to some maximum veloc-ityatacertaingradientlevelabovewhichthewindvelocityremainsconstant. Theshapeofthevelocityprofileabovethegrounddependsontheroughness characteristicsoftheterrain,suchasflatopencountry, woodedhillycountry-side, ora largecity center. It can be expressedby thepower-law formula wherethevalueof theexponentndependsontheterrain and z isthe elevation abovethe ground level. Sincethestandardheightforwind-speedrecordinginstrumentsis30ft,the powerformulaisusedtocorrectstandard-heightvelocityreadingsforother heights above ground with anygiventerrainprofile (seeFig.1.2). Thevelocity(dynamic)pressurerepresentingthetotalkineticenergyof the movingairmassunitattheheightof 30ftabovethegroundonaflatsurface perpendicular to thewindvelocity isgivenbythe equation: q30= pV2/2= (t)0.00238 (5280/3600)2 V ~ o= 0.00256 V ~ o where V30 = basicwindspeedat30ftheightabovethegroundinmph,maximum asmeasuredontheloca tionovera certain periodof time Q30= basic wind velocitypressure at 30 ftheight abovethe ground inp ~ f p = 0.00238 slugs per ft3 ,airmass density at atm pressure, and 60F. ThemagnitudeofthebasicwindvelocityV30 usedindeterminationofthe designpressureQ30dependsonthegeographicallocationof thejobsite.The windpressureQ30isusedtocomputetheactualwinddesignloadsonpressure vesselsandconnectedequipment.However,sincethewindvelocityVisin-fluencedbytheheightabovethegroundandterrainroughness andthepressure Qitselfisinfluencedbytheshapeofthestructure,thebasicwindvelocity pressureQ30hastobemodifiedfordifferentheightsabovethegroundlevel anddifferent shapes of structures. Indoingsoeithertheolder,somewhatsimplerstandardASAA58.l-19SSor thenewrevisedANSIAS8 .1-1972aregenerallyused,unlessthe client's specifi-6PRESSUREVESSEL DESIGNHANDBOOK J! " "0 C 2 '" , 0 .D L: -" z 15001200V 900V 30 f-..V =wind open,flat coastalregion;rough,woodedarea;largecity centersvelocity flat,open countrytown;largecity, suburban areas 900It 117 1200 It 1/4.5 1500 It 1/3 gradientlevel exponent n Fig.1.2.Windvelocitypromesoverthreebasicterrainroughnesscharacteristics.Ifthe windvelocityinflat,opencountryatthestandardheight30ft isV 30= 60mph,atthe gradientlevel900ftthewindvelocityis V=60(900/30)1/7",100mph. Inthesameregion,the gradientwindvelocityinthe suburban area isthe same,i.e.,V=100 mph,andthegradientlevelis1200ft;hencethewindvelocityatthe standard levelinthe suburban area is V30 =100(30/1200)1/4.5=44mph. cationsdictateotherwise.Althoughtheformerstandardisobsolete,itwas extensivelyusedformanyyearsanditisstillusedinsome codes.It istherefore quiteprobablethatforsometimetocomethedesigrferwillencounterdesigns wherethewindloadswerecomputedonthebasisof the oldstandard. Alsothe maybecomeengagedintheconstructionof pressurevesselsforpetro-chemicalplantsinforeigncountrieswherelong-termwindvelocitydataare lackingandthedesignproceduresarespecifiedmoreinlinewiththenow obsolete specification. Windloads asComputedinAccordancewithASASpecificationA58.1-1955 Theprocedureforcalculationof theminimumdesignwindloadnormaltothe surfaceof the structure isasfollows. DESIGNLOADS7 1.Thegeographicalareaof the jobsiteislocatedonthewindpressuremap, (seeFig.1 of thestandard).Thebasicwindpressure pisselected. It isbasedon the maximumregionalmeasured wind velocity V 30(excluding tornado veloci ties) andincludestheshapefactor1.3forflatsurfaceandthegustfactor1.3for heightsupto500ftabovethe ground. No distinction ismadeforterrain rough-ness or meanrecurrence intervalof the highest wind inthe area. 2.ThewinddesignpressuresPz,correspondingtothebasicwindpressurep, forvariousheightzonesabovethegroundaregiveninTable3of thestandard. Theyincludeaheightfactorbasedontheseventh-root lawto expressthevaria-tion of the wind velocitywith the height abovethe ground . 3.TheshapefactorBforroundobjectsisequalto0.6andisappliedtothe designpressurepz.Theshapefactors,astheyappearinthestandardareactual shapecoefficientsdividedbytheflatsurfacefactor1.3whichwasincludedin thebasic windpressure p. 4.If thewindwardsurfaceareaprojectedontheverticalplanenormaltothe directionof thewindisAft2,thentheresultant of the windpressure loadover thearea P wisassumedto act at the areacentroid and isgivenby Pw =ABpz lb. Thewindpressureforcesareappliedsimultaneously,normaltoallexposed windwardsurfacesofthestructure.Theminimumnetpressure(Bpz)inthe aboveformulaforcylindrical verticalvesselsisnot lessthan13psf for LID';;; 10 and18psf forLID;;' 10,whereListheoveralltangent-to-tangent lengthof the vesseland D isthe vesselnominal diameter. Windloads asComputedinAccordancewithANSIA58.1-1972 Theeffectivewindvelocitypressures on structures qF and onparts of structures qpinpsf atdifferentheightsabovethegroundarecomputedbythefollowing equations: where qF =Kz GFq30 qp = KzGpQ30 K. = velocitypressurecoefficient depending on thetypeof the terrain surface (exposure)andtheheightabovetheground z.To simplify computation, heightzonesof constant windvelocity canbe assumed.For instancethe pressureqFat100ftabovethegroundcanbeappliedoverthezone extending from75to 125ftabovethe ground. GF= gustresponsefactorforstructure Gp = gust responsefactorforportion of structures. 1 8PRESSUREVESSELDESIGNHANDBOOK Thevariablegustresponsefactorswereintroducedtoincludetheeffectof sudiienshort-termrisesinwindvelocities.Thevaluesof GpandGpdepend on thetypeof theterrainexposureanddynamiccharacteristicsof structures.For tallcylindricalcolumns,forwhichgustactionmaybecome significant, detailed computationsofGpareusuallyrequiredasshownintheAppendixtothe standard.SincetheequationforG pcontainsvalueswhichdependonthefirst naturalfrequency[ofthecolumn,whichinturndependsonthevesselwall thicknessttobecomputedfromthecombinedloadings(wind,weight,and pressure),theadditionalmathematicalworkinvolvedinsuccessiveapproxima-tionmayrender thisstandard lessattractive than thepreviousone. Theqp and qp valueshaveto befurthermodified by a net pressurecoefficient Cf fordifferent -geometric shapes of structures.. If theprojectedwindwardareaof thevesselsection on a verticalplanenormal tothewinddirectionisAftz,thetotaldesignwindload Pwona vesselsection may becomputed by equation: Theminimumnetpressure(Cfqp)intheaboveformulashouldbenotless than15psfforthedesignof structuresand13psf forstructuralframes.The windloadsonlargeappurtenancessuchastopplatformswithtrolleybeams, piping,etc.canbecomputedinthesamemanner,usingappropriateCf and qp withallowancesforshieldingeffectandmustbeaddedtothewind load acting ontheentire vessel. Thethreestandardterrainroughnesscategoriesselectedareasshownin Fig.1.2. a.ExposureA:centers oflarge cities, rough and hilly terrains; b.ExposureB:rolling terrains, wooded areas, suburban areas; c.ExposureC:flat,open grasscountry, coastal areas. Mostlargepetrochemical plants willbelong to category C. Theprocedureforcomputingtheminimumdesignwindloadonanenclosed structure such asa tall column can be summarizedasfollows. I.UsingascriteriatheantiCipatedservice lifeof the vesselandthe magnitude ofthepossibledamageincaseof failure,thebasicwindspeedV30 isselected fromFig.1or2o[ thestandardfortheparticular joblocationandmodified by special localconditions; seealsoAppendix Alof this book. 2.The basic windpressure q30= 0.00256 y2is computed. 3.Theeffectivewindvelocitypressureqpisgivenbyqp =K.Gp q30'The heightzonesof constantwindvelocitiesareselectedand K.determinedfrom Fig.A2of thestandardforeachzone. The gust responsefactor Gp , which does DESIGNLOADS9 notchangewiththeheightabovetheground,iscomputedfromtheequation Gp= 0.65 + 1.95 (a..jJi). 4.Usingthenet pressurecoefficient C, fromTable15of the standard andthe projectedareaAinft2,thetotalwindloadonthevesselsectionsof constant winddesignpressuremay be evaluated by equation Pw= C,Aqp. Computation of theProjectedArea A Itisnotpossibleforthedesignertoevaluatetheprojectedwindward area Aof atowerandallappurtenancesaccurately.Whenavesselisbeingdesignedonly themainfeaturessuchastheinsidediameter,overalllength,nozzlesizes, num-ber of manholes, etc. areknown and a complete layout is unavailable. Toarriveatsomereasonableapproximationof theprojectedarea,someas-sumptionsbasedonpastexperiencemustbemade.Anapproximatelayout sketchofthevesselwithallprobableplatforms,ladders,andconnectedpiping canbemadeandwithresultingwindloads,windshearsandwindmomentsat differentheightsabovethegroundcanbecomputed.Unlessthevesseliscom-parativelysimple,forinstanceashortverticaldrumwithatopplatformasin Fig.5.8, this approach istime-consuming and not really justified. AnapproachtocomputingAwhichisoftenusedandisrecommendedhere istoincreasethevesseldiameterDtotheso-calledeffectivevesseldiameterto approximatethecombined designwind load: De=(vesselo.d. + twiceinsulation thickness) X Kd Thecoefficien t Kdisgivenin Table1.1. Therequiredprojected area Awillthen beequal to whereH. = lengthof theshellsectioninthezoneof theuniform windvelocity. However,theeffectivediameterDecanbederived by a simpleprocedlSrewhich allows the designerto adjust Deaccording tothe actual standard vessellayout. Table1.1. VESSELOUTSIDEDIAMETER INCLUDINGINSULATION lessthan 36in. 36to 60 in. 60 to 84in. 84to108 in. over108 in. Source:Ref.6. COEFFICIENT Kd 1.50 1.40 1.30 1.20 1.18 1 I 10PRESSUREVESSELDESIGNHANDBOOK insulation thickness I;J o.d. , insulationI thickness caged ladder 1 J Fig.1.3.Assumed column layout fordetermination of the effectiveDe. Accordingtoanassumedtypicalsectionof aprocesscolumnshown inFig.1.3 the principal parts contributing to the total wind load are asfollows: 1.Vesselshelloutside diameter withtwicetheinsulationthickness, if any. 2.Adjustedplatformarea.Assuminghalf of theplatform,3ft6in.wideat eachmanhole,spacedat15ft,theequivalentincreaseinthevesseldiameter (42X18)/(15X12) =4.2 in. 3.Cagedladder.Assumeonecagedladderrunningfromthetop of thevessel totheground. Theincreaseinthe column diameter canbetaken as12 in. 4..Piping.Assumethelargestpipeinthetopthirdof thecolumnrunningto theground level. Alltheaboveitemscanandshouldbeadjusted accordingto the actual standard layout asused. Forexample,theeffectivediameterof a6-ftdiametercolumnwithI-in. wall thickness,2in.insulation,anda6in.nozzleinthetopthirdwithI-in.insula tion iscomputed asfollows: De= (vesselo.d. + 2X insulationthickness) + (pipe o.d. + 2X insulationthickness) + (platform) + (ladder) = (74 + 4) + (6.625+ 2) + 4.2 + 12=102.8in.= 8.6 ft Thefactor Kd=102.8/78= 1.31isinagreement with the value inTable1.1. DESIGNLOADS11 Theformulaabovedoesnotincludespecialattachedequipmentsuchasheat exchangersorlarge-topoversizedplatformwithliftingequipment,whosewind loads andmoments arecomputed separatelyandaddedtotheabove. Example1.1.Determinethewindloadsactingontheprocess columnshownin Fig.1.4withanaveragewallthicknessofIin.,insulationthickness1.5in., locatedinthe vicinityof Houston, Texas, using: a.ASAA58.11955 b.ANSIA58.11972. a.EffectivediameterDe= KdX o.d. =1.30 X 6.4 = 8.3ft.FromFig.Iof the standardthebasicwindpressureisp= 40psf.FromTable3of thestandard the winddesignpressures are pz= 30 psf,elevation 40 psf, 50 psf, 60 psf, o to 30 ftaboveground 30 to 49ft 50 to 99ft 100to 499ft Windloadsinpounds peronefootof columnheight, W = B X DeX Pz, are WI= 0.60 X 8.3X 30 =150 lb/ft W2= 0.60 X 8.3X 40 = 200 Ib/ft W3= 0.60 X 8.3X 50 = 250 lb/ft W4= 0.60 X 8.3X 60 = 300 lb/ft. b.Selected:100yearsrecurrenceinterval;type-Cexposure; dampingJactor ~= 0.01; fundamentalperiodof vibrationT =1 sec/cycle. FromFig. 2of the standard:basicwindvelocity isV30=100 mph;q3o= 25.6 psf; designpressures are qF = Kz GF q30' FromFig.A2of thestandard: K30= 1.0,Kso= 1.2,K9s =I.40,K12o =1.50. GustresponsefactorisGF = 0.65+ 1.95 (uvP) = 0.65+ 1.95X 0.332=1.297 forenclosed structures: uvP =1.7 [T(2h/3)[0.785 P F / ~+ S/(l+ 0.002d) 1/2= 0.332 1 12PRESSURE VESSEL DESIGNHANDBOOK 6'i.d. l

Q) w4 = 300 IIb/tt I 60' ::0, t 1 ;e I 0 '"NIII f @ I 50' @ :!: WindloadShea,Windmoment Pw(Ib)O(lb).MOb-ft) 6,0006,00060,000 10,000 16,000500,000 2,50018,500672,500 25'B-t-t2 8 NII f 3,000 21,500972,500 1,00022,5001,082,500 25' t 3 :!: -f-B-0(1) IIt,k 'i 10' 3,000 25,5001,562,500 1,500 L 27,0001,825,000 Fig.1.4.Conditions and resulting wind loads forExample1.la. where, from Fig.A3_T(2h/3) = T(80)= 0.149 Fig. A4.1.l2(K30)1/2V30/f=1.12X 1 X 100/1=112 ---+ P =0.092 Fig,AS.0.88fh/(V30 yK 120) = 0.88X 1.0X 120/100y'f.5 = 0.862andh/e = .hid = 120/6.5= 18.5--F= 0.1 Fig.A6.structure sizefactor:S = 1.0 forh= 120. therefore qF = 25.6 X 1.297 X Kz = 33.2 Kz, qF30= 33.2 XI ,: 34 psf qFso= 33.2 X 1.2 =40 psf QF9S= 33.2 X 1.4 =47psf Windloads inpounds perfootof column height,W= Cf X DeX qF, are WI=W30=0.66 X 8.3X 34 = 1861b/ft DESIGNLOADS13 6' i.d. W el.120' Q) r :!: B-....'" NWindloadShea,Windmoment Pw (lb)o (lb)MOb-ft) 12.850 60't,IIf 70' () 12,850321.250 2,19015,040460,700 :!: @ B-e. t2N5,475 II:f @ 20,515905,140 1,09521,6101,010.455 t3 3,720 25,3301,479.855 t,k'i1,860 _0'_ (1) 27.1901,742,455 Fig.1.5.Conditions and resulting wind loads forExampleLIb. W2=WSO=0.66 X 8.3X 40 =2191b/ft W3=W9S=0.66 X 8.3X 47 =257 lb/ft. Themethodofdeterminingwindloadsonvesselsof twoormorediameters isthesameasforavesselof auniformdiameter.Whentheconicalsectionisnomorethan10percentof thetotalheight,cylindricalsectionscan beassumedtoextendtothemid-heightof theconicalsection.Otherwise,the transitionsectionshouldbeconsidered asa separate section. 1.S.EARTHQUAKELOADS GeneralConsiderations Seismicforcesonavesselresultfromasuddenerraticvibratorymotionof the groundonwhichthevesselissupportedandthevesselresponsetothismotion. Theprincipalfactorsinthedamagetostructures aretheintensity andthedura-tionoftheearthquakemotion.Theforcesandstressesinstructuresduringan 1 14PRESSURE VESSELDESIGNHANDBOOK earthquakearetransient,dynamicinnature,andcomplex. An accurate analysis isgenerallybeyond the kind of effortthat canbeafforded ina designoffice. Tosimplifythedesignproceduretheverticalcomponentof theearthquake motionisusuallyneglectedontheassumptionthattheordinary structures pos-sessenough excess strength inthe verticaldirectionto be earthquake resistant. Thehorizontalearthquakeforcesactingonthevesselarereducedtothe equivalentstaticforces.Earthquake-resistantdesignislargelyempirical,based onseismiccoefficientsderivedfromtheperformanceof structuressubjected inthepasttosevereearthquakes.Thefundamentalrequirementsetforthin buildingcodesisthatthestructuresinseismicriskzonesmustbedesignedto withstandacertainminimumhorizontalshearforceappliedatthebaseof the vesselinanydirection.Havingassignedaminimumvaluetothebaseshear basedonthepastexperience,theproblemwhicharisesishowtoresolvethis shearintoequivalentstaticforcesthroughouttheheightof thevesselinorder todeterminetheshearandthebendingmomentsinthestructureatdifferent elevationsaswellastheoverturningmomentatthebase.Theresultdepends inlargepartonthedynamicresponseof thestructure,whichmaybeassumed either rigidor flexible. Fordesignpurposesitissufficientandconservativetoassumethatthevessel isfixedatthetop of itsfoundation; noprovisionisusuallymadeforanyeffects of the soil-structureinteraction. SeismicDesignof aRigidCylindricalVessel Thestructureanditsfoundationareassumedtoberigidandtheassumedearth quakehorizontalaccelerationofthegroundaistransmitteddirectlyintothe vessel.Theterm rigidisused hereinthesenseof having no deformations. Eachsectionofthevesselwillbeacteduponbyahorizontalinertialforce equaltoitsmassandmultipliedbythehorizontalaccelerationa of thequake movement, t:.P= t:.W(a/g), actingatthecenterofthegravityof thesection.Theoverturningmomentat anarbitraryelevationisequaltot:.Ptimesthedistanceofthecenterof gravity ofthevesselsectionabovethesectionplane(seeFig.1.6). Theirresultant Pe is assumedtoactatthecenterof gravityoftheentirevesselandisgivenbythe equation Pe = Ma= (a/g)W = cW where g= gravitational acceleration W = operating vesselweightduringtheearthquake DESIGNLOADS15 loadsshears Fig.1.6.Earthquakeloadsandshearsforarigidcolumnof auniformcrosssectionand weight. Table 1.2.Values of the Coefficient c. ITEM Vessel Equipment attached to vessel Source:Ref.4. ZONE0 o o ZONEIZONE2 0.050.10 0.250.50 ZONE3 0.20 1.00 c = a/g,anempiricalseismic coefficient, depending onthe seismiczonewhere the vesselislocated. The usual valueof c, whenused, isgiven in Table1.2. Theoverturningmomentat thebase Mbisequalto Pe timestheelevation h of the center of gravityof the vesselabovethe vesselbase: Thesimplerigid-structure approach wasused in early building codes. For a short heavyvesselorahorizontaldrumontwosupports this designprocedure iseasy to apply and probably justified. However, it cannot be reasonably appliedto tall, slender process columns, regardlessof their dynamic properties. Seismic DesignofFlexible TallCylindricalVessels Thesuddenerraticshift during an earthquake of the foundation under aflexible tallcylindricalvesselrelativetoitscenterof gravity causes the vesselto deflect, sincetheinertiaofthevesselmassrestrainsthevesselfrommovingsimultane-16PRESSUREVESSELDESIGNHANDBOOK alL 'T2HI !0.5H mode1 C =0.560 mode2 C =3.51 mode 3 C =9.82 .1[nEil" Naturalfrequencyf= 'f = C~ j wherewis theuniformweightof beam per unit length. I Fig.1.7.Sketchofmodeshapesforacantileverbeamofauniformcrosssectionand weight. ouslywithitsfoundation.Thevibrationinitiatedbytheinducedelastic deflec-tion isthen graduallyreduced by damping or partial yielding in the vessel. Fromexperienceandtheoreticalstudiesitcanbeassumedthatastructure withalongerfirstperiodof vibration T and higher damping willbe subjected to . lesstotal damagethan a structure with a shorter T andsmaller damping capacity, provided that it hasthe strength to withstand the sustained deflections. Atall,slendervesselrepresentsadistributedsystemofmultipledegreesof freedomandwillbeexcitedintoatransversevibratorymotion,consistingof relativelysimpledeflectioncurvescalledmodes,withthefirstfundamental modepredominant.Eachmodehasauniqueperiod of vibration. The firstthree modesforacantileverbeamof uniformcrosssectionandweight are illustrated inFig.1.7.Sincethevesselwilltrytovibratewith acombinationof natural frequencies,withthefirstfrequencypredominant,theresultantmotioncan becomecomplexanddetermined by superposition. The forceon a vessel section causedbyvibrationisthen equaltotheproductof its mass and the vector sum of theaccelerationsassociatedwitheachmode.Eachpointunder vibration can . experienceamaximumdynamicshear.Obviously,aninvolved,detailedmathe-maticalanalysis,basedonstillincompletefielddataderivedfromobservations of thepast earthquakes would not be justified in the practical design. Forpracticaldesignpurposesthebuildingcodes[3,5] * requireallfree-standingstructuresinseismiczones to be designed and constructed to withstand theminimumlateralforceVappliedatthe base inany horizontal direction and *Numbers inbrackets refer to the references in AppendixA9: DESIGNLOADS17 equal to the product of the weight and empirical coefficients: V=ZKCW, where W = normaloperatingweightmostlikelytoexistatthetimeof apossible future earthquake; Z= earthquakezonefactor,anumericalcoefficientdependantuponthe seismiczoneinwhichthestructure is located and found from the seismic zoningmapsinref.3(seealsoAppendixAI);forlocationinzone1, Z= 0.25; zone 2,Z = 0.50; zone 3,Z = 1; K= structurecoefficient,anumericalcoefficientrelatedtotheinherent resistanceof astructuretypetothedynamicseismicforces; Kisbased ontheearthquakeperformancerecordofthetypeof thestructure; for cylindricalvesselssupportedatbasetheKvalueistakenasequalto2; forother structures Kmay vary from 0.67 to 3.00; C = flexibilityfactor,anumericalcoefficientdependingontheflexibilityof thevesselandgivenbytheequation C = 0.05 fTI/3 ,where T isthefunda-mentalperiodof vibration of the vessel, in seconds, in thedirection under consideration;forT < 0.12s,thefactor C isusuallytaken as0.1 0; a great accuracyof Tisnotneededincomputing thecoefficientCintheabove equation,sinceCisinverselyproportionaltothecuberootofT,and hencedoesnotchangeappreciablywithsmallvariationsofT,andthe assumedfixityatthebase willtendto make the calculated T smaller than thetrueperiod(seeSection4.7fortheprocedureusedtocomputethe basicperiod of vibration of tall, slender, self-supporting process columns). ThebuildingcodesprescribethedistributionofthebaseshearVoverthe height of the structures inaccord withthe triangular distribution equation n Fx= (V - Ft )wxhx f L w/h/ I-I where Ft = 0.004 V(hnfD8?..;0.15 Visaportion of V assumed concentrated at theuppermostlevelhn of the structure to approximate the influence of higher modes. For most towers Ft =0.15 V, since hnfD8 > 6.12; F/, Fx= the lateral forceapplied at levels hi, hx, respectively; WI,WX= thatportionofW whichislocatedatorisassignedtolevelsi,x, respectively; D8= the plan diameter of the vessel. For check, the total lateral shear at thebase isV = Ft + 1:?_1Fl. 18PRESSUREVESSELDESIGNHANDBOOK Theforceactinglaterallyinanydirection on any appendage connected to the vesselisgivenby the equation whereWpistheweightof theappendageandCpistakenequalto0.2.Force Fpisapplied at the center of gravity of the attached equipment. Sincethehighermodalresponsescontributemainlyonlytothebaseshear, butnottotheoverturningmoments,thebasemomentMbandthemoments Mxatlevelshxabovethebasecanbereducedbymeansof reductioncoef-ficients J and Jx,and aregivenby the following equations: Mb=J(Fthn+tFlhl)'where0.45 300) thin wall cylindrical shells under axial pressurewillfailby bucklingatacriticallongitudinalcompressivestressunder proportionatelimit.Atlowerratiosof Roltthe cylindrical shells will failrather byplasticyieldingof thematerialwhenthecriticalcollapsestressiscloseor equal to the yieldstress. Thegeneralequationforthecritical(minimum)elastic buckling stress forthe cylindricalshellswiththeendssimplysupportedcan bederived using the strain energy method[131] : OLc=NLcit=EtlR [3(1- v2)]1/2 OLc= 0.605 Et/R. and forj,= 0.3 NLc isthecriticalaxialcompressiveloadinlb/in. of circumference.Thelength of a half sine waveof the deflected shell equals to: Lw= 1.72 (Rt)1/2. The critical compressivestressoLc, in contrast to overall orinterstiffener collapse stress under lateral pressure, does not depend on the unsupported shell length L. MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS55 Belowthe critical stressOLc(or load NLc) the cylinder is ina stable equilibrium, meaning, that the stressthroughout the cylinder isbelow elastic limit.If the load isremoved, the cylinder returns into original unstrainedcondition. At the critical stressthecylinderisinneutralequilibrium.Atanyfurtherincreaseof NLcthe shelldeflectionincreasesnotinproportiontotheload.Thestrainedcylinder passesintounstableequilibrium,causedby inelasticactionatsomesectionof the shell, leading to collapse by any small disturbance. The elastic strains at some pointsintheshellincreaseintoplasticstrains.It isintuitivelyobviousthatthe collapsewillbeinitiatedatthesection with the largest imperfections, geometric ormaterial. However,thetestresultsshowthat the failuresin buckling of fabricatedcylin-dricalshellsoccuratstressesmuchlowerthanthetheorypredicts,between40 to60percentof thepredictedtheoreticalload.Toadjusttheformula with the experimental resultsanempirical constant isintroduced: OLc= 0.605K Et/R wherethe empirical factor Kforshells with L/R < 5 can betaken as K= I- 0.901(1- e-a)anda =0.0625 (R/t)1/2 . If theaverageof test values 0.5istaken asequal to K then: OLc= 0.3 EtIR. TheallowableCodecompressivestressforaxiallyloadedcylindricalshellsis the sameasthe allowablestressfor spherical shells: Sa= 0.0625 Et/Rfortheelasticrange and Sa= B (readfromthe Code material charts) forplastiCrange. Clearlythelimitingstressintheaboveformulaistheyieldstressof thecon-structionmaterial.However,if safetyfactorequal to10, asusedinCodedivi-sionI,isappliedagainstthecriticaltheoreticalstressOLcthenthisallowable compressive stress will govern in a considerable rangeof Rolt. The total load cor-responding to the critical stressoLcis Fc= rrDtoLc' 3.4.SPHERICALSHELLSANDHEMISPHERICALHEADS Wheneverprocessdesignorstorageconditionspermitorhighdesignpressure requires,asphericallyshapedvesselisused.Althoughitismoredifficultto 56PRESSUREVESSELDESIGNHANDBOOK fabricatethanacylindricalshell,itrequiresonlyhalft h ~wallthicknessof the cylindricalshellunderthesamepressure,withminimumexposedsurface . .Onlarge-diametercylindricalvessels,hemisphericalheadswillintroduce negli-giblediscontinuitystressesat junctures.Asphericalshellisdevelopedbyrota-tion of a circlearound anaxis(seeFig.3. I 0). SphericalShellsorHeadsUnderInternalPressure. Bothprincipalradiiarethesameandthestressesat and aLarethe same.From theequation specifyingthestatic equilibrium at section a-ainFig.3.1 0, (aLcosa)t2rrRcos a = rrR2cos2 aP weconcludethat aL=PR/2t. But aL/R + atfR =P/t, so that at =PR/2t =aL TheCode stress formula, basedonthe insideradius andthe joint efficiency E: SE = PRd2t + 0.1 Port = PRd(2SE - 0.2P), whereO.1P isthe correctionfactor.TheradialgrowthflRis axisof R cos O!Fig.3.10. MEMBRANE STRESS ANALYSIS OFVESSEL SHELL COMPONENTS57 where Eisthemodulus of elasticity. BothstressesaLandatareuniformacrosstheshellthickness intension. The discontinuitystressesatthecylinder-hemisphericalhead junction can(per Code Fig.UW-13.1)beminimizedbyadesigntaperbetweentheheadandthecylin-dricalshelltosuch anextent that the stress formulaforthe spherethickness can beusedtocomputethethicknessofthehemisphericalheads.Iftheradial growthofthecylinderandconnectedhemisphericalheadwerethesamethe discontinuitystresswouldbeeliminated.However,thethicknessoftheshell wouldnot befullyutilized. If flRe= flRsthen or ts= teO- v)/(2- v) = OAlte, where ts= thickness of the hemispherical head te= thickness of the cylindrical shell. SphericalShellsandHeadsUnderExternalPressure Themembranestressesinsphericalshellsorheadsunderexternalpressurecan becomputedfromthestressformulasfortheinternalpressureP,substituling -PforP.However,asinthecaseof thincylindricalshells,thin-wallsphcrical shellsorheadswillfailbybucklinglongbeforetheyieldstressintheshellis reached. ToestablishthemaximumallowableexternalpressureP ~inthedesignof sphericalshells,hemisphericalheadsorsectionsof a sphericalshelllargeenough tobeabletodevelopcharacteristiclobesatcollapse,theconvenientCodc procedure hasto beused, asdescribedbelow. SphericalshellshaveonlyonecharacteristicratioRo/t, whereRoistheout sideradiusandtistheshellthickness.Thetheoreticalcollapsingexternalpres sure P ~forperfectlyformedsphericalshells isgivenby P ~= (t/Rof 2E/[3(1- v2)]1/2 andforv = 0.3, P ~=I .2 I E(t/Ro)2 . For commercially fabricatedshells P ~can be aslow as P ~= 0.3E(t/R)2 . 1 58PRESSURE VESSEL DESIGNHANDBOOK ForthinfabricatedshellswithpermissIbleoutofthefollowing equationisused, including the safety factorforthe allowable externalpressure: InordertousethematerialCodechartsforcylindricalshellstodeterminethe allowableexternalpressure forsphericalshellsintheelastic-plasticregion, thefollowingadjustmentincomputationsmustbemade.Thematerialcharts wereplottedwithSe/2= PaDo/t= B(ordinate)againstSe/E = Aaabscissa, whereSeisthecriticalstressinacylindricalshella1'collapseandPa isthe maximumallowableexternalpressureforcylindricalshells.Theabscissafora sphericalshellcanbe computed from giving = 0.125/(Ro/t) = A, where isthecriticalstressinasphericalshellatcollapse andisthe maxi-mumallowableexternalpressureonaspherical shell.However, sincea spherical shellcansustaintwiceashighpressure at the same strain asa cylindrical shell in tangentialdirection,thefactorBmustbeadjustedcorrespondingly,i.e.,multi-pliedby2 sothat 2B =and the maximum external pressurefor spherical shells and headsisthen given by the equation = B/(Ro/t). Thepermissibleaxialbuckling loadforanarbitraryshellof revolutionis safely approximatedbythatof aspericalshellwitharadiusequalto themaximum tangential radius of curvature Rt of the shell at the location. Example3.2.AsphericalcoversubjectedtointernalpressureP hasaflanged opening,asshowninFig.3.11.Auniformlydistributedforceq(lb/in.)is appliedatthetopringflangebyaconnectedpipe. Compute membranestresses inthe coverdueto the internalpressure andthe forceq. (a)Membranestressduetotheaxisymmetricalloadq.Theequilibrium equation inthe axialdirectionat angle is q2rrRsino/2nRsin = al- tsinor aL=q sino/tsin2 .MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS59 Fig.3.11. aLismaximum at = 0' From (alR) + (at/R) = Pitfor P = 0, obtain The shear and bending stresses at the section at aredisregarded. (b)Membrane stresses dueto the internalpressure Pare at=aL =PR/2t. BothmembranestressesfromP andqhavetobesuperposed.Theringflanges onthecoverrepresentagrossstructuraldiscontinuitywherelargesecondary bendingandshearstresseswilldevelop.Toevaluatethemamoreinvolved analysiswouldberequired.However,awayfromtheringflangesatadistance greater than (Rt)1/2 , the simple membrane stresses will be significant. 3.5.SEMI ELLIPSOIDALHEADS Semiellipsoidalheadsaredevelopedbyrotationof asemiellipse.Headswith.a 2: 1ratioof majoraxisRtominoraxish (Fig.3.12)arethemostfrequently usedendclosuresinvesseldesign,particularlyforinternalpressuresabove150 psiandforthebottomheadsof tall,slender columm. Inthe followinganalysis, semiellipsoidalheadsaretreatedasseparatecomponentswithnorestraintsat edges under uniform pressure P. UnderUniformInternalPressure SincebothRLandRt varygraduallyfrompointtopointontheellipsethe stressesaLandatvarygraduallyalso.Themainradiiof curvatureRLandRt 60PRESSUREVESSELDESIGNHANDBOOK . aregivenby Rt =[R4/h2+ (l- R2/h2) x2 Jl/2 RL= R: h2/R4 . Atpoint1,Rt =RL=R2/handatpoint2, RL=h2/RandRt =R. Thestresses aLandataretheprincipalstresses,withnoshearingstressonthesidesof the differentialelement. Thelongitudinalstress aLcan befoundfromthe equilibrium equation written forthe latitudinal section A-Ainthe vertical (axial) direction: Atpoint1, andatpoint 2, knucklering aL=PR/2t. axisof "rotation y __-l__~__________~__-f__~__~ ~ ~_____ x R t= RaIsin cp, tangential radius of curvature. R L= longitudinal (meridional)radiusof curvature. Fig.3.12.Geometry of a semi ellipsoidalhead. MEMBRANE STRESSANALYSIS OFVESSELSHELL COMPONENTS61 Thetangential stress at can bedetermined from (atfRt) + (alRd =P/torat = (PRt/t)[l- (Rt/2Rd] AtpointI, and at point 2, at =PRt/2t =PR2/2th=aL Fromtheequation forat at point 2 it can beseenthat as long as R2/2h2 < 1, at remains tensile; if R2/2h2 > I or R > IAlh, at becomes negative, in compression. For standard 2: 1 ellipsoidal heads with R= 2h, at = (PR/t)[1- (4h2/2h2)]= -PR/t. Theradialdisplacement at point 2 is t:.Rispositivefor(R/h)2+ v < 2andnegativefor(R/h)2+ v> 2. Sincethere arenodiscontinuitiesinauniformlythickellipsoidalheadtheonlygrossdis-continuityisthejunctureoftheheadtothecylindricalshell,whichincreases the stresses intheknuckleregion. Membranetensilestress,if usedaloneindesign for various R/h ratios without includingtheeffectsofthediscontinuitystressesatthehead-shelljunction, wouldresultintooIowa headthickness.Tosimplifythedesignprocedurethe Coderelatesthestressdesignformulaforthethicknessof ellipsoidalheadsto thetangentialstressofthecylindricalshellof theradiusR,modifiedbythe empiricalcorrectivestressintensificationfactorK,whichisbasedonmany tests. The Code equation forthe maximum allowablestress inthe head becomes SE = (PDi/2t) K + OJP, wherethefactora.IPmodifiesthestressforusewiththeinsidediameterDj andfactorK =[2 + (D/2h)2] /6andEistheweldjointefficiency.For2: 1 ellipsoidalheadsK= 1andtheheadthicknessisverynearlyequaltothatof 62PRESSUREVESSELDESIGNHANDBOOK theconnectedcylindricalshell.WithlowdiscontinuHystressesatthehead-shelljunctionthestandard2: 1ellipsoidalheadisasatisfactoryconstruction .at allpressure levels. Inlarge,thin-wallheadswithratios R/t > 300andR/h;;;' 2.5,afailureinthe knuckleregionduetothetangentialstress0t incompressioncanoccureither throughelasticbuckling(circumferentialwrinklesinthemeridionaldirection withoutanywallthinning)atastressmuch lessthan the yield stress or through plasticbuckling(atlowerratiosR/h).Themaindividingparameterhereisthe ratioR/t.Thedivisionlinebetweenthetwomodesoffailureisnotclearly definedandhasatransition,elastic-plasticrange,wherebothtypesof failure canoccur.Thecombinationof tensile,longitudinalstress0Landcompressive, tangentialstress tbecomessignificantduringhydrotests,whenmostfailures occur. Unfortunately,thereisnocompletelyreliableanalysisavailableforpredicting .abucklingfailureof asemie11ipsoidalheadunderint(lrnalpressureatthistime. However, thereader willfindsomeguidelines intherefs. 22, 30, 31,61, and 62. UnderUniformExternalPressure Themembrane stress distribution inanellipsoidal headdueto pressureacting on theconvexsidecanbecomputedbysubstituting -P for P inthe spherical stress formulaforanequivalent radius of the crown sectionof the head, unless buckling governs. Thedesignpressureontheconvexsideof theheadisusuallymuch smaller in magnitudethanontheconcaveside,andtheCodedesignprocedure(UG33d) hastobeusedindeterminingadequatethicknessof thehead. Theprocedure i ~ . basedontheanalogybetweenthemaximum allowablecompressive stress inthe crownregionof theheadwithanequivalentcrownradiusRandthemaximum allowablecompressivestressintheexternallypressurized spherical shell withthe sameradius.Nobuckling willoccur intheknuckleregionbecause of theinduced hightensiletangential stress beforedeformation inthe crownregionoccurs. 3.6.TORISPHERICALHEADS Torispht'Iicalheadshaveameridianformedoftwocirculararcs,aknuckle sectionwithradiusr,andasphericalcrownsegmentwithcrownradiusL(see Fig.3.13). .UnderUniformI nternalPressure ThemaximuminsidecrownradiusforCodeapprovedheadsequalstheoutside diameteroftheadjacentcylindricalshell.Underinternalpressurethiswould MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS63 givethe same maximum membrane stress inthecrownregionasin the cylindrical shell.Themostcommonlyusedandcommerciallyavailabletorisphericalhead type iswith the minimum knuckleradius equal to 6 percent of Lj Inspiteof the similaritybetweensemiellipsoidalandtorisphericalheads,thesuddenchangein theradiusofcurvature(point a inFig.3.13)fromLtor introduceslargedis-continuitystresseswhichareabsentinthestandard semiellipsoidal heads. How-ever,sincetheyarelessexpensiveto fabricateandthedepth of dish His shorter thaninellipsoidalheads,torisphericalheadsarequitefrequentlyusedforlow designpressures300psi).Theknuckleregionisquiteshort,andthedis-continuityforcesatpointahavelargeinfluenceonthediscontinuitystresses atpoint2,thehead-cylinder junction.Also,bendingstressesina knucklewith sharpcurvaturewillbedistributedmorehyperbolicallyacrossthewallthick-nessthanlinearly,muchasinacurvedbeam.Thelocalplasticstrainsinduced byhighdiscontinuitystressesatpointatendtocausetheknuckleradiusto mergemoregraduallyintothecrownradius,thusformingaheadwitha better shapeto resisttheinternal pressure.. Asinthedesignof semiellipsoidalheads,to simplifytheprocedureof finding anadequateheadthicknessttheCodeintroducesanempiricalcorrection axisofrotation spherical(crown) c H= depth of dish t= corroded thickness sin'" = (R- r)/(L- r) Fig.3.13.Geometry of a torispherical head. 64PRESSUREVESSELDESIGNHANDBOOK factorMintotheformulaformembranestressinthecrownregion,tocom-pensateforthediscontinuitystressesattheshell-headjunction.TheCode formulaforthemaximum allowablestress inthe headbecomes SE = PL jM/2t + O.1P or t =PLjM/(2SE - 0.2P), where M = ! [3+ (L;/r;)1/2] forrj=0.06L j, M = 1.77andE isthe weld joint efficiency. MembranestressesduetoinsidepressureintheknuckleatpointQinFig. 3.14 arecalculatedasfollows: therefore (atfL) + (aL/r) =P/t (atfL) + [(PL/2t)/r]=P/t, at = (PL/t)[l- (L/2r)]. Theaboveatisthemaximumcalculatedmembranecompressivestressinthe knuckleatpoint Q,whileinthe spherical cap the stressesareboth intension and equaltoaL= at = PL/2t.Theactualcompressivestressat viillbeaffectedby thetensilestressinthe adjacent spherical segment, and thefinalaverageatpoint Qcould beestimated[22]asanaveragestress equal to R Fig.3.14.Membrane' pressurestressesinthe knuckle of a torispherical head. MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS65 at = (PL/4t)[3- (L/r)]. Sincethetangentialcompressivestressintheknuckleregionof atorispherical headismuchlargerthanthatinasemiellipsoidal head, thepossibilityof failure wouldseemtobehigher.Large,thin-waJItorisphericalheadsareknown to col-lapsebyelasticbuckling,plasticyielding,orelastic-plasticyieldinhydrotests. Becausethemoduliof elasticityforordinary and high-strength steels arealmost thesame,thereisnoadvantagetousinghigh-strengthsteelforlarge-diameter, thin-walltorispherical heads. Topredicta possiblefailureunder internal pressurethefollowing approximate formulaforthecollapsepressureof atorisphericalheadwithlargedoltratios can beused[29,107] : Pc/ay =[0.43+ 7.56 (r/d)](t/L) + 34.8[1- 4.83(r/d)](t/L)2- 0.00081 where Pc=collapsepressure, psi a y= yieldstrengthof theusedmaterial, psi r = knuckleradius,in. d=vesseldiameter, in. t= headthickness,in. L = crownradius, in. ForstandardtorisphericalheadswithLj =doandrj= 0.06do' theequation can bewritten Pc= ay ([0.8836 + 24.7149(t/do)] (t/do) - 0.00081}. Theaboveequationcouldbeusedtoestimatethecollapsing pressureinsemi-ellipsoidalheadsifvaluesforrandLapproximatingcloselytheellipseare substituted. Semiellipsoidalheadswith D/2h = 2 havetorispherical properties equivalent to a torispherical head with L/D = 0.90 and riD = 0.17[2]. Theuseoftheaboveempiricalformula[107)inpreferencetotheformula fromreference29, Pc/al=[0.33 + 5.5(r/d)](t/L) + 28[1- 2.2(r/d)](t/L)2- 0.0006, isjustifiedherebecausethecomputedPcisclosertothetestresults.Also,in practicetheactualheadthicknessesarealwaystaken1/16 in.to1/8 in.thicker (nominalthickness)thantheminimumthicknessestakenincomputationsto accountforthepossiblethinningof plateinsomelocations.Thiswouldmake the Drucker's formulatoo conservative. 66PRESSUREVESSEL DESIGNHANDBOOK Example 3.3.Determinethe maximwn allowable internaLpressure for a standard torispherical head, do= 10 ft,t = 0.25 in. minimwn, Oy= 32,000 psi. dolt = 120/0.25 =480. Codeallowablepressure is Codeyield is P = 2SEt/(L/M + 0.2t) = (2 X 15000 X 0.25)/(120 X 1.77+ 2X 0.25) = 35psi Py = 35X32/15 = 74 psi Buckling pressure is Pc= 32,000 {(0.25/120)[0.8836 + 24.7149X 0.25/120]- 0.0008l} = 36.5psi. If theentirevesselhadtobesubjectedtoahydrotestof P = 35X1.5= 53 psi, anincreaseinthe headthicknesswould benecessary. UnderUniformExternalPressure Asinthecaseof semiellipsoidalheadstheCodeprocedureforcomputingthe maximumexternalallowablepressureP usestheanalogybetweenthecompres-sivestressinthecrownregionof thehead withtheallowablecompressivestress inthesphere of equivalentradius, andmust befollowed(UG33e). 3.7.CONICALHEADS UnderUniformInternalPressure Aconicalheadisgeneratedbytherotationofastraightlineintersectingthe axisof rotationatanangleorwhichisthehalf apexangleof theformedcone. IftheconicalshellissubjectedtoauniforminternalpressureP theprincipal stressa tat section a-ainFig.3.15canbedeterminedfromthe equations a doc + ot/Rt = Pitandat = PRt/t = PR/t cos or. MEMBRANE STRESSANALYSIS OFVESSELSHELL COMPONENTS67 axis ofrotation '"=half apex angle Rt = R/cos "',tangentialradius of curvature RL = ~ ,longitudinalradiusof curvature Fig.3.15.Geometry of a conical head. The equilibrium condition intheverticaldirectionatsection a-ayields 0L= PRI2t cos or. The abovestress formulasarethe cylindricalformulaswhere Rhasbeenreplaced by Rlcos or. TheendsupportingforcetaL=PR/2 cos or lb/in.atsectiona-ainFig.3.15is showninthemeridionallineasrequiredif onlymembranestressesareinduced intheentireconicalhead.Inactualdesign,wheretheconicalheadisattached to a cylindricalshell, the supporting forcePR/2 lb/in. iscarried by the cylindrical shell,asshowninFig.3.16.Thisarrangementproducesanunbalancedforce (PRtanor)/2pointing inwardandcausing a compressivestress inthe region of the junction.Obviously,thelargertheangleor,thebiggertheinwardforce.This inwardforcehastobetakenintoconsiderationwhendiscontinuitystressesat Fig.3.16.Foree diagram at eone-cylinder junction. 68PRESSUREVESSELDESIGNHANDBOOK cone-cylinderjunctionsareinvestigated.Theangleais;.therefore,limitedin Codedesignto30degrees,andthejunctionhastobe 'reinforcedper Coderules (UA5bandc).Otherwisethe Codeusesthemembranestressformulaforconical shells(UG32g)todeterminethemaximumstressandthicknessofaconical shellwiththe joint efficiency E: SE = PD;/2tcos a+ 0.6P, where0.6Pisthecorrectionfactoraccountingforusingtheinsidediameter Diinthestressformula.If aspecialanalysisispresentedthehalf apexanglea canexceed30degrees.However,beyond a> 60 degreesthe conicalshellbegins to resemblea shallow shellandfinallya circularplate. Theradialgrowth at section a-ais t:,.R= R [(atfE)- (vaL/E)]= R2 P[1- (v/2)] /tE cos a, where E isthemodulus of elasticity. Therotationof themeridianat section a-ais (J = 3PRtan a/2Et cosa. UnderExternalPressure Therequiredthicknessof theconicalheadoraconicaltransitionsectionunder externalpressureisdeterminedbythesameCodeprocedureasforcylindrical shells,wheretheactuallengthof the conical section LinFig.3.17 isreplaced by 1---t--:7lines of support stiffener ct.-+---D L=outside diameterof the largecylinder Ds= outside diameterof the smaller cylinder Fig.3.17.Conical shellreducer with outside stiffeners. MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS69 anequivalentlengthLe= (L/2)[1+ (DL/Ds)]ofacylindricalshellofthe diameterDL(Le= H/2forconicalheads)andtheeffectivewallthickness te= t cos a(UG33f). Usuallythesharpcone-small-diametercylinder junction, point AinFig.3.17, requiresreinforcementfordiscontinuitystresses(U A8c).The stiffening ringfor thesmallcylinder(necessitatedbyexternalpressure)shouldbeplacedasclose topoint Aasfabricationallows,with the maximum a = (Rsts)I/2/2. Thedimen-sion(b+ Le) shouldbenolargerthantheCodeallowabledistancebetweenthe stiffenersforthe largercylinder, with DLand teforthe external pressure, unless additionalreinforcementatpointBisrequired;wherethereisadditionalrein forcementthemaximumdistancebisequalto(RL tdl/2/2.Theresultantdis-continuitymembranestressesatthepointBareintensionandopposethe compressivemembranestressesduetoexternalpressure.Theforcecomponent (PRtana)/2inFig.3.17pointsoutward.Thehalf apexangleaisherelimited to60degrees.Theinterestedreaderwillfindthedevelopmentof theCodede signmethodforreducers under external pressure andother loads inref.126. Forcesona Ring at Juncture of TwoConeFrustums underExternal Moment. TwointersectingconefrustumsinFig.3.18aresubjecttoanexternal moment M.TheverticalcomponentV atanypointmon the reinforcing ring isequal to [22] : V=Mc/I=MRcos(J/rrR3 =M cos(J/rrR2Ib/in. and TI=M cos 8/rrR2 cos al T2= M cos 8/rrR2 cos a2 Since a2> aI, the resultant radial force H on theringtension sidewillbe: H = T2sin a2- TIsin al H = M(tan a2- tan al) cos8/rrR2 I b/in. ThemaximumforceH on the ring occurs at8 = 00 and1800,ontheassumption that the ring resiststhe loadwithoutbending. Theoutward radial force H induces acircumferential forceFin the ring equal to: HRd8= Fd8/2+ Fd(J/2and F =HR=M(tan a2- tan al) cos(J /rrRIbsper arch lengthIin. 1 70PRESSUREVESSELDESIGNHANDBOOK Reinforcingring Cone2 "-r.2 Sectiona-a Fig.3.18.Two conical frustums witha ringai theSincetheradialforceshavenotangential component the tangential shears in the junctureSisdueto the increment in thecircumferential force F in a differential elementRdO.FromFig.3.19atpointmwegettangentialforceactingon the ring: S = M(tan 0:2- tan 0:1 )[cos (/I- d/l/2) - cos (/I + dO/2)] /rrR which forsmall anglesreducesto: S = M(tan 0:2- tan o:d sin/ld/l /rrRperdifferential arc/I' and S =[M(tan 0:2- tan O:I)J rrR 2]sin tJ in lbsper arc oflength 1 in. Theaboveforceshavetobetakenintoaccountwhensizingthereinforcing ringandtheconnectingwelds.Therequiredcross-sectional areaof theringis: a = H max. R/Sa where Saisthe allowable stress of the ring material. MEMBRANE STRESS ANALYSIS OFVESSEL SHELL COMPONENTS71 y F cos(8 -d812) Fig.3.19. Clearlyan assumptionis madehere that the unbalanced force H is counteracted bythering,andtheconicalshellsarestressed by membrane forces only, in ten-sion and compression. If the coneIisreplaced by a cylindrical shell then theangle 0: 1= O. 3_8.TOROIDALSHELLS Atoroidisdevelopedbytherotation of a closed curve, usuallya circleabout an axispassingoutsidethegeneratingcurve.Whileanentiretoroidalshell,suchas anautomobiletire,israrelyutilizedbyitself inthedesignof pressurevessels, segments of toroidal shellsarefrequentlyused asvesselcomponents. Membrane StressAnalysisof a ToroidwithCircular CrossSectionunderI nternalPressure Figure3.20presentsthegeometryofatoroid.Tofindtheprincipalstresses, longitudinalaLandtangentialat, aring-shapedtoroidalsectionisisolatedand theequilibriumconditionbetweeninternalpressureP andmembranestressaL intheverticaldirectionisexpressed asfollows: and aL=P(R2 - R5)/2tR sin4> =(Pr/t)[(R + Ro )/2R]. 72PRESSUREVESSELDESIGNHANDBOOK Section a-a Rt = R/sin , tangentialradiusof curvature RL = r,meridionalradius of curvature Fig.3.20.Geometry of a toroidal shell. AtpointI, whereR= Ro- r, a L =(PrI2t)[(2Ro- r)/(Ro- r)] Atpoint 2, whereR= Ro+ r, a L = (PrI2t) [(2Ro+ r)/(Ro+ r)] Atpointb,whereR=Ro, aL= Prlt. Notethatbythegeometryof atoroid,aListhelongitudinalstress despitethe factthatitistangentialtothecircle.Themeaningof aLandatforthetoroid isthereverseof thatfora straight cylinder. Fromtheequation MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS73 [atl(RlsincJ]+ (aLlr)= PIt thestress at canbedetermined: at= P(R- Ro)/2t sincJ>or at= Prl2t. Tosummarize,bothstressesaL(variable) andat (constant) areintensionand theyaretheprincipalstresses. Stress aLatpoint b isequivalentto themaximum stress in a straight cylinder. A more accurate analysis is offered inref.23. 3.9.DESIGNOFCONCENTRICTORICONICALREDUCERS UNDERINTERNALPRESSURE Foratransitionsectionbetween two coaxial cylindrical shellsof differentdiam etersaconicalreducer witha knuckleatthe larger cylinder anda flare(reintrant knuckle)atthesmallercylinderisveryoftenpreferredtoasimpleconicalsec-tionwithoutknuckles.Themainreasonforthisistoavoidhighdiscontinuity stressesatthejuncturesduetotheabruptchangeintheradiusof curvature, particularly at high internal pressures (> 300psig).Thiscan be further aggravated byapoor fitof thecone-cylinder weld joint. Theconical transition sectionwith knuckleshasbothcircumferentialweldjointsawayfromdiscontinuitiesand usuallyabetteralignmentwiththecylindricalshells; however, it ismoreexpen siveto fabricate. Theknuckleatthelargecylindercanconsistof aringsectioncutoutof an ellipsoidal,hemispherical,ortorisphericalheadwiththesamethicknessand shapeasrequiredforacomplete head. Moreoften, both knuckles arefabricated ina formof toroidal ringsof the sameplatethickness asthe conical section. InthefollowingdiscussiontherequiredradiifortheknucklerLandtheflare rsarecomputedusingtheprincipalmembranestressesasgoverningcriteriafor the casewherethe sameplatethickness isusedforthe en tirereducer. TheCodespecifiesonlythelowerlimitfortheknuckleradius'L:"rLishall notbelessthanthesmallerof 0.12 (R Li + t)or3t whilershasnodimensional requirements"(Fig.UG36).The stresses inthe conical shellsection atpoint1 in Fig.3.21are givenby and from 74PRESSUREVESSELDESIGNHANDBOOK smallcylinder flare aconical section " 1-'----1---R, ----0< T.l. PRe L=-1--_____ R,_---'''''''____2 ~ t . . : : k L ~ R 1=(R L- TL + TLCOSa) = R L[1-~~(l - cosa)] L1= ~=RL[1- (TL/RL)(1- cos a)] cos Qcos OfR2= (Rs + TS - Tscos a) = Rs [1+ ;: (1- cosa)] L _R2_[1 + (Ts/RsH1- cos a)] 2- ---Rs COS Of COS Q = PRJt, PRe at =r;-0,= PRe12t, Fig.3.21.Membrane stresses in conical transition sectionwith knuckles. All dimensin 1.0the shell-plate midsurface in corroded condition.0s are MEMBRANE STRESSANALYSIS OFVESSEL SHELL COMPONENTS75 wehave and the cone thickness MinimumKnuckleRadius 'L Theknuckleatpoint1 inFig.3.22willbesubjectedtothesamelongitudinal membrane stress 0Lasthe conical section onthe assumption tc= tKL: and from wehave SinceL 1willinpracticebealwayslargerthan2rL,atwillbenegative(in compression)membranestressandmaximumat1.Theinwardcomponentof 0L,0Lsina,willbepartiallybalancedbytheinternalpressureonthevertical projectionof theknucklering. Theprincipal radius of curvature L 1at pointIis sharedbyconicalandknucklesectionsalikeandfixedbythegeometryof the cone.Thesecondradiusof curvaturerLcan 'nowbecomputedintermsof RL assumingtc= tKL'Fromstressesatand0Lbyinspectionthetangentialstress Fig.3.22. 1 76PRESSUREVESSELDESIGNHANDBOOK willgovernandcanbeexpressed interms of RLatpointt: or at =(PL1 /tKL)[1- (Ld2rd]=(PR1/tKLcos a)[1- (Rl /2rLcosa)] = (P/tKLcos a)(RL- rL+ rLcos a) .[(2fLcos a- RL+ rL- fLcosa)/2fL cosa] =-(PRL/tKL){[(RLfrL) + (rLsin2 a/Rd - 2]/2 cos2 a} Inreplacing -at withthemaximum allowablecompressivestress Satwo criteria mustbemet:first,thesubstitutionshouldbeinaccordancewiththeintentof theCoderules;and,second,theknuckleradiusfLshouldbekepttothemini-mumrequired, sincethelarger fL, thehigherthefabricationcost. .Thestress-at isthemaximumcalculatedcompressivemembrane stress inthe knuckle.However,sincethereiscontinuitybetweentheknuckle andthe conical section,thefinaltangentialstressintheknuckleatpointIwillbesmallerand couldbeestimatedastheaverageof atintheconeandknuckle[22].It also hastoberememberedthatthestressesat andaLdonotrepresenttheentire stressprofile,butareonlystresscomponentsusedasdesigncriteriainabsence of amoreaccurateanalysis.Thediscontinuitystresseswouldhavetobesuper-imposedto obtainthe complete state of stress. Sincea knuckleisa sectionof a torus which inturn approaches a cylinder with anincreasingradiusRoinFig.3.20,themaximum allowablecompressive stress Sashouldbethesameasrequiredforcylinders. With smallratios fLftKLasused infabrication,thefailurewouldoccurratherinlocalyieldingthanelastic buckling.Basedontheabovereasoningitwouldseemacceptableto useforthe maximumallowablestress(2SaE) (R. + ,.). SeeFig.3.24. (F.+ FL)cos (Jsin -y = a 2.(RL- 'L) < (R. + ,.). Not shown. D lOs,beyond which it dropsto 0.3-0.5. ThetotaldragforceDconsistsof thefriction component Dr, predominant at lowerRevalues,andthepressurecomponentDpduetothepressureresultant onthecylinder.Usually,Dpforbluntbodies willbemuch largerthan theforce Dr.Thedragforceisdealtwithinthedesignofpressurevesselsasthewind load. Development of theForce on a Cylinder Transverse to theDirection of Flow Becauseoffrictionaboundarylayerofretardedfluidformsonthecylinder surface.A separation of theflowoccurs neartherearof thecylinder and a wake, aturbulentregionwith alowerflowvelocitythanthesurrounding freestream, develops.Withhigh Re, theseparationpointsof the flow move forwardtoward thetransversediameter of thecylinder andthewakewidth increases(Fig. 4.14). Byfrictionthehigher-velocityenclosinglayerssetthelowervelocityfluid immediatelybehindtheseparationpointsinrotation. Two symmetrical vortices withoppositerotationalvelocitiesdevelop,andasthewindvelocityincreases, thefrictionforcebetweenthefreestream andvorticesincreases. Atsomepointonevortexbreaksawayandpassesdownstream, disintegrating inthefreestream[Fig.4.14(d)].-UptothecriticalvalueofRe < 5 XlOsthevorticeswillformandshed alternatelyon either side of thecylinder, givingrisewhileformingto a transverse forceL(psf)onthecylinder,atafrequencyofvortexshedding Iv.Beyond thiscriticalRe valuetheseparationpointsmovebackward,thewakecontracts and becomes entirely turbulent, and thevortex periodicityvanishes. VonKarmanwasthefirsttoanalyzetheformationof suchvorticesmathe-matically,andfoundthatmaximumstabilityintheflowof vorticesoccurred whenthevorticeswereshedalternatelyfromthesidesof acylinder,asshown inFig.4.15. Thevorticesmoving downstream at a speed somewhat lessthanthat of thefreestream forma so-called VonKarmanvortex street or trail. ThecrossthrustforceL, likethedrag force, can beexpressed conveniently as 110PRESSUREVESSELDESIGNHANDBOOK DESIGNOF TALL CYLINDRICAL SELF-SUPPORTING PROCESSCOLUMNS111 o 1 v v,=O.86V h=1.3d Fig.4.15.Vortex trail in the wake of a cylinder. a part of the stagnationpressure q = pV2/2: L= CL pV2/2(psf of theprojected area). Unfortunately,incaseofacylindernofirmvalueof C Lhasbeenestablished. ValuesforCL rangingfrom0.2toashigh1.7aregivenintechnicalsources. CalculatedCLvaluesbasedonfieldobservationsareusuallylow,whilethe valueofCL = 1.7isbasedonamathematicalanalysis.Valuesof CL between 0.2to1.0appeartobegenerallyacceptedandusedinpractice;avalueof 1.0 isconsideredconservative.Forpracticalengineeringcomputations,valuesof CL = 0.4-0.6 arerecommended here. Critical WindVelocity Thefirstoriginalinvestigationof thevortex forming and shedding phenomenon wasdone in1878 by Strouhal, who foundthefollowingrelationship: s = Ivd/V where S = adimensionlessparameter, called the Strouhal number Iv = shedding frequencyof vortices, cps d= diameter of thecylinder perpendicular to theflowof thewind, ft V = wind velocity, fps. Thevalueof Svarieswiththevaluesof Re. However,intherangeof Re values between103 and105 thevalueof Sremainsnearlystable,equaltobetween 0.18and0.21,andisgenerallytakenasequalto0.20incomputations.For valuesof Re > 105 the Strouhal number increasesrapidlyto 0.35. ThefirstcriticalwindvelocityVIisthewindvelocityatwhichthe shedding frequencyofwindvorticesI visequaltothefirstnaturalfrequencyof the 112PRESSUREVESSELDESIGNHANDBOOK columnf.ItcanbecomputedfromtheStrouhalnumber,substituting I = l/T for Iv: VI= Id/0.2 (fps) VI= 3.40d/Tmph. Thesecond critical wind velocity can betaken asV2=6.25 VI' BasicVibrationalPrinciples; MagnificationFactor Whenthevibrationofstackswasfirstobserveditwasascribedtoresonance. Accordingtothistheory,if thefrequency Iv of thecrossthrustforceLcoin-cideswithoneof thenaturalfrequencies Iof thecylinder,howeveritmaybe supported, a forcedresonant vibrationresults. Thestressesinducedbysuchoscillationscanbuildup and increaseoverthose calculatedfromstatic wind load analysis. Atthispointsomeconsiderationof thebasicvibrationalprinciples applicable tothisproblemisappropriate.Atallvessel,acantileverbeam, representsa sys-temwithadistributedmassof infinitely manyvibrational freedoms and natural periodsofvibration,eachofthemaccompaniedbyadistinctmodeorvibra-tionalcurve.However,theprevailingperiodof vibrationwillbethelongest, calledfundamentalorfirst,whichisof maininterest.Thegeneralconclusions pertainingtotheresonancecasecanbederivedbyusingasystemwitha single degreeof freedom, asshown inFig.4.16. Thedifferentialequationfordamped,forcedharmonicmotionofasystem withasingledegreeoffreedomexpressesthedisplacementxofmassmbe-tweenitsequilibriumpositionanditsinstantaneouslocationasafunctionof timet: mx + cx + kx = F cos wI spring K>;---+-equilibrium position dashpot Fig.4.16.Systemwithasingledegreeoffreedomsubjectto viscousdamping and externally imposed hamonic force. DESIGNOF TALL CYLINDRICAL SELF-SUPPORTING PROCESS COLUMNS113 where m = Wig isthe mass c = dampingfactor,resistanceto motion inpounds when thevelocityisequal to one inch per second, lb-sec/in. k = scaleof aweightlessspring,forceinpounds requiredto deflectthespring by one inch, Ib/in. Fcos wt = harmonicperiodicforceimpressedonthesystem,lb.Itismaxi-mumwhenI= O.ItcouldalsobetakenasF sinwt, withtheminimum whent = 0, depending on thedesiredapproximation w= thecircular frequencyof the impressedforceF, rad./sec. Thegeneralsolutionof thisdifferentialequationconsistsoftwopartssuper-posed on each other. I.Thecomplementaryfunctionisthegeneralsolutionof thehomogeneous equation(theright-handsidesetequaltozero).Withtwoindependentcon-stants,thecomplementaryfunctionrepresentsthefreedamped vibrationof the system and isgivenby: whereWd=[(kim) - (c/2m)2] 1/2 ,isthe damped natural circular frequencyand CI andC2 areconstantswhichdependontwoinitialconditions.Forsmall dampingcoefficientsc,asinthecaseof processcolumns, wdcanbereplaced byWn= (k/m)I/2 ,thenaturalcircularfrequencyfora vibrating system with no damping and no impressed forceanddescribedby the equation mx + kx = O. Freevibrationdisappearsbecauseof damping.Verylightdampingwillallow manyoscillationstooccur, buttheywillgraduallylesseninamplitudeandbe-comenegligible.However,it isquitepossiblethattheintensityof theinduced freevibration,forinstanceduringanearthquake,issuchthatitwillcause damageto thevesselbefore it has the time to dieout. 2.Theparticularintegralisthesimplestpossiblesolutionoftheequafion, with no integration constant, specifying the forcedoscillation of the system: cos(wt - c/, where Cc= 2 (mk)1/2isthecriticaldampingfactor,representingthedividingline betweenoverdamping(c/cc> 1,nonvibratingmotion)andunder-114PRESSUREVESSELDESIGNHANDBOOK damping(c/ce < 1,harmonicvibration).ThedampingfactorCof a tall, slender pressurevesselisusuallyonlyfewpercent oj: Cc F/k = Xsisthedeflectionof thesystemdueto themaximum impressedforce F whenacting asa static force = cons tan t phaseangle. Themaximumactualamplitudexofforcedvibrationcanbeubtainedby multiplying the statical deflection Xsby the fraction calledthemagnificationfactor.Forsmallvaluesof c/ce themaximum valueof M.F. willbereachedverynearly at W= Wn: Maximum M.F.= ce/2c. InpracticalcomputatIOnsthedampingcoefficientsCand Ce of the system are notknown,notreadilymeasurable,anddifficulttoestimatewithsufficient accuracy.Forfreevibrationwithdampinglessthancritical,theratioof two consecutivemaximum amplitudes is(seeFig. 4.17) _expHc/2m)[t+ (21T/Wd)]} - exp[-(c/2m) t] =exp (-1Tclm Wd)= e-b where 8 isknown asthe logarithmic decrement and isgivenby 8= 1TC/mwd. Expressedinterms of c/ce, and substituting forWdand mk =8 = (1Tc/m)[k/m- (c/2m)2]1/2= 21T(c/ce)/[1- (c/Ce)]112 x Fig. 4.17. DESIGNOF TALL CYLINDRICAL SELF-SUPPORTING PROCESSCOLUMNS115 and for small values of c/ce, Themagnification factoratresonancecannow be expressed asfollows: Maximum M.F. = 1T18. Someinvestigatorsconsiderthisvaluetoohightoapplyagainstcylindrical structures,andsuggestthatamaximumM.F.equalto1/8ismoreinlinewith their actual measurements. LogarithmicDecrement [, Dampinginasystemcanalsobedefinedastherateatwhichthematerial absorbsenergyunderacyclicload.Theenergyisdissipatedthroughheatby internaldampingduetomicroscopicplasticactioninthematerial.Damping of avibratingcolumnwouldinclude,inadditiontointernaldamping,friction lossesbetweentheshellandair,anyenergylossesduetoresistanceofthe external piping and internals, and conditions at thebasesupport. Sincexn+l/xn = e-b :, 1 - 8, thelogarithmicdecrement8canbeexpressed by a simpleratio, whichcanbeinterpretedasthepercentagedecaypercycleintheamplitudeof afreelyvibratingvesselandthusprovidesameansof measuringthevibration decayrateofacolumninfield,sincethevalueof twoconsecutiveamplitudes xnand Xn+1can bedetermined byfieldmeasurements. However,8lacksfullsignificance,sinceitdescribes conditions awayfromthe pointofresonanceandissusceptibletoerrorsinmeasurement.Mostavailable data,however, arepresented interms of 8, although its reliablevaluesforty,piCal process towers and stacks arenot as extensive asit would bedesirable. Valuesof8forweldedtowersandstacksandmagnificationfactorsbased onfieldmeasurementsaregiveninTable4.3.Unlessotherwisespecified,the valuesof 8andM.F. in columns IIandIIIcanbeusedinmostapplications. Sinceadeflectionisdirectlyproportionaltotheappliedforce,themaximum forceperunitareaof theprojected surface of a cylinder atresonance can beexpressed by L= (CL X M.F.) pVi/2 (psf). For thefirstestimatea valueof (CL X M.F.) = (0.5X 60) can beused. Table4.3.AverageValues for [jand M.F. IIIII SOFTSOILSSTIFF SOILSROCK,VERYSTIFF 6M.F.6M.F.6M.F. Tallprocess columns0.126250.080400.05260 Unlined stacks0.105300.052600.03590 lined stacks0.314100.105300.07045 Source:Ref.48. Conflict betweenForced and Self ExcitedVibration Theories Allvalues essentialto theaerodynamic design analysis of towersand stacks, such asthecoefficient CL, the Strouhal number S, and particularly the damping data, aresubjectsof widedifferencesof opinion among various investigators. Further-more,thequestionhasbeenraisedwhetherforcedvibrationtheoryshouldbe usedasthebasicdesign-governingtheoryintheaerodynamicdesignof stacks andtowers. Accordingtoforcedvibrationtheory thealternating transverseforceacting on acylinderinanairfloworiginatesfromvortexshedding,whichisindependent ofthemotionofthecylinder.Thisforceexistsatallwindvelocitiesatwhich thevonKarmanvorticesareformed.Sincethefrequencyof forcedvibrations isequalto thefrequencyof the impressed force, this would mean, that the stack wouldvibratetosomedegreeatafrequencyproportionaltowindvelocity.At thecriticalwindvelocitythefrequencyofvortexshedding fvequalstothe naturalfrequencyof thestackortower fandresonanceoccurs.Theprincipal reasonthistheoryhasnotbeengenerallyacceptedisthat"thefrequenciesof stackvibrationsasobservedinthefieldhavenearlyalwaysbeenfoundtobe approximatelyconstantatthevalueofthenaturalfrequenciesofthestacks. Thisisinpoor agreement with forcedvibrationtheory. Toexplainthisdiscrepancyothertheorieshavebeenadvanced,particularly self-excitationtheory.A self-excitedvibrationshouldtakeplaceonlynearthe naturalfrequencyofthestructure.Inself-excitedvibrationtheimpressed alternatingforcesustainingthemotion iscreated by the motion itself and, when themotionof theobjectstops,theimpressedforcedisappears.Once excited to vibrationthecylindercontrolsthefrequencyof theimpressedforce[36,47] . Vibrationwouldcontinuethroughawiderangeof windvelocities,andapar-ticularcriticalwindvelocitywouldnotexist. If theamount of energy extracted bythecylinderfromtheairexceedsdamping, the amplitUdeof oscillations will increaseuntil,inpractice, aneqUilibriumbetweendamping andthe energy input develops.However,afullsatisfactorytheoryforastackoscillatingwithself-excited vibrations hasnot beenmathematically worked out by its proponents. DESIGNOF TALL CYLINDRICAL SELF-SUPPORTING PROCESS COLUMNS117 It wouldseemthatatlow-Reflowsthecylinderisdynamicallystable,sub-jectonlytoforcedvibrations; atthe higher-Re flowsa possibility of self-excited vibrationexists.Fromalargenumberof fieldobservationsit canbeconcluded thatthefirstpeakamplitudesappearatthecriticalwindvelocityV1,cor-respondingtoa Strouhalnumberof approximately0.2pointingouttheforced vibrationtheoryasthebasicexcitation.Theformationofsecondarypeak amplitudesathigherwindvelocitiesinsomecasesmayleadtomodificationof forcedvibrationtheory,particularlyafterthebehaviorofstreamsaround elastically deflecting cylinders athigher Re values isbetter known. Foravesseldesignerthemostsignificantfactisthatthepeakvibrationsas predicted by forcedvibrationtheory are not inconsistent withthe observed data and can forma basis for the mathematical checking of a stack or tower forwind-induced vibrations. DesignMethodBasedonForcedVibration Thefirstcritical wind velocityV1 isgivenby V1=3.40d/Tmph wheredistheoutsidediameterof thevesselinfeet, andT isthefirstperiodof vibration insecondpercycle. The second critical velocity canbe takenasV 2= 6.25 V 1 Themaximum unitpressure Ltransverseto the winddirection atresonanceon theprojected cylinder areais L= (CL X M.F.)pVV2 psf. Assumingthatwindpressureandtowermasseffectsareconcentratedinthe vesseltopsection(!- totof Hdependingonexposure)andcolumnstiffness effectsarelimitedtothevesselbottomsection,theactionofthetotalpres-sureactingonacylindricaltallslendervesselatresonancecanthen be approxi-mated by its resultant equivalent static forceat thetop of the column: F = (CL XM.F.)(pVV2)(d X H/3). WhenV1 isinmph, F = 0.OO086(CL XM.F.)(d X HXVf). TheequivalentforceFcanbeusedincomputingthemaximumnominal stressesatvariouselevationsoftheshellandalsothemaximumresonance amplitudeatthetopof thevessel.Theinducedstressesmustbesuperposed on 118PRESSUREVESSEL DESIGNHANDBOOK thestressesduetoweightandoperatingpressure.Thetotalcombinedstress shouldnotexceedthemaximumallowablestressfortheShellmaterial.If the combinedstressesaretoohighanda possibility of fatiguefailureexists, changes inthe designhaveto bemade. Novibrationalanalysiswouldbecompletedwithoutanattempttoevaluate theeffectsof fatigue.Thestressinduced inthe bottom shellsection, welds, and skirtsupportiscompletelyreversedinbendingwiththestressrangeS R=2S. Thefollowingrelationshipbetweenthenumberof cyclestofailureNandthe stressrange S Risusedasa fatiguecurve[74] : wherenandKarematerialconstantsand~isthestressintensificationfactor. Forcarbon steelwith anultimate strength of 60,000 psi, n =5 and K=780,000. Thevalueof ~willvaryaccording the typeof construction, weldand inspection. Somesuggestedvaluesfor ~are: ~= 1.2fora shell plate with a smooth finish ~= 1.8forbutt-weld joints ~= 3forfilletweldorincompletepenetrationgrooveweldwiththerootun-sealed betweenthebottom head and the support skirt. ThefatiguelifeexpectencyLeinhoursofvibrationinasteadywindatthe resonant velocity willbegivenby Le =NT/3,600. Thesafeservice lifeof continuous vibration is Le/S.F. Noprecisecriteria forthe safety factorS.F. can begiven.However, the minimum safeservicelifeshouldbeatleastequaltothesumof probabletimeperiods of steadywindof resonantvelocityat the job site. A safety factor of 10-15, asap-pliedto cycles to failure,could beaccepted forthistype ofload and structure. Fromtheabovediscussioniti& quiteobviousthat,withsomanyassumed variablefactors,theresultsof designcomputationscanonlybeinterpretedas approximations, the starting point for the evaluation of thevessel behavior. Corrective Measures against Vibration Themostcommonlyusedpreventivemeasureagainst vibration isa spiral vortex spoilerweldedaroundthetopthirdofthestack.Anothercorrectivemeasure foranexistingstructuresubjectto excessivevibrationwouldbeinstallationof DESIGNOF TALL CYLINDRICAL SELF-SUPPORTING PROCESSCOLUMNS119 permanentguys.Specialexternaldampingdevicesareexpensive,impractical, andinpracticenotveryeffective.Theydonotseemto represent a satisfactory solution. As in any mechanical design the cheapest and most effective vibration-preventive measurescanbeaccomplishedby a careful analysisduring thedesign stage, even at a higher initial cost. Ovaling Sincemostlyunlinedstacksareaffecteabythisresonancephenomenononlya briefdiscussionwillbeofferedhere_Thecrossaerodynamicforcescaninduce oscillating deformations of the upper section of stacks called ovaling or breathing. Thepressurepulsationswhichcauseovalingofastackoccuroncepervortex formation.If thenaturalfrequencyof acylindricalshelltaken asa circular ring coincideswiththevortexsheddingfrequency,theshellwillhavetendencyto flattenperiodicallyintoanellipse,withthedirectionof themajoraxisvarying fromperpendicular to paralleltothewind direction. The lowestnatural period of a cylinder taken asa ring isgivenby T= 2rr/wn = (2rr/2.68)(mR4/EI)I/2 where m = themassperunitlengthofthering;forsteel,m = 1in.XIin.X tX 0.28/386) E = modulus of elasticity, 27X106 psi 1= t3/12,moment of inertia, in.4. Substituting theabovevalues, weget where disthe mean cylinder diameter infeet andthickness t in inches. Theresonantwindvelocity whichtheoretically wouldinduceovaling is VI=3.40d/2TorVI= 1,120t/dmph. Reinforcingringswith a required section modulus areadded inthetopthird of a stacktosecureitfromovaling_A spiralvortexspoilercan be usedinplaceof therings(124] . Limiting Values for aVibrationalAnalysis Itwouldbeadvantageoustobeabletodetermineinadvancefromasingle, simpleparameterwhetheravibrationalanalysisisnecessary.Unfortunately,no.----_"""""."""."'.... ",... _______________________________________________ 120PRESSUREVESSELDESIGNHANDBOOK suchparameterisavailable.Mostinvestigationshavebeendoneontallstacks, whichcanbestraightcylindrical,tapered,orhalf-tapered; linedwithguniteor unlined;andofweldedorrivetedconstruction.Theweightisnearlyevenly distributed along the height, and there isno connected process piping involved. Empiricalparametersintendedforstacks,thereforecannotbe blindly applied totalltowers.Asalreadymentioned,processcolumnsarenotafflictedwith cross-windvibrationsveryoften.However,giventhetrendtohigher,slender vessels,thepossibilityofvibrationwilllikelybeof increasingimportance.To obtainmorereliablecriteria,moreresearchlaboratoryworkandexperimental workonfull-scale,field-erected, high vesselsareneeded. Thefollowingcanbeusedasgeneralguidelinesindecidingwhetheravibra-tional analysis isrequired. LTheupperlimitof thecriticalwindvelocitiesV IandV2canbelimitedto 60mph,sinceaverysmallpossibilityof asustainedwindvelocitybeyondthis limitwillexist.Thisvaluealsoisashighasanyknownwindvelocityatwhich crossvibrations havebeenrecorded. 2.IfV Iisgreaterthanthedesignwindvelocityusedinstaticpressurecom-putations,no furthercheck isrequired. Only if VIfallsin therangeof prevailing wind velocities atthe sitearea isa further investigation justified. 3.Thelimiting minimumheight-to-diameterratios Hid forvibrational analysis are: unlined stacks: line.dstacks process columns: Hid;;:'13 Hid;;:'15 Hid;;:'15 4.Possiblecriteria,relatingtototalweightW (lb),heightH (ft), andaverage diameterof thetophalf of thevesseld (ft) aresuggestedinref. 49to establish theneedfora vibrational analysis, asfollows: W/Hd2 ,;;;; 20, 20 < W/Hd2 ,;;;; 25, 25< W/Hd2, analysismust beperformed analysis should beperformed analysisneed not beperformed. 5.Accordingtoanoftenusedruleof thumb,completevibratoryanalysisof a stackoratowerisnotrequiredif thetotal forceonthe stack or tower caused bythewindof critical velocityVI(V2if it is lessthanthewind designvelocity) doesnot exceed-hof theoperating (corroded) weightWoor t pV2 Hd/Wo ,;;;; 0.067. DESIGNOF TALL CYLINDRICAL SELF-SUPPORTINGPROCESS COLUMNS121 Intheaboveformula,Histhetotalheightofauniform-diameterstackor tower.For a flaredstack,H can bereplaced by an equivalent height He=H (cylinder) + H/2(cone). For lined stacks, W0includes the weightof the internal lining. 4.7.FIRSTNATURALPERIODOFVIBRATION Thefirstnaturalperiodof vibrationTof tallslendervesselsisanimportant criterionindesignforwindorearthquakeloads.Foratall,slendercylindrical towerof uniformdiameterandthickness,equivalenttoafixed-endcantilever beam, T isgiven (in seconds per one complete cycle) by T= (1/0.560) (wH4 IgEI) 1/2 where g = 32.2 ft/sec2 ,gravitational acceleration E = modulus of elasticity, 29X106 psi H = total height infeet 1= (rrd3 t/8) (1/12), moment of inertia of the section, ft4 w = weightperfootof the vessel, Ib/ft t = shell thickness, in. d= shell mean corroded diameter, ft. Substituting the abovevaluesinto the equationforTgives T= (2.70/105)(Hld)2{wdlt)I/2sec/cycle IfthevesseloperatesathighertemperaturetheperiodTatthenewoperat-ingtemperaturecanbefoundfromT' = T X (29 X106 IE'at temperature )1/2 . Inpractice,tallprocessvesselshaveeitherastepped-downshellthicknessor/ andsectionsofdifferentdiameters,representingasystemwithunevenlydis-tributedmassandflexibility.Insuchsystems,Rayleigh'smethodisusedto de-terminethefirstnaturalperiodT.AlthoughRayleigh'smethodappliesonlyto undampedsystems, it yields the fundamentalperiodTwith a sufficientaccuracy formost engineering problems[1371. If thevibrationof acolumn isassumedto be undamped harmonic, the sumof theelasticpotentialenergyP .E.andthekineticenergyK.E.remainsconstant. MaximumkineticenergyK.E.occurs,whenthesystempassesthroughthe equilibriumpositionandtheelasticpotentialenergyiszero.Themaximum 122PRESSURE VESSEL DESIGNHANDBOOK elasticenergyP.E.occurs,whensystemisatmaximumdisplacement, withzero kinetic energyK.E.Both maximum energies mustbe equaL Maximum K.E. = MaximumP ..E. For a simple, one-mass, vibrating system, and WV1/2g= Fy/2 W(wny)2 /2g = Fy/2 where Y= amplitude, themaximum deflection of the center of gravityof themass V= maximumvelocityofthemass;equaltoYWnforasimpleharmonic motion Wn= angularnatural undampedfrequencyof the system, rad/sec F = initial acting force W = weightof themass g = 32.2ft/sec2 ,gravitational acceleration. Inordertoapplythe aboveequations to a cantilever beam, the following simpli-fyingassumptions haveto bemade. I.Thedistributedweightsofthebeamsectionsareassumedtobeconcen-tratedor"lumped"atcentersof gravityof thebeamsectionswithunchanged stiffnessalongthelengthofthebeam(seeFig.4.18).Thisassumptioncon-siderablysimplifiesthecomputationsforthedeflectionsYofthecentersof gravity.Thegreaterthenumberofsections,thehigherthefinalaccuracy achieved. W,Ya, Yb, Y c=deflections of centers of gravity a, b, cof individual sections WI, W2, W3= operating weights of individual sections /1'/2'/3 = moments of inertia of individualsections Fig.4.18. DESIGN OF TALL CYLINDRICAL SELF-SUPPORTINGPROCESSCOLUMNS123 2.Inorderto computethemaximumdeflectionsof thecenters of gravityof thelumpedmassesthemostprobablevibrationcurveof the system forthe first periodhastobeassumed.Theclosertheassumedcurvetotheactualcurve, themoreaccuratetheresultingT.However,evenif theselectedcurvediffers fromtheactual