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ST BILITY OF C STELL TED BE M WEBS

Sevak DemirdjianMarch 999

Department o Civil Engineeringand pplied Mechanics

McGill UniversityMontreal Canada

thesis submitted t the faculty of Graduate Studies ndResearch in partial fulfilment of the requirements of theDegree of Master of Engineering

Sevak Demirdjian


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BSTR CT

A study on the web-buckiing behavior of castellated beams is descnbed in this thesisBoth elastic and plastic methods of anaiysis are utilized to predict the faiiure modes othese beams.Interaction diagrams predicting formation of plastic mechanisrns. yietding of thhorizontal weld length and etastic bucfling anaiysis using the finite element method arcorrelated with a number of experimentai test results iom previous studies given in thliterature.

Test-to-predicted ratios for a total of 42 test be ms ranging from 45 to 60 openings arcomputed with the plastic and elastic methods of anaiysis, and a mean of 1.086 ancoefficient of variation of 0.195 re obtained. A parameter stu y covenng a wide range o60 castellated be rn geomevies is perfonned to derive elastic buckling coefficients undepure shear and bending forces. An elastic buckling interaction diagram is then definedwhich along with the diagrarns utilized in the plastic analysis, can e used to predict thelastic buckling and plastic failure loads under ny given moment-to-shear ratio.To incorporate the effect of plasticity associated with buckling, expressions are derived toimprove the previous theoretical models used, y cornbining both elastic and plastiresults. This results in an irnprovement in the coefficient of variation of the test-topredicted ratios for the 60' beams considered from 0.170 to 0.137.


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R ~ S U M Dans la cadre de la prsente thse. une tude su le voilement de l me des poutreaioures a t effectue. Les modes de rupture de ces poutres et les chargescorrespondantes sont evalus par des anaiyses de plasticit et d lasticit.es charges estimes par les diagrammes d interaction pour la formation d un mcanisme

de rupture. pour la rupture du joint de soudure horizontai par coulement. et pour levoilement de l me prdit par analyse par lment finis. sont compares aux rsultats deplusieurs tudes antrieures.Les rapports ntr les rsultats exprimentaux pour 42 poutres avec 45 60

d ouvertures et les prdictions par les mthodes d analyse de plasticit et d lasticit ont obtenus, et une moyenne de 1.086 et un coefficient de variation de 0.195 ont tobtenues. Une tude paramtrique sur les coefficients de voilement lastique de l me at effectue pour des charges en cisaillement pur et en flexion. pour un grand nombre depoutres ajoures avec des ouvertures de 60 . Un diagramme d interaction pour levoilement lastique de l me a t dvelopp. Ce diagramme est utilis en combinaisonavec les diagrammes pour la formation d un mcanisme de rupture pour estimer la forcede cisaillement par rapport au moment de flexion. correspondant la formation d unmcanisme de rupture et au voilement lastique de l meL effet de la plasticit lors du voilement de l h e est ensuite inclus dans les expressionsthoriques. Cette addition rduit l cart-type de 0.170 0.137 sur les prdictionsthoriques pour les poutres ajoures avec des ouvertures de 60 .


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CKNOWLEDGMENTS

would like to express y sincere gratitude to Prof. R G Redwood for his constanguidance. encouragement and help throughout the course of this project.

Sprcial thanks are due to Prof. G McClure for al1 her help throughout the course ot thisproject. and to al1 her guidance and advising throughout my graduate level studies.

The support of Fonds des Chercheurs et l aide la recherche FCAR) is greatlyacknowledged.

uould like to thank m y parents Krikor and Alice, and m y brother H m 9 or their intnitesupport and encouragement for al1 these years. Finally would like to acknowledge yuncle Joseph Bedrossian. for his valuable knowledge and help for man- yerirs.


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T BLE OF CONTENTSABSTRACT...................................... .... ................................................... iRSUM ................................................................................................CWOWLEDGMENTS .......................................................................

........................................................................BLE OF CONTENTS iv.

................................................................................IST OF FIGURES vilLIST OF T BLES .................................. .......................................... ixNOTATIONS ................................. ...........................................................x

.................................................................H PTERONE ntroduction 11 1 Introduction.................................................................................... 1

..............................................2Failure Modes of Castellated Beams2 1 Vierendeel or Shear Mechanism .........................................

1.2 2 Flexural Mechanism............................................................ 62 . 3 Lateral Torsionid Buckling ................................................ 72 4 Rupture o f Welded Joints.................................................... 9

............................................................2.5Web Post Buckling 101 2 6 Web Post Buckling Due To Compression ........................ 13

.......................................................................3 Research Program 1 4............................................3.1 Objective and Scope of Work 14

..........................................................3.2 Outline of the Thesis 15CH PTER TWO Methods of Analysis............................................... 16

2.1 Genera ......................................................................................... 16


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.............................................................................2 Plastic Analysis 16........................................................................3 Mid-Post Yielding 1 9...................................................................4 Buckling Analysis .

................................................................5 Finite Element Analysis 21..............................................................................5.1 General 2 4

.......................................................5 2 nput File Preparation 27...............................................................5 -3 M odel Geometry -28

2.5.4 Constraints ........................................................................ 282.5.5 Loads........................................... ................................... 29

....................................5.6 Buckling Analysis .........................32.......................................................................................6 Summary 34

.................................................HAPTER THREE Literature Review 35.........................................................................................- General 35

3.2 Literature Review......................................................................... 35.......................................2.1 Redwood and Demirdjian 1 998) 36

...................................................................2.2Zaarour 1 996) 36................................2.3 Gaiambos, Husain and Speirs 1975) 37

..................................................2 4 Husain and Speirs 1973) 3 83 2 5 Husain and Speirs 1971) .................................................. 39

...................................................2 6 B a d e and Texier 1968) 39..................................................................2 7 Halleux 1967) 40

...........................................................2.9 Sherboume 966 4 1


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................................................2. O Toprac and Cooke 1959) 42.............................2.1 Altifillisch. Toprac and Cooke 1957) 4

........HAPTER OUR Reconciliation o f And ysi s With Test Results 524.1 General ........................................................................................ 524.2Comparative Data ........................................................................ 5 2

................................................................................. 3 Cornparisons 3......................................................................................4 Discussion 5 7

CHAPTER FIVE Generalized Analysis and Design Considerations 625.1 General . -625 2 Loading on General Models ......................................................... -63. Elastic Buckling interaction Diagram........................................... 67

............................................................................4Parameter Smdy 735 . 5 Previous Parameter Study ............................................................ -73

.........................................................6 Shear Buckling Coefficients 7 65 .7 Flexural Buckling Coefficien ts .................................................... - 7 85 -8 Effect of Inelasticity on Ultimate Strength ...................................79

CHAPTER SIX Conclusion.................................................................. - 8 4REFEWNCES ........................................................................................ 87A P P E N D I X A inite Element Input FileA P PENDIX B Detailed Test-To-TheoryResultsA P P E N D I X C Elastic and Plastic Theoretical Computations


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LISTO FIGURES

xurmwmFigure 1 1 Castellated Beams ..................................................................... 1

3Figure 1.2 Zig-Zag Cutting Dimensions of Rolled Beams ........................ .Figure 1 3 Castellated Beam Section Properties.................... ...................4Figure 1.4 Castellated Beam Section Properties with Plates at Mid-Depth 4Figure 1.5 ParalleIogram Mechanism ........................ . .........................6Figure 1.6 Lateral Torsional Buckling......................................................... 8Figue 1 7Weld Joint Rupture ..................................................................... 9Figure 8Web Post Buckling .................................... ......... 2cHAmmnwFigure 2 1 Interaction Diagram.................................................................. 18Figure 2.2 Free-Body Diagram ................................................................ 20

3Figure 2.3 Predicted Web-Post Buckling Moments .................................. 3Figure 2.4 a) Mode1 used By Zaarour and Redwood 1 996) .................... 26Figure 2.4 b) Non-Composite Mode1 Used by Megharief 1 997) ............26Figure 2.5 Finite Element Mode1 .............................................................. 30Figure 2.6 Pure Bending and SheadMoment Arrangement ................. 3 1

3Figure 4.1 Test Arrangement of Beam H ................................................... JFigure 4.2 Interaction Diagram Demonstrating Theoretical Methods ......54


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................igure 5 1 Two Hole FEM Model Under Vertical Loads nly 64..................igure 5.2 Three Hole FEM Model Under ure Shear Forces 65

Figure 5 3 Three Hole FEM Model Under Pure Bending Moments .........66..........................................igure 5.4 Three and Four Hole FEM Models 69

Figure 5.5 Zaarour and Redwood 996 ................................................. -70Figure 6 Husain and Speirs 1973 ....................... . .................. 7Figure 5.7 Husain and Speirs 197 ) ................................................ 7 1Figure 5.8 Altifillisch Cooke and Toprac .............................................. 72Figure 5.9 Shear Buckling Coefficient Redwood and Demirdjian 1998 75

. ................igure 5 I0 Modified Pure Shear Buckling Coefi cient Curves 77Figure 5 1 1 Buckling Coefficient Curves Under Pure Bending Forces.... 7 9

................................igure 5.12 Elastic and Plastic Interaction Diagrams 8......igure 5 1 3 Comparison of Test Results With Proposed Expressions 83


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LlST OF TABLES

............................................able 3 1 Redwood and Demirdjian 1998) 44...................................................able 3.2 Zaarour and Redwood 996) -44

Table 3 3 Galarnbos Husain and Speirs 1975) ......................................... 45...........................................................able 3 4 Husain and Speirs 1 973) 46...........................................................able 3 5 Husain and Speirs 1 971) 7...........................................................able 6Bazile and Texier 1968) 47

..........able 3 7 Halleux 1967) ................................................................. 8.....................................................................able 3 8 Sherbourne 1966) 49

Table 3 9 Toprac and Cooke 1959) .......................................................... 5......................able 3 10 Attifillisch. Cooke and Toprac 1957) ..............5

c K M n m mTable 4.1 Surnmary of Test to Theoretical Predictions ............................. 38-Table 5.1 Summary o Resuits under Pure Moment Forces ...................... 6 7

.........................able 5 2 Summary of Results Under Pure Shear Forces 68.......................................................................able 5 3 Statistical Results 82


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Ar.4b

bd

4d b

4

covDOFE

FEFEM

FG

GDh

o

h

NOT TIONSre of fl ngere of web

width of one sloping edge of the holewidth of flangedepth of the origin l be m sectiontot l depth of c stell ted be m sectiondepth of bottom tee sectiondepth of top tee section

compression forcecoefficient of v ri tiondegree of freedommodulus of el sticitylength of welded jointfinite element n lysisfinite element methodyield stressstiffiiess matrixdifferenti l stiffness m trixheight of one sloping edge of holeheight of holeheight o pl te


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moment of inertiadepth of top tee section excluding fl ngebuckling coeffkientflexur l buckling coefficient

shear buckling coefficientlength of be mbending momentelastic buckling moment under pur bending forceselastic mom nt to cause web buckling

critical momentplastic momentcritic l moment based on be m test resultsyield momentmoment to fonn flexwal mechanismultimate momentconstant forceelastic section modulusdistance from center line to centerline of adjacent castellation holestension forcethickness of the flangethickness of the webdisplacement vector


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modified displacement vectorShear forceelastic buckling shear under pure shear forcescritical shea to cause we bucklingshear obtained fiom elastic anaiysishorizontal shear forcecritical value of hcntical shear based on beam test resultsplastic shear

shear Obtained fiom plastic analysisvertical shear force to cause rnid post yieldingvertical shear force to fom plastic mechanismultimate shear to cause we bucklingapplied loaddistance fiom top of th fl nge to centroid of tee sectionplastic section modulus of castellated beamfull section plastic modulusfactor utilized in plastic analysisfactor utilized in plastic analysisangle of castellationcritical stressexpansion ratio

s


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factor appiied to shear y iel stresseigen valueeigen vectorpoisson s ratioaspect ratio


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C H P T E R O N E

1 IntroductionSincc rht. Second World L\.*ar.man> attcmpts ha\-e been made b structural eny inee r s: i d ne\\ ways to dccreass the cost o f steel structures. Eue to limitations on mrisimumcillowable deflsctions. the high strength prupert ies of structural i t ~ t l annut ~ 1 ~ \ 3 > utilized t best adwxage. As result. se\.ernl ne\ rne~l-iodsha\.e been airned ILincrcnsing the stiffncss of steel mcmbers w-ithout an - ncreasri in i te iylir O&' the steerq ui re d. Cnstellated bams w r e one o f th s solutions Fig 1.1 1

Fig 1.1 CastslIatzd Beams


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Castellated or expanded) beams are fabricated from wide flange 1-beams.The web of thesection is cut by flarne along the horizontal x x avis along a zigzag pattern as shown inFig 1 2

Figure 1.2 Zig-Zag Cutting Dimensions of Rolled Be m sThe two halves are then welded together to produce a beam of greater depth withhexagonal openings in the web Fig. 1.3 , or rectangular plates may be inserted betweenthe two parts. producing octagonal holes Fig. 1.4). The resulting beam has a Iargesection modulus and greater bending rigidity than the original section. without anincrease in weight. However, the presence of the holes in the web will change thestructural behavior of the beam from that of plain webbed beams. Experirnental tests oncastellated beams have shown that beam slenderness. castelhtion parameters and theloading type are the main parameters, which dictate the strength and modes of failure ofthese beams.Casteilated bearns h ve been used in constmction for many years. Today. with thedeveloprnent of automated cutting and welding equipment. these beams are produced inan alrnost unlimited number of depths and spans suitable for both light and heavyloading conditions. n the past. the cutting angle of castellated beams ranged from 45 o


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7 but currently, 60 has become a fairly standard cuning angle. although 45 sectionsare also available. It should be noted that these are approxirnate values. actuai angles willv ry slightly from these to accommodate other geometrical requirements. s roof or floorbeams. joists. or purlins, these sections may replace solid sections or tmss members.Their aesthetic attributes produce an attractive architectural design fe ture for stores,schools and service buiidings. In structures with ceilings. the web op nings of thescmembers provide a passage for e sy routing and installation of utilities and airconditioning ducts.Typically. the dimensions of a castellated beam are defined a s follows referring to

Figs. 1.2 to 1.4):

d = d h h , For no plates. h,=O

Expansion ratio,

where. d original beam depthh depth of cuth height of plateb width of sloping edge of holed, depth of top tee section


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Figure 1 3 astellated beam sectionproperties

v

Figure 1 1 astellated beam section propertieswith plate rit mid epth


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decrease. The location of this failure will occur at the opening under greatest shearingforce. or if several openings are subjected to the s m maximum shear. then the one withthe greatest moment will be the critical one.

Plastic HingesFigure 1.5 ParalleIogram Mechanism

1 2 2 lexural MechanismUnder pure bending. provided the section is compact (at lsast Class 2 CSA 1991 . thetee sections above and below the openings yield n tension and compression until thtybecomc fully plastic. This mode of failure was reported in the lcorks ot Toprric and Cook1 959) and Halleux 1967). They conciuded that yieiding in the tee sectioris ribo\ c and

bclow t he openings of a castellated be m was similar to that of a solid beam under purebendiny forces. Thus. the maximum in-plane carrying capacity of a castellatrd beam

under pure moment loading was determined to be = Z xE ; wliere Z s tlie fullsection plastic modulus taken through the vertical enteriine of r hole.


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1 2 3 Lateral TorsionaCBucklingAs in soiid web beams, out of plane movement of the b r n without any web distonionsdescribes this mode of failure. Lateral torsional buckling as s h o w in Fig.1 6 is usuallyassociated with longer span beams with inadequate lateral support to the compressionflange. The reduced torsionai stifiess of the web, as a result of relatively deeper andslender section properties, conmbutes to this buckling mode. Nethercot and erdal1982) investigated this mode of failure. h e y concluded that web openings had

negligible effect on the overail lateral torsional buckling behavior of the beams theytested. Furthemore, it was suggested that design procedures to determine the latera

buckling strength of solid webbed bearns could e used for castellated be ms providedreduced cross sectional properties are used.


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Fia . 1 6 Lateral Torsional uckling Redwood smirdjian 998


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1 2 4 Rupture of elded JointsThe mid depth weld joint of the web post between two openings rnay rupture whenhorizontal shear stresses exceed the yield strength of the welded joint Fig. 1.7 . Husainand Speirs 1971) investigated this failure mode by testing six beams with short weldedjoints. This mode of failure depends upon the lengtb of the welded joint e). Thehorizontal length of the openings is equai to the weld length. and if the horizontal lengthis reduced to decrease secondary moments. the welded throat of the web-post becomesmore vulnerable to failure in this mode.

Weld RuptureFigure 1 7 Weld joint Rupture

As mentioned in 1.2.1, formation of a Vierendeel mechanism is likely to occur in be mswith long horizontal hole lengths and hence long welds). n the other hand short weldlengths are prone to cause failure of the welded joints as the horizontal yield stress isexceeded. Dougherty 1993) found a reasonable balance of these twvo failure modes. ysuggesting the following geometry


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lWeld length e = and for a 60 cutting angle with no plates.3

Therefore. opening pitch = 2 6 e = 24 0.289+ = 1-08h,, 1.1 h,,This concept has been demonstrated in many of the current available Castelite StandardBeam Geometry sections. Castetite Steel Beam Design Manual 1996).

1.2.5 Web Post bucklingThe horizontal shear force in the web post is associated with double curvature bendingover the height of the post. As shown in Fig. 1.8 one inclined edge of the opening wili bstressed in tension, and the opposite edge in compression and buckling will cause atwisting effect of the web post along its height. Several cases of web post buckling havbeen reported in the literature: Sherbourne 1966), Haileux 1967). Bazile and Texie1968 .

Many analytical studies on web post buckling have also been reported to predict the webpost buckling load due to shearing force. Based on finite difference approximation for anideally elastic-plastic-hardeningmaterial glan and Redwood 1976) produced sonisgraphical design approximations for a wide range of beam and hole geometries; sornecorrelations between experimental and non-linear finite element analysis FEMestimations were found in the works of Zaarour and Redwood 1996). Delesque 1968


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used an energy method to solve an elastic buckling problem by treating the wsb post as avariable section rectangular bearn in double curvature bending, susceptible to lateraltorsional buckling. However. Zaarour and Redwood 1 996 found large differences in theresults obtained frorn Blodgett s method in cornparison to their test results and finiteelement approximations they used. Blodgett s method is therefore not used in this project.In recent works of Redwood and Demirdjian (1998). approximations of buckling loadsw r derived based on elastic finite eiement analysis and good correlations tmveenexperimental and theoreticai estimations were found. This work showed that the results ofAglan and Redwood 976 should not e used for very thin webs. This mode of failureand these theoretical results are discussed in greater detail in subsequent chapters.


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Fig 1 8 U eb Post Buckling (Redwood rP Demirdjian 998


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1 2 6 Web Post uckling Due to Compressionconcentrated load or a reaction point applied directly over a web-post causes thi

failure mode. This mode was reported in the expenrnents conducted y Toprac and Coo1959). Husain and Speirs 1973). Buckling of the web post under large compressio

forces is not accornpanied by twisting of the post. s it would be under shearing forceSuch a failure mode could be prevented if adequate web reinforcing stiffeners arprovided. A strut approach was proposed in the works of Dougherty 1993). whicsuggests that standard column equations could be used to determine the strength of thwe post located at a load or a reaction point.


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1 3 RESEARCH PROGRAM

1 3 1 Objective nd Scope of orkThe objective of the current research is to study failure of castellated b e m s \cithpanicular emphasis on web post buckiing. The goal is to make use of the availabie elastia n d plastic analysis methods. and derive expressions that will predict critical shear forcecausing web-post buckling.This thesis uses many previous experimental results to provide cornparisons \\-iththeoretical approximations. and thus validation of the suggested methods described.

T h e first part of the research program focuses on the theoretical methods o f analysis to beused to predict failure loads of castellated beams. These methods include plastic analysisof the Vierendeel mechanism and for yielding of the mid-post joints. The finite elemenmethod is used to perform elastic buckling analysis and predict critical loads of al1 testbeams. A thorough literature search then follows to list al1 relevant experimental data tobe compared with theoretical methods. Correlations between experimental and theoreticalresults are then made.Thc second part of the thesis tocuses on general design considerations and thus is aimedat the principal objective o the research. Elastic buckling modes are investigated underdifferent moment to shear M N ) ratios. Well-defined relationships. based on pure shearand pure bending forces to cause web buckling. are developed to predict eiastic bucklingloads under an M N ratios. Results of elastic buckling and mechanism yielding loads arethen combined and fitted curves are derived to predict ultimate shearing forces causins


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web-post buckiing. To apply these expressions in a more general fashion. a parametricstudy investigating the behavior of a wide range of casteltated beam geometries isdeveloped. and buckiing coefficients under pure shear and bending forces are derived.Sugested predictions are then tested against actual test results, and good correlations areobtained.

1 3 2 Outiine of the ThesisThe thesis is divided into s x sections. M e r a bnef introduction to castellated beams and

their modes of ailure of Chapter 1 Chapter 2 focuses on several theoretical methods ofanalysis to predict modes of failure of castellated beams. hese methods include plasticanalysis, web-post yielding at mid-height, buckiing analysis, and finite elementapproximations. Chapter 3 contains a surnmary of relevant test data provided by previoustesting and available in the Merature. Relevant information on each test beam istabulated. Theoretical approaches described in Chapter are tested against actualexperimental test beams, and reconciliation of analysis with test results is the topiccovered in Chapter 4.Chapter focuses on design considerations for castellated beams. Relationships definingelastic buckling under ny N ratio are developed. A parametric study. as wel asexpressions estimating shear force causing buckling are derived. Results of suggestedmethods are tested against actual experimental test results, and correlations between testsand theories are made. Concluding rem are summarized in Chapter 6.


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CH PTER TWMETHODS OF N LYSE

2.1 GeneralSeveral theoretical approaches are considered to analyze the yielding and buckling fiduremodes of castellated bearns Plastic anaiysis of the Vierendeel mechanism failure, r s rvellas anaiysis of mid web post yielding are sumrnaiized. Elastic finite element buckiinganalysis is used to predict uckling loads. Finite element mode1 generation as well asbuc kling analysis in the MSCMASTRAN finite element package are descri bed.

2.2 Plastic AnalysisThe construction of n interaction diagram relating shear force and bending moment atmid-length of an opening has been described y Redwood 1983). This diagram can beused to study failure caused by the formation of a Vierendeel mechanisrn formed by thedevelopment offour plastic hinges at the re-entrant comers of the tee section. above andbelow the hole. or the beam to anain this plastic failure, the web and flanges areassumed CObe stable and withstand the high shear load until plastic hinges are fonned atthe reentrant corners of n opening in high shear region. As the load increases, primaryand secondary stresses resulting fiom combined effect of shear and moment forces lead tocomplete yield at th four corners thus forming plastic hinges. This analysis is based onthe assurnption of perfectly plastic matenal behavior with yielding according to Von


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Mises criterion. A typical interaction dia- is shown in Fig. 2 1 The shear andmoment values have been non-dirnensionalized y division of the section's ti1y plasticshear and moment capacities.The di gr m can e constructed using the following results:

A I YJS Where shear area A, dg .

To generate the curve, k is varied between and 1 Below the value 1 the curvebecomes vertical. For given beam characteristics and hole location subjected to a load. aradial line can be dr wn fiom the origin to intercept the interaction diagram for thecorresponding shear-to-moment ratio VIM). The horizontal and vertical coordinates ofthe intercepted point then predict the shear and moment values to cause yield nlechanismfailure.


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Interaction DirammSpecimen 10 5a

YieidTheoryTest Resuit

igure 2.1 Interaction Diagram Redwood and emirdjian 998))


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2.3 iMid-Post Y eldingIt is possible for yielding of the web-pst at mid-height to occur before failure due t

formation of shear mechanism takes place. This mode of failure occurs particularly tobeams with closely spaced openings with low moment-to-shear ratio. The vertical sheaforce to cause mid-post yielding is defined through.

and the basic approach to define this relationship Hosain and Speirs 197 is derived byusing equilibrium equations from the free ody diagram of castellated beam section ashown in Fig. 2.2.The horizontal shear force, V can e expressed as

when the vertical shear force V nd V are equal, thenv x where,d x - 2 y ,

V is defined as the difference between the two horizontal forces C and C7This equation is based on the assurnption that the line of action of forces I and C-> reacting at the centroid of the tee section above the openings.The web post will yield when the minimum weld-post re is subjected to the shear y i d d

Due to the maximum shear stress k i n g at the throat, the yielding is contained. and i t m


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be expected that strain-hardening will develop leading to a significantly higher failureload than that given by Eqn 2.1. In the work of Husain and Speirs 197 the sliear y ieldstress has been measwed directly and is significantly higher thanb sed on F~ 4 3 n view of this the yield stress used for this mode oflater increased by a factor P as discussed in Chapter 4.

the expected value

Mure only will be

V Q

Figure 2 2 Free-body diagram of castellated be rn


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2.4 Buckling AnalysisBased on a finite difference bifrcation analysis of the web post ueated as a beamspanning between the top and bottom of the openings, graphical results relating criticalmoments in the p s t to different beam opening geometries were developed by Aglan andRedwood 1976 . The materiai was considered to be an elastic-perfectly plastic line rstrain-hardening material. For different hole height to minimum width ratios. criticalmoments in the post at the level of the top and bottom of the opening, divided by thatsection s plastic moment capacity A , = 0 . 2 5 ~s f were presented. s shown inFig. 2.3.F0r.a given beam, the value of MJM is first read from Fig. 2.3. By multiplying thegiven ratio by the section s plastic capacity M,, as given above. the horizontal shear

w,racting at the minimum weld length is calculated as h From the free bodyh,,hdiagram of Figure 2.2, the Vfl ratio is given by . Therefore. the verticald y ,

2 .~.l,,, d, 2 y , )shear force to cause buckling n the web-pst is then derived as VL ~ h , ,et,where yield on the smallest web-pst cross-section -- is an imposed upper limit oni

V,,. In the work of Zaarour and Redwood (1996), who tested 12 castellated beamssatisfactory predictions were obtained with the glan and Redwood (1976) approach.However, in more recent work (Redwood and Demirdjian 1998). tests of very thin


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webbed castellated beams showed that the graphical results such as shown in Fig. 2 provided unsafe predictions. a result that w s believed to be due t the assumed restrainconditions at the top and bottom of the w eb p st . The method of Aglan and Redwoo1976 is therefore not considered M e r n this study.


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?> .04@-

O ? dt =lO23

Figure 2.3 Predicted Web Post Buckling Moments for 4=60UAglan and ed~ood 976


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2.5 FINITE ELEMENT N LYSIS

2.5.1 GeneralThe finite element method has previously been used to pertorm buckling analyses oncastellated beams and is also used in this project. This section therefore describes thesofhare used and the specifics of the application to castellated beams.In previous work (Zaarour and Redwood, 199 and Meghariefand Redwood, 1997) FE Mstudies of the buckling of web-posts in composite and non-composite beams were foundto give good approximations of test resuits (2-10 variations). Both studies utilized the

finite element package MSCRV STR N developed y the MacNeal SchwindIeCorporation (Caffiey and Lee 1994). The same package is used in the current researchwith the objective to utilize FEM as a reiiable tool to simulate experimental tests andgenerate web p s t buckling loads.

Zaarour and Redwood (1996) studied buckling of thin webbed castellated beams based ona single web-post model, as show in Fig. 2.4 a). Mesh refmement was based on theconvergence of web p s t buckling loads in cornparison to several experimental testresults. Megharief and Redwood (1997) investigated the behavior of web-post buckiingof composite castellated beams. Their mode1 consisted of full flanges web and transversestiffeners and the model comprised two complete web openings s shown in Fig 2.4 b).This larger model was needed in order to incorporate the shear co ~ e c t i o netween steelsection and slab, and hence the composite action on the bearn. he model used in the


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current research is similar to the non-composite be m mode1 utilized y Megharief anRedwood 1997) as shown in Fig.2.4 b), however, based on the different needs in thcurrent work, more refmed meshes and a greater number of openings are used. adiscussed subsequently. The following sections describe the particular steps necessary tuse the MSCNASTRAN system and the details o f the generation o f the models.


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Fig 2.4 a) Model used by Zaarour nd Redwood 1996

Fig 2.4 b) Non Composite Model used by Megharief and Redwood 1997


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2 5 2 Input F i le P repamt ionElastic finite element bifurcation analysis w s carried out for al1 test beams. nalysisin MSCNASTRAPJ is submitted in an input file, which consists of three major sections:Executive control. Case control and ulkdata. SampIe input file is given in Appendix AExecutive C on tro l Sectioa is the first required group of statements to detne the type ofanalysis, time allocation nd system diagnostics.Case Control Section specifies a co~lection f grid point numbers or element numbersto be used in the analysis. Requests output selections and loading subcases.ulk Data Entry contains al1 necessary data for describing the structural model.

Includes geometric locations of grid points constraints, element connections, elementproperties and loads.To prepare a detailed description of a model, the following classes of input data must beprovided:Gromerry: locations of gnd points and the orientations of the coordinate systemElrrneni connectivity: identification number of grid points to which each dernent isc o ~ e c t e d .Efement properties definition of the thickness and the bending properties of eachelement.Materialproperties definition of Young's modulus and Poisson's ratio.

C onsrrninis: specifications of boundary and symmetry conditions to constrain free-bodymotion that will cause the anaiysis to fail.Louds: definition of extemally applied loads at gridpoints.


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2 5 3 Mode Geomety andType of ElernentsA skeleton model based on a given beam geometry is first developed through defining thex. y. and z coordinates of each grid point. Grid points are used to define the geometry of astructure. to which finite elements are attached. Each gnd point possesses six possibledegrees of fieedom DOF) about the x y and z-axes three translations (T1. T2 T3) andthree rotations (RI, W 3 , which constrain the grids to displace with the loadedstructure.As the geometry of the structure is defined. the grid points are connecteci by finiteelements. Two-dimensional CQUAD4 isotropic, tinear elastic (MAT1 membrane-bending quadrilateral plate elements were chosen to define the finite elements of themodel. CQUAD4 element input card is defined through four grid points whose physicallocation determines the length and width of the element. By assigning a materialidentification number in the CQUAD4 input card. ail essential material properties.membrane, bending, thickness, shear and coupling effects of the elements are defined inthe shell element input property card PSHELL). imilady, linear elastic properties of thematerial, modulus of elasticity, Poisson s ratio are defined in the MAT1 data entry inputcard by assigning a property identification number in the PSHELL entry card.

2.5.4 ConstraintaSingle point constraints (SPC) are used to enforce a prescribed displacement (componentsof translation or rotation) on a grid point. The degrees of fieedom in MSC/NASTEWN


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are defined as numbers 1. 2 3.4, 5, and 6. corresponding to three translation. T l , T2 T3and three rotational degrees of freedom. RI. R2. X The propenies of CQUADelements used in modeling the web, flanges and the stiffeners had zero normal twisitingstiffness. One way to ensure non-singularity in the st if ie ss matrix and to account for theout of plane rotational stiffhess or the sixth degree of fieedom R3) s through AUTOSPCand KGROT commands in the Bulk Data Entry, a s recommended in the manuals. In ailmodels K6ROT w s taken as 10,000. This v lue is a fictitious number assigned tosuppress singularities associated with the normal degrees of freedom. Values of 100.10.000 or 100,000 are recommended by the manuals, however, a value of 10.000 w stested to provide acceptable results. Fig. 2.5 shows a typical mesh. this one comprisingtwo openings. The model is supported at the bottom lefi-hand corner where constraints 2and 3 are applied; these prevent movement in the vertical and out of plane directions.Displacements in the and z directions at the upper and lower flange to web intersecringnodes at the right end are restricted by constraints 1 and 3. to prevent t igid body rotationabout the z-axis. These constraints simulate symmetry of half the span of a simplysupported beam geometry. Out of plane displacements are prevented on the perirneter ofthe web.

2 5 5 LoadsShearing forces were applied to the models by assigning two transverse negative ydirection) loads at the right hand end, as shown in Fig. 2.5. Moment loads were appliedby applying two equal and opposite x-direction) concentrated horizontal loads at the lefi-


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iland end at the flange-to-web intersections Fiy. 2.6 . Thus sliear and moment couldssigned in any desired combination.

Fig. 2.5 Finite Elernent Mode1


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Fig 2 6 Pure ending nd SheadMoment Arrangement


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to create unstable equilibriurn conditions. will be the factor to c a w buckling.163 t r l l G ~ l W 0

This requires the solution of an eigenvalue problern:[ -tlG,I{v 0

The solution is nontrivial. q different fiom zero only for specific values of q that wouldmake the matrix [ qGD]singular.The product of the first load intensity factor or the first eigenvalue q with the applied loadwould give the frst buckling load of the rnodel, and the eigenvector


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CHAPTERTHRLITERATURE:REVIEW

3.1 Generaln investigation of previous literature on non-composite castellated beam tests w s

conducted fiom which d t was obtained in order to make comparisons betweenexperimental and theoretical results in later chapters. or each test beam the sectionproperties geometry and experimental arrangements were studied and relevant data aresummarized in tables at the end of this chapter.

3.2 Literature eviewReviews on non-composite castellated beams have been extensively reported in thei terature. owever generally accepted design methods have not been established due to

the complexity of castellated beams and their associated modes of failure. An outline ofprevious expenmental work on castellated beams is reported here with the objective ofdescribing only the main features of each investigation. The data and t st results for thebeams described are the subject of detailed n lysis in subsequent chapters of this thesis.The test programs are described in reverse chtonological order.


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3 2 1 Redwood and Demirdjiaa (1998)Four castellated beams, two identical ones with four openings 10-5(a), IO-5 b). a thirdwith six openings 10-6) and a fourth with eight openings (10-7), al1 with identical crosssectional properties. were tested. The main focus of the experiment was to investigate thebuckling of the web post between holes and to study any effects of moment-to-shear ratioon the mode of failure. Simple supports and a central single concenuated load were usedfor al1 specimens. Al1 beams were provided with bearing stiffeners at support and at loadpoints. Mean flange and web yield stress values were obtained from tensile coupon tests.ased on the experimental ultimate loads, except beam 10-7. which failed by lateral

torsional buckling, buckling of the web post was the observed mode of failure of al thesebearns. Bearn 10-7 is omitted fiom further consideration in this project, since interest is inweb buckling only. The buckling mode involved twisting of the post in oppositedirections above and below the mid-depth. Ultimate load values were given s the peaktest loads. Test conditions were then sirnulated by elastic finite element analysis. andgood predictions of the buckling loads were reported (4- 14 variations).

3.2.2 Zaarour 1995)Fourteen castellated beams fabncated rom 8,10,12, and 14 inch light beams Bantamsections manufactured by Chaparral Steel Company) were tested. Six of these had in.

(50.8 mm high plates welded between the two beam halves at the web-post mid-depth.The objective of the experiments was to study the buckling of the web post between


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openings. Simple supports and a central single concentrated load were used for al1specimens. Al1 beamswere provided with bearing stiffeners at support and at Ioad points.Average flange and web yield stresses were obtained from tensile coupon tests for eachsize of beamThe reponed ultimate strengths were based on peak load capacities of the bearns. ebpost buckling was observed in the failure of 10 cases. and in two cases. local buckling ofthe tee-section above the openings subjected to greatest bending moments occurred. Twolateral torsional buckling modes were also observed; these have been omitted tiomfurther consideration shce interest is in web buckling only. FEM analysis was also usedto predict web-ps t buckling load.

3 2 3 Galambos usain and Speirs 1975)Four castellated beams fabrcated fiom W 1 x sections 1 in deep. 1 pounds per footsee Table 3.1 for dimensions)) were tested to validate a numerical analysis approach to

determine the optimum expansion ratio based on both elastic and plastic methods ofanalysis. Al1 beams were simply supported and were subjected to a concentrated load atmid-span. The span and weld lengths were kept constant, but the depths were variedbased on different expansion ratios. Ultimate loads were recorded. but no furtherdiscussion about the modes of failure was given.


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3.2.4 Husain and Speirs 1973)Bearns fabricated fiom twelve 10 15 beams alternative designation for W OX i5 weretested to investigate the effect of hole geometry on the mode of failure and ultimatestrength of castellated beams. Specimens A-2 B-1, C and D were subjected to twoconcentrated point loads, and the rest of the beams had a single concentrated load at mid-span. Al1 beams were simply supponed and adequate lateral bracing and full depthbearing stiffeners were provided except for beams C and D where partial depth stiffenerswere used . The loads were based on the ultirnate load values obtained during theexperiments.

Specimens A-1, A-2, nd B-3, failed by the formation of plastic hinges at the re-entrantcorners of the opening where both shear and moment forces are acting. As for SpecimensG 1 G-2, with flanges of Canadian Standard S16.1-94 class 1 section properties. and G-3, a class 2 section, yielding of the flanges n the region of high bending moment iead toflexural failure. The class section properties were calculated for some beams in ananempt to investigzte if any local buckiing possibilities were present. Bearns 8-2 C, and

failed prematurely due to web buckling directly under the point of load application.Similar failure was exhibited by eam B-1 that failed by web buckling under theconcentrated load before a Vierendeel mechanism had fonned. hus beams B-1. B-2. C.nd D were omitted fiom m e r tudy.


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3.2.6 Husain and Speirs 1971)The main focus of this experiment w s to study the yielding and rupture of urlded jointsof castellated beams. The experimental investigation consisted of testing sis simplysupported beams under various load systems. single concentrated point load \vasapplied to b e m s E 2. E 3. F-1 and F 3 and two concentrated loads were used for beamsE-1 and F 2. Full depth-bearing stiffeners and sufficient lateral bracings wr.Etrt: pro~idedopre\.ent premature buckling. The reponed fin l results were caiculated on the b a is o fdirectI). measured yield and ultimate shear stress values. The measured shcar stresses\iwe significantly higher than values which wvould have been expected tiom tsnsilecoupon tests. probably as a result of strain hardening. The prediction of ultimate strengtlibased on web-post yield (see Section 2.3) can therefore be espected ro bc. wrpconsenative. Sudden weld rupture accornpanied by violent strain energj- reIease \vas thecommon mode of failure for all beams.

3.2.7 Bazile and Texier 1968)T ~ v oeries of beams. four HEAS6 and three IPE270 sections (for dimensions see Table3 1 kvere tested to failure. The objective of the experiment was to de\.elop a trtherunderstanding of different beam characteristics and properties. geometn and espansionratios of castellated beams. The simply supported beams were tested under eightunifonnly distributed concentrated loads. Three test loads. P l . P L and P3 wrtt reponedto describe the different phases of the load-deflection diagrarn of each beam. Loads PI


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and P define sudden changes in slope and P w s the ultimate load. Flange and webyield stresses were obtained iom beam coupon tests and full depth stitTeners wereprovided at support reaction points. Beams A, B and E failed under web buckling in thezone of maximum shear. The beams F and G failed by lateral torsional buckling and werethus omitted from M e r tudy herein. Beams C and D had deep 200rnm) plates at mid-depth, and were reported as faiiing by web-pst buckling. Estimated strengths of the postso f these two bems, using the colwnn strength formula of CS 1994) assuming widthsequal to the maximum and minimum actual widths, bracket the uitimate test value of theconcentrated load. It is therefore evident that these were compression buckling failuresunder the action of the concentrated loads acting directly above the unstiffened web-posts. Since this mode s not ki n g studied, these two beams were not considered M e r .

3 2 8 Halleux 1967)Five types of beams with different geometricai properties, l fabricated fiom the IPE300rolled steel sections, were tested to destruction under two equal concentrated loadsapplied at the third-span points. The experimental faiiure load was based on theintersection of the tangent to th tinear part of the load vs. deflection diagram with thetangent to the h o s t horizontal part of the curve. Measured yield stresses are notreported. Calculations in the reference are basedon the yield stress of the material, that is,24 kglmm (235 MPa , and it is later stated that yield stresses determined from umeponedtensile tests were significantly higher than the above-mentioned value. Therefore, due to


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the uncertainty in the yield stresses the reported results must be treated circumspectly.3 2 9 Sherbourne 1966)This test program was designed to investigate the interaction of shear and moment forceson the behavior of castellated beams under varying load conditions. The test arrangementconsisted of simply supported be ms with fidl depth bearing stiffeners under load andreaction points. Seven tests were perfonned which ranged from pure shear to purebending loading conditions. Load-deflection curves are given in the paper. rom these theultimate loads and Ioads obtained fiom the intersection of tangents to the initiai linearp rt and to the almost linear pst-yield part were obtained. Beam El , subjected to a single

concentrated load at mid-span, failed through extensive yielding of the b o a t at middepth of the post between the first and second hole opening. Beam E2 w s designed toinvestigate the effect of pure moment, and was subjected to two concentrated point loads.Failure of this be m however, w s outside the central control section and was associatedwith extensive yielding in the end zones experiencing both shear and moment forces. Thehole closest to the load w s the most severely damaged. eb buckling w s the mode offailure of specimen E3 in the zone of maximum shear, under the two point loadingsystem. Specimen E4 was designed to study the effect of pure shear across the centralopening. The deflection curve demonstrates considerable strain-hardening, and webbuckling was the observed mode of faiiure. Beams L 1 L2 and L3 were tested under purebending moments. The first two were reported to fail by flexurai mechanisms. L3 w salso reported to fail by fiexural mechanism, however, lateral torsional buckling was alsoassociated with the failure mode.


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3 2 10 Toprac and ooke 1959)Nine castellated beams fabricated fiom 8B1 rolled sections were tested to destruction.The objectives of the investigation were to study the stniciural behavior in elastic andplastic ranges. to study load carrying capacity and modes of failure, to compare observedresults with theoretical calculations, and to determine an optimum expansion ratio forsuch bearns. Loads were applied at four concentrated points and failure loads werereported as the ultimate loads. Well-defined yield stress values were obtained throughcoupon tests and adequate bearing stiffeners were provided under reaction points.Specimens A and C failed through excessive lateral buckling and are omitted from furtherstudy. The iiltimate load of specimen B was recorded, but no frther details were given.s for specimen D which had a ciass 2 web tee stem section, web throat, tee section and

compression flange yielding progressed in the shear span. As the maximum load wasreached. yield at the top low moment hole corner and at web-post mid-depth was evident.Yielding and buckling of the compression fl nge in the pure bending region was thefailure mode of Bearn E. Local buckling of the compression flange in the constantmoment region was d s o the observed failure mode of specimen F; however, as the loadwas hrther increased, the beam buckled laterally. A Vierendeel mechanism in the regionof highest shear was the mode of failure of specimen G. Specimen H with a classlange section, failed through buckling of he compression fiange in the constant momentregion. Specimen 1 with a class 1 web tee stem section failed through a Vierendeelrnechanism in the highest shear region.


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3 2 1 ~ltfilliscb ooke and Toprac 1957)The objective of the investigation was to study the structural behavior of castellatedbeams both in the elastic and plastic ranges, and to study their strength and mode offaiIure. Three joists fabricated from 1 B 1.5 shapes with equal spans and simple supportsand with varying positions of two symmetrical concentrated loads were used. Varyingexpansion ratio, beam depths hole and web-post geometries were studied for each ofthese tests. Test loads were reported as the ultimate loads obtained during theexperiments. Beam A was provided with full bearing stiffeners under each load. It failedthrough extensive yielding of the tee section and local compression flange buckling in theregion of constant moment. The flange to width ratio of beam A corresponded to a class 2section.Bearn B consisted of three tests. In the first two, B and B2 loads were in the elasticrange in order to venQ theoretical stress and deflection analyses. The third test. B3involved loading to destruction, but was omitted fiom M e r study because of theinadequacy of lateral bracing system.Bearn C was provided with shon bearing stiffeners, approximately half beam depthbelow the load points. The first two tests were in the elastic r nge and the third wasloaded to destruction. The filure mode of this beam involved yielding of the web at thetop low-moment corner of the opening in the shear sp n nearest the load applicationpoint. followed by local buckling of the compression flange at the other end of theopening. The flange had a Class 2 section properties. Yielding of the throat was alsonoticed.


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T BLE 3.1 Redwood Demirdiian (1998

T BLE 3.2.a aarour Redwood ( 1996)

TABLE 3.2.b Zaarour Redwood 1996 continued)

or , refer to descriptionof footnotes on page 1


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TABLE 3 2 c Zaarour Redwood ( 1996) continued)

T A B L E 3.3.a Galambos Husain Speirs 1975)253 75 302 65 354.58101.60 101-60 101 605 . 8 4 5.84 5 . 8 46.86 6.86 6 . 8 6N.A. 53 40 153.40N.A. 100 89 303-59N A 335 45 425 45N A 39 9 59 3

333 .43 333 43 3 3 3 . 1 3

T A B L E 3 3 b Galambos Husain Speirs 19 75 ) continued)BEAiMd,bitu

l

ljos aO hF

H-3 P H-4340.6 403 -35101.60 101 60

83 5.846.86 6.86152.40 152.30176.58 302 5

425 35 425 4555.68 68 3338.67 333.43


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TABLE 3 5 a Husain S~eirs197

T A B L E 3 5 b Husain S~e irs197 ) continued

T A B L E 3.6.a Bazile Tesier 1968

B E A Mdb atf i leh,,s aO hF,

. - -F- 1 F 2 F-3

38 O0 38 O0 38 .O010 1.60 101.60 101.605 .33 5 33 5 -336 .8 3 6 . 8 3 6.8350.55 50 55 50.55

254.00 254.00 254.00247.65 217.65 347 6560.00 60.00 60.00248.2 248 2 248 .2


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T BLE 3 6 b Bade Texier ( 1968) continued)

T B L E 3 7 a Halleus ( 1967) Series

T B L E 3 7 b Halieux ( 1967)Series continued)


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TABLE 3 7 c Hallei 1967 Series 2 continued)

TABLE 3 7 6 Halleux 1967 Series 2 continued)

T BLE 3.8 Sherbourne 1 966


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TABLE 3 9 a Tovrac Cooke 1959)

TABLE 3.Y.b Toprac & Cooke 1 959 continued)

TABLE 3 9 c Toprac i Cooke ( 1959 continued)BEAMdb; l1,t,.ilei lh,,aP h .

F y bF tiana

G H J330.20 295.9 354.33 200 9 1100.33 100.3 100 .33 100.334.72 3 35 4.70 4.705 18 5.16 5-13 5 1 176 20 38.1 38.10 N A

264 16 194.3 309 .63 K..A.4 16 56 370.5 1 385 .83 N . A .

45 45 45 N A2 9 6 . 4 1 296 4 296 41 N A296.4 296 4 296 4 1 S i\


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T BLE 3.10 Altfillisch Cooke Toprac 1957)A B.

Al dimensions are inmmb Angle in degrees.ield Stress F inMpa


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CHAPTER FOURRECONCILIATION OF ANALYSISWT TEST R SULTS

1 1 GeneralThe results of the previous research work on castellated beams described in Chapter arecompared in this chapter with the methods of anaiysis described in Chapter 2 Al1 shearand bending moment loads are non-dimensionalized by dividing by the plastic shear ormoment capacity of the section to facilitate numerical cornparirom, and a governingmode of failure is predicted. Correlations between test results to theory are then reported.

1 2 Comparative DataThe complete set of data for al1 78 bearns tested in the references of Chapter 3 are givenin Tables 3.1 to 3.10. Of these, 21 were eliminated fiom fiuther consideration becausethey failed by modes other than those k i n g considered in this project. The remaining 57beams are considered in this chapter. For reasons discussed below, more of these beamshad to be removed fiom consideration. For the remainder the predicted and measuredultimate loads are compared. A summaxy of these results is given in Table 4.1.Detailed computations for each of the four predicted failure modes Vierendeel andhorizontal web-pst yield mechanisms, flexural mechanism and FEM buckiing analysis)are given for each beam in Appendix B Because of the varying moment-ta-shear ratios ateach hole in a beam al1 holes must be considered independently, and the most critical one


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for each failure mode must be identified.Constructionof the interactiondiagrams representing plastic failure mechanism s was firscarried out. For the given beam arrangement shown below Fig. 4 l ) , such a diagram isdemonstrated inFig 4.2.

m D L I Q nFigure 4 1 TestArrangemento Beam i Toprac Cooke 1959


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ho 1

Figure 4.2 Interaction Diapram Demonstrating Theoretical Methods of Analyses.

The radial lines represent the MN ratios for each of the openings in one-half of the spanwith the two holes under pure bending being represented on the vertical axis as hol s 7and 8. The M N ratio at the centerline of each opening is used. For each openingtheoretical predictions of V N and M/M are obtained from the intersections of the radiallines with the interaction diagrarn representing Vierendeel and flexural plasticmec hanisms.On the diagram are also plotted the predicted ilure loads corresponding to mid-postyielding V,,,N,,, and the buckling load predicted by FEM. The first of these is based on


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Eqn 2.1 with the shear yield stress taken as f3F443 This has a constant value for al1 web-posts nd plots on Fig. 4.2 as a vertical Iine WO lines corresponding to two values ofare shown). Elastic FEM results are given. although it is recognized that this bucklingusually involves inelastic action. The influence of plasticity is considered in Chapter 5.and is neglected t this stage as good results with elastic analysis have been reported byRedwood and Demirdjian 1W8 , and initially the simplest sotution was sought.Based on the typical FEM modelarrangementsof Section 2.5, a two-hole model with 816elements, as shown in Fig. 2.5 was chosen to sirnulate the behavior of a web-post underhigh shearing force. This represents a half-span of a beam with four holes and w ssubject to the restraints and other details outlined in Section 2 5 Only vertical loads wereused and the model is subjected to constant shear force with some small bending forceswhich were considered to be negligible insofar as they wodd affect the buckling loadsee Redwood and Demirdjian 1998). These FEM results are plotted on the interactiondiagram as two points with ordinates representing the moments at the two holes used inthe model. Thus it is implicitly assumed that moment has negligible effect; thisassumption is examined in detail in Chapter 5.

4.3 CornparisonsAl1 modes of failure for each hole in a beam are identifiable on a diagram such s Fig.4.2. The triangies represent the loading V and M) at each hole for a given load on thebeam values given in fact correspond to the failure load). s load is applied to the beam.


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these points cm be considered as expanding proportionally outward from the origin. Thcritical hole is the one for which the plotted point first reaches the failure envelope. andthe mode would be identified by the part of the envelope attained. This may alternativelybe interpreted s identiQing the failure hole as that one for which the ratio of test load topredicted load is a maximum

The results shown in Fig. 4.2 are affected by the analysis for the horizontal web-posshear yield mode which, s discussed in Chapter 2, is known to be quite conservative. Ithese results i.e. the vertical dashed lines) are ignored it can be seen that a flexuramechanism failure is predicted at holes 7 and 8: hole 6 is almost at the point of failure ina Vierendeel mechanism mode, and holes 5 and 4 in the same shear span re farther fiomthe failure surface. Holes 1 and 3 are loaded well below the Vierendeel mechanismload, and are far below the elastic buckiing load. The observed failure mode was that opure bending, as predicted by the above reasoning. I f the horizontai yield mode had eenconsidered holes 1 2 and 3 would have been criticai with both predicted failure loadslower than observed). It seems clear that in this case the horizontal yield mode was norelevant; in effect the verticai line should be shifted to the right to reflect a higher sheayield stress than 1. ~ S F43There is some evidence that the effective shear stress at rnid-depth of the post at failure ivery high compared with the expected value F,/ /). Husain and Speirs 197 directlymeasured the shear yield stress of notched specimens fabricated from ASTM A 6 stee

nominal F,=36 ksi 248 MPa and for a number of specimens the average value w s41.6 ksi 287 MPa . The tensile yield stress was not reported, and so some uncertainty


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exists as to the enhancement above F44 that this represents. However. if it is assumethat the A 6 web material had a real t ende yield stress of about 53 ksi 365 MPa suchi values have been measured for A 6 steels in the 1960-70 period, see Redwood anMcCutcheon 1969)) then the measured shear yield is 1.35 (=41.6+(53/ 13)) times thaexpected value of J J ~ reater enhancement would occur if the estimate of the tensilyield w s too high On this basis. it has been asswned throughout that the effective sheayield stress at the mid-depth of the posts is 1.35 times ~ 4 d 3 hus the factor is taken a1.35. In the example of Fig. 4.2, it ppe rs that even this enhancement is insufficient toreflect the effective shear in th test beam.Following the above procedure, test-to-predicted Ioad ratios were computed for each tesbeam. Certain tests had reported maximum test loads, while others derived their failurloads fiom the intersection of tangents of the two curves of load vs. deflection diagrarnWhenever applicable, both reported loads are used for comparisons.

4.4 DiscussionIn generai, the numencd results indicate good correlation with test results. .Most of thecases with poor correlation, as indicated in Table 4.1, are those for which yield stresvalues were not given, and nominal values have been used. These beams are identified byasterisks, and are noted in the literature review of Section 3.2.Excluding the identified beams for which F, is not known, the mean and the coefficien

of variation COV) of the test-to-predicted ratios for al1 other beams are 1.127 and 0 2 2 5


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Thes e a re based o n the ul t imate loads; if the tangential Ioad is used wh ere available. thesenum bers becorne .O86 and 0.195.Of t he 57 beams listed app roxim ately half 29) had the mode of failure predictedcorrect ly. Of the others some test modes were not defined 4), in others modes areidentified as flange buckling when a yield mechanism may have been imminent oal ready developing S ) , in others. the uncertainty concerning the shear capacity of theweb-post affects the prediction. and for most of the remaining cases there were onlysmall differences between the failure load for the predicted mode and that of anal ternat ive mode.

Table 4.1 Summary of Test and Theoretical PredictionsReference

RedwoodDemirdjian 1998)

ZaarourRedwood 1996)

B e a m

10-5a10-Sb10-6

8

8-28-38-4

10-10-2

T e d t h e o r yUl t imate

oads.O431.1371.132

1.105

0.7930.9 50.646

0.9670.847

TesthheoryTangentialLoads

ModeTestWebBucklingWebBuckiing

Flange andTee Buckling

hearMechanismW ebBucklingSheariMechanismW ebBucklingW ebBucklingW ebBuckling

o failureTheory

WebBucklingWebBucklingWeb

BucklingShear-MechanisrnShearMechanisrnSheariMec hanismW ebBuckling

Shear~MechanismWebBuckling


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Reference

alambos, HusainSpeirs (1975)

Husain Speirs( 1973)

Husain Speirs1971)

Beam

10-310-412-1

12-2

Test theoryTangentiai

Loads' Testltheory

UltimateLoads0.9500.8 130.953

966

13-32-4

H-2H 3

H-3PH-4A-1A-2

B-3

G

G 2

G-3

E 1E 2

\VsBuckling

W e bBucklingN A

N.A .N.A.

N..

ShearMechanism

ShearMechanism

SheorMechanisrn

ShearMechanisni

ShearMec hanism

ShearMechanism

Mid-PostY ieldingMid-PostYielding

ModeTestW e b

BucklingWeb

BucklingWebBuckIingLVeb

Buckling

U c bBuckling

L\ e bBuckling

S iearblechanismS iear

.MeclianismShsar

kleciianisrnShear

MechanismShear

iLlechanisrnShear

.MecliariisrnSh e u

MechanismMid-PestYieldingMid-PostYielding

SliearMschanismMid-PostYieldingMid-PostY iclding

of failureTheon-Cb-sb

BucklingL *e b

Buck1 ingShearMechanismShear

.Meclianisrn0.8570.840

1.001.O87.O62

1.1861.1361.259

1.196

1.3141.1461.208

1.9601.81 l

.O51.158

1.137

1.1730.990

.O46


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Minimumyield stress values o f the conespon ding beams were defined. Th e nominalyield stress o f 48MPa 36ksi) was used to co mp ute these ratios.

Re ference

Toprac Cooke1959)

Altfillisch,TopracCooke 1957)

Actual yield stress values of these beams were not reported. M inimu m yield stress

Beam

E 3E 4L 1

L 2

L-3

D

E

H

A

TesthheoryUltimateLoads1.700

1.6131.O63

1 O43

1.1 13

0 956

1 277

1.425

1.2181.8080 887

1.122

value of 35 MPa 24kg/rnm2)was used to compute these ratios.

TestltheoryTangentialLoads1.423

1 442O63

1O43

1.1 13

ModeTestWebBucklingWeb

BucklingFlexuralMechanismFlexuraiMechanismFlexuralMechanismL.T.B?)Flange

BucklingFlangeBucklingShearMechanism

FlangeBuckiingShearMechanism

FlangeBucklingFlangeBuckling

of failureTheoryShearMechanismShear

hlechanismFiexuralMechanismFlexural.MechankmFlexuralMechanismFlexwal

MechanismShearM e c h a ~ s mShearMechanismMid PostYieldingShearMechanisrnFlexuralMechanisrnShearMechanism


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CH PTER FIVE

GENERALIZED N LYSIS ND DESIGN CONSIDER TIONS

5 1 GeneralIn the FEM analyses considered so far. the only loading condition treated approximatepure shear. and furthemore the mode1 has been limited to one cornpx-ising only twopenings. In this chapter more cornplete models are examined and moment-to-shear ratio\ . a ~ - i n grom pure shear to pure bsnding are considered. In addition. the analysis has deaonly with elastic buckling behavior. and the impact of inelasticit . is esamined.In section 5.2. the loading used to create any moment-to-shear ratio is described and isection 5.3 modeIs containing up to four openings are considered under pure shear as tveas pure bending. The effect of moment-CO-shear ratio is then considered for four tesbeams representative of a wide range of castellated beam geometries. These results arused to establish a generai forrn of interaction diagram to define elastic buckling loads o

castellated beams under ny shear to moment ratio. Having established this form isections 2.3 5.7 a parameter study deriving web buckling coefiicients co~ering widrange of geometries s performed. The use of these elastic results. in conjunction uith thplastic analyses is examined in section 5.8 with the aim of developing inelastic bucklinequations. These are then compared with relevant test results.


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5 2 Loading on eneralModelsTo study the behavior of models under various shear to moment ratios, severaMSCM STR N elastic fmite element buckling anaiysis n ns were necessary. To creatpure shear and pure bending forces, as well as various V M ratios. different loadingpatterns had to be irnposed on the finite element model descnbed in Chapter 2.In order to produce pure shear force conditions at any point within the length of themodel, the two vertically concentrated static loads Fig.5.I) used in the analysedescribed in section 2.4 must be supplemented with forces producing a counter-clockwiscouple. This couple was created by applying equal and opposite horizontal forces at the

top and bonom web-to-flange intersection points at the left hand end of the model, ashown in Fig. 5.2. In the several models considered below these forces could be adjustedto provide pure shear at any desired point e.g. the hole centerlines). Similarly, with thevertical loads removed, a clockwise couple applied by such horizontal forces on the lefend of the beamwas used to simulate pure moment conditions, as shown in Fig. 5.3. nycombination of shear and moment forces could be generated by cornbining these verticaand horizontal loads in m y esired proportion.The deformed shapes under vertical loads and under pure shear conditions as shown inFigs. 5 1 and 5.2, demonstrate the same buckling pattern of the post, with slight rwistingof the flange to accomodate the double curvanue bending efiect over the hieght of thepost. Under pure bending conditions however, the region above the middle openingresisting the compression force is buckled, with large twisting of the flange toaccomodate the buckled shape.


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Fig 5 2 Three Hole FEM Mode1 Under Pure Shear orces


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Fig 5 3 Three Hole FEM Model nder pure Bending Moments


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5 3 Elastic Buckling Interaction Diagramue to the presence of the stiffener on the left end and the applied constraints on the righ

of the model, it was thought that the stiffened web posts adjacent to these ends mi@provide restraint to the rotations of the imer web-post of the tw hole model. To ensurethere is no such restmint, models consisting of three and four holes were alsoinvestigated. Both pure bending and pure shear forces were considered for two. three andfour hole models, al1 under the same boundary and loading conditions. These analyseswere canied out for four of the test beams described in the literature. These were beam G-2 from Husain and Speirs 1973), beam B-1 fiom Altifillisch Toprac and Cook 1957).beam F 3 Husain and Speirs 1971) nd b a r n 10 3 from Zaarour and Redwood 1996).These four beams were found to have the diverse properties representing a wide range ofcastellated beam geometries.Resuits for pure bending are expressed s the beam buckling moment s a ratio of theplastic moment and are given in Table 5.1. The three and four hole models producesimilar buckling moments and these were lower than for the two hole rnodel.

Beams 2 Hole Mode1 3 Hole Mode1 4 Hole Mode1

Table 5 1 Summary of Results Under Pure Moment Forces.Similar numerical simulations were conducted to investigate the behavior of 2 3. and 4hole models under pure shear conditions Fig. 5.4). The chailenge here w s to determineat which hole zero moment forces should be ed orce d to produce the pure shear condition.


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As indicated in Table 5.2, several analyses were done to create the zero moment forcecondition at different holes. Al1 the holes of the two and three hole models were tested,and only minor difierences in the results were obtained. For the four hoie model only thetwo interior holes had imposed the zero moment conditions and again only minordifferences are evident, with no trend discernible between the rnodels with differentnumbers of holes. The differences in the critical buckling shear loads of 2 3. and 4 holernodeIs were less than 3 .

In view of these results and to be consistent in subsequent analyses, the three hole modelwas chosen to represent al1 further FEM analyses in this study. It sbould be noted that

MBeam

10 3B

under pure shear loading the different models produced only marginally different results

Table 5.2. Surnmaryof Results nder Pure Shear Forces.

and the two hole mode1 utilized for the analysis of Chapter 4 was thus conftnned to be

3 ole Mode1Hole Mode1

satisfactory for that application.

at holeVcrkW

39.848 1.40

at holeVcrW

39.0983.25

4 Hole Mode1at hole 2

VcrOcN39.0083 .25

at holecrW

38.6282.58

at hole 2VcrkW

40.0982.46

at holeVcrW

38.6681.36

at hole 3VcrW

39.9982.42


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Fig 5 4 Three and Four ole FEM Models


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A complete interaction di gr m for elastic buckling w s obtained for each of the fouselected beams using the three hole model. The results are shown in Figures 5 5 5.6. 3.7nd 5.8 the two ordinates of the elastic FEM results plotted for each V p ratio refer to

the M N atio for the first two holes of the model).I t m be seen that under pur bending, plastic failure occurs at much lower loads th n thebuckling loads. Under pure shear, buckling loads may range from much lower to muchhigh r v tues th n the plastic failure load. The resuits shown on these diagrams will bdiscussed below.

Interaction DiagmmBeam 1 3

Figure 5.5 Zaarour and Redwood 1996)


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InteractionDiagramBeam G-2

igure 5 6 Husain nd Speirs 1973

Interaction DiagnmBeam F 3

- ield TheoryBasUCFRd

= 2

RH TheoryBastic FEM

n = 2

igure 5 7 Husainand Speirs 9 )


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Interaction OiagnmBeam B

Figure 5.8 Altifillisch, Co ok and Toprac 1 957)

The interaction buckIing relationships c m e approximated by a curve defined by :

with Mo and V corresponding to pure shear and bending con ditions respectively. Severadif ferent values o f n were exarnined. The curve found to best represent the FEA resultsfor the fu l range of M N was f o u i d t correspond to n 2 In this way. given -Mc nd Vvalues, a relationship defining the buckling behavior under any M N ratio is established.


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5 1 Parameter StudyHaving established a generai expression defining the elastic buckling beha~iorocastellated beams under any M N ratio. a pararneter study relating the behavior of beams

with different geometries under pure shear and pure bending conditions was carried out.Elastic finite element analysis was performed on 27 beams to derive elastic web bucklingcoef tkients under pure shear and pure bending conditions. The beams were designed andselected to present various ratios and proportions of castellated bearn geometries. Therelevant parameters were considered to be the ratio of hole height to minimum web-postwidth. hJe. and the ratio of minimum web-post width to web thickness. eh,,. Because ofthe wide range of possible beam and castellated hole geornetries. the pararneter study hadto be of limited scope. and thus the following computations are restricted to castellationswitli a hole edge siope of 60 to the horizontal.

5 1 Previous Parameter StudyIn a previous study by Redwood and Demirdjian (1998). a pararneter study to tind theelastic buckling loads under high shear loading was carried out that incorporated a \\ iderange of bearn characteristics. The study assumed elastic behavior throughout. The midpost weld was assumed to be ull penetration and had the s rns thickness and materialproperties as the web. The flange was included in the mode because of its importance inrestraining web rotations, but conservative estimates of flange dimensions were used fort h e general case. These assumed that the flange was only as thick as the web. and thef'lange width was that of a Canadian Standard S16.1-94 class 3 section. Thus:


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shear on the beam VJV. The product o f the ratio V N and the vertical shear force tocause buckling gave the criticai horizontal shear force V,,,, in th web-post at the welded

4 Sjoint. V,,, was then related to vertical shear V through Eqn 2.3) derivedd x - 2 ~from the free body di gr m of Figure 2.2. where y defines the line of action of thelongitudinal force resultant acting in the the tee section. which was taken as being at thecrntroid. This was verified by comparing this va lue with that given by the FEM for the 27beams used in the parameter study. An average ratio of 0.983 with coefficient of variationof 0.02 was found. suggesting that the centroid provided a close approximation.dalues of k obtained from the parameter study are shown in Figure 5.9.

k curves

Figure 5.9. Shear uckling Coefficient Redwood Demirdjian 1998)The vertical shear that will cause web post buckling can therefore be obtained by

-reading the value of k fiom Figure 5.9


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-using equation 5.2 to find the horizontal shear in the web post-using equation 2 3 to uansformV to vertical shear V

These curves cover a wide range of castellated beam geometries with 60 openings.Through linear interpolation between the two series of curves. the buckling coefficient fora wide range of beam geometries c m be detemined.

5 5 Shear Buckling oef f~c ients k.)The previous parameter study was refined in the current research to correspond tobuckling under pure shear, and to make the flange modelling slightly more conservative.A new study to incorporate pure bending is descnbed in section 5 6 For the currentresearch the selected mode1 consisted of three holes s FEM results revealed its betterperformance under bending moment: although no irnprovement was noted for pure shear.consistency between models for the two load cases was considered desirable. The flangedimensions assumed were modified so that the width w s based on the assumption of r

2 l45),Class section. i.e. 6 . Clause 1 2 Canadian Standard S16.1-94 ~vhere\vas taken as 350 MPa. This reduced the flange widths, making the tlange restraintslightly more conservative. Narrower flange widths would make al1 cases consenfativeascompared to class 3 section. which w s found to be slightly unconservative for somecompact sections.Under pure shear conditions, two vertical forces were applied on the nglit end at the levelof the flanges and two horizontal counter clockwise coupling forces were applied on the


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lefi end at the flange to web intersecting nodes to counter the overturning etlct of thvcrtic lly applied forces. Thus there were no bending moments at the centre o f the spa(Fig. 5.2).

Fig. 5.10 shows the results o f the analyses for web-buckling coefficient k due to purshear. There were minor variations between the results of the new and the previous studdue to the minor modelling changes. For beams with hJdZ 0.5. el\ 15 is plotted othe curves to demonstrate the slightly greater dependency of e /4 th n evident in thprevious study. Furthemore. there are minor differences between the shape of the cun efor h /d =O S From the F M studies of different models in Section 5.3 differences up to3 can be expected between the hvo and three hole models. and this together with thflange modelling change explains the differences between the results shown in Figs 5.and 5.10.

k curves

Figure 5.10. Modified Pure Shear buckling coefficent Curves


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5 7 Flexural uckling Coefficients kb)To derive an expression for web buckling due to pure bending moment forces. the same

series of beams under the sarne conditions were subjected to two horizontal clockwise

coupling forces.M,r where S is the section rnodulus of the unperforated section. andaking a =

assuming that the area of the web resisting the compression force is jt, a coefficient k isdefined by

k ximpli @ing by incorporating into kb[ 2 1 P ) ]

This tlexural buckling coefficient. kb is given in Fig. 5.1 for a given variety ofcastellated beam geometries. Almost constant kb values are maintained in the Jdo = 0.74unti1 the lines curve downward, indicating that hole height to minimum width h,/e) ratiohas very little effect on the overall beam buckling behavior under pure bending forces.The k values Vary less th n the k curves, indicating that the flexural buckling load is notsensitive to the ratio of hole height to minimum width hJe).While camparing the two


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series of bearns larger buckling coefficients under pure moment conditions were foundfor the senes of bearns Mth larger tee sections hJd,=.5. but the behavior w s reversedunder pure shear conditions, where beams with lower tee sections with hJd =0.74 hadhigher k coefficients.Thus b sed on a given beam geometry, the critical moment to cause elastic buckling issimply caiculated using equation 5.5).

Figure 5 11 BuckIing Coefficient Curves Under ure Bending Forces

5.7 Effect of Inelasticity on Ultimate StrengbSince buckling usually involves inelastic action, the influence of plasticity is consideredin this section to improve the already mentioned methods of analysis and derive generalexpressions incorporating oth elastic and inelastic buckling actions.

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