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STABILITY OF CASTELLATED BEAM WEBS Sevak Demirdjian March 1999 Department of Civil Engineering and Applied Mechanics McGill University Montreal, Canada A thesis submitted to the faculty of Graduate Studies and Research in partial fulfilment of the requirements of the Degree of Master of Engineering O Sevak Demirdjian

Stability of Castellated Beam Webs

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Page 1: Stability of Castellated Beam Webs

STABILITY OF CASTELLATED BEAM WEBS

Sevak Demirdjian

March 1999

Department of Civil Engineering and Applied Mechanics

McGill University Montreal, Canada

A thesis submitted to the faculty of Graduate Studies and Research in partial fulfilment of the requirements of the Degree of Master of Engineering

O Sevak Demirdjian

Page 2: Stability of Castellated Beam Webs

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Page 3: Stability of Castellated Beam Webs

ABSTRACT

A study on the web-buckiing behavior of castellated beams is descnbed in this thesis.

Both elastic and plastic methods of anaiysis are utilized to predict the faiiure modes of

these beams.

Interaction diagrams predicting formation of plastic mechanisrns. yietding of the

horizontal weld length and etastic bucfling anaiysis using the finite element method are

correlated with a number of experimentai test results fiom previous studies given in the

literature.

Test-to-predicted ratios for a total of 42 test beams ranging from 45" to 60" openings are

computed with the plastic and elastic methods of anaiysis, and a mean of 1.086 and

coefficient of variation of 0.195 are obtained. A parameter study covenng a wide range of

60" castellated bearn geomevies is perfonned to derive elastic buckling coefficients under

pure shear and bending forces. An elastic buckling interaction diagram is then defined.

which along with the diagrarns utilized in the plastic analysis, can be used to predict the

elastic buckling and plastic failure loads under any given moment-to-shear ratio.

To incorporate the effect of plasticity associated with buckling, expressions are derived to

improve the previous theoretical models used, by cornbining both elastic and plastic

results. This results in an irnprovement in the coefficient of variation of the test-to-

predicted ratios for the 60' beams considered from 0.170 to 0.137.

Page 4: Stability of Castellated Beam Webs

RI~SUMÉ

Dans la cadre de la présente thèse. une étude sur le voilement de l'âme des poutres

aiourées a été effectuée. Les modes de rupture de ces poutres et les charges

correspondantes sont evalués par des anaiyses de plasticité et d'élasticité.

Les charges estimées par les diagrammes d'interaction pour la formation d'un mécanisme

de rupture. pour la rupture du joint de soudure horizontai par écoulement. et pour le

voilement de l'âme prédit par analyse par élément finis. sont comparées aux résultats des

plusieurs études antérieures.

Les rapports entre les résultats expérimentaux pour 42 poutres avec 45' à 60'

d'ouvertures et les prédictions par les méthodes d'analyse de plasticité et d'élasticité ont

été obtenus, et une moyenne de 1.086 et un coefficient de variation de 0.195 ont été

obtenues. Une étude paramétrique sur les coefficients de voilement Çlastique de l'âme a

été effectuée pour des charges en cisaillement pur et en flexion. pour un grand nombre de

poutres ajourées avec des ouvertures de 60'. Un diagramme d'interaction pour le

voilement élastique de l'âme a été développé. Ce diagramme est utilisé en combinaison

avec les diagrammes pour la formation d'un mécanisme de rupture pour estimer la force

de cisaillement par rapport au moment de flexion. correspondant à la formation d'un

mécanisme de rupture et au voilement élastique de l'âme .

L'effet de la plasticité lors du voilement de l ' h e est ensuite inclus dans les expressions

théoriques. Cette addition réduit l'écart-type de 0.170 à 0.137 sur les prédictions

théoriques pour les poutres ajourées avec des ouvertures de 60'.

Page 5: Stability of Castellated Beam Webs

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Prof. R.G. Redwood for his constant

guidance. 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'this

project. and to al1 her guidance and advising throughout my graduate level studies.

The support of Fonds des Chercheurs et l'aide à la recherche (FCAR) is greatly

acknowledged.

1 uould like to thank my parents Krikor and Alice, and my brother H m 9 for their intïnite

support and encouragement for al1 these years. Finally 1 would like to acknowledge my

uncle Joseph Bedrossian. for his valuable knowledge and help for man- yerirs.

Page 6: Stability of Castellated Beam Webs

TABLE OF CONTENTS

ABSTRACT ......................................-....--................................................... i

.. RÉSUMÉ ................................................................................................

... AC WOWLEDGMENTS ....................................................................... iii

........................................................................ TABLE OF CONTENTS iv

. . ................................................................................ LIST OF FIGURES vil

LIST OF TABLES ................................... .......................................... ix

NOTATIONS .................................-........................................................... x

................................................................. CHAPTER ONE : Introduction 1

1 . 1 Introduction .................................................................................... 1

............................................. 1.2 Failure Modes of Castellated Beams 5

1 .2.1 Vierendeel or Shear Mechanism ......................................... 5

1.2 -2 Flexural Mechanism ............................................................ 6

1 2 . 3 Lateral Torsionid Buckling ................................................ -7

1 2 .4 Rupture of Welded Joints .................................................... 9

........................................................... 1.2.5 Web Post Buckling 10

1.2.6 Web Post Buckling Due To Compression ........................ 13

....................................................................... 1.3 Research Program 1 4

........................................... 1.3.1 Objective and Scope of Work 14

......................................................... 1.3.2 Outline of the Thesis 15

CHAPTER TWO : Methods of Analysis ............................................... 16

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

Page 7: Stability of Castellated Beam Webs

............................................................................ 2.2 Plastic Analysis 16

....................................................................... 2.3 Mid-Post Yielding 1 9

.................................................................. 2.4 Buckling Analysis . 1

............................................................... 2.5 Finite Element Analysis 21

............................................................................. 2.5.1 General 2 4

....................................................... 2 -5 -2 Input File Preparation 27

.............................................................. 2.5 -3 Model Geometry -28

2.5.4 Constraints ........................................................................ 28

2.5.5 Loads ........................................... .. .................................... 29

................................... 2.5.6 Buckling Analysis ......................... 32

...................................................................................... 2.6 Summary 34

................................................. CHAPTER THREE : Literature Review 35

......................................................................................... 3- l General 35

3.2 Literature Review ......................................................................... 3 5

...................................... 3.2.1 Redwood and Demirdjian ( 1 998) 36

.................................................................. 3.2.2 Zaarour ( 1 996) 36

............................... 3.2.3 Gaiambos, Husain, and Speirs (1975) 37

.................................................. 3.2.4 Husain and Speirs ( 1 973) 38

3.2.5 Husain and Speirs (1971) .................................................. 39

................................................... 3 .2.6 B a d e and Texier ( 1 968) 39

................................................................. 3.2 -7 Halleux ( 1967) -40

.......................................................... 3.2.9 Sherboume ( i 966) 4 1

Page 8: Stability of Castellated Beam Webs

............................................... 3.2. I O Toprac and Cooke (1959) 42

............................ 3.2.1 I Altifillisch. Toprac and Cooke (1957) 43

........ CHAPTER FOUR : Reconciliation o f Andysis With Test Results 52

4.1 General ........................................................................................ -52

4.2 Comparative Data ........................................................................ 52

- - ................................................................................ 1.3 Cornparisons 2 3

..................................................................................... 3.4 Discussion 57

CHAPTER FIVE : Generalized Analysis and Design Considerations .... 62

5.1 General .............................................................. -62

5.2 Loading on General Models ......................................................... -63

5 . 3 Elastic Buckling interaction Diagram ........................................... 67

............................................................................ 5 -4 Parameter Smdy 73

5.5 Previous Parameter Study ............................................................ -73

......................................................... 5 -6 Shear Buckling Coefficients 76

5.7 Flexural Buckling Coefficients .................................................... -78

5 -8 Effect of Inelasticity on Ultimate Strength ................................... 79

C HAPTER SIX : Conclusion .................................................................. -84

REFEWNCES ......................................................................................... 87

APPENDIX A : Finite Element Input File

AP PENDIX B : Detailed Test-To-Theory Results

APPENDIX C : Elastic and Plastic Theoretical Computations

Page 9: Stability of Castellated Beam Webs

LIST OF FIGURES

ixurmwm Figure 1.1 Castellated Beams ..................................................................... 1

3 Figure 1.2 Zig-Zag Cutting Dimensions of Rolled Beams ......................... .-

Figure 1 -3 Castellated Beam Section Properties .................... .. .................... 4

Figure 1.4 Castellated Beam Section Properties with Plates at Mid-Depth 4

Figure 1.5 ParalleIogram Mechanism ........................ ..... ......................... 6

Figure 1.6 Lateral Torsional Buckling ......................................................... 8

Figue 1 -7 Weld Joint Rupture ..................................................................... 9

Figure 1 -8 Web Post Buckling .................................... ... .......... 1 2

cHAmmnw Figure 2.1 Interaction Diagram .................................................................. 18

Figure 2.2 Free-Body Diagram ................................................................ -20

3' Figure 2.3 Predicted Web-Post Buckling Moments .................................. -3

Figure 2.4 (a) Mode1 used By Zaarour and Redwood (1 996) .................... 26

Figure 2.4 (b) Non-Composite Mode1 Used by Megharief (1 997) ............ 26

Figure 2.5 Finite Element Mode1 .............................................................. -30

Figure 2.6 Pure Bending and SheadMoment Arrangement ................ ..... 3 1 - - 3 Figure 4.1 Test Arrangement of Beam H ................................................... >J

Figure 4.2 Interaction Diagram Demonstrating Theoretical Methods ...... 54

Page 10: Stability of Castellated Beam Webs

. ................. Figure 5.1 Two Hole FEM Model Under Vertical Loads Only 64

.................. Figure 5.2 Three Hole FEM Model Under Pure Shear Forces 65

Figure 5.3 Three Hole FEM Model Under Pure Bending Moments ......... 66

.......................................... Figure 5.4 Three and Four Hole FEM Models 69

Figure 5.5 Zaarour and Redwood ( 1 996) .................................................. -70

Figure 5 -6 Husain and Speirs ( 1973) ....................... ....,... .................. 71

Figure 5.7 Husain and Speirs ( 197 1 ) ................................................ 7 1

Figure 5.8 Altifillisch, Cooke and Toprac .............................................. 72

Figure 5.9 Shear Buckling Coefficient Redwood and Demirdjian ( 1998) 75

. ................ Figure 5 I0 Modified Pure Shear Buckling Coeficient Curves 77

Figure 5.1 1 Buckling Coefficient Curves Under Pure Bending Forces .... -79

................................ Figure 5.12 Elastic and Plastic Interaction Diagrams 80

...... Figure 5.1 3 Comparison of Test Results With Proposed Expressions 83

Page 11: Stability of Castellated Beam Webs

LlST OF TABLES . .............................. ................ Table 3.1 Redwood and Demirdjian ( 1998) , 44

................................................... Table 3.2 Zaarour and Redwood ( 1 996) -44

Table 3.3 Galarnbos Husain and Speirs (1975) .......................................... 45

........................................................... Table 3.4 Husain and Speirs (1 973) 46

........................................................... Table 3.5 Husain and Speirs ( 1 971) 47

........................................................... Table 3 -6 Bazile and Texier ( 1968) 47

.......... Table 3.7 Halleux ( 1967) ................................................................. 48

..................................................................... Table 3.8 Sherbourne (1966) 49

Table 3.9 Toprac and Cooke (1959) .......................................................... 50

...................... Table 3.10 Attifillisch. Cooke and Toprac (1 957) - .............. 51

c K M n E m m

Table 4.1 Surnmary of Test to Theoretical Predictions ............................. 38 - Table 5.1 Summary of Resuits under Pure Moment Forces ...................... 67

......................... Table 5.2 Summary of Results Under Pure Shear Forces 68

....................................................................... Table 5.3 Statistical Results 82

Page 12: Stability of Castellated Beam Webs

Ar

.4,

b

b r

d

'4

d b

4

C

cov

DOF

E

e

FEA

FEM

F,

G

GD

h

h o

h,

NOTATIONS

area of flange

area of web

width of one sloping edge of the hole

width of flange

depth of the original beam section

total depth of castellated beam section

depth of bottom tee section

depth of top tee section

compression force

coefficient of variation

degree of freedom

modulus of elasticity

length of welded joint

finite element analysis

finite element method

yield stress

stiffiiess matrix

differential stiffness matrix

height of one sloping edge of hole

height of hole

height of plate

Page 13: Stability of Castellated Beam Webs

moment of inertia

depth of top tee section excluding flange

buckling coeffkient

flexural buckling coefficient

shear buckling coefficient

length of beam

bending moment

elastic buckling moment under pure bending forces

elastic moment to cause web buckling

critical moment

plastic moment

critical moment based on beam test results

yield moment

moment to fonn flexwal mechanism

ultimate moment

constant force

elastic section modulus

distance from center-line to centerline of adjacent castellation holes

tension force

thickness of the flange

thickness of the web

displacement vector

Page 14: Stability of Castellated Beam Webs

modified displacement vector

S hear force

elastic buckling shear under pure shear forces

critical shea to cause web buckling

shear obtained fiom elastic anaiysis

horizontal shear force

critical value of Vh

cntical shear based on beam test results

plastic shear

shear O btained fiom plastic analysis

vertical shear force to cause rnid-post yielding

vertical shear force to fom plastic mechanism

ultimate shear to cause web buckling

applied load

distance fiom top of the flange to centroid of tee-section

plastic section modulus of castellated beam

full section plastic modulus

factor utilized in plastic analysis

factor utilized in plastic analysis

angle of castellation

critical stress

expansion ratio

sii

Page 15: Stability of Castellated Beam Webs

factor appiied to shear y ield stress

eigen value

eigen vector

poisson's ratio

aspect ratio

Page 16: Stability of Castellated Beam Webs

CHAPTER ONE

1. I Introduction

Sincc rht. Second World L\.*ar. man>- attcmpts ha\-e been made b' structural enyineers w

: i d ne\\ ways to dccreass the cost of steel structures. Eue to limitations on mrisimum

cillowable deflsctions. the high strength pruperties of structural i t ~ t l cannut ~ 1 ~ \ 3 > ' : be

utilized to best adwxage. .As 3 result. se\.ernl ne\! rne~l-iods ha\.e been airned .IL

incrcnsing the stiffncss of steel mcmbers w-ithout an)- increasri in iteiylir O&' the steel

rquired. Cnstellated béams w r e one of these solutions ( Fig. 1.1 1.

r

1

Fig. 1.1 CastslIatzd Beams

Page 17: Stability of Castellated Beam Webs

Castellated (or expanded) beams are fabricated from wide flange 1-beams. The web of the

section is cut by flarne along the horizontal x-x avis along a "zigzag" pattern as shown in

Fig. 1.2.

Figure 1.2 Zig-Zag Cutting Dimensions of Rolled Bems

The two halves are then welded together to produce a beam of greater depth with

hexagonal openings in the web (Fig. 1.3), or rectangular plates may be inserted between

the two parts. producing octagonal holes (Fig. 1.4). The resulting beam has a Iarger

section modulus and greater bending rigidity than the original section. without an

increase in weight. However, the presence of the holes in the web will change the

structural behavior of the beam from that of plain webbed beams. Experirnental tests on

castellated beams have shown that beam slenderness. castelhtion parameters and the

loading type are the main parameters, which dictate the strength and modes of failure of

these beams.

Casteilated bearns have been used in constmction for many years. Today. with the

developrnent of automated cutting and welding equipment. these beams are produced in

an alrnost unlimited number of depths and spans. suitable for both light and heavy

loading conditions. In the past. the cutting angle of castellated beams ranged from 45" to

Page 18: Stability of Castellated Beam Webs

70° but currently, 60" has become a fairly standard cuning angle. although 45" sections

are also available. It should be noted that these are approxirnate values. actuai angles will

vary slightly from these to accommodate other geometrical requirements. As roof or floor

beams. joists. or purlins, these sections may replace solid sections or tmss members.

Their aesthetic attributes produce an attractive architectural design feature for stores,

schools and service buiidings. In structures with ceilings. the web openings of thesc

members provide a passage for easy routing and installation of utilities and air

conditioning ducts.

Typically. the dimensions of a castellated beam are defined as follows (referring to

Figs. 1.2 to 1.4):

d, = d + h + h , (For no plates. h,=O)

4 Expansion ratio, y = - d

where. d = original beam depth

h = depth of cut

h, = height of plate

b = width of sloping edge of hole -

d, = depth of top tee section

Page 19: Stability of Castellated Beam Webs

Figure 1.3 Castellated beam section properties

I - 4 v

Figure 1.1 Castellated beam section properties with plate rit mid depth

Page 20: Stability of Castellated Beam Webs

1.2 Failure Modes of Castellated Beams

To date, experimental studies on castellated beams have reported six different modes of

failure (Kerdal & Nethercot 1984). These modes are closely associated with beam

geornetry. web slendemess, hole opening, type of loading, and provision of lateral

supports. Under given applied transverse or coupling forces, failure is Iikely to occur by

one the following modes: Vierendeel or shear mechanism. flexurai mechanism. lateral

torsional buckling, rupture of welded joints, web post buckling in shear and compression

buckling.

1.2.1 Vierendeel or Sbear Mecbanism

This mode of failure is associated with high shear forces acting on the beam. Formation

of plastic hinges at the reentrant corners of the holes deforms the tee section above the

openings to a parailelogram shape (Fig. 1.5). This mode of failure was first reported in

the works of Altifillisch (1957), and Toprac and Cook (1959). Bearns with relatively

short spans with shdlow tee sections and longer weId lengths are susceptible to this mode

of failure. Shorter spans can carry higher loads leading to shear becoming the goveming

loac!. When a castellated beam is subjected to shear, the tee sections above and below the

openings must carry the applied shear, as well as the pnmary and secondary moments.

The primary moment is the conventional bending moment on the beam cross-section. The

secondary moment, also known as the Vierendeel moment. results from the action of

shear force in the tee sections over the horizontal length of the opening. Therefore. as the

horizontal length of the opening decreases, the magnitude of the secondary moment will

Page 21: Stability of Castellated Beam Webs

decrease. The location of this failure will occur at the opening under greatest shearing

force. or if several openings are subjected to the s m e maximum shear. then the one with

the greatest moment will be the critical one.

Plastic Hinges

Figure 1.5 ParalleIogram Mechanism

1.2.2 Flexural Mechanism

Under pure bending. provided the section is compact (at lsast Class 2 (CSA 1991)). the

tee sections above and below the openings yield in tension and compression until thty

becomc fully plastic. This mode of failure was reported in the lcorks ot'Toprric and Cook

( 1 959) and Halleux (1967). They conciuded that yieiding in the tee sectioris ribo\.c. and

bclow the openings of a castellated beam was similar to that of a solid beam under pure

bendiny forces. Thus. the maximum in-plane carrying capacity of a castellatrd beam

under pure moment loading was determined to be = Z'xE; wliere Z' is tlie full

section plastic modulus taken through the vertical çenteriine of ri hole.

Page 22: Stability of Castellated Beam Webs

1.2.3 Lateral-TorsionaCBuckling

As in soiid web beams, out of plane movement of the b a r n without any web distonions

describes this mode of failure. Lateral torsional buckling as s h o w in Fig. 1.6. is usually

associated with longer span beams with inadequate lateral support to the compression

flange. The reduced torsionai stifiess of the web, as a result of relatively deeper and

slender section properties, conmbutes to this buckling mode. Nethercot and Kerdal

(1982) investigated this mode of failure. ï h e y concluded that web openings had

negligible effect on the overail lateral torsional buckling behavior of the beams they

tested. Furthemore, it was suggested that design procedures to determine the lateral

buckling strength of solid webbed bearns could be used for castellated beams provided

reduced cross sectional properties are used.

Page 23: Stability of Castellated Beam Webs

Fia,. 1.6 Lateral Torsional Buckling (Redwood & Dsmirdjian 1998 )

Page 24: Stability of Castellated Beam Webs

1.2.4 Rupture of Welded Joints

The mid depth weld joint of the web post between two openings rnay rupture when

horizontal shear stresses exceed the yield strength of the welded joint (Fig. 1.7). Husain

and Speirs (1971) investigated this failure mode by testing six beams with short welded

joints. This mode of failure depends upon the lengtb of the welded joint (e). The

horizontal length of the openings is equai to the weld length. and if the horizontal length

is reduced to decrease secondary moments. the welded throat of the web-post becomes

more vulnerable to failure in this mode.

Weld Rupture

Figure 1.7. Weld joint Rupture

As mentioned in 1.2.1, formation of a Vierendeel mechanism is likely to occur in beams

with long horizontal hole lengths (and hence long welds). On the other hand. short weld

lengths are prone to cause failure of the welded joints as the horizontal yield stress is

exceeded. Dougherty (1993) found a reasonable balance of these twvo failure modes. by

suggesting the following geometry:

Page 25: Stability of Castellated Beam Webs

hl Weld length e = - and for a 60' cutting angle with no plates. 3

Therefore. opening pitch s = 2(6 + e ) = 24) 0.289 + - = 1-08 h,, 1.1 h,, ( 3 This concept has been demonstrated in many of the current available Castelite Standard

Beam Geometry sections. (Castetite Steel Beam Design Manual 1996).

1.2.5 Web Post buckling

The horizontal shear force in the web-post is associated with double curvature bending

over the height of the post. As shown in Fig. 1.8, one inclined edge of the opening wili be

stressed in tension, and the opposite edge in compression and buckling will cause a

twisting effect of the web post along its height. Several cases of web post buckling have

been reported in the literature: Sherbourne (1966), Haileux (1967). Bazile and Texier

( 1968).

Many analytical studies on web post buckling have also been reported to predict the web-

post buckling load due to shearing force. Based on finite difference approximation for an

ideally elastic-plastic-hardening material Aglan and Redwood (1976) produced sonis

graphical design approximations for a wide range of beam and hole geometries; sorne

correlations between experimental and non-linear finite element analysis (FEM)

estimations were found in the works of Zaarour and Redwood (1996). Delesque ( 1968)

Page 26: Stability of Castellated Beam Webs

used an energy method to solve an elastic buckling problem by treating the wsb post as a

variable section rectangular bearn in double curvature bending, susceptible to lateral

torsional buckling. However. Zaarour and Redwood (1 996) found large differences in the

results obtained frorn Blodgett's method in cornparison to their test results and finite

element approximations they used. Blodgett's method is therefore not used in this project.

In recent works of Redwood and Demirdjian (1998). approximations of buckling loads

were derived based on elastic finite eiement analysis and good correlations btmveen

experimental and theoreticai estimations were found. This work showed that the results of

Aglan and Redwood ( 1 976) should not be used for very thin webs. This mode of failure

and these theoretical results are discussed in greater detail in subsequent chapters.

Page 27: Stability of Castellated Beam Webs

Fig. 1.8 U'eb Post Buckling (Redwood rP: Demirdjian 1998 >

Page 28: Stability of Castellated Beam Webs

1.2.6 Web Post Buckling Due to Compression

A concentrated load or a reaction point applied directly over a web-post causes this

failure mode. This mode was reported in the expenrnents conducted by Toprac and Cook

(1959). Husain and Speirs (1973). Buckling of the web post under large compression

forces is not accornpanied by twisting of the post. as it would be under shearing force.

Such a failure mode could be prevented if adequate web reinforcing stiffeners are

provided. A strut approach was proposed in the works of (Dougherty 1993). which

suggests that standard column equations could be used to determine the strength of the

web post located at a load or a reaction point.

Page 29: Stability of Castellated Beam Webs

1.3 RESEARCH PROGRAM

1.3.1 Objective and Scope of Work

The objective of the current research is to study failure of castellated b e m s \cith

panicular emphasis on web post buckiing. The goal is to make use of the availabie elastic

and plastic analysis methods. and derive expressions that will predict critical shear force

causing web-post buckling.

This thesis uses many previous experimental results to provide cornparisons \\-ith

theoretical approximations. and thus validation of the suggested methods described.

The first part of the research program focuses on the theoretical methods of analysis to be

used to predict failure loads of castellated beams. These methods include plastic analysis

of the Vierendeel mechanism and for yielding of the mid-post joints. The finite element

method is used to perform elastic buckling analysis and predict critical loads of al1 test

beams. A thorough literature search then follows to list al1 relevant experimental data to

be compared with theoretical methods. Correlations between experimental and theoretical

results are then made.

Thc second part of' the thesis tocuses on general design considerations and thus is aimed

at the principal objective of the research. Elastic buckling modes are investigated under

different moment to shear ( M N ) ratios. Well-defined relationships. based on pure shear

and pure bending forces to cause web buckling. are developed to predict eiastic buckling

loads under an' M N ratios. Results of elastic buckling and mechanism yielding loads are

then combined and fitted curves are derived to predict ultimate shearing forces causins

Page 30: Stability of Castellated Beam Webs

web-post buckiing. To apply these expressions in a more general fashion. a parametric

study investigating the behavior of a wide range of casteltated beam geometries is

developed. and buckiing coefficients under pure shear and bending forces are derived.

Suçgested predictions are then tested against actual test results, and good correlations are

obtained.

1.3.2 Outiine of the Thesis

The thesis is divided into six sections. M e r a bnef introduction to castellated beams and

their modes of failure of Chapter 1, Chapter 2 focuses on several theoretical methods of

analysis to predict modes of failure of castellated beams. ïhese methods include plastic

analysis, web-post yielding at mid-height, buckiing analysis, and finite element

approximations. Chapter 3 contains a surnmary of relevant test data provided by previous

testing and available in the Merature. Relevant information on each test beam is

tabulated. Theoretical approaches described in Chapter 2 are tested against actual

experimental test beams, and reconciliation of analysis with test results is the topic

covered in Chapter 4.

Chapter 5 focuses on design considerations for castellated beams. Relationships defining

elastic buckling under any M N ratio are developed. A parametric study. as welI as

expressions estimating shear force causing buckling are derived. Results of suggested

methods are tested against actual experimental test results, and correlations between tests

and theories are made. Concluding rem& are summarized in Chapter 6.

Page 31: Stability of Castellated Beam Webs

CHAPTER TWO

METHODS OF ANALYSE

2.1 General

Several theoretical approaches are considered to analyze the yielding and buckling fidure

modes of castellated bearns. Plastic anaiysis of the Vierendeel mechanism failure, ris rvell

as anaiysis of mid web post yielding are sumrnaiized. Elastic finite element buckiing

analysis is used to predict buckling loads. Finite element mode1 generation as well as

buc kling anal ysis in the MSCMASTRAN finite element package are descri bed.

2.2 Plastic Analysis

The construction of an interaction diagram relating shear force and bending moment at

mid-length of an opening has been described by Redwood (1983). This diagram can be

used to study failure caused by the formation of a Vierendeel mechanisrn formed by the

development of four plastic hinges at the re-entrant comers of the tee section. above and

below the hole. For the beam to anain this plastic failure, the web and flanges are

assumed CO be stable and withstand the high shear load until plastic hinges are fonned at

the reentrant corners of an opening in high shear region. As the load increases, primary

and secondary stresses resulting fiom combined effect of shear and moment forces lead to

complete yield at the four corners thus forming plastic hinges. This analysis is based on

the assurnption of perfectly plastic matenal behavior with yielding according to Von

Page 32: Stability of Castellated Beam Webs

Mises criterion. A typical interaction dia- is shown in Fig. 2.1. The shear and

moment values have been non-dirnensionalized by division of the section's tùi1y plastic

shear and moment capacities.

The diagram can be constructed using the following results:

A, I;Y v =JS Where shear area A, = dg&.

To generate the curve, k, is varied between O and 1. Below the value 1. the curve

becomes vertical. For given beam characteristics and hole location subjected to a load. a

radial line can be drawn fiom the origin to intercept the interaction diagram for the

corresponding shear-to-moment ratio (VIM). The horizontal and vertical coordinates of

the intercepted point then predict the shear and moment values to cause yield nlechanism

failure.

Page 33: Stability of Castellated Beam Webs

Interaction Diramm Specimen 10-5a

I

j -YieidTheory

A Test Resuit

Figure 2.1 Interaction Diagram (Redwood and Demirdj ian ( 1 998))

Page 34: Stability of Castellated Beam Webs

2.3 iMid-Post Y ielding

It is possible for yielding of the web-pst at mid-height to occur before failure due to

formation of shear mechanism takes place. This mode of failure occurs particularly to

beams with closely spaced openings with low moment-to-shear ratio. The vertical shear

force to cause mid-post yielding is defined through.

and the basic approach to define this relationship (Hosain and Speirs 197 1 ) is derived by

using equilibrium equations from the free body diagram of castellated beam section as

shown in Fig. 2.2.

The horizontal shear force, V, can be expressed as

when the vertical shear force V, and V, are equal, then

- v x s where,

- (d x - 2 y , )

V, is defined as the difference between the two horizontal forces C , and C7,

This equation is based on the assurnption that the line of action of forces CI and C-> are

acting at the centroid of the tee section above the openings.

The web post will yield when the minimum weld-post area is subjected to the shear yidd

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

Page 35: Stability of Castellated Beam Webs

be expected that strain-hardening will develop leading to a significantly higher failure

load than that given by Eqn. 2.1. In the work of Husain and Speirs ( 197 1 ) the sliear y ield

stress has been measwed directly and is significantly higher than

based on F ~ 4 3 . In view of this the yield stress used, for this mode of

later 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 bearn

Page 36: Stability of Castellated Beam Webs

2.4 Buckling Analysis

Based on a finite difference bifùrcation analysis of the web post ueated as a beam

spanning between the top and bottom of the openings, graphical results relating critical

moments in the p s t to different beam opening geometries were developed by Aglan and

Redwood (1 976). The materiai was considered to be an elastic-perfectly plastic linear

strain-hardening material. For different hole height to minimum width ratios. critical

moments in the post at the level of the top and bottom of the opening, divided by that

section's plastic moment capacity, A(, = 0 . 2 5 ~ (s - ef were presented. as shown in

Fig. 2.3.

F0r.a given beam, the value of MJM, is first read from Fig. 2.3. By multiplying the

given ratio by the section's plastic capacity M,, as given above. the horizontal shear

w, 'r acting at the minimum weld length is calculated as Vh = - . From the free body h,,

Vh S diagram of Figure 2.2, the Vfl ratio is given by - = . Therefore. the vertical d$y,

2 .~.l,,,(d, - 2 y , ) shear force to cause buckling in the web-pst is then derived as VL, =

~ h , ,

et, where yield on the smallest web-pst cross-section -- is an imposed upper limit on f i

V,,. In the work of Zaarour and Redwood (1996), who tested 12 castellated beams.

satisfactory predictions were obtained with the Aglan and Redwood (1976) approach.

However, in more recent work (Redwood and Demirdjian 1998). tests of very thin

Page 37: Stability of Castellated Beam Webs

webbed castellated beams showed that the graphical results such as shown in Fig. 2.3

provided unsafe predictions. a result that was believed to be due to the assumed restraint

conditions at the top and bottom of the web-pst. The method of Aglan and Redwood

( 1976) is therefore not considered M e r in this study.

Page 38: Stability of Castellated Beam Webs

#? >

1 ; 0.0

i 0.4 @-

O.? dt,=lO 20

O 30

Figure 2.3 Predicted Web-Post Buckling Moments for 4=60U (Aglan and Red~ood 1976)

Page 39: Stability of Castellated Beam Webs

2.5 FINITE ELEMENT ANALYSIS

2.5.1 General

The finite element method has previously been used to pertorm buckling analyses on

castellated beams and is also used in this project. This section therefore describes the

sofhare used and the specifics of the application to castellated beams.

In previous work (Zaarour and Redwood, 1 996 and Megharief and Redwood, 1 997) FE AM

studies of the buckling of web-posts in composite and non-composite beams were found

to give good approximations of test resuits (2-10% variations). Both studies utilized the

finite element package MSCRVASTRAN developed by the MacNeal SchwindIer

Corporation (Caffiey and Lee 1994). The same package is used in the current research

with the objective to utilize FEM as a reiiable tool to simulate experimental tests and

generate web p s t buckling loads.

Zaarour and Redwood (1996) studied buckling of thin webbed castellated beams based on

a single web-post model, as show in Fig. 2.4(a). Mesh refmement was based on the

convergence of web p s t buckling loads in cornparison to several experimental test

results. Megharief and Redwood (1997) investigated the behavior of web-post buckiing

of composite castellated beams. Their mode1 consisted of full flanges, web and transverse

stiffeners and the model comprised two complete web openings as shown in Fig. 2.4(b).

This larger model was needed in order to incorporate the shear co~ec t ion between steel

section and slab, and hence the composite action on the bearn. The model used in the

Page 40: Stability of Castellated Beam Webs

current research is similar to the non-composite beam mode1 utilized by Megharief and

Redwood (1997) as shown in Fig.2.4(b), however, based on the different needs in the

current work, more refmed meshes and a greater number of openings are used. as

discussed subsequently. The following sections describe the particular steps necessary to

use the MSCNASTRAN system and the details of the generation o f the models.

Page 41: Stability of Castellated Beam Webs

Fig. 2.4(a) Model used by Zaarour and Redwood ( 1996)

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

Page 42: Stability of Castellated Beam Webs

2.5.2 Input File Prepamtion

Elastic finite element bifurcation analysis was carried out for al1 test beams. h analysis

in MSCNASTRAPJ is submitted in an input file, which consists of three major sections:

Executive control. Case control and %ulk data. SampIe input file is given in Appendix A.

Executive Control Sectioa: is the first required group of statements to detïne the type of

analysis, time allocation and system diagnostics.

Case Control Section: specifies a co~lection of grid point numbers or element numbers

to be used in the analysis. Requests output selections and loading subcases.

Bulk Data Entry: contains al1 necessary data for describing the structural model.

Includes geometric locations of grid points, constraints, element connections, element

properties and loads.

To prepare a detailed description of a model, the following classes of input data must be

provided:

Gromerry: locations of gnd points and the orientations of the coordinate system

Elrrneni connectivity: identification number of grid points to which each dernent is

co~ec t ed .

Efement properties: definition of the thickness, and the bending properties of each

element.

,Material properties: definition of Young's modulus and Poisson's ratio.

C'onsrrninis: specifications of boundary and symmetry conditions to constrain free-body

motion that will cause the anaiysis to fail.

Louds: definition of extemally applied loads at grid points.

Page 43: Stability of Castellated Beam Webs

2.5.3 Mode1 Geomety and Type of Elernents

A skeleton model based on a given beam geometry is first developed through defining the

x. y. and z coordinates of each grid point. Grid points are used to define the geometry of a

structure. to which finite elements are attached. Each gnd point possesses six possible

degrees of fieedom (DOF) about the x, y, and z-axes, three translations (T 1. T2. T3) and

three rotations (RI, W. R3), which constrain the grids to displace with the loaded

structure.

As the geometry of the structure is defined. the grid points are connecteci by finite

elements. Two-dimensional CQUAD4 isotropic, tinear elastic (MAT 1 ) membrane-

bending quadrilateral plate elements were chosen to define the finite elements of the

model. CQUAD4 element input card is defined through four grid points whose physical

location determines the length and width of the element. By assigning a material

identification number in the CQUAD4 input card. ail essential material properties.

membrane, bending, thickness, shear and coupling effects of the elements are defined in

the shell element input property card (PSHELL). Similady, linear elastic properties of the

material, modulus of elasticity, Poisson's ratio are defined in the MAT1 data entry input

card by assigning a property identification number in the PSHELL entry card.

2.5.4 Constrainta

Single point constraints (SPC) are used to enforce a prescribed displacement (components

of translation or rotation) on a grid point. The degrees of fieedom in MSC/NASTEWN

Page 44: Stability of Castellated Beam Webs

are defined as numbers 1. 2, 3.4, 5, and 6. corresponding to three translation. T l , T2. T3.

and three rotational degrees of freedom. RI. R2. RX The propenies of CQUAD4

elements used in modeling the web, flanges and the stiffeners had zero normal twisiting

stiffness. One way to ensure non-singularity in the stifiess matrix and to account for the

out of plane rotational stiffhess or the sixth degree of fieedom (R3) is through AUTOSPC

and KGROT commands in the Bulk Data Entry, as recommended in the manuals. In ail

models K6ROT was taken as 10,000. This value is a fictitious number assigned to

suppress 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 was

tested to provide acceptable results. Fig. 2.5 shows a typical mesh. this one comprising

two openings. The model is supported at the bottom lefi-hand corner where constraints 2

and 3 are applied; these prevent movement in the vertical and out of plane directions.

Displacements in the x and z directions at the upper and lower flange to web intersecring

nodes at the right end are restricted by constraints 1 and 3. to prevent t-igid body rotation

about the z-axis. These constraints simulate symmetry of half the span of a simply

supported beam geometry. Out of plane displacements are prevented on the perirneter of

the web.

2.5.5 Loads

Shearing forces were applied to the models by assigning two transverse (negative y

direction) loads at the right hand end, as shown in Fig. 2.5. Moment loads were applied

by applying two equal and opposite (x-direction) concentrated horizontal loads at the lefi-

Page 45: Stability of Castellated Beam Webs

iland end at the flange-to-web intersections (Fiy. 2.6). Thus sliear and moment could be

assigned in any desired combination.

Fig. 2.5 Finite Elernent Mode1

Page 46: Stability of Castellated Beam Webs

Fig. 2.6 Pure Bending and SheadMoment Arrangement

Page 47: Stability of Castellated Beam Webs

2.5.6 Buckling Analysis

The type of analysis to be performed in MSCINASTRAN is specified in the Executive

Case Control section in the input file using the SOL command with the CEND delimiter

to represent the end of this section.

Linear buckiing analysis is defined through SOL 105 command. Two loading conditions

must be defined in the case Control section. Subcase 1 will define the static load

condition applied to the system, and subcase 2 selects the method of eigen value

extraction method.

The equilibriurn equations for a structure subjected to a constant force may be written as

P l {u) = {Pl

where G is the stifniess rnatrix, u the displacement vector, and P the applied load vector.

To include the differential stiffbess effects, [Go] the differential stifiess matrix is

introduced that results fiom including higher-order ternis of the strain-displacement

relations (these relations are assumed to be independent of the displacements of the

structure associated with an arbitrary intensity of load).

Hence, by introducing q as an arbitrary scalar multiplier for another "intensity" of Ioad.

the equilibrium equation becomes,

( [G] q[GD] ){u*) = (qP) where u* is the modified displacernent

vector resulting fiom displacements under an intensity of Ioad, and fiom differentiaI

sti ffness effects.

By perturbing the structure slightly at a variety of Ioad intensities, the "intensity" factor q

Page 48: Stability of Castellated Beam Webs

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:

[G -tlG,I{v) = 0,

The solution is nontrivial. (q different fiom zero) only for specific values of q that would

make the matrix [G -qGD] singular.

The product of the first load intensity factor or the first eigenvalue q with the applied load

would give the fïrst buckling load of the rnodel, and the eigenvector <p, the bucicied shape.

The requirements for an eigen value solution in MSCMASTRAN are defined in the Bulk

Data Entry. By using the EIGB entry, and specifying a set identification number for the

model. the range of interests of eigenvalue limits is determined. Two methods of

eigenvalue extraction methods are avaiiable in the software invoked by the cornmands:

INV and S M . The S W method is an enhanced version of the iNV method. Lt uses

Snirm sequence techniques to ensure that ail roots in the specified range have k e n found.

I t is suggested that S W is a more reliable and more efficient method than the INV

method, and hence is used in al1 computations. PARAM entry is another statement used

to account for AUTOSPC command to constrain dl singularities on the stifiess matrix

as described in Section 2.5.4.

Limitations of SOLLOS required small deflections in the prebuckied c ~ ~ g u r a t i o n and

stresses to be elastic and linearly related to strains. The two conditions were fully

satisfied.

Page 49: Stability of Castellated Beam Webs

Buckling modes resulting from the analyses were examined carefully in each case.

Unrealistic buckling modes were sometimes O btained. for example buc kling on the

tension side of the beam under pure bending. and in each such case the associated

eigenvalue was negative. and was rejected. Under pure shear. the two identical symmetric

modes were associated with positive and negative eigenvdues of almost equal magnitude,

and in some cases the negative one was msrrginally iower than the positive one. The

lowest value was accepted.

2.6 Summary

In this chapter the severai methods of analysis used later in this thesis have been

described. Further details, especially of the FEM applications. are described when

particular applications are discussed in the following chapters.

Page 50: Stability of Castellated Beam Webs

CHAPTER THREE

LITERATURE: REVIE W

3.1 General

An investigation of previous literature on non-composite castellated beam tests was

conducted fiom which data was obtained in order to make comparisons between

experimental and theoretical results in later chapters. For each test beam, the section

properties, geometry and experimental arrangements were studied and relevant data are

summarized in tables at the end of this chapter.

3.2 Literature Review

Reviews on non-composite castellated beams have been extensively reported in the

Li terature. Ho wever, generally accepted design methods have not been established due to

the complexity of castellated beams and their associated modes of failure. An outline of

previous expenmental work on castellated beams is reported here with the objective of

describing only the main features of each investigation. The data and test results for the

beams described are the subject of detailed analysis in subsequent chapters of this thesis.

The test programs are described in reverse chtonological order.

Page 51: Stability of Castellated Beam Webs

3.2.1 Redwood and Demirdjiaa (1998)

Four castellated beams, two identical ones with four openings 10-5(a), IO-5(b). a third

with six openings (10-6) and a fourth with eight openings (10-7), al1 with identical cross

sectional properties. were tested. The main focus of the experiment was to investigate the

buckling of the web post between holes and to study any effects of moment-to-shear ratio

on the mode of failure. Simple supports and a central single concenuated load were used

for al1 specimens. Al1 beams were provided with bearing stiffeners at support and at load

points. Mean flange and web yield stress values were obtained from tensile coupon tests.

Based 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 al1 these

bearns. Bearn 10-7 is omitted fiom further consideration in this project, since interest is in

web buckling only. The buckling mode involved twisting of the post in opposite

directions above and below the mid-depth. Ultimate load values were given as the peak

test loads. Test conditions were then sirnulated by elastic finite element analysis. and

good 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 (Bantam

sections manufactured by Chaparral Steel Company) were tested. Six of these had 2 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

Page 52: Stability of Castellated Beam Webs

openings. Simple supports and a central single concentrated load were used for al1

specimens. Al1 beams were provided with bearing stiffeners at support and at Ioad points.

Average flange and web yield stresses were obtained from tensile coupon tests for each

size of beam.

The reponed ultimate strengths were based on peak load capacities of the bearns. Web

post buckling was observed in the failure of 10 cases. and in two cases. local buckling of

the tee-section above the openings subjected to greatest bending moments occurred. Two

lateral torsional buckling modes were also observed; these have been omitted tiom

further consideration shce interest is in web buckling only. FEM analysis was also used

to predict web-pst buckling load.

3.2.3 Galambos, Husain and Speirs (1975)

Four castellated beams fabrïcated fiom W 10x 1 5 sections ( 10 in deep. 1 5 pounds per foot

(see 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 of

analysis. Al1 beams were simply supported and were subjected to a concentrated load at

mid-span. The span and weld lengths were kept constant, but the depths were varied

based on different expansion ratios. Ultimate loads were recorded. but no further

discussion about the modes of failure was given.

Page 53: Stability of Castellated Beam Webs

3.2.4 Husain and Speirs (1973)

Bearns fabricated fiom twelve 10% 15 beams (alternative designation for W 1 OX i 5 ) were

tested to investigate the effect of hole geometry on the mode of failure and ultimate

strength of castellated beams. Specimens A-2, B-1, C and D were subjected to two

concentrated 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 depth

bearing stiffeners were provided (except for beams C and D where partial depth stiffeners

were used). The loads were based on the ultirnate load values obtained during the

experiments.

Specimens A-1, A-2, and B-3, failed by the formation of plastic hinges at the re-entrant

corners of the opening where both shear and moment forces are acting. As for Specimens

G- 1. G-2, with flanges of Canadian Standard S 16.1-94 class 1 section properties. and G-

3, a class 2 section, yielding of the flanges in the region of high bending moment iead to

flexural failure. The class section properties were calculated for some beams in an

anempt to investigzte if any local buckiing possibilities were present. Bearns 8-2, C, and

D failed prematurely due to web buckling directly under the point of load application.

Similar failure was exhibited by Beam B-1 that failed by web buckling under the

concentrated load before a Vierendeel mechanism had fonned. ïhus, beams B-1. B-2. C.

and D were omitted fiom m e r study.

Page 54: Stability of Castellated Beam Webs

3.2.6 Husain and Speirs (1971)

The main focus of this experiment was to study the yielding and rupture of urlded joints

of castellated beams. The experimental investigation consisted of testing sis simply

supported beams under various load systems. A single concentrated point load \vas

applied to b e m s E-2. E-3. F-1 and F-3 and two concentrated loads were used for beams

E-1 and F-2. Full depth-bearing stiffeners and sufficient lateral bracings wr.Etrt: pro~ided to

pre\.ent premature buckling. The reponed final results were caiculated on the ba is of

directI). measured yield and ultimate shear stress values. The measured shcar stresses

\iwe significantly higher than values which wvould have been expected tiom tsnsile

coupon tests. probably as a result of strain hardening. The prediction of' ultimate strengtli

based on web-post yield (see Section 2.3) can therefore be espected ro bc. wrp,

consenative. Sudden weld rupture accornpanied by violent strain energj- reIease \vas the

common mode of failure for all beams.

3.2.7 Bazile and Texier (1968)

T ~ v o series of beams. four HEAS60 and three IPE270 sections (for dimensions see Table

3.1 ) kvere tested to failure. The objective of the experiment was to de\.elop a tùrther

understanding of different beam characteristics and properties. geometn and espansion

ratios of castellated beams. The simply supported beams were tested under eight

unifonnly distributed concentrated loads. Three test loads. P l . P L and P3 wrtt reponed

to describe the different phases of the load-deflection diagrarn of each beam. Loads PI

Page 55: Stability of Castellated Beam Webs

and P2 define sudden changes in slope and P3 was the ultimate load. Flange and web

yield stresses were obtained fiom beam coupon tests and full depth stitTeners were

provided at support reaction points. Beams A, B and E failed under web buckling in the

zone of maximum shear. The beams F and G failed by lateral torsional buckling and were

thus omitted from M e r study 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 posts

of these two bems, using the colwnn strength formula of CSA (1994) assuming widths

equal to the maximum and minimum actual widths, bracket the uitimate test value of the

concentrated load. It is therefore evident that these were compression buckling failures

under the action of the concentrated loads acting directly above the unstiffened web-

posts. Since this mode is not k i 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, dl fabricated fiom the IPE300

rolled steel sections, were tested to destruction under two equal concentrated loads

applied at the third-span points. The experimental faiiure load was based on the

intersection of the tangent to the tinear part of the load vs. deflection diagram with the

tangent to the h o s t horizontal part of the curve. Measured yield stresses are not

reported. Calculations in the reference are based on the yield stress of the material, that is,

24 kglmm'(235 MPa), and it is later stated that yield stresses determined from umeponed

tensile tests were significantly higher than the above-mentioned value. Therefore, due to

Page 56: Stability of Castellated Beam Webs

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 forces

on the behavior of castellated beams under varying load conditions. The test arrangement

consisted of simply supported beams with fidl depth bearing stiffeners under load and

reaction points. Seven tests were perfonned which ranged from pure shear to pure

bending loading conditions. Load-deflection curves are given in the paper. From these the

ultimate loads and Ioads obtained fiom the intersection of tangents to the initiai linear

part 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 mid-

depth of the post between the first and second hole opening. Beam E2 was designed to

investigate the effect of pure moment, and was subjected to two concentrated point loads.

Failure of this beam however, was outside the central control section and was associated

with extensive yielding in the end zones experiencing both shear and moment forces. The

hole closest to the load was the most severely damaged. Web buckling was the mode of

failure of specimen E3 in the zone of maximum shear, under the two point loading

system. Specimen E4 was designed to study the effect of pure shear across the central

opening. The deflection curve demonstrates considerable strain-hardening, and web

buckling was the observed mode of faiiure. Beams L 1, L2, and L3 were tested under pure

bending moments. The first two were reported to fail by flexurai mechanisms. L3 was

also reported to fail by fiexural mechanism, however, lateral torsional buckling was also

associated with the failure mode.

Page 57: Stability of Castellated Beam Webs

3.2.10 Toprac and Cooke (1959)

Nine castellated beams fabricated fiom 8B10 rolled sections were tested to destruction.

The objectives of the investigation were to study the stniciural behavior in elastic and

plastic ranges. to study load carrying capacity and modes of failure, to compare observed

results with theoretical calculations, and to determine an optimum expansion ratio for

such bearns. Loads were applied at four concentrated points and failure loads were

reported as the ultimate loads. Well-defined yield stress values were obtained through

coupon tests and adequate bearing stiffeners were provided under reaction points.

Specimens A and C failed through excessive lateral buckling and are omitted from further

study. The iiltimate load of specimen B was recorded, but no fùrther details were given.

As 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 was

reached. yield at the top low moment hole corner and at web-post mid-depth was evident.

Yielding and buckling of the compression flange in the pure bending region was the

failure mode of Bearn E. Local buckling of the compression flange in the constant

moment region was dso the observed failure mode of specimen F; however, as the load

was hrther increased, the beam buckled laterally. A Vierendeel mechanism in the region

of highest shear was the mode of failure of specimen G. Specimen H, with a class 2

îlange section, failed through buckling of the compression fiange in the constant moment

region. Specimen 1, with a class 1 web tee stem section failed through a Vierendeel

rnechanism in the highest shear region.

Page 58: Stability of Castellated Beam Webs

3.2.1 1 ~ltfilliscb, Cooke and Toprac (1957)

The objective of the investigation was to study the structural behavior of castellated

beams both in the elastic and plastic ranges, and to study their strength and mode of

faiIure. Three joists fabricated from 10B 1 1.5 shapes with equal spans and simple supports

and with varying positions of two symmetrical concentrated loads were used. Varying

expansion ratio, beam depths, hole and web-post geometries were studied for each of

these tests. Test loads were reported as the ultimate loads obtained during the

experiments. Beam A was provided with full bearing stiffeners under each load. It failed

through extensive yielding of the tee section and local compression flange buckling in the

region of constant moment. The flange to width ratio of beam A corresponded to a class 2

section.

Bearn B consisted of three tests. In the first two, B1 and B2, loads were in the elastic

range in order to venQ theoretical stress and deflection analyses. The third test. B3,

involved loading to destruction, but was omitted fiom M e r study because of the

inadequacy of lateral bracing system.

Bearn C was provided with shon bearing stiffeners, (approximately half beam depth)

below the load points. The first two tests were in the elastic range and the third was

loaded to destruction. The fàilure mode of this beam involved yielding of the web at the

top low-moment corner of the opening in the shear span nearest the load application

point. followed by local buckling of the compression flange at the other end of the

opening. The flange had a Class 2 section properties. Yielding of the throat was also

noticed.

Page 59: Stability of Castellated Beam Webs

TABLE 3.1 Redwood & Demirdiian ( 1998)

TABLE 3.2.a Zaarour & Redwood ( 1996)

TABLE 3.2.b Zaarour & Redwood (1 996) (continued)

For ', refer to description of footnotes on page 5 1.

Page 60: Stability of Castellated Beam Webs

TABLE 3.2.c Zaarour & Redwood ( 1996) (continued)

TABLE 3.3.a Galambos Husain & Speirs ( 1975)

253.75 302.65 354.58 101.60 101 -60 101 -60 5.84 5.84 5.84 6.86 6.86 6.86 N.A. 1 53.40 153.40 N.A. 100.89 303-59 N.A. 335 -45 425.45 N.A. 39.9 59.3

333.43 333 -43 333.13

TABLE 3.3.b Galambos Husain & Speirs (1975) (continued) BEAiM

I

d,"

bi " l tu

4 il e

ljo"

s a

O h F,

H-3 P H-4 340.6 1 403 -3 5 101.60 101 -60 5 -83 5.84 6.86 6.86

152.40 152.30 176.58 302.5 1 425.35 425 -45 55.68 68 -3

338.67 333.43

Page 61: Stability of Castellated Beam Webs

TABLE 3.4.a Husain & Speirs ( 1973)

TABLE 3.4.b Husain & Speirs (1 973) (continued)

TABLE 3.4.c Husain & S~eirs ( 1973) (continuedl

Page 62: Stability of Castellated Beam Webs

TABLE 3.5.a Husain & S~eirs ( 197 1 1

TABLE 3.5.b Husain & S~eirs ( 197 1 ) (continued 1

TABLE 3.6.a Bazile & Tesier ( 1968)

BEAM d2 a

b;. a

L tfi l e h,," s a

O h F,

, . - -

F- 1 F-2 F-3 1

38 1 .O0 38 1 .O0 38 1 .O0 10 1.60 101.60 101.60 5.33 5.33 5 -3 3 6.83 6.83 6.83 50.55 50.55 50.55

254.00 254.00 254.00 247.65 217.65 347.65 60.00 60.00 60.00 248.2 1 248.2 1 248.2 1

Page 63: Stability of Castellated Beam Webs

TABLE 3.6.b B a d e & Texier ( 1968) (continued)

TABLE 3.7.a Halleus ( 1967) Series 1

TABLE 3.7.b Halieux ( 1967) Series 1 (continued)

Page 64: Stability of Castellated Beam Webs

TABLE 3.7.c Hallei (1 967) Series 2 (continued)

TABLE 3.7.6 Halleux ( 1967) Series 2 (continued)

TABLE 3.8 Sherbourne ( 1 966)

Page 65: Stability of Castellated Beam Webs

TABLE 3.9.a Tovrac & Cooke (1959)

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

TABLE 3.9.c Toprac gi Cooke ( 1959) (continued) BEAM

d, " b;. il 1, t,.il

eil

h,," a

( P h . F y b

F, tianaç '

G H 1 J 330.20 295.9 1 354.33 200.9 1 100.33 100.3 3 1 00.33 100.33 4.72 3.35 4.70 4.70 5.18 5.16 5-13 5.1 1 76.20 38.1 O 38.10 N..A

264.16 194.3 1 309.63 K..A. 4 16.56 370.5 1 385.83 N . A .

45 45 45 N.A. 296 .4 1 296.4 1 296.4 1 N.A. 296.4 1 296.4 1 296.4 1 S..i\.

Page 66: Stability of Castellated Beam Webs

TABLE 3.10 Altfillisch, Cooke & Toprac (1957) A B C . I

" Al1 dimensions are in mm. b Angle in degrees. ' Yield Stress F, in Mpa.

Page 67: Stability of Castellated Beam Webs

CHAPTER FOUR

RECONCILIATION OF ANALYSIS WTH TEST RESULTS

1.1 General

The results of the previous research work on castellated beams described in Chapter 3 are

compared in this chapter with the methods of anaiysis described in Chapter 2. Al1 shear

and bending moment loads are non-dimensionalized by dividing by the plastic shear or

moment capacity of the section to facilitate numerical cornparirom, and a governing

mode of failure is predicted. Correlations between test results to theory are then reported.

1.2 Comparative Data

The complete set of data for al1 78 bearns tested in the references of Chapter 3 are given

in Tables 3.1 to 3.10. Of these, 21 were eliminated fiom fiuther consideration because

they failed by modes other than those k i n g considered in this project. The remaining 57

beams are considered in this chapter. For reasons discussed below, more of these beams

had to be removed fiom consideration. For the remainder the predicted and measured

ultimate 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 and

horizontal 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 at

each hole in a beam, al1 holes must be considered independently, and the most critical one

Page 68: Stability of Castellated Beam Webs

for each failure mode must be identified.

Construction of the interaction diagrams representing plastic failure mechanisms was first

carried out. For the given beam arrangement shown below (Fig. 4. l ) , such a diagram is

demonstrated in Fig. 4.2.

m - D L I Q n

Figure 4.1 Test Arrangement of Beam Fi (Toprac & Cooke 1959)

Page 69: Stability of Castellated Beam Webs

P " hob 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 span.

with the two holes under pure bending being represented on the vertical axis as holes 7

and 8. The M N ratio at the centerline of each opening is used. For each opening,

theoretical predictions of V N , and M/M, are obtained from the intersections of the radial

lines with the interaction diagrarn representing Vierendeel and flexural plastic

mec hanisms.

On the diagram are also plotted the predicted îàilure loads corresponding to mid-post

yielding (V,,,N,,, and the buckling load predicted by FEM. The first of these is based on

Page 70: Stability of Castellated Beam Webs

Eqn 2.1 with the shear yield stress taken as f3F 443. This has a constant value for al1 web-

posts, and plots on Fig. 4.2 as a vertical Iine (WO lines corresponding to two values of P

are shown). Elastic FEM results are given. although it is recognized that this buckling

usually involves inelastic action. The influence of plasticity is considered in Chapter 5.

and is neglected at this stage as good results with elastic analysis have been reported by

Redwood and Demirdjian (1 W8), and initially the simplest sotution was sought.

Based on the typical FEM model arrangements of Section 2.5, a two-hole model with 8 16

elements, as shown in Fig. 2.5, was chosen to sirnulate the behavior of a web-post under

high shearing force. This represents a half-span of a beam with four holes, and was

subject to the restraints and other details outlined in Section 2.5. Only vertical loads were

used and the model is subjected to constant shear force with some small bending forces

which were considered to be negligible insofar as they wodd affect the buckling load

(see Redwood and Demirdjian 1998). These FEM results are plotted on the interaction

diagram as two points with ordinates representing the moments at the two holes used in

the model. Thus it is implicitly assumed that moment has negligible effect; this

assumption is examined in detail in Chapter 5.

4.3 Cornparisons

Al1 modes of failure for each hole in a beam are identifiable on a diagram such as Fig.

4.2. The triangies represent the loading (V and M) at each hole for a given load on the

beam (values given in fact correspond to the failure load). As load is applied to the beam.

Page 71: Stability of Castellated Beam Webs

these points c m be considered as expanding proportionally outward from the origin. The

critical hole is the one for which the plotted point first reaches the failure envelope. and

the mode would be identified by the part of the envelope attained. This may alternatively

be interpreted as identiQing the failure hole as that one for which the ratio of test load to

predicted load is a maximum.

The results shown in Fig. 4.2 are affected by the analysis for the horizontal web-post

shear yield mode which, as discussed in Chapter 2, is known to be quite conservative. If

these results (i.e. the vertical dashed lines) are ignored it can be seen that a flexural

mechanism failure is predicted at holes 7 and 8: hole 6 is almost at the point of failure in

a Vierendeel mechanism mode, and holes 5 and 4 in the same shear span are farther fiom

the failure surface. Holes 1, 2 and 3 are loaded well below the Vierendeel mechanism

load, and are far below the elastic buckiing load. The observed failure mode was that of

pure bending, as predicted by the above reasoning. If the horizontai yield mode had been

considered holes 1, 2 and 3 would have been criticai (with both predicted failure loads

lower than observed). It seems clear that in this case, the horizontal yield mode was not

relevant; in effect the verticai line should be shifted to the right to reflect a higher shear

yield stress than 1 . ~ S F 443.

There is some evidence that the effective shear stress at rnid-depth of the post at failure is

very high compared with the expected value F,/'/). Husain and Speirs ( 197 1 ) directly

measured the shear yield stress of notched specimens fabricated from ASTM A36 steel

(nominal F,=36 ksi (248 MPa)) and for a number of specimens the average value was

41.6 ksi (287 MPa). The tensile yield stress was not reported, and so some uncertainty

Page 72: Stability of Castellated Beam Webs

exists as to the enhancement above F 443 that this represents. However. if it is assumed

that the A36 web material had a real tende yield stress of about 53 ksi (365 MPa) (such

hi& values have been measured for A36 steels in the 1960-70 period, see Redwood and

McCutcheon (1969)) then the measured shear yield is 1.35 (=41.6+(53/'13)) times that

expected value of F J J ~ . Greater enhancement would occur if the estimate of the tensile

yield was too high. On this basis. it has been asswned throughout that the effective shear

yield stress at the mid-depth of the posts is 1.35 times ~ 4 d 3 . Thus the factor P is taken as

1.35. In the example of Fig. 4.2, it appears that even this enhancement is insufficient to

reflect the effective shear in the test beam.

Following the above procedure, test-to-predicted Ioad ratios were computed for each test

beam. Certain tests had reported maximum test loads, while others derived their failure

loads fiom the intersection of tangents of the two curves of load vs. deflection diagrarn.

Whenever applicable, both reported loads are used for comparisons.

4.4 Discussion

In generai, the numencd results indicate good correlation with test results. .Most of the

cases with poor correlation, as indicated in Table 4.1, are those for which yield stress

values were not given, and nominal values have been used. These beams are identified by

asterisks, 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 coefficient

of variation (COV) of the test-to-predicted ratios for al1 other beams are 1.127 and 0.225.

Page 73: Stability of Castellated Beam Webs

These are based on the ultimate loads; if the tangential Ioad is used where available. these

numbers becorne 1 .O86 and 0.195.

Of the 57 beams listed, approximately half (29) had the mode of failure predicted

correctly. Of the others, some test modes were not defined (4), in others modes are

identified as flange buckling when a yield mechanism may have been imminent or

already developing ( S ) , in others. the uncertainty concerning the shear capacity of the

web-post affects the prediction. and for most of the remaining cases, there were only

small differences between the failure load for the predicted mode and that of an

alternative mode.

Table 4.1 Summary of Test and Theoretical Predictions

Reference

Redwood & Demirdjian (1998)

Zaarour & Redwood (1996)

Beam

10-5a

10-Sb

10-6

8- 1

8-2

8-3

8-4

10- 1

10-2

Tedtheory Ul timate

Loads 1 .O43

1.137

1.132

1.105

0.793

0.9 15

0.646

0.967

0.847

Testhheory Tangential

Loads

Mode Test

Web Buckling

Web Buckiing

Flange and Tee Buckling

Shear Mechanism

Web Buckling

Shear iMechanism

Web Buckling

Web Buckling

Web Buckling

of - failure ,

Theory

Web Buckling

Web Buckling

Web Buckling

Shear -Mec hanisrn

S hear Mechanisrn

S hear iMec hanism

Web Buckling

Shear ~Mechanism

Web Buckling

Page 74: Stability of Castellated Beam Webs

Re ference

Calam bos, Husain & Speirs (1975)

Husain & Speirs ( 1 973)

Husain & Speirs (1971)

Beam

10-3

1 0-4

12-1

12-2

Test'theory Tangentiai

Loads

' Testltheory Ultimate

Loads 0.950

0.8 13

0.953

O -966

13-3

1 2-4

H-2

H-3

H-3 P

H-4

A-1

A-2

B-3

G- 1

G-2

G-3

E-1

E-2

4

\Vs b Buckling

Web Buckling

N.A.

N.A.

N.A.

N..%

S hear Mechanism

S hear Mechanism

S heor Mechanisrn

Shear Mechanisni

S hear Mec hanism

Shear Mechanism Mid-Post Y ielding Mid-Post Yielding

Mode Test

Web Buckling

Web Buckling

Web BuckIing

LVeb Buckling

U'c b Buckling

L\'e b Buckling

S liear blechanism

S liear .Meclianism

Shsar kleciianisrn

Shear Mechanism

S hear iLlechanisrn

S hear .Mecliariisrn

S h e u Mechanism

Mid-Pest Yielding Mid-Post Yielding

Sliear Mschanism

Mid-Post Yielding Mid-Post Y iclding

of failure Theon-

Cb-s b Buckling

L!*e b Buck1 ing -

Shear Mechanism

Shear .Meclianisrn

0.857

0.840

1.001

1 .O87

1 .O62

1.186

1.136

1.259

1.196

1.314

1.146

1.208

1.960*

1.81 l*

1 .O5 1

1.158

1.137

1.173

0.990

1 .O46

Page 75: Stability of Castellated Beam Webs

T

Reference

Bazile & Texier ( 1 968)

S hear iMechanism

S hear iMechanism

3

5

Series 2

Sherbourne (1965)

Testltheory Ultimate

Loads 1.809*

1.397*

2.135*

1.530*

1.3 14

--

1.1 16

0.942

Beam

E-3

F- 1

F-2

F-3

A

-

B

E

Halleux (1967) Series 1

2.090* *

1.504**

1

3

3B

5

E- l

E-2

2.82 i **

3 .OOO* *

1

IB

TesU'theory Tangent ial

Loads

- ..

Flesural Mechanism

FlesuraI blschanism

2.854**

2.181**

2.058**

1.576**

1.503

1.630

Slierir blechanism

S hear Mechanism

Mode Test

Mid-Post Y ielding Mid-Post Y ielding $1 id-Post Y ielding

Flesural Mec tianisii~

Web Buckling

. -- -.

We b Buckling

We b Bucklitig

Sliear Mechrinism

S lierir klcchanism

O!' fai 1 ure Theor?.

~Mid-Post Yielding [Mid-Post Yielding Mid-Post Irit.lditig

S liear blcchanism

S liear - Mechanisin -- -

Shear Mechanism

Flesural h~lèclianism

1.226

1 2 8 4

Sherir h4echanisni

S hear Meciianism

S hear Mechanism

Flesural Meclianism hl id-Post Y ielding Mid-Post Y ielding

S I i a r .Mecfianism

S fierir ~Mechanisrn

Sliear blechanism

S liear Meclianisin

S Iierir LMechrinism

S hear iblechrinism

Page 76: Stability of Castellated Beam Webs

*Minimum yield stress values of the conesponding beams were defined. The nominal

yield stress of 248 MPa (36ksi) was used to compute these ratios.

Re ference

Toprac & Cooke (1959)

Altfillisch, Toprac & Cooke (1957)

** Actual yield stress values of these beams were not reported. Minimum yield stress

Beam

E-3

E 4

L- 1

L-2

L-3

D

E

G

H

1

A

C

Testhheory Ultimate

Loads 1.700

1.613

1 .O63

1 .O43

1.1 13

0.956

1.277

1.425

1.218

1.808

0.887

1.122

value of 235 MPa (24kg/rnm2) was used to compute these ratios.

Testltheory Tangential

Loads 1.423

1.442

1 .O63

1 .O43

1.1 13

Mode Test

Web Buckling

Web Buckling Flexural

Mechanism

Flexurai Mechanism

Flexural Mechanism (L.T.B?) Flange

Buckling

Flange Buckling

S hear Mechanism

Flange Buckiing

Shear Mechanism

Flange Buckling

Flange Buckling

of failure Theory

S hear Mechanism

S hear hl echanism

Fiexural Mechanism

Flexural .Mechankm

Flexural Mechanism

Flexwal Mechanism

Shear M e c h a ~ s m

Shear Mechanism Mid-Post Yielding

S hear Mechanisrn

Flexural Mechanisrn

Shear Mec hanism

Page 77: Stability of Castellated Beam Webs

CHAPTER FIVE

GENERALIZED ANALYSIS AND DESIGN CONSIDERATIONS

5.1 General

In the FEM analyses considered so far. the only loading condition treated approximates

pure shear. and furthemore the mode1 has been limited to one cornpx-ising only two

openings. In this chapter more cornplete models are examined and moment-to-shear ratios

\ . a~ - ing from pure shear to pure bsnding are considered. In addition. the analysis has dealt

only 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 in

section 5.3 modeIs containing up to four openings are considered under pure shear as tvell

as pure bending. The effect of moment-CO-shear ratio is then considered for four test

beams representative of a wide range of castellated beam geometries. These results are

used to establish a generai forrn of interaction diagram to define elastic buckling loads of

castellated beams under any shear to moment ratio. Having established this form. in

sections 2.3-5.7 a parameter study deriving web buckling coefiicients co~er ing a wide

range of geometries is performed. The use of these elastic results. in conjunction uith the

plastic analyses is examined in section 5.8 with the aim of developing inelastic buckling

equations. These are then compared with relevant test results.

Page 78: Stability of Castellated Beam Webs

5.2 Loading on General Models

To study the behavior of models under various shear to moment ratios, several

MSCMASTRAN elastic fmite element buckling anaiysis nins were necessary. To create

pure shear and pure bending forces, as well as various V M ratios. different loading

patterns 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 the

model, the two vertically concentrated static loads (Fig.5.I) used in the analyses

described in section 2.4 must be supplemented with forces producing a counter-clockwise

couple. 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, as

shown in Fig. 5.2. In the several models considered below these forces could be adjusted

to provide pure shear at any desired point (e.g. the hole centerlines). Similarly, with the

vertical loads removed, a clockwise couple applied by such horizontal forces on the lefi

end of the beam was used to simulate pure moment conditions, as shown in Fig. 5.3. Any

combination of shear and moment forces could be generated by cornbining these verticai

and horizontal loads in m y desired proportion.

The deformed shapes under vertical loads and under pure shear conditions as shown in

Figs. 5.1 and 5.2, demonstrate the same buckling pattern of the post, with slight rwisting

of the flange to accomodate the double curvanue bending efiect over the hieght of the

post. Under pure bending conditions however, the region above the middle opening

resisting the compression force is buckled, with large twisting of the flange to

accomodate the buckled shape.

Page 79: Stability of Castellated Beam Webs

Fig. 5.1 Two Hole FEM Mode1 under Vertical loads onl?.

Page 80: Stability of Castellated Beam Webs

Fig. 5.2 Three Hole FEM Mode1 Under Pure Shear Forces

Page 81: Stability of Castellated Beam Webs

Fig. 5.3 Three Hole FEM Model Under pure Bending Moments

Page 82: Stability of Castellated Beam Webs

5.3 Elastic Buckling Interaction Diagram

Due to the presence of the stiffener on the left end and the applied constraints on the right

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 two hole model. To ensure

there is no such restmint, models consisting of three and four holes were also

investigated. Both pure bending and pure shear forces were considered for two. three and

four hole models, al1 under the same boundary and loading conditions. These analyses

were 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) and 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 of

castellated beam geometries.

Resuits for pure bending are expressed as the beam buckling moment as a ratio of the

plastic moment and are given in Table 5.1. The three and four hole models produce

similar buckling moments and these were lower than for the two hole rnodel.

1 Beams 1 2 Hole Mode1 1 3 Hole Mode1 1 4 Hole Mode1

I 1 I I

Table 5.1. Summary of Results Under Pure Moment Forces.

Similar numerical simulations were conducted to investigate the behavior of 2. 3. and 4

hole models under pure shear conditions (Fig. 5.4). The chailenge here was to determine

at which hole zero moment forces should be edorced to produce the pure shear condition.

Page 83: Stability of Castellated Beam Webs

As indicated in Table 5.2, several analyses were done to create the zero moment force

condition 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 the

two interior holes had imposed the zero moment conditions and again only minor

differences are evident, with no trend discernible between the rnodels with different

numbers of holes. The differences in the critical buckling shear loads of 2, 3. and 4 hole

rnodeIs were less than 3%.

In view of these results and to be consistent in subsequent analyses, the three hole model

was chosen to represent al1 further FEM analyses in this study. It sbould be noted that

M = O Beam

10-3 B- 1

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

Table 5.2. Surnmary of Results Under Pure Shear Forces.

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

3 Hole Mode1 2 Hole Mode1

satisfactory for that application.

at hole 1 Vcr (kW 39.84 8 1.40

at hole 1 Vcr (W 39.09 83.25

4 Hole Mode1 - at hole 2

Vcr OcN) 39.00 83 .25

at hole 2 Vcr (W 38.62 82.5 8

at hole 2 Vcr (kW 40.09 82.46

at hole 3 Vcr (W 38.66 81.36

at hole 3 Vcr (W 39.99 82.42

Page 84: Stability of Castellated Beam Webs

Fig. 5.4 Three and Four Hole FEM Models.

Page 85: Stability of Castellated Beam Webs

A complete interaction diagram for elastic buckling was obtained for each of the four

selected beams using the three hole model. The results are shown in Figures 5.5. 5.6. 3.7

and 5.8 (the two ordinates of the elastic FEM results plotted for each V N p ratio refer to

the M N ratio for the first two holes of the model).

I t c m be seen that under pure bending, plastic failure occurs at much lower loads than the

buckling loads. Under pure shear, buckling loads may range from much lower to much

higher vatues than the plastic failure load. The resuits shown on these diagrams will be

discussed below.

Interaction Diagmm Beam 103

Figure 5.5 Zaarour and Redwood (1 996)

Page 86: Stability of Castellated Beam Webs

Interaction Diagram Beam G-2

Figure 5.6 Husain and Speirs ( 1973)

Interaction Diagnm Beam F 3

-, , , Yield Theory

BasUCFRd

n = 2

, , , , Y RH Theory

Bastic FEM

n = 2

Figure 5.7 Husain and Speirs ( 1 97 1 )

Page 87: Stability of Castellated Beam Webs

Interaction Oiagnm Beam B

Figure 5.8 Altifillisch, Cook and Toprac ( 1 957)

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

with Mo and V, corresponding to pure shear and bending conditions respectively. Several

dif'ferent values o f n were exarnined. The curve found to best represent the FEA results

for the fuIl range of M N was fouid to correspond to n = 2. In this way. given -Mc, and V,,

values, a relationship defining the buckling behavior under any M N ratio is established.

Page 88: Stability of Castellated Beam Webs

5.1 Parameter Study

Having established a generai expression defining the elastic buckling beha~ior of

castellated 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 buckling

coeftkients under pure shear and pure bending conditions. The beams were designed and

selected to present various ratios and proportions of castellated bearn geometries. The

relevant parameters were considered to be the ratio of hole height to minimum web-post

width. hJe. and the ratio of minimum web-post width to web thickness. eh,,. Because of

the wide range of possible beam and castellated hole geornetries. the pararneter study had

to be of limited scope. and thus the following computations are restricted to castellations

witli a hole edge siope of 60° to the horizontal.

5.1 Previous Parameter Study

In a previous study by Redwood and Demirdjian (1998). a pararneter study to tind the

elastic buckling loads under high shear loading was carried out that incorporated a \\.ide

range of bearn characteristics. The study assumed elastic behavior throughout. The mid-

post weld was assumed to be Full penetration and had the sarns thickness and material

properties as the web. The flange was included in the mode! because of its importance in

restraining web rotations, but conservative estimates of flange dimensions were used for

the general case. These assumed that the flange was only as thick as the web. and the

f'lange width was that of a Canadian Standard S 16.1-94 class 3 section. Thus:

Page 89: Stability of Castellated Beam Webs

V '

Two series of beams were considered, each with a constant hole height-to-beam depth

ratio. For each senes. the relevant parameters were selected to be the hole height to

minimum web-post width, hJe, and the ratio of minimum web-post tvidth to web

thickness, e/t, The castellations had hole edge slopes of 60" to the horizontal. without

intermediate plates at mid-height. This angle is representative of present industry standard

cutting angles.

The FEM model consisted of two holes and was identical to that used for the analyses

described above in Chapter 4. Thus loading was primarily a sliear load. with tu-O vertical

loads applied at one end at the level of the flanges. with the model supported venically by

a point load at the other end.

Ln the study the critical horizontal web-post shear force dong the welded joint was found

using FEM. and then the corresponding vertical shearing force on the beam was found.

lncorporating the principal parameters by writing horizontal shear force at buckling as

a non dimensional shear buckling coefficient k was derived as

X: =

The tlnite elernent analysis gave the ratio of shearing force in the web post to the vertical

Page 90: Stability of Castellated Beam Webs

shear on the beam, VJV. The product of the ratio V,N and the vertical shear force to

cause buckling gave the criticai horizontal shear force V,,,, in the web-post at the welded

4, S joint. V,,, was then related to vertical shear V through - = . (Eqn 2.3) derived " d x - 2 ~ $

from the free body diagram of Figure 2.2. where y, defines the line of action of the

longitudinal force resultant acting in the the tee section. which was taken as being at the

crntroid. This was verified by comparing this value with that given by the FEM for the 27

beams used in the parameter study. An average ratio of 0.983 with coefficient of variation

of 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 Buckling 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

Page 91: Stability of Castellated Beam Webs

-using equation 5.2 to find the horizontal shear in the web post

-using equation 2 -3 to uansform V,,, 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 for

a wide range of beam geometries c m be detemined.

5.5 Shear Buckling Coeff~cients (k.)

The previous parameter study was refined in the current research to correspond to

buckling 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 current

research the selected mode1 consisted of three holes as FEM results revealed its better

performance under bending moment: although no irnprovement was noted for pure shear.

consistency between models for the two load cases was considered desirable. The flange

dimensions assumed were modified so that the width was based on the assumption of ri

2(l45), Class 1 section. i.e. 6, = . (Clause 1 1 2. Canadian Standard S 16.1-94) ~vhere F,

E \vas taken as 350 MPa. This reduced the flange widths, making the tlange restraint

slightly more conservative. Narrower flange widths would make al1 cases consenfative as

compared to class 3 section. which was found to be slightly unconservative for some

compact sections.

Under pure shear conditions, two vertical forces were applied on the nglit end at the level

of the flanges and two horizontal counter clockwise coupling forces were applied on the

Page 92: Stability of Castellated Beam Webs

lefi end at the flange to web intersecting nodes to counter the overturning etlèct of the

vcrtically applied forces. Thus there were no bending moments at the centre of the span

(Fig. 5.2).

Fig. 5.10 shows the results of the analyses for web-buckling coefficient k, due to pure

shear. There were minor variations between the results of the new and the previous studp

due to the minor modelling changes. For beams with hJdZ - = 0.5. el\, = 15 is plotted on

the curves to demonstrate the slightly greater dependency of e/4, than evident in the

previous study. Furthemore. there are minor differences between the shape of the cunes

for h,/d, =O.S. From the FEM studies of different models in Section 5.3 differences up to

3% can be expected between the hvo and three hole models. and this together with the

flange modelling change explains the differences between the results shown in Figs. 5.9

and 5.10.

k" curves

Figure 5.10. Modified Pure Shear buckling coefficent Curves

Page 93: Stability of Castellated Beam Webs

5.7 Flexural Buckling 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. and Taking a,, = - S

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

defined by

k x ' S impli @ing by incorporating into kb

[ 2 ( 1 - P ' )]

This tlexural buckling coefficient. kb. is given in Fig. 5.1 1 for a given variety of

castellated beam geometries. Almost constant kb values are maintained in the h Jdo - = 0.74

unti1 the lines curve downward, indicating that hole height to minimum width (h,/e) ratio

has very little effect on the overall beam buckling behavior under pure bending forces.

The k, values Vary less than the k, curves, indicating that the flexural buckling load is not

sensitive to the ratio of hole height to minimum width (hJe). While camparing the two

Page 94: Stability of Castellated Beam Webs

series of bearns, larger buckling coefficients under pure moment conditions were found

for the senes of bearns Mth larger tee sections hJd,=û.5. but the behavior was reversed

under pure shear conditions, where beams with lower tee sections with hJd,=0.74 had - higher k, coefficients.

Thus based on a given beam geometry, the critical moment to cause elastic buckling is

simply caiculated using equation (5.5).

I

Figure 5.11. BuckIing Coefficient Curves Under Pure Bending Forces

5.7 Effect of Inelasticity on Ultimate Strengîb

Since buckling usually involves inelastic action, the influence of plasticity is considered

in this section to improve the already mentioned methods of analysis and derive general

expressions incorporating both elastic and inelastic buckling actions.

Page 95: Stability of Castellated Beam Webs

The construction of interaction diagrarns for elastic buckling can now be performed for

any beam with 60" openings, and follows the procedure used for the four bearns as

discussed in Section 5.3. Elastic buckling values of shear (V,) and moment (M,) can be

computed from the k, and kb curves. By dividing the resuits by the plastic shear and

moment capacities of the section, such a diagram can be plotted on the sarne a.xes as the

yield mechanism interaction diagram. (see Fig. 5.12)

Interaction Diagnm 6eam 6-2 Huaain & Speim (t973)

Figure 5.12 Elastic and Plastic Interaction Diagrams

On this diagram, radial lines fiom the origin for each hole of the test bearns were then

drawn and from each line a plastic and elastic buckling shear capacity is obtained at the

intersection points. For each test beam, the two governing shear values were thus

obtained. the plastic mechanism and elastic buckling shears. V,, and V,,.

Page 96: Stability of Castellated Beam Webs

To obtain an estimate of the ultimate shear load of the test beams which incorporates the

possible interaction of elastic buckling and yielding failure modes. the following two

cases were considered:

From equations for inelastic lateral buckling of beams (Clause 13.6. CSA 1994).

( Oc:) . If M. is replaceci by V,, and My by V . , we c m wï te .VI, = 1.15 M, 1-

Alternarively, fiom colvmn strength equations (CS& 1994) C , = AF, + A'" 1;. the

following expression is proposed,

y, = y,, (1 + k2")-;

where h. is now interpreted as and n is a coefficient based on fitting to test results.

The equations were then plotted and compared against acnial test results for the 60"

casteilated beams (summary of results is given in Appendix C). To plot the results in a

non dimensional form wtiile maintaining consistency, it was convenient to divide Vu by

V,, , as indicated in Fig. 5.13.

Based on the results of 17 test beams with 60' holes and relevant failure modes. both

equations 5.6 and 5.7, with n taken as 4.0 in the latter, were found to provide similar

Page 97: Stability of Castellated Beam Webs

predictions of the test results. The following statistics apply to the two predictor

equations

For these 17 beams the simplified approach taken in Chapter 4, in which the predicted

strength was taken as the lower of the yield strength and the elastic buckiing (FEM)

strength, produced a mean of 1 .O96 and COV of 0.170. The increased mean value for the

two equations is expected, since both will predict a lower value than the lowest of the

yield and elastic buckling strengths. It should also be noted that for use in equations 5.6

and 5.7, the elastic buckling strengths were computed using the generalized buckling

interaction equation 5.1, whereas the computations in Chapter 4 were based on exact

rnodeiing of each bearn. The lower COVs represent an improvement in the prediction if

equations 5 .O and 5.7 are used.

As s h o w in Fig. 5.13, the four bearns with of about 0.5 reported by Sherbourne (1 968)

show significant overstrength compared with the predictions. The reason for this is not

clear. but it may be noted that the actual beam cross-section dimensions were not given.

and nominal values have been used in the calculations.

COV 0.137 0.148

TesVPredicted Eqa. 5.6 Eqn. 5.7

Table 5.3 Statistical Results

Mean 1.1 13 1.166

Page 98: Stability of Castellated Beam Webs

VuNpl, MuIMpl Vs Lambda

1

Lambda

- -

Figure 5.13. Cornparison of Test Results With Proposed Expressions

Page 99: Stability of Castellated Beam Webs

CHAPTER SIX

CONCLUSION

6.0 Conclusion

The objective o f this research prograrn was to study the failure of castellated bean~s with

particular emphasis on web-buckling. Several theoretical rnethods predicting fom~ation of'

plastic mechanisms, yielding at mid-depth o f web-posts and eiastic buckling anafyses

u-ere correlated with the results of a number of physical tests of castellated beams

reported in the Iiterature.

Since web buckling usually involved inelastic action. the effect of plasticity was

considered in conjunction with elastic FEM results. to modify the theoretical nlodeIs used

initially.

.4 parameter study for a wide range of castellated beam geornetries rias performed to

deri\*e elristic web buckling coefficients under pure shear and pure bending forces. These

results established elastic buckling interaction diagrams. For any given M N ratio. results

obtained from elastic and plastic interaction diagrams were established.

The following remarks on the behavior of castellated beams are based on the several

theoretical models used incorporating both elastic and plastic analyses. and their

cornparisons with physical test results.

- Results obtained from the interaction diagrams based on plastic anaiysis used to predict

84

Page 100: Stability of Castellated Beam Webs

shear or flexural mechanisms were found to give generaily satisfactory predictions. This

diagram is designed based on the properties of a given beam. However, it does not

account for yielding of the web-pst, or web-buckiing.

- Y ield stress developed at the minimum horizontal width of the mid-post, equation 2.3.

was found to be conservative. A factor of P = 1.35 was applied to the sheaf yield stress to

account for the stta in hardening eEect expected to be developed at this section. Much

higher failure loads were then obtained compared with those given by the initiai stress

limit equation, and this led to more realistic resuits.

- Elastic buckling analysis with FEM models could be correlated with experimental

results, and therefore was used to perfonn various parameter studies. However. it was

considered necessary to take into account the effect of plasticity on the buckling loads. To

do this, the following steps were taken:

- Given the eiastic criticai buckling loads under pure shear and pure

bending (V,, b) loads, a curve of shape (MMO)" + (VN,)" = 1 with n=2

was fitted to define the buckling loads under any VA4 ratio.

- A parameter study was performed to denve the buckling coefficients

under pure shear and pure bending conditions covering a wide range of

castellated beaxn geometries. This study in conjunction with the elastic

Page 101: Stability of Castellated Beam Webs

FEM buckling curves. gave the elastic buckling loads of a variety of

castellated beams under any M / V ratio.

- Expressions incorporating both elastic and inelastic behavior of web

buckling gave better approximations of the buckling loads. with

coefficient of variations fiom O. t 90 to O. 137.

- The design considerations and computations incorporating the effect of elasticity and

plasticity on the buckling loads is limited to 60U castellated beam geometries. Extension

to other beam geometries is desirable.

Page 102: Stability of Castellated Beam Webs

REFERENCES

Aglan. A.A., and Redwood, R.G. 1974. Web buckling in castellated beams. Proc. Instn. Civ. Engrs, London, U.K., Part 2, Vol. 57, pp 307-320.

Altifillisch, M.D., Cooke, B.R., and Toprac, A.A., 1957. An investigation of open web expanded beams. Welding Research Council Bulletin, Series No.47. pp 77s-88s.

Bazile, A., and Texier, 5.1968. Essais de poutres ajourées (Tests on castellated beams). Constr. Métallique, Paris, France, Vo1.3, pp 12-25.

Caffiey, J.P., and Lee, J.M. 1994. MSCMASTRAN: Linear static analysis user's guide, V68. The Macneal-Schwendler Corporation, Los Angeles, California, USA

Canadian Institute of Steel Construction. 1995. Handbook of steel construction, 2nd edition. Universal Off'set Limited, Markham, Ontario, Canada

Galambos, A.R., Husain, M.U., and S p i n W.G. 1975. Optimum expansion ratio of castellated steel beams. Engineering Optimization, London, Great Britain, Vol. 1. pp 213- 22s.

Halleu, P. 1967. Limit anaiysis of castellated steel beams. Acier-Stahl-Steel, 325, 133- 144.

Husain, M.U., and Speirs, W.G. 1971. Failure of castellated beams due to rupture of welded joints. Acier-Stahl-Steel, No. 1.

Husain, M.U., and Speirs, W.G. 1973. Experiments on castellated steel beams. J. American Welding Society, Welding Research Supplement, 52:8, pp 329s-3423.

Kerdal, D., and Nethercot, D.A. 1984. Failwe modes for castellated beams. Journal of Constructional Steel Research, Vol. 4, pp 295-3 15.

Megharief, J.D. 1997. Behavior of composite castellated beams. M. Eng. Thesis. Department of Civil Engineering and Applied Mechanics, McGill University.

Raymond, M., and Miller, M. 1994. MSC/NASTRAN: Quick reference guide, V68. The Macneal-Schwendler Corporation, Los Angeles, California, USA.

Redwood, R.G. and McCutcheon, J.O. 1969. Beam tests with unreinforced web openings, Journal of the Structural Division, ASCE, Vo1.94, No.ST1, 1-1 7.

Page 103: Stability of Castellated Beam Webs

Redwood, R.G. 1968. Ultimate strength design of beams with multiple openings. Preprint No. 757, ASCE Annuai Meetings and National Meeting on Structural Engineering. Pittsburgh, Pa, U.S.A..

Redwood. R.G., and Cho, S.H. 1993. Design of steel composite beams with web openings. Journal of Constructionai Steel Research, 25: 1&2.23-42.

Redwood R.G., and Demirdjian S. 1998. Castellated beam web buckling in Shear. Journal of Stmc?ural Engineering, American Society of Civil Engineers, 124(8): 1202- 1207.

Sherboume, A.N. 1966. The plastic behavior of castellated bearns. Proc. 2"" Commonwealth Welding Conference. Inst. Of Welding, No. C2, London. pp 1-5.

Toprac, A.A., and Cooke, B.R. 1959. An experimental investigation of open-web beams. Welding Research Council Bulletin, New York. Series No.47, pp 1 - 10.

Ward, J.K. 1990. Design of composite and non-composite cellular beams. The Steel Cofistniction Institute.

Zaarour, W.J. 1995. Web buckling in thin webbed castellated beams. M.Eng. Thesis. Department of Civil Engineering and Applied Mechanics. McGill University.

Page 104: Stability of Castellated Beam Webs

APPENDIX A

Finite Element Input File

This Appendis contains a sample input file to construct the 2 hole Finitr: Element mode1

and perforrn Elastic Buckling Analysis.

Page 105: Stability of Castellated Beam Webs

S ........................................................ $ - : A r E X E C U T I V E C O N T R O L S E C T I O N S !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! S s Elastic Buckling analysis of "Castellated Beam" s 2 Hole !Mode1 of reference Beam 10-3 (Zaarour and Redivood ( 1996)) S S S SOL 105 Tl M E=9OO CEND S S .............................................. S. B:. C A S E C O N T R O L S E C T I O N S .............................................. S TITLE = beam 10-3 S SET 1 = 1.73.78.83.88.93,16 1.162.163,164.165.238.233.238.253.258.333.348. 3 73.378.427.428.429.430.43 1.499.504.509.5 14.5 19,664,669,674.679.683.769. 774.799.803 S S ECHO = NONE FORCE = I SPC = I O SPCFORCE = ALL

STRESS(PL0T) = ALL DISPLACEMENT(PL0T) = ALL

S S SUBCASE I SPC = 10 LOAD = I O DlSP = ALL FORCE = ALL S SUBCASE 2 SPC = I O METHOD = 100 FORCE = ALL DlSP = ALL S S BEGIN BULK PARA M.POST.0 PARAM.KfiROT. 10000.0 PARAM.AUTOSPC.YES EIGB. IOO.S1NV,-5.0,5.0,.3,3.,+EIGB +EIGB.MAX S S THIS SECTION CONTAINS BULK DATA FOR SE O

Page 106: Stability of Castellated Beam Webs

S S GRID 1 O 0.0 0.0 0.0 O GR!D 2 O 28.956 0.0 0.0 O GRID 3 O 339.34 0.0 0.0 O GRlD 4 O 368.3 0.0 0.0 O S . s*************************************** s The coordinates for 1057 grid points are defined. s * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S .

GRlD IO57 O 705.56 12- 185.99 -35.3 1 O

S 8 16 elements are thus defined through grid points s************************************

CQUADJ 816 Z 867 617 1033 1057 S S THIS SECTION CONTAINS THE LOADS. CONSTRAINTS. AND CONTROL BULK DATA SENTRIES S s S MATI. 1.200000 ... 0.333333 MAT I .2700000...0.333333 M A T 1 .;.2OOOOO...O.333 333 S S PSHELL. 1.1.3.6068.1 PSHELL.2.2.4.445,2 PSHELL.3.3.9.525.3 s S FORCE. 10.609.,5000..0.0.-1.0.0.0 FORCE. 10.6 17..5000..0.0.-1.0.0.0 S S SPC 10 17 235 SPC 10 609 13 SPC 10 617 13 SPC I O 9 3 SPC I O 10 3 SPC I O 115 3 SPC 10 116 3 SPC 10 117 3

Page 107: Stability of Castellated Beam Webs
Page 108: Stability of Castellated Beam Webs

SPC IO SPC IO SPC 10 SPC 10 SPC I O SPC 10 SPC 10 SPC IO SPC 10 SPC IO SPC 10 SPC IO SPC 10 SPC 10 SPC 10 sec 10 SPC IO SPC IO SPC IO SPC I O SPC 10 SPC 10 SPC 10 SPC 10 SPC 10 SPC IO SPC 10 S ENDDATA

Page 109: Stability of Castellated Beam Webs

APPENDIX B

Detai led Test-to-Theory Results

This Appendix contains detailed Test-to-Theory computations for al1 the beams Iisted in

Table 4.1. For each test beam. each hole until mid-span is studied. Al1 results are

transformed to shear and moment forces, and are non-dimensional. Reported ultimate test

load (V ,,,, N p ) and (M,,,/M,), elastic FEM buckling (VJV,). Shear mechanism (Vl,N,).

yielding of the horizontal joint (VyhNJV,), and flexural mechanism (M,,,/M,) ratios are al1

calculated. Ratios of test results to the predicted failure modes are then computed. aiid

maximum ratio on each row is calculated. The predicted failure mode is derived based on

the ratio selected by the maximum of ail the ratios of Test-to-Theory on each row.

Page 110: Stability of Castellated Beam Webs

, ' , : 1 : El E, i 1

: ai"/ Pi 3E: 8 18 i a . m'cv Q D I

. % , ' W ~ W ! F hl'-: 3; * kib! h i ? ! : o i p 1 s i O :o. 2: k k ? ,

si , ! . , SI i ; di 0 0 O . I l !

I g ~ j m Pi . , ,m - 1 $ ! Cr) C r ) I i ' (O El qIq1 . E ! 9 '9 1 c 3 - h !

>-: ; ' 0 , O I . si , . O O ' > El 0 "!'" o 1 O

Page 111: Stability of Castellated Beam Webs
Page 112: Stability of Castellated Beam Webs

-

C I 5 m' ln in.- a0 o o l m a01 a X X;5; C n Q i C n a Q )

0 0 0 0 0 "'v)!? rr, ,

al * ml* * L " I c > ~ I ~ ? ~ 5 O1OiO'Ol 9' , Of00O 5 0 0 0 0 0

m b V) hl! - QuaD hl 0 h - r ~ " (DJb hl - r-r-oco m e - CD * ' ? ? e * C? Y , ? w m ( O a o a 0 0 0 0 O 0 0 0 0 w- >- >- 0 0 0 0 0

b -

F - F - CD'* a (O 00000 a1 raO ta) I Q

0) (9:q4<4'Cc1 O 1 % % 8 % , a ) a D a q a o Q r - r - r - b r -

5 0'0 0'01 >- 1

0 ~ 0 0 - 3' 0 0 0 0 0 a

a * $ a. a - - - b m q ( V Q g * C V I ' O I O l v - V ) U 3 r n O g N <D (O b v r:? q i q i p ~ ' q -9 o Q , a D m ( D

0,o O 01 Ei O 00,o O O ~ c q c o aE I 0 0 0 0 0

r" C ~ , Q ~ - , = I 1 si W O C V - QD !9 1 - l b o r n a - si' a Q S & C Z

2 O ; f l i q : U ? i 21 O F - * -5: :ha o\olo 0 : , $! , 0 0 0 0

cumU-)S s' P ~ O o O O

h l ; h l l m CUI * pi cr) 0 cwm < - 7 - F

.-!FJ"sF/ , F - - ' v n

0 :O io , O ! '3"' 0 0- 0 ; o '

P' SI

Page 113: Stability of Castellated Beam Webs

- eam iole

1 .- - 2 3 4

.. - 5 -- -

- - lean - --.- vole --

1 2 3 4

Bar(

hole

1 2 3 4

-

Max ratio -- -

0.847 0.847 0.847 0.847 0 847 - --

Max ratio

- - -

0.95C 0.95(1 0.95(1 O. 950

Max ratio

O 812 0.812 O 812 O 812

4 0 s m p

over VtrslNp

0.415 1.245 2.085 2.915 3.745

M,e,4Mp - - --

avec vu.&

O. 606 1 818 3.023 4 235

over

VirstNp

O 577 1 744 2 897 4 051

test over heoq

0.817 - - -

test over :heoq

0.950

test over theoq

0.813

)tediction

buckling . -

mdictior

buckling

~redictior

buckling

Page 114: Stability of Castellated Beam Webs
Page 115: Stability of Castellated Beam Webs

Max ratio --. - -

O. 840 0.840 0, . 840 -

Max ratio

I

1.338 1.409 j.592 1.785

MtesdM, over

v t e s f l p . . .. .

0.609 .. .

1.827 3.055 . . . .

k J M P . -. - . . - -

over L / V p

0.662 1.986 3.310 4.641 5.966

test ove1 heor

0.84(

test over heoq

1 .O01

predictioi

buckling - - -

predictioi - - .

shear mech.

Page 116: Stability of Castellated Beam Webs

-(

y , a E b 1 -1 (q ad E . r 3 m l , , z 51 F

F i

-1 . -' z

q as 1 a), a z: =? > S. QO a 5. % 1 a)

O

Page 117: Stability of Castellated Beam Webs

Max ratio . .

0.976 0.976 0.976 1.052 1,106 - . .

Max ratio --- .

O. 862 0.91C -

1.136 . .

Max ratio

1.259 0.474

test over heory

1 .l86 -- - -

test ovet heory - .

1.136 . .

test over heory

1.259

predictior

shear mech.

predictioi

shear mech.

predictior

shear mec h.

Page 118: Stability of Castellated Beam Webs
Page 119: Stability of Castellated Beam Webs
Page 120: Stability of Castellated Beam Webs
Page 121: Stability of Castellated Beam Webs
Page 122: Stability of Castellated Beam Webs

- 3ear holt

1 2' 3 4- -- - 5 6 - 7 8

ref:

lean hole -- --

1 2 3

. --

Y?"

0.283 0.283 or000

Max ratio

1 .O20 1.020 1.020 1.020 1 .O20 1 .oz0 1 .O20 1 .O46

Max ratio -

1.960 1 960 1 .O39

over

V,,#,

0.268 0.804 1 .%O 1 .876 2.412 '2.948 3.484 4.020

tesUM over

ItesUV

0.495 1.484 infinite

test over :heor

1 .O16 - . - -

test ove1 heor)

1.960

~redictior

shear mech.

irediction

mid-post

Page 123: Stability of Castellated Beam Webs

hole 7 tesW VcrN tesWy . . - tesuMy - - Max tesUM ratio over

Csflv, - . . . -- k.4- - - - -- Max W a s i l M p

ratio over - -- - v , e s f l ,

-

Mp kN.n - - .* . . .-

105.: -.

. .

ml! kN.w -. - .. - .

109.1

kN.m . - .

109.1

test over iheory - --

1 .8l 1 - -

test - - -- over heory -- - - -

1.809

test ove1 heory

1.497

predictioi

mid-post

predictioi - - - - - -

mid-posl - - -

predictioi

mid-post

1

Page 124: Stability of Castellated Beam Webs
Page 125: Stability of Castellated Beam Webs

. . , a: , 'm'm'm9- c ~ o J - ! ~ ,

si ! N I N I N / C Y I N ~ P ~ C V ~ C V ~ W - r ' W a * , ' w 9

1 . S l F i S , S i S , - ? I ~ i ~ ! W 1 N 1 C V i N . C V CV CV hl '+ 7 5 , T . Y y T ' y >a; ! IO 10 ;O IO IO ;O /O !O j T1

h

O 0:o 0'0 O O O , 8 -

LD CO s! , dbib;wim'Fi a!,!, : : $: r3 m rn m ' a N in

E : i , ~ ~ ~ ~ ~ ~ ~ ; ~ ! r i ~ , o , . E i 0 4 0 4 N,N .CV . - - O Q) Q> Q).a O - 0

y ! ! 8j8!8;0[010~ T: : O O ; O ' O O O ~ ài ~ ~ ~ l ~ i ~ ' ~ l ~ ~ ~ ~ ~ j

a) I Q D aD joD 'P -,a0 ep - . Y 7 . 7 ' - - - F

:a) * !a3 tQD IoD :al 1 Q i * ! W : ( D (D W ( D ' C D (O (D

! ,8 /8 iO;1818181Pi~i , , O .O ;O !O O O O O Ob - 0, ca; .b C0.w. w,nl -nl ïüa *. CV - r i t %!a a~ OD Cu . o . N i C V b - ' - O O 0 * ri !> i . o : o ; o , o . ~ O O

I . I I : . I . , , j s - m . ! 1 l , g ! ~ ! : ; . .

O , ! : - I C V ; * I 9 l m ! < D i b : < o ! I ! I I ; I . I ? 2 . - C V : r n ! O rC) (O'b ai

; : S l ~ i : , I I ,

Page 126: Stability of Castellated Beam Webs
Page 127: Stability of Castellated Beam Webs

- bear holt

1 2 - 3 4 5 - - 6 - -.

bean -- - hols

1 2 3 4 5 6

- Max ratio

3.00c 2.945 2,943 2,953 0.830 D.838

Max ratio - -

1.845 1.845 1.845 !. 090 1.742 1.742

l

1

M,es4M1 over

VI*,fl,

O 360 1.075 1.795 2.510 infinite infinite - -

M,.,4MF - - over

v..fl, - -

O 433 1.296 2 162 3 027 infinite infinite

test ove1

theor

3.00(

test over

theo~

2.090

predictioi

s hear mech. . . . .

predictior . . .

shear mech.

Page 128: Stability of Castellated Beam Webs
Page 129: Stability of Castellated Beam Webs

Max ratio

- "

2.854 2.822 2.821 0.907 O. 907

Max ratio

1.728 2.181 1 725 1 364 1.364

Mt @ S m ,

over

Vtes&

0.547 j .636 2r729 infinite infinite

4 e * m p

over V i e r i N p

0.574 1.359 2 866 infinite infinite

test ovet theor

-- .

2.854 - -. -

test over heoq

2.181

aredictioi

shear mech. ...

~redictior

s hear mech

Page 130: Stability of Castellated Beam Webs

- bea ho1

1 2

*-- - 3 4 5

- .-

lear . --. . rok

-

- - 1 2 3 4 5

-

over Vte.1N

0.493 1.479 2.461 infinite infinite . -

- test over

theor]

2.058 - .

test ove1 heory

1 Si6

predictia

shear mech.

vedictioi - -

shear mech

ratic

2.051 2.051 2. OS( 1.08( I .O&

Max rtio -

.O43 ,136 .576 .547 .547

Ove1 v!../v?

0.603 1.809 3.015 infinite infinite

Page 131: Stability of Castellated Beam Webs
Page 132: Stability of Castellated Beam Webs

- ~ear holr

1 -

2 -- - 3

-- - 4 -

~i' . .

L2 L 3

ean !oie - - -

1 2 3 4

-

1

ential Loads

-

Max ratio

1.21C 1.214 1-61 3 1.606

106 1 ,O4 1.11

Max rtio

.225

.225 226 225

MI,S,IM

over

v t e , f l ,

0.283 0.850 0.567 0.000

infinite infinite infinite

k S m f

ovec

O 283 0.850 1416

test over heoq

1.613

1 .O63 1 .O43 1.113

test ovec ieory

1.226

predictio

shear mech.

flexural f~exura~ flexural . . mech.

~redictioi

shear mech

Page 133: Stability of Castellated Beam Webs
Page 134: Stability of Castellated Beam Webs
Page 135: Stability of Castellated Beam Webs
Page 136: Stability of Castellated Beam Webs

Man ratic

1.214 1 ,211 1.211 3.771 3.87: j. 99. I .oor i .ooi

Max ratio

1 .8O8 l.8Oîl L9OO 1 953 1,821

-

1 Mt,s/M, over

V t e s f l p

0.429 1.281 2.134 5.550 6.404 7.257 infin~te infinite -.

MtedM, over

v l e p p

0.580 1.741 5.099 6.259 infinite

test ove1 :heoi

t ,211 -- .

.

. .

test . .

over heoq

1.808 .-

predictio

mid-post . .

predlctioi

shear mech.

Page 137: Stability of Castellated Beam Webs
Page 138: Stability of Castellated Beam Webs

APPENDIX C

Elastic and Plastic Theoretical Computations

This Appendix contains al1 caIculations in deriving the Buckling loads under the effect of

Inelasticity on the Ultimate Strength.

Page 139: Stability of Castellated Beam Webs
Page 140: Stability of Castellated Beam Webs

r -: gi i - . - 1 a'- Y) Y)'

S t t t p n . bbf '

5 s d! ):St - - a m 1 i f . h l m m ? ? - a 12 1 Ccs a 0 Q D Q D Q D

- m u