effective width and ultimate strength of continuous composite beams. · 2020. 4. 2. · composite...

EFFECTIVE WIDTH AND ULTIMATE STRENGTHOF CONTINUOUS COMPOSITE BEAMS.

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Authors MOUSSA, ALBERT ELIAS.

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Moussa; Albert Elias

EFFECTIVE WIDTH AND ULTIMATE STRENGTH OF CONTINUOUS COMPOSITE BEAMS

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EFFECTIVE WIDTH AND ULTIMATE STRENGTH OF

CONTINUOUS COMPOSITE BEAMS

by

Albert Elias Moussa

A Dissertation Submitted to the Faculty of the

. DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 987

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examina.tion Conunittee, we certify that 'tole have read

the dissertation prepared by ALBERT ELIAS MOUSSA

entitled "EFFECTIVE WIDTH AND ULTIMATE STRENGTH OF ------~~~~~~~~~~~~~~~~~~~~~-------------------

CONTINUOUS COMPOSITE BEAMS"

and reconunend that it be accepted as fulfilling the dissertation requirement

for the Degree of Doctor of Philosophy

~1J~~

~M-ti;'k~ &:~ ~~L". ~Ut..~

12 /)'~

191/7

Date

Date

Date 1/ J/ff71 •

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

Date

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made 3vailable to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgme~: the proposed ~se of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED: ~-;;~~

DEDICATION

To my wife, Sonia, and my daughters, Christine and Carmen

iii

ACKNOWLEDGMENTS

I wish to express my sincere appreciation and gratitude to my

advisor, Professor Reidar Bjorhovde, for his guidance, valuable assis

tance, and continued encouragement throughout this study. The present

form of this dissertation has been possible due to his careful review of

the manuscript and helpful comments.

Special thanks are due to Professors C. S. Desai, M. Ehsani,

T. Kundu, and P. Kiousis for their cooperation and helpful comments.

The greatest thanks and appreciation are due to my wife, Sonia,

and my children for their patience, help, and love.

I am partic~laLly thankful to my friends, H. Al-Safade and

S. Hardash, for helping me in preparing the illustrations.

Grateful appreciation is extended to my mother-in-law, Helani

Lazkani, for being with us, encouraging, and taking care of my family.

Thanks are also extended to my father and mother, brothers, sisters, and

friends for their love and encouragement.

Finally, I wish to thank the Department of Civil Engineering for

providing financial support throughout my graduate work.

iv

1.

2.

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS

LIST OF TABLES

ABSTRACT

COMPOSITE FRAMES IN BUILDINGS .

BACKGROUND AND SCOPE

2.1 Background ... 2.2 Scope of Study.

2.2.1 Limitations 2.2.2 Items of Study

Page

viii

xx

xxii

1

10

10 14 14 15

3. PREVIOUS STUDIES 16

3.1 Introduction ..... 3.2 Simply Supported Composite Beams 3.3 Composite Beams in the Negative Moment Region

4. ANALYSIS OF COMPOSITE FLOOR SYSTEMS ..

16 17 21

31

4.1 Introduction. . • . . . 31 4.2 Plate and Beam Governing Differential Equations 32 4.3 Application of Plate and Beam Differential Equations to

a Composite Floor System . . . . . . . • . . . . . 36 4.4 Finite-Element Model of a Composite Floor System. . 41

4.4.1 Finite-Element Model for the Concrete Slab 41 4.4.2 Finite-Element Model for Steel Beam, Column, and

Stud Shear Connectors . 4.4.3 Special Considerations

5. NONLINEAR ANALYSIS MODEL

5.1 Introduction .. 5.2 Yield Criterion 5.3 Numerical Method of Nonlinear Analysis.

6. CORRELATION BETWEEN THEORY AND TEST RESULTS .

6.1 6.2

Introduction . Description and Results of Tested Specimens

v

52 64

67

67 69 72

75

75 75

TABLE OF CONTENTS--Continued

6.3 Finite-Element Modeling of the Test Specimens 6.4 Comparison of Experimental and Analytical Results

7. EVALUATION OF COMPOSITE FLOOR SYSTEMS ....

7.1 7.2 7.3

Description of Composite System Models . Shear Connector Design . . . . . • . . • • . . . Modeling of the Composite Floor Systems of Groups I, II, and III •.•.•..•......•. 7.3.1 Material and Element Properties

8. EFFECTIVE WIDTH CRITERIA BASED ON NONLINEAR ANALYSIS

8.1 Introduction .•..•............. 8.2 Analytical Criteria for Calculating the Effective

8.3 8.4 8.5 8.6 8.7

8.8

8.9

Width . . . . . . . . • : . . . . . . . Current Effective Width Criteria . . . • . . . . . . Variation of the Effective Width Along the Girders . Variation of the Effective Width at Mid-Span . . . . Variation of the Effective Width at Supports . . . . Influence of Other Variables on the Effective Width at Mid-Span . . ...........•...

'.

8.7.1 Influence of b/~ Ratio ..•.......... 8.7.2 Influence of Beam-to-Column Connections ..... 8.7.3 Influence of Steel Beam-to-Slab Stiffness Ratio. 8.7.4 Influence of the Slab Thickness .•.....•• 8.7.5 Influence of the Slab Reinforcement Ratio, p Influence of Specimen Variables on the Effective Width a t the Support . . . . • . . • . . • . . . . . . . • . . 8.8.1 Influence of b/~ Ratio ....... . 8.8.2 Influence of the Type of Beam-to-Column

Connection . . . . . . . . , . . . . . 8.8.3 Influence of Steel Beam-to-Slab Stiffness Ratio . 8.8.4 Influence of the Slab Thickness ........ . 8.8.5 Influence of the Slab Reinforcement Ratio ... . Proposed Criteria for Calculating the Effective Width in a Continuous Composite System . . . . . 8.9.1 At Mid-Span. . . . ...••... 8.9.2 At the Supports ....

9. BEHAVIOR OF COMPOSITE BEAMS IN CONTINUOUS COMPOSITE FLOOR SYSTEMS •.

9.1 Introduction 9.2 Moment Resistance in the positive and Negative Moment

vi

Page

76 95

107

107 113

116 116

121

121

123 125 128 159 199

240 240 250 254 257 257

260 260

269 273 276 276

279 279 280

282

282

Regions • . . . .. ...•......... . .. 282

10.

9.3 9.4

TABLE OF CONTENTS--Continued

9.2.1 9.2.2 9.2.3

Composite Beams with Fixed Ends . • . . . . . . . Composite Beams with Semi-Rigid End Connections . Influence of Slab Thickness on the Ultimate Moment Resistance . . . . . . . . .

9.2.4 Influence of Reinforcement Ratio on Moment Capacity . . . . .

Ductility of Composite Beams . . . . . . Deflection of Composite Beams

Ultimate

SUMMARY AND CONCLUSIONS . .

11. RECOMMENDATIONS FOR FURTHER RESEARCH

NOMENCLATURE

REFERENCES

vii

Page

283 299

308

308 311 312

329

333

334

337

Figure

1.1

1.2

1.3

1.4

1.5

2.1

2.2

4.1

4.2

4.3

4.4

LIST OF ILLUSTRATIONS

Composite System in a Building . . . . .

Some Typical Types of Composite Beams in Buildings

Composite Beams

Composite F'loor System with Formed Steel Deck

Stress Distribution in a Wide Slab Supported on a Steel Beam and the Effective Width, b . . . • . . • . • . . .

e

Beam Moment Distributions for Various End Connections in Braced Frames . . . . . . . . .. . . . . .

Collapse Mechanism and Moment Distribution for a Continuous Beam at Ultimate Gravity Load .

Unit Internal Stress Resultants for a Plate Element

Plan of a Composite Floor System

Transfer of Girder Slab Forces

Assumed Cracking Model for the Slab

4.5 Behavior of the Equivalent Material for the Shell Elements

4.6 Typical Concrete Stress-Strain Curves

4.7 Idealized Stress-Strain Curve for 60 Ksi Reinforcing Steel

4.8 Assumed Distribution of Tensile Stresses in the Slab of a

4.9

4.10

4. 11

Composite Section

Finite-Element Idealization of a Composite Floor System

Deflection and Lateral Load Idealization of a Stud in a Real Structure and in the Finite-Element Model . .

Beam Element Representing a Stud in the Real Structure

viii

Page

2

4

5

6

8

12

13

33

37

39

43

45

46

47

50

56

57

59

Figure

4.12

4.13

S. 1

6.1

6.2

6.3

6.4

6.5

LIST OF ILLUSTRATIONS--Continued

Beam Element Representing a Stud in the Finite-Element Mode 1 . . . . . . . • . . . . . . . .

Arrangement of Reinforcing Bars Around a Column in a Negative Moment Area . . . • . . • . .

General Response of Nonlinear Behavior

Details of Tested Specimens

Moment-Rotation Curves for Specimen with Flexible Beam-toColumn Connection (CB2), with and Without Composite Action

Moment-Rotation Curves for Specimen with Rigid Connection (CB3), with and Without Composite Action . . . . . .

Moment-Rotation Curves for Specimen with Semi-Rigid Beamto-Column Connection (CBS), with and Without Composite Act ion . . . . . . . . .. ...... . . .

Finite-Element Discretization of a Specimen

6.6 Specified Boundary Conditions for Flexible and Semi-Rigid

6.7

Connections

Equivalent Stress-Strain Relationship for the Shell Element which Represents the Concrete Slab (Specimen CB3) . . . . . . . . . . . • • . . .• • . . .

6.8 Stress-Strain Relationship for the Material of the WSx20

6.9

6.10

6. 11

6.12

6.13

Steel Beam . . . . . . . .

Stress-Strain Relationship for the Material of the Studs Elements . . . . . . . . . . . . . . . . .

Analytical Definition of Connection Rotation

Deflection Measurement for Test Specimens

Comparison Between Experimental and Analytical MomentRotation Curves for Specimen with Flexible Beam-to-Column Connection . . . . . . • . . . . . . . . . . . . . . .

Comparison Between Experimental and Analytical MomentRotation Curves for Specimen with Rigid Beam-to-Column Connection . . . . . . . . . . . • . . . . . . . . . .

ix

Page

62

66

68

77

80

81

82

85

88

91

92

94

96

97

98

99

Figure

6.14

6.15

6.16

6.17

7.1

7.2

7.3

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9


Comparison Between Experimental and Analytical MomentRotation Curves for Specimen with Semi-Rigid Beam-toColumn Connection . . . . . . . . . . . . . . .

Comparison Between Analytical and Experimentally Determined Deflections of Specimen CB2 . . . . .

Comparison Between Analytical and Experimentally Determined Deflections of Specimen CB3 . • . . .

Comparison Between Analytical and Experimentally Determined Deflections of Specimen CBS . . . . .

Typical Plan of a Composite Floor System, Showing Types of Beam-to-Column Connections and Panel Dimensions

Finite-Element Model of Specimen 3 of Group II .

Boundary Condition of Specimen 3 of Group II . .

Membrane Stress Distribution Due to Composite Action .

Effective Width According to the AISC Specifications [4] .

Variation of (b /2)/(b/2) at Yield Load for the Composite Specime~ 2, Group Beam of II . . . . . . · ·

Variation of (b /2)/(b/2) at 0.533 of the Ultimate for the composiEe Beam of Specimen 2, Group II · · Variation of (b /2)/(b/2) at 0.666 of the Ultimate for the ComposiEe Beam of Specimen 2, Group II · · Variation of (b /2)/(b/2) at 0.767 of the Ultimate for the composiEe Beam of Specimen 2, Group II · · Variation of (b /2)/(b/2) at 0.867 of the Ultimate for the composiEe Beam of Specimen 2, Group II

Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam 5f Specimen 2, Group II ..

Load

· · · Load

· · · Load

· · · Load

· · ·

·

·

·

·

Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specime~ 3, Group II ........... .

x

Page

100

101

102

103

108

117

118

122

127

129

130

131

132

133

134

135

xi

LIST OF ILLUSTRATIONS--Continur~6.

Figure Page

8.10 Variation of (b /2)/(b/2) at 0.522 of the Ultimate Load for the composiEe Beam of Specimen 3, Group II · · · · · · 136

8.11 Variation of (b /2)/(b/2) at 0.627 of the Ultimate Load for the composiEe Beam of Specimen 3, Group II · · 137

8.12 Variation of (b /2)/(b/2) at 0.784 of the Ultimate Load for the composiEe Beam of Specimen 3, Group II · · · · · · 138

8.13 Variation of (b /2)/(b/2) at 0.888 of the Ultimate Load for the composiEe Beam of Specimen 3, Group II · · · · 139

8.14 Variation of (b /2)/(b/2) at Ultimate Load for the Composite

ef

. Beam 0 Spec1men 3, Group II · · · · · · 140

8.15 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specime~ 1, Group II · · · · · · · · 144

8.16 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam 6f Specimen 1, Group II .. · · · · 145


8.18 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam 6f Specimen 4, Group II · · · · · · 147


8.20 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam 6f Specimen 5, Group II · · · · · · 149

8.21 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specime~ 6, Group II · · · · . . · · · · · · 150

8.22 Variation of (b /2)/(b/2) at 0.712 of the Ultimate Load for the ComposiEe Beam of Specimen 6, Group II · · · · 151

8.23 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam 6f Specimen 6, Group II .. · · · · 152

8.24 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specime~ 7, Group II · · · · . . · · · · · · 153

xii


Figure Page

8.25 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam :Sf Specimen 7, Grol.'.p II . . · · · · 154

8.26 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specime~ 8, Group II · · · '. · · · · · · · 155

8.27 Variation of (b /2)/(b/2) at Ultimate Load for the Composite

ef

. 8, Group II 156 Beam 0 Spec~men · · · · · · 8.28 Variation of (b /2)/(b/2) at Yield Load for the Composite

Beam of Specime~ 9, Group II · · · · · · · · 157

8.29 Variation of (b /2) / (b/2) at Ultimate Load for the Composite Beam :Sf Specimen 9, Group II · · · · · · 158

8.30 Variation of b /b at Mid-Span for the Girder of · 1 e I 160 Spec~men , Group · · · · · · · · · · · · · · · · ·



8.33 Variation of b /b at loUd-Span for the Girder of · 1 e II 163 Spec~men , Group · · · · · · · · · · · · · · · · ·

8.34 Variation of b /b at Mid-Span for the Transverse Beam of Specimen 1,

e 164 Group II · · · · · · · · · · · · · · · · ·

8.35 Variation of b /b at Mid-Span for the Girder of · 2 e II 165 Spec~men , Group · · · · · · · · · · · · · · · · ·

8.36 Variation of b /b at Mid-Span for the Transverse Beam of · 2 e II 166 Spec~men , Group · · · · · · · · · · · · · · · · ·

8.37 Variation of b /b at Mid-Span for the Girder of Specimen 3, Gr:Sup II · · · · · · · · · · · · · · · · · 167

8.38 Variation of b /b at Mid-Span for the Transverse Beam of · 3 e II 168 Spec~men , Group · · · · · · · · · · · · · · · · ·

8.39 Variation of b /b at Mid-Span for the Girder of Specimen 4, Gr6up II · · · · · · · · · · · · · · · · · 169

xiii


Figure Page

8.40 Variation of b /b at Mid-Span for the Transverse Beam of e

Specimen 4, Group II · · · · · · · · · · · · · · · · · 170

8.41 Variation of b /b at Mid-Span for the Girder of Specimen 5, Gr5up I! · · · · · · · · · · · · · · · · 171

8.42 Variation of b /b at Mid-Span for the Transverse Beam of · 5 e I! 172 Spec1men , Group · · · · · · · · · · · · · · · ·

8.43 Variation of b /b at Mid-Span for the Girder of Specimen 6, Gr5up I! · · · · · · · · · · · · · · · · · 173

8.44 Variation of b /b at Mid-Span for the Transverse Beam of Specimen 6, Gr5up I! · · · · · · · · · · · · · · · · · 174

8.45 Variation of b /b at Mid-Span for the Girder of · 7 e I! 175 Spec1men , Group · · · · · · · · · · · · · · · · ·

8.46 Variation of b /b at Mid-Span for the Transverse Beam of · 7 e I! 176 Spec1men , Group · · · · · · · · · · · · · · · · ·


8.48 Variation of b /b at Mid-Span for the Transverse Beam of e

Specimen 8, Group I! · · · · · · · · · · · · · · · · · 178


8.50 Variation of b /b at Mid-Span for the Transverse Beam of · 9 e I! 180 Spec1men , Group · · · · · · · · · · · · · · · · ·

8.51 Variation of b /b at Mid-Span for the Girder of · 1 e II! 181 Spec1men , Group · · · · · · · · · · · · · · · ·

8.52 Variation of b /b at Mid-Span for the Girder of · 2 e II! 182 Spec1men , Group · · · · · · · · · · · · · · · ·

8.53 Variation of b /b at Mid-Span for the Girder of · 3 e II! 183 Spec1men , Group · · · · · · · · · · ·

8.54 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 1, Group I · · · · . . · · · 184

xiv


Figure Page

8.55 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 2, Group I · · · · · · · · · 185

8.56 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 3, Group I · · · · · · · · · 186

8.57 Slab Membrane Stress Distribution at fHd-Span for the Girder of Specimen 1, Group II · · · · · · · · · · · · 187

8.58 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 2, Group II · · · · · · · · · · · · 188








8.66 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 1, Group III · · · · · · · · · 196

8.67 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 2, Group III · · · · · · · · · 197

8.68 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 3, Group III · · · · · 198

8.69 Variation of b /b at the Support for the Girder of . 1 e Spec~men , Group I . . . . . . . . . 200

xv


Figure Page

8.70 Variation of b /b at the Support for the Girder of Specimen 2, Gr5up I · · · · · · · · · · · · · 201

8.71 Variation of b /b at the Support for the Girder of · 3 e I 202 Spec~men , Group · · · · · · · · · · · · ·

8.72 Variation of b /b at the Support for the Girder of · 1 e II 203 Spec~men , Group · · · · · · · · · · · · ·

8.73 Variation of b /b at the Support for the Transverse Beam of Specimen 1,eGroup II · · · · · · · · · 204

8.74 Variation of b /b at the Support for the Girder of Specimen 2, Gr5up II · · · · · · · · · · · · · 205

8.75 Variation of b /b at the Support for the Transverse Beam of Specimen 2,eGroup II · · · · · · · · · 206


8.77 Variation of b /b at the Support for the Transverse Beam of Specimen 3,eGroup II · · · · · · · · · · · · '. 208


8.79 Variation of b /b at the Support for the Transverse Beam of Specimen 4,eGroup II · · · · · · · · · · · · · 210

8.80 Variation of b /b at the Support for the Girder of · 5 e II 211 Spec~men , Group · · · · · · · · ·


8.62 Variation of b /b at the Support for the Girder of · 6 e II 213 Spec~men , Group · · · · · · · · ·



xvi


Figure Page



8.87 Variation of b /b at the Support for the Transverse Beam of Specimen 8,eGroup II · · · · · · · · · · · · 218



8.90 Variation of b /b at the Support for the Girder of Specimen 1, Gr6up III · · · · · · · · · · · · · 221

8.91 Variation of b /b at the Support for the Girder of Specimen 2, Gr6up III . · · · · · · · · · · · · · 222

8.92 Variation of b /b at the Support for the Girder of . 3 e 223 Spec~men , Group III . · · · · · · · · ·

8.93 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1 , Group I · · · · 224

8.94 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 2, Group I · · · · 225

8.95 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 3, Group I · · · · 226

8.96 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1, Group II · · · · · · · 227




xvii


Figure Page

8.100 Slab Membrane Stress Distribution at the Support for the

8.101

8.102

8.103

8.104

8.105

8.106

8.107

8.108

8.109

8.110

8.111

8.112

8.113

8.114

8.115

Girder of Specimen 5, Group II ...

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 6, Group II ...

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 7, Group II ...

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 8, Group II . . .

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 9, Group II . .. . .•.

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1, Group III ....

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 2, Group III

Slab Membrane Stress Distribution at the Support for the Girder of Specimen 3, Group III

b /b vs. b/£ at Mid-Span for the Specimens of Group I e

b /b vs. b/£ at ~id-Span for the First Three Specimens of e

Group II

b /b vs. b/£ at Mid-Span for the Specimens of Group III e

b /£ vs. b/£ at Mid-Span for Specimens 1, 2, and 3 of e

Groups I, II, and III

b /b at Mid-Span vs. the Degree of Fixity at Support (Specimens 3, 5, 6, 7, and 8; Group II) ....

b /£ at Mid-Span vs. the Degree of Fixity at Support (Specimens 3, 5, 6, 7, and 8; Group II) .•...

b /£ at Mid-Span vs. Beam-to-Slab Stiffness Ratio for e. 12 d3 f d Spec~mens , , an 0 Groups I, II, an III ...•

b /b vs. b/£ at the Support Sections for the Specimens of e

Group I

231

232

233

234

235

236

237

238

244

245

246

249

252

253

256

265

Figure

8.116

8.117

8.118

8.119

8.120

8.121

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9


b /b vs. b/~ at the Support Sections for the First Three S~ecimens of Group II ...... .

b /b vs. b/~ at the Support Sections for the Specimens of e

Group III ... . . . . . . . . . .

b /~ vs. b/~ at the Support Sections for the First Three S~ecimens of Groups I, II, and III . . . •...

b /b at the Supports vs. the Degree of Fixity at the sijpports (Specimens 3, 5, 6, 7, and 8; Group II)

b /~ at the Supports vs. the Degree of Fixity at the sijpports (Specimens 3, 5, 6, 7, and 8; Group II)

b /~ at the Supports vs. the Beam-to-Slab Stiffness Ratio f6r the First Three Specimens of Groups I, II, and III ..

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 1 , Group I

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 2, Group I

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 3, Group I

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 1 , Group II · · · g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 2, Group II · · · g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 3, Group II · · · g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 1 , Group III

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 2, Group III

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 3, Group III

g g

xviii

Page

266

267

268

271

272

275

284

285

286

287

288

289

290

291

292

Figure

9.10

9.11

9.12

9.13

9.14

9.15

9.16

9.17

9.18

9.19

9.20

9.21

9.22

9.23

9.24


Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 5, Group II

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 6, Group II ...

g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 7, Group I'r

g g


g g


g g

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 9, Group II ..•

g g

Moment Due to Steel Beam Forces (M + Tge) at the Support vs. Maximum Rotation for Specimen ~, Group I ...... .

Moment Due to Steel Beam Forces(M + Tge) at the Support vs. Maximum Rotation for Specimen 1, Group II ....•.

Moment Due to Steel Beam Forces (Mg + Tge) at the Support vs. Maximum Rotation for Specimen 2, Group III ....

Mid-Span Load-Deflection Curve for Specimen 1, Group I

Mid-Span Load-Deflection Curve for Specimen 3, Group II

Mid-Span Load-Deflection Curve for Specimen 5, Group II

Mid-Span Load-Deflection Curve for Specimen 2, Group III .

Typical Gravity Load Distribution on Girders

Gravity Load on Specimen 1, Group I

xix

Page

300

301

302

303

309

310

313

314

315

316

317

318

319

324

326

Table

4.1

4.2

6.1

6.2

7.1

7.2

7.3

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

LIST OF TABLES

Recommended p. for the Slabs of Composite Beams m~n

Maximum Reinforcement Ratio Based on the ACI Code [51]

Material Properties of Test Specimens . • . . .

Yield and Ultimate Moments for Test Specimens .

Group I of Composite Floor Systems

Group II of Composite Floor Systems .

Group III of Composite Floor Systems

Influence of b/~ Ratio on the Effective Width at Mid-Span (Group I Specimens) . .. .•... . .....

Influence of b/~ Ratio on the Effective Width at Mid-Span (Group II Specimens) . • . . . . . • . . . .

Influence of b/~ Ratio on the Effective Width at Mid-span (Group III Specimens) . . . . . . . . . . ..... .

Influence of Beam Length on the Effective Width at Mid-Span

Influence of Support Conditions on the Effective Width at Mid-Span . . . • . • • . . . . . . . . . . . . .

Influence of Steel Beam-to-Slab Stiffness Ratio on the Effective Width at Mid-Span · · · · · · . · · · . Influence of the Slab Thickness on t.he Effective Width at Mid-Span . . . . . . . . . · · · · · · . · · · · Influence of the Reinforcement Ratio on the Effective Width at Mid-Span . . . . . .. ..... .....

Influence of b/~ Ratio on the Effective Width at the Support (Group I Specimens) · · · · · · · · · · Influence of b/~ Ratio on the Effective Width at the Support (Group II Specimens) · · · · · · · · ·

xx

Page

53

54

79

83

111

112

114

241

242

243

248

251

255

258

259

261

262

LIST OF TABLES--Continued

Table

8.11 Influence of b/~ Ratio on the Effective Width at the Support (Group III Specimens) . . . . . . . . . •

8.12 Influence of Beam Length on the Effective Width at the Support . . . . . • . . . . . . . •

8.13 Influence of the Support Conditions on the Effective Width at the Support . • • . . . . . • . . •

8.14 Influence of Steel Beam-to-Slab Stiffness Ratio on the Effective Width at the Support . . . . . . . . .

8.15 Influence of the Slab Thickness on the Effective Width at the Support . . . . . . . . . . . . . . . . . .

8.16 Influence of the Slab Reinforcement Ratio on the Effective Width at the Support . . . . . . . . . .

9.1 Ultimate Moment Capacity at Mid-Span for Fixed-Ended Beams . • . . .

9.2 Ultimate Moment Capacity at the Support for Fixed-Ended

9.3

9.4

9.5

9.6

Beams • . . . .

Mid-Span Ultimate Moment Capacity for Composite Beams with Semi-Rigid End Connections • . • . .

Support Ultimate Moment Capacity for Composite Beams with Semi-Rigid End Connections . . . . . . . •

Load-Deflection Data for Composite Beams

Service Load Support Moments

xxi

Page

263

264

270

274

. 277

278

294

295

304

305

320

323

ABSTRACT

The primary objective of this study has been to examine the

behavior of continuous composite beams. The investigation has been

based on a nonlinear analysis of 15 composite floor systems. New design

criteria have been proposed to calculate the effective width at mid-span

and supports, at yield and at ultimate loads. New design equations have

been developed to calculate the ultimate strength of a composite section

in both positive and negative moment regions. Ductility and deflection

characteristics have been studied and equations are proposed to calcu

late the deflection at service load.

xxii

CHAPTER 1

COMPOSITE FRAMES IN BUILDINGS

Composite construction currently is becoming more and more com

monly used in structural systems for buildings. This is due to the

inherent economy and efficiency of the joint action between the steel

elements and the concrete--the concrete provides added stability for the

steel elements under compression, and the steel works most effectively

where tension occurs. It has long been common practice to take advan

tage of these characteristics for the beam elements of the frame; com

posite columns and complete framing systems are now being utilized as

well.

Typically, composite frames are made up of composite floor sys

tems and the columns which support the floors. The columns mayor may

not be composite, although the latter applies to a true composite frame.

In current design, the frames are normally treated as planar structures,

as shown in Figure 1.1. The girders which frame into the columns are

assumed to incorporate parts of the concrete slab, to form the major

composite frames. These usually carry the largest share of the gravity

load. Depending on the specifics of the structural system, the frames

in the orthogonal direction and other components will carry part of the

gravity load; the lateral loads are either taken by moment-resistant

frames or some other form of structural components.

1

M j C il or c ompos it F 0 rnmo

1\, IllY !'\, )Y I~, lY ./

I'

II 'I'. [L/ '", /

~'" IL/ ~/ r'\ b"

'''", / vy '''" V

"'" V /v .,Iv

" ,I "- "-I' 'I ',,- /1' "-

" v/L "'~ / , Y IL ~

I"" / "", VV I" VV V "-" " ./ "' ./ ./'

1// ~ 1I/ '~ lk~ ~ th re .Y Gravity Tributary A a

( a)

JL ~ ~, h J 1'1 J .~ .Jr ,~ ~ .",

\'7 .~ ~ u -~ ~ ~ .,~ ~ _t .~ -g

0/1 -~. {, -b .'\ .1L J C _1'1.. .it.- .t .~ .

DL+LL .:!L . ?7. JJ J. ~ _ok. J, ./r ~ ~ _It .lL

~'L .:!. ~ -~ ..A J J _k .-st ~ J -~

-~ ~ .J, 6 ~ . ok ... ./r ~, ;, ~, _6

,~ ~ ~ ~ <1. -,~. ,', .} ~ . \~ ~L .q

I I 'I I TrITT (b ) 11/1/ " 'II

Figure 1.1 Composite System in a Building. -- (a) Plan of a composite floor in a building showing gravity load distribution. (b) ~raming system.

2

3

The applied load on buildings consists of gravity and lateral

loads, where dead and live loads belong to the former, and wind and

earthquake loads to the latter category. It will be assumed in the fol-

lowing that the buildings are braced by shear walls or trussed bays to

carry the lateral loads, whereas the gravity loads will be carried by

series of composite frames.

Composite frames can be constructed in many ways; however, this

study will be restricted to those which consist of hot-rolled (or simi-

lar welded built-up) cross-sections that are connected to the reinforced

concrete slab by stud shear connectors. The columns will be assumed to

be non-composite.

Figure 1.2 gives some details of typical composite beams as they

are used today. The solid slab that is shown in Figure 1.2a is seldom

used now, due to the improved ~conomics of the hollow-core system of

Figure 1.2b. The steel deck can be used with the ribs oriented perpen-

dicular or parallel to the steel beam axis, as shown in Figure 1.3.

Extensive research and development have provided the design criteria

that are now used in the current code of practice [1-6].

The shear connectors which tie the steel section to the concrete

slab are shown in Figures 1.2 and 1.4. These resist the shear force

between the slab and the steel beam, and also prevent uplifting between

the slab and the beam. The interaction that is provided in this fashion

depends on a portion of the slab that is commonly known as the effective

slab width.

The effective slab width, b , is defined as that width of the e

slab which has equivalent longitudinal stresses and sustain a force

Solid Slab

rt~e P~otectton Mate~tul __ ~

( a )

ro~med Steel Deck

(b)

Figure 1.2 Some Typical Types of Composite Beams in Buildings. __ (a) Composite beam with solid slab. (b) Composite beam with formed steel deck.

4

5

Ca)

Cb)

Figure 1.3 Composite Beams. -- (a) Deck. ribs perpendicular to the steel beam. (b) Deck ribs parallel to the steel beam.

7

equal to the actual one in the slab. As shown in Figure 1.5, the inten-

sity of the extreme fiber compressive stress, a , which is at a maximum x

over the steel beam, decreases nonlinearly as the distance from the sup-

porting beam increases. The area under the stress diagram along the

slab is replaced by an equivalent uniform stress across the effective

width of the slab [7,8].

Extensive research over the past 35 years has focused mainly on

determining the behavior of isolated (i.e., individual) simply supported

composite beams. Thus, the effective slab width in the positive moment

region, the behavior and properties of the shear connectors, and the

service load and ultimate strength characteristics of the composite beam

are well understood [1-19]. The current design criteria [4,6] are,

therefore, almost entirely based on data for individual beams.

The composite beam criteria have been used in the design of non-

sway frames, and it is usually assumed that the beam-to-column connec-

tions are pinned. On this basis, the beams have been designed as simply

supported, and the beam reactions are transferred to the columns with

small eccentricities. This makes for an easy computational solution,

although it is clear that a smaller steel section might be used if any

negative moment capacity could be realized for the connections. On the

other hand, this would entail dealing with the concrete slab cracks that

could occur in the negative moment region, along with the potential for

larger long-term deflections and similar serviceability problems. This

will be addressed in detail in a later chapter.

Before improvements such as those that are outlined above can be

utilized, however, it is necessary to determine the details of the

FT:'14 r----- ---- -----I I

Membrnne Stress Distri/aution

t«------Total 51 ala Wtdth-----t::>t

8

Figure 1.5 Stress Distribution in a Wide Slab Supported on a Steel Beam and the Effective Width, b .

e

behavior and strength of composite beams in the negative moment region.

In particular, effective slab width criteria must be developed along

with data on the factors that ~re important for this; for example, the

amount of longitudinal reinforcing steel in the slab is likely to be of

primary importance. These and other of the primary problems of the

negative moment behavior and strength of composite beams in frames form

the major thrust of this study.

9

CHAPTER 2

BACKGROUND AND SCOPE

2.1 Background

In a bare steel frame structure, the overall behavior of the

frame depends on the characteristics of the beam-to-column connections,

and the moment distribution in the beam is tied to the types of

connections.

The basic types of beam-to-column connections are generally

known as flexible (Type 2) [4], semi-rigid (Type 3), and fully rigid

(Type 1).1 The classification depends on the amount of relative rota-

tion that is allowed to take place between beams and columns.

In frames having flexible beam-to-column connections, the size

of the beams is controlled by the positive moment at mid-span. In

frames with rigid connections, the beam sizes are governed by the nega-

tive moment at the ends. In the case of semi-rigid connections, the

partial restraint of the rotation at the beam ends can be used to reduce

the amount of positive moment. Under gravity loads, the flexibility of

the semi-rigid connections will be at an optimum if it can be adjusted

to equalize the end and mid-span moments.

1. The new AISC specifications for load and resistance factor design (LRFD) [6] identify only two types: Type FR (fully restrained) and Type PR (partially restrained). This has been done in recognition of the fact that perfectly flexible (i.e., pinned) connections do not exist in real structures.

10

11

Figure 2.1 shows the gravity load moment distributions in a sub

assemblage of a multi-story frame, using various types of beam-to-column

connections. For an elastic analysis of a frame, it is clear that semi

rigid connections usually give the most desirable moment distribution,

which in turn produces the most economical beam cross-sections; however,

the current American allowable stress design specifications [4] do not

have specific criteria for semi-rigid connections, although the type is

recognized as one form of construction. This is primarily due to a

sparsity of data on the moment-rotation characteristics of these connec

tions, and also because current practical frame analysis techniques

cannot incorporate such types. This is changing rapidly, however, and

several design firms now do perform frame design on the basis of semi

rigid concepts, using the new AISC LRFD specifications [6].

In plastic design, full use of the beam strength may be achieved

at the collapse load; therefore, for a beam with rigid end connections,

the moment distribution at the collapse load will be similar to the

elastic moment distribution for a beam having optimum semi-rigid connec

tions. Thus, the maximum negative and positive moments are equal at the

collapse load, as shown in Figure 2.2; however, this requires that the

connections possess sufficient rotation capacity to allow the develop

ment of a plastic hinge mechanism in the frame. This is of particular

concern in the case of complex frames, where a large number of hinges

are required to form before the ultimate load is reached. The rotation

capacity of the first hinge to form is especially important, as is the

req~irement that no loss in moment capacity takes place for large

rot.ations.

(a)

(b)

(c )

Figure 2.1 Beam Moment Distributions for Various End Connections in Braced Frames. -- (a) Flexible connections; beams are assumed to be simply supported. (b) Rigid connections; maximum moment~ are t~e negative moments. (c) Semi-rigid connections; M ~ M

max max

12

13

Ca)

( 10 )

Figure 2.2 Collapse Mechanism and Moment Distribution for a Continuous Beam at Ultimate Gravity Load. -- (a) Collapse mechanism. (b) Moment distribution.

14

The concerns regarding moment-rotation characteristics and rota

tion capacities are among the primary reasons that plastic design has

not achieved widespread design usage in the profession.

2.2 Scope of Study

In composite framing systems, the behavior of composite beam-to

column connections is further complicated, due to additional parameters

that influence their response. Thus, the degree of interaction between

the steel beam and the concrete slab, the amount of reinforcement in the

negative moment region, the effective widths in the positive and nega

tive moment regions, local buckling in the steel beam, and the long-term

effects of creep and shrinkage can be important considerations in com

posite frames.

This study has been devoted to determining the effective slab

width in the positive and negative moment regions at yield and ultimate

loads. The influence of the degree of rigidity of the steel beam-to

column connection also will be examined, along with its influence on the

ultimate moment distribution in the positive and negative moment

regions. The limitations and the detailed study items are summarized in

the following sections.

2.2.1 Limitations

1. The study is restricted to frames where the composite beams con

sist of solid reinforced concrete slabs that are attached to

wide-flange steel beams by headed stud shear connectors.

2. The interaction between the steel beam and the concrete slab is

assumed to be 100 percent.

3. The cross-section of the steel beam is assumed to be compact.

4. The beams of the frame are assumed to be in interior bays, and

the adjacent bays have the same or similar dimensions.

2.2.2 Items of Study

1. The variation of the effective width along the length of the

beam will be examined for load levels through service and

failure.

15

2. The effective width at beam mid-span and at the support will be

determined at yield and ultimate loads.

3. Yield and ultimate moment capacities at mid-span and at the sup

port of the composite beams will be found.

4. Mid-span beam deflections and the moment-rotation behavior at

the supports will be determined for loads through yield and

ultimate, including the effects of beam-to-column connections of

different moment capacities.

The parameters which are considered in this study are:

1. The ratio of the beam spacing to the beam span.

2. The ratio of the steel beam stiffness to that of the reinforced

concrete slab.

3. The amount of slab reinforcement.

4. The concrete slab thickness.

5. The type of beam-to-column connection.

The investigation will be performed through a nonlinear, finite

element analysis of several uniformly loaded composite floor systems.

CHAPTER 3

PREVIOUS STUDIES

3.1 Introduction

The behavior of composite framing systems has been studied

through experimental and theoretical research'on individual members of

the composite frames, such as simply supported composite beams, continu

ous composite beams, and composite beam-to-column connections. In these

investigations, it ~as usually assumed that the behavior of a composite

frame in the positive moment region would be similar to the behavior of

a simply supported composite beam having a span length equal to the dis

tance between the inflection points of the beams in the frame. The pri

mary characteristics of simply supported beams, therefore, including

their service and ultimate load behavior, effective slab criteria, shear

connector distribution, and cracking behavior, have been adopted for

designing the composite beams in the posit:Lve moment regions of frames.

The behavior of composite frames in the negative moment regions

has been examined through studies of continuous composite beams and com

posite beam-to-column connections. The results of these studies, such

as the effectiveness of the shear connectors in the negative moment

region, the effective concrete slab width, the effectiveness of the

reinforcing bars, local buckling of the steel beams, and the cracking

behavior in the slab, have been used as guidelines in the development of

design methods for composite frames.

16

17

The above simplifications of the overall behavior of composite

frames are reasonable from a practical engineering standpoint, in view

of the complex behavior and analysis that are required for complete

frames. The interaction of the members of the structure and its influ

ence on the behavior is missing. This is due to the complexity and cost

of performing realistic studies of composite framing systems. Theoreti

cal modeling is needed, therefore, along with a few selected tests that

can be used to verify the results.

The research that is described later is intended to study the

nonlinear behavior of composite frames in the positive and negative

moment regions, and to determine the effective slab widths by taking

into account the continuity of the structure. The studies that are

relevant to the work in these areas are reviewed in the following

sections.

3.2 Simply Supported Composite Beams

The experimental and theoretical studies which have been carried

out on simply supported composite beams illustrate their behavior and

positive moment failure modes [1-19].

Three modes of behavior are known to govern the positive moment

regions. The first one is shear failure, which takes place when the

connection between the slab and the steel beam is insufficient to

develop the ultimate moment capacity of the member. The second mode is

crushing of the concrete slab; this may occur if the degree of yielding

in the steel beam is insufficient, and is classified as a brittle fail

ure of the composite beam. The third mode involves the failure of a

18

major portion of the steel beam through extensive yielding; this is a

ductile failure.

Many types of mechanical shear connectors have been used to

transfer the shear force and to resist any uplift between the steel beam

and the concrete slab. A complete study on the effectiveness of several

types of connectors was given by Viest [9]. It was found that flexible

shear connectors, such as studs and channels, perform much better than

the rigid types, due to their ability to allow for the redistribution of

the internal stress resultants in the cross-section and along the beam

span; for example, this led to the development of the uniform stud

spacing criteria for composite beams.

The required number of shear connectors is usually determined by

dividing the ultimate force of the steel beam or the concrete slab by

the ultimate connector capacity. This is a simplified approach \'lhich

was developed on the basis of the studies of a number of researchers.

It was found that the ultimate strength of hooked or headed stud shear

connectors can be given by the equation [10]

Q = 0.5 A ~~ < A F u sc VLC~C - sc U

where

Qu

= ultimate shear capacity of the connector,

A = cross-sectional area of connector, sc

f' = concrete compressive strength, c

E = modulus of elasticity of concrete, and c

F = ultimate tensile strength of the connector steel. u

(3.1)

19

The left-hand side of equation 3.1 represents the shear ultimate limit

state of the connector; the right-hand side is the tensile ultimate

limit state.

The ultimate tensile force in the steel beam is

T = A F (3.2) max s y

where A = cross-sectional area of the steel beam, F = yield stress of s y

the steel, and the ultimate force in the concrete is

C = 0.85 fib t (3.3) max c e c

where b = effective width of the concrete slab -and t = concrete slab e c

thickness.

The ultimate flexural capacity and corresponding overall behav-

ior of composite beams have been demonstrated by a number of experimen-

tal studies [11-14]. Slutter and Driscoll [11] found that slip of the

shear connectors did not affect the ultimate moment capacity as long as

it was less than the one that accompanies the failure of an individual

connector.

In determining the ultimate strength of a composite section, the

concrete is usually assumed to carry compressive stress only, and ten-

sion is carried only by the steel beam. Formulas for computing the

ultimate moment capacity are given by Hansell et al. [3] and Salmon and

Johnson [8]. One of the primary variables is the effective width of the

slab. The magnitude of the effective width, which is used in deter-

mining the ultimate moment capacity, traditionally has been based on

linearly elastic solutions, probably because of the difficulty of

incorporating the nonlinear characteristics of materials as diverse as

concrete and steel into a theoretical evaluation.

20

The first experimental and analytical investigation of the

effective width was conducted by Mackey and Wong at the University of

Hong Kong in 1961 [15]. Using simply supported beams with continuous

double spiral shear connectors, the test results indicated that the

effective width was larger than the effective width which is predicted

by plane stress theory. An experimental study by Hagood, Guthrie, and

Hoadley [16] was aimed at determining the effective width in a series of

three composite beams. Their results confirmed the conclusions of

Mackey and Wong.

Adekola [17] formulated a theory for effective width of compos

ite beams based on deflection considerations and the use of the trans

formed cross-section. He also studied the influence of partial inter

action. It was found that the effective width increases with an

increasing degree of interaction, because larger portions of the con

crete will participate in the stiffness of the composite T-beams. He

also concluded that the effective width formulas in the British design

specifications [18] were too conservative.

Vallenilla and Bjorhovde [19] studied the effective width of

composite beams with steel deck, based on deflection considerations and

using statistical analysis of a large number of tests of composite beams

with formed steel deck. They found that the length of the beams and the

degree of interaction are the major parameters for the effective width.

Two sets of effective width crit:eria were proposed, basing the require

ments on the degree of interaction. Observations were also made to the

21

effect that the current AISC design rules [4] tended to underestimate

the beam deflections.

Grant, Fisher, and Slutter [5] conducted a major study of com-

posite beams with formed steel deck. Seventeen simply supported compos-

ite beams, made of lightweight concrete and steel deck connected by

3/4-inch headed studs, were tested. The aim of the study was to verify

whether the effective width of the concrete slab as defined by the

(16tc

+ bf

, requirement of AISC [4] cCDld be used in calculating the

flexural capacity of the composite beam. The results showed that this

approach was very conservative. Subsequent theoretical and experimental

studies have shown that the effective width is independent of the slab

thickness [20].

3.3 Composite Beams in the Negative Moment Region

The effectiveness of the shear connectors and the interaction

between the concrete slab and the steel beam in the negative moment

region were first studied by Siess and Viest [21]. Two continuous two-

span composite beams were tested, one of which had shear connectors in

the positive as well as in the negative moment region, and the other

using shear connectors only in the positive moment region. The longi-

tudinal slab reinforcement in the two specimens was identical. The test

results and the strain measurements at the center support showed that

complete interaction existed in the negative moment region for the first

specimen; and only partial interaction was found for the other specimen.

The study demonstrated that the shear connectors do have an effect in

the negative moment region. Advantage, therefore, can be taken of the

composite behavior near the supports of continuous composite beams.

22

Johnson, Greenwood, and Van Dalen [22] also studied, the behavior

of stud shear connectors in the negative moment region. Fifteen pushout

tests were performed; the reinforcement in the concrete slab was put

into tension by stressing the end of the reinforcing bars and a pushout

force was applied. The test results showed that the load-slip relation

ship and the ultimate load of the studs are affected by the transverse

slab reinforcement, the cracking pattern, the mean tensile strain in the

slab, and the effects of uplift. The authors recommended that the s~uds

in the negative moment region (or when the slab is in tension) should be

designed for 80 percent of their ultimate load; however, this was based

on tension pushout tests only, and the 20 percent reduction in the ulti

mate capacity of studs may be too conservative for connectors in actual

beams.

The importance of the slab steel reinforcement ratio at the

interior supports of continuous composite beams has been examined by

several researchers. Slutter and Driscoll [11] tested a two-span con

tinuous composite beam. The results confirmed the importance of the

longitudinal reinforcement ratio and the shear connectors in the nega

tive moment region. It was found that increasing the longitudinal rein

forcement over the supports increases the collapse load and controls

cracking of the concrete slab.

Park [23] tested four two-span continuous composite beams, each

connected to the concrete slab by channel connectors. The major purpose

of this study was to determine the effect of the longitudinal

23

reinforcement ratio on the ultimate negative bending moment. The nega-

tive reinforcement ratio was varied from 5.56 to 0.184 percent. The

investigation examined service load as well as ultimate load behavior,

including unloading effects. The final failure was precipitated by

insufficient transverse slab reinforcement; this led to longitudinal

splitting.

The beams with smaller reinforcement ratios had cracks becoming

visible at early stages; for example, the beam with the smallest rein-

forcement ratio developed the first visible crack over the interior sup-

port at a load equal to only 14 percent of the ultimate. The primary

conclusion drawn from the study of Park [23] was that simple plastic

theory can be used in the design of continuous composite beams in the

negative moment region.

Van Dalen [24] conducted a study to determine the optimum area

of longitudinal reinforcement in the negative moment region. Seventeen

composite beams were tested as double cantilevers, with the concrete

slabs placed on the tension side. The main conclusions were: 1) local

buckling was the most common failure made; 2) the optimum longitudinal

reinforcement area in the negative moment region can be given as

where

A F 0.2 < r yr < 0.4

A F s ys

A = area of reinforcing bars over the interior support, r

F = yield stress of the reinforcing steel, yr

A area of the steel beam, and s

F = yield stress of the steel beam; ys

(3.4)

24

and 3) the shear connectors in the negative moment region can be assumed

to share the tensile force equally. The connectors, therefore, can be

uniformly distributed between the supports and the inflecti?n points.

Finally, it was noted that bottom transverse reinforcement not less than

0.33 percent should always be provided to prevent longitudinal slab

splitting.

A minimum reinforcement ratio over the supports has also been

proposed to limit the crack width at service load [25-27]. Tachibana

[25] proposed a minimum value of 1.5 percent, in order to limit the

crack widths to 0.2 mm at service load. Fisher, Daniels, and Slutter

[26], however, showed that this could be impractical in many cases, and

suggested instead that a ratio of 1.0 percent would be sufficient to

control cracking and to provide a more favorable stress distribution in

the reinforcing bars at service load. Other recommendations were made

by Randl and Johnson [27], including an equation for calculating the

minimum reinforcement ratio. This was given as

::: 0.17

where

(f )2/3 cu

f Y

P = minimum reinforcement ratio, cr

f = cube strength of concrete (Ksi), and cu

f = yield strength of reinforcing bars (Ksi). y

An additional equation is given by Johnson and Allison [28].

Ductility and rotation capacity in the positive and negative

moment regions are among the most important considerations when applying

simple plastic theory to continuous composite beam design. The

25

ductility requirements of composite beams in the positive moment region

were studied by several r~searchers [29-31].

Rotation capacity in the negative moment region is ~lso essen

tial; this is obtained by preventing local buckling from taking place in

the lower flange (and the web) of the steel beams before plastic hinges

develop in the negative and positive moment regions. The negative

moment rotation capacity and compactness criteria were examined in sev

eral research projects [32-34].

Rotter and Ansourian [29] developed an expression to measure the

ductility in the positive moment region of composite beams. It was

found that the material properties as well as the geometry of the cross

section of the composite beam influence the ductility in the positive

moment region. Further experimental studies by Ansourian [30,31] con

cluded that the use of simple plastic theory in the design of continuous

composite beams required having a compact section to allow sufficient

rotation capacity in the negative moment region. Secondly, the cross

sectional dimensions should be such that concrete slab crushing would

not occur before a plastic hinge was developed in the positive moment

region. Finally, an adequate shear connection was required to prevent a

shear failure. It was shown that the rotation capacity was inadequate

in composite beams with small slabs or low-strength concrete, large

steel beams or high yield stress, because the concrete slab was crushed

before the steel beam yielded.

Compactness of the steel beam in the negative moment region was

first studied by Climenhage and Johnson [32]. The authors tested 18

double cantilever specimens with rigid joints. They found that the

26

parameters which influenced the rotation capacity of the section in the

negative moment region were: 1) the yield stress of the reinforcing

bars and the steel beam, 2) the flange and web width-to-thickness ratio,

and 3) the ratio of the force in the slab reinforcement to the force in

the steel beam. The authors also presented compactness criteria that

apply to rigid joints. It was observed that local buckling first

occurred in the steel beam flange, followed by web deformations until

the load started dropping off. Longitudinal web stiffeners, therefore,

were recommended.

Hamada and Longworth [33,34] studied local flange buckling and.

lateral buckling in the negative moment region of continuous composite

beams. It was shown that: 1) lateral buckling will control if the com

pression flange is stiffened by a cover plate, 2) the moment producing

local flange buckling will decrease significantly as the flange slender

ness and the beam span increase, 3) the failure modes will be affected

significantly by the amount of longitudinal slab reinforcement ratio,

and 4) an increase in the slab reinforcement ratio will decrease the

curvature at the point where local buckling occurs. The authors also

recommended limits on the flange slenderness to prevent local buckling

before plastic hinge development.

Garcia and Daniels [35] performed preliminary studies to deter

mine the effective width in the negative moment region of continuous

composite beams. They concluded that the effective width is almost

equal to that of the positive moment region.

An analytic~l elastic study by Ansourian [36] was aimed at

determining the effective width along a continuous composite beam. It

was concluded that the effective width of such beams varies along the

span, and the important factors were found to be:

27

1. The ratio of the panel proportions, b/~, where b is the distance

between adjacent girders and ~ is the span length.

2. The type of loading (distributed or concentrated).

3. The relative stiffness of the steel beam and the concrete slab.

Ansourian observed that item 2 was very important; item 3 had minor

influence.

The author found that the effective width at mid-span equals

0.27 times the beam spacing for a square panel having distributed load.

This will increase to be equal to the beam spacing as the length of the

span increases. For the same square panel with a concentrated load at

mid-span, the effective width drops to 0.18 times the beam spacing, but

this will also increase. to 1.0 as the beam length increases. Comparing

these with the current design criteria [4,6] shows that the AISC speci

fication formulations for determining the effective width give conser

vative design for uniformly distributed load and unconservative design

for concentrated loads.

In the negative moment region, where the support acts as a con

centrated load, Ansourian concluded that the effective width should be

determined as for a concentrated load. He also proposed spitable design

criteria.

Ansourian noted that cracks in the negative moment region

increase the effective width in the positive moment region; he found the

increase to be about 11 percent. It was also observed that stiffening

28

the supports by a transverse beam decreases the effective width at mid-

span by nearly 11 percent for uniform type loading.

Fahmy [37] studied the effective width in the positive moment

region of a beam-to-column connection. He found that the main factors

which influence the value of b were the slab length-to-width ratio and e

column width-to-slab width ratio. It was shown that b increases along e

with these two ratios. The occurrence of slip and a decrease of the

interaction between the steel beam and the slab will increase the effec-

tive width slightly, although the change is too small to warrant being

considered in design. The effective width was found to have a minimum

value at the column face. Using these and other data, the author recom-

mended that the same b be used in elastic and plastic frame analysis. e

Heins and Horn [38] analytically studied the effective width at

ultimate load for multigirder composite systems, using a finite-

difference approach. They realized that at ultimate load the axial ten-

sion force in the steel beam and the compression force in the concrete

slab might not be in equilibrium. Thus, computing the effective width

at mid-span of the edge girders using the axial force in the steel beam

was found to give larger values than when it was calculated on the basis

of the slab axial force; however, this was found to be reversed for the

interior girders. The effective width was determined to be very much

influenced by the panel aspect ratio, b/~; as b/~ increases, b for an e

interior girder also goes up, whereas it decreases for an edge girder.

This effect is attributed to the load distribution, where for large

aspect ratios the contribution of the edge girder is smaller than when

b/~ is reduced.

29

The effects of the type of connections in composite frames on

the ultimate moment carrying capacity have been studied by several

researchers [39,40]. Johnson and Hope-Gill [39] showed that there is a

need for new criteria to design semi-rigid connections. The findings

indicated that the semi-rigid connections can be easily designed to

carry the total plastic moment of the steel beam, M , at less cost than p

with a fully rigid joint. This can be done by providing enough rein-

forcing bars in the negative moment region of the concrete slab. The

tension force in the slab reinforcement, A F , multiplied by the disr yr

tance between the center of gravity of the steel bars and the center of

rotation of the steel beam gives the ultimate resisting moment of the

semi-rigid joint. The results also indicated that the semi-rigid con-

nection has a larger rotation capacity than the fully rigid one; thus,

local buckling in the steel beam will be less critical. Simple plastic

design, therefore, was recommended as the suitable approach for beams

with semi-rigid connections.

Van Dalen and Godoy [40] tested simple, rigid, and semi-rigid

composite beam-to-column connections. It was found that a composite

beam-to-column connection with 0.46 percent longitudinal reinforcement

ratio in the negative moment region has an ultimate moment capacity that

ranges from 1.5 to 6 times the ultimate moment of a corresponding bare

steel connection. The moment-rotation curves of the tested specimens

indicated that the composite beam-to-column connections had almost the

same ultimate moment capacity as the corresponding bare steel connec-

tions; however, the rotation of the composite connection with a simple

joint between the steel elements had almost four times the rotation of

30

one with a rigid joint, and two times the rotation of one with a semi

rigid joint. The authors stated that the moment capacity of each speci

men was equal to or larger than the theoretical plastic moment of the

composite beam. Simple plastic theory, therefore, was recommended as

suitable for the design of composite connections; however, the effects

of the type of joints between the steel members on the effective width

have not been examined.

Work is currently being conducted at The University of Minnesota

[41] and The University of Texas at Austin on the response characteris

tics of certain composite beam-to-column connections. In particular,

these studies aim at developing moment-rotation criteria for such ele

ments, for use in the analysis and design of composite frames; however,

the results are not yet available.

CHAPTER 4

ANALYSIS OF COMPOSITE FLOOR SYSTEMS

4.1 Introduction

As described earlier, a composite floor system usually consists

of a concrete slab, with or without a steel deck. The slab is supported

by steel beams and girders, and is attached to them by flexible shear

connectors. Figure 1.4 shows an example of a typical system.

The design of this continuous system is usually simplified to

the extent that it is treated as a group of isolated single composite

T-beams, where each steel beam or girder is assumed to act together with

a certain portion of the concrete slab. Together, the assembly of indi

vidual composite members forms the composite floor [4,6). In practice,

the structural engineer determines the dimensions of the concrete slab

and the steel beam, based on traditional beam theory. Among other

things, he assumes that plane sections remain plane during bending. ~he

behavior of the continuous system, however, which really behaves in a

two-way fashion, may be significantly different from the individual

beams. This is especially true when the system is loaded to a point

where the response is no longer elastic.

It is important, therefore, to acquire a full understanding of

the behavior of a continuous composite floor system. In particular, the

interaction between the steel and the concrete and the effective partic

ipation of the concrete slab are phenomena that need to be fully

31

32

analyzed and understood, including the nonlinear range of behavior. The

latter consideration is essential for ultimate strength and ductility

criteria.

A determination of the effective width of the slab which takes

into account the continuity and the boundary conditions of the slab and

the steel beam is needed to provide improved analysis and design cri-

teria. This requires a rigorous evaluation of the continuous composite

floor system, which involves solving the plate bending and membrane dif-

ferential equation simultaneously with the beam-column differential

equation, and satisfying the boundary conditions at the interface.

4.2 Plate and Beam Governi~ Differential Equations

In general, the concrete slab in the composite floor system acts

as a plate subjected to bending and membrane forces. Figure 4.1 shows

the internal stress resultants for an element dxdy of a plate supporting

a perpendicular distributed load, q. The elastic properties of the slab

material are generally not the same in all directions, since the slab

with its reinforcing steel is actually anisotropic; however, for the

case of plane stress in the xy plane, which applies to the reinforced

concrete slab, the material can be assumed to be orthotropic. This

means that the constitutive relationships in the orthogonal directions

are the same. Taking these as the coordinate planes, the stress and

strain relationship can be expressed as follows [41]:

(a)

(b)

(c)

Mxy~

Nxy ;

~

N x

q(d)c.dy)

J." --t-- - + ---I-

dO x I

Myx My

I

-+-I

~

yx ...

I

-+-I

,

I

dx dx

dM x M,,+ dx

'" d x

dN x Nx + clx dx

dN xy Nxy+ dx dx

dN dN x --yx- cly dy !Ny+ ~ dy

33

Figure 4.1 Unit Internal Stress Resultants for a Plate Element. -(a) Shear forces. (b) . Bending and twisting moments .. (c) Membrane forces. From Timoshenko and Woinowsky-Kreiger [42] •

34

a E' E" 0 e: x x x

a = E" E' 0 e: (4. 1 ) Y Y Y

T 0 0 G Yxy xy xy

where a , a , and T are the stresses in the xy plane and E , E , and x y xy x y

yare the corresponding strains. E' E' E", and G are the con-xy x' y' xy

stants which define the elastic properties of the material; these are

given by:

E E'

x = x (1 - 'V 'V ) x Y

E E' = Y.

y (1 - 'V 'V ) x Y

E" = 'V E = 'V E Y x x y

where

E = modulus of elasticity in the x-direction, x

E = modulus of elasticity in the y-direction, y

'V = Poisson's ratio in the x-direction, x

'V = Poisson's ratio in the y-direction, and y

G = shear modulus of elasticity. xy

(4.2a)

(4.2b)

(4.2c)

Assuming that plane sections remain plane during bending, the relation-

ships between strains and curvature are:

E a2w = - z x ax 2

(4.3a)

e: a2w = - z y ay 2

(4.3b)

35

where w is the displacement in the z-direction. Using the above rela-

tionships to find the stresses a , a , and T , and the moments M , M , x Y xy x y

and M in the slab, these are subsequently substituted in the differenxy

tial equation of equilibrium for the element shown in Figure 4.1 to give

the governing differential equation [38,42), as follows:

= q + N x

where the plate bending stiffness terms are

where

E t3

D x

= x 12(1 - V V

x y

')

E t.J

D = Y. Y 12 (1 - V V x y

H = D1 + 2D xy

D =VD =VD 1 Y x x Y

D xy

N x

= Gt

3

12

N Y

)

)

F = Airy stress function,

t = slab thickness, and

N xy

=

q = distributed perpendicular load per unit area.

(4.4)

(4.5)

(4.6)

(4.7)

(4.Ep

(4.9)

(4.10)

36

Specific values of the plate bending stiffness terms for rein-

forced concrete slabs are given in Timoshenko and Woinowsky-Kreiger

[42] . The membrane forces N , N , and N can be obtained by solving x y xy

the strain compatibility relationship in terms of the Airy stress func-

tions; thus:

1 a4F 1

---+(G E '"I 4 Y oX

v x

E x

where all of the terms have been defined previously.

(4.11)

Equations 4.4 and 4.11 represent only the response of the slab,

which is a part of the composite floor system. These ~quations have to

be solved together with the governing differential equation of the steel

beam to satisfy the interface boundary conditions. The steel beam is

governed by the general beam-column differential equation, which is

given by [43]

E I s s

The parameters of equation 4.12 are:

E = modulus of elasticity of the steel; s

I = moment of inertia of the steel beam; s

(4.12)

P = axial force, applied at the centroid of the steel beam; and

q = distributed load per unit length.

4.3 Application of Plate and Beam Differential Equations to a Composite Floor System

Figure 4.2 shows the layout of a typical composite floor system

of a building. In the following evaluations, this type of a floor

37

I I I II II I I II II I I II II I I II II

- ::Ike I ---- - - - ---~------- - --»r---- n - ---- - - - --..... rr ----------"fT--I I II I I I II 1 I I I I I I I II I I I II I II II I I I II 1 II II I II II 1 II II I I I Stee 1 Be am II I

---'~Cl - - - -- - -- - - ~c-- - --- -- --.:fr/:----rr- ----- ----~ -- --- -- --- ff--II II~lumn:: I I L. I I I I II al I I I I 11 ~ II II II I I II II t!J1I II I I II II II alII II II !! II II II (1)11 11

--']JrrCI ----------~Cl __________ ~----111 ----------n----------ff--I I II 11 I I II 11 II II II I I II II

Figure 4.2 Plan of a Composite Floor System.

38

system will be the physical representation of the models that are

developed.

Figure 4.3 illustrates the forces that develop at the slab-beam

interface of a composite floor system due to the applied loading. An

analytical solution for this case requires solving equations 4.4, 4.11,

and 4.12 along with satisfying the boundary conditions. For a simple

system with simple boundary conditions, and assuming that the slab has

the same r.einforcement ratio in the x- and y-directions, equations 4.4,

4.11, and 4.12 can be solved by using, for example, a series solution.

This was done for the case of a semi-infinite slab that is rigidly

attached to a steel beam (i.e., slip is assumed not to take place), and

an effective width was calculated on this basis [44,45].

The general solution of equations 4.4 and 4.11 for an isotropic

plate, loaded symmetrically and simply supported in bending, including

slabs [44], is given by the following stress and deflection functions:

F = L sin ay[A cosh ax + B x sinh ax] n n

(4.13) n

w = L sin ay[q + P cosh ax + Q x sinh ax] n n n

n

where

n = 1 , 2, 3, 5, ... , nTI

a = L

(4.15)

4 =

4gL qn 5

(nTI) D (4.16)

L dimension of the plate in the y-direction;

q = vertical load per unit area;

Nx+dN x

--- -NXy+dNXyt(~ __________ -=====~~==~

Slab Mid-Surface

!..q'f-Shear Connector ~>x,u

v,y I ~z.w m «<; I ! II-

Beam Centro i d

Figure 4.3 Transfer of Girder Slab Forces. -- From Ansourian [36] and Heins and Horn [38].

LV I.D

40

Et3

D = ---"':;"';;"'---12 (1 _ \)2)

(4.17 )

and the constants A , B , P , and Q in the series have been derived to n n n n

satisfy equilibrium and the compatibility conditions between the slab

and the steel beam [44]. This approach is very complex, however, even

for the simplest case. This is because the stress and deflection func-

tions have to be found by a trial-and-error procedure.

For more complex boundary conditions, approximate solutions such

as a finite-difference approach have been used to solve the governing

diff~rential equations [38,46]; however, the finite-difference method

requires much work to set up the linear system of equations and to sat-

isfy the boundary conditions at the plate-to-girder interface [38].

Specifically, it was assumed that the strain at the top of the girder

must be equal to the strain at the bottom of the slab, based on elemen-

tary beam bending theory. In the case of realistic composite members,

however, it is obvious that it is important to consider the slip between

the steel beam and the concrete slab. When this occurs, plane sections

will no longer remain plane for the assembly as a whole. In addition to

the complexity of satisfying the boundary conditions with the linear

system of equations, this limitation makes the finite-difference method

impractical to use when solving for cases of general geometry and bound-

ary conditions.

The finite-element approach has been used successfully in the

elastic analysis of composite floor systems [47]. The general theory

and uses of this method have been described in detail for numerous

applications [48,49], and will not be repeated here. Basically, it has

41

been found that a fine mesh of finite elements for the slab and the beam

appears to give the best solution, irrespective of the complexity of the

boundary conditions. In the following analysis of a compos~te floor

system, the finite-element approach has been adopted as the tool to use

in the analysis of the effective width in the positive and negative

moment areas of a composite beam.

4.4 Finite-Element Model of a Composite Floor System

A finite-element model similar to the one used by Ansourian [47]

has been adopted for this study; however, Anscurian's model did not con-

sider cracking in the concrete slab, the flexibility of the shear con-

nectors, and the degree of fixity of the connection between the steel

beam and the column. All of these factors will be included in the fol-

lowing analysis. In addition, this study also will include an evalua-

tion of the nonlinear behavior of the composite system, for which the

finite-element solution is the most practicable one. This will provide

the numerical solution of a highly nonlinear structure, including the

ultimate strength and behavior. To ease the complexity of the develop-

ment, the elements of the composite floor will be evaluated separately.

4.4.1 Finite-Element Model for the Concrete Slab

The concrete slab of a composite floor system is subjected to a

combi.nation of axial and bending stress resultants. The slab" there-

fore, will be represented most accurately by a group of quadrilateral,

isoparametric thin shell elements, which have combined bending and

membrane stiffness. Three-dimensional, thick prismatic shell elements

42

could be used, but this would result in a large amount of work in terms

of data preparation for the solid elements, a substantial increase in

the computer time, and considerable time in manipulating the computer

output. Most importantly, however, the th~ck shell elements will pro

duce only a minimal improvement in accuracy [47].

The material properties of the flat thin shell element were

derived from the properties of the concrete and the reinforcing steel

bars, which together form the slab in a composite system. In the analy

sis, it was assumed that the concrete strength in tension is negligible,

and that any tension will be carried by the reinforcing steel bars only.

This is also the most realistic approach to considering cracking of the

concrete slab.

The model which was used to derive the properties of the equiva

lent new material for the shell elements is illustrated in Figure 4.4.

This prismatic element, which is assumed to represent the slab in the

composite section, is sandwiched between two stiff plates, and uni-axial

forces are applied until failure.

The assumptions which are used in the analysis can be enumerated

as follows:

1. The reinforced concrete slab of the composite floor system is

assumed to have the same reinforcement ratio in the x- and y

directions and in the positive and negative moment regions.

Thus, the material of the slab is assumed to be isotropic.

2. The cracks in the concrete slab will extend to the mid-surface

of the slab, due to bending and membrane action.

43

Assumed Cracked Concrete Thickness = t/2

~ __ R~einforcin9 Steel Bars ."'i

1......1 Stiff Plate

~ I"r------L ----q;....--c:=j

Uncracked.Concrete Thickness = t/2

Figure 4.4 Assumed Cracking Model for the Slab.

44

3. The cracked concrete is ignored in the pr~smatic element, and

the uni-axial forces will be carried by the uncracked concrete

and the reinforcing bars.

4. The equivalent material, which consists of the uncracked con-

crete and the reinforcing bars, has a linearly elastic-perfectly

plastic stress-strain characteristic, as shown in Figure 4.5.

In the figure, a is the equivalent yield stress and E is the y

equivalent modulus of elasticity for the reconstituted homo-

geneous slab.

5. It can be seen from Figures 4.6 and 4.7 that nOL~al weight con-

crete with a failure strength of up to f' = 6000 psi reaches its c

peak strength at a strain of 0.002, which is on the same order

of magnitude as the yield strain of the reinforcing bars that

are commonly used in slabs (G~ade 60 steel). It was assumed,

therefore, that the steel stress at failure will reach yield

(F ) and the concrete stress will reach f'. It should be noted y c

that in the actual structure the slab element is going to have

axial load as well as bending moment. Nevertheless, this model

is proven to be accurate to describe the average behavior of the

reinforced concrete slab.

If the element in Figure 4.4 undergoes a displacement ~L at the

ultimate load, then the corresponding force will be

F = F + F (4.18) s c

where

45

(1

----,---------

~--------------------------------~---~ £

Figure 4.5 Behavior of the Equivalent Material for the Shell Elements.

6~----------'------------r----------~-----------'

......

Concrefe stroin, in./in.

Figure 4.6 Typical Concrete Stress-Strain Curves. -- From Winter and Nilson [50].

46

47

60 KSI

~--~~------------------------------------~£ 0.002

Figure 4.7 Idealized Stress-Strain Curve for 60 Ksi Reinforcing Steel. -- From Winter and Nilson [50].

48

F E: E A = flL E A s 5 sr sr L sr sr

(4.19)

A ~L

A c

E c

F = E: E = c c c 2 L c 2 (4.20)

and

A = area of gross concrete cross-section, c

A = area of reinforcing bars, sr

E = modulus of elasticity of the concrete, c

E = modulus of elasticity of the steel bars, sr

E: = longitudinal strain of the reinforcing bars, and s

E: = longitudinal strain of the concrete. c

flL and L are as defined in Figure 4.4. From equations 4.19 and 4.20,

the total force in the model can be expressed as

A F = ~L (E A + E c

L sr sr c 2

The average stress in the steel and concrete is

F a = -----"'-----av A + (A /2)

sr c

(4.21)

(4.22)

which is equal to E(~L/L), where E is the equivalent modulus of elastic-

ity. Consequently,

F . - ~L

A + (A /2) = E ~ sr c

substituting for F from equation 4.21 gives

E A + E (A /2) ~L sr sr c c L A + (A /2)

sr c

and, therefore,

- ~L = E

L

49

E = E A sr sr

A + sr

+ E (A /2) c c

(A /2) c

(4.23)

The equivalent modulus of elasticity of the reinforced concrete slab is

then

E = E + 2PE c sr

1 + 2P

where P is the reinforcement steel ratio, given as A /A. sr c

(4.24)

To determine the equivalent yield stress, a , the reinforcing y

bars were assumed to be distributed over the cracked concrete area.

This had been previously defined as extending through one-half of the

slab thickness, giving the cracked area as A /2. The equivalent yield c

stress then becomes

A F + (A /2)f' a sr yr c c =

Y A c

(4.25)

or

f' F

c a = + y yr 2

(4.26)

where F is the yield stress of the reinforcing steel in the slab. yr

To apply equations 4.24 and 4.26; a minimum amount of reinforce-

ment steel should be used. In actual practice, this depends on engi-

neering calculations that provide a sufficient amount of steel to carry

the tension stresses in the slab.

An approximate method is used to determine the minimum rein-

forcement ratio, P . , as follows. The maximum tension force in the mLn

slab of a composite section, as shown in Figure 4.8, will be equal to

k1--------b-------~

. ['A+d p'." 'a' .. Q'.~" ~'d" •.••. , '0 r<" F ', .. ",.,.,. .. 7.,;.,).~ .. ·· ... ·4)9' ... i .... ~+-lBil---~> S " 4 .' , : ~,':' ~ ,:41 . _':.'" ,,. : ,f.t ' '~ """"., u .' • t .. ,., .. ,' ........ ·'·.4CI.: .••.•. O • " , ••• ~" ~.,' ," : ," ..... ~ ,,' Q ' •. ~. "... .": .,. I,.~ , ". ',' " ." ,- '., ., " ; 6",:, .' :i: ,,'.0 fI, '.=.0 ': " e Q.". , , 6·' A' .<J.~:.: ". ,,;' ~ , ",., , ., , ".. .. ,. , "" : .. .) '.. .,,,"" .,,, .. " .... ''/' ... -:-:,',

Figure 4.8 Assumed Distribution of Tensile Stresses in the Slab of a Composite Section.

50

51

F = A F (4.27) s sr yr

The line of action of F coincides with the centroid of the steel shape. s

This force is assumed to be the resultant of the tensile stresses that

are distributed as shown in Figure 4.8.

From the above assumptions, the following can be developed:

cr 3d' t

A F = Y b sr yr 2 t

(4.28)

or

A 3 d'

F = cr --A sr yr y 2 t c

(4.29)

where d' = the distance from the top of the concrete slab tG the cen-

troid of the reinforcing bars, F = yield stress of .the reinforcing yr

bars, and band t are defined in Figure 4.8. For realistic slab sizes,

the ratio d'it usually lies between 0.1 and 0.2 [50]. Assuming an aver-

age value of d'it = 0.15, the value of cr becomes y

2A F cr = sr yr

y 3(0.15) A c

or

cry = 4.44 PFyr (4.30)

The value of the equivalent yield stress cr from equation 4.30 y

should not be larger than that given by equation 4.26. Equating the two

expressions and solving for the resulting minimum reinforcement ratio,

P . , gives ml.n

f' c

pF + - = 4.44 pF yr 2 yr

and, therefore,

52

f' c

Pm in = 6.88 F yr

(4.31 )

Table 4.1 gives the values of P. for different concrete and m~n

steel strengths. These appear to be reasonable, especially when com-

pared with the recommendations of other researchers [26-28,50,51].

The value of P that produces concrete slab failure by crushing

of the concrete along with simultaneous yielding of the reinforcing

steel is defined as p balance, Pb

. According to the ACI Code [51], the

maximum reinforcement ratio, p , is taken as 0.75 Pb

' and is given in max

Table 4.2. A recommended practical value of P is P /2 [51]. Commax

paring the values of p in Table 4.1 to the values of p in Table 4.2, it

can be seen that the former are approximately equal to 0.5 times the

values of p of Table 4.2; this agrees with the recommendations of the

ACI Code [51].

As stated in thr> .. '. ,sumptions, in mode ling the s lab of the com-

posite floor system, it is assumed that p has the same value in the

positive and negative moment regions. This is considered reasonably

realistic for the slab. All of the plate elements, therefore, have the

same material properties in the negative and positive moment regions.

4.4.2 Finite-Element Model for Steel Beam, Column, and Stud Shear Connectors

The main and transverse beams, the column, and the stud shear

connectors are represented by conventional beam elements. These take

into account axial, flexural, shear, and torsional stiffnesses. The

properties of the beam and the column elements are derived directly from

the properties of the cross-sections of the primary members. Some

Table 4.1 Recommended p . for the Slabs of . m~n

Compos~te Beams.

Concrete for Steel Grades a

Pmin Strength

60b

a

b

(Ksi) 40 70

3 0.011 0.0073 0.0062 4 0.0145 0.0097 0.0083 5 0.0182 0.0121 0.0104 6 0.022 0.0145 0.012

For P larger than the values in this table, equation 4.26 should be used to calculate a ; otherwise, equation 4.30 should be u~ed.

Grade 60 reinforcing steel is the most common for slabs.

53

Table 4.2 Maximum Reinforcement Ratio Based on the ACI Code [51]. -- P =

max 0.75 Pb .

Concrete for Steel Grades a

Pmax Strength

(Ksi)

3 4 5 6

a From

2.

3.

40 60 70

0.0277 0.0159 0.0127 0.03 0.0201 0.016 0.0408 0.0235 0.0189 0.0458 0.0264 0.0212

Winter and Nilson [50] and ACI [51] :

f' c 87,000

= 0.75 (0.85 81 F -8-7"::",0"'-0:""';0;"";"';;+-F) y y

Practical recommended p = p 12, or max p = 0.18 fllF .

c y

P = 200/F .. min y

54

modifications, however, are applied to the properties of the beam ele

ments which are used to represent the studs.

55

Figure 4.9 shows the complete discretization of a composite

floor system. The steel beams and the column are modeled by lines of

beam elements. The studs are modeled by a row of vertical beam ele

ments, linking the beam axis to the mid-surface of the slab. In this

model, the length of the elements which represent the studs is much

longer than the actual length of the studs. The axial and bending

stiffnesses of the model elements are increased to account for the addi-

tional length.

The increase of the stiffnesses of the modeled elements is· based

on maintaining the slip and the eccentricity of the steel beam with

respect to the slab as close as possible to those of the real structure.

The slip between the steel beam and the concrete slab is related to the

horizontal deflection of the studs, and the eccentricity of the steel

beam with respect to the slab is related to both horizontal and vertical

deflections; therefore, the bending and axial stiffnesses of the study

model are increased to give deflections equal to the real ones.

In the actual structure, a stud will assume a deflection shape

similar to the one shown in Figure 4.10a. This has been sUbstantiated

through numerous full-scale tests. The stud is assumed to be fixed at

the top flange of the steel beam, while the head will have a rotation

equal to zero. This is because the diameter of the stud head is much

larger than the body, and the concrete around the stud head, therefore,

prevents it from rotating.

~Quadrilateral Flat Shell Element at 51 ab Mid-Surface

Transverse

Beam Axis

Column Axis

Figure 4.9 Finite-Element Idealization of a Composite Floor System.

56

r--., ... -r" ~----'7.\ \ <.- I \~ \ ~ Ll

·==-1

(a)

Beam

.6 2

.LfL-r-"l ~ p I S 1 I \ I I \ \ \ \

R

\ \ I 1

Stud is Fixed at The Axis of The Steel Beam

(c)

Stud is Fixed at The Top of Steel Beam

(b)

Figure 4.10 Deflection and Lateral Load Idealization of a Stud in a Real Structure and in the Finite-Element Model. --

57

(a) Deflection of a flexible shear connector. (b) Stud in a real structure. (c) Stud in a finite-element model.

58

The shear forces that are exerted by the slab on a stud may be

distributed trapezoidally. An infinitesimal slip, however, will change

the distribution to a shape that will b~ very close to uniform, as shown

in Figure 4.10b.

In the finite-element mod21, the studs will be assumed to be

fixed at the axis of the steel beam, as shown in Figure 4.10c. The

other end condition is the same as in the real structure, but the shear

force from the slab in the finite-element model will be a concentrated

load at the common node of the stud and the slab elements, as shown in

Figure 4.10c.

To keep the slip and the eccentricity of the beam with respect

to the slab as close as possible to reality, the two structures shown in

Figures 4.10b and 4.10c should have the same deflection, i.e., ~1 = ~2.

The deflection ~1 of the stud head in the real structure due to

the shear stress in the slab, which is assumed to be uniformly distrib-

uted along the stud, as shown in Figure 4.11a, is calculated using an

energy approach. This is accomplished as follows.

The strain energy of the eq~ivalent stud structure as shown in

Figure 4.11b is given as

1 ~1

M2dx U = J (4.32)

2EI1 0

where

E = modulus of elasticity,

11 = moment of inertia for the actual cross-section of the stud,

~1 = the actual length of the stud, and

e =U =V =0 a a a

( a)

59

Figure 4.11 Beam Element Representing a Stud in the Real Structure. -(a) Deflection configuration. (b) Equivalent determinant structure. (c) Moment distribution. (d) Imaginary applied force F to calculate ~1'

60

M = function representing the moment distribution along the stud

due to the applied load.

Since the rotation at the end B of the stud is equal to zero, as indi-

cated in Figure 4.11b, the derivative of the strain energy with respect Q

to the moment at B will be zero; thus,

= 0

or

aM M ar:1 dx = 0

B

The moment along the beam can be expressed as

M = 2

wx 2

(4.33)

(4.34)

where w is the uniformly distributed shear. Taking the derivative of M

with respect to MB and substituting into equation 4.33 gives an expres-

sion for MB as

8M 1

8MB =

1 )/,1 2

J (-wx

MB)dx 0 --+ = EI1 0

2

After integration, this will be

Solving for MB gives

61

(4.35)

substituting equation 4.35 in 4.34 gives the moment distribution along

the beam as shown in Figure 4.11c. The deflection 61

at the end B will

then be equal to the derivative of the strain energy with respect to the

imaginary force F applied at B, as shown in Figure 4.11d; therefore,

or

.Q,1

f o

The moment function M becomes

2 wx

M = - Fx - 2 +

(4.36)

(4.37)

which gives oM/ox = - x. Substituting this into equation 4.36 and

solving for 61

gives

2 (_ wx

2

2 w.Q,l

+ 6 ) (- x) dx·

After integration, this will give

4 w.Q,l

24El1 (4.38)

The deflection 62

for the beam element is obtained in the same

manner, as shown in Figure 4.12. This is used to represent the studs in

the finite-element model, and is found to be

( a)

P

~'I------~-~-=~ j M~

(c )

JF ~~ _______ -<_ ... __ P...ll~) (P 12 ) /2

)(

(d)

Figure 4.12 Beam Element Representing a Stud in the Finite-Element Model. -- (a) Deflection configuration. (b) Equivalent determinant structure. (c) Moment distribution. (d) Imaginary applied force F to calculate ~2.

62

63

/::"2 = PQ, 3

2

12E12 (4.39)

where

Q,2 = length of the stud in the finite-element model, equal to the

distance between the steel beam axis and the centerline of the

concrete slab;

12 = modified moment of inertia of the stud in the finite-element

model; and

P = actual shear force in the study, equal to wQ,l.

Equating equations 4.38 and 4.39 and solving for 12 give?

pQ, 3 2

12E12

substituting for P = w1

Q,1 gives

pQ, 3 1

pQ, 3 2

Solving for 12 gives

where all the terms have been defined previously.

(4.40)

The eccentricity of the steel beam with respe~t to the slab is

also related to the axial deformation of the studs; therefore, the two

structures in Figures 4.10b and 4.10c should have the same axial stiff-

ness. Hence,

(4.41)

64

(4.42)

where Kl and K2 are the axial stiffnesses. Kl must be equal to K2 to

obtain the same axial displacements:

(4.43)

and, therefore,

(4.44)

where Ai = cross-sectional area of the stud; A2 = modified cross

sectional area of the stud, for use in the finite-element model; and £1

and £2 have been defined earlier.

A finite-element program for the analysis of composite floor

systems was developed, but this had certain limitations with respect to

loading and boundary conditions, and also had only linear capability.

For this reason, the general-purpose, finite-element program NASTRAN

[52] was used in the analysis, incorporating the previous developments

regarding the properties of the elements which were used. In addition,

a nonlinear analysis was conducted to study the behavior at ultimate

load.

4.4.3 Special Considerations

The composite floor systems which will be considered in this

study are symmetric with respect to the axes of the main and transverse

beams; therefore, only one-quarter of the slab will be discretized and

analyzed, as shown in Figure 4.9.

65

The boundary conditions along the edges will be derived from the

symmetry conditions; however, near the column edges, a gap between the

concrete slab and the steel column will be assumed. This i~ because the

concrete slab is discontinuous around the column, and a crack is likely

to open up due to the negative moment in that region. In practice, the

concrete slab reinforcement may be bent around the column, as shown in

Figure 4.13. The tensile stresses near the flanges of the column,

therefore, will be redistributed once a small crack has formed. The

only way to consider the slab as continuous is to make the reinforcing

bars continuous and anchored to the flange of the steel column by

welding or bolting. This, however, is an expensive and possibly imprac

tical solution; therefore, in the analysis, a gap is assumed to take

place in the element ABCD, which is shown in Figure 4.9

Chapter 5 details the nonlinear analysis that has been used for

the composite floor system elements.

Reinforcement Bars

Concrete Slab Column

Crack Crack

Transverse Steel Beam Main Steel Girder

Figure 4.13 Arrangement of Reinforcing Bars Around a Column in a Negative Moment Area. -- The bars in the other direction are arranged in the same manner.

66

CHAPTER 5

NONLINEAR ANALYSIS MODEL

5.1 Introduction

General representation of nonlinear behavior is usually

described by a load-displacement relationship for the member or struc

ture; such can also be characterized by a nonlinear stress-strain curve

as shown in Figure 5.1. Nonlinearity is caused by several factors.

Apart from the state of stress, the nonlinearity reflects the type of

loading, material properties, and discontinuities such as cracks or

holes.

In engineering structures, two kinds of nonlinearity are the

most common. These are geometric and material nonlinearities. For the

former, the equilibrium equations must be written with respect to the

deformed structure. In material nonlinearity, these equations must be

based on material properties that depend on stresses/s"trains that are

not known prior to the analysis. In this study, only material nonlin

earity will be taken into account, since it is assumed that the struc

tural and member deformations are sufficiently small to neglect second

order effects of this type. The material used in this study is assumed

to display linearly elastic-perfectly plastic stress-strain response.

The following discusses the basic yield criterion that has been

used in the development of the model of the composite member. A brief

description is also given of the numerical solution technique.

67

Q

Linear

Non 1 i ne ar

q

( a)

a Linear

Non 1 i near

~----------------------------------~~ E

(to )

Figure 5.1 General Response of Nonlinear Behavior. -- (a) Load displacement behavior. (b) Stress-strain behavior. Frc~

Desai and Siriwardane [53].

68

69

5.2 Yield Criterion

The von Mises yield criterion has been used for both the steel

beam and the slab of the composite members. It is recognized that this

is particularly suitable for members of ductile materials such as steel.

Although the Tresca yield condition is more conservative for use with

the slab, it was decided not to use it in this study. This was due to

the fact that the structure is symmetric and torsion will not take place

in loading; therefore, both Tresca and von Mises would give close

yielding conditions for symmetric loading. In addition, this was

prompted in part by the decision to model the reinforced slab as a homo-

geneous steel-like material, with properties as detailed in Chapter 4.

The von Mises criterion, therefore, has been used for all materials in

the composite member.

Details of the von Mises approach and its background are avail-

able in the literature [53-55]. Briefly, according to the criterion,

yielding begins in a component when the elastic distortion energy

exceeds the distortion energy required to cause yielding in a tension

test specimen of the same material. Based on this definition, the von

Mises yield condition can be expressed as follows in terms of the normal

stresses (0 , 0 , 0 ) and shear stresses (T , T , T ) [52]: x y z xy yz zx

1 [(0 _ 0 ) 2 + (0 2 x Y Y

2 + 6(T

xy

o ) 2 + (0 z z

2 -2 + T )] = 0

zx y (5.1)

where 0 is the yield stress in simple tension. For the case of plane y

stress, as in a slab element, this equation becomes:

70

(5.2)

The compressive and tensile stresses in the slab elements have

both been considered, because the concrete slab is assumed to have

enough reinforcement to carry the tensile stress. As discussed in

Chapter 4, the reinforced concrete slab material is assumed to be

cracked to the mid-surface, and an equivalent yield stress, a , has been y

derived, as given in equations 4.26 and 4.30 for the transformed mate-

rial of the slab elements.

In the beam elements, the normal stress due to bending moment

and axial force causes yielding to occur in the cross-section. Torsion

has not been considered because of the symmetry of the member and

loading. The beam, therefore, will yield when the normal stress, ax'

is equal to the yield stress for the material of the steel beam.

The incremental plasticity theory relates increments of stress

to increments of strain during the load-deformation process. During

incremental loading, the strain, dE .. , is assumed to be the sum of l.J

incremental elastic and plastic strains, e

and p

and be dE .. dE .. , can l.J l.J

expressed as

dE.. = dE ~. + dE~. ( 5 • 3 ) l.J l.J l.J

The incremental plastic strain can be given as

P dE .. l.J =A~

da .. l.J

(5.4)

using the normality rule. In equation 5.4, A is a positive infinitesi-

mal scalar proportionality factor, a .. is the stress tensor, and Q is l.J

called the plastic potential function [53].

71

For plastic material such as steel, the plastic function, Q, is

assumed to be identical to a yield function, F. This is expressed in

terms of the principal stresses. The stress function may be written in

the form [53]

F=~-K=O 2D

(5.5)

where K is a material constant equal to 0 //3 for simple tension tests, y

and J2D

is the second invariant of the deviatoric stress [53-55]. J2D

is conveniently given in terms of the principal stresses 01

, O2

, and 03

as

(5.6)

Thus, substituting for Q in equation 5.4 by F gives the well-known as so-

ciative flow rule

p de: ..

~J =A~

dO .. ~J

(5.7)

For linearly elastic-perfectly plastic material during yielding,

the stress point must remain on the yield surface. In other words, at

yielding, the yield function must .satisfy

dF = 0 (5.8)

Using this equation, the incremental stress equation can be expressed as

where [Cep

] represents the inelastic constitutive matrix. Derivation of

this matrix is presented by Desai and Siriwardane [53]; for linearly

elastic-perfectly plastic material, it is given as

dO .. = 2GdE .. + KdIlO .. - ( ~J ~J ~J

GS dE S .. mn mn ~J

K2 (5.1 Oa)

72

or in matrix notation as

(5.10b)

where

S .. = deviatoric stress tensor = 2GE .. , ~J ~J

E .. = deviatoric strain tensor E .. - (11/3)0 .. ,

~J ~J ~J

G = the shear modulus,

11 = first invariant of the strain tensor = E .. = Ell + E22 + E33

, ~J

K = bulk modulus, and

0 .. = Kroneker delta = 0 for i ~J

:I j and 1 for i = j.

5.3 Numerical Method of Nonlinear Analysis

In nonlinear structural analysis, the loads are applied to the

structure in increments. This causes yielding to spread gradually in

the components of the structure. For each load increment, the equilib-

rium equation must be solved and the incremental stresses and strains

found. The internal stresses at any load stage must satisfy the equa-

tion of equilibrium along with the yield and boundary conditions. The

procedure is repeated for each load increment until a sufficient number

of plastic hinges have formed to cause the failure of the structure.

Thus, equilibrium is no longer maintained. In the numerical solution,

this is reflected by the lack of convergence of the iteration procedure.

The basic approach of this method is summarized in the

following. Details of the procedure are given by numerous sources

[e.g., 52]:

1. The total load, QT' on the structure is divided into increments,

6Q .• ~

73

2. The initial stiffness matrix is formed on the basis of an ini-

tial point of stress, using a tangent stiffness form. This is

given as

[K]O = J [B]T[C]O[B]dV (5.11) v

where

[K]O = the initial stiffness matrix of the structure,

[B) = strain-displacement transformation matrix, and

[C]o = initial stress-strain matrix.

3. The equilibrium equation of the structure, which is

(5.12 )

will be solved for the first load increment to give the incre-

mental displacement, {~q}l. This, in turn, gives the total dis-

placement as

(5.13)

where {q}O is the initial displacement, which is usually equal

to zero.

4. The incremental strain then can be calculated from equation 5.14

as

{ M: } 1 = [B) {~q} 1 (5.14)

which leads to the total strain, given as

(5.15)

where {c}O is the initial strain.

5. The incremental stress is found from

(5.16)

and the total stress becomes

74

(5.17)

6. After finding the stress and the strain, the stress-strain

matrix, [C]O' is revised to [C]l' using the constitutive

relationship.

7. Steps 2 through 6 are iterated until convergence (yield func-

tion = 0) is obtained.

8. Steps 2 through 6 are repeated for each succeeding load incre-

ment. This incremental solution can be described by the general

equations

[K]. 1 {l\q}. = {l\Q}. l.- l. l.

{q}i = {q}o + E {l\q}i

{E}i = {E}O + E {l\E}i

{ali = {a}o + E {l\a}i This procedure is repeated as long as the solution is

converging.

(5.18)

(5.19)

(5.20)

(5.21)

In the NASTRAN program [52], the above incremental method is

used, as well as the initial-strain method and the initial-stress method

[49,52] •

CHAPTER 6

CORRELATION BETWEEN THEORY AND TEST RESULTS

6.1 Introduction

To verify the validity of the finite-element model, results of

composite beam-to-column connection tests were compared with the compu

tational findings for the same connections. In particular, the data of

Van Dalen and Godoy [40] were used for this evaluation. Details of the

finite-eleme~t analysis of the tested specimens are given in the fol

lowing sections.

6.2 Description and Results of Tested Specimens

Van Dalen and Godoy [40] tested several composite beam-to-column

connections to study their strength and rotational behavior. The speci

mens had been designed with two main variables in mind; these were the

reinforcement steel ratio in the concrete slab and the type of connec

tion between the steel beam and column. In particular, it was of inter

est to determine the effect of various degrees of rotational restraint,

using flexible, rigid, and semi-rigid connections.

Three connections of the Van Dalen and Godoy study were of spe

cial interest to this study, because they contain the three types of

beam-to-column connections that are needed to verify the finite-element

model. The properties and results of these tests, therefore, were used

in a realistic evaluation of the performance of the finite-element

model.

75

76

Details of the relevant test specimens (CB2, CB3, and CBS of the

Van Dalen and Godoy study) are shown in Figure 6.1. All of them had a

slab reinforcement ratio of p = 0.8 percent. The beam-to-column con-

nections for the test specimens were flexible, rigid, and semi-rigid,

respectively. The flexible connection was made by using seat and top

angles, the rigid connection was made by welding, and the semi-rigid

connection was made by top-welded plate and seat angle. The material

properties of the specimens are summarized in Table 6.1. During the

testing, the load was applied to the specimens as a concentrated line

force, as shown in Figures 6.1a and 6.1c.

The test results of primary interest of these specimens are the

moment-rotation curves of the beam-to-column connections. The rotation

of the beam-to-column connection was defined as the change in the angle

between the centerline of the steel beam and the centerline of the steel

column. Figures 6.2-6.4 show the moment-rotation curves of specimens

CB2, CB3, and CBS, respectively, with and without composite action.

Table 6.2 gives the values of the yield and ultimate moments for the

specimens.

6.3 Finite-Element Modeling of the Test Specimens

Due to symmetry in loading and geometry, only one-quarter of

each assemblage needed to be modeled, as indicated by the cross-hatched

area of Figure 6.1a. This consists of one-quarter of a column, half of

a steel beam, half the number of shear connectors, and one-quarter of

the concrete slab.

77

, , Lines of Loading

A

H 8X2B

, , 48.03

H 8X20 -- .'--'f--'& ---

B B

L .---f-

I I

--+--+---I

~.j I

.. I

L A , ,

.1 76.53 ~ l'

( a)

Figure 6.1 Details of Tested Specimens. -- (a) Plan view of composite connection specimens. (b) Section A-A of composite connection specimens. (c) Section B-B of composite connection specimens. From Van Dalen and Godoy [40].

, ,

, , 4B.03 '\--

~~'--------------------~I

, , 33.9B

Line of Loading

W BX20

Location

(b )

, , 8.5B 33.98

, , 0.24 Clear

W BX20

, ,

78

:0<3 Bur s

03 @ 6

3/4 Cover

0.748 ~~

, ,

t " 4-4•016

0.27~'t TI l " 2.52

" 1

LX4X3Xl/2 W 8X20

( c )

Figure 6.1--Continued.

0.394 ---J>~ 'k-

79

Table 6.1 Material Properties of Test Specimens. -- From Van Dalen and and Godoy [40] .

Steel Beam Stud Shear Average Concrete Reinforcing Connectors

Strength (Ksi) Strength Bars (Ksi) (Ksi) Designation

F F f' (Ksi)

F c F F F of Specimen y u y u y u

CB2 45.54 70.34 6.57 70.0 108.8 50.0 60.0

CB3 45.54 70.34 6.4 70.0 108.8 50.0 60.0

CB5 45.54 70.34 6.18 70.0 108.8 50.0 60.0

d""Io

0

~ ~

@. <Q=O

~ '=J

~ f: @

e a ~

1121121

A Bare Steel Con.

~ Composite Con.

Spec. CB2

1121 2121 3121 4121

Rotatio~ Crad ~ 10-3 )

Figure 6.2 Moment-Rotation Curves for Specimen with Flexible Beam-to-Column Connection (CB2) , with and Without Composite Action. -- From Van Dalen and Godoy [40].

co o

~

1121121 0

~ t f-~

A Bare Steel Con.

@" y A I\. ~ Composite Con.

.."..

~ """ ~ 50 ~ m IE 0 ~ d..

Spec. CB3

~\~ il~ ~I I~

5 10

Rot-cation (&"~d ~

15

10-3 ) 20

Figure 6.3 Moment-Rotation Curves for Specimen with Rigid Connection (CB3) , with and Without Composite Action. -- From Van Dalen and Godoy [40].

00 .......

~

0

~ ~

!'21. ..... ~ '=P

~ ib @

fS 0} ~

100

A Bare Stee 1 Con.

~ Composite Con.

Spec. CBS

.-----.; ,-., 'I ,~-,--. I'" I

5 10 15 20

Rot at ton (r ad x 12),-3)

~!

Figure 6.4 Moment-Rotation Curves for Specimen with Semi-Rigid Beam-to-Column Connection (CBS), with and Without Composite Action. -- From Van Dalen and Godoy [40].

co I\.)

Table 6.2 Yield and Ultimate Moments for Test Specimens. -- Adapted from Van Dalen and Godoy [40, Table 1].

Designation of Specimen

CB2

CB3

CB5

Type of Steel Beam-toColumn Connection

Flexible

Rigid

Semi-rigid

Steel only a

COmposite

Steel only a

Composite

a Steel only

Composite

P (%)

0.8

0.8

0.8

M yc

b

(K-ft)

26.55

57.53

87.03

20.65

87.03

M uc

c

(K-ft)

19.91

120.22

75.23

115.8

44.25

119.50

G d uc

-3 (rad x 10 )

95

36

9

10

66

14

Reason for Termination of Test

Failure of top angle

Capacity of load cell reached

Failure of top weld


Excessive elongation of top plate


a The steel beam, W8x20, has a yield moment of M K-ft. Y

= 62.70 K-ft, and an ultimate moment of M u

= 70.0

bIll yc

cl1

uc d

Guc

yield moment of the connection.

ultimate moment of the connection.

rotation at ultimate moment.

co LV

84

The finite-element discretization of the specimen is shown in

Figure 6.5. The mesh size which is used in the analysis of the compos

ite assemblage was selected to satisfy the boundary conditions near the

steel column and the location of the applied load. In addition, the

mesh size and layout are important to ensure adequate convergence of the

numerical solution.

Near the steel column, the dimensions of the element which

represents an opening in the concrete slab have been chosen to be

approximately equal to h/2 x bf/2, where h is the height of the column

cross-section and bf

is the width of the column flange. These choices

were made due to the assumption that the concrete in this area will be

ineffective, since no reinforcing bars can pass through the flanges or

the vleb of the steel shape. For the W8x20, h/2 is equal to 4.075

inches, and bf/2 is equal to 2.65 inches. The size of the element near

the steel column, therefore, becomes 4x3.25 inches. ,The mesh becomes

coarser as the distance to the column axis increases. Common element

nodes were provided at the location of the applied load, to identify the

exact location of this load.

The convergence of the numerical solution using the above mesh

size and layout was studied by re-analyzing the same specimen with a

mesh size equal to 1.5 times the first one. The results were almost

identical (for example, there was a difference in vertical deflections

of less than one percent). Although the coarser mesh could be used with

some savings in computer time; it was decided to utilize the finer dis

cretization, since this would give a more accurate representation of the

performance of the composite assemblage.

Quadrilatera-I Shell Elements

With Thickness t=4.016 on Slab Mid-Surface ~---- -- ------ ""

. . . . ..

. Y~~·~:.2~

'"

. .

.. .. "--" .. ..

U"'9y::9z zz0

9

-r-

Line of Beam Elements V=9~""ez=0

Figure 6.5 Finite-Element Discretization of a Specimen.

'1.28

s.as

11.56

CD U1

86

The column cross-section axes are the axes of symmetry for the

specimen; therefore, the on1X deformation the steel column can exhibit

is the axial deformation, which induces only a rigid body motion for the

whole system. It was not necessary, therefore, to discretize the column

itself. Thus, the boundary condition at node 9 of Figure 6.5, which is

the joint between the steel elements, has been chosen to be flexible,

rigid, or semi-rigid, according to the moment-rotation curves for the

corresponding test specimens.

The load was applied to the specimen as a line load, as indi-

cated in Figure 6.1a. For the model, this was discretized as concen-

trated loads at nodes 11, 20, 29, 38, 47, 56, and 65, and the magnitude

of each load was calculated on the basis of the tributary area of the

adjacent elements. For example, the applied load at node 29 (see

Figure 6.5) is equal to

(3.25 + 4.0)/2 = x p 24.016

where P is the total applied load.

The boundary conditions were modeled to satisfy the symmetry of

the structure and the joint conditions between the steel members. Due

to symmetry, all nodes were considered restrained against torsion.

Thus, the nodes along the axes of symmetry were restrained to ensure the

continuity of the slab. For example, the nodes along the )c-axis, such

as nodes 10-17 (see Figure 6.5), were restrained against movement in the

y-direction and against rotation around the x-axis; however, nodes 27,

36, 45, 54, 63, and 72, which are located on the y-axis, were restrained

against movement in the x-direction and against rotation about the

y-axis.

87

The boundary conditions at the joint between the steel members

were represented by specifying the proper restraints at node 9. This

depends on the type of beam-to-column connection that has been used.

Consequently, for specimen CB2, which had a flexible connection, the

boundary conditions at node 9 were specified to satisfy both the condi

tions of a simply supported end and to resist a moment, as shown in

Figure 6.6. This moment represents the actual capacity of the flexible

steel beam-to-column connection, and its value was taken from the test

results for the corresponding bare steel connection. Data for the

latter are given in Figures 6.2-6.4 and Table 6.2.

For the flexible connection, it was found that a restraining

moment of 19.91 K-ft would be suitable. Used in the nonlinear analysis,

this moment was applied in increments in the same fashion as the applied

load, which was increased step by step until the ultimate load of the

specimen was reached.

The above procedure was also followed for the other two speci

mens (CB3 and CBS), with some minor changes. For specimen CB3, the

boundary conditions at node 9 were replaced by a fixed end, which pre

vents all displacements and rotations. For specimen CBS, the only

change was the value of the restrained end moment at node 9. This value

was much higher than for the flexible case, as shown in Table 6.2, and

was found to be 44.25 K-ft. It is readily understood that the boundary

condition at node 9 can be made to represent any type of beam-to-column

\

Boundury Conditions Along Thoso Elemonts Are:

Boundary Conditions Along These Elements Are:

Figure 6.6 Specified Boundary Conditions for Flexible and Semi-Rigid Connections.

88

89

connection, as long as actual connection rigidity data are available to

determine the restraint of the connection.

An accurate solution of this problem requires a nonlinear

elastic-plastic spring to represent the moment-rotation curve and the

rigidity of each connection. This can be a very complex solution

approach, and an approximate procedure is generally preferable. The

solution that was used in this study gave very good results, and did not

affect the ultimate moment capacity of the specimen.

The material properties of the elements were taken from the test

data, as given in Table 6.1. The procedure that was formulated in

Chapter 4 to determine the equivalent yield strength and modulus of

elasticity of the composite system can now be used to find the necessary

properties of the finite-element models. Taking specimen CB3 as an

example, the quadrilateral shell element has a reinforcement ratio of

0.8 percent, with a yield strength of 70 Ksi for the reinforcing bars

and ultimate concrete stress of f' = 6.4 Ksi. The equivalent modulus of c

elasticity is found from equation 4.24 as

E + 2 E E =

c sr 1 + 2p

The modulus of elasticity of the concrete, E , is calculated by the ACI c

proposed equation for normal weight concrete as [51]

E = 57,000 ~ c c

(6.1)

For a concrete strength of 6400 psi, equation 6.1 gives a value for E c

of 4560 Ksi. The equivalent modulus of elasticity for the shell ele-

ment, which represents the reinforced concrete slab, will then be E =

4945 Ksi, using equation 4.24.

90

Since the reinforcement ratio of O.B percent is less than the

Pm in of eq~ation 4.31 (this gives a value of 1.3 percent), the equiva

lent yield stress is calculated by equation 4.30 as a = 2.49 Ksi. The y

values of E and a are used in the linearly elastic-perfectly plastic y

stress-strain relationship of the model, as shown in Figure 6.7.

The stress-strain relationship for the elements that represent

the WBx20 steel beam is shown in Figure 6.B. Since one-half of the

steel beam is modeled, only one-half of its cross-sectional area and

one-half of its moments of inertia have been used in the analysis.

The properties of the beam elements that represent the studs, as

shown in Figures 6.5 and 6.6, have been modified according to equations

4.40 and 4.44. The actual stud has a diameter of 0.3937 inches, which

gives a moment of inertia of 117.93x10-5

in.4 and an area of

-3 2 121.737x10 in .• The modified moment of inertia can be calculated

from equation 4.40 as

where 22 = 6.0B in. is the distance from the centerline of the steel

beam to the centerline of the slab, as shown in Figure 6.5. 21 = 2.01

in. is the actual length of the stud, or one-half the concrete slab

thickness, whichever is smaller. This gives a moment of inertia of I2 =

0.0655 in.4.

The modified cross-sectional area for the stud element can also

be calculated by substituting 21

, 22

, and the actual stud area into

equation 4.44. This gives a value of

(J ( Ks t)

-3 e; ycl2l. 51213 Xl &2l

91

Figure 6.7 Equivalent Stress-Strain Relationship for the Shell Element which Represents the Concrete Slab (Specimen CB3).

a (Ks i )

-3 £: ya 1 • 57)( 1 fZl

Figure 6.a Stress-Strain Relationship for the Material of the Wax20 Steel Beam.

92

93

= 0.3386 in.2

Since one-quarter of the specimen was used for the finite-element model,

only one-half of the actual number of the studs was incorporated. As

shown in Figure 6.5, the number of beam elements which represents the

studs is eight, and the stiffness of the 11 studs, therefore, has been

distributed evenly among the eight beam elements. The material of the

studs is assumed to be linearly elastic-perfectly plastic, as shown in

Figure 6.9.

After defining all of the element properties and their material

stress-strain relationships, the input data for each of the models shown

in Figures 6.5 and 6~6 were prepared according to the user's manual for

the NASTRAN program [52]. Solution 66 of the NASTRAN program, which

represents a nonlinear analysis, was then used for tpe analysis of each

specimen. The ultimate load was defined as the one after which no

further load could be applied to the specimen. It is noted that the

final failure always was accompanied by the formation of a plastic hinge

at the centerline of the composite connection.

Determining the proper load steps required several evaluations.

It was found that a linear analysis gives a good indication at what load

the structure will enter the nonlinear range. Based on such data and

some judgment on the part of the analyst, the load steps could then be

determined. It is recommended to use small initial load steps to ensure

good convergence, especially when the stresses in the elements of the

structure approach yield.

a (I<~ 1)

Fy=50 Ksi

Ecz 29f2l12)0 Ksi

-3 e: y= 1 .724 1 X 1 rn

e:

Figure 6.9 Stress-Strain Relationship for the Material of the Studs Elements.

94

6.4 Comparison of Experimental and Analytical Results

95

The results of the nonlinear analysis of the three specimens are

given in the form of moment-rotation curves and load-deflection curves,

these are then compared with the corresponding test data [40]. It is

noted that, in the tests, the relative rotation between the beam and the

centerline of the column was calculated by using several dial gages.

These gages measured the slip at the bolted beam-to-seat angle inter-

. face, or the deformation in the column flange, depending on the type of

connection. In addition, a dial gage that was mounted at a level 6.5

inches above the bottom flange of the beam and at a distance of 4.25

inches from the face of the column measured the displacement of the beam

at that level. In the analysis, the angle of rotation for the connec-

tion was calculated by dividing the deflection at the point where the

main dial gage was mounted by its distance from the column axis, S, as

shown in Figure 6.10.

The deflection of the specimens was measured at the tip of the

cantilever, as shown in Figure 6.11, and plotted against the total

applied load.

Figures 6.12-6.14 compare the experimental and the analytical

moment-rotation curves for specimens with flexible, rigid, and semi-

rigid connections, respectively. Also, Figures 6.15-6.17 compare the

test data and the results of the finite-element analysis for the load

deflection curves of the three specimens, respectively.

The finite-element results for the specimen with a rigid connec-

tion and the specimen with a semi-rigid one are in good agreement with

e

Boom Aulo

Locution of t-1nln Diol Gogo

, ,

Anglo of

Rotlltlon

Column Ruin

Figure 6.10 Analytical Definition of Connection Rotation.

96

p p

Bellm A)(ts Column A)(ts

Figure 6.11 Deflection Measurement for Test Specimens. -- From Van Dalen and Godoy [40].

97

i""'\ 100 0

cC"" ~

Finite Element

~ A Experimental Test.

--~ "OJ

~ g: m re fiJ ~

Spec. CB2

10 20 . 30 40 50

RotatilOln (rClcl x 10'"""3)

Figure 6.12 Comparison Between Experimental and Analytical Moment-Rotation Curves for Specimen with Flexible Beam-to-Column Connection.

1," .

o.D OJ

~ 100 0

~ t / f) Finite Element ~

@.. ir A Experimental Test

"I?'"

~ ~

~ 50 ~ m IE 0 ~

Spec. CB3

5 10 15 20

Ret at ion (~ad ~ 1 f2r-3 )

Figure 6.13 Comparison Between Experimental and Analytical Moment-Rotation Curves for Specimen with Rigid Beam-to-Column Connection.

1.0 1.0

d'9> 100 .. yll

t /' ~ Finite Element ~ ,

@.. J7 A Expe r f ment a 1 Test

"""" ~ "b:P

9D 50 g: @

e c ~

I tI

Spec. CB5

5 10 15 20

Rotat.ion (~ad ~ 10-3 )

Figure 6.14 Comparison Between Experimental and Analytical Moment-Rotation Curves for Specimen with Semi-Rigid Beam-to-Column Connection. .....

o o

~

00 fil.. C=3 ~

""" 30 ~Fin1te Element

1Ql A Expe r i ment a 1 Test

cg 01

G'="

c;co=>

cg

Spec. CB2 ~ ©)

C--

.2 • 4 .6 .8 1 1 .2 1 • 4

lOlefUectio6'l (XNo)

Figure 6.15 Comparison Between Analytical and Experimentally Determined Deflections of Specimen CB2.

I-'

o I-'

50

.:F"'\

00 40 a. H ~

"'=J'

~ ct; (iJ

c===>

= r.?J

{,.»

til c=

• 1 .2 .3 04

D@f1G3(c;t.U©lfll

~ Finite Element

A Experimental Test

Spec. CB3

.5

(INti )

.6 .7 .8

Figure 6.16 Comparison Between Analytical and Experimentally Determined Deflections of Speciment CB3. >-> o N

50

~

tOO 40 (l. M ~

~ 30+ ~~ <? Finite Element

"'g / (1)- I::i Experimental Test ClJ

20 to P'='

""""" ~ 10 ~ 0 tJl Spec. CBS F

o 1 .2 .3 .4 .5 as .7 . 8

109fl~~tftQ)G'1 (INa)

Figure 6.17 Comparison Between Analytical and Experimentally Determined Deflections of Specimen CBS.

o LV

104

test data. The major differences appeared in the final stages of the

loading. It appears that, during testing [40], loading of specimens CB3

and CBS was stopped because the capacity of the load cell had been

reached; however, that load was considered to be the ultimate, because

the strain in the steel beam was larger than the yield strain. In the

finite-element analysis, the loading was stopped because a plastic hinge

had formed in the steel beam, and no more load could be carried.

The ultimate capacities of the composite connections as found in

the finite-element analysis were less than the test values, by approxi

mately 8 percent for the specimen with a rigid connection and 16 percent

for the specimen with a semi-rigid connection. This indicates that the

theoretical evaluation tends to give conservative results.

Figures 6.16 and 6.17 show that the deflections at the tip of

the beam of specimens CB3 and CBS were small. This may be an indication

that strain hardening does not play a major role in the difference

between the test results and the finite-element analysis. The differ

ences can be related to the actual location of the center of rotation of

the steel beam. Thus, in the finite-element analysis, the center of

rotation was at the level of the steel beam axes, whereas in the actual

test the center of rotation would be closer to the compression flange as

the degree of the fixity becomes less. The importance of the location

of the center of rotation comes from the fact that the axial membrane

force in the steel beam due to the composite action will act at the

center of rotation. Consequently, when the center of rotation is close

to the compression flange, the beam axial force moment arm, which is the

distance between the center of rotation and the center of the concrete

105

slab, will be larger than in the case of having the center of rotation

at the beam axis. Also, the beam axial force will contribute to

increasing the moment capacity of the composite connection by the amount

of its magnitude times its eccentricity with respect to the mid-surface

of the concrete slab; therefore, the test results gave a higher moment

capacity for the composite connection than the finite-element analysis.

The analytical results for the specimen with a flexible connec

tion (CB2) are shown in Figures 6.12 and 6.15. The difference between

the finite-element analysis and the test was larger than for those of

the other two specimens. The analytical ultimate moment capacity of the

connection was about 26 percent less than the test results. This can be

attributed to several causes. First, the center of rotation for the

flexible connection may be closer to the seat angle. Secondly, due to

strain hardening, the deflection for the flexible' connection specimen

was much larger than that of the other two specimens, as shown in

Figures 6.15-6.17. This may indicate that strain hardening was reached

for some elements of the flexible connection. This was not considered

in the finite-element analysis. Finally, the rigidity of the stud shear

connectors may have played a role. The finite-element analysis showed

that plastic hinges formed in all of the stud elements. This indicates

that the studs did not have enough rigidity to allow a larger rotation

and consequent higher moment capacity. This problem can be controlled

by providing enough studs to develop the full composite action between

the concrete slab and the steel beam.

The results of the analytical study, therefore, can be regarded

as providing realistic and good agreement with the test data. In

106

consequence, it was decided to utilize the finite-element model in the

analysis of a range of composite floor systems, to study the effective

width in the positive and negative moment regions as well as the ulti

mate capacity of the composite connection. This will be described in

the following chapter.

CHAPTER 7

EVALUATION OF COMPOSITE FLOOR SYSTEMS

7.1 Description of Composite System Models

Three groups of composite floor systems were chosen for the

analysis. Each group used the same size of steel beam for all of its

specimens. In Group I, a W16x36 steel beam was chosen, which is assumed

to represent a relatively lightweight girder in a composite floor

system. A W21x50 steel beam was used for the specimens of Group II,

which is assumed to represent medium-sized girders. For heavier compos

ite beams, a W27x94 steel beam was chosen, to be used in the specimens

of Group III.

The plan of a typical composite floor system with details and

dimensions, as well as the location of the beams and connections that

will be analyzed, is shown in Figure 7.1. The main girders and the

transverse beams are assumed to have the same cross-section. The joints

of the main girders to the column flanges are assumed to be rigid or

semi-rigid, and the joints of the transverse beams to the webs of the

columns are assumed to be simple, for all of the specimens. This

assumption is derived from the fact that in a real structure the joints

between the girder and the strong axis of the column will carry moment

regardless of how the structure has been designed. For the transverse

beam, however, which is connected to the minor axis of the column, the

connection resisting moment will be very small, because of the

107

b b +~------~'I,------~t

All Beom:: Have Tho Some Craso-Soctlan

Trllnovoroo Glrdor

Dlocrotlzod Portion of Tho S I Ilb.

Pinned Ends

F"1~od OR Semi-Rigid End::

~,~--b----~i,~------~+

108

L

L

~-,-

Figure 7.1 Typical Plan of a Composite Floor System, Showing Types of Beam-to-Column Connections and Panel Dimensions.

109

flexibility of the column in the minor direction and the use of a flex-

ible connection; therefore, a simple connection is assumed for the

transverse beam and a rigid or semi-rigid one for the girder. More

details about the specimens will be given for each individual group.

Group I consists of three specimens. Each of them uses the

W16x36 steel beam, a span length of 20 feet, a slab thickness of 6.0

inches, and a reinforcement ratio of one percent. The joints between

the main steel girder and the columns for all of the specimens in this

group are assumed to be rigid. The only variable for this group is the

main girder spacing, b, giving spacing-to-span ratios, b/~, of 0.8, 0.6,

and 0.4. This generates different steel beam-to-concrete slab stiffness

ratios, defined as

where

b =

D =

E = c

K = r

main

E I s s Db

girder

E t 3

c c

12{1 - V 2) c

spacing,

concrete modulus of elasticity,

E = steel beam modulus of elasticity, s

(7.1)

(7.2)

I = moment of inertia of the steel beam about the strong axis, s

t = concrete slab thickness, and c

V = Poisson's ratio for the concrete (assumed to be 0.17). c

The specimens are assumed to be made with a concrete of strength

equal to 4.0 Ksi, reinforcing bars with a yield s"tress of 60 Ksi, and

110

steel beams with a yield stress of 36 Ksi. Using these material prop-

erties, the resulting stiffness ratios for the various models have been

computed as shown in Table 7.1.

As an illustration, the calculations that provided the data for

Model No. 1 will be shown. For this specimen, b/~ = 0.8. The value of

the concrete modulus of elasticity is found to be 3605 Ksi, using

equation 6'.1, and a value of 66,821 K-in. is found for D, using equa-

tion 7.2. Substituting E and D into equation 7.1 gives a stiffness c

ratio of K = 1.01. r

Group II consists of nine specimens, designed to study the

effects of varying the b/~ ratio, the slab thickness, the type of steel

beam-to-column connection, and the slab reinforcement ratio. All of the

specimens have the same steel beam, W21x50, with a span length of 25

feet. The first three specimens of Group II have a slab thickness of 6

inches and a one percent reinfor~ement ratio. They are designed to

examine the effects of varying the b/~ ratio between 0.8, 0.6, and 0.4.

These generate steel beam-to-concrete slab stiffness ratios that range

from 1.78 to 3.55, as shown in Table 7.2. Model No.4 is the same as

Model No.3, except that the slab thickness has been reduced to 4

inches. This was done to study the effect of the slab thickness.

Models 5 through 8 were designed exactly as Model No.3, except for the

type of beam-to-column connection. Intended as the basis for an exami-

nation of the influence of the degree of restraint of the connection,

semi-rigid connections capable of providing resisting moments of 0.1,

0.3, 0.5, and 0.7 M are used, as shown in Table 7.2. The variable M p P

is the plastic moment capacity of the steel girder. Finally, Model

Table 7.1 Group I of Composite Floor Systems. -- Steel beam W16x36.a

Transverse beam has the same cross-section of the girder for each specimen.

Steel ,Q,b

Steel Beam-to- b e Specimen Section

t b

C Column Connection K d c e

No. A36 (in. ) (ft) (ft) b/,Q, Restraint (in. ) p r

1 W16x36 6 20 16 0.8 0.01 Fixed 1.013 60

2 W16x36 6 20 12 0.6 0.01 Fixed 1.35 60

3 W16x36 6 20 8 0.4 0.01 Fixed 2.025 60

aW16x36 has I = 448.0 in.4, I = 24.5 in.4, and A = 10.6 in. 2 . x y

b,Q, = the span length of the main girder.

c b . d . = glr er spaclng.

dK = stiffness ratio of steel beam to concrete slab = E I /Db, D r s s

E t 3/[12(1 - V2)]. c c

e b was calculated according to the AISC speci~ications [4]. e

Condition Controls

of b e

L/4

L/4

L/4

I-> ...... I->

Table 7.2 Group II of Composite Floor Systems. -- Steel beam W21x50.a


Condition Steel

t Q,b Steel Beam-to- b f Controls

Specimen Section c bC

Column Connec~ion K e e of b No. A36 (in. ) (ft) (ft) b/Q, (in. )

e p Restraint r

1 W21x50 6 25 20 0.8 0.01 Fixed 1. 78 75 L/4 2 W21x50 6 25 15 0.6 0.01 Fixed 2.37 75 L/4 3 W21x50 6 25 10 0.4 0.01 Fixed 3.55 75 L/4 4 W21x50 4 25 10 0.4 0.01 Fixed 8 70.5 16t+b

f 5 W21x50 6 25 10 0.4 0.01 0.1 M 3.55 70.5 L/4 6 W21x50 6 25 10 0.4 0.01 0.3 MP 3.55 70.5 L/4 7 V121x50 6 25 10 0.4 0.01 0.5 MP 3.55 70.5 L/4 8 W21x50 6 25 10 0.4 0.01 0.7 MP 3.55 70.5 L/4 9 W21x50 6 25 10 0.4 0.015 0.3 MP 3.55 70.5 L/4

Q a = 984.0 in.

4 = 24.9 in. 4 2

W21x50 has I , I , and A = 14.7 in. . x y

bQ, = the span length of the main girder.

c b girder spacing.

dThe major girder end restraint varies from fixed (capable of resisting the total plastic moment, M , of the steel beam) to semi-rigid (capable of resisting 0.1 M ). P

P

e K = stiffness ratio of steel beam to concrete slab = E I /Db, D r s s

fb was calculated according to the AISC specifications [4]. e

E t 3/[12(1 - V2)].

c c

......

...... N

113

No.9 in Group II was designed with a reinforcement ratio of 1.5 percent

rather than one percent, to study the effects of this variable.

Group III consists of three models using a W27x94 steel beam

with a span length of 30 feet. The concrete slab thickness and the

reinforcement ratio are kept the same as in Group I, leaving only the

b/t ratio as the variable, with values of 0.8, 0.6, and 0.4. This gen-

erates steel beam-to-concrete slab stiffness ratios that range from 4.93

to 9.855, as shown in Table 7.3.

It is seen that the steel beam-to-concrete slab stiffness ratio

in the three groups ranges from 1.013 to 9.855. It is believed that

this covers the full range of practical sizes in composite floor

systems.

7.2 Shear Connector Design

The girders in the composite floor systems of Groups I, II, and

III were designed for 100 percent interaction between the slab and the

steel beam, using regular 3/4"x3" stud shear connectors. The required

number of connectors was calculated according to the AISC specifications

[4], and they were uniformly distributed along the beam. Details of

this calculation are shown below as they apply to Group I.

In Group I, the steel beam, W16x36, has a yield stress of 36 Ksi

and a cross-sectional area of 10.6 in.2

. The total horizontal shear to

be resisted by the connectors under full composite action in the posi-

tive region is the smallest value calculated from the following

equations [4]:

Vh

= 0.85 f'A

c c 2

(7.3)

Table 7.3 Group III of Composite Floor Systems. -- Steel beam W27x94.a


Condition Steel

.Q,b Steel Beam-to- b f Controls

Specimen Section t

bC

Column Connec~ion of b c K e e" A36 (in. ) (ft) (ft) b/.Q, (in. )

e No. p Restraint r

1 W27x94 6 30 24 0.8 0.01 Fixed 4.93 90 L/4

2 W27x94 6 30 18 0.6 0.01 Fixed 6.57 90 L/4

3 W27x94 6 30 12 0.4 0.01 Fixed 9.855 90 L/4

aW27x94 has I = 3270 in. 4

124 in. 4

and A = 27.7 in. 2 , I , .

x y

b.Q, the span length of the main girder.

c b girder spacing.

d The major girder end restraint varies from fixed (capable of resisting the total plastic moment, M , of the steel beam) to semi-rigid (capable of resisting 0.1 M ). p

p

e K = stiffness ratio of steel beam to concrete slab = E I /Db, D = E t 3/[12(1 - V2)].

r s s c c

fb was calculated according to the AISC specifications [4]. e

..... ..... ~

115

A F = 2J...

2 (7.4)

where f', A , A , and F have been defined previously. c c s Y

To find the concrete area, A , the effective width, b , is c e

needed. In the following, it was determined according to the AISC spec-

ifications, and the resulting values are given in Tables 7.1, 7.2, and

7.3 for the models of Groups I, II, and III, respectively. For the

specimens in Group I, b was governed by the L/4 criterion, making it e

equal to 60 inches. Substituting this value into equations 7.3 and 7.4,

with f' = 4.0 Ksi, A c c

= b t e c

. 2 .~n. , A

s = 10.6 in.

2, and F

y

gives the smallest value of Vh

as 1908 Kips.

= 36 Ksi,

In the negative moment region, the total horizontal shear was

calculated from the equation [4]

V' = h

A F sr yr 2

(7.5)

where A and F equal the area and the yield stress of the reinforcing sr yr

bars in the slab, respectively. Substituting A = pb t and F = 60 sr e c yr

Ksi into equation 7.5 gives a value of Vh

of 108.0 Kips.

According to the AISC specifications [4], the required number of

studs in the positive moment region should be distributed evenly between

the inflection points. Also, the required number of studs in the nega-

tive moment region should be distributed evenly between the inflection

points and the supports. In practice, however, it is more common to

have the studs spaced uniformly along the full length of the beams;

therefore, applying these rules to the design of the models, the total

horizontal shear in both the positive and negative moment regions has

116

been added to produce a shear design force of 298.8 Kips. This is to be

resisted by evenly spaced studs.

These specimens used 3/4-inch diameter studs with a length of 3

inches. With a normal weight concrete strength of 4.0 Ksi, the allow-

able shear capacity per stud is taken from the AISC specifications as

13.3 Kips [4, Table 1.11.4]. This value was multiplied by a reduction

factor of 0.8 to account for the possibility of using a steel deck in

the floor system, as well as for the combination of the studs of the

positive and negative moment regions. In this fashion, the total

required number of studs was found to be 28.

7.3 Modeling of the Composite Floor Systems of Groups I, II, and III

The finite-element model, which was used in the analysis of the

specimens of Groups I, II, and III, was based on the theory developed in

Chapter 4. As an example, Figure 7.2 shows the finite-element discreti-

zation of one-quarter of an interior panel of Model No. 3 of Group II,

and Figure 7.3 shows the boundary conditions that represent the symmetry

and the connections between the beams and the column. Figures 7.2 and

7.3 are self-explanatory, and the modeling details will not be repeated

here. Models similar to the one shown in Figure 7.2 have been used

throughout the a~alysis of the specimens.

7.3.1 Material and Element Properties

The material and element properties for the finite-element model

that is shown in Figures 7.2 and 7.3 were calculated in the same manner

as was done for the finite-element model that was used to analyze the

Ouodrilnternl riot Sholl Element

ot Mid-Surfoce, t ... S.0 IN.

..

Beom Element Rapre~ontlng

1/2 of Tho Stud:::: Along L/2

..

Moln Girder Axis

W 21X50

Tron::::veroo Boom A~I~

W21X50

This Element Hu~ Not Besn Conoldorad Due TOrmlntlting of The Reinforcing Bars. .

Figure 7.2 Finite-Element Model of Specimen 3 of Group II. ...... ...... --.J

Increasing Uniform Distributed Land Applied to Tho Structuro Un til F a I I u r e

v",ex=s

Alone This Axis

UcSY"'12i ( a) 1

Sz c l2l

V"'9x=8

Along Th I s Ax Is

U"'Vo t<-J=!2I

U"Sy=8

Along Thlo A~to

U"V"'W"'0

u ... v=w=a 9x"'9yc 92 m 8 <l'----..:.... __

" 13

"\. F'I xod End

~MY ~ (bJ

I Boundary Condition

For Semi-Rigid Connoctlon

Figure 7.3 Boundary Condition of Specimen 3 of Group II. -- (a) With fixed-end connection. (b) With semi-rigid connection.

>-> >-> CD

119

test specimens. The following is a brief recapitulation of the modeling

principles (the details are given in Chapter 6) :

1. The properties of the quadrilateral shell element were derived

as follows:

a. The actual reinforcement ratio, p = 0.01, is compared to the

minimum one given by equation 4.31, which yields Pmin

=

0.0097.

b. Since p actually is more than p . , equation 4.26, therem1n

fore, should be used, rather than equation 4.30, to calcu-

late the equivalent yield stress. In this case, the value

is a = 2.6 Ksi. Y

c. The equivalent modulus of elasticity is calculated from

equation 4.24 to be E = 4103 Ksi. Also, the material is

assumed to be linearly elastic-perfectly plastic.

2. The main and transverse girders are assumed to be made of 36 Ksi

steel with linearly elastic-perfectly plastic behavior. Since

only one-half of the beam cross-section is modeled, only one-

half of the corresponding stiffnesses is utilized in the

analysis.

3. The stud shear connectors also are assumed to be made of a

linearly elastic-perfectly plastic material with a yield stress

of 50 Ksi. The properties of the stud element were derived as

follows:

a. Since one-half of the beam cross-section needs to be

modeled, only one-half of the required number of studs

between mid-span and the support needs to be incorporated.

120

b. The moment of inertia for each stud is modified according to

equation 4.40, and is calculated to be 2.776 in.4.

c. The cross-sectional area for each stud is also modified by

equation 4.44; it is found to be equal to 1.976 in.2

.

d. Finally, the total stiffnesses of the modeled studs have

been distributed to the beam elements that represent them in

Figure 7.2 according to the tributary area of each beam

element.

The analysis mentioned above was used for modeling each specimen

in Groups I, II, and III. The input data were prepared according to the

NASTRAN program user's manual [52], and a nonlinear analysis was per

formed. Each specimen was loaded by a uniformly distributed load until

failure. This generally took place when plastic hinges were formed in

the negative and positive moment regions of the modeled beam.

CHAPTER 8

EFFECTIVE WIDTH CRITERIA BASED ON NONLINEAR ANALYSIS

8.1 Introduction

The tension and compression membrane stresses that are developed

in a composite section due to the interaction between its components

increase the bending stiffness significantly. Figure 8.1 gives an

illustration of these concepts.

The composite structural action is developed when the slab acts

together with the steel beam in order to resist bending. In general,

the stress distribution in a wide slab that is supported on a steel

beam, as shown in Figure 8.1a, is not uniform across the slab width.

The intensity of the compressive stress, a , usually is a maximum over x

the steel beam, and decreases nonlinearly as the distance from the sup-

porting beam increases.

For design considerations, the approximate moment capacity of

the section can be obtained by using an equivalent or effective width of

the slab, where the stresses can be considered uniform. The area of the

actual stress distribution across the slab, therefore, can be replaced

by an equivalent area of constant stress, a , over the effective width, x

b , as shown in Figure 8.1a. This width is what has been defined as the e

effective width of the composite beam [7,8].

121

( a)

r-- - - - ::::-;;";;;;;;"'F'"~~

I I I I I

,··.·.a. "6' ", ...... ~ •••• • 'c2Io' ..... 6 .. '0' •• Q •.•••. ,A ••• ' -.:;l'o·

.:: : '. -: • : • '. : ' •• CI.' ••• '. : : : 'l1I.' : :. : •• '~. ' :. :.: : •• ~.' : :' • ' ••• '. : '. :,' : ::4" ' .. ::: " . .,. ...... : • A •• : •• 0'. . ... .:.. .• ~ . : : p" • ~ •• ': ci: . '. . .;. .' .' :. :~ .. : : '. 4'.' . '.' : .. .c. ••.•• ',"1>

~~,----------------~----------------~.~

'··.4 '.'c.' •... : ....... .':: 'l>;"~' ••. ,I!)' • ••••••••••• Q. •• • ':.~ ••• '.' •• ':.' :6·.:.·.·.0.· .... <=> •••• 4',' .<11 •••••• ,c, : •• '

Ms "\

... --';

\

r-- Cs

e

Tg

(!o ) .J Me

Figure 8.1 Membrane Stress Distribution Due to Composite Action. -(a) Cross-section. (b) Elevation.

122

8.2 Analytical Criteria for Calculating the Effective Width

123

Traditionally, the effective width is developed on the basis of

the stress distribution along the beam span and across the width of the

slab. It is generally obtained by integrating the longitudinal stress

function at the top or mid-surface of the plate, and dividing by the

maximum stress across the slab. This procedure considers the effective

width as the equivalent width of slab having a constant stress, and has

been adopted by several researchers [15,16,36-38,45].

Mathematically, the effective width can be expressed by the fol-

lowing equations which all give the same basic solution:

b = e

b/2 2 f

o (a ) dy

x

(a ) x max

(8. 1 )

where b = spacing of the steel girders and a = longitudinal stress at x

top surface of the slab [38], and

b = e

b/2 2 f (a) dy

o x

(a ) x max

(8.2)

where a is the longitudinal stress at the middle surface and (a) is x x max

its peak value [45]. The third approach gives

t/2 b/2 2 f f (a ) dydz

-t/2 0 x

b = t/2

(8.3) e

f (ax) y=o dz -t/2

where t is the slab thickness and (a) 0 is the longitudinal stress in x y=

the slab at the steel beam centerline [37].

124

Similar expressions have been used in this study to evaluate the

effective width. First, the longitudinal stresses have been integrated

across the slab height to obtain the stress in linear units, across the

slab width. This can be expressed by the following equation:

t/2

= f -t/2

(a ) dz x

The effective width can then be calculated from the expression

(8.4)

(B.s)

The only difference between equations 8.3 and 8.5 is that equa-

tion B.3 considers (a n) to be in the slab above the centerline of XiV max

the steel beam, while in equation 8.5 (a n) has to be taken equal to XiV max

the maximum value across the slab. In the linear range, (a n) usu-Xx, max

ally occurs in the slab above the centerline of the steel beam, and

equations B.3 and B.s will produce the same values. When the structure

enters the plastic range, however, the area of the slab above the steel

beam will yield mostly due to bending and slab stress redistribution;

therefore, the membrane force in the slab elements above the steel beam

will be redistributed over the adjacent slab elements that have not

yielded. Consequently, in this case, (a n) may not assume its maxi-Xx, max

mum value above the centerline of the steel beam, but it will be close

to that area. Equations B.3 and B.s, therefore, will yield different

values.

In evaluating the effective width for an isolated T-beam, the

compressive force, C , that is developed in the slab, as shown in s

125

Figure 8.1b, should be equal to the tensile force, T , which is develg

oped in the steel beam. In the case of a multigirder composite system,

however, the compressive slab force and the tensile girder force may not

be equal, due to the behavior of the statically indeterminant system

[38]. This finding has been considered in this study.

The effective width, therefore, has been calculated by consid-

ering C as well as by considering T. Equation 8.5 gives the effective s g

width based on the slab axial force, C , and equation 8.6 gives the s

effective width on the basis of the steel beam force, T , determined as g

T b = __ =9_-

e (OxQ,) max (8.6)

In evaluating the effective width throughout this study, equa-

tions 8.5 and 8.6 have been used and their results compared. It will be

shown that they give almost the same values for the effective width in

the linear range of behavior; however, different values result when the

structure enters the inelastic range. The analytical data will be pre-

sented after a brief discussion of the current practical criteria for

calculating the effective width.

8.3 Current Effective Width Criteria

The current design rules for composite beams use the effective

width concept to determine the portion of the slab that acts compositely

with the steel member. The AISC specifications [4] state that the por-

tion of slab that acts compositely with the steel beam is governed by

the smallest of the following:

126

1. b = L/4 (8.7) e

2. b b = 2b' + bf

(8.8) e

3. b = 16t c + b f (8.9)

e

where

L = steel beam span, which is defined as the distance center-to-

center of the supports;

b' = one-half the clear distance to the adjacent beam;

b = steel beam spacing;

bf

= steel beam flange width; and

t = thickness of the concrete slab. c

Figure 8.2 illustrates the definitions of these variables.

The provisions for composite beams in the recently adopted AISC

specifications for load and resistance factor design (LRFD) [6] use the

first two conditions (equations 8.7 and 8.8) for calculating the effec-

tive width; however, equation 8.9, which relates the effective width to

the slab thickness, has been eliminated, because it was based on an

early study of shear lag and stability of thin steel plates. Other cri-

teria reflect the special conditions that apply to one-sided slabs, as

is found with spandrel beams.

The effective width criteria of the AISC specifications [4,6]

are based on linE!ar analysis. In addition, they are independent of

loading and boundary conditions, and it is assumed that b is constant e

along the beam span. It is also the same for simply supported and con-

tinuous beams.

127

..

b b

Figure 8.2 Effective Width According to the AISC Specifications [4].

128

In this study, the variation of the effective width in the elas-

tic and inelastic ranges of loading will be examined as it is influenced

by different support conditions.

8.4 Variation of the Effective Width Along the Girders

The variation of the effective width along a beam has been

determined by analyzing the behavior of the composite member. at 10 to 12

locations between the mid-span and the support. The calculations were

based on equations 8.5 and 8.6, where the former is governed by the

axial force in the slab and the latter by the axial force in the steel

beam.

Detailed data and evaluations are given for the specimens of

Group II, to illustrate the effective width variation within the group

that covers most of the variables in this study. For specimens 2 and 3

of this group, which had b/~ ratios of 0.6 and 0.4, respectively, the

variation of the effective width along the span is shown in

Figures 8.3-8.14. These show the variation of the effective width along

the beam for different load stages, starting with an initial (small)

load and incrementing up to and including the ultimate.

It was consistently found that the variation of the effective

width along the beam span followed the same pattern at all load stages

up to yielding. This is as expected. The figure that represents the

variation of the effective width at the yield load, therefore, can be

considered representative for the pre-yielding range of behavior.

Figures 8.3-8.14 and the computations also make the following additional

points:

, ! -

I

~ ~-I

~lnnDd E:nd

L -I

I I

-

~ ~ ~t ,T I -.,-

~ Il'--

- m " . IVB•2sB N

Ci'1

"" I • LIl .. , ~

m.3 1 8.236

_ 1 -, a B.2SG , \ \

"'\J " .... "':""'" c-. -- m tn ... • /' .

/ N I 0.323 m \-

r:< r- B.2L ..... \ "", I l' - . ;.., -

-

-

-

~~ , 25

I ~.~ - -

------ BOGod an Beorn A~tal Land I ,BnGed on Slob A~tnl Lend I

Lend lovel co Yield (fa.427 Pu)

- -I I

k

~, \/~ I -

..

I-~ 61 . Cal

"" fJl

IJiI;

8.251 /}-- - ~ /\ ~ 0.2L 7 ~

wI \ • CDI ' W /

-* r-f-s .. ........... A.I ' ... _ . tD F'luod End

"" m

Qz:ad .... _______ ---t,Iw

. I . ttl ...

W

"" "" "'I - .. '---- L-

- - -- 1...

Figure 8.3 Variation of (be /2)/(b/2) at Yield Load for the Composite Beam of Specimen 2, Group II. -Refer to Table 7.2.

.... N ~

, 25

it'

m . N

"" I1J Booed on Beom A~lnl Land

I " ; Booed on 510b Ald 01 Lond I Ji,n r

<3./ I _' 0.235 I Load Loval ~ 1lI.533 Pu

<r'~ '\ ftJ.2®S --+--jf

Ptnnocl End I

l

J

"'\J

lr. ..

"'\J

UI ..

m .

8.251 #t-I-•

ta.2L 1"

m 1\ • I \ wI,

1-1r---'m -• I

.D>

m CAt CD '*-+~ ... ,'" J0/ .... -I mill ,",./ - - - - - - - __ r.-~m .

w ~ m

6) .

\ F"t::od End /

I

III

",

III --L.\...J.- :::n:t:.-x

Figure 8.4 Variation of (b /2)/(b/2) at 0.533 of the Ultimate Load for the Composite Beam of Specimen 2, Gro~p II. -- Refer to Table 7.2.

.... w o

1 ! T

.....

~t! ~ ,

25 ----.,....-----,..J...-J1Id m 1"-1 . N CJ'l r Boood an Beam Axial Lond

-,1,1,1----

I Ul,

J- 121.235 '"

I { ~ V - l' "

,BOGod on Slob A"fnl Lond"

Lond lovel m B.SGG Pu

'" UI ...

Pinned

I .....

Ul ...

121.251 ,;; ___ _

m CAl m b

w €&)

CD

.f..-,b6:1 .

--+-- - ~' Jli - :l±t-

Figure 8.5 Variation of (b /2)/(b/2) at 0.666 of the Ultimate Load for the Composite Beam of Specimen 2, Groijp II. -- Refer to Table 7.2.

... w ...

I I '

I

~ -.......

I

I

Pinnotl! End l

I

,

-

Figure 8.6

S.236 ~ 14 ~ 25' L ~i r

" J: . . I, tEl

• - - - -III 9.26 \ . N . 7 U1 ... ----- Bnmocl on Boom A~lnl La od

'" I

· ,Boood on Slob A~tal U1

Lood I ,

"" S.3aQ Lond Lovel C3 fa.767 Pu 0.23

...,;

- 'I - - - - .-, ; fil.28S , 1 -. .' I I

\

" m

" .....

· " h m . Ul

.,-- . .e. ., N m .. "" / I CD

$~ m .

.' 9.318 . .

8.16L .b I , .c:. -

, m I .J).

\ I T .b :l . , il:>' - - , -8.257 -~T -. i\ 0.2L7

I~ m m I \ . .

;~ -- .... _--/ \ U1 .c:. / m en ~I m \ fli) w F'I~od End , I I

'" N . , .,.----------- r

· CD II'$

Ul, .f.l> k)

1 - - - - - --- >- - ....

Variation of (b /2)/(b/2) at 0.767 of the Ultimate Load for the Composite Beam of . 2 e Speclmen , Group II.

~

...... w l\.)

I I •

-I

.J--

Plnnod End , -

I

I

I -

Figure 8.7

~'I~' /f\ 25 L r- i

.::J I

ffTEt - - - -

~ ~ . N r- ------ Boood on Boom A~lol Load

'" . I

~ I

U1 .Boood on Slob A~I~1 Load ..

S.3 I _-+_LO'd lavel a 12I.8S7 Pu

} . 8.372 t

- .-I B.303 . 'C

I I

\ , m . '"

, " IS) ~ -- . -. ... .

U1 ",'" N en .. /" ,.. IS)

I~ , 8.31

. 61

! tao lfiLI-. A I

. \

U\ • -C>

\ " I 7r U\

I~ I , -Lt. - ,..l S.2L I S.27 / / , m m .

~ ---,,,' \ . . / m CJ'I f'tuod End

~I m A

'" \ m I N I

. w \ Ul ..

IS) ,-------- "* t>J \ " ' .. "

. " - - -- - - :--

Variation of (b /2)/(b/2) at 0.867 of the Ultimate Load for the Composite Beam of Specimen 2, Gro~p II. >->

w w

I ! _ f

r

I

/' -"-I

Ptnnodl End i

r

I

- ..

Figure 8.8

8.394 ~ ~ ~ 25' I ~ a 1'1

. Lt.! . I

- - - I La--- -

V 1"--

€D . N r- Snood an 80D~ A~lol Lood , ------

" '~ . lI'I", ,BoGocl on Slob A~lal Lend I

Lond Lovel CI Ultlmnta -L 0.363 J ~ - - - --f f.?l.3S4 I 8.32 I I I

" .

" ....... m --- I-: • . ...." (JJ'I N I-: .. ,,"'" CD

~

~~( ~

'" m - m / ~ .

I r m . tn

I I 0.1SL

tn, ~ N

"\ L C1J I " I Of Ul I 1,.,.L1~ ~ - - - -

8.34 I '1 r~/l> I- II / . -/- '" m m .

- __ / I . . I ) ~

. , 12) en fl:$od End 1 m , N I en I ..... m. U) A . . . \

til", W N , ~ N U) .... -----

Cii'.I N \ "."

\ '" '" .... '" , - - - - - --- I- - '-

Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 2, Group II. e ...

w ,!::.

~ ~~ 1\ N -, tr:. -rr"-Ui iJ8.146 I

:. - > I rJJki- _ _ _ ~I"{r-_-:

I ~ l/ -_____ Sn:Dod on So om A~ to 1 Lo od I

UI, 'I 1\ 1Sl011l01il Oil Slo~ Altlol LOud - 0.13 I , I

-tE' "I Load lovell C1 Vleld (12).444 Pu)

:-;1'- - ~ -40;------ - - - --, S.lSI- -

In. ':, Ul~ ~ \.r- h lSI /' , r S. 2L Ui l I ~:e. I _ ' ~'I _ I 1 _ ~ 1 Ui ~ ,..1> ;- , _ ir.r m 1\ ; m - - .. .. C>f - 1/1>

8.169 7 ~/\ UJ / --- N I .... -- (1)

/ (9 --, ~, -----___ ,-, N f'lued Enol UI • I ..... ,... '""I -- ,

~~ ~ - --+- - --, ~lf1InocJ E:nG1 I, I

Ui / 25

I ' ~V ' I

i- . rr- -, - i - Ji~ -j

Figure 8.9 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam or Specimen 3, Group II. -- Refer to Tabie 7.2.

... w U1

E-I

Ul ..

~ .'"

_<l./ l_e.134 ~r-~\ B.&66"----------

F-I

m.159

" ,-" m UI " • w

U1

.. N ., lUJ

// r eJ.2l , I 1

tn. til) "IJ

01'

-~

1-

m

s 1 11m , \ .

• \ UI

~ / ' .... - . ~ ~--""_&" ------+J

nl

-1,}>

f"I.o~ End J -1 7 1>

.. I:c. .. CIJ

~-.m

It[ P'"nod End -+ III III -

'. U'l ..

F--~

Figure 8.10

, . / 25

r~L-- ,J,F-=I :rrr- I

Variation of (b /2)/(b/2) at 0.522 of the Ultimate Load for the Composite Beam of Specimen 3, Groijp II. -- Refer to Table 7.2.

w lJ'

~

fi-~~===+=~~~ ~'Ilj ------ 8000d on Boom A~lnl Lond

"* ~ L= 1iJ. 13 7 i !!llooo d on S1n1ll Flu I ~ I La ad

~_' Load lovel c IiJ 627 P "

\ !?l.171 • u , ~ III

U1 .. /

F- I

" cg ".

".

. N til r

=~-

Fa.2L~ I'

m til, ClD m

0:169 I ~l~ 1; /\ ... ,,_ JJ~ .... --, / - - - - - CD ~ - ~ . • • U1 .. ~ N

\wH~ ",(.rr'"

f"tltod £n~

-~'k-- HI III

Plrmocl End "

U1 / 25

~ '-..J> I/lr

· .. I . 1--1 f?- l :t-. : Figure 8.11 Variation of (b /2)/(b/2) at 0.627 of the Ultimate Load for the Composite Beam of

e Specimen 3, Group II. ....

W '-l

-rr-I

, -- I I '1-- "'-0-- - - - --i

~" ,,6,.1 ~.~ 1\

J 7 __ .____ 18nO(lH.a on 180 no A~ I n I Lo nd '

~''\ ------ ,Snomd on Slob A~lol lond B.lqS I I

~ "r L.eod Love I c 0.784 Pu <f '" '\ - T - - - --

, 0.19!

', .... m ~ ,r~ tn, ,,--' ~ m \ m --, m / Ul • ••

I' I r 8. 2L I. til I en I A . -1 to :: OJ I,t>

I-- :l-*- - ..>;: - - -" ki~il- -

I 1\ m I 1/1> S.lS4 T.!<- I \ ~ .

m' / \ N • "'-'-'" ,_ _ I» f'htod End

Ul w 1 m -------~"'[<- I 'N' 1

-~- '" --~ ---f-- - --I £=ltrnnoci E:ntil I" I

Ul, i. I 25 ----------~-----------------------------------~~-iV

I

1=:-::-*===1i~]}- - - . - - lLi; . -1 I ...... :...

Figure 8.12 Variation of (b /2)/(b/2) at 0.784 of the Ultimate Load for the Composite Beam of . 3 e Speclmen , Group II. .....

w co

.6lA ~ ~~. /1" -ii-

~I 8. I q I 1

I III I t-- - ........ rii-ff - - - .. ~,~ ---I I I

~ ------ Snood an Sooo A~lol Lom~ U'J.. I~ 6oootl'l on Slob fh;lol Lootll,

~ ~.!52 I '\ Lend level m 9.988 Pu ~~'- '. - - - - --4;J.... , s. 19 [,.

", lfl.. .~ -, ~ ~ ~,(~ -, ~

/1 UP en 1 b I' 1 U1 W "-I CD

J I· r N en I . N .I ........ P 1--- a .... '-- - - - ~ :v--

8:153 ~- /\ ~ ~ / 1/1) ~ ,\~, en

Ul ... _-_..... \ ~ ~ F"llfOc:l End .. m I m '~I - . t-o • ,.-' ---------"f I

W en _ N -..,'-_ _O_~_____ _ _ __ I Plnnod e:n(~ I 4' I

Ul L / 25

I .. OJ I

;-- ,r'--- - i - -rr -i

Figure 8.13 Variation of (b /2)/(b/2) at 0.888 of the Ultimate Load for the Composite Beam of Specimen 3, Groijp II. 0->

W ~

~8.133 J~ t-- -

Ui v

..

~ ~

Soorod on Boom AHlol Lond

• ISloorod on S 1 8~ fh: t 01 La orAl'

Load Lovel c Ulttmate

--

4'",L ----41+1111---

. 9. tBl

..,A...-.;~ 1" ~

Ui, CD f.i')

I

Fhoc:3 Ene:! 8~ ISS I ?~--L/\ ~; l~

!" \ ...... -------1 . J::> ", m

III -·m----- -

~ t Plnno~ End .. / ,

r- l' ~~- -=====

, 25

J[ I --=I

Figure 8.14 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 3, Group II. Refer to Table 7.2. ~

01'>0 o

141

1. Equations 8.5 and 8.6 give almost the same values for the effec-

tive width along the beam span, for all loads up to and

including yield.

2. Following yielding, equations 8.5 and 8.6 give progressively

more different values for the effective width. The differences

are caused by the lack of equilibrium between the tensile force

in the steel beam and the compressive force in the concrete slab

at the section where the computations are made. The lack of

equilibrium is caused by the stress redistribution that occurs

after yielding of the elements that are highly stressed, which

are located at the mid--span and support sections.

3. As expected, the effective width at the inflection points

approaches zero for all load levels. This is in contradiction

to the results of Ansourian [36J, who found that b becomes e

infinite at these locations. This is not an important point,

however, since the moment is zero at the inflection points, and

no interaction between the steel beam and the concrete slab is

needed.

4. The inflection point location changed slightly as the load

increased beyond the yield value. For example, in specimen 2 of

Group II, the inflection point was located at a distance of

0.2 L from the support at yield load, and it changed to a point

0.16 L from the support at ultimate load, as shown in Figures

8.3 and 8.8. The amount of change depended on the span length.

5. The effective width at a location between the inflection point

and the column support was smaller than at either segment end,

142

as can be seen in Figures 8.3-8.14. This is due to the stress

concentration that results from ignoring the concrete near the

column, assuming an opening in the slab at this location.

6. The effective width at a load close to the ultimate value some

times will reach a peak value at a location close to the inflec

tion point, as can be seen in Figures 8.7, 8.8, and 8.14. This

happens when the beam has yielded at mid-span, and some of the

membrane forces are redistributed to the adjacent elements. For

example, Figure 8.7 indicates that the effective width has a

maximum value somewhere between mid-span and the inflection

point. In this particular case, the calculation was based on

the axial force in the steel beam. It is clear that the steel

beam has yielded at mid-span, and the adjacent elements have

started to carry part of the axial load that is supposed to be

carried by the beam section at mid-span. At the peak value of

the effective width, the steel beam, therefore, has a larger

axial force than the one in the slab, causing equation 8.6 to

give a higher value for the effective width than the one calcu

lated by equation 8.5.

In the same manner, equation 8.5 sometimes gives a peak

value near the inflection point, such as when yielding occurs in

the slab elements at mid-span, and the membrane forces are

redistributed to the adjacent elements. This effect can be seen

in Figure 8.8.

7. The magnitude of the effective width is a function of the

applied load level, once yielding has commenced.

143

Figures 8.3-8.14 show that the effective width increases as the

load increases beyond the yield level; however, it is not prac

tically feasible to incorporate effective width criteria that

vary along the beam span as well as with the load level. The

effective width variations for the other specimens of Group II,

therefore, are given only at the yield and ultimate loads, as

can be seen in Figures 8.15-8.29.

8. Figures 8.3-8.29 give the variation of the effective width along

the girder spans as well as along the transverse beams. The

study has focused on the effective widths along the girders,

because these are more generally applicable; however, these

results also apply t,o the transverse beams.

9. Finally, after detailing the variation of the effective width

along the beams of nine specimens of Group II, it was obvious

that it would be impractical as well as unnecessary to give all

of the details about the effective width at each point along the

span for all of the composite beams. The focus, therefore, was

narrowed to analyzing the conditions at the two most important

points, namely, at mid-span and at the support. The variation

of the effective width for the specimens of Groups I, II, and

III that is presented in the following section deals only with

the data that have been developed for these two beam span

locations.

, 9. 2?S ~ ~ A[,A 25 I

I - i' !L I I ,!! ~ - - - - -I

eEl . ill.2SS -" / r-

laDd ------ SnGod an Sonm A~tDI - I I I S anGo~lan SID~ A~IDI Lond ,

1-Land lovel e YIeld (121.379 Pu)

8.2~S . - 1i - - - - --

f a.2S? I

I

Plnnoc:il End

I

I I

I

Figure 8.15

I I , , \ - \

GJ \. ... ..... -.:: ... ---....... /'0.355 I , )-~

- , -j--:'-! ,

" -a.3DI :/ m I . w .... N 'j-

m . N r

S. lSL I

~:hI'-1 -,~ ~ ~. , ... ~~

'i CJ CD . . N rn.. - m ru m

--"I

i\ , , ,

I I

fYm . . NN mm

':\L~l/ , E'l

6.1 • . . to) (oJ m - c (oJ

~ - - . - . 1m O.2~ I .

/ to) I m Fillod End I '- .... _------ ~Q

Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 1, Group II. -- Refer to Tabl~ 7.2.

....... ~ ~

, I 8.326 ~7'R' ~ 25 J I - - - - - - ' I=:E::J

~ I J ------ Dnood on lIoao Au I u I Lond .... -

~~ Bnood@n Slob R~'ol Lond

I -6J I 13 549 I I , • ~ Lond Lovel c Ultimate

~ ~m.273 " _ _1 _ _ _

<J ' ; 9.36S ----- - --I I if I , I

...... l, ~ m ........... ::: m

I ... /"; I ~ ,!1 i-:r" I r- liB I e I S)

t=' t "nod I!:nd I' :c..' !D ~ '" 8. 1~~L N ei),..,-

rL - ' - iT _ .. • _ &oJ _ ~ _ :;;: i4t;~~

* t I-II " . r"" I O. lJaq 1 / I 1\ · CD G.173L

Figure 8.16

0<- /, _ • /

~. ',/ ,:. ~ f"'~oril End . I ".~ GJ , ~ N ", .e.. W \ _-- I--w ~ ~-- \ .,

\ /' \ I \ I \ I \ I ,I

Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 1, e

Group II. .t::. U1

~--I

U1 .. B. n28

-:r \ ;r B. 156

~: \

tn , 61 .. , .

N r S.2L

1 F- ,., G!...-.- _ ..-

~. 17 ,- :r l / \, .. I'

I --,_ I ....... G;J

tn .. .

~ -,-rr-I

------ '8oo~d on Boom Rxlol Lond

,SootDd on Slob fhelo) Laodl l

Load lovel c Yfold (8.5 Pu)

-:=J I

III .-

~r-enl liD m

m rlJ.2L . ---"'=" hto cl En d /

en -- -I-I'fi2 ----- .e. II

I

-I:. en ,,-_ N --.0.-------+-U1 ...

1=-I

Figure 8.17

,

I~ 25

t=-~

Plnnocil Er..:i r

v . -r-Variation of (b /2)/(b/2) at Yield Load for the-Composite Beam of Specimen 4, Group II. -- Refer to Tabie 7.2. .....

,J::. ()I

~ ~-

tEl

v ,f\ ..

~8.124

~ 111

7".

Bnomcl on Beam A~iDl lond

IBoomd on Slob A~lol lan~11

Loud Lavel c Ulttmate -----+I-HII- -

-~ I

~ (111 r S-; .1:-::6:-::8---

U'I \ .. '

b:: -J _ r-:;J±t. tIL . I .A

--- I _~ "p. s. 17L 1-

m . en CDI w I. I

___ rrTnt

13. 148 I /4-.J-: j' m m ... )/ \ . . . s. 19l -::r

I

F" t uod End /

t/'t>

tfi ..

t-U'i ..

E-I

Figure 8.18

m ,__ \ m ~

~ \ " lC IN II Im

II I

v -liJ:fU--1iL

• f.!J .o!l>

.. Pinned e::nd I 25

-~ I I

JlL

Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 4, e

Group II. ... ~ --.J

A~ /1' --rtf""

------ IBoGod ~n ~oo~ AHlol Load til

~ I~oood on Slob A~aol Leod

•

II <L,.-J_9_~;16S", Lotldll0~el "" Vleld (0.375 PU)II~ __

~ \ S.197 \

Ul', G:! ~ .... .

N r S.2L ,

~ . Cil. -,t,. I, I,.

""""''''''Y 1(0)

~~

'"

J~ - -,f

S.2LI III I-- -

a.151 IIi- - ~ /', J~ ..... to, ,.... U1 m ...... W I -__ m . ..... b, ""----W , I

CD \ / 'I ,UI ... /

-:1- - --t-H-- -------

SOOI-R1e1d (8.1~p) / . Ul ~

til ~~ "'nHod End ~

F 1 H jI-

, I!I--25

1=--Figure" 8.19 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 5, Group II.

-- Refer to Table 7.2. ..... of:> CD

~ ~ -rrr-I

I ftJ:. 183 -rrr-I 1 'I I

t-- ,; rihl-- - I I V - - M ';'1 ~,~ ------ Boood an Boom A~lol Land

8.152 .1 '\ il~oood on 510b Fhd 01 Laod I

<l/ ~ _ Lond Level"" Ultlmnte

<f" , . a. 261 - - - - -

\ ' IS) .r r. r--~.I m " · ~ ~ , " .. h::<'\,) 51 -- i£l m -; ~~ w •• • m

/,.. ~. r 0.25L /, ~I ~ '" • •

I • 'I w '" OJ '. U1 1--_ ..... :i-.}- _ • _ N I CD I

01.1138 }7 - ,/', m - ; - ,j"M':""::jn}-; w " . I .... ------.. w m N ~ .' Ul b I • I Ul Sam I-~ I E3 • c:3 Hi). H~j/3' .

'~, Sf -+-1 I r,-- " - , -'\ : - - --

I \ ' I ,

Ui, -, I 25 •

>t

\ i I \ I t--~F=- - I

,;1'\ Plnnod End i - - 1-1

Figure 8.20 Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 5, e

Group II. .!>o \D

mA ~ r . /

--, r- -iT' tEl I 0. 1313 • I

1 r 1 'I 1 I' - . l:;}-+_ - - - .-i

6

I

~~ / I

.\ ------ Bnood en Bon~ A~!ul Load

U1 .. 0.16

Bnood on Slob A~lnl Lond •

V _1 ; Load level E> Yield (8.458 Pu)

4'" ~ - - - --

S.18S \

J~ \/~'" " CJ1 .... m m ... ..

I 1~ . ~

. / G.2L Ci'J. I

W

I .' m CD

/ I ~ en I~ t- - , '..I":: - - - - .. :15<i J,f- - 6j \ m I I"

• I , . .<-- (11) I ' .... til m .......... N/ -- ~~ Socl-RI £11 cl Uil. :3Kj13 )/' • U1 • I , wI -----------:: .. W

, • tD

... /

...:

--~- - - - - --

I Plnnod Enol I , I U1 J 25 ..

I / I

1--- _C- - - --1 I •

.-. I

I ..J.i. -

Figure 8.21 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 6, Group II. ~

-- Refer to Table 7.2. '-' U1 o

1=-mt ~ 001.29 Ul , L r _ * rrr ,

-~ I ~

I

w, ------ IBOOOd on Boom A~tol Land

~ l B. IS. ,Boood an 51 ob A~I 01 Load" ~I ~ Lond Level 0 0.712 ~ _ I \ 0.199 ----J+.J-

\ U1 \ -t-, , r-------

" 8.2L en, -, lD

'_ ;\, OJ II.IBL ~. m I ......... Ul •

"', Ul -__ Ul / ',/ ----, .... Socl-~IG·d UJ~ Iii "' _\./ --t-- --tiL--

, 25' II

UI ~911 .. .. 'Ui I v ___

UJ .. L Plnnod £n~

~4 ~~~-~~~~~~ l!L I . t--~

Figure 8.22 Variation of (b /2)/(b/2) at 0.712 of the Ultimate Load for the Composite Beam of . e Specimen 6, Group II. ......

In ......

l:::, ~ t ..( r- -rr;-, 0 .• 1 I •

I -- rtr:r- - - - - ---i

,f\ I

------ Bnood on Boom A~tol Land en

... ,Booed on Slub fhdnl Land, -6.1~El

~~ '1 • \ Lond Lovel c Ultlmnte '--- ~ a.254Jr - - - --

~~r~ \ m I. \, • ·"m :-;; m

Ul ... - . . .. ,-' m c:il, til ,,' ta.25L ,

/ r ~

1 ~

I I i ' '1 r.n co I~ 1-- --7 - - "" , /1>

e'. 185 J - I ' m 8. taL I "-

r m '- . . N I -------hW

Soot-RIGid (G.3M~~ CD . I m til til m , 10 , .. Ct.' ' N •

c--,

.... --+-! ~:--

"-~ - .-,- ;

U1 ..

I I::- -I

Figure B.23

\ , I \ I , • •

\ I 25 , / ,

\ i I

I:~ 4 - - - --;

~ITr Plnnod [nd • I I

Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 6, Group II. e

...... U1 ~

I

~i\ ru-~ ; S.192

)=-

<l/ <r' ------

S.14 - "I

\

~ -T"h-

Bnood on Beom A~lul Lond

,Bueed on Slu~ A~'ol Lond ll Lond lovel c Vteld (9.5 Pu)

------- III

I

-==4

,H'~ .t;J.

\~~ U1 , .

... -/, CD / r

+- /It'. m ~ . , ~ 8. 2L~,.

~-~8.2L2...

Sec.-Rigid (~.sM~);I

,~'J/ =J7"&

I rl

6'.1521':\ lE~ - I........ ~

U1 I"..... I ---... ,', I ----.... ,,-

-,\'-- III - ----+--- III

-1

, 25 finned End

U1 I

I 1=- ::3

-I r

.. II "L'}--

lit

Figure 8.24 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 7, Group II. -- Refer to Tabie 7.2.

I->

U1 W

~~ lA ~ ~ -- - r--

Ul I 8.11G I

I r " .

1 I \, - , - - - - .-J I ~ V I

1"- ,1\ ------ Bnood on Beam A~lnl Land U1 .. ,1300001 on Slob Alttol Lend I

0.,145 ~./ Lonc1 Level c Ultimate

'r <i~ -, - - - --

, r 8.227

\ 6) J:- 1i\(r \ . UI '. - 19 m .. ...

r-:W . "&n ., I r El. 225L ~ Oi,

I ,,/ W

I _

I 1 m "'" 1,.1 t- . r:r .~ - - ,

¥ ~. -, ,/I 8~ 173 -,i-, I' m J-s.153L",:

I ' • fg '- .e. SOOI-~tE:llfil (s.s~'P)/ I . - I ~~----- ~

U1 .. J~ . I " UI

I

"'" I I

I'!.- I --+-'. I \ - I --\ I , \ I

U1 \v' I

, . .. 25

I J ~- Il

E - - - - ---f

I , , /?CflInoc3 E.:n~

I I ilL--


Group II. ..... U1 ~

1=-I

UI ..

eiJ~ ~ .00146 r .. -:"'1!

~~ -,;r

Bocod on Bonrn A~tol LOQcl

'Bncod on Slob A~lnl Lond.

~J_~.1~3 Loaf lovell Cl Yield (0.42 PU)::!

\ lZl.iSS \

\

'" \/tj m mf-t _ UI .. N r

U1 ..

8.2L o

UI CD m

, is) I ,

I ' Ul I....... en ,;I' -,. ... :-------- ....

I

-----W. cL. I III

Sao I-~ I Ell d (In 0 jJtti-;.,/ -m o

m --f. - - -N----

W II ---t-- III

I

.I Ii

U1 ~ Pinno~ tn~ ,

25

-1 I

m 0

.e. U1

-

, :Y~i pt----t-l-=t - - liL~

7t>"

Figure 8.26 Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 8, Group II. -- Refer to Tabie 7.2.

..... U1 U1

~L ~ ,8.11 r I I

~ -,t \=-----, ~:i=.

I

------ 'Saood on Boom A~lal Loud Ul .. , fa. 1 il9 I -- I Boood on 510b Alt t n 1 Land,.

V I ' I Lond Love) .,. U)tlmnta

~ ~- -\ I 8.225 III \ \

Ul , m .

" N /""'w,r S. 22L '" I I~ . 1

.. ,4

lSI

m .Do' N ~ ~

• Ul CltI N _ Ul

inf

(9.7MP! I- -8. lS21=>'1~~ ... J",- ll~

SOOI-~IBIt1 , .

II --

UI J: ~ I ---~----- ~ .. _ (D I

--7.' - - '\, - -1-/-1-1_'--- --+' - III

" , Plnnod

--.z. 1

Figure 8.27

"

TI

, , rr ~ -~

End , ' I 25 UI .. il[7

•

Variation of (b /2)/(b/2) at Ultimate Load for the Composite Beam of Specimen 8, e Group II.

.... U1 en

U1 ...

I

F= s. &55

U1 ...

-J-f-- -

U1 ...

E-,

Figure 8.28

1\ ~~~~~

Booed on Boom A~tul Lcod

...k-.--___ _

m . 0').

~I - -It.

8.2L I )-~-f9 I:/'\ . w CD

, J I~ ',,------- .J;:;: ~ I til) I - I ...... J.c./

"<1\ / So~t-~IG.d (6.3~~)

Ill) III

'I f"innoQ1 End J. I

, 25

Fll/ J~_

TIl

III

m w CD

..., J ',j> -,/1>

Variation of (b /2)/(b/2) at Yield Load for the Composite Beam of Specimen 9, Group II. -- Refer to Table 7.2.

...... U1 -...J

l~ A ~ t I - .-- -jTl-

I eJ. 187 I I

i 1 I I ~-

, t-r- - - - - --of

1~ V I

,r'\ ------ B~ood on BOurn A~lnl LOud Ul .. I Bnood on S 1 nb IFhd n 1 La nd I

9.157

* ~ Lond Loval c Ultfmate

1 ~- ~ - - - - --, 9.25

\ ------:1\ (~ \ S m Ul ' .. .

N 6:J • • . ..

.... I :--. • m ~

I UI r 90 2SL L :' ~ ~ / en I / [l- II N 1'-.,» E-

':~ -- _ '-- -

~r - .-71>. I -

Y I " m . 8.17L' !?i. ISS I ,~ • r

-- W N I -----r,' m Sool-~Inld (603M2'/ U1 .

I I Ul .. AI S • w I ., ::- I

- , - - - - --, \ I

I \ I , , \ I 2S •

Ul , .. I

-\ / I . F-~

- I - - - ---I J Plelnoc;l En~

, ~"- 1 I


Group II. I->

U1 (XJ

159

8.5 Variation of the Effective Width at Mid-Span

The variation of the effective width at mid-span for the girders

of the specimens of Groups I, II, and III is illustrated in

Figures 8.30-8.53. The data for b for the transverse beams of Group II e

are also given. In addition, the yield load (Y.L.) is shown in all of

the figures. The following summarizes the findings:

1. As indicated previously, equations 8.5 and 8.6 give results that

are in close agreement for loads up to the yield level.

2. Equation 8.5 gives a smaller effective width than equation 8.6~

at the moment the first yield occurs in the slab. This is

because the area adjacent to the mid-span of the slab starts to

pick up the additional axial forces that are required to cause

yielding of the steel beam at this point. In this case, the

axial force in the steel beam is larger than the one in the

slab. This occurred in all of the girders that had fixed ends

(see Figures 8.30-8.33, 8.35, 8.37, 8.39, and 8.51-8.53).

3. Equation 8.6 gives a smaller value for the effective width when

the first yield occurs in the steel beam. This took place in

all of the girders that had pinned or semi-rigid end connections

(see Figures 8.34, 8.36, 8.38, and 8.40-8.50).

Figures 8.54-8.68 show the actual effective width and the longi-

tudinal stress distribution at mid-span for all of the girders of Groups

I, II, and III. The stress distribution ucross the slab has been inte-

grated across its depth, using equation 8.4, and the units, therefore,

.n '\.

G)

~

Figure 8.30

1 '-b. Based Beam Axial Force on

<> Based on 51 ab Axial Force

. 5 -

A ./'.. A ~ ~ v v

~ 'I ~ ~

1 1 1 1 1 1

Y. L.I I I 0 I I I

0 . 5 1

P/PQJJ

Variation of b /b at Mid-Span for the Girder of Specimen 1, Group I. -- Refer to Table 7.1. e

:

I I

I I

0"1 o

.!Q) , Q) ~

Figure 8.31

1 '-b. Based Beam Axial Force on

0 Based on 51 ab Axial Force

~ ..... . 5 I-

~ /\. /\. A A V

v v -v ..... "t'

I I I I I I I I I

Y. L.I J I I 0 I _IJ I

0 . 5 1

P.I'PU


...... 0'1 ......

..@

" @D ..@

Figure 8.32

1 b. Based on Beam Axial Force

0 Based on Slab Axial Force

I . 5 1 1 I 1 1 1 1 1 1 1 1 I

~ 1 Y • L.I

0 0 . 5 1 I

P/PU I I


()I N

~ , ill

.12l

Figure 8.33

1 l-f). Based on Beam Axial Force


. 5 -

~~~~ ~ ~ ~ , , , , , t ,

Y. L.' I I 0 I I , I

0 . 5 1

P/PU

Variation of b /b at Mid-Span for the Girder of Specimen 1, Group II. -- Refer to e

Table 7.2.

I

I I I ! • ,

J

..... m (.oJ

."g

" ill ..@

Figure 8.34

b. Based on Beam Axial Force

. 5 ~ 0 Based on 51 ab Axial Force

/\ A /\~ -v v v v ~

.. 1\ 1\ 1\ .. /\ 1-.. -/\. ..

~ .......... ~ v v v

.25 ~ 1\ 1.. 1 "'-..... ..... '1"

.. <-.>

I

l . Y.L .

0 I I I I I I I I I I 0 . 5 1

!P/PQJ

Variation of b /b at Mid-Span for the Transverse Beam of Specimen 1, Group II. -- Refer to Table 7.2. e

0'1 ~

..,g

'" QJ .1Ql

Figure 8.35



.5

1 1 1 1 1 1 1 1 1

Y. L.I 0

0 . 5 1

P/PU

Variation of b /b at Mid-Span for the Girder of Specimen 2, Group II. -- Refer to . e Table 7.2. ....

(jI U1

b. Based on Beam Rxial Force , . 5 0 Based on Slab Rxial Force

~ , 01» .25

.,©j 1 1 1 1 1 1 1 1 1

Y. L.I 1

0 0 . 5 1

P.I'PU

Figure 8.36 Variation of b /b at Mid-Span for the Transverse Beam of Specimen 2, Group II. -- Refer e -to Table 7.2 .. .....

0"1 0"1

..@ , C)

..l9l

""""'= =-a

Figure 8.37

1 t- b. Based Beam Axial Force on

0 Based on Sl ab Ax i al Force

-A

A A A ~ /\. '1( v v V'

~ ~

. 5 I l- I

I

r I I I I I I I I I I I

Y. L.I I

I

0 I ! 0 . 5

J P/PU

Variation of b /b at Mid-Soan for the Girder of Specimen 3, Group II. -- Refer to e -

Table 7.2. . ~

0"1 ~

-6 Based on Beam Axia1 Force

.5 r- 0 Based 51 ab Axia1 Force on

.,g , OJ .25 t-

.!!l A A -k>

/\. A V ..., A ..., ...,

A ~

I 1\ 1\ 1\ ~

~

I' , , f-

, Y. L.' ,

I I I 0 I I I I I

0 . 5 1

P/PItJ

Figure 8.38 Variation of b /b at Mid-Span for the Transverse Beam of Specimen 3, Group II. -- Refer to Table 7.2. e

I->

(j'\ CD

1 - l:J. Based Beam Axial Force on

0 Based on 51 ab Ax i a 1 Force

A

A A A

v v

A A V 'I(

..,g 'T'

I

" . 5 r- I

® I

~

Figure 8.39

I I I I I I I I I

l- I I

0 I I I Y. L.I

~ I I I I

0 . 5 1

P/PU


Table 7.2.

J

...... (j) 1.0

~

" (j) ~

Figure 8.40

fj, Based on Beam Axial Force

.5 f- 0 Based 51 ab Axial Force on

.25 f-

A A A A A A A -v v v v I

v

1\ .. A . .. .. '-' 1 ...... L-> ...... --=

I I ,

Y. L.' , 0 I I I I

0 . 5 1

P/PW


I

....... -..I o

.Ell

" (]) .,.g

Figure 8.41

1 !-~ Based Beam Axial Force on


A A A A A A A 0--0 v '7 v v

A v X V

A A I ~ .... ....

I .5 '-I I I I I I I I I I I I I

0 I Y. L.: , I _I , I

0 .5 1

P/PU


Table 7.2.

,

...... -..J ......

JQl

" e1Jl ~

Figure 8.42


. 5 c- O Based Slab Ax i a 1 Force on

.25 -A A -v v

A A /'\ A V v v v " , 1\ 1\ 1\ I A . . ~ ~ ~I --=- L..> ....,

~ = I , , , ,

Y. L.' , 0 I I I I

, 0 . 5 1

P/PUl

Variation of b /b at Mid-Span for the Transverse Beam of Specimen 5, Group II. -- Refer . e to Table 7.2.

I

..... --.J N

..,g '\. m

..g

Figure 8.43

1 l-II Based Beam Rxial Force on

<> Based on 51 ab Rxial Force

/\. A A A A A /\. /\.

': v v v v V

')( I ~

& ~ I ~ L...>

I .5 L-I I I I I J J I J . J J J I

0 ~. L.: I • I

0 .5 1

P/PllJ

Variation of b /b at Mid-Span for the Girder of Specimen 6, Group II. -- Refer to Table 7.2. e

,

I

...... --.J W

.J'il

" OJ .,g

Figure 8.44

II Based on Beam Axial Force

. 5 f-- 0 Based on 51 ab Axial Force

.25 -A

V A A.

A. A A v v -v

v v 1 ' A A .. .. A ~ "f <..> <-> --= <-> ~

'-' , , , , I

Y. L.' , 0 , I I , I

0 . 5 1

P/PIUI


I I

I

i

:

~

>-> ..-J ~

,..g '\, m

..!Q!

Figure 8.45

1 - ,6. Based Beam Axial Force on


/\. h.. /\ /\. A A /\. /\. ::: v v v v

~ '! A

~ 'I - ~

.5 I l- I

I I I I I I I I I I I I

0 L I I Y. L.I

I I

0 . 5 1

P/PU


...... --.J U1

~ , OJ ~

Figure 8.46

6 Based on Beam Axial Force

.5 - 0 Based 51 ab Axial Force on

.25 I-

A V

A A /\. v v

/\. T v

v " A ,.. A

A .... .... L..Jo -= ..... I

I I I I

0 I

Y. L.i I I I

0 . 5 1

P/PlJJ


~

~ en

..,g

" (ill c©l

Figure 8.47

1 l- f). Based Beam Axial Force on

0 Based on 51 ab Rxial Force

A A A A A A # A A v -v v v v

'I( X 'Ie ':. A

~ ~ ~ ~ L...> <...>. =- .

I . 5 l- I I I I

i I I

I I I I I I

- I I

Y. L.' I I 0 I I I I , I I I I

0 . 5 1

P/PUl


I

..... -...J -...J

JQl

'\. OJ

.J§!

Figure 8.48

-Do Based on Beam Axial Force

. 5 - 0 Based on Slab Axial Force

-

.25 I-

~ ~ 0 : ! : -As 6 ,

- , , , - Y. L.' ,

I I 0 I I I I

0 . 5 1 i

. P/PIIJ

----- -

Variation of b /b at Mid-Span for the Transverse Beam of Specimen 8, Group II. -- Refer e

to Table 7.2. 1-0

~ CD

.!Ll

" Gil J:lj

Figure 8.49

1 -b, Based Beam Axial Force on

0 Based on 5 lab Axial Force

A A A A A A A A.L\

I '::" v v v v V v v

. 'j ~

.5 f--I I I I I I I I I I I I I I

(21 I I Y. L.: I I I I

12) . 5 1

P/PU


0->

-..J 1.0

.J?l , m

..,@

Figure 8.50

,

/:). Based on Beam Rxia1 Force

. 5 I- <> Based on 51 ab Rxia1 Force

.25 '-A. A

"'V' v A. A

A A V -v v I A .. .. .. .. ..

L...> -=- L...> -= ~ ""t' 1 1 1 1 1

f- Y. L.1

0 , , ~ 1 1 i_I , , , I

0 . 5 1

P/PU


I I

I-"

co o

.1Qj

'\. G:I

.,@

Figure 8.51

1~ " Based on Beam Ax I a I Force I

o Based on Slab Axial Force

.5 t ~~ ~ ~ ~

~ I I I I I

f- I I

o I Y. L.: I I I

0 . 5 1

P/PU

Variation of b /b at Mid-Span for the Girder of Specimen 1, Group III. -- Refer to e

Table 7.3. ...... CD ......

1 l::!. Based on Beam Rxial Force

0 Based on Slab Rxial Force

"g q, . 5 @

.. {Ql

" , , , , , Y. L.'

0 0 . 5 1

P/PU

Figure 8.52 Variation of b /b at Mid-Span for the Girder of Specimen 2, Group III. -- Refer to Table 7.3. e ....

co f\.J

"g ., Gil

c!21

~-Figure 8.53

1 l::l Based on Beam Axial Force


I . 5 , , , I , I , , , I , , ,

Y. L.' 0

0 . 5

P/P~

Variation of b /b at Mid-Span for the Girder of Specimen 3, Group III. -- Refer to e

Table 7.3. ..... CD w

Tg 0 Tho Axlul rorco in tho Boom

Busod on Slob Axfal Lend

------ Busad en Boom Axf~l Lend

8.814 KI'XN.

: ...... 6," • _,0

Tgc46.2 Kips , ,

~~, ____________ ~b~c~1~9~2~·®~ ____________ ~i

/0 0 /10 Cl" fa.327

b a .l'lo Q 0.313

( a)

8.26t1 K/IN. ...

(b)

100

/10 c 0.44

100

/10 C) 0.504

-I c

T i be ~~

, I I I I

• ,0, '0 A', ,'0· .....

L<J 16)(36

, , ~ S.1a

1.63 K/IN.

~6.ri'

Figure 8.54 Slab Membrane Stress Distribution at Mid-Span for the Girder of Spe~imen I,-Group I. -- (a) At yield load. (b) At ultimate load.

184

(a)

( 10 )

T9 Q Tho Axfol Forco fn tho Boom

Buood on Slob Axfol Lend

Bosod on Boom Axfol Lond

)~ I'be

I __ ---i~'--------~m~.~B~B~6-K~/~I~N~. /2).134

Ea.435

Il:I c /10 "" 8. 4 1 ~

be/fo a 9.398

T9=159.641(fps

10 /10 0 0.49 o /0' /10'" m.SS! o

~-J 16)(36

, ,

: •••••• ;,. •• : .... :. ;'.':' • •. ..,. ... : •.••.. "" .......... lJ. .• : •••. ,'. :.' -+ " -T S.~ ~.J 1 S)C36

Figure 8.55 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 2, Group I. -- (a) At yield load. (b) At ultimate load.

185

(a)

(b )

T9 Q Tho R~inl Force in tho Benm

Bused on Slub Axial Loud

Bnood on Boom Axial Lood

_ e . 95? K/ IN.

fZJ.36 I(/IN.

~'_--11

T9"",52.28 t( I lOG lSX36

, , I jl~, __________ ~b_=9~6~.~e~ ________ ~~1

10 /10 c::I 8.59 e c::I 121.569

~'~~------·_~~I __ b~e_·~_~ __ ~~~~'~~ __ ~2~.~5~1 __ K_/_I_N_. 'p'

1.25 IO'IN. \ __ -1

!'r.": .' :'.' y . .. : ••.. ::.i.,': ........... "'.: ... : .. : .. .:.. .. ;: ... :.'~.:' . .... : ••• ,~. •••• • y ~

+ #I -t s.ra

Figure 8.56 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 3, Group I. -- (a) At yield load. (b) At ultimate load.

186

(a)

(b )

Tg ~ Tho Axlul forco In tho Boom

Busod on Slob Axlnl Lond

Busod on Boom Axlo! Lond

0.09 K/IN.

!o 0 /10 c 0 • :3 1 :3

lDo/lo c 0.308

, ,

2.45 K/IN. 1t. +.1-----=-----'k-

/

0.397 K/IN.

1.034 K/IN.

. ': : Jl. to. • ',0..' '0 ... : .....: 0. •• .' • '0' 16: • •. ' ....... ' .' o.r .... '0 4;0 • ~ " ""t G.12.l

Tgc2S3.B Kips

to /10 CI 0.423 c

10 ,/10 C> 0. 4 q 8

W 2DCSe

Figure 8.57 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 1, Group II. -- (a) At yield load. (b) At ultimate load.

187

( a)

(b )

18 CI Tho Axinl force in tho Boom

Bnood on Slok:! Ax in 1 Lond

------ Bnood on iSoom Axlnl Lond

~ ~(

I be ~r

0.163

,;,..·.0' .-,':' ..... :'. 'A. ~ - ......... " •• ~' .•••••• ," ... ",. .• ... z ..... l.;A

T9083.71 I(fps

~~,, ___________ b_a_l_B_0_.0 __________ ~,~

k:! /10 c:I ta.39 o

b c /Io t::I ta.393

ta.S K/IN.

190257.66 I( t po

0.5

0.SSCJ

W 21 ~C5f2)·

1.183 t(/IN.

2.501 K/IN.

, , ;t~ 5.0


188

(b )

Tg c Tho A~tol force In tho Boom

Bnood on Slob Ax'ol Lond

Bnood on Boom A~fnl Lond

1\ be • "

V 0.452 '·VIN. II ~. I

~

I! . .... . . . ... . .: ':.' . .. :t:, '0' '" ",:" • : ' •• ' • A' . .. . .....

·rg c 78.1212 Kips

1. 17

19=231. ras K (lOS

100

//0 t:I 0.596

/oe/Io "" 0.671

., .' ... '

It~ 21X50

-L... , ,

~

1.16 K/IN. k

I I I

.4.' .. :. '~I • •.• ' •••• '0 ,f t s . i

+ " +5.9


189

,

T9 c Tho Axlul roreo In tho Boo~

Bused on 51nb Axtol Land

Bnnod on

~-1 .23 K/IN.

G.402 K/IN II ~ I

I :".!I,'; .... ~ ~. " , '4 ,::,' :'6' 1

19£183. 1 KIlOS

b /100= 0.589 e b /ba 0.564 e

,r:.

-

/ I

I I I

,'.:, :~: ::: ,~ :". = ,', '6' ,: ,':.' ,',~::I

~a

!m.

21)(50

# ,

J I'

( a)

*/0", ~---------~~~~~------~.,~ . 2.38 K/IN. ~, ,

~ /

1 I

-~ ~ I

. 12 J</IN. I 1 I I

~l I I I I : I

j I , .. ~: ....... .' .. . . : : . :-' ',,~, , : : ".p." ...... _dO ........ ':.~ .. ".' 4 .... ".0 .~ ••. :1 .. 0 ..... A'

T9=226.94 KilOS ~J 21 )(50

- - I... b /b e 0.693 o

io /10 era. 794 o (b)

4 " +4.e

.

;t4.m"


.

190

19 a Tho Axlol forco In tho aoo~

Boood on Slob Axlnl Lond

A?J.492 ~

Bnood on Boom Axlol Lond

~be -- I

/ K/IN. I --------I

I I

I I I .. . . . . ~ ,'. ~: "' . . '. ... Q •• • '.' '. ','4lo I ... : .. . .10:." ...... ' '..G" • '"

.42 Kfpo

,I , 10 0 //0

Io o //D

(a)

1.84 K/IN.

~,--~

c:a 9.614

"" 0.58

1

. ' , . , ...

W 21)(50

- :... ~ .. 10=120.0

*be

/

• 1- 156 K/IN.

, .,' "'J ... '-i .. ' ..... .. .. .. .. -:F6.a" -" .

~

~L I' 4.091 1</1N.

•• , •• to' - ••• '. 'j!' •••••• :'~"; •• ', : :.o. ..... ~.: ,',<) .. " :e: ... : .. _ "'..... ~" ..... "' ...... ",'" .. Q'" .. •• .. .... ..~. .. , .. .. ...... 0 ...... \ -t " -+6.0

( /0 )

/0 a /10 c:J IZJ. £)4:3

100

/10 C3 /i3.395

Figure 8.61 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 5, Group II. -- (a) At yield load, (b) At ultimate load.

191

( a)

T9 to Tho Aulnl force In tho Boom

Basad on Slob Axlnl Lond

Bnsed on Bonm Axlol Lond

lbe

'~

0.59 K/IN. I ~ ~

I I

I

.. " : ~ .". ~ .. .... " " _6 ". "_ ~ . . "4 ~': .: '.:.;. : : ; : ~ p .... ••• .' ··0·· .. : •••••••• 4l : : ".

1 gCl9?42 KipS W 21)(58

.... .... , .

1.<!l12 K/IN.

I I I 1 16 • et ' . . " '. "". '.1 .". ".: .,;,.: " ... \. '.

~

J b=128.S ~1'--------------~----------~ bc/to "'" 0.697

b a /10 to (2) • 575

*be ~ ............... --.......... ~' .................... ----~lf .q.12l9 K/IN.

1.84

Tgc191.1 Kips

10\:/10 Q 0.645

bo/to c 0.399

//~ ......................... --.................... --------..... -

(b)


192

Tg c Tho Axtn1 Force in tho Bourn

BU30d on Slob AXiol Lood

------ Bnood on Boom Axtnl Lond

.~ 1 .496 t(/IN. I~

9.61 &</IN i~ I ~ I

I I I I _L ' , t . . '.'~ ' ... '.' ',' G •• : '" ·.··:.:4 ',I S.fll :~.:: I::: ~ .~.:' :":'&: ,':'<fJ:.:; :j. ... : ,,- ~' • ••• ,.4 ., ••• ' • /Jj' ,' ......

TgcsUJ2.42 Kips W 21)(Sfll

-I- , ,

~~ /o""'12t2l.12l

:.~

10 0 /10 ClI 0.6

Io e /Io CI 0.571

(a.)

ibe 'L 3.9'7 K/IN. I

" if"

, , • • '. 6. • ' •• ' _, .6." .' ,'., .,', .":' :,~,' .,' • .' ••• t). . • ." '4' .·A·.·.·.··· .. ·.·o.· u ••• .I};. ••••• : ••• ~ ••• : ••

6.0

Tom19S.26 f(ips

- 10.0

/10 c 0.638

10 0 /10 c 0. LJ3

L-.! 21)tS0

Figure 8.63 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 7, Group II, -- (a) At yield load, (b) At ultimate load,

193

Tg c Tho Axinl force in tho Boom

Bonod on Slob R~tnl Lond

BnDod on Boom Rxtnl Lond

.... L 1.15 K/IN. I"

B.·45 K/IN. ~

,'-_--rl~-----~ , I

I I I .. .. 1 • • • '.' I> '. ..·s. ! ..•. p

~ ,b' • •• '., ' •••••• ,;. .... ' •• : .d " .. .,.,' :13: .... ............ ~ • ;1

- . . ~ • •• " : ..••• -4. -.- . 'AI S.0

Tgc77 • 52 ~(i lOs W 21)(5121

.... !...

~, b Cl 128.0 J j;;

/0 0 /10 r::I El.ses

ba/Io C3 0.561 (a)

~ 'tbe

'It- 3.74

: '. , ••••••••• ~ •••••••••• '.~'.' : ••••• ,"'.' •••• '. :A.' .• ':. ,'.Doo .': " '. '. ,'. ' •• ': ' ..... " _ ~ "-T -•••• ' .' c; • .." • I •• • " A . ... ". A

Toc217.G5 '(Ipc

100

/10 c 0.642

(b ) /0 <3 /10 C> 0 • 485

/;J 2 1>( 5 fa

CO'IN.


194

( a)

Tg m Th~ A~tDl 'orce in the Bcom

Bnood on Slnb Axtnl Lond

------ Bnood on Boom Autnl Lond

TbG :-1 • ,

Ir lZJ.583 K/IN. I ~ )

! I

i ' :: :' :' : .. ,: ',' ' .. : . A.: ',':'. Co: .~ :;.:..:, ~ :: :.~. :.:.; ',4. ' .....

Tgc96.G Kipo frJ 21XSI2I

0010.. ' . . ~~ to c 12lZl.1Zl

Ie /10 ca 121. 606 e

to /Io c 0.575 a

foe

r-- loti K/IN.

I . I I

! l " .0 •• 'A: ....... -~ 6.0 . . " ' .... ' ...

'1--

... ti.2B K/IN.

(b )

' .... ' .. '

"' ... ' .............. ~"

Tgc137.38 Kip:::;

ill /foc 0.G3 o

b 0 /b => ra • 365

4 .......... .

#'

....... ~ ... '" ' .... ' ... e', 10 ......... ' ...

I. ~ .... : ' •• II. ~ II ••••• " .0,', •

W 21X50


195

T9 ~ Tho A~tul rorcQ fn tho Boom

Bnsod on Slnb Axful Lond

... _---- Bosed on Bo~m Ax1nl Lond

~to . e " , "f

Iff

• 1 ! 2 K/IN. l I

1.24 " 5 K IN .

--' VI I ' , 1_·· .. :·· ........ ; .... :., .·'·9· .. '. :" .•. a ..... D· :. ••••• '""' •• 0 •• ~ ::t: S • 0

T9"" 11 1 • !:3 Ktps trJ 27)(94 -- . , ~o2ea.12l

~i---------------~

( a)

b 0 /10 "'" ra. 3 1 7

10 0 /10 CI 8.31

2.6 K/IN.

8.38 K/1N.

+-+-----.,

, ,

I I I I I I I

•• , ....... " • .:... '. ! .0' ..... ,",.0. or· : ••• ' •• 'r" •• : .' '0. * S.12l

(b)

19""387.15 Ktpc

10 0 /10 "" 8.45

Ii.l /10 "" 0.517 o

Figure 8.66 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 1, Group III. -- (a) At yield load. (b) At ultimate load.

196

Te "" Tho A~ in 1 rDf'CO in t.ho Bcom

Bnood on Sl Db Ax 101 Lond

------ Bneod on B60m Ax t B 1 Land

~ !bc

T 1.'19 K/IN.

0.205 K/IN.

( a)

(b)

••••• ,0, ~W.,:., 0' .. ' ~. 0' 0'. . ... :·6· •• ·•··.· =* 6.0' W 27~(9t1

, , b o 21S.S L

...lb-k --.----=.-----~'" b /b 0 fa. t104 . o b /to a 8.395 o

3.11 K/IN. ~b~.e_' _____ '~I'~~ __ ~l~.~S~S~K~/~I_N_. ----------~~-+----------~l

I I I I I I I I

;,,'. ; , ....... ' .. "... .: .... ..... '.: '" . : .. ,... ." '... .. ,..... ~ 6. 0 ' Tget:} 11. ~3 Kfpc

!o /10 c 9.S2Q c 10

0/10 CD fa. S 12

W 27)(94

Figure 8.67. Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 2, Group III. -- (a) At yield load .

. (b) At ultimate load.

197

Tg "" Tho Rxtol Ferce in tho Boom

Boscd on 510b Fh: in 1 Lond

------ Bnood on Boom Ax i nl Lond

~ 1be " f 1.SS K/IN.

0.602 K/IN

~:. ~ ':.: '~~::'''''":..'.,: :" ...... : ''':''~ :.;:.e:r: ...... ~.;;..~: ;.·.d .... 0°

T8""135.12 Kips

, , I 10 0 144.0 J ~:----_-----Ar

Io e /b= fa. 583

Ioa /b o 8.5? ( a)

3.56 K/IN. 3.51 K/IN. ~---------------------------

r-r-----~~~~------~-, I I I I I I I I 1.S8 K/IN. I I

I I I I I

198

:+6.';'

(b)

he/be 0. S8?

to 0 /10 1:3 0 • a 1 S

W 27}C94

Figure 8.68 Slab Membrane Stress Distribution at Mid-Span for the Girder of Specimen 3, Group III. -- (a) At yield load. (b) At ultimate load.

199

are Kips per inch. The cross-sections represent the conditions at the

yield and ultimate loads.

The figures show that the maximum longitudinal stress in the

concrete slab at the yield load occurs in the element along the y-y axis

of the steel beam cross-section. This may not be true at the ultimate

load, however, especially when the slab assumes large dimensions, as in

the models of Group III.

Figures 8.66-8.68 indicate that the maximum longitudinal stress

in the concrete slab at the ultimate load does not occur at the location

along the y-y axis of the steel beam cross-section. This is because the

slab elements above the steel beam have yielded due to the large moment

in the slab and, consequently, the adjacent element will carry the addi

tional membrane stresses due to stress redistribution.

8.6 Variation of the Effective Width at Supports

Figures 8.69-8.92 show the variation of the effective width at

the supports for the girders in Groups I, II, and III, in addition to

giving the same data for the transverse beams of Group II.

Figures 8.93-8.107 show the effective width and the stress distribution

in the cross-section at the support for the girders of Groups I, II, and

III at yield and ultimate loads.

The observations that were made with regard to the effective

width variation at mid-span (see section 8.5) were found to apply at the

supports as well. Details of the variation, therefore, will not be

given, but a comparison between the two sets of data will be given in

the following, to provide a complete understanding of the behavior of

.JQl

" OJ ~

Figure 8.69

1 6 Based on Beam Axial Force


.5

, , , , , , , , Y. L.'

0 0 . 5

P/PU

Variation of b /b at the Support for the Girder of Specimen 1, Group I. -- Refer to e

Table 7.1. N o o

.,.g '\. m

.,.©l

Figure 8.70

1 /). Based on Beam Axial Force


. 5

1 1 1 1 1 1 1 1

Y. L.I 0

0 . 5 1

P/PU

Variation of b /b at the Support for the Girder of Specimen 2, Group I. -- Refer to Table 7.1. e

'" o ......

.,©J , m ~

Figure 8.71

1

. 5

o o

I I I I I 1 1 1 I 1

Y. L.I

~ Based on Beam Axial Force


.5 1

P/Pu

Variation of b /b at the Support for the Girder of Specimen 3, Group I. -- Refer to Table 7.1. e

N o N

JQ1

'\. m

.JQl

Figure 8.72

1 -b. Based Beam Axial Force on

0 Based on Sl ab Axial Force

.5 f-

A 0 000 A 0 A A /'\ /'\

v

y

~~ I I

A I z:; er- A h....A. I L-' '-" ~'-"~

I I I I

0 I I Y. L.: I I I I

0 . 5 1

P/PtlJJ

Variation of b /b at the Support for the Girder of Specimen 1, Group II. -- Refer to 7

e Table .2.

N o w

.£J1

" (1l) ~

Figure 8.73

==

b. Based on Beam Rxial Force

.5 0 Based on Slab Rx i a 1 Force

~ I

.25 I I I I I I I I I I I

Y. L.I I

0 0 . 5 1

P/py

Variation of b /b at the Support for the Transverse Beam of Specimen 1, Group II. -e

Refer to Table 7.2. tv o ..".

.!ll '\. m

.!l

Figure 8.74

1 - 6 Based on Beam Axia] Force


.5 l-

A v

A A /\ .... T .A

1 . 1 . ......

LS 1 1 1 1 1 1

0 Y. ~.: I I I

0 .5 1

P/PUI

Variation of b /b at the Support for the Girder of Specimen 2, Group II. -- Refer to e

Table 7.2.

I

I I I

I I I I I I I

N o U1

~~ \~

Qj) ~

Figure 8.75

lJ. Based on Beam Axial Force

. 5 c-O Based on Slab Axial Force

I

1\ 1\ 1\ . 11.-.25 '-' ..... ..... 1 A A. A

v v v l' , , , , , , , , Y. L.'

0 I I I , , . I

0 . 5 1

P/P~

Variation of b /b at the Support for the Transverse- Beam of Specimen 2, Group II. -Refer to Table

e7.2.

I

N o 0'1

.n ,~

(!j) ~~

Figure 8.76

1 I-~ Based Beam Axial Force on


. 5 I-

A A A A V

i x v ~

I I

..A I A

'-' L...>

I I I I I I I

0 I Y. L.: I I I I

0 . 5 1

P/PU

Variation of b /b at the Support for the Girder of Specimen 3, Group II. -- Refer to Table 7.2. e

N o -J

~-

.,.@ '\

(]) J~

Figure 8.77

l:1 Based on Beam Axial Force

.5 ....:... 0 Based 51 ab Axial Force on

.25 f-

" " " . . /\. J: A ~ ~ ~ v '( v V v ~ , ,

I I

Y. L.'

12) I I : I I I

12) . 5 1

P/Py ------

Variation of b /b at the Support for the Transverse Beam of Specimen 3, Group II. -Refer to Table

e7.2.

N o CD

JQ1

" (]l .c©1

Figure 8.78


0 Based on 5 lab Ax i al Force

. 5

I I I I I I I I I I I

Y.L. 0

0 .5 1

P/PU


Table 7.2. N o ~

"""""""'--==-

.,@

" OJ ~

Figure 8.79

D. Based on Beam Axial Force

. 5 I-0 Based 51 ab Axial Force on

.25 -

A A

A 'A ';: L..> . A V Y V

...... -n ,.. I

v

I I I

Y. L.I

0 I I I I I

I

0 . 5 1

P/PU


e7.2.

,

N ...... o

1

..@

" .5 Oil

.,@

o o

1 1 1 1 1 1 1 1 1

Y. L.I

l:l Based on Beam Axial Force


.5 1

P/PIIJ

Figure 8.80 Variation of b /b at the Support for the Girder of Specimen 5, Group II. -- Refer to Table 7.2. e

I\.)

......

......

Jg '\.

(]) c©

Figure 8.81


. 5 - 0 Based 51 ab Ax i a 1 Force on

.25 f-

A " ......

" " " A A

~ A AI ';:. -<;: A A V V V , V V v V

A , , , Y. L.'

I . ,

I 0 • I. • 0 . 5 1

P/PU


Refer to Table 7.2. r-J ..... tv

1

·.!2l \. .5

CJjl ~

o o

1 1 1 1 1 1 1 1 1

Y. L.I



. 5 1

P/Pu

Figure 8.82 Variation of b /b at the Support for the Girder of Specimen 6, Group II. -- Refer to e

Table 7.2. I\) -Vol

£,1l

" OJ ~

Figure 8.83


. 5 - 0 Based 51 ab Axial Force on

.25 f-

A " -~

" A . A. --=-X x ~ ~ ~

"-'

v v

I v A /\. A A V V v v , , ,

Y. L.' , 0 I I , • • I

0 . 5 l

P/PU


e7.2.

rv ...... ~

1 ~ Based on Beam Axial Force


~ '\. . 5 m

.~ I I I I I I I I I I

Y.L. 121

121 . 5 1

P/PU

Figure 8.84 Variation of b /b at the Support for the Girder of Specimen 7, Group II. -- Refer to Table 7.2. e

f\J ...... lJ1

..... ~ "' ..

OJ .k~

Figure 8.85

IJ. Based on Be am Ax i a I Force

. 5 i- 0 Based Slab Axial Force on

. 25 t-A A A -v ~

A A A v " A v

-A-A :::: A ~ a a ~

"T" L...l L...l L...l

I I I I I

y, L"

0 • I I

0 . 5 1

P/PQJ

Variation of b /b at the Support for the Transverse Beam of Specimen 7, Group II. -

Refer to Tablee7.2.

I

N ~

(j')

1 -6 Based Beam Rxial Force on

0 Based on Sl ab Rxial Force

JbJl ~ ',\ .5 f- ~

v v

(iJ A .A ""-i .... v

""Ibll

Figure 8.86

- oa I I I I I I I I I I

0 I Y.L,.: I I I

0 . 5 1

P/PUl


Table 7.2. tv .... ~

.b,. Based on Beam Axial Force

. 5 c- O Based on 51 ab Axial Force

.. ~ q, (lJ .25 '-

.. @

Figure 8.87

A 1\ " A . :r :;:;: A .x -=- ~L1

A 'X -::: 'r v v v 'V V~~ , , , ,

Y. L.' , 0 I I I

0 . 5 1

P/PIUJ


e7.2.

,

I I

I

I I

I

tv ..... CD

1

.£Ll ~\ . 5

ill "g

o o

, , , , , , , , , Y. L.'



.5 1

P/Pu

Figure 8.88 Variation of b /b at the Support for the Girder of Specimen 9, Group II. -- Refer to Table. 7.2. e

N .--ill

• .@ , Gll

.~

Figure 8.89

'"


. 5 f- <> Based 51 ab Axial Force on

.25 !-

A A II.

A J. A ~-= ..... ..... ~

A ';: A A /\ A A V i v v v v V

A

1 1 1

Y. L.I 1

0 I I I I I I

0 .5 1

P/PtlJ


Refer to Table 7.2. I\J I\J o

1 -D. Based Beam Axial Force on

<> Based on Slab Ax i a 1 Force

.JQl

" .5' f-

a» • .fQJ

II. " A A A A 0 0 0 v I ....,.

~ .J( V L...>

I I -A-I I I I I

0 Y. L.: , I . , I

0 . 5 1

P/PU

Figure 8.90 Variation of b /b at the Support for the Girder of Specimen 1, Group III. -- Refer to Table 7.3. e

f'V f'V -

..frdl ,~

(}ll .£dl

Figure 8.91

1 f-b. Based Beam Axial Force on

0 Based on Sl ab Axial Force

. 5 '-

A A A A A A A AA A v "( v V V v v ~vv

I I I -- 1>.~

I -= I I I I

0 I Y. L.: I I I

0 .5 1

P.I'Pu

Variation of b /b at the Support for the Girder of Specimen 2, Group III. -- Refer to e

Table 7.3. N N N

1 i- f). Based Beam Axial Force on


.Jfl A~ '\. . 5 -

"- A A A A. A A h OJ v ,.., ..... = <=" V V V

..@ I I I ~ I I I I I I I I

0 I I Y. L.: I I

0 .5 1

\ P/PU

Figure 8.92 Variation of b /b at the Support for the Girder of Specimen 3, Group III. -- Refer to Table 7.3. e

N N W

( a)

(b)

T9 c Tho Axtnl Force in tho Bonm

Bnsod on Slub Axtnl Land

------ Bnood on Boom Axtnl Land

T '0

0.99 K/IN. ~ I e 'I-,r~-----------------------------

0.156 K/IN.

\a ' ..... :.,'.

TgC169.5t3 t(ipc

n-----.",JI!:-.-----..

~/ •••••.... : c. ' •• _ '.' .:.: ....... :. '., ".<>" ...... Ii.. ...... !" ..•.

, , 10 0 192.0

~~------------------------~~

100

/10 CJ G.356

lao/b c 0.366

1.31 K/IN.

0. q K/IN

\ :.----

t I

~,,~ ______ I~b_e ____ ~'f

. :::- :\". ' ......... .••.. : ..... •••.• .. ':.·.A··f .. :~.:·.::.,. •

T9090.35 Kips

be/b "'" 0.53

Io e /Io c 0.36

~ ~ :=t Ii.ta

1.18 K/IN.

, ,

Figure 8.93 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1, Group I. -- (a) At yield load. (b) At ultimate load.

224

T9 c Tho A~ I u 1 F'orco In 'tho Beom

Bnood on 51 nla A~ 101 Loud

------ 3000 cI on Boom Ax i 01 Lo oc!

i· ~e ~ . " 1.373 K/IN.

B.2ItH K/IN.

T9=74.66

....ok- ' , +6.0

~~, __________ ~b=~14_4~._0 __________ ~i~

(a)

100

/10 c 8.367

&:'0/10 t:I e. 378

~I _______ ~+b~e~ _____ 4'1, I 1 ':15 K/IN 1"- I . 1.34 K/ N. ~.~~------'------------~ ~-------------------------ta.S46 K/IN.

'. '.be •• :. '~' •• '.~ .' '. e.: ...... :'1" ••••• ::. ",.: eO'; : P~. ',' . • ',b, .. ' eO ,.:6

10 0 //0 c 0.60

b /10 era. 496 o

, , :t 6.0


225

(b)

Tg 0 Tho Axtnl Force In tho Boom

Bnnad an Slnb Axfnl Lood

------ BnGod an Bcorn Axfnt Lond

0.42 K/IN.

~'-----

klo/b t!:I rEI. 64

10 /10 C3 tao S4 a

~". '., '. '.:. ~ :'.': .. - "

W 16)'36

.. ..

1.B4 I(/IN.

c.19 K/IN.


226

(a)

(b )

TS "'" Tho Altlnl -forco in 'tho Boom

B.tlcod on Slob Alt to 1 Lood

------ Bnsad on Boom Ax t n 1 Lood

tOe 'I- "}

1.14 K/IN.

/2).23 K/IN. \.

....... "0.:"'.° .... "0

TgclfZH.94 Klpc

JV' I

.' .,A,' • • ' •• • '.tl. •••••••••• ':'" ........ '. ". ~ •••••• <t- •••• ~.'

I /;J 2l)C5~ , ,

bc24~.0 L ··I~------------~----------~' --, ~

"

, , :t 6.0

2.295 I(/IN.

,0 ;..' :: ••• ,G. .',. • .. ..d .... 0 ..... ' : •••• .' •• :.' eo • '0 '.. • ',", • " ... : .. to eO • ,a." ; •

bo/be> ta. q 1?

loa/be 0.243

W 21}!S8

Figure 8.96 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1, Group II. -- (a) At yield load. (b) At ultimate load.

227

T8 ~ Tho A~inl force in ~ho Boom

Bnnod on Slob A~tnl Lood

------ Boood on Boam A~tol Lond

..i I' I'

V"

1 Il fa V' • 273 K/IN • 1 I I

( cd

~ 1

............ 1 1 I r

t·~······, " . -.. ". .......... ~ ..... .... ~ .. : ... , .........

T9""111.69 Kips

10 /10 m fa. 315 o

100

/10 "" 121.371

.1

. ~J 2 UCS0

.10.

228

1 .645 K/IN.

' , " . ... =t: s •ra

2.54 K/IN.

(fa )

0.72 K/IN.

. ';' , ',.', .. '.' .-'.. ", ...... .".,.. . ~ " .. : .... : ............. ' '. '., .. ~

Tgt:::149.14 Kipn

bo/b Q 0.527

100

/10 r::: 9.328


( a)

19 ~ The A~tul torce tn ~ho Boom

Bn~od on Slob A~fnl Loud

------ Bu~od on Bourn A~fnl Lond

~ • be 1.915 K/IN.

• ... • " •• • eA .:".:'.: ".Il.: .... ;, ......... .... :.0 •• ; :-:.: .:·.·.4::· .:: .... ::;.~.: ,".'::

T9c103.17 Kips

. , 10=120.0

10 e /10 CD fa • q 5

10 e /b t:> e . q 5

4';: , 3.Gl K/IN. A

-I : I I I I I

I I I , I I

--------I , I I

r-- I , I I I I I I

229

I

=l , ,

(fa )

.' .. . . . . . 6 ..... ...

3.45 Kips

100

/10 co 0.54

ho/to Q B.308

"l . . .. .. .,. . : ,.,.. • 1iI.'" .~ .. .,

' . .p. <I S.S . ~ " .z,; . ... . .

l..J 21X50

.1


T9 c The Axlnl forco In the BlOom

Bnood on Slnb Axtnl Lond

Boood on Be am

V

V 0.52 f</IN. I

\ --' I I I

2 05 KI'IN . . i I I I I I I

.;':. '. : : :(1' •• :. ' •• l!,:' ..... ~ . ". :;. ... '. : •••• "1 ••••••••• '.", : : ... • ltl· .', .•• ~ ••••. , .':'J ....,·4.0 :t " T90109 .95 ~Ips trJ 21X50

-"- . , ..,~

/0"'120.13 ~

- /00

//0 CI 0.45

bel /10 0 ta.447

(a) ~~ foe l\, ,

~ 4.31 K/IN .

.; :" .. .' :.' !" .. : ,0. C::..: .. '~·:."o eo ".0"0: :~:.:'"o:· .6,0" .- "0. "o'~ "0.°.: ;,.

T90133.47 KIps

/0 C'.l',,/o c:lI 0. 457

/0 e /10 c::J 121. 2 G . (b)


230

T9 c Tho Ax~nl Forco tn tho Bourn

Bnood on Slob Axfnl Lond

BOGod on Boom

'~'------~/-_~_~_~-_~~'~r ____________ ~2~.~9~K~/~1~N~ .

• ';,.::,' ,',. ,~, : ' ,,~,' ': ' : : ':" ',,',: ,;., ,:: ::~' :',:', ".0' ,', ", ',' ,4, ':,': ,~ ',': ':.':: -:t 6 • rei '

t'4 2 n(S0

, , I '/oCl12e.3 J 4~~-----------~--------------------+~

bo/b tl:I fa.384

100

/1:1 c 0.385

(a)

. 3.36 K/IN •

!

v --I r----

/ I I I I I I

.45 K/IN.I I , 1 ___ I I r-- I I·

I I I I I I

I I .. ". ' <1>: ' , ... . ',:0."::':' ::::'. :", .:.:s ........... 6: ' ," " '.I +6.8' .... .'.

,0 •• • 0° ~. , ... " , , ,,' ~"

Tgc341.1 Ktpo ItJ 20(50 - Io o /b c:I fa.S8

_I-

(10 )

Figure B.100 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 5, Group II. -- (a) At yield load. (b) At ultimate load.

231

(a)

Tg c Tho Axfnl rerce fn tho Scnm

Bonod on Slob Axtnl Land

Bonod on

Tg=154.23 Kips

b /Iocs fa. 396 o

10 a/lot:J e. 397

BOllm.Axiol Lond

}\be -;\~' ----t-+-' ----./f-

f:J 21)(50

, , Io m 120.flJ

!"loe ~~----------t--r----------~jf

3.24 K/IN.

T " -i- 6.0

3.381(/IN. ~r--------------------------

r - - - r------="'i~-r------~~- - -, I I

1.46

I I I I

K/IN.I ! ---""\, I I

T90345.0 C<1ps

Io c /Io o 9.579

(b) De/loCI (d. 853

W 21)(50

I I I I , ,

6.0

232

Figure 8.101 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 6, Group II. -- (a) At yield load.

-- (b) At ultimate load.

Te c The Axin1 force In tho Boom

Bnead on Slnb Axlnl Lond

Bnsod on Boom

233

3.125 f(/IN.

ta.S69 KI'IN.

- •• • 'el •• a, .Q

" : : 6 I" •• - •••• """. '.

, ,

~\ 10 0 /10 = 0.41

10 0 /10 t:l 8.41

( a) ~l, 3.58 K/IN.

I

1

r---

/ ---.,

I I I I I I

.36 K/IN. I I

~ ---- I

\ I I ~ I I

I I I I I I I I t " : .I> ....... ~ ..... ,1> •••••••• ' .... " ••• ' ....... ;\ 6.121

~ : : . . o"A. : •• : •• oS' : •• :. : .' ....... : '.' .' .. ' .:a'. : . : ... ~

Tg=326.54 f< i lOG IrJ 2t)tsra

-- 100

/10 '" 0.517 ... '-

(b )


Ta c Tho A~lill forco In tho Boom

Bil~od on Slob A:t t n 1 Leila

------ BIl~ed on Benm fl~ f n I Lond

( a)

1.4

"'j, 4>e 'F 1.974 K/IN.

: .• ' ~. ...... ".:.'. '.&>. ". : .' ".: ~ ~":".' .- 0 •• ~ ..... .0' .' ' • .0 •••• • -,: ... ~ ••• Q ....... ~ ••• • 0.° .... : •• e •• eo •••• ~.: •••••••• tt;;

~" 21 X50

1 •

• 1 IoCl1212l.t2I i ;"C"'"\i --------------~'~

be/b 0 13.454

100

/10 Cl 13.452

3.594 I(/IN. ~ r-

~ -r

I I I I I I I

K/IN. --+- I I \' I

I I I I I I I I I 4 6 •

01' • .A eo : .:. : .~ . .': .~ .&: :. e.:. ~ .: .. ; ~ . :.: ~. "0- ",.6.- •• 6 • • •• • •••• !l:. -. \ : I

..... • "6 •••• ,. ••• .:0..

TgCl272.1 Kfp::: W 20(59

- Ioo/b r:::I 8.525 -!..

(b )


234

19 0 Tho Axlol Forco in ~ho Boom

Bnood on Slob Axtnl Lood

0.542

Bnsocl on Boom

• 0" '0 '~ •• : • '0' ,0 b* ••••• : :'.~ •••••• '0' 0" 'do,',: •..•• :..q ,0, ~'.' r: . e •• • •• ' o'·p. eO ,to·· eO .,., eo. to, a, e. eo ..... ,:\ ••• ' • ....tI.

19"'152.27 Kfps W 21){S8

, ,

3.19 KI'IN.

~~~ __________________ b_O_l_2_0_._0 __________________ ,~~ .

100

/10 C1 0.397

Ca) ro fa /10 c f2l • :3 9 8

'k

r----

K/IN.! V .4B ~ I~

( 10 )

I I I I

:~ .... : .. :.~:.: .• :.:.~.::: .. ~ .. :.. T9",3SB.0 I<t lOG

- 100

/10 CJ 0.562

b /10 c 0.943 G

\: .. 4,'

-

" I' 3 • 54 t<1'1 N • I ........

--'1 I I I I I

I I I I I I --t " . , 'tt:1 • 6.121 "" , " ','0' ' .. ,' ,.', , , "

':, ',0:, " , ,",,' '6" , " '" ,0 :

!rJ 21){SI2l

--


235

(a)

(b)

T9 0 Tho A~lol forco In tho Boom

Bilsod on S 1010 Axlnl Lond

------ !Bosed on Boom Axftll Lend

>k ~be ',," I

0.271 KI'IN.

••••• '. • -............ , •••• ~ '0' •

.to .1'10 0 ra. 3S o

10 1'10 0 Ill. 35 ! o

~I' Ibe

'I-

9.473 KI'IN.

T90251.12 Ktpo

b /1:1 c 0.347 o

b /10'" f2J. 225 (3

~'J

I I I I I I I I I I I I ',' t· •

27~94

, , -4 S • 1ll

3.873 1(I'IN.

, , ,0,. 00 ... '0·

Figure 8.105 Slab Membrane Stress Distribution at the Support for the Girder of Specimen 1, Group III. -- (a) At yield load. (b) At ultimate load.

236

237

Ta c Tho Ax i n 1 tor-co In tho Boom

Bn~od on 51 nb Rxlnl LOlld

------ Blasod on Boom Axial Lond

b .

'" '1 be '~ r: 1.984 K/IN.

............ <;I •••• ",' .... -:. -·.···· •• :~I·..,··,· ... · -.. :.·.,.. .. 1 .. :"'!: ~.6.ra'·'

T90161.5 I( f ps ftJ 27)(94

. , '~

/00216.0 'f

100

/10 c:J 0.37

to (3 /10 c .ra.377 tbe ( a) 'F ,,1- Q.7S K/IN. ...

121.53 K/IN.

.13'.' ...... : .... ";. •• '0' ...... ""'- ... :.: .~ ...... :'0 ..... ~ •••• ,: -1 ' , '" \ • ..:.a ~ • . . "'. ..... . G.0 r

IrJ 27)(SQ

(10 )


( a)

( 10 )

T9 CI The Axlol force In tho Boom

Bnood on S 1 ilb Ax 101 Land

------ lEh:lI3od on Boom Axlol Land

be r , 2.59 K/IN.

f2J.GS t</IN.

19""1'72.5 Klpo ~J 2'7~94

I I

4~ _________ b_O_14_4_._B ________ ~'~I~

b /10 C1 9.454 o bQ/b c:z el.4S3

~I------~------~ 5.32 K/IN.

238

~----------------------------

I I I I I I I I

1.'71 K/IN. I I I I I I I

:A': to' ,: .~.~ : :'0 :A." • : : It· .: 4,: I ,0 to .'~ •••• ~ •• ", to t.: :d·.to : .,: •••• ':":'0'

100

/10 q 9.51

bc/b c fa.33

ftJ 2'7~94


239

composite floor systems. For simplicity, the comparisons will be made

for each specimen group separately.

The girders of Group I had fixed end supports (fully moment

resistant beam-to-column connections). The first yield took place in

the steel beam, and the effective width that was based on the beam axial

force, therefore, was smaller than the one based on the slab force.

Examples are given in Figures 8.69-8.71. At mid-span, the first yield

occurred in the slab, and the conditions, therefore, are the opposite.

The above observations can also be made for the first four spec

imens of Group II, since these also had fixed ends. The resulting

effective width variation at the supports is shown in Figures 8.72,

8.74, 8.76, and 8.78. In comparison, the mid-span effective width vari

ation is shown in Figures 8.33, 8.35, 8.37, and 8.39.

For the rest of the specimens of Group II, as well as for the

transverse beams, the end connections between the girders and the col

umns were semi-rigid, and simply supported for the transverse beams.

The axial force capacities, therefore, are high at the supports. In

these specimens, the first yield at the supports occurred in the slab,

which therefore gives effective widths by equation 8.5 that are lower

than those of equation 8.6. This is illustrated in Figures 8.80-8.89.

At mid-span, the reverse is the case, because the steel beam yields

first due to the large moment. The results are shown in

Figures 8.41-8.50. These conditions also apply to the transverse beams,

since they had simply supported ends.

The specimens of Group III had fully moment-resistant beam-to

column connections. The observations that were made regarding Group I,

240

therefore, apply here as well. The effective width variation at the

supports is shown in Figures B.90-B.92, and the variation at mid-span is

shown in Figures 8.51-B.53.

On the basis of the above discussion, it is clear that the

girder and beam boundary conditions are very significant for the behav-

ior of a composite floor system. It also means that the effective width

is strongly influenced by the support conditions, especially after first

yield in the composite system. It is important, however, to observe

that the effective width that is based on the slab axial force is the

one that should be used in practice-oriented applications, because it

gives a better representation of the conditions in the slab. Also,

since the analysis was performed for an indeterminant structure and the

aim was to find the effective width at ultimate load, the actual slab

value should be considered.

B.7 Influence of Other Variables on the Effective Width at Mid-Span

B.7.1 Influence of b/~ Ratio

The effective width at mid-span at yield and ultimate loads has

been calculated from equations B.5 and B.6 for the first three specimens

of Groups I, II, and III. These have b/~ ratios between 0.4 and O.B.

The results are given in Tables B.1-B.3, and are further illustrated in

Figures B.10B-B.110. The following points may be made:

1. At ultimate load, the effective width that is calculated by

equation B.5 gives a conservative value.

Table 8.1 Influence of b/~ Ratib on the Effective Width at Mid-Span (Group I Specimens).

b /b e

at Yield b /b e

at Ultimate

Specimen b/~ Byb Byc By By No.

a Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

1 0.8 0.327 0.313 0.437 0.504

2 0.6 0.414 0.398 0.49 0.551

3 0.4 0.59 0.569 0.673 0.716

aSpecimens 1, 2, and 3 of Group I had the following in common: W16x36, p = 0.01, ~ = 20 ft, slab thickness = 6.0 in., and fixed ends.

bEquation 8.5 gives the effective width based on the axial force in the slab.

CEquation 8.6 gives the effective width based on the axial force in the beam.

241

Table 8.2 Influence of b/~ Ratio on the Effective Width at Mid-Span (Group II Specimens).

b /b e

at Yield b /b e

at Ultimate

Specimen b/~ Byb Byc By By No.

a Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

1 0.8 0.313 0.308 0.423 0.48

2 0.6 0.39 0.393 0.5 0.564

3 0.4 0.587 0.562 0.596 0.671

aSpecimens 1, 2, and 3 of Group II had the following in common: W21x50, p = 0.01, ~ = 25 ft, slab thickness = 6.0 in., and fixed ends.



242

Table 8.3 Influence of b/~ Ratio on the Effective Width at Mid-Span (Group III Specimens).

b /b e

at Yield b /b e

at Ultimate

Specimen b/~ Byb Byc By By a

No. Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

1 0.8 0.317 0.31 0.45 0.517

2 0.6 0.404 0.395 0.524 0.612

3 0.4 0.583 0.569 0.687 0.816

aSpecimens 1, 2, and 3 of Group III had the following in common: W27x94, p ~ 0.01, ~ ~ 30 ft, slab thickness ~ 6.0 in., and fixed ends.



243

1

~

" .5 OJ

.IQ)



~::, " " ':::, .....

............ :::: ............ ~ ........................

..................... ~ ................... _--..... _--

Load Level = Ultimate

----- Load Level = Yield

o o

Figure 8.108

. 5

b.l'L

b /b vs. b/~ at Mid-Span for the Specimens of Group I. e

1

r-v ..,. ..,.

1

..©l " .5

(J] n

o o

Figure 8.109


o Based on Slab Rxial Force

Load Level =.Ultimate


. 5

ro,/L

b /b vs. b/~ at Mid-Span for the First Three Specimens of Group II. e

1

t\.) ..,. lJ1

1

jQl , .5

ill .JQl


<> Bas e don Slab Ax i a 1 For c e

~~, .... ,

.... "" .... "" "'"" "~~ ~"!::..'::::.~ ....

M_~~_-. --~~~~~



o o

Figure 8.110

. 5

b/L

b /b vs. b/£ at Mid-span for the Specimens of Group III. e

1

I'V .r:::. (j)

247

2. The ratio of b /b decreases as b/£ increases. Since the span is e

the same for all of the specimens within a group, this indicates

that the value of b /b decreases as the slab width increases for e

a beam with a constant span.

To show the influence of the span of the beams on the effective

width, Table B.4 gives a comparison between the effective widths for the

first three specimens of Groups I, II, and III. The results are further

illustrated in Figure B.lll. Major findings can be observed as follows:

1. It is obvious that the span length is a primary influence on the

effective width. This is in agreement with previous studies.

2. The effective width at the yield load for the first three speci-

mens of each group is found to be between £/4.3 and £/3.9.

Figure B.lll shows that a straight line can be passed through

all of the points that represent b /£ at yield load, without e

introducing a significant error. Consequently, the effective

width at yield can be expressed by the following formula:

b ; e

£ £ +

4.3 24 b £ - 0.4) (B.l0)

Equation B.l0 is almost in complete agreement with the current

practical design criteria [4,6]. This confirms that the spec i-

fication formulation is elastically based.

3. The effective width at the ultimate load increases with the b/£

ratio. The increase is substantial for specimens with b/£

ratios greater than 0.4, and may be related to several factors.

A conservative equation for the effective width at ultimate load

Table 8.4

Specimen No.

1

2

3

Influence of Beam Length on the Effective Width at Mid-Span. -- The effective width used in this table is based on the axial force in the slab.

Group I Grou12 II Grou12 III b /9- b /9- b /9-

e e e b/9- 9- At At 9- At At 9- At At

Ratio (ft) Yield Ultimate (ft) Yield Ultimate (ft) Yield Ultimate

0.8 20 0.261 0.35 25 0.25 0.338 30 0.253 0.36

0.6 20 0.25 0.294 25 0.234· 0.3 30 0.242 0.314

0.4 20 0.236 0.269 25 0.235 0.238 30 0.233 0.275

f\.)

~ (l)

1 .-

J " .5.

GD ..!rll

o o

---- Load Leve1 = Yie1d ~ Group I Specimens

- - Lo ad Leve 1 = U 1 t i mate <> Group II Specimens

Proposed Equations o Group III Specimens

::====::~~ __ --~---_--<ru 8..~ 2 :-=~ ~ __ ::::::::.:_::::::;-==-::::::.;:===:.,~==!l;1------------== ___ A G=- -:....... - - - - = c$lr------

I l i i i i i I 1 . 5 1

b/L

Figure 8.111 b /£ vs. b/£ at Mid-Span for Specimens 1, 2, and 3 of Groups I, II, and III. e

t\.)

~ \0

250

may be based on a straight line, as shown in Figure B.111.

Thus, the following simplified equations are proposed for the

computation of the effective width at ultimate load~

b Q,

for E. < 0.4 = 4 Q,- (B.11) e

Q, Q, b 0.4) for b > 0.4 b = -+- (--

e 4 5 Q, Q, (B.12)

B.7.2 Influence of Beam-to-Column Connections

The influence of the type of beam-to-column connections on the

effective width at mid-span has been examined for specimens 5-B of

Group II. These used semi-rigid connections with different rigidities.

The results are presented in Table B.5 and Figures B.112 and B.113.

Figure B.112 gives the ratios of b /b vs. the degree of fixity e

of the beam-to-column connection. The latter is expressed in terms of

the ratio M /M , where M is the moment that can be resisted by the con-e p c

nection and M is the plastic moment of the beam. Figure B.113 gives p

the ratio of b /Q, vs. the ratio of M /M • e c p

Both figures show that the effective width at mid-span decreases

as the degree of fixity increases. It was found that the difference in

the effective width for a specimen with fixed ends and another with

simply supported ends is about 5 percent at the yield load and B percent

at the ultimate load. In this evaluation, the effective width was cal-

culated on the basis of the axial force in the slab.

To account for the type of beam-to-column connection, another

term should be added to equations B.10-B.12. In this manner, the

expression

Table 8.5 Influence of Support Conditions on the Effective Width at Mid-Span.

Specimen No.

a

3

5

6

7

8

b/Q. Ratio

0.4

0.4

0.4

0.4

0.4

Support b Restraint

Fixed

0.1 M P

0.3 M P

0.5 M P

0.7 M P

b /b at Yield e By

Eq. 8.5

0.587

0.614

0.603

0.6

0.586

By Eq. 8.6

0.562

0.58

0.573

0.564

0.56

b /Q. at e

From Eq. 8.5

0.235

0.245

0.241

0.24

0.234

Yield

From Eq. 8.6

0.225

0.232

0.229

0.226

0.224

b /b at Ultimate e

By Eq.8.5

0.596

0.643

0.645

0.638

0.642

By Eq. 8.6

0.671

0.395

0.39

0.43

0.485

b /Q. at Ultimate e From

Eq. 8.5

0.238

0.257

0.258

0.255

0.257

From Eq. 8.6

0.268

0.158

0.156

0.172

0.194

a The specimens had the same variables (W21x50, Q. = 25, b = 10, P = 0.01, t 6.0") except for the degree of fixity at the supports.

bThe girder end restraint varies from fixed to semi-rigid (capable of resisting 0.1 M ). p

r-v U1 .....

1

..@

"' .5 (]) ~

o o



---------~---------~--7------A------- ----fr---------b------- __ ~---------~-----~-----

Load Level ~ Ultimate

----- Load Level ~ Yield

. 5

Mc/Mp

1

Figure 8.112 b /b at Mid-Span vs. the Degree of Fixity at Support (Specimens 3, 5, 6, 7, and 8; G?oup II).

N LT1 N

Based on Slab Axial Force

. 5 I- b/L = 0.4

..J

'" m .251- s- -~ __ ~ ~ ______________ ~ .J2}

Figure 8.113



o ~ i i i ~

o . 5

Mc/IM~

b Ii at Mid-Span vs. the Degree of Fixity at Support (Specimens e

Group II).

I I

1

3, 5, 6, 7, and 8;

N U1 W

254

Q, M

(1 -c

65 M (B.13)

P

should be added to equation B.10 and

Q, M (1 -

c 50 M

(B.14)

P

should be added to equations B.11 and B.12. This gives effective width

equations that incorporate end connection as well as beam stiffness, and

they assume the forms:

1. At yield:

Q, Q, ( b Q, b 0.4) (1 = + +- -

e 4.3 24 Q, 65

2. At ultimate:

M Q, Q, b b (1

c 0.4 = + - for Q, ~ e 4 50 M

P

Q, Q, ( b Q, b 0.4) ( 1 = + + 50 e 4 5 Q,

B.7.3 Influence of Steel Beam-to-Slab Stiffness Ratio

M c -

M P

M c

M (B.15)

P

(B.16)

b for i" > 0.4 (B.17)

The effect of the steel beam-to-slab stiffness ratio on the

effective width at mid-span is illustrated by the data in Table B.6 and

Figure B.114. It can be seen that b /Q, decreases as the stiffness e

ratio, K , increases. r

For each group of specimens, increasing the stiffness ratio from

the minimum to the maximum value of K results in a decrease of the r

effective width of B percent at yield load and 30 percent at the ulti-

mate load. This may be caused by many factors, although it is believed

255

Table 8.6 Influence of Steel Beam-to-Slab Stiffness Ratio on the Effective Width at Mid-Span. -- The effective width used is based on the slab axial force.

Group Specimen K a b /9., e

r No. No. At Yield At Ultimate

I 1 1.013 0.261 0.35 2 1. 35 0.25 0.294 3 2.025 0.236 0.269

II 1 1. 78 0.25 0.338 2 2.37 0.234 0.3 3 3.55 0.235 0.238

III 1 4.93 0.253 0.36 2 6.57 0.242 0.314 3 9.85 0.233 0.275

aK = beam-to-slab stiffness ratio.

r

1 I-

.dJ " . 5 ~

IDJ .,@

o o

Load Level = Yield b. Group I Specimens

Load Leve~ = Ultimate o Group II Specimens

Based on Slab Axial Force o Group III Specimens

~~ G-

~

~ G-------8----- __________ ~

I I I I

5 10

Krr

Figure 8.114 b /£ at Mid-Span vs. Beam-to-Slab Stiffness Ratio for Specimens 1, 2, and 3 of Groups I, II, and III.

N U1 0'1

257

that the slab width and the beam span are the major contributors. For

this reason, the previously recommended ~quations have been retained.

8.7.4 Influence of the Slab Thickness

The effect of the slab thickness on the effective width at mid

span has been studied by comparing specimens 3 and 4 of Group II, as

shown in Table 8.7. Taking the effective width based on the slab axial

force, at the yield load the effective width for the specimen with a

slab thickness of 4 inches was 4 percent less than that of the specimen

with a 6-inch slab. At the ultimate load, however, the effective width

for the specimen with the smaller thickness was 15 percent la~ger than

the effective width of the specimen with the large slab thickness.

These results indicate that the effective width at yield load is

independent of the slab thickness. At the ultimate load, however, the

specimen with the smaller thickness had larger effective width to be

able to carry the equivalent axial force in the steel beam.

Even though larger effective width at the ultimate load was

achieved for the slab with smaller thickness, no recommendation is made

to increase the effective width. This is to be on the conservative side

for design.

8.7.5 Influence of the Slab Reinforcement Ratio, p

The effects of the slab reinforcement ratio have been studied by

comparing specimens 6 and 9 of Group II, as illustrated in Table 8.8.

It can be seen that the effective widths at yield and ultimate loads are

almost the same for both specimens, with a difference of less than 2.5

percent.

Table 8.7 Influence of the Slab Thickness on the Effective Width at Mid-Span. -- Specimens 3 and 4 of Group II were identical except for the slab thickness.

b Ib e

t At Yield At Ultimate Specimen c

bill, By By By By No. (in. ) Ratio Eq. 8.5 Eq. 8.6 Eq.8.5 Eq. 8.6

3 6.0 0.4 0.587 0.562 0.596 0.671

4 4.0 0.4 0.563 0.589 0.683 0.794

At Yield By By

b I~ e

At Ultimate By By

Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

0.235 0.225 0.238 0.268

0.225 0.236 0.273 0.318

l\.)

U1 (l)

Table 8.8 Influence of the Reinforcement Ratio on the Effective Width at Mid-Span. -Specimens 6 and 9 of Group II were the same except for the slab reinforcement ratio.

b /b e

b /£ e At Yield At Ultimate At Yield At Ultimate

Specimen b/£ By By By By By By By By No. P Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6 Eq.8.5 Eq. 8.6 Eq.8.5 Eq. 8.6

6 0.01 0.4 0.603 0.573 0.645 0.39 0.241 0.229 0.258 0.156

9 0.015 0.4 0.61 0.572 0.630 0.365 0.244 0.229 0.252 0.146

I\.)

U1 U)

260

As long as the slab has enough reinforcing steel to carry the

tension forces, the increase in the reinforcement ratio does not result

in an increase of the effective width; however, the ultimate load might

increase.

8.8 Influence of Specimen Variables on the Effective Width at the Support

8.8.1 Influence of b/~ Ratio

Studying the effective width at the support was handled in the

same manner as was done for the conditions at mid-span. Tables 8.9-8.12

and-Figures 8.115-8.118 illustrat~ the findings. The same general com-

ments of section 8.7 are found to be applicable to Figures 8.115-8.117.

In Figure 8.118, all of the previous results have been combined

and the proposed equations to calculate the effective width are shown.

At yield load, the curves that represent b /~ vs. b/~ are very close, e

and a straight line can easily be used, as shown in Figure 8.118. The

proposed equation to calculate the effective width at yield load is

expressed by

b e = 1

+ 12.5 b 4~ )~ (8.18)

At ultimate load, the curves that represent b /~ vs. b/~ are far e

apart, although they can be represented by nearly parallel straight

lines. This indicates that the stiffness of the steel beam provides a

major contribution to the effective width at the support.

In consideration of the above findings, the following equations

are intended to cover light, intermediate, and heavy steel beams:

Table 8.9 Influence of b/£ Ratio on the Effective Width at the Support (Group I Specimens).

b /b at Yield b /b at Ultimate e e

Specimen b/£ Byb Byc By By a

Ratio 8.5 8.6 8.5 8.6 No. Eq. Eq. Eq. Eq.

1 0.8 0.356 0.366 0.53 0.36

2 0.6 0.367 0.378 0.60 0.496

3 0.4 0.416 0.427 0.640 0.54

aSpecimens 1, 2, and 3 of Group I had the following in common: W16x36, p = 0.01, £ = 20 ft, slab thickness = 6.0 in., and fixed ends.



261

Table 8.10 Influence of b/£ Ratio on the Effective Width at the Support (Group II Specimens).

Specimen No.

a

1

2

3

b/£ Ratio

0.8

0.6

0.4

b /b at Yield e

Byb Eq. 8.5

0.368

0.375

0.45

Byc Eq. 8.6

0.372

0.377

0.45

b /b at Ultimate e

By Eq. 8.5

0.428

0.46

0.54

By Eq. 8.6

0.243

0.3

0.348

aSpecimens 1, 2, and 3 of Group II had the following in common: W21x50, p = 0.01, £ = 25 ft, slab thickness = 6.0 in., and fixed ends.



262

Table 8.11 Influence of b/£ Ratio on the Effective Width at the Support (Group III Specimens) .

b /b at Yield b /b at Ultimate e e

Specimen b/£ Byb Byc By By No.

a Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

1 0.8 0.35 0.351 0.347 0.225

2 0.6 0.370 0.377 0.378 0.245

3 0.4 0.454 0.463 0.51 0.33

aSpecimens 1, 2, and 3 of Group III had the following in common: W27x94, p = 0.01, £ = 30 ft, slab thickness = 6.0 in., and fixed ends.



263

Table 8.12

Specimen l~o •

1

2

3

Influence of Beam Length on the Effective Width at the Support. -- The effective width used in this table is based on the axial force in the slab.

GrouE I GrouEII GrouE III b II). b II). b II).

e e e b/l). I). At At I). At At I). At At

Ratio (ft) Yield Ultimate (ft) Yield Ultimate (ft) Yield Ultimate

0.8 20 0.285 0.424 25 0.294 0.342 30 0.28 0.278

0.6 20 0.22 0.36 25 0.225 0.276 30 0.222 0.227

0.4 20 0.17 0.256 25 0.18 0.216 30 0.182 0.204

N (j) .t:>

1 I-

..,g , .5 I-

0) ..@

o o

Figure 8.115

6 Based on Beam Rxial Force


: --~ A. -....,~i ~======~==~=======--



I I I I

. 5

b/L

b /b vs. b/~ at the Support Sections for the Specimens of Group I. e

~

1

N (j) U1

1 t- 6 Based on Beam Axial Force

t- <> Based on Sl ab Ax i a 1 Force

..@ , .5 t-

G.1 .,©j

G__ ~ ~ ...... ~~~~

-$- .. ==a_.:._c:a~ A-



o o I I ~

.5 1

tbJ/L

Figure 8.116 b /b VS. b/~ at the Support Sections for the First Three Specimens of Group II. e

N 0' 0'

1 t-

~ , .5 I-

ID .J1

o o

Figure 8.117

6 Based on Beam Rxial Force

o Based on Slab Rxial Force

A __

~--::::::::::::::



I I I

.5

"'~- - = S iiI _ = ~

L..l

I .L

b/L

I

b /b vs. b/£ at the Support Sections for the Specimens of Group III. e

I 1

I'V m .....

1

..JJ ~\ . 5

OJ .(Q1

o

Figure 8.118

---- Load Level = Yield .6 Group I Specimens

--- Load Level = Ultimate 0 Group II Specimens

Proposed Equations 0 Group III Specimens

--A_------8 --

o . 5 1

~/L

b /£ vs. b/£ at the Support Sections for the First Three Specimens of Groups I, II, a~d III.

I\J (J) CD

269

1. For steel beams not heavier than W16x36:

be = (0.12 + 0.35 ~ )~ (B.19)

2. For steel beams not heavier than W21x50 but heavier than W16x36:

be - (0.1 + 0.3 ~ )~

3. For steel beams heavier than W21x50:

b be = (O.OB + 0.25 I )~

B.B.2 Influence of the Type of Beam-to-Column Connection

(B.20)

(B.21)

Table B.13 and Figures B.119 and B.120 demonstrate the effects

of varying the beam and restraint. It was found that the effective

width at yield load increases as the degree of fixity increases at the

supports. To account for this, the term

M 0.03 (1 - M

C )~ P

should be subtracted from equation B.1B.

(B.22)

At the ultimate load, however, the above effect is reversed.

Thus, the effective width decreases as the degree of fixity increases.

The expression

M 0.016 (1 - M

C )~ P

(8.23)

therefore, should be added to the effective width equations at the ulti-

mate load. This gives effective width equations that incorporate end

connection as well as beam stiffness, and they assume the forms:

Table 8.13 Influence of the Support Conditions on the Effective Width at the Support.

Specimen No.

a

3

5

6

7

8

b/Q, Ratio

0.4

0.4

0.4

0.4

0.4

Support b Restraint

Fixed

0.1 M P

0.3 M P

0.5 M P

0.7 M P

b /b at e By

Eq.8.5

0.45

0.385

0.403

0.435

0.452

Yield

By Eq. 8.6

0.45

0.384

0.403

0.435

0.454

b /Q, at Yield e

From Eq. 8.5

0.18

0.154

0.161

0.174

0.181

From Eq. 8.6

0.18

0.154

0.161

0.174

0.181

b /b at Ultimate e

By Eq. 8.5

0.54

0.58

0.58

0.517

0.525

By Eq. 8.6

0.348

0.845

0.85

0.76

0.63

b /Q, at Ultimate e From From

Eq. 8.5 Eq. 8.6

0.216 0.139

0.232 0.338

0.232 0.34

0.207 0.304

0.21 0.252

a The specimens had the same variables (W21x50, Q, 25, b = 10, P = 0.01, t = 6.0") except for the degree of fixity at the supports.

bThe girder end restraint varies from fixed to semi-rigid (capable of resisting 0.1 M ). p

I\.)

~

o

1 6 Based on Beam Axial Force


on '\. .5

(j) Q 9 -----~ _________ A ___ ------~--- ______ A----

-~

Figure 8.119



o o . 5 1

b'1c/Mp

b /b at the Supports vs. the Degree of Fixity at the Supports (Specimens 3, 5, 6, 7, and e

8; Group II) . N ~ .....

..JI '\

. 5 f-

Based on Slab Axial Force

b/L = 0.4

(TI) .251- o 0 .,@

Figure 8.120

~ 0 0

~---------~---------~---------~--------------~ r



0 I I I 0 . 5 1

Mc/Mp

b /~ at the Supports vs. the Degree of Fixity at the Supports (Specimens 3, 5, 6, 7, and e

8; Group II) . N -..J N

1. At yield load:

b = e

1 b ~ 12.5 + 4~ )~ - 0.03 (1 - M )~

P

2. At ultimate load for steel beams not heavier than W16x36:

b Mc be = (0.12 + 0.35 I )~ + 0.016 (1 - M )~

P

273

(8.24)

(8.25)

3. At ultimate load for steel beams not heavier than W21x50 but

heavier than W16x36:

b M be = (0.1 + 0.3 I )~ + 0.016 (1 - M

C )~ P

4. At ultimate load for steel beams he~vier than W21x50:

b Mc be = (0.08 + 0.25 £ )~ + 0.016 (1 - M )£

8.8.3 .Influence of Steel Beam-to-Slab Stiffness Ratio

p

(8.26)

(8.27)

The influence of the stiffness ratio, K , on the effective width r

at the support is indicated by the data in Table 8.14 and by

Figure 8.121. It was found that the effecti.ve width at the support is

more sensitive to the value of K , at yield load as well as at ultimate r

load, than the effective width at mid-span.

For speci~~ns of Groups I and II, increasing the stiffness ratio

from the minimum to the maximum value of K results in about a 40 perr

cent decrease of the effective width at both the yield load and the

ultimate load. For specimens of Group III, this decrease in the effec-

tive width becomes 35 percent at yield load and 25 percent at ultimate

load. This decrease in the effective width is caused by many factors,

274

Table 8.14 Influence of Steel Beam-to-Slab Stiffness Ratio on the Effective Width at the Support. -- The effective width used is based on the slab axial force.

Group Specimen K a b 19.-e r

No. No. At Yield At Ultimate

I 1 1.013 0.285 0.424 2 1. 35 0.22 0.36 3 2.025 0.17 0.256

II 1 1. 78 0.294 0.342 2 2.37 0.225 0.276 3 3.55 0.18 0.216

III 1 4.93 0.28 0.278 2 6.57 0.222 0.227 3 9.85 0.182 0.204

aK = beam-to-slab stiffness ratio.

r

1 I-

..Jl " .5 I

Oll ..!Ql

I-

I-

o o

Load Level = Yield


Based on 51 ab Ax i al

,~ &, ~ , ~-----o ~,

'6

I i I

Force

I I

5

ll. Group I Specimens

<> Group II Specimens 4l

0 Group III Specimens

-=-8--- ____________ ~

I _I I I

10

Kir

Figure 8.121 b /~ at the Supports vs. the Beam-to-Slab Stiffness Ratio for the First Three Specimens of Groups I, II, and III.

N -..J l11

276

such as the steel beam size, the span, and the slab width. All these

factors have been considered in developing the effective width equa-

tions; therefore, the previously recommended equations have been

retained.

8.8.4 Influence of the Slab Thickness

The data in Table 8.15 demonstrate that the effective width at

the yield load was the same for specimens 3 and 4. At the ultimate

load, however, the specimen with the smaller thickness was found to have

an effective width 15 percent less than the one for the specimen with

the thicker slab. This result is the opposite of what was found for the

mid-span section. Since no reduction in the effective width has been

introducej due to any slab thickness effects, a factor a is introduced

to account for the slab thickness in calculating the effective width at

the supports at the ultimate load.

The factor a is a reduction factor that has been derived on the

basis of a straight line passing through two points; a is set equal to

1.0 for a slab thickness of 6.0 inches and 0.85 for a slab thickness of

4.0 inches. Consequently, a can be given as

a t (0.55 + 0.45 6

where a < 1 for all slab thicknesses.

8.8.5 Influence of the Slab Reinforcement Ratio

(8.28)

Table 8.16 gives a comparison of the effective width at the sup-

ports for two specimens with different reinforcement ratios. The maxi-

mum difference in the effective width for the two specimens is about

Table 8.15 Influence of the Slab Thickness on the Effective Width at the Support. -Specimens 3 and 4 of Group II were identical except for the slab thickness.

b /b e b /£

e t At Yield At Ultimate At Yield At Ultimate

Specimen c

b/£ By By By By By By By By No. (in. ) Ratio Eq.8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

3 6.0 0.4 0.45 0.45 0.54 0.348 0.18 0.18 0.216 0.139

4 4.0 0.4 0.45 0.448 0.457 0.26 0.18 0.179 0.183 0.104

I\J -..J -..J

Table 8.16 Influence of the Slab Reinforcement Ratio on the Effective Width at the Support. -- Specimens 6 and 9 of Group II were the same except for the slab reinforcement ratio.

Specimen No.

6

9

p

0.01

0.015

At Yield b/~ By By

b /b e

At Ultimate By By

Ratio Eq. 8.5 Eq. 8.6 Eq. 8.5 Eq. 8.6

0.4 0.403 0.403 0.58 0.85

0.4 0.405 0.4 0.562 0.843

b /~ e

At Yield At Ultimate By By By By

Eq . 8. 5 Eq . 8. 6 Eq. 8.5 Eq. 8.6

0.161 0.161 0.232 0.34

0.162 0.16 0.225 0.337

tv ~ CD

279

3 percent; therefore, it can be seen that, as long as the slab has

enough reinforcement to carry the tensile force, the reinforcement ratio

is not a factor for the effective width.

8.9 Proposed Criteria for Calculating the Effective Width in a Continuous Composite System

8.9.1 At Mid-Span

The effective width at mid-span has been derived for the load

stages of yield and ultimate. At the yield load, it is recommended that

the effective width be calculated by the following equation:

t Mc 0.4) + 65 (1 - M

P

A further simplification of this equation leads to the following:

M

be = 0.23t + 0.04b - 0.015 MC

t p

(8.29)

(8.30)

At the ultimate load, the following equations are recommended:

A

M

be = 0.25t + 0.02t (1 - MC

P

b for t ~ 0.4

M

O 04) t (1 _ c • + 50 M

b for t > 0.4

P

further simplification leads to the following equations:

M b

b = 0.27t - 0.02 -.£t for "i ~ 0.4 e M

p

M b

b = 0.19t + 0.2b - 0.02 -.£t for "i > 0.4 e M

p

(8.31)

(8.32)

(8.33)

(8.34)

280

8.9.2 At the Supports

The effective slab width at the supports has also been derived

for the yield and ultimate stages. At the yield load, the following

equation is recommended:

b Mc be = (0.08 + 0.25 I )£ - 0.03 (1 - M )£

A further simplification leads to

M be = 0.05£ + 0.25b + 0.03 M

C £

p

p (8.35)

(8.36)

At the ultimate load, it is necessary to utilize three equa-

tions, depending on the size of the steel beam:

1. For a composite beam with a steel beam having a moment of

inertia equal to or less than that of the shape W16x36, the

effective width is

b Mc be = a[(0.12 + 0.35 £ )£ + 0.016 (1 - M )£] (8.37)

p

2. For a composite beam with a steel beam having a moment of

inertia greater than the one for a W16x36 but less than or equal

to a W21x50, the effective width is

M be = a[(O.l + 0.3 ~ )£ + 0.016 (1 - M

C )£]

p (8.38)

3. For a composite beam with moment of inertia greater than the one

for a W21x50, the effective width is

M b 'c

be = a[(0.08 + 0.25 £ )£ + 0.016 (1 - M )£] p

(8.39)

281

The factor a is given by

t a = 0.55 + 0.45 6 < 1

A furt~er simplification of equations 8.37-8.39 leads to the following:

M b = a(0.14£ + 0.35b - 0.016 M

C £)

e (8.40)

p

.M b = a(O.12£ + 0.3b - 0.016 M

C £)

e (8.41)

p

M b = a(O.l£ + 0.2Sb - 0.016 M

C £) e

(8.42) p

The effective width at the ultimate load that is calculated by equa-

tions 8.40-8.42 will be equal to or greater than the effective width at

the yield load, as given by equation 8.36.

CHAPTER 9

BEHAVIOR OF COMPOSITE BEAMS IN CONTINUOUS

COMPOSITE FLOOR SYSTEMS

9.1 Introduction

In this chapter, the behavior of a composite beam in a continu-

ous composite floor system will be examined. The moment resistance at

mid-span and at the support have been plotted vs. the applied load for

all of the models of Groups I, II, and III. Data for studying the rota-

tion capacity at the support as well as the deflection at mid-span of

continuous composite beams are also presented.

9.2 Moment Resistance in the Positive and Negative Moment Regions

The moment resistance of a composite section in the positive and

negative moment regions can be expressed by the following equation:

M = M + T e + M g g s

where

M = moment in the steel girder, g

T = axial force in the steel girder, g

e = (h + t )/2, g c

h = depth of the steel girder, g

t = concrete slab thickness, and c

(9. 1 )

M = moment in the concrete slab, whose width equals the effective s

width.

282

283

Determining the values of M , T , and M at any load stage is g g s

achieved through a complex procedure, especially if methods other than

the numerical approaches of the finite-element technique ar~ utilized.

This is because the primary variables depend on many factors, such as

the boundary conditions and the behavior of the material that forms the

composite section. Using the finite-element method, these variables can

be determined at any load stage.

Practically, the most important values for M , T , and Mare g g s

those associated with the ultimate load level. The magnitudes of these

variables at the ultimate load, therefore, will be expressed in a gen-

eral form, as derived from actual numerical data.

9.2.1 Composite Beams with Fixed Ends

The first three specimens of Groups I, II, and III utilized

moment-resistant, beam-to-column connections for the steel beam. The

moment due to the first two terms of equation 9.1 vs. the applied load

has been plotted for each model of these groups, as shown in

Figures 9.1-9.9, including the yield and ultimate values. The moment

resistance of the slab has not been included, for two reasons. First,

it was intended to show the increase in the bending moment capacity due

to the beam axial force that results from the interaction between the

steel beam and the slab. This has been illustrated by normalizing the

total moment with respect to the plastic moment of the steel beam.

Second, the ultimate moment of the concrete slab is easy to calculate

and to add to the other two terms in equation 9.1.

2

1 • 5

n.

b. Mid-Span Section

o Support Section

Specimen l,Group I

() 0

~ 1,

" a::

Figure 9.1

. 5

o 0

I I I I I I Y.L.

. 5 1

P/Pu

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 1, Group I. -- Refer to Table 7.1. g g

tv ro ~

2

1 . 5

gl,. 2: 1 , ~

. 5

(21 (21 Y.

.5

b. Mid-Span Section

o Support Section

Specimen 2,Group I

1

P/Pu

Figure 9.2 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 2, Group I. -- Refer to Table 7.1. g g

l\.)

co \J1

P ..... :>1: ' ... ~

2

1 • 5

1 L

. 5

o 0 Y.!....

I I I I I I I

.5

II Mid-Span Section

o Support Section

Specimen 3,Group I

1

P/Pu

Figure 9.3 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for -Specimen 3, Group I. -- Refer to Table 7.1. g g

t\)

co (j)

2

1 • 5

Q.

~ Mid-Span Section

o Support Section

Specimen 1.Group II

~ 1

" ~

Figure 9.4

.5

(2) 0 Y.L.

I I I I I I

.5 1

P/py

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load fo~ Specimen 1, Group II. -- Refer to Table 7.2. g g

N 00 -....J

2

1 • 5

!Q..

l:l Mid-Span Section

o Support Section

Specimen 2,Group II

:?!: 1 ,~

~

Figure 9.5

.5

o 0 Y.L..

I I I I I I I

.5 1

P/Pu

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Spec"imen 2, Group II. -- Refer to Table 7.2. g g

I\..J CD CD

2

1 • 5

n. ~ 1 '\~ 2C

.5

o 0 Y .. L.

I I I I I I

.5

b. Mid-Span Section

o Support Section

Specimen 3.Group II

1

P/Pu

Figure 9.6 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 3, Group II. -- Refer to Table 7.2. g g

N co 1.0

2

1 • 5

~

II Mid-Span Section

¢ Support Section

Specimen l,Group III

~ 1 '\. ~

.5

121 0 .5 1

P/!Pu

Figure 9.7 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 1, Group III. -- Refer to Table 7.3. g g

(\J

~ o

2

1 • 5

~

b. Mid-Span Section

o Support Section

Specimen 2,Group III

~ 1

" ~

Figure 9.8

.5

o 0 Y.l.

I I I I I I

. 5 1

P/Pu

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 2, Group III. -- Refer to Table 7.3. g g

N ~ ~

2

1 .5

Q.. ~ 1 , ~

.5

12) 12) ~.L.

I I I I I I

.5

l:!. Mid-Span Section

o· Support Section


1

P/Pu

Figure 9.9 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 3, Group III. -- Refer to Table 7.3. g g

f\.)

~ f\.)

293

Figures 9.1-9.9 demonstrate the following points:

1. All of the specimens yielded at a load between 0.4 and 0.5 of

the ultimate one. The yield load (Y.L.) is defined. as the load

at which yielding first occurs in any portion of the specimen

(slab or steel beam).

2. In the linear range of loading, the moment at the supports was

60 to 75 percent higher than that at mid-span. The first plas-

tic hinge, therefore, formed at the supports.

3. The support sections for all of the specimens provided enough

rotation capacity and ductility to allow the subsequent forma-

tion of plastic hinges at mid-span.

4. The ultimate capacity of the mid-span sections was higher than

that at the supports, by an amount between 5 and 11 percent.

5. The axial force contribution to the ultimate moment capacity of

the composite section is substantial. Thus, at the supports it

contributed to an increase of 35 to 42 percent of the steel beam

plastic moment, and at the mid-span sections this increase was

between 40 and 55 percent of M . P

Details of the ultimate moment calculations are shown in

Table 9.1 fer mid-span sections and in Table 9.2 for support sections.

It should be noted that the moment contribution of the concrete slabs is

not included in these data.

The new variables in Tables 9.1 and 9.2 are presented in the

last two columns of the tables. These are P and p, where P is the y y

yield load for the column and p is an axial load that has been derived

Table 9.1 Ultimate Moment Capacity at Mid-Span for Fixed-Ended Beams.

Specimen T M M +T e M +T e M

Group g g e g g 9 9 --.S!. No. No. (Kips) (K-ft) (ft) (K-ft) M M

P P

I 1 157.83 142.7 0.911 286.48 1.492 0.743 2 159.64 141.77 0.911 287.2 1.496 0.738 3 168.84 136.85 0.911 290.66 1.520 0.713

II 1 245.4 231.1 1.118 505.46 1.532 0.70 2 251. 19 227.01 1.118 507.84 1.538 0.69 3 231. 85 227.7 1.118 486.91 1.475 0.69

III 1 399.9 618.83 1.372 1167.5 1.40 0.742 2 411.336 613.96 1.372 1178.31 1.413 0.74 3 426.62 609.1 1.372 1194.42 1.432 0.73

T --.S!. P

Y

0.413 0.418 0.442

0.463 0.475 0.438

0.401 0.412 0.427

T --.S!. P

1.116 1.116 1. 116

1.14 1.138 1.06

1.08 1.096 1.12

N W .t>-

Table 9.2 Ultimate Moment Capacity at the Support for Fixed-Ended Beams.

Specimen T M M +T e r.1 +T e M Group g g e g g 9 9 ~

No. No. (Kips) (K-ft) (ft) (K-ft) M M P P

I 1 96.35 177.07 0.911 264.845 1. 379 0.922 2 96.5 177 .10 0.911 265.01 1. 38 0.922 3 110.05 170.1 0.911 270.35 1.41 0.89

II 1 134.0 309.52 1.118 459.332 1. 392 0.937 2 149.59 298.65 1. 118 465.89 1.411 0.91 3 133.45 308.91 1. 118 458.11 1.39 0.936

III 1 251.12 773.5 1.372 1118.04 1.34 0.927 2 251. 7 773.88 1.372 1119.21 1.342 0.927 3 252.24 774.25 1.372 1120.32 1.343 0.928

T ~ P

Y

0.252 0.253 0.288

0.253 0.282 0.252

0.252 0.252 0.253

T ~ P

1.15 1.157 1.156

1. 23 1. 21 1. 22

1.176 1.18 1.186

N ~ U1

296

from equation 2.4-3 of the AISC specifications [4]. This equation is

given as

M --E£ = M

1.18 (1 - }- ) y

(9.2) P

and in these computations the value of M is substituted for the column g

plastic moment, M pc

The actual value of the axial load in each steel

beam, T , is then compared to P and p. s y

A careful examination of Table 9.1 leads to the following

conclusions:

1. The steel beam in a composite floor system acts as a beam-

column; the axial force, T , provides a considerable contribug

tion to the overall moment capacity of the composite section.

2. The steel beam bending moment, M , lies between 0.7 and 0.74 M , g P

and the axial load, T , assumes values between 0.4 and 0.45 P • g Y

These values were found to be true for all of the specimens

which had steel beam sizes between W16x36 and W27x94.

3. By applying the interaction equation for a short beam-column

(equation 9.2), to calculate the axial force when the section is

loaded by M = M , it was found that the actual axial force was pc g

6 to 12 percent larger than p of equation 9.2. This confirms

that the steel beam acts as a short beam-column. The difference

between the axial load from the finite-element analysis and the

one from equation 9.2 can be attributed to the approximations

that were used in deriving equation 9.2. It is noted that this

equation represents an approximate average of the relationship

between M and p for a range of typical W-shapes [4]. Since pc

297

equation 9.2 is partly based on experimental data, however, the

discrepancy between T and p will be considered acceptable. g

4. Based on the observations of items 1, 2, and 3 above, equa-

tion 9.3 is proposed for use in calculating the ultimate moment

capacity in the positive moment region of a continuous composite

beam with fixed ends:

M = 0.7 M + pe + M up P us

(9.3)

Substituting for M equal to 0.7 M in equation 9.2 and solving pc p

for p in terms of P gives p = 0.41 P. Equation 9.3 then becomes y y

M = 0.7 M + 0.4 P e + M (9.4) up P Y us

In this equation, M , P , and e have been defined previously. M is p y us

the ultimate moment capacity of a concrete section with a thickness

equal to the slab thickness and a width equal to the effective width.

In calculating the ultimate moment of the slab, it should be realized

that the axial force of 0.4 P is also applied to the slab. This, howy

ever, is generally less than Pb

, which is the nominal axial load capac-

ity for the reinforced concrete cross-section at the balanced strain

condition. Considering the axial force equal to zero and calculating

M on the basis of flexure only will give an acceptable as well as a us

conservative design. In the case when Pb

is less than 0.4 Py

' the

moment capacity of the slab should be calculated on the basis of beam-

column formulations.

The data in Table 9.2 for the ultimate moment capacity at the

supports provide the following findings:

298

1. The steel beam moment contribution in forming a plastic hinge at

the support is higher than at mid-span. The value of M was g

around 0.92 M for most of the models. This is bec~use the end p

conditions are fixed and a large moment is required to prevent

any rotation.

2. The axial load contribution in all of the specimens is about

0.25 P. T, however, is larger than the load p, as calculated y g

from equation 9.2, by an amount between 15 and 23 percent;

therefore, to be on the conservative side, T will be assumed to g

be equal to the value of p that satisfies equation 9.2.

3. Equation 9.5 is recommended for use in computing the ultimate

capacity at the supports of a continuous composite beam having

fixed ends. Thus:

M = 0.92 M + pe + M un p us

(9.5)

substituting M = 0.92 M in equation 9.2 and solving for p in cp p

terms of P gives p = 0.22 P. Equation 9.5 is then modified to become y y

M = 0.92 M + 0.22 P e + M (9.6) un p y us

In using this equation, the effective concrete slab was assumed to have

a reinforcing steel ratio capable of resisting the tensile force of 0.25

P , in addition to the required reinforcement for the slab moment, M y us

In developing equations 9.4 and 9.6 for use in calculating the

ultimate moment capacity in the positive and negative moment regions of

a continuous composite beam, the traditional assumption of plane sec-

tions remaining plane has not been used. These equations, therefore,

reflect the actual behavior of composite beams.

9.2.2 Composite Beams with Semi-Rigid End Connections

299

The results for models 5-8 of Group II have been used in deter-

mining the ultimate moment capacity of a composite section in the posi-

tive and negative moment regions of continuous composite beams with

semi-rigid end connections. Figures 9.10-9.13 show the mid-span and

support moments in relation to the applied load, for specimens with end

connection capacities of 0.1, 0.3, 0.5, and 0.7 M , respectively. p

Tables 9.3 and 9.4 give the details of the ultimate moment calculations.

The following summarizes the major findings:

1. The mid-span ultimate moment capacity for all of the specimens

is the same, regardless of the degree of fixity for the speci-

mens. This is as expected. Moreover, comparing these specimens

with model 3 of Group II, which has fully moment-resistant end

connections, demonstrates that all of the specimens reach the

same ultimate moment. This indicates that the ultimate moment

at mid-span is independent of the support conditions; however,

the deflection at mid-span and the rotation at the supports for

the specimens with a small degree of fixity are much larger than

those for the models with a large degree of fixity.

2. Table 9.3 shows that the steel beam for all of the specimens

with a degree of fixity (expressed in terms of the connection

moment capacity divided by the beam plastic moment) less than or

equal to 0.5 had an actual moment of 0.8 M. Substituting for p

M 1M = 0.8 in equation 9.2 gives a value of 0.32 for pip. pc p y

The actual beam axial load, T , was about 15 percent larger than g

2

1 .5

Q..

i). Mid-Span Section

o Support Section

Specimen 5.Group II

2!: 1

" ~

Figure 9.10

.5

o 0

I I I I I I Y.L.

.5 1

P.I'PtJ

Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 5, Group II. -- Refer to Table 7.2. g g

.w o o

2

1 . 5

~

b. Mid-Span Section

o Support Section

Specimen 6.Group II

~ 1 '\. 2C

Figure 9.11

. 5 I I I I I I I I o lr 'I!.L.,

o .5 1

P/PQJ


(.oJ

o .....

2

1 • 5

In. ~ 1 '\. ~

.5

o 0 Y.L.

.5


o Support Section

Specimen 7,Group II

1

P/Pu


Figure 9.12

w o I\)

2

1 • 5

~ ~ 1 , :?t:

.5

(2) 0 Y.

.5

/). Mid-Span Section

o Support Section

Specimen 8,Group II

1

P/Pu

Figure 9.13 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 8, Group II. -- Refer to Table 7.2. g g

w o w

Table 9.3 Mid-Span Ultimate Moment Capacity for Composite Beams with Semi-Rigid End Connections. -- Specimens for Group II.

Specimen M a T M M +T e M +T e M T T

c 9 9 e 9 9 g g J J J No. M (Kips) ( K-ft) (ft) (K-ft) M M P P

P P P Y

5 0.1 193.4 266.5 1.118 482.72 1.463 0.81 0.366 1.158

6 0.3 191.1 268.12 1.118 481.8 1.46 0.812 0.361 1.159

7 0.5 199.26 262.5 1.118 485.23 1.47 0.80 0.377 1.154

8 0.7 217.66 250 1.118 493.34 1.495 0.76 0.41 1.149

aM 1M is the ratio of the semi-rigid connection moment to the plastic moment of die ~teel beam.

w o ~

Table 9.4 Support Ultimate Moment Capacity for Composite Beams with Semi-Rigid End Connections. -- Specimens for Group II.

Specimen M a T M M +T e M +T e M T T

c g g e g g 9 9 ~ ~ ~ No. M (Kips) (K-ft) (ft) (K-ft) M M P P

P P P Y

5 0.1 354.4 33.0 1.118 429.22 1.30 0.1 0.67 0.732

6 0.3 346 99.0 1.118 485.83 1.472 0.3 0.654 0.88

7 0.5 320.5 165.0 1.118 523.32 1.586 0.5 0.606 1.05

8 0.7 272 .1 213.625 1.118 517.83 1.569 0.65 0.514 1.14

aM 1M is the ratio of the semi-rigid connection moment to the plastic moment of tge ~teel beam.

w o lJ1

p from equation 9.2.

assumed equal to p.

T , therefore, will be conservatively g

The equation

306

M ~ 0.8 M + 0.32 P e + M up P Y us

(9.7)

therefore, is recommended to calculate the ultimate moment

capacity in the positive moment regions for continuous composite

beams with semi-rigid connection capacities less than 0.7 M . P

(It is noted that this equation and equation 9.4 give almost the

same results; the differences are not more than 5 percent.)

3. Specimen 8 had a connection rigidity of 0.7 M , and the behavior p

was even closer to the one with fixed ends. Also, since equa-

tions 9.4 and 9.7 give almost the same results, the former is

recommended for use with composite beams having semi-rigid con-

nections with rigidities equal to or larger than 0.7 M . P

4. The ultimate moment capacity at the supports for the specimens

with semi-rigid connections was found to be equal tQ or larger

than the one for the specimen with fixed ends. The only excep-

tion was specimen 5, which had about 7 percent smaller ultimate

moment capacity. The contribution of the beam axial load in

increasing the ultimate moment was the major factor. Comparing

the actual value of the axial load, T , with that derived from g

equation 9.2 gives a ratio of T /p less than 1.0 for models 5 g

and 6. In the finite-element analysis, the steel beam yielded

in both of these models. As a conservative approach to design,

however, the smallest value of the axial force, which is T , is g

used in calculating the ultimate moment capacity. The value of

307

T is expressed in terms of-the axial force from equation 9.2 as g

T = ap, where the factor a < 1 is given by g

a = M

c (0.65 + 0.7 M

P

< 1.0 (9.8)

This factor represents a conservative straight line that passes

through the two points with coordinates (M 1M = 0.1, T Ip = c p 9

0.732) and (M 1M = 0.3, T Ip = 0.88), as indicated in c p g

Table 9.4. Consequently, the equation that is recommended for

calculating the ultimate support moment capacity of a composite

beam with semi-rigid connections of rigidity M 1M < 0.7 becomes c p-

M = M + ape + M un c us (9.9)

where M is the semi-rigid connection moment restraint, 8 is c

given by equation 9.8, and p is the axial load according to

equation 9.2, using M = M . pc c

5. For semi-rigid connections with a rigidity M 1M > 0.7, the folc p

lowing moment equation applies:

M = 0.92 M + pe + M un c s

(9.10)

Equation 9.10 must be used for members of this type, reflecting

the fact that they are virtually indistinguishable from beams

with fully moment-resistant end connections.

6. It should be noted that the amount of reinforcement in the con-

crete slab must be sufficient to carry the tensile axial load,

calculated by equation 9.2, in addition to the reinforcement

that is needed for the slab moment, M us

In case the tensile

load is large, the slab moment will be small, and it is conser-

vative to assume M equal to zero. us

308

9.2.3 Influence of Slab Thickness on the Ultimate Moment Resistance

To examine the influence of varying the slab thickness, the

strength and behavior of specimens 3 and 4 of Group II were compared.

As e}cpected, it was found that the steel beam behaved identically for

both specimens at mid-span and at the support. For example, the ratios

of M 1M at mid-span were 0.69 and 0.697 for specimens 3 and 4, respecg p

tively, and assumed a value of 0.935 at the support for both specimens.

Also, comparing Figures 9.6 and 9.14 showed that specimen 3, which had a

slab thickness of 6 inches, reached an ultimate load that was about 4.5

percent hi9her than that of specimen 4, .which had a slab thickness of 4

inches. This difference can be attributed almost entirely to the dif-

ference in internal moment arms, e. The previously developed equations,

therefore, are applicable for all slab thicknesses. This is because the

slab thickness is included as a factor in e and M , which are two of us

the variables of these equations.

9.2.4 Influence of Reinforcement Ratio on Ultimate Moment Capacity

The results for models 6 and 9 of Group II have been compared to

examine the influence of the reinforcement ratio on the ultimate moment

capacity. The ratio was one percent for specimen 6 and 1.5 percent for

specimen 9. It \-las found that the contribution of the beam moment, M , g

and its axial load, T , to the ultimate moment capacity was the same for g

both specimens. For example, the M 1M g p

assumes a value of 0.81 at mid-

span for model 6 and 0.82 for model 9; it was equal to 0.3 for both

specimens at the supports. Also, comparing Figures 9.11 and 9.15 showed

· I

2

1 • 5

Q..

b. Mid-Span Section

o Support Section

Specimen 4,Group II

2!: 1 , 2::

. 5

.5 1 o 0 Y.L.

P/Pu


Figure 9.14

\..oJ o ~

2

1 • 5

Q. a: 1 , ~

.5

o 0 Y oIL.

I I I I I I I I

.5


o Support Section

Specimen 9,Group II

1

P/Pu

Figure 9.15 Composite Section Moment Due to Steel Beam Forces (M + T e) vs. Applied Load for Specimen 9, Group II. -- Refer to Tabel 7.2. g g

w

o

311

that both specimens had the same moment at mid-span, due to the contri-

bution of the beam forces, and specimen 9 had about 3 percent higher

moment capacity than specimen 6 at the support.

The above findings lead to the conclusion that as long as the

concrete slab has sufficient reinforcement to carry the axial tensile

force the reinforcement ratio will not have a significant influence on

the behavior of the composite section; however, the concrete slab moment

(M ) will be larger for the specimen with a higher reinforcement ratio. us

This has been taken care of by calculating the ultimate moment capacity

of the concrete slab by itself, M , which is then added to the contrius

bution of the steel beam forces.

9.3 Ductility of Composite Beams

The ductility of the composite beams has been examined by

showing the capability of the support section to rotate after its plas-

tic hinge has been formed. It was found that all of the specimens that

had fully moment-resistant steel beam-to-column connections provided

enough rotation capacity at the supports to allow a third plastic hinge

to be formed in the beam (at mid-span) •

The models that had semi-rigid end connections developed the

first plastic hinge at mid-span. Sufficient additional deformation was

developed to allow additional plastic hinges to be formed at the sup-

ports. These behavioral characteristics are demonstrated by the data in

Figures 9.1-9.15.

It is noted that the ductility requirements have not been

studied in detail; however, to show the ductility of the composite beams

312

used in this study, the moment at support vs. the maximum rotation along

the beam was plotted for each model. The results indicate a great deal

of similarity between all of them, and Figures 9.16-9.18 give examples

of the findings. It can be seen that the rotation at the ultimate load

is between 2 and 2.5 times the rotation at the first plastic hinge.

Consequently, the results demonstrate that plastic design is fully

applicable to the design of continuous composite beams. Further studies

are needed on the ductility of composite beams under large-displacement

seismic forces.

9.4 Deflection of Composite Beams

Examples of the mid-span deflection for composite beams are

shown in Figures 9.19-9.22. These are typical of the results for all of

the models. Table 9.5 gives the ultimate and yield loads and the corre

sponding beam deflections, in addition to the deflection at a load equal

to 60 percent (0.6) of the ultimate one for the models of Groups I, II,

and III. As can be seen from Figures 9.19-9.22, the load-deflection

curve is linear to the first yield, and then becomes increasingly

nonlinear.

The deflection of a composite beam is related to the following

factors:

1. Stiffness of the steel beam.

2. Stiffness of the concrete slab and its reinforcement ratio, p.

3. Support conditions for the steel beam. These require a full

understanding of the rotation behavior of all types of composite

connections.

~ :;r:

" a:

2 Specimen 2,Group I

1 . 5 1.38

----~--------------1~ A A A ~

~ Plastic Hinge Forms at Support 1 __ 2.y / Mp

---------

I I I I

.5 I I I I I I 1 I

o 0 .5 1

a/au

Figure 9.16 Moment Due to Steel Beam Forces (Mg + Tge) at the Support vs. Maximum Rotation for Specimen 2, Group I. -- Refer to Table 7.1.

w

w

!!.:l. :r:

" ;r:

2 Specimen 2,Group II

1 . 5 1 .39 ___ r---- ---=-&..n-- ~A~----~A------ -~8

,JjY/Mp __ ..e--------

1 \ Plastic Hinge Forms at Support

.5

o ~ .5 {

a/au

Figure 9.17 Moment Due to Steel Beam Forces (Mg + Tge) at the Support vs. Maximum Rotation for Specimen 2, Group II. -- Refer to Table 7.2. w ....

.t>

t&. ~ , 2::

Figure 9.18

2

1 . 5

1

. 5


l!:. ~ ~--f.l!:.r----t.l!:.:r--~3~ _______________ ~\ l!:. .5 __ _ ----

___ JI'_ ~y/Mp ------- Plastic Hinge Forms at Support

o V h I I o .5 1

a/eu

Moment Due to Steel Beam Forces (M + T e) at the Support vs. Maximum Rotation for

Specimen 2, Group III. -- Refer togTablg 7.3. w ...... LT1

1

o! --0.. .5

o

Servi ce Load

o .5

ojou

Specimen 1,Group I

1

Figure 9.19 Mid-Span Load-Deflection Curve for Specinlen 1, Group I. -- Refer to Table 7.1.

w ...... (j)

1

~ Ii:' . 5

o (2)

Servi ce Load

. 5

6/6u

Specimen 3,Group II

1

Figure 9.20 Mid-Span Load-Deflection Curve for Specimen 3, Group II. -- Refer to Table 7.2. w ~

--.J

r'------=-==~==--==----==-----=-----=---------===-==------====-==----------------------~

1

~ -- . 5 ~

o

Se r vic e La ad

o .5

6/ou

Specimen 5,Graup II

1

Figure 9.21 Mid-Span Load-Deflection Curve for Specimen 5, Group II. -- Refer to Table 7.2.

w ...... (»

1

~ '- .5 ~

o 0

Service Load

.5

6/6u


1

Figure 9.22 Mid-Span Load-Deflection Curve for Specimen 2, Group III. -- Refer to Table 7.3. LV ...... \D

Table 9.5 Load-Deflection Data for Composite Beams.

Group No.

I

II

III

Specimen No.

1 2 3

1 2 3 5 6 7 8

1 2 3

p a u

(Ksf)

1. 50 1.90 2.575

1. 27 1.50 1.914 1.385 1.966 2.00 2.033

1. 30 1.633 2.225

b Y.L. U.L.

0.4 0.421 0.427

0.38 0.427 0.444 0.375 0.458 0.5 0.42

0.385 0.429 0.472

First Yield Occurs in

Steel beam Steel beam Steel beam

S':::eel beam Steel beam Steel beam Concrete slab Concrete slab Concrete Slab Steel beam

Concrete slab Concrete slab Steel beam

o c y

(in. )

0.213 0.234 0.24

0.258 0.282 0.265 0.304 0.359 0.368 0.262

0.253 0.293 0.314

'd o 0.6

(in. )

0.329 0.347 0.357

0.426 0.407 0.366 0.504 0.50 0.461 0.433

0.40 0.419 0.408

a p = ultimate uniform distributed load on slab (Ksf). u

b y • L./U. L . = ratio of yield load to ultimate load.

Co = deflection at yield load. y

do = deflection at load equal to 0.6 of the ultimate one. 0.6

eo = deflection at the ultimate load. u

o e u

(in. )

0.78 0.844 1.154

1.02 1.07 1.10 1. 395 1. 35 1. 21 1.126

0.93 1.021 1.064

320

321

4. Occurrence of slip between the slab and the beam.

5. Degree of interaction between the beam and the slab. This can

be represented by the stiffness of the studs.

6. The actual load distribution between the girders and the trans

verse beams. This depends mainly on the b/~ ratio and on the

support conditions.

All of the above factors make an exact deflection calculation

for continuous composite beams very difficult, even in the linear range.

An approximation method for finding the service load deflection, there

fore, has been developed on the basis of the deflection data for all of

the models in Groups I, II, and III, utilizing the following

assumptions:

1. The service load is assumed to be not more than 0.6 of the ulti

mate load. The value of 0.6 is the ratio of the service load to

the ultimate load in plastic design of steel structures [4]. At

this level of loading, the specimens will have some local

yielding, because most of them yield at a load between 40 and 50

percent of the ultimate one, as shown in Table 9.5; however,

Figures 9.19-9.22 show that the load-deflection curv~s up to a

load level equal to 0.6 of the ultimate one can be approximated

by a straight line, with negligible error.

2. At service load, the effective width is assumed to be constant

along the full length of the composite beam, equal to the effec

tive width at mid-span at the yield load. Based on this

322

assumption, the moments of inertia for the composite beams have

been calculated, and are listed in Table 9.6.

3. The load on the composite beams is assumed to be distributed

according to their tributary area, as shown in Figure l.la.

4. Support conditions for the composite beams are assumed to be

semi-rigid, because it was found that a rigid steel beam-to-

column connection will not produce fixed-ended behavior for the

overall composite beam. The major objective, therefore, is to

find the negative moment at the support, since this has a pri-

mary influence on the final deflection.

5. The deflection at mid-span is calculated by superimposing the

deflection due to the gravity load on a simply supported compos-

ite beam and the opposite deflection due to the restraining

moments of the semi-rigid connections at the supports.

The negative moments at the semi-rigid end connections have been

calculated to satisfy the deflection at the service load. The general

equation for the deflection at mid-span for a beam loaded by gravity

load and end moments Ma and Mb is given by the well-known expression

v (9.11)

where va = deflection at mid-span of a simply supported beam due to the

gravity load, and Ma and Mb = moments at the supports.

The gravity load on the composite girders is distributed trape-

zoidally, as shown in Figure 9.23. vo' the deflection due to this Load,

has been derived from basic structural analysis as

Table 9.6 Service Load Support Moments.

vO• 6 - vo M = M M I vo a b Eq. 9.16 tr

Group Specimen w Negative Negative M .1

F.E . No. No. (in. -) (K/in. ) ex = al'l (in. ) (in. ) (K/in. ) y (%)

I 1 1607 1.2 0.4 0.846 0.517 3,346 0.78 + 5.5 2 1600 1.14 0.3 0.914 0.567 3,654 0.79 + 14.5 3 1585 1.03 0.2 0.907 0.55 3,511 0.77 + 12

II 1 3249 1.27 0.4 1.081 0.655 5,486 0.77 + 2.0 2 3228 1.125 0.3 1.092 0.685 .5,700 0.79 + 1.2 3 3206 0.957 0.2 1.017 0.652 5,388 0.80 2.0

III 1 8788 1.56 0.4 1.018 0.618 9,722 0.77 - 11.0 2 8728 1.47 0.3 1.094 0.675 10,546 0.78 8.5 3 8667 1.335 0.2 1.089 0.681 10,566 0.78 7.2

II 5 3242 0.9333 0.2 0.981 0.476 3,978 0.615 + 15.5 6 3227 0.9833 0.2 1.038 0.546 4,542 0.66 + 10.5 7 3222 1.0 0.2 1.057 0.596 4,950 0.71 + 4.4 8 3213 1.0165 0.2 1.078 0.645 5,342 0.75 + 0.5

aM = moment at the supports of the composite beams due to the steel beam forces (~.~.+ M ) obtained from the finite-element analysis at load equal to 0.6 times the ul~imategload.

a - 1

(.oJ

N (.oJ

a ~~,,----~

UJ a ~---'l,

Figure 9.23 Typical Gravity Load Distribution on Girders.

324

325

16 4) + 25 a (9.12)

where a = a/~; and a, ~, and ware defined in Figure 9.23.

The moment at the supports can be obtained by substituting for

v = 00.6 into equation 9.11, as given by Table 9.5, using Vo from equa-

tion 9.12 and letting Ma = Mb for symmetric loading. As an example,

this calculation will be illustrated for model 1 of Group I. The girder

span of this model is 20' = 240", and the slab has a width of b = 192".

The gravity load, therefore, will have a trapezoidal shape, as shown in

Figure 9.24, with a value of a = b/2 = 96".

The load w is found as w = 0.6(P /144)b = 1.2 K/in., where P is u u

given in Table 9.5. Substituting into equation 9.12 for a = 96/240, ~ =

240", w = 1.2 K/in., I = I = 1607 in.4, and E = 29,000 Ksi gives Vo = tr

0.846". Substituting into equation 9.11 for v = 00.6 = 0.329", Ma = Mb ,

and the above values of vo' E, ~, and ~ gives Ma = Mb = - 3346 K-in.

The calculations for the rest of the specimens have been carried out in

a similar manner; the results are given in Table 9.6.

The moment at the supports of the composite girders, which has

been calculated on the basis of the deflection criteria, is compared to

the support moment for a fixed-ended beam. The moment equation for the

latter is

M = w~2 2 3 12 (1 - 20. + a )

whereas the composite beam end moment was found to be equal to

M a

w~2 2 3 Y 12 (1 - 20. + a )

(9.13)

(9.14)

326

, , , , 96 1.2 K/IN 96

~r-, --41

, , )

240 ~~,--------------------------------~*

Figure 9.24 Gravity Load on Specimen 1, Group I.

327

where y is a reduction factor that is calculated for each specimen and

listed in Table 9.6. As can be seen from this table, y assumes values

between 0.77 and 0.8 for the models having fixed ends; it v~ries between

0.615 and 0.75 for end connection rigidities, M 1M , between 0.1 and c p

0.7. Assuming a value of y = 0.78 for M 1M = 1, a conservative c p

straight line equation for y is expressed by

M c Y = 0.16 + 0.18 M P

(9.15)

Thus, the moment for the composite beam at service load can be expressed

as

M a

M c

= - (0.6 + 0.18 M P

Wn2 2 3 )V (1-20. +0.)

12 (9.16)

The moments given by equation 9.16 have been compared to the

value found as M + T e, at 0.6 times the ultimate load, which is given g g

by the finite-element analysis. The differences between these two

moment magnitudes are also shown in Table 9.6. It is seen that the dif-

ference assumes a maximum value of about ~ 15 percent, which must be

considered as acceptable, considering the complexity of the problem.

Summarizing, the deflection calculations provide the following

observations:

1. The negative moment due to the service load, which is equal to

or less than 0.6 times the ultimate load, should be calculated

by equation 9.16.

2. The.deflection of a simply supported beam due to the gravity

load should be calculated by equation 9.12.

328

3. The two deflections found in 1 and 2 above are superimposed to

give the final deflection, using equation 9.11.

CHAPTER 10

SUMMARY AND CONCLUSIONS

This study has examined the behavior and strength of continuous

composite beams in composit.e floor systems. The primary objectives

were:

1. To study the effective width in the positive and negative moment

regions at the yield and ultimate loads.

2. To calculate the ultimate strength of a composite beam at the

mid-span and at the support.

3. To study the ductility of continuous composite beams, and the

use of plastic design for such members.

4. To calculate the deflection of continuous composite beams at and

beyond the service load.

To conduct this study, 15 composite floor systems with several

variables (b/~ ratio, span length, slab thickness, reinforcement ratio,

type of steel beam-to-column connection, steel beam size) have been

analyzed by using a nonlinear finite-element solution technique. The

findings are summarized as follows:

1. A finite-element model has been developed to represent the com

posite floor system. In this model, the cracks in the concrete

slab are assumed to extend up to mid-height of the slab. The

slab is treated as an isotropic material with an equivalent

yield stress and modulus of elasticity that are based on the

329

properties and dimensions of the concrete and the reinforcing

bar steel. A minimum reinforcement ratio of one percent has

been used in this model, as expressed in equation 4.31.

330

2. The flexural and axial stiffnesses of the stud shear connectors

have been modified to take into account their actual behavior in

the structure. This model reflects the slip between the steel

beam and the concrete slab that may occur as a result of the

axial and lateral deformations of the studs.

3. The finite-element models have been verified by analyzing tested

specimens and comparing the theoretical and experimental

results. Very good agreement was found, as discussed in

Chapter 6.

4. Based on stress analyses, new criteria have been developed for

the effective width at the wid-span and at the support of a com

posite beam, at the yield and ultimate loads. These are given

by equations 8.30, 8.33, 8.34, 8.36, and 8.40-8.42.

5. The most important factors for the effective width were found to

be the beam span, the b/~ ratio, the steel beam size, and the

support conditions. The beam span length was the most signifi

cant. The recommended effective width criteria have considered

all of these parameters.

6. The slab thickness influence on the effective width was found to

be negligible; however, it was included in calculating the

effective width at the support.

7. New design equations are proposed to calculate the ultimate

moment capacity of a composite beam at mid-span and at support,

331

as given in equations 9.4, 9.6, 9.7, 9.9, and 9.10. These equa

tions were developed on the basis of actual (analytical) behav

ior. The assumption that plane sections will remain plane has

not been used.

8. The ultimate moment capacity was found to be almost independent

of the type of connection between the steel beam and column;

however, different equations are proposed to calculate the ulti

mate moment capacity based on the support conditions. This is

because these were found to influence the values of the axial

load and the moment which cause a plastic hinge to form in the

steel beam.

9. Continuous composite beams with all types of beam-to-column con

nections were found to have sufficient ductility to achieve a

proper plastic collapse mechanism. Plastic design, therefore,

can be used for continuous composite beams.

10. The support moment of a continuous composite beam, under a ser

vice load that does not exceed 60 percent of the ultimate load,

should be found by the recommended moment equation (9.16). This

moment comes in addition to that of the gravity load, and should

be used for calculating the deflection of continuous composite

beams.

11. Comparing the criteria that have been developed in this study

with those of the current specifications found that new and

original equations are developed for calculating the effective

width at mid-span and at support based on the ultimate load cri

teria. Also, original equations for calculating the ultimate

332

strength and the deflection become available for the design of

continuous composite beams.

CHAPTER 11

RECOMMENDATIONS FOR FURTHER RESEARCH

Further research on continuous composite beams should address

the following topics:

1. The effect of the degree of interaction on the ultimate strength

and the effective width.

2. The influence of a steel deck on the behavior of the composite

beams.

3. Rotation characteristics of composite beam-to-column connections

with different types of joints between the steel members.

4. The response of composite beams to seismic load effects,

including ductility requirements.

333

NOMENCLATURE

A Area of slab reinforcing bars sr

b Distance between adjacent girders

b Effective width of the concrete slab e

e Distance between the centerline of the steel girder and the

centerline of the concrete slab of a composite section

E Equivalent modulus of elasticity of the reinforced concrete slab

E Modulus of elasticity of concrete c

E Modulus of elasticity of steel girder s

E Modulus of elasticity of the steel bars sr

f' Concrete compressive strength c

F Yield stress of the slab reinforcing bars yr

h Depth of the steel girder g

I Moment of inertia of the composite section tr

K Steel beam-to-slab stiffness ratio r

~,L Girder span length

M Composite section moment due to the girder forces (T e + M ) g g

M Semi-rigid connection moment capacity c

M Moment in the steel girder of a composite section g

M Steel girder plastic moment p

M Moment in the concrete slab, whose width equals the effective s

width

M Ultimate moment capacity of the composite section in a negative un

moment region

334

335

M Ultimate moment capacity of the composite section in a positive up

moment region

M Ultimate moment capacity of the concrete slab, whose width us

equals the effective width

p Steel beam axj.al force that satisfies the beam-column interac-

p

p u

p y

t c

T g

U.L.

Y.L.

o u

o y

o 0.6

tion equation 9.2 [4, equation 2.4-3)

Applied load

Ultimate applied load

Steel girder axial load capacity

Concrete slab thickness

Axial load in the steel girder of a composite section

Ultimate applied load

Simply supported beam deflection due to gravity load

Deflection due to service load (assumed to be equal to 0.6 of

the ultimate load)

Yield load of the composite section

Reduction factor

Reduction factor

Reduction factor

Deflection due to the applied load

Deflection due to ultimate load

Deflection due to yield load

Deflection due to service load (assumed to be equal to 0.6 of

the ultimate load)

o Maximum rotation along the girder due to the applied load

o Maximum rotation along the girder due to the ultimate load u

336

p Slab reinforcement ratio

Minimum slab reinforcement ratio

Equivalent yield stress of the reinforced concrete slab

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