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Buckling behaviour of steel and composite beams at elevated temperatures
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Buckling behaviour of steel and composite beams at elevated temperatures
Ronny Budi Dharma
https://hdl.handle.net/10356/12064
https://doi.org/10.32657/10356/12064
BUCKLING BEHAVIOUR OF STEEL AND COMPOSITE BEAMS AT ELEVATED
TEMPERATURES
2007
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Buckling Behaviour of Steel and Composite Beams at Elevated Temperatures
Ronny Budi Dharma
School of Civil and Environmental Engineering
A thesis submitted to the Nanyang Technological University in fulfillment of the requirement for the degree of Doctor of Philosophy
2007
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
The author would like to express his sincere appreciation to his supervisor, Associate
Professor Tan Kang Hai, for his invaluable supervision, guidance, and support. The
author would also like to extend his gratitude to Singapore Millennium Foundation for
providing the scholarship.
This research was funded by ARC 5/03 project entitled “Mitigation of Progressive
Collapse of Tall Buildings” from the Ministry of Education, Singapore. In addition, the
author would also like to acknowledge Corus South East Asia for supplying the
structural I-beams and TTJ Design and Engineering for fabricating the steel beams.
He wishes to thank Dr. Huang Zhanfei, Dr. Yuan Weifeng, and Mr. Qian Zhenhai for
their comments and helpful discussions. Special thanks should also extend to all
FERGAN members, the staff of Construction Technology Laboratory, especially Mr.
David Tui, Mr. Chelladurai Subasanran and Mr. Phua Kok Soon, and all those who have
given valuable advice and comments throughout the study. Finally, he is indebted to his
parents and Ms. Fenita Naviria for their unceasing moral support.
i
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
CONTENTS
CONTENTS
1.4 Enclosure Fire Behaviour and Standard Fire Exposure 5
1.5 Structural Response and Design in Fire 7
1.6 Objective and Scope 9
1.7 Organization 11
2.3 Local Buckling 21
2.3.2 Local Buckling Behaviour at Ambient Temperature 27
2.3.3 Local Buckling & Ductility in Fire – Importance & Motivations 35
ii
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CONTENTS
3.1 Introduction 39
3.2.1 Stress-Strain Relationships 40
3.2.2 Yield Strength 45
3.2.3 Elastic Modulus 47
3.2.5 Coefficient of Thermal Expansion 48
3.3 Concrete Properties at Elevated Temperature 50
3.3.1 Stress-Strain Relationships 50
3.3.2 Thermal Strain 52
3.5 Load-Slip Relationship of Shear Stud at Elevated Temperature 54
CHAPTER 4: LATERAL TORSIONAL BUCKLING OF
UNRESTRAINED STEEL BEAMS
4.1 Introduction 57
4.2.1 Unrestrained Beams: (a) Perfectly Straight and (b) Crooked 58
4.2.2 BS5950-Part 1:2000 Approach for LTB 64
4.2.3 Eurocode3-Part 1.1:2005 Approach for LTB 67
4.3 Lateral Torsional Buckling at Elevated Temperature 71
4.3.1 Review of Various Approaches 71
4.3.2 Alternative Approach at Elevated Temperatures 72
4.3.3 Rankine Approach at Elevated Temperatures 79
4.4 Numerical Analysis and Comparisons of Different Approaches
at Elevated Temperature 81
iii
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CONTENTS
4.4.3 Comparisons and Discussions of Various Approaches 86
4.4.4 Worked Example 91
BUCKLING FAILURE AND DUCTILITY
5.2.1 Design of Beam Specimens 96
5.2.2 Instrumentation 101
5.2.4 Geometric Imperfection Measurements 108
5.2.5 Material Properties 111
5.3.1 General Observations 113
5.3.2 Temperature Effects 116
5.4 Experimental Programme on Composite Steel Beams 122
5.4.1 Design of Composite Steel Beam Specimens 123
5.4.2 Instrumentation 126
5.4.4 Geometric Properties Imperfection 127
5.4.5 Material Properties 129
5.5 Results and Discussions on Composite Steel Beam Tests 132
5.5.1 Temperature Developments and Distribution 133
5.5.2 General Observations 139
iv
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CONTENTS
BUCKLING FAILURE AND DUCTILITY
6.1 Introduction 150
6.2 Overview and Validation of Steel Beam’s FE Model 150
6.2.1 Overview of Numerical Modelling 151
6.2.2 Validation of Finite Element Model at Ambient Temperature 152
6.2.3 Validation of Finite Element Model at Elevated Temperature 155
6.3 Parametric Study on Rotational Capacity of Steel Beams 159
6.3.1 Sensitivity of Initial Geometric Imperfection 160
6.3.2 Influence of Temperature 162
6.3.3 Influence of Flange Slenderness 165
6.3.4 Influence of Web Slenderness 167
6.3.5 Influence of Effective Length 168
6.3.6 Influence of Steel Grade 170
6.4 Overview and Validation of Composite Beam’s FE Model 172
6.4.1 Overview of Numerical Modelling 172
6.4.2 Validation of Finite Element Model 174
6.5 Parametric Study on Rotational Capacity of Composite Beams 178
6.5.1 Influence of Temperature 179
6.5.2 Influence of Flange Slenderness 181
6.5.3 Influence of Web Slenderness 182
6.5.4 Influence of Reinforcement Area 183
6.5.5 Influence of the Number of Shear Studs 184
6.5.6 Influence of Effective Length 185
6.6 Isothermal versus Transient Response 186
v
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CONTENTS
RELATIONSHIP
7.2 Development of Moment-Rotation Design Model for Steel Beams 192
7.2.1 Non-Linear Pre-Peak Region 192
7.2.2 Horizontal Plateau Region 194
7.2.3 Unloading Region 195
7.3 Plastic Collapse Mechanism Model for Steel Beams 200
7.4 Plastic Collapse Mechanism Model for Composite Beams 216
CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS
8.1 Conclusions 223
8.1.2 Lateral Torsional Buckling of Unrestrained Steel Beams 224
8.1.3 Experimental Investigation on Local Buckling
Failure and Ductility 225
8.1.4 Numerical Analysis on Local Buckling Failure and Ductility 228
8.1.5 Modelling of Moment-Rotational Relationship 230
8.2 Recommendations 231
Appendix A2: Derivation of Slenderness Ratio Expression
in BS5950-1 (BSI, 2001) 252
vi
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CONTENTS
Appendix A3: Data and Results on the Numerical Analysis of
Lateral Torsional Buckling 254
Appendix B1: Longitudinal Imperfection Measurements 260
Appendix B2: Photographs from First Series of Tests – Steel Beams 267
APPENDIX C
Appendix C2: Detailed Design of Composite Beam Specimens 280
Appendix C3: Photographs from Second Series of Tests – Composite Beams 284
vii
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SUMMARY
SUMMARY
The objective of this research was to study the buckling behaviour of steel and
composite beams at elevated temperatures. Both global buckling in the form of lateral
torsional buckling and local buckling were studied. The first part of the study focused on
the lateral torsional buckling behaviour of bare steel beams with an aim to develop
design approaches for laterally unrestrained steel beams at elevated temperature. The
second part focused on the local buckling behaviour which limits the ductility of beams.
Both bare steel beams and composite deck slab with re-entrant steel decking were
considered for the second part to investigate the ductility issue related to inelastic
behaviour in the hogging moment regions under fire conditions and to propose the
model of the moment-rotational relationships.
Numerical analysis using MSC.MARC Mentat and published test results were used to
study the lateral torsional buckling of steel beams at elevated temperatures.
Subsequently, a general approach, different from the current approach, called an
alternative approach was suggested. Besides, a simple analytical approach, based on
Rankine principle, was applied to estimate the lateral torsional buckling failure of steel
beams in fire. Both proposals were shown to provide a good correlation with the
numerical and test results.
The investigation of the local buckling behaviour at elevated temperatures comprised of
both experimental and numerical investigation. The experimental investigation
consisted of two series of tests, namely, investigation on steel beams as the first series
and investigation on composite beams as the second series. The numerical investigation
involved fairly extensive parametric studies using the numerical model which had been
validated with test results. Finally, the analytical models for the moment-rotational
relationships under fire conditions were proposed.
viii
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LIST OF TABLES
LIST OF TABLES
Part 1.2 (CEN, 2005) 43
Table 3.2 Reduction Factors for Stress-Strain Relationship of Steel at
Elevated Temperatures for EC3 Material Model (CEN, 2005) 44
Table 3.3 Strength Reduction Factors and Strain Limits of Concrete at
Elevated Temperatures for Eurocode 4:1.2 Model (CEN, 2005) 51
Table 3.4 Eurocode 4:1.2 Reduction Factors for Cold Worked
Reinforcement at Elevated Temperatures (CEN, 2005) 54
Table 3.5 Parameters for Load-Slip Relationship Model of Headed
Shear Stud in Composite Beams with Re-entrant
Steel Decking (Zhao & Kruppa, 1995) 56
Table 4.1 Comparisons of BS5950 and EC3 Approach for LTB 70
Table 4.2 Boundary Conditions at End Nodes 83
Table 4.3 Comparison between FE Predictions and Elastic Theoretical
Bifurcation Solution for Mid-span Point Load Case 83
Table 4.4 Comparisons between FE Predictions and Elastic Theoretical
Bifurcation Solution for Uniform Moment Case 83
Table 4.5 Comparisons between FE Predictions and Experimental Results
by Kitipornchai and Trahair (1975) 84
Table 4.6 Various Section Sizes, Lengths and Loadings
for Numerical Analyses 85
Table 4.7 Comparison of Test Results (Vila Real et al., 2003)
with Various Approaches 87
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LIST OF TABLES
Table 4.8 Comparison of FE Results with Various Approaches 87
Table 5.1 Details of First Series of Test Specimens 99
Table 5.2 Locations of Thermocouple Wires 101
Table 5.3 Measured Cross-Section Dimensions of First Series of Specimens 109
Table 5.4 Summary of Maximum Longitudinal Imperfection 111
Table 5.5 Tensile Test Results of First Series of Specimens
at Ambient Temperature 111
Table 5.6 Summary of First Series of Test Results 113
Table 5.7 Details of Second Series of Test Specimens 126
Table 5.8 Measured Cross-Section Dimensions of Structural Steel 128
Table 5.9 Tensile Test Results of Second Series Structural Steel 129
Table 5.10 Concrete Test Results 131
Table 5.11 Relative Temperature Profiles of Tested Composite Beams 134
Table 5.12 Summary of Second Series of Test Results 140
Table 5.13 Comparisons of S1-2 and C1 Test Results 149
Table 6.1 Summary of Validation Results of Steel Beams
at Ambient Temperature 153
at Elevated Temperature 156
Table 6.3 Summary of Reference Beams for Parametric Studies 160
Table 6.4 Various Forms of Initial Imperfection 161
Table 6.5 Summary of Composite Beam Validation Results 175
Table 7.1 Statistical Analysis Results of the Ultimate Rotation
Regression Model 197
Table 7.2 Summary of Internal Virtual Works of Collapse Mechanism
(Gioncu & Mazzolani, 2002) 207
PCM Model 221
x
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LIST OF TABLES
Table A.1 FEA Data and Results for S275 at 400°C 254
Table A.2 FEA Data and Results for S355 at 400°C 254
Table A.3 FEA Data and Results for S275 at 500°C 255
Table A.4 FEA Data and Results for S355 at 500°C 255
Table A.5 FEA Data and Results for S275 at 600°C 256
Table A.6 FEA Data and Results for S355 at 600°C 256
Table A.7 FEA Data and Results for S275 at 700°C 257
Table A.8 FEA Data and Results for S355 at 700°C 257
Table A.9 FEA Data and Results for S275 at 800°C 258
Table A.10 FEA Data and Results for S355 at 800°C 258
xi
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LIST OF FIGURES
LIST OF FIGURES
Figure 1.1 Typical Temperature Development in an Enclosure Fire 5
Figure 1.2 ISO 834 Fire Curve 6
Figure 2.1 Section Classifications of Beams 30
Figure 2.2 Standard Moment-Rotation Curve of Plastic/ Compact Beams 31
Figure 2.3 Local Buckling Failures in Cardington Fire Test 37
Figure 3.1 Idealisation for Stress-Strain Relationship 42
Figure 3.2 Bilinear-Elliptical Idealisation for Stress-Strain Relationship 42
Figure 3.3 Basic Formulation of Stress-Strain Relationship of Steel
at Elevated Temperatures (CEN, 2005) 43
Figure 3.4 Comparison of Test Data (Kirby & Preston, 1988) and
EC3:1.2 Stress-Strain Relationships 45
Figure 3.5 Two Different Concepts of Defining Yield Stress 46
Figure 3.6 Variation of Strength Reduction Factor with Temperature 47
Figure 3.7 Thermal Strain of Steel as a Function of Temperature 49
Figure 3.8 Eurocode 4:1.2 Model of Stress-Strain Relationship of Concrete
at Elevated Temperatures (CEN, 2005) 51
Figure 3.9 EC2 and EC4 Strength Reduction Factors of Concrete 52
Figure 3.10 Thermal Strain of Concrete as a Function of Temperature 53
Figure 3.11 Load-Slip Relationship Model for Headed Shear Stud in Composite
Beams with Re-entrant Steel Decking (Zhao & Kruppa, 1995) 56
Figure 4.1 Buckling of Simply Supported I-Beam 59
Figure 4.2 Buckling and Yielding of I-Beams (Trahair et al., 2001) 62
Figure 4.3 Test Results for Hot-Rolled Beams at Ambient (Trahair et al., 2001) 64
Figure 4.4 Comparison of BS5950 and EC3 Approach at Ambient Temperature 70
xii
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LIST OF FIGURES
Figure 4.5a Comparison of Test Results and EC3:1.2 Approach at 200°C 73
Figure 4.5b Comparison of Test Results and EC3:1.2 Approach at 300°C 74
Figure 4.5c Comparison of Test Results, FEA and EC3:1.2 Approach at 400°C 74
Figure 4.5d Comparison of Test Results, FEA and EC3:1.2 Approach at 500°C 74
Figure 4.5e Comparison of Test Results, FEA and EC3:1.2 Approach at 600°C 75
Figure 4.6 Different Rates of Degradation of Tangent Modulus 77
Figure 4.7 Variation of Non-Linearity Factor with Temperature 78
Figure 4.8 Comparisons of Various Approaches for S275 at 400°C 79
Figure 4.9 LTB Design Curves at Various Temperatures
using Rankine Approach 81
Figure 4.10 Different Types of Nodes at End Cross-Section 83
Figure 4.11 Lateral Torsional Buckling Failure of Numerical Model 85
Figure 4.12 Comparisons of Test Results with Various Approaches at 200°C 88
Figure 4.13 Comparisons of Test Results with Various Approaches at 300°C 88
Figure 4.14 Comparisons of Test & FE Results with Various
Approaches at 400°C 89
Figure 4.15 Comparisons of Test & FE Results with Various
Approaches at 500°C 89
Figure 4.16 Comparisons of Test & FE Results with Various
Approaches at 600°C 90
Figure 4.17 Comparisons of FE Results with Various Approaches at 700°C 90
Figure 4.18 Comparisons of FE Results with Various Approaches at 800°C 91
Figure 5.1a Simplified Substitute Member for Hogging Moment Region 96
Figure 5.1b Standard Beam Arrangement 97
Figure 5.2 General Layout of First Series of Test Specimens 98
Figure 5.3 Reference Gridlines of Test Specimens 100
Figure 5.4 Schematic Positions of LVDTs 102
Figure 5.5 General View of Test Set-Up 103
xiii
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LIST OF FIGURES
Figure 5.6 Plan View and Section View of Test Set-up 104
Figure 5.7 Front View and Side View of Test Set-Up 105
Figure 5.8 Adjustable Fork Support Systems 106
Figure 5.9 Mid-span Roller Systems 106
Figure 5.10 Lateral Restraint Systems 107
Figure 5.11 Positions of Longitudinal Geometric Imperfection Measurement 110
Figure 5.12 In-Plane Imperfection of S2-1 110
Figure 5.13 Out-of-Plane Imperfection of S2-1 110
Figure 5.14 Stress-Strain Relationship of Coupon Type A 112
Figure 5.15 Stress-Strain Relationship of Coupon Type T 112
Figure 5.16 Local Buckling in S3-2 and S4-1 Specimen 114
Figure 5.17 Lateral Torsional Buckling in S2-1 and S2-2 Specimen 115
Figure 5.18 Local and Lateral Torsional Buckling in S3-1 Specimen 115
Figure 5.19 Load-Rotation Responses of S2-1 and S2-2 Specimen 116
Figure 5.20 Moment-Rotation Response of S3 Series 117
Figure 5.21 Load-Deflection (mid-span) Response of S3 Series 118
Figure 5.22 Moment-Rotation Response of S1 Series, S3-2 and S3-3 119
Figure 5.23 Load-Deflection (mid-span) Response of S1 Series, S3-2 and S3-3 119
Figure 5.24 Moment-Rotation Response of S3-2, S3-3 and S4 Series 120
Figure 5.25 Load-Deflection (mid-span) Response of S3-2, S3-3 and S4 Series 121
Figure 5.26 Moment-Rotation Response of S1 and S2 Series 122
Figure 5.27 Load-Deflection (mid-span) Response of S1 and S2 Series 122
Figure 5.28 Layout of C1 Specimen 125
Figure 5.29 Composite Beam before Testing 126
Figure 5.30 Stress-Strain Relationship of Flange Plate 129
Figure 5.31 Stress-Strain Relationship of Web Plate 130
Figure 5.32 Stress-Strain Relationship of Reinforcement 130
Figure 5.33 Configuration of Push-Out Test Specimen and Test Set-Up 132
xiv
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LIST OF FIGURES
Figure 5.35 Temperature Developments and Distribution of C1 135
Figure 5.36 Temperature Developments and Distribution of C2 136
Figure 5.37 Temperature Developments and Distribution of C3 137
Figure 5.38 Temperature Developments and Distribution of C4 138
Figure 5.39 Temperature Distributions of CB3 and PB2
during Cardington Fire Test 139
Figure 5.40 Moment-Rotation Response of C1 141
Figure 5.41 Moment-Rotation Response of C2 141
Figure 5.42 Moment-Rotation Response of C3 142
Figure 5.43 Moment-Rotation Response of C4 142
Figure 5.44 Local Buckling Failures of Second Series of Specimens 143
Figure 5.45 Mid-Span Concrete Cracking 144
Figure 5.46 Load-Rotation Response of C4 144
Figure 5.47 Proposed Temperature Distributions for Analysis 147
Figure 5.48 Comparisons of Cardington and Current Test Failure Modes 147
Figure 5.49 Moment-Rotation Comparisons of S1-2 and C1 149
Figure 6.1 Boundary Conditions at Both Supports 152
Figure 6.2 Typical Finite Element Mesh of a Steel Beam 152
Figure 6.3 Moment-Rotation Comparisons of FEA and S3-1 Test Results 154
Figure 6.4 Moment-Rotation Comparison of FEA and
Tests by Lukey & Adams (1969) 154
Figure 6.5 Failure Modes Comparison of FEA and Test 155
Figure 6.6 Moment-Rotation Comparisons of FEA and Tests
at Elevated Temperature 158
at Elevated Temperature 158
xv
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LIST OF FIGURES
at Elevated Temperature 159
Moment-Rotational Relationship 161
Figure 6.10 Effect of Temperature on Moment-Rotational Relationship (RB2) 163
Figure 6.11 Effect of Temperature on Moment-Rotational Relationship (RB4) 163
Figure 6.12 Variation of Rotational Capacity with Temperature 165
Figure 6.13 Variation of Stress-Strain Relationship Parameter kE/ky
with Temperature 165
Moment-Rotational Relationship 167
Moment-Rotational Relationship 168
Figure 6.16 Effect of Effective Length on Moment-Rotational Relationship 170
Figure 6.17 Comparisons of Failure Mode with Different Effective Length 170
Figure 6.18 Effect of Steel Grade on Moment-Rotational Relationship 171
Figure 6.19 Moment-Node Displacement Plot of RB1-S235 and RB1-S450 172
Figure 6.20 Typical Finite Element Mesh of a Composite Beam 174
Figure 6.21 Moment-Rotation Comparisons of Composite Beam
FEA and Tests 177
Figure 6.22 Failure Modes Comparison between FEA and Tests 178
Figure 6.23 Comparison of Crack Pattern between FEA and Tests 178
Figure 6.24 Temperature Distributions for Parametric Studies 179
Figure 6.25 Influence of Temperature on the Moment-Rotation Response 180
Figure 6.26 Comparisons of Failure Modes at Different Temperatures 181
Figure 6.27 Influence of Flange Slenderness on
the Moment-Rotation Response 182
Figure 6.28 Influence of Web Slenderness on the Moment-Rotation Response 183
xvi
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LIST OF FIGURES
the Moment-Rotation Response 184
Figure 6.30 Influence of Number of Studs on the Moment-Rotation Response 185
Figure 6.31 Influence of Effective Length on the Moment-Rotation Response 186
Figure 6.32 Influence of Effective Length on the Failure Modes 186
Figure 6.33 Moment-Rotation Response of RB2 at Various Temperatures 188
Figure 6.34 Temperature-Rotation Response of RB2 at
Various Applied Moments 188
Figure 6.35 Support’s Rotational Response of RB2 during Fire 189
Figure 7.1 Longitudinal and Cross-Section Discretization 194
Figure 7.2 Statistical Plots from the Regression Model 197
Figure 7.3 Moment-Rotation Design Model for Steel Beams
at Elevated Temperature 199
Figure 7.4 Validation of Steel Beam Design Model at Elevated Temperature 200
Figure 7.5 Post-Critical Curve of Plastic Collapse Mechanism 201
Figure 7.6 Illustration of Plastic Collapse Mechanism 201
Figure 7.7 Various Plastic Collapse Mechanism Models
(Gioncu & Mazzolani, 2002) 203
Figure 7.8 Plastic Collapse Mechanism Model (Gioncu & Petcu, 2001) 204
Figure 7.9 PCM based Moment-Rotation Model (S3-2) 211
Figure 7.10 Validation of Proposed Plastic Collapse Mechanism Model 212
Figure 7.11 Comparisons of FEA and PCM Predictions 213
Figure 7.12 Plot of Ratio of Web to Flange Slenderness-PCM Predictions 214
Figure 7.13 Plot of Web Slenderness-PCM Predictions 215
Figure 7.14 FE Results and PCM Predictions at Various
Flange Slenderness Ratios 216
within Parameter Limits 216
xvii…
Ronny Budi Dharma
https://hdl.handle.net/10356/12064
https://doi.org/10.32657/10356/12064
BUCKLING BEHAVIOUR OF STEEL AND COMPOSITE BEAMS AT ELEVATED
TEMPERATURES
2007
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
Buckling Behaviour of Steel and Composite Beams at Elevated Temperatures
Ronny Budi Dharma
School of Civil and Environmental Engineering
A thesis submitted to the Nanyang Technological University in fulfillment of the requirement for the degree of Doctor of Philosophy
2007
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
The author would like to express his sincere appreciation to his supervisor, Associate
Professor Tan Kang Hai, for his invaluable supervision, guidance, and support. The
author would also like to extend his gratitude to Singapore Millennium Foundation for
providing the scholarship.
This research was funded by ARC 5/03 project entitled “Mitigation of Progressive
Collapse of Tall Buildings” from the Ministry of Education, Singapore. In addition, the
author would also like to acknowledge Corus South East Asia for supplying the
structural I-beams and TTJ Design and Engineering for fabricating the steel beams.
He wishes to thank Dr. Huang Zhanfei, Dr. Yuan Weifeng, and Mr. Qian Zhenhai for
their comments and helpful discussions. Special thanks should also extend to all
FERGAN members, the staff of Construction Technology Laboratory, especially Mr.
David Tui, Mr. Chelladurai Subasanran and Mr. Phua Kok Soon, and all those who have
given valuable advice and comments throughout the study. Finally, he is indebted to his
parents and Ms. Fenita Naviria for their unceasing moral support.
i
ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library
CONTENTS
CONTENTS
1.4 Enclosure Fire Behaviour and Standard Fire Exposure 5
1.5 Structural Response and Design in Fire 7
1.6 Objective and Scope 9
1.7 Organization 11
2.3 Local Buckling 21
2.3.2 Local Buckling Behaviour at Ambient Temperature 27
2.3.3 Local Buckling & Ductility in Fire – Importance & Motivations 35
ii
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CONTENTS
3.1 Introduction 39
3.2.1 Stress-Strain Relationships 40
3.2.2 Yield Strength 45
3.2.3 Elastic Modulus 47
3.2.5 Coefficient of Thermal Expansion 48
3.3 Concrete Properties at Elevated Temperature 50
3.3.1 Stress-Strain Relationships 50
3.3.2 Thermal Strain 52
3.5 Load-Slip Relationship of Shear Stud at Elevated Temperature 54
CHAPTER 4: LATERAL TORSIONAL BUCKLING OF
UNRESTRAINED STEEL BEAMS
4.1 Introduction 57
4.2.1 Unrestrained Beams: (a) Perfectly Straight and (b) Crooked 58
4.2.2 BS5950-Part 1:2000 Approach for LTB 64
4.2.3 Eurocode3-Part 1.1:2005 Approach for LTB 67
4.3 Lateral Torsional Buckling at Elevated Temperature 71
4.3.1 Review of Various Approaches 71
4.3.2 Alternative Approach at Elevated Temperatures 72
4.3.3 Rankine Approach at Elevated Temperatures 79
4.4 Numerical Analysis and Comparisons of Different Approaches
at Elevated Temperature 81
iii
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CONTENTS
4.4.3 Comparisons and Discussions of Various Approaches 86
4.4.4 Worked Example 91
BUCKLING FAILURE AND DUCTILITY
5.2.1 Design of Beam Specimens 96
5.2.2 Instrumentation 101
5.2.4 Geometric Imperfection Measurements 108
5.2.5 Material Properties 111
5.3.1 General Observations 113
5.3.2 Temperature Effects 116
5.4 Experimental Programme on Composite Steel Beams 122
5.4.1 Design of Composite Steel Beam Specimens 123
5.4.2 Instrumentation 126
5.4.4 Geometric Properties Imperfection 127
5.4.5 Material Properties 129
5.5 Results and Discussions on Composite Steel Beam Tests 132
5.5.1 Temperature Developments and Distribution 133
5.5.2 General Observations 139
iv
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CONTENTS
BUCKLING FAILURE AND DUCTILITY
6.1 Introduction 150
6.2 Overview and Validation of Steel Beam’s FE Model 150
6.2.1 Overview of Numerical Modelling 151
6.2.2 Validation of Finite Element Model at Ambient Temperature 152
6.2.3 Validation of Finite Element Model at Elevated Temperature 155
6.3 Parametric Study on Rotational Capacity of Steel Beams 159
6.3.1 Sensitivity of Initial Geometric Imperfection 160
6.3.2 Influence of Temperature 162
6.3.3 Influence of Flange Slenderness 165
6.3.4 Influence of Web Slenderness 167
6.3.5 Influence of Effective Length 168
6.3.6 Influence of Steel Grade 170
6.4 Overview and Validation of Composite Beam’s FE Model 172
6.4.1 Overview of Numerical Modelling 172
6.4.2 Validation of Finite Element Model 174
6.5 Parametric Study on Rotational Capacity of Composite Beams 178
6.5.1 Influence of Temperature 179
6.5.2 Influence of Flange Slenderness 181
6.5.3 Influence of Web Slenderness 182
6.5.4 Influence of Reinforcement Area 183
6.5.5 Influence of the Number of Shear Studs 184
6.5.6 Influence of Effective Length 185
6.6 Isothermal versus Transient Response 186
v
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CONTENTS
RELATIONSHIP
7.2 Development of Moment-Rotation Design Model for Steel Beams 192
7.2.1 Non-Linear Pre-Peak Region 192
7.2.2 Horizontal Plateau Region 194
7.2.3 Unloading Region 195
7.3 Plastic Collapse Mechanism Model for Steel Beams 200
7.4 Plastic Collapse Mechanism Model for Composite Beams 216
CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS
8.1 Conclusions 223
8.1.2 Lateral Torsional Buckling of Unrestrained Steel Beams 224
8.1.3 Experimental Investigation on Local Buckling
Failure and Ductility 225
8.1.4 Numerical Analysis on Local Buckling Failure and Ductility 228
8.1.5 Modelling of Moment-Rotational Relationship 230
8.2 Recommendations 231
Appendix A2: Derivation of Slenderness Ratio Expression
in BS5950-1 (BSI, 2001) 252
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CONTENTS
Appendix A3: Data and Results on the Numerical Analysis of
Lateral Torsional Buckling 254
Appendix B1: Longitudinal Imperfection Measurements 260
Appendix B2: Photographs from First Series of Tests – Steel Beams 267
APPENDIX C
Appendix C2: Detailed Design of Composite Beam Specimens 280
Appendix C3: Photographs from Second Series of Tests – Composite Beams 284
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SUMMARY
SUMMARY
The objective of this research was to study the buckling behaviour of steel and
composite beams at elevated temperatures. Both global buckling in the form of lateral
torsional buckling and local buckling were studied. The first part of the study focused on
the lateral torsional buckling behaviour of bare steel beams with an aim to develop
design approaches for laterally unrestrained steel beams at elevated temperature. The
second part focused on the local buckling behaviour which limits the ductility of beams.
Both bare steel beams and composite deck slab with re-entrant steel decking were
considered for the second part to investigate the ductility issue related to inelastic
behaviour in the hogging moment regions under fire conditions and to propose the
model of the moment-rotational relationships.
Numerical analysis using MSC.MARC Mentat and published test results were used to
study the lateral torsional buckling of steel beams at elevated temperatures.
Subsequently, a general approach, different from the current approach, called an
alternative approach was suggested. Besides, a simple analytical approach, based on
Rankine principle, was applied to estimate the lateral torsional buckling failure of steel
beams in fire. Both proposals were shown to provide a good correlation with the
numerical and test results.
The investigation of the local buckling behaviour at elevated temperatures comprised of
both experimental and numerical investigation. The experimental investigation
consisted of two series of tests, namely, investigation on steel beams as the first series
and investigation on composite beams as the second series. The numerical investigation
involved fairly extensive parametric studies using the numerical model which had been
validated with test results. Finally, the analytical models for the moment-rotational
relationships under fire conditions were proposed.
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LIST OF TABLES
LIST OF TABLES
Part 1.2 (CEN, 2005) 43
Table 3.2 Reduction Factors for Stress-Strain Relationship of Steel at
Elevated Temperatures for EC3 Material Model (CEN, 2005) 44
Table 3.3 Strength Reduction Factors and Strain Limits of Concrete at
Elevated Temperatures for Eurocode 4:1.2 Model (CEN, 2005) 51
Table 3.4 Eurocode 4:1.2 Reduction Factors for Cold Worked
Reinforcement at Elevated Temperatures (CEN, 2005) 54
Table 3.5 Parameters for Load-Slip Relationship Model of Headed
Shear Stud in Composite Beams with Re-entrant
Steel Decking (Zhao & Kruppa, 1995) 56
Table 4.1 Comparisons of BS5950 and EC3 Approach for LTB 70
Table 4.2 Boundary Conditions at End Nodes 83
Table 4.3 Comparison between FE Predictions and Elastic Theoretical
Bifurcation Solution for Mid-span Point Load Case 83
Table 4.4 Comparisons between FE Predictions and Elastic Theoretical
Bifurcation Solution for Uniform Moment Case 83
Table 4.5 Comparisons between FE Predictions and Experimental Results
by Kitipornchai and Trahair (1975) 84
Table 4.6 Various Section Sizes, Lengths and Loadings
for Numerical Analyses 85
Table 4.7 Comparison of Test Results (Vila Real et al., 2003)
with Various Approaches 87
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LIST OF TABLES
Table 4.8 Comparison of FE Results with Various Approaches 87
Table 5.1 Details of First Series of Test Specimens 99
Table 5.2 Locations of Thermocouple Wires 101
Table 5.3 Measured Cross-Section Dimensions of First Series of Specimens 109
Table 5.4 Summary of Maximum Longitudinal Imperfection 111
Table 5.5 Tensile Test Results of First Series of Specimens
at Ambient Temperature 111
Table 5.6 Summary of First Series of Test Results 113
Table 5.7 Details of Second Series of Test Specimens 126
Table 5.8 Measured Cross-Section Dimensions of Structural Steel 128
Table 5.9 Tensile Test Results of Second Series Structural Steel 129
Table 5.10 Concrete Test Results 131
Table 5.11 Relative Temperature Profiles of Tested Composite Beams 134
Table 5.12 Summary of Second Series of Test Results 140
Table 5.13 Comparisons of S1-2 and C1 Test Results 149
Table 6.1 Summary of Validation Results of Steel Beams
at Ambient Temperature 153
at Elevated Temperature 156
Table 6.3 Summary of Reference Beams for Parametric Studies 160
Table 6.4 Various Forms of Initial Imperfection 161
Table 6.5 Summary of Composite Beam Validation Results 175
Table 7.1 Statistical Analysis Results of the Ultimate Rotation
Regression Model 197
Table 7.2 Summary of Internal Virtual Works of Collapse Mechanism
(Gioncu & Mazzolani, 2002) 207
PCM Model 221
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LIST OF TABLES
Table A.1 FEA Data and Results for S275 at 400°C 254
Table A.2 FEA Data and Results for S355 at 400°C 254
Table A.3 FEA Data and Results for S275 at 500°C 255
Table A.4 FEA Data and Results for S355 at 500°C 255
Table A.5 FEA Data and Results for S275 at 600°C 256
Table A.6 FEA Data and Results for S355 at 600°C 256
Table A.7 FEA Data and Results for S275 at 700°C 257
Table A.8 FEA Data and Results for S355 at 700°C 257
Table A.9 FEA Data and Results for S275 at 800°C 258
Table A.10 FEA Data and Results for S355 at 800°C 258
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LIST OF FIGURES
LIST OF FIGURES
Figure 1.1 Typical Temperature Development in an Enclosure Fire 5
Figure 1.2 ISO 834 Fire Curve 6
Figure 2.1 Section Classifications of Beams 30
Figure 2.2 Standard Moment-Rotation Curve of Plastic/ Compact Beams 31
Figure 2.3 Local Buckling Failures in Cardington Fire Test 37
Figure 3.1 Idealisation for Stress-Strain Relationship 42
Figure 3.2 Bilinear-Elliptical Idealisation for Stress-Strain Relationship 42
Figure 3.3 Basic Formulation of Stress-Strain Relationship of Steel
at Elevated Temperatures (CEN, 2005) 43
Figure 3.4 Comparison of Test Data (Kirby & Preston, 1988) and
EC3:1.2 Stress-Strain Relationships 45
Figure 3.5 Two Different Concepts of Defining Yield Stress 46
Figure 3.6 Variation of Strength Reduction Factor with Temperature 47
Figure 3.7 Thermal Strain of Steel as a Function of Temperature 49
Figure 3.8 Eurocode 4:1.2 Model of Stress-Strain Relationship of Concrete
at Elevated Temperatures (CEN, 2005) 51
Figure 3.9 EC2 and EC4 Strength Reduction Factors of Concrete 52
Figure 3.10 Thermal Strain of Concrete as a Function of Temperature 53
Figure 3.11 Load-Slip Relationship Model for Headed Shear Stud in Composite
Beams with Re-entrant Steel Decking (Zhao & Kruppa, 1995) 56
Figure 4.1 Buckling of Simply Supported I-Beam 59
Figure 4.2 Buckling and Yielding of I-Beams (Trahair et al., 2001) 62
Figure 4.3 Test Results for Hot-Rolled Beams at Ambient (Trahair et al., 2001) 64
Figure 4.4 Comparison of BS5950 and EC3 Approach at Ambient Temperature 70
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LIST OF FIGURES
Figure 4.5a Comparison of Test Results and EC3:1.2 Approach at 200°C 73
Figure 4.5b Comparison of Test Results and EC3:1.2 Approach at 300°C 74
Figure 4.5c Comparison of Test Results, FEA and EC3:1.2 Approach at 400°C 74
Figure 4.5d Comparison of Test Results, FEA and EC3:1.2 Approach at 500°C 74
Figure 4.5e Comparison of Test Results, FEA and EC3:1.2 Approach at 600°C 75
Figure 4.6 Different Rates of Degradation of Tangent Modulus 77
Figure 4.7 Variation of Non-Linearity Factor with Temperature 78
Figure 4.8 Comparisons of Various Approaches for S275 at 400°C 79
Figure 4.9 LTB Design Curves at Various Temperatures
using Rankine Approach 81
Figure 4.10 Different Types of Nodes at End Cross-Section 83
Figure 4.11 Lateral Torsional Buckling Failure of Numerical Model 85
Figure 4.12 Comparisons of Test Results with Various Approaches at 200°C 88
Figure 4.13 Comparisons of Test Results with Various Approaches at 300°C 88
Figure 4.14 Comparisons of Test & FE Results with Various
Approaches at 400°C 89
Figure 4.15 Comparisons of Test & FE Results with Various
Approaches at 500°C 89
Figure 4.16 Comparisons of Test & FE Results with Various
Approaches at 600°C 90
Figure 4.17 Comparisons of FE Results with Various Approaches at 700°C 90
Figure 4.18 Comparisons of FE Results with Various Approaches at 800°C 91
Figure 5.1a Simplified Substitute Member for Hogging Moment Region 96
Figure 5.1b Standard Beam Arrangement 97
Figure 5.2 General Layout of First Series of Test Specimens 98
Figure 5.3 Reference Gridlines of Test Specimens 100
Figure 5.4 Schematic Positions of LVDTs 102
Figure 5.5 General View of Test Set-Up 103
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LIST OF FIGURES
Figure 5.6 Plan View and Section View of Test Set-up 104
Figure 5.7 Front View and Side View of Test Set-Up 105
Figure 5.8 Adjustable Fork Support Systems 106
Figure 5.9 Mid-span Roller Systems 106
Figure 5.10 Lateral Restraint Systems 107
Figure 5.11 Positions of Longitudinal Geometric Imperfection Measurement 110
Figure 5.12 In-Plane Imperfection of S2-1 110
Figure 5.13 Out-of-Plane Imperfection of S2-1 110
Figure 5.14 Stress-Strain Relationship of Coupon Type A 112
Figure 5.15 Stress-Strain Relationship of Coupon Type T 112
Figure 5.16 Local Buckling in S3-2 and S4-1 Specimen 114
Figure 5.17 Lateral Torsional Buckling in S2-1 and S2-2 Specimen 115
Figure 5.18 Local and Lateral Torsional Buckling in S3-1 Specimen 115
Figure 5.19 Load-Rotation Responses of S2-1 and S2-2 Specimen 116
Figure 5.20 Moment-Rotation Response of S3 Series 117
Figure 5.21 Load-Deflection (mid-span) Response of S3 Series 118
Figure 5.22 Moment-Rotation Response of S1 Series, S3-2 and S3-3 119
Figure 5.23 Load-Deflection (mid-span) Response of S1 Series, S3-2 and S3-3 119
Figure 5.24 Moment-Rotation Response of S3-2, S3-3 and S4 Series 120
Figure 5.25 Load-Deflection (mid-span) Response of S3-2, S3-3 and S4 Series 121
Figure 5.26 Moment-Rotation Response of S1 and S2 Series 122
Figure 5.27 Load-Deflection (mid-span) Response of S1 and S2 Series 122
Figure 5.28 Layout of C1 Specimen 125
Figure 5.29 Composite Beam before Testing 126
Figure 5.30 Stress-Strain Relationship of Flange Plate 129
Figure 5.31 Stress-Strain Relationship of Web Plate 130
Figure 5.32 Stress-Strain Relationship of Reinforcement 130
Figure 5.33 Configuration of Push-Out Test Specimen and Test Set-Up 132
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LIST OF FIGURES
Figure 5.35 Temperature Developments and Distribution of C1 135
Figure 5.36 Temperature Developments and Distribution of C2 136
Figure 5.37 Temperature Developments and Distribution of C3 137
Figure 5.38 Temperature Developments and Distribution of C4 138
Figure 5.39 Temperature Distributions of CB3 and PB2
during Cardington Fire Test 139
Figure 5.40 Moment-Rotation Response of C1 141
Figure 5.41 Moment-Rotation Response of C2 141
Figure 5.42 Moment-Rotation Response of C3 142
Figure 5.43 Moment-Rotation Response of C4 142
Figure 5.44 Local Buckling Failures of Second Series of Specimens 143
Figure 5.45 Mid-Span Concrete Cracking 144
Figure 5.46 Load-Rotation Response of C4 144
Figure 5.47 Proposed Temperature Distributions for Analysis 147
Figure 5.48 Comparisons of Cardington and Current Test Failure Modes 147
Figure 5.49 Moment-Rotation Comparisons of S1-2 and C1 149
Figure 6.1 Boundary Conditions at Both Supports 152
Figure 6.2 Typical Finite Element Mesh of a Steel Beam 152
Figure 6.3 Moment-Rotation Comparisons of FEA and S3-1 Test Results 154
Figure 6.4 Moment-Rotation Comparison of FEA and
Tests by Lukey & Adams (1969) 154
Figure 6.5 Failure Modes Comparison of FEA and Test 155
Figure 6.6 Moment-Rotation Comparisons of FEA and Tests
at Elevated Temperature 158
at Elevated Temperature 158
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LIST OF FIGURES
at Elevated Temperature 159
Moment-Rotational Relationship 161
Figure 6.10 Effect of Temperature on Moment-Rotational Relationship (RB2) 163
Figure 6.11 Effect of Temperature on Moment-Rotational Relationship (RB4) 163
Figure 6.12 Variation of Rotational Capacity with Temperature 165
Figure 6.13 Variation of Stress-Strain Relationship Parameter kE/ky
with Temperature 165
Moment-Rotational Relationship 167
Moment-Rotational Relationship 168
Figure 6.16 Effect of Effective Length on Moment-Rotational Relationship 170
Figure 6.17 Comparisons of Failure Mode with Different Effective Length 170
Figure 6.18 Effect of Steel Grade on Moment-Rotational Relationship 171
Figure 6.19 Moment-Node Displacement Plot of RB1-S235 and RB1-S450 172
Figure 6.20 Typical Finite Element Mesh of a Composite Beam 174
Figure 6.21 Moment-Rotation Comparisons of Composite Beam
FEA and Tests 177
Figure 6.22 Failure Modes Comparison between FEA and Tests 178
Figure 6.23 Comparison of Crack Pattern between FEA and Tests 178
Figure 6.24 Temperature Distributions for Parametric Studies 179
Figure 6.25 Influence of Temperature on the Moment-Rotation Response 180
Figure 6.26 Comparisons of Failure Modes at Different Temperatures 181
Figure 6.27 Influence of Flange Slenderness on
the Moment-Rotation Response 182
Figure 6.28 Influence of Web Slenderness on the Moment-Rotation Response 183
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LIST OF FIGURES
the Moment-Rotation Response 184
Figure 6.30 Influence of Number of Studs on the Moment-Rotation Response 185
Figure 6.31 Influence of Effective Length on the Moment-Rotation Response 186
Figure 6.32 Influence of Effective Length on the Failure Modes 186
Figure 6.33 Moment-Rotation Response of RB2 at Various Temperatures 188
Figure 6.34 Temperature-Rotation Response of RB2 at
Various Applied Moments 188
Figure 6.35 Support’s Rotational Response of RB2 during Fire 189
Figure 7.1 Longitudinal and Cross-Section Discretization 194
Figure 7.2 Statistical Plots from the Regression Model 197
Figure 7.3 Moment-Rotation Design Model for Steel Beams
at Elevated Temperature 199
Figure 7.4 Validation of Steel Beam Design Model at Elevated Temperature 200
Figure 7.5 Post-Critical Curve of Plastic Collapse Mechanism 201
Figure 7.6 Illustration of Plastic Collapse Mechanism 201
Figure 7.7 Various Plastic Collapse Mechanism Models
(Gioncu & Mazzolani, 2002) 203
Figure 7.8 Plastic Collapse Mechanism Model (Gioncu & Petcu, 2001) 204
Figure 7.9 PCM based Moment-Rotation Model (S3-2) 211
Figure 7.10 Validation of Proposed Plastic Collapse Mechanism Model 212
Figure 7.11 Comparisons of FEA and PCM Predictions 213
Figure 7.12 Plot of Ratio of Web to Flange Slenderness-PCM Predictions 214
Figure 7.13 Plot of Web Slenderness-PCM Predictions 215
Figure 7.14 FE Results and PCM Predictions at Various
Flange Slenderness Ratios 216
within Parameter Limits 216
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