Collapse Performance Assessment of Steel-Framed Buildings Under Fires

Department of Civil and Environmental Engineering Stanford University

Report No.

The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus.

Address:

The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020

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ii ABSTRACT

ABSTRACT

The main objective of this research is to investigate the collapse performance of steel-framed buildings under fires and to contribute to the development of methods and tools for performance-based structural fire engineering. This research approach employs detailed finite element simulations to assess the strength of individual members (beams and columns) and indeterminate structural sub-assemblies (beams, columns, connections and floor diaphragms). One specific focus of the investigation is to assess the accuracy of beam and column strength design equations of the American Institute of Steel Construction (AISC) Specification for Structural Steel Buildings. The simulation results show these design equations to be up to 60 % unconservative for columns and 80-100 % unconservative for laterally unbraced beams. Alternative equations are proposed that more accurately capture the effects of strength and stiffness degradation at elevated temperatures. About eight hundred simulations are performed to verify the proposed equations, accompanied by studies on members with different steel strengths and section sizes. The assessment technique for individual members is then extended to examine fire effects for indeterminate gravity frame systems, including forces induced by restraint to thermal expansion and nonlinear force redistribution due to yielding and large deformations. Structural sub-assemblies are devised to examine indeterminate effects of gravity-framing in a 10-story building, which is representative of design and detailing practice in the United States. Three types of sub-assemblies are considered, including an interior gravity column, a composite floor beam, and an exterior column-beam assembly. The sub-assembly models include the restraining effects of floor framing that surrounds (both horizontally and vertically) the localized compartment fire. The sub-assembly simulations support the following observations and conclusions: (1) the rotational end restraint provided by the columns above and below the fire story have a significant stabilizing effect on gravity columns in the fire zone (providing up to a 40 % increase in strength above the pin-ended condition at 400 °C), (2) vertical restraint of the heated column, by floor framing above the fire story, does not significantly impact the strength limit state of the columns in the fire zone (3) short of designing the building system with special redundant load paths, thermal

iv ABSTRACT insulation is essential to avoid progressive collapse of highly-stressed gravity columns during building fires (4) thermal insulation requirements for beams can be reduced while preserving collapse resistance through enhanced connection details that are insulated, employ slotted holes to permit thermal elongation, and incorporate thermally protected reinforcing bars in the slab. These studies and conclusions are limited to evaluation of collapse safety and do not address aspects related to post-fire repairs and loss assessment. Uncertainty in the collapse behavior under fires is evaluated considering variability in the gravity loading and structural response parameters. Using the statistical information to quantify the random variables, the collapse probability of the column, beam and beam- column sub-assemblies is assessed by the mean-value first-order second-moment (FOSM) method. The collapse probability is conditioned with respect to the scaled intensity of fire compartment gas temperature, which is treated as independent variable. These studies indicate that the variability in the high-temperature steel yield strength is the most significant factor in the uncertainty assessment. The studies further show that for the design fire temperature, the probability of column failure ranges from 4 % to 38 % (β = 0.3-1.8) for designs based on the AISC strength provisions (with φ = 0.9). These probabilities reduce to 0.5 % to 3 % (β = 1.9-2.6) based on the proposed equations (with φ = 0.9).

v ACKNOWLEDGEMENTS

This work was funded by the Fulbright graduate student fellowship and the John A. Blume Earthquake Engineering Center.

This report was originally published as the Ph.D. dissertation of the first author.

The authors would like to thank Professors Sarah Billington, Helmut Krawinkler, Jack Baker and Eduardo Miranda, for their advice on this research.

The authors gratefully acknowledge Dr. Liang Yu and Professor Karl H. Frank at the University of Texas at Austin provided the essential test data of high strength bolts under elevated temperatures. Professor Emeritus Brady R. Williamson provided priceless research papers and reports. Dr. Barbara Lane and Dr. Susan Lamont helped the heat transfer simulation with their expertise. Scott Hamilton worked on risk assessment and framework of structural fire engineering with Professor Deierlein. Professor Paulo Vila Real at University of Aveiro in Portugal kindly provided the most recent draft of Eurocode. Professor Richard Liew at the National University of Singapore also provided his research papers and proceedings of past fire workshops. Dr. Ryoichi Kanno at Nippon Steel kindly arranged the use of the test data of steel at elevated temperatures performed by the Japan Iron and Steel Federation. Corus (British Steel) Swinden Laboratories provided their test data on steel beams at elevated temperatures. Karen Greig, Head Librarian at Engineering Library at Stanford University, obtained papers regarding structural fire engineering.

vi TABLE OF CONTENTS

Chapter 1 Introduction 1 1.1 Overview 1 1.1.1 Background and Focus of This Research 1 1.1.2 Performance-Based Fire Engineering 2 1.1.3 Role of Structural Fire Engineering 3 1.1.4 Behavior of Steel Structures Exposed to Fire 4 1.1.5 Domains for Limit-state Evaluation 5 1.1.6 Disaster of the World Trade Center 6 1.1.7 Uncertainties in Structural Fire Engineering 6 1.2 Objectives 7 1.3 Scope 8 1.4 Organization 9

Chapter 2 Overview of Steel Structures Exposed to Fire 11 2.1 Past Fire Disasters 11 2.1.1 Fires on Steel Structures 11 2.1.2 Broadgate Phase 8 13 2.1.3 One Meridian Plaza 15 2.1.4 World Trade Center Building 7 16 2.1.5 Windsor Building 20 2.1.6 Cardington Fire Test 23 2.1.7 Summary of Past Fire Disaster Review 25 2.2 Mechanical Properties of Steel under Elevated Temperatures 25 2.2.1 Experimental Results 25 2.2.1.1 Experiments by Harmathy and Stanzak 26 2.2.1.2 Experiment by Skinner 28 2.2.1.3 Experiments by DeFalco 29 2.2.1.4 Experiments by Fujimoto et al. 32 2.2.1.5 Experiments by Kirby and Preston 33

vii TABLE OF CONTENTS

2.2.1.6 Comparison of the Experiments 35 2.2.2 Equations of Stress-strain Curves 38 2.2.2.1 Eurocode Stress-strain Curves 38 2.2.2.2 AS4100 Stress-strain Curves 40 2.2.2.3 AIJ Stress-strain Curves 41 2.2.2.4 AISC Stress-strain Curves 44 2.2.2.5 Comparison of the Equations of Stress-strain Curves 45 2.2.3 Experiments by JISF 49

Chapter 3 Analysis of Individual Members 51 3.1 Summary 51 3.2 Introduction 51 3.3 Basis of Member Strength Evaluations 53 3.3.1 Steel Properties under Elevated Temperatures 55 3.4 Finite Element Simulation Model 57 3.5 Column Strength Assessment 61 3.5.1 AISC Column Strength Equations 62 3.5.2 EC3 Column Strength Equations 62 3.5.3 Assessment of Column Strengths 64 3.5.4 Proposed Column Strength Equations 67 3.5.5 Column Test Data 68 3.5.6 Influence of Yield Strength and Section Geometry 69 3.6 Beam Strength Assessment 70 3.6.1 AISC Beam Strength Equations 70 3.6.2 EC3 Beam Strength Equations 72 3.6.3 Proposed Beam Strength Equations 73 3.6.4 Assessment of Beam Strengths 74 3.7 Beam-Column Strength Assessment 78 3.7.1 AISC Beam-Column Strength Equations 79 3.7.2 Proposed Beam-Column Strength Equations 80 3.7.3 EC3 Beam-Column Strength Equations 80 3.7.4 Assessment of Beam-Column Strengths 80 3.8 Summary and Conclusions 83

viii TABLE OF CONTENTS

3.9 Limitations and Future Research 84

Chapter 4 Analysis of Gravity Frames 87 4.1 General 87 4.1.1 Overview 87 4.1.2 Benchmark Office-type Building Design 88 4.1.3 Failure Mechanisms and Sub-assembly Analysis Models 90 4.1.4 Time-temperature Relationships in Localized Fire 92 4.1.5 Organization of Chapter 4 93 4.2 Evaluation of Interior Column Sub-assembly 93 4.2.1 Summary 93 4.2.2 Introduction 93 4.2.3 Analysis Model 96 4.2.3.1 Modeling of System 96 4.2.3.2 Modeling of Column 98 4.2.3.3 Modeling of Constraint Springs 101 4.2.4 Evaluation of Critical Temperatures 106 4.2.5 Comparison between Design Equations and Sub-assembly Simulations 111 4.2.6 Improvement of Structural Robustness 112 4.2.7 Conclusions 114 4.3 Evaluation of Beam Sub-assembly 115 4.3.1 Summary 115 4.3.2 Introduction 115 4.3.3 Analysis Model 116 4.3.3.1 Modeling of System 116 4.3.3.2 Modeling of Steel Beam 117 4.3.3.3 Modeling of Concrete Slab 118 4.3.3.4 Modeling of Bolted Connection 119 4.3.3.5 Modeling of Longitudinal Constraint by Floor Framing 126 4.3.4 Evaluation of Behavior and Limit-state 128 4.3.4.1 Performance of Typical Design 128 4.3.4.2 Performance of Alternative Design 130 4.3.4.3 Effect of Longitudinal Constraint 133

ix TABLE OF CONTENTS

4.3.5 Conclusions 134 4.4 Evaluation of Exterior Column Sub-assembly 135 4.4.1 Overview 135 4.4.2 Analysis Model 136 4.4.2.1 Modeling of System 136 4.4.2.2 Modeling of Bolted Connection 139 4.4.3 Evaluation of Behavior and Limit-state 142 4.4.3.1 Basis of Simulations 142 4.4.3.2 Simulation Results 142 4.4.3.3 Alternative Connection Design 144 4.4.4 Conclusions 145 4.5 Overall Limit-state Evaluation 146 4.6 Conclusions of Gravity Frame Analysis 147

Chapter 5 Probabilistic Assessment 149 5.1 Overview 149 5.2 Structural Uncertainties in Fire Engineering 149 5.2.1 Summary of Statistical Data 149 5.2.2 Variability of Yield Strength of Steel 151 5.2.3 Variability of Longitudinal Spring Stiffness for Interior Column 153 5.2.4 Variability of Shear Strength of Bolts 156 5.2.5 Variability of Longitudinal Strength of Springs for Bolted Connections 160 5.2.6 Variability of Deformation Capacity of Bolted Connections 160 5.2.7 Variability of Time-temperature Relationships in Compartment Fire 162 5.3 Probabilistic Studies 165 5.3.1 Sensitivity of Critical Temperatures to Uncertainties 165 5.3.1.1 Sensitivities in Interior Column Sub-assembly Study 166 5.3.1.2 Sensitivities in Beam Sub-assembly Study 167 5.3.1.3 Sensitivities in Exterior Column Sub-assembly Study 168 5.3.2 Collapse Probabilities of Sub-assemblies given Temperatures 170 5.3.3 Reliability of AISC-LRFD Fire Equation 172 5.3.4 Conclusions 178

x TABLE OF CONTENTS

Chapter 6 Conclusions 181 6.1 General 181 6.2 Summary 182 6.2.1 Steel Properties at Elevated Temperatures 182 6.2.2 Past Fire Disasters 183 6.2.3 Member-based Strength Study 183 6.2.4 Benchmark Building Study 184 6.2.5 Probabilistic Studies 184 6.3 Major Findings and Conclusions 185 6.3.1 AISC Member-based Design Criteria 185 6.3.2 Effect of Residual Stress and Local Buckling 185 6.3.3 Proposed Design Criteria for AISC 186 6.3.4 Steel-framed Building under Localized Fire 186 6.3.5 Longitudinal Constraint of Interior Column 187 6.3.6 Longitudinal Constraint of Beam 188 6.3.7 Properties of Bolted Connections 188 6.3.8 Evaluation of Structural Uncertainties 189 6.3.9 Probabilistic Studies 189 6.4 Design and Analytical Modeling Recommendations 190 6.4.1 Design Recommendations 190 6.4.2 Analytical Modeling Recommendations 191 6.5 Future Work 192 6.5.1 Member-based Strength Evaluation 192 6.5.2 Performance Evaluation of Steel Buildings under Fires 192

Appendix A Supplemental Studies on Individual Members 195 A.1 Tangent Modulus Theory 195 A.1.1 Flexural Buckling 195 A.1.2 Lateral Torsional Buckling 200 A.2 Modeling Comparison of Individual Members 204 A.2.1 Fiber Model 204 A.2.2 Effect of Local Buckling 205 A.2.3 Post Buckling Strength 206

xi TABLE OF CONTENTS

A.3 Effect of Uncertain Conditions 209 A.3.1 Overview 209 A.3.2 Non-uniform Temperature Distribution 209 A.3.3 Imperfections 212 A.3.4 Boundary Conditions 213 A.3.5 Steel Properties 216 A.4 Other Miscellaneous Studies 218 A.4.1 Temperature Distribution of Composite Beams 218 A.4.2 Modeling Comparison of Composite Beam 220 A.4.3 Effect of Heat Conduction 222

Appendix B Reference Equations 225 B.1 Conversion of Units 225 B.2 Symbols 226 B.3 Design Equations of Steel at Elevated Temperatures 228 B.3.1 Eurocode 3 228 B.3.2 AS4100 231 B.4 Time-temperature Relationships 231 B.4.1 Parametric Fire Curve 231 B.4.2 Step-by-step Steel Temperature Simulation 236 B.5 FOSM 238

Appendix C JISF Experiment 241 C.1 Summary 241 C.2 Data Conditions 241 C.3 General 241 C.4 JISF Stress-strain Curves 242 C.5 Comparison of the Test Data with AIJ and EC3 250 C.6 Statistical Study 254

Bibliography 259

Symbols 267

xii LIST OF TABLES

Chapter 2 Overview of Steel Structures Exposed to Fire Table 2-1 Past major fire disasters of steel buildings 12 Table 2-2 Chemical composition of steels by Harmathy and Stanzak (Wt%) 26 Table 2-3 Chemical composition of steel by Skinner (Wt%) 28 Table 2-4 Chemical composition of steel by DeFalco (Wt%) 32 Table 2-5 Chemical composition of steel by Fujimoto et al. (Wt%) 32 Table 2-6 Chemical composition of steel by Kirby and Preston (Wt%) 34 Table 2-7 Comparison of steel experiments at elevated temperatures 36 Table 2-8 Coefficients in AIJ equations for stress-strain curves 43

Chapter 3 Analysis of Individual Members Table 3-1 Stress-strain reduction factors in EC3 57 Table 3-2 Steel section data 64 Table 3-3 Measured and calculated strengths of column tests 68

Chapter 4 Analysis of Gravity Frames Table 4-1 Section sizes (mm) 90 Table 4-2 Section sizes of columns in 5- and 20-story buildings (mm) 110 Table 4-3 Critical temperatures with different number of building stories 110 Table 4-4 Values of reduction factor of bolt strength 124 Table 4-5 Comparison of the critical temperatures 132 Table 4-6 Effect of the constraint stiffness to the critical temperatures 134 Table 4-7 Limit-states for exterior column sub-assembly model 144 Table 4-8 Critical time, steel temperatures and failure mechanisms of sub-assemblies 146

Chapter 5 Probabilistic Assessment Table 5-1 Statistical data for uncertainties 150 Table 5-2 Symbols regarding statistical properties of steel strength 152 Table 5-3 Ratios of upper and lower bounds of factors for the vertical spring 155

xiii LIST OF TABLES

Table 5-4 Combinations of factors for vertical spring of interior column 156 Table 5-5 Mean and c.o.v. of shear strength of bolts 160 Table 5-6 Band of influential factors for fire simulation 163 Table 5-7 Maximum temperatures in variation of fire simulation (°C) 165 Table 5-8 Critical temperature with various constraint stiffness 168 Table 5-9 Variability of the collapse probability with respect to gas temperature 172 Table 5-10 Comparison of expected, nominal, and factored load (N/m2) 173

Appendix A Supplemental Studies on Individual Members Table A 1 Combinations of non-uniform temperature distributions 210 Table A 2 Section sizes of beam tested by Wainman and Kirby (mm) 219 Table A 3 Thermal properties of steel column 223

Appendix B Reference Equations Table B-1 Conversion of temperature units 225 Table B-2 Conversion of length and force units 225 Table B-3 Conversion of pressure units 226 Table B-4 Symbols in AISC and Eurocode 226 Table B-5 Parameters and conditions for parametric fire curves 234 Table B-6 Thermal properties of steel and fire insulation 237

Appendix C JISF Experiment Table C-1 Number of tests for each steel type and temperatures 243 Table C-2 Elastic modulus and yield strength (Gr.50) defined in AIJ, EC3, and AISC 250 Table C-3 Mean and coefficient of variation of 1 % and 2 % strength 256

xiv LIST OF FIGURES

Chapter 1 Introduction Figure 1-1 Assessment strategy 8

Chapter 2 Overview of Steel Structures Exposed to Fire Figure 2-1 Photos of Broadgate fire 14 Figure 2-2 Photos of One Meridian Plaza fire 16 Figure 2-3 Floor plan and damages of WTC 7 17 Figure 2-4 Fires observed from the east and north face of WTC 7 18 Figure 2-5 Probable global collapse mechanism of WTC 7 19 Figure 2-6 Exterior view of Windsor Building before and after the fire 21 Figure 2-7 Detailed photos of Windsor Building fire 22 Figure 2-8 Photos of the Cardington Fire Test 24 Figure 2-9 Floor framing and test locations 24 Figure 2-10 Stress-strain curves by Harmathy and Stanzak (1970) 27 Figure 2-11 Stress-strain curves by Skinner (1972) 29 Figure 2-12 Stress-strain curves by DeFalco (1974) 31 Figure 2-13 Stress-strain curves by Fujimoto et al. (1980, 81) 33 Figure 2-14 Stress-strain curves by Kirby and Preston 35 Figure 2-15 Comparison of stress-strain curves in experiments 37 Figure 2-16 Stress-strain curves defined by EC3 39 Figure 2-17 Stress-strain curves defined by AS4100 41 Figure 2-18 Stress-strain curves defined by AIJ 42 Figure 2-19 Reduction ratios in the stress-strain curves defined by AIJ 44 Figure 2-20 Stress-strain curves defined by AISC 45 Figure 2-21 Comparison of stress-strain curves (equations and experiments) 47 Figure 2-22 Eurocode stress-strain curve (500 °C) with hardening at large strains 48 Figure 2-23 Comparison of normalized stress-strain curves (equations and experiments) 49 Figure 2-24 Comparison of stress-strain curves by JISF, EC3, and AIJ 50

xv LIST OF FIGURES

Chapter 3 Analysis of Individual Members Figure 3-1 Comparison of temperature and load control analyses 54 Figure 3-2 Stress-strain response at high temperatures as defined by EC3 56 Figure 3-3 Shell finite element mesh and boundary conditions 59 Figure 3-4 Load versus displacement response from FEM simulations under ambient and elevated temperatures 59 Figure 3-5 Influence of residual stresses (W14×90 Gr. 50 column at 500 °C) 60 Figure 3-6 Critical compressive strengths of W14×90 Gr.50 column 66 Figure 3-7 Percentage error in the calculated compression strength of W14×90 Gr.50 column at 500 °C 67 Figure 3-8 Comparative assessment of column compression strength at 500 °C 69 Figure 3-9 Critical bending moment strengths of W14×22 Gr.50 beam 76 Figure 3-10 Percentage error in the calculated bending moment strength of W14×22 Gr.50 beam at 500 °C 76 Figure 3-11 Comparative assessment of beam bending moment strength at 500 °C 77 Figure 3-12 Critical axial load and moment strengths of W14×90 Gr.50 (λ=60) beam-column 82 Figure 3-13 Comparative assessment of beam-column strengths at 500 °C 83

Chapter 4 Analysis of Gravity Frames Figure 4-1 Floor plan of benchmark building design 88 Figure 4-2 Details of column-beam shear tab connections 89 Figure 4-3 Possible failure mechanisms (column line 3) 91 Figure 4-4 Sub-assembly analysis models 91 Figure 4-5 Time-temperature relationships in a fire simulation 92 Figure 4-6 Analysis model of a column with constraint springs 95 Figure 4-7 Analysis model for column buckling collapse mechanism 97 Figure 4-8 Preliminary model for interior column 99 Figure 4-9 Axial load carrying capacity of the interior column at elevated temperatures 99 Figure 4-10 Comparison of column strength with different models 100 Figure 4-11 Post buckling deformation of shell element model 101 Figure 4-12 Analysis model of beams for vertical spring stiffness of floor structure 102 Figure 4-13 Rotational properties of shear-tab connections 103 Figure 4-14 Analysis model of floor structure for in-plane stiffness calculation 104

xvi LIST OF FIGURES

Figure 4-15 Longitudinal constraint stiffness of beams 105 Figure 4-16 Vertical resistance of floor structure 106 Figure 4-17 Resistance of the system under elevated temperatures 108 Figure 4-18 Vertical displacement of the interior column under elevated temperatures 109 Figure 4-19 Critical axial strength of W14×90 (4 m) based on member-based and sub-assembly approaches 111 Figure 4-20 Options for strengthened connections 112 Figure 4-21 Total vertical load carrying capacity with strengthened connection for Beam a and b 113 Figure 4-22 Vertical displacement of the buckled column with improved beam connection 113 Figure 4-23 System of finite element composite beam model in floor framing 117 Figure 4-24 Temperature distribution of composite section 118 Figure 4-25 Compressive stress-strain curve of concrete 119 Figure 4-26 Gravity load supporting systems of beams at elevated temperatures 119 Figure 4-27 Detail of beam connection 120 Figure 4-28 Single shear bolt test by Yu (2006) 120 Figure 4-29 Load-displacement relationships of single shear connections by Yu (2006) 121 Figure 4-30 Maximum single shear strength of A325 bolts by Yu (2006) 122 Figure 4-31 Force-displacement relationship model of bolted connection 123 Figure 4-32 Reduction factor of bolt strength by ECCS 124 Figure 4-33 Comparison of force-displacement relationships of bolted connection between analysis model and test data by Yu (2006) 126 Figure 4-34 Analysis model for constraint stiffness 127 Figure 4-35 Mid-span displacement and modeling comparison 128 Figure 4-36 Post peak-strength evaluation of bolted connection 129 Figure 4-37 Proposed design options for bolted connections 130 Figure 4-38 Performance of composite beams with alternative design options for the connections 131 Figure 4-39 Alternative connection detail of secondary beams to prevent shear failure at elevated temperatures 132 Figure 4-40 Influence of the longitudinal constraint stiffness 133 Figure 4-41 Failure mechanisms simulated with exterior column sub-assembly 136 Figure 4-42 System of exterior column sub-assembly model 138

xvii LIST OF FIGURES

Figure 4-43 Lateral constraint by floor slab with membrane action 138 Figure 4-44 Comparison of compartment fire for exterior column sub-assembly simulations 139 Figure 4-45 Detail of exterior column connection 140 Figure 4-46 Comparison between analysis model and test data by Yu (2006) of longitudinal force-displacement relationships of beam-column bolted connection 141 Figure 4-47 Time-temperature relationships in a fire simulation 142 Figure 4-48 Displacement for the exterior column sub-assembly model 143 Figure 4-49 Connection details between external column and beam 145

Chapter 5 Probabilistic Assessment Figure 5-1 Variation of tested steel strength under elevated temperatures 153 Figure 5-2 Variation of vertical spring properties 156 Figure 5-3 Shear strength of bolts at elevated temperatures 157 Figure 5-4 Shear strength of bolts normalized with ECCS strength 159 Figure 5-5 Uncertainty of deformation capacity of bolted connection 162 Figure 5-6 Variations of time-temperature relationships 164 Figure 5-7 Sensitivity of the critical temperature of interior column sub-assembly 166 Figure 5-8 Sensitivity of the critical temperature of beam sub-assembly 167 Figure 5-9 Sensitivity of the critical beam temperature of exterior column sub- assembly 169 Figure 5-10 Time-temperature relationships in a fire simulation 171 Figure 5-11 Collapse probability of sub-assemblies 171 Figure 5-12 Load and resistance relationships in AISC-LRFD 173 Figure 5-13 Probability of failure of W14×90 column at 500 °C with varied length 175 Figure 5-14 φ factors for 0.47 % (β = 2.6) probability of failure of W14×90 column at 500 °C with varied length 175 Figure 5-15 Probability of failure of W14×90 (L = 4 m) column with varied temperatures 176 Figure 5-16 φ factors for 0.47 % (β= 2.6) probability of failure of W14×90 (L = 4 m) column with varied temperatures 177

xviii LIST OF FIGURES

Appendix A Supplemental Studies on Individual Members Figure A-1 Strain level and residual stress 197 Figure A-2 Critical strength of columns by tangent modulus theory 198 Figure A-3 Stress-strain curves with the average tangent stiffness in section 199 Figure A-4 The critical strength of W14×90 column 200 Figure A-5 Lumped fiber model 201 Figure A-6 Critical moment by tangent modulus theory 203 Figure A-7 Comparison of the critical moment by analyses and tangent modulus theory 204 Figure A-8 Integration points in fiber model section 205 Figure A-9 Effect of imperfection for local buckling 206 Figure A-10 Post buckling strength (W14×90, Gr.50, L=4m) 207 Figure A-11 Post-buckling behavior for LTB 208 Figure A-12 Non-uniform temperature distribution modes 210 Figure A-13 Sensitivity of critical strength to non-uniform temperature distribution 211 Figure A-14 Sensitivity of critical strength to non-uniform temperature distribution for the weak axis 212 Figure A-15 Sensitivity of critical strength to imperfections 213 Figure A-16 Sensitivity of critical strength to boundary conditions at 500 °C 215 Figure A-17 Sensitivity of critical strength to boundary conditions at 20 °C 216 Figure A-18 Sensitivity of critical strength to steel properties at 500 °C 217 Figure A-19 Sensitivity of critical strength to steel properties at 20 °C 218 Figure A-20 Temperature distribution of composite section 218 Figure A-21 Beam experiment by Wainman and Kirby (1988) 219 Figure A-22 Recorded temperatures in three sections by Wainman and Kirby (1988) 220 Figure A-23 Comparison between analysis and test by Wainman and Kirby (1988) 221 Figure A-24 Study model for heat conduction 222 Figure A-25 Temperature increase by heat conduction 224

Appendix B Reference Equations Figure B-1 Section axes in AISC and Eurocode 226

xix LIST OF FIGURES

Appendix C JISF Experiment Figure C-1 JISF stress-strain curves (SM490A Plate) 244 Figure C-2 JISF stress-strain curves (SM490A Wide Flange) 245 Figure C-3 JISF stress-strain curves (SM490AW Plate) 246 Figure C-4 JISF stress-strain curves (SN490C Plate) 247 Figure C-5 JISF stress-strain curves (STKN400B) 248 Figure C-6 JISF stress-strain curves (STKN490B) 249 Figure C-7 Comparison of stress-strain curves (up to 2.5 % strain) 252 Figure C-8 Comparison of stress-strain curves (up to 10 % strain) 253 Figure C-9 Comparison of stresses at 1 % strain 255 Figure C-10 Comparison of stresses at 2 % strain 255 Figure C-11 JISF paper (page 1) 257 Figure C-12 JISF paper (page 2) 258

CHAPTER 1 INTRODUCTION

1 INTRODUCTION

1.1 OVERVIEW

1.1.1 Background and Focus of This Research

Traditional building-code design provisions for fire resistance in steel-framed buildings are highly prescriptive and empirically based. As a result, structural engineers have both limited means and opportunities to devise, assess and implement alternative solutions for fire resistance that may be more cost-effective than conventional solutions. So-called performance-based approaches seek to change this by offering more transparent and scientifically-based methods to assess impact of fires on buildings. Performance-based fire engineering and design encompasses a broad range of expertise and considerations, which span far beyond the discipline of structural engineering. Performance-based fire engineering and design has received considerable attention in recent years, as evidenced by major specialty conferences (e.g., SFPE 2004), books (e.g., SFPE 2007, Custer and Meacham 1997) and many published papers. For example, in related research at Stanford University, Hamilton and Deierlein (2004) have explored the parallels between performance-based approaches for structural design to resist fire and earthquakes. This research is intended to contribute to one aspect of performance-based structural fire engineering involving the development of models and criteria to assess the collapse performance of steel-framed structures at elevated temperatures. The specific focus is on evaluating the strength limit state of gravity framing systems, which are likely to be the most vulnerable components of steel-framed buildings subjected to fire. The research employs the development, calibration and application of detailed nonlinear analyses to investigate the strength limit states of individual steel members and sub-assemblies of members subjected to combined gravity loads and elevated temperatures. In addition to assessing the response of

1 CHAPTER 1. INTRODUCTION conventional steel building details, this research also examines alternative structural design and details to improve collapse safety.

1.1.2 Performance-Based Fire Engineering

In the progression from prescriptive design toward performance-based design, structural engineers are taking more responsibility for assessing structural performance and relating its implications to key stakeholders, including building owners, building code-officials, and society at large. Performance-based approaches allow more flexibility in the structural design, since they relieve engineers of the mandate to follow prescriptive design requirements. Specifically, performance-based approaches may relieve the prescriptive design provisions that require specific thermal insulation on steel members to limit steel temperatures during fires. Past experience shows that this fire insulation works well; however, the prescribed insulation requirements usually do not distinguish between alternative fire exposures and differences in structural behavior for different buildings. For example, the standard fire curve, which is the time-temperature relationship commonly used for evaluating the fire resistance of materials, does not represent actual flashover fire characteristics in steel-framed buildings. Rather, it is intended for qualification testing of structural components and insulating materials, where the limit state criteria does not necessarily relate to real behavior in buildings. While such approaches were a practical necessity when computer analysis technology was less developed and structural simulation under fire conditions was difficult, the over-reliance on empirical testing using the standard fire curve has become an obstacle to more thoughtful and case-specific design. In contrast, a more rational and scaleable framework for design should enable the use of simulation methods to assess structural response to fire, including evaluation of the inherent uncertainties in the fire and its effects on the structure (e.g., Hamilton and Deierlein 2004). Such an approach in structural fire engineering (or performance-based fire engineering, PBFE) makes it possible to develop a fire-resistant structural design by explicitly evaluating the behavior of the buildings under fires. This approach is especially useful for buildings that are not addressed well by prescriptive approaches, such as high-rise buildings or buildings with unique functions and/or configurations. For such buildings, PBFE can enable one to simulate explicitly the structural behavior under fires, and then determine the required thermal insulation (or other protective measures) to ensure that the building has the desired level of performance. It is reported that conventional fire insulation can add up to 30 % to

2 CHAPTER 1. INTRODUCTION the construction cost for steel building frames (Lawson, 2001). Thus, there is a potentially significant economic motivation to design buildings using PBFE rather than common prescriptive requirements. Depending on the design philosophy and goals, financial benefit- cost analyses may show that it is more cost-effective to allow certain levels of structural damage in extreme and rare fires. Alternatively, more stringent requirements may be appropriate to further reduce the risk of structural collapse where it has significant implications on life safety. In order to use such a design philosophy, methods and criteria are needed to simulate realistically structural behavior under fires.

1.1.3 Role of Structural Fire Engineering

Overall, fire protection engineering involves many engineering fields such as materials, mechanical equipment, chemistry, human behavior, heat transfer, statistics, and structures. Each field has its unique relevance to fire safety, including efficient measures to control both the risk of fire ignition/growth and possible resulting impacts of fire. Approaches to control fire damage are generally categorized as either active or passive measures. Mechanical or human interventions are active measures, such as sprinklers, fire alarms, or detection systems. Passive measures are incorporated with built-in systems such as fire insulation on structural members or fire-rated room partitions, which create fire compartments that inhibit fire spread. Active measures are especially important for controlling the early stages of the fire, limiting fire growth, and reporting the fire to fighting personnel. Passive measures are important in the case that these active measures fail and the fire fully develops into a so called “flashover fire”. Passive measures are the main focus of structural fire engineering, though the performance requirements for passive systems may depend on active systems in the buildings. Simulations required to evaluate structural behavior under fires include (1) simulation of fire behavior, (2) simulation of heat transfer to the structure, and (3) simulation of structural behavior. The primary focus of structural fire engineering is to assess structural behavior. Structural temperatures can be simulated in some advanced analyses; however, simulation of the fire behavior itself is generally outside the scope of structural engineering. Interactions between these simulations are relatively limited and it is generally assumed that each component of the analysis (fire, thermal, and structural) can be performed independently. This is advantageous as it allows for structural behavior under fire to be simulated based on either peak temperature or time-temperature relationships in the structural members. Steel

3 CHAPTER 1. INTRODUCTION temperatures (either peak values or time-varying values) and be related to parametric fire curves using straightforward heat transfer analyses. While structural simulation is only a part of the overall process necessary to evaluate building safety against fire, research on structural behavior is important because structural collapse is potentially devastating. Depending on the circumstances, human, economic, and physical loss caused by a structural collapse can overwhelm the damage caused by the initial fire.

1.1.4 Behavior of Steel Structures Exposed to Fire

One of the ultimate goals of structural fire engineering is to simulate behavior and limit state under fires. As discussed further in the next section, the risks and safety of the structures under fires can be evaluated in terms of the alternative metrics of strength (load resistance), temperature or time. Whichever metric is used, the primary behavioral effect in structural assessment is the degradation in stiffness and strength of structural materials at high temperatures and the potential for localized structural failure to trigger global collapse. Thermal expansion is also a significant issue in structural fire engineering, in addition to the material deterioration at elevated temperatures. Effects of thermal expansion vary depending on the longitudinal constraint of heated members. Under elevated temperatures, longitudinal elongation is induced when the constraint is relatively low; while compressive axial force is induced when the constraint is high. There has been some debate whether or not thermal expansion is critical at the structural limit state, because thermally induced force tends to eventually decrease at this limit state with the deteriorated material under the elevated temperatures. These discussions are inconclusive and further study is needed. Three-dimensional (3D) effects are more significant for structural behavior under fires, as compared to other types of extreme loadings such as wind and earthquakes. This is because initially localized structural damage in fires spreads three dimensionally to the connecting members. Cast-in-place concrete slabs and composite beam slab systems are typically used in steel buildings. It is known that this composite effect significantly enhances performance of steel frames under fire conditions. Temperatures of the concrete slab under fire conditions are generally lower than steel members and the strength degradation of reinforced concrete is much less. Furthermore, concrete slab systems potentially have high load carrying capacity under large deformation, due to catenary action. However, evaluating this enhanced

4 CHAPTER 1. INTRODUCTION performance is difficult, because of the complexity of how the composite system behaves under large deformations. Specifically, simulating the behavior of shear stud connections and the interaction between concrete slabs and steel beams is difficult and further research is needed. The behavior of bolted and welded connections between members is also influential to overall frame behavior. Strength deterioration in connections is more severe than that of steel members, making it possible for connection failure to be critical under fire conditions. Also, large deformations of beams can induce significant tensile forces under catenary action, and the strength of typical shear-tab connections may not be large enough to support these forces.

1.1.5 Domains for Limit-state Evaluation

Evaluation of structural limit states under fire conditions can be performed in one of three domains: time, temperature, and strength. Evaluation of the structural limit state in the time domain is most closed associated with requirements for evacuation or fire fighting activities, which are calculated as a function fire development and suppression times. In the temperature domain, the collapse performance is evaluated in term of the critical temperatures in the steel members. This domain has the advantage of enabling the structural performance to be evaluated independent of fire growth behavior. Limit states calculated in the time and temperature domains can be directly converted once the relationship between the time and temperature during the fire is provided. Critical strength (i.e. maximum applied load level that the structure can carry) is calculated under a specified constant temperature in the strength domain. For a specified maximum temperature, the critical strength is calculated and compared to the applied gravity load assumed in the design. This approach is advantageous in terms of numerical analysis, since loads and displacements are common control variables used in structural analysis software. On the other hand, time and temperature can only be accounted for indirectly in analysis or by using specialized analysis software. Structural performance can be evaluated in either of these domains, and the domain should be properly selected to meet the purpose of the analytical simulation and performance evaluation.

5 CHAPTER 1. INTRODUCTION

1.1.6 Disaster of the World Trade Center

Since the terrorism attack and collapse of the World Trade Center buildings on September 11th of 2001, in New York City, behavior of steel buildings exposed to fire has been a popular topic of study and debate. Behavior of individual members and connections had been the focus of much of the research before the disaster, and there are still many research needs for element-based studies. However, the complete collapse of three major buildings (WTC towers 1 and 2 and the 47 story WTC 7 building) highlighted the importance of understanding the overall structural system performance. It is generally accepted that redundancy is desirable in structures; and this is especially true for structural fire design. This concept follows the “fail-safe” concept, which implies that a loss in the load carrying capacity of some members will not lead to global building collapse. Surrounding elements of the damaged structure should provide an alternative load carrying path. Therefore, redundancy can be provided by statically indeterminate structures; however, even highly indeterminate structures do not necessarily ensure the presence of alternative load carrying paths that can resist progressive collapse. Past discussions regarding redundancy have often remained abstract, and have rarely resulted in specific fire design recommendations.

1.1.7 Uncertainties in Structural Fire Engineering

Fires are similar to earthquakes, being rare events with high consequence. This characteristic makes uncertainty assessment a key subject of this research. There are many uncertain factors including fire occurrence and behavior in the overall fire risk assessment. From the structural fire engineering point of view, there are many uncertain aspects of the loads and strengths. Load and Resistance Factor Design (LRFD) is designed to deal with uncertainties and lead to a design with an acceptable probability of failure. The LRFD method for structural fire engineering is still developing, in part because the acceptable level of probability of failure under fires has not been explicitly defined. Development of fire hazard analysis models is especially needed for this purpose in addition to the development of structural analysis technology. Controlling the probability of failure is one of the most important goals of performance-based design. Since some of the statistical information regarding structural responses needed for uncertainty assessment is not readily available,

6 CHAPTER 1. INTRODUCTION

engineering assumptions or judgments are used in this research when appropriate to enable probabilistic assessment of failure of steel buildings under fire conditions.

1.2 OBJECTIVES

The objectives of this research are summarized in following points: (1) Synthesize and interpret current design specifications for structural fire engineering for steel buildings, and contribute to developing structural fire design methodologies based on performance-based design concepts. (2) Advance knowledge to systematically evaluate fire-induced collapse performance of steel framed buildings under fires. (3) Investigate the member-based strength criteria at elevated temperatures defined in the design specifications of American Institute of Steel Construction (AISC, 2005), and assess the accuracy of these provisions relative to the assessment of strength limit- states simulated with rigorous finite element analysis. Where appropriate, propose improved member-based strength design criteria, whose accuracy is validated by analytical simulations. (4) Assess performance of gravity framing in an archetypical steel-framed building under localized fire, and explore improved design concepts and details, including analytical validation. (5) Investigate variability and uncertainties in the important aspects in the structural performance evaluations under fires. Probabilistically assess member-based strengths and building performance. Use these findings to develop a basis for probabilistic risk assessments in structural fire engineering.

Meeting these objectives requires integration of past research to draw practical implications on design practice. Integration is necessary to cover various subjects of structural fire engineering, including analysis of members and frames, and simulations from fire behavior to structural failure. Knowledge from not only structural fire engineering, but also other fields such as earthquake engineering, will be integrated. Regarding practical significance, the directions of this research was selected to focus on topics that are expected to provide findings and conclusions that will be of practical use in the engineering profession.

7 CHAPTER 1. INTRODUCTION

The significance of frame analysis in structural fire engineering is to evaluate numerically possible alternative load carrying paths using rigorous analytical simulations. Showing processes and results of frame analysis based on research of individual members and details is greatly influential to practical structural fire design. In other words, this work is to evaluate concretely and objectively structural reliability and redundancy. The ultimate goal is to develop and apply rigorous analytical simulations to systematically evaluate the collapse limit-state for buildings of various framing configurations and fire scenarios.

1.3 SCOPE

The objectives described in the previous section are pursued using the approach shown in Figure 1-1, which shows research development from deterministic to probabilistic assessment, including the overlap of structural and fire simulations. The vertical axis in the deterministic assessment shows the flow of the structural performance assessment from member-based strength studies to performance analyses of frames. This shows not only the development of the analytical models, but also the flow from statically determinate to indeterminate structures. The horizontal axis shows the sequence of simulations from fire to structural behavior, which are carried out for a benchmark office-type building. These two axes are first studied deterministically. Then probabilistic risk assessment is introduced to examine and quantify the effects of uncertainty in the process.

Figure 1-1 Assessment strategy

8 CHAPTER 1. INTRODUCTION

Fire and structural simulations are studied for fully developed (flashover) fires. Post fire behavior, thermal transient effects, structural dynamic behavior, creep, and rate dependent effects are excluded from the scope of this research. Steel properties at elevated temperatures defined in Eurocode 3 (EC3, 1995) are evaluated based on available test data in Chapter 2, and are adopted for structural analyses. The critical strengths are calculated for individual members under specified temperatures using finite shell element models considering material and geometric nonlinearity. Critical strengths are parametrically studied, considering specified temperatures with variable member length, member sizes, and steel strength. Sub-assembly analysis models are created for the benchmark building simulations using finite shell elements and inelastic constraint springs for boundary conditions. Properties of these inelastic boundary springs are carefully developed to represent realistic building behavior under fires. Time-temperature relationship for fire is adopted from Eurocode 1 (EC1, 1991). The maximum temperatures of steel members are calculated by a one-dimensional heat transfer approach described by Buchanan (2002). Structural stability during the fire is evaluated by comparing the maximum induced temperatures to the critical temperature of frames, calculated using structural simulation. In the probabilistic study, dead and live load, and material properties are considered as random variables. Sensitivity of the limit-state to each random variable is studied. Probabilistic collapse assessment given magnitude of gas temperatures is performed by utilizing the mean-value first-order second-moment (FOSM) approach.

1.4 ORGANIZATION

This dissertation is divided into six chapters. Chapter three and sections in Chapter four are designed to be self-contained because they have been or are being planned to be published as individual journal papers. As a result, there may be some repetition of the material. Chapter two provides an overview of the behavior of steel structures exposed to fire including a review of past fire disasters and experimental data for steel properties at elevated temperatures. Chapter three includes a member-based strength study utilizing finite shell element models. Alternative design equations for individual steel members under elevated temperatures are proposed for use in the AISC specification for design of steel buildings (AISC, 2005). Appendix A also contains supplemental studies on the behavior of individual

9 CHAPTER 1. INTRODUCTION

members at elevated temperatures. Chapter four describes the collapse assessment of a benchmark office building, which includes evaluation of time-temperature relationships using parametric fire curves and analyzing sub-assemblies of the building structure. Some design recommendations are also suggested. The simulations in Chapter three and four are performed deterministically. Chapter five extends these deterministic simulations to probabilistic assessment. Uncertainties are reviewed from past studies or obtained from existing experimental data. A proposed framework for probabilistic assessment is presented and applied to illustrate examples for member-based and system-based collapse limit-state checks. Summary, conclusions and future work are discussed in Chapter six.

CHAPTER 2 OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2 OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2.1 PAST FIRE DISASTERS

Both experimental studies and analytical simulations are essential in the evaluation of the behavior and performance of steel structures under fire conditions. Experimental investigations of steel under elevated temperatures have been carried out in many different forms from the steel material levels to individual members and finally frame assemblies. While relatively greater numbers of tests have been performed for material and individual members, tests for frames are limited due to the technical and financial difficulties. However, frame tests are very helpful to investigate the characteristic behavior of indeterminate structural systems under fires such as redistribution of forces and thermally induced effects. In addition to laboratory tests, the performance of real buildings that have experienced fires provides important and helpful information about the system behavior. Other reports, such as Wang (2002), provide summaries of past experiments on steel frame assemblies under elevated temperatures. This section will focus on several case studies on the behavior of actual buildings during and after fire disasters.

2.1.1 Fires on Steel Structures

Table 2-1 summarizes past major fire disasters for twelve steel-framed buildings. The buildings are all office occupancy and most of them are high-rise, where fire fighting is difficult and there is a potential risk of the spread of fire. The only buildings that experienced total collapse are World Trade Center (WTC) towers 1 and 2, and building 7. Although the fire duration lasted more than 12 hours in some of buildings (e.g. Alexis Nihon Plaza, One Meridian Plaza and Parque Central) and there was almost no fire protection at Broadgate Phase 8 due to its stage in the construction process, these buildings did not totally collapse. The potential strength of steel structures under fire conditions can be seen from these case studies.

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Table 2-1 Past major fire disasters of steel buildings

Building # of Dura Nature of Location Date Reference name stories tion Structural damage Connection bolts failure, One New New York, NIST, 50 8/5/70 6 hr causing beam falling at York Plaza USA 2002 33-34th floor Alexis Montreal, Partial collapse at 11th NIST, 15 10/26/86 13 hr Nihon Plaza Canada floor 2002 First Los Angeles, No collapse USFA, Interstate 62 5/4/88 3.5 hr USA Burnout of 12-16th floor TR022 Bank Broadgate During construction London, UK 14 6/23/90 4.5 hr SCI, 1991 Phase 8 No collapse Mercantile Churchill Credit Plaza, unkno No collapse NIST, 12 1991 Insurance Basingstoke, wn Burnout of 8-10th floor 2002 Building UK One Philadelphia, No collapse USFA, Meridian 38 2/23/91 19 hr USA Burnout of 22-29th floor TR049 Plaza WTC New York, FEMA, 110 9/11/01 1.5 hr Total collapse Tower 1 USA 403 WTC New York, FEMA, 110 9/11/01 1 hr Total collapse Tower 2 USA 403 New York, Partial collapse of 4 FEMA, WTC 5 9 9/11/01 8 hr USA stories and 2 bays 403 New York, NIST, WTC 7 47 9/11/01 4-8 hr Total collapse USA 2004 Reinforced concrete and Parque Caracas, steel structure Moncada, 56 10/17/04 17 hr Central Venezuela No collapse 2005 Burnout of 34-56th floor Reinforced concrete and Windsor 18-20 steel structure NILIM, Madrid, Spain 32 2/12/05 Building hr Partial collapse at top ten 2005 floors

Details of fire behavior, fire protection, and structural damage for some of the listed buildings (Broadgate Phase 8, One Meridian Plaza, World Trade Center building 7 and Windsor Building) are discussed in Sections 2.1.2 to 2.1.5. Details about the WTC towers 1 and 2 are not described here since there have been many reports on the buildings (FEMA 403; NIST, 2005) and the collapses were triggered by airplane attacks that are fundamentally different from other fires.

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2.1.2 Broadgate Phase 8

On 23rd June 1990, a fire broke out in partially completed 14-story steel building in Broadgate development in central London, UK (SCI, 1991). The fire began in a large contractor’s hut on the first floor at about 12:30 am. There were no automatic fire detection systems or sprinklers in operation, and fire protection had not been installed to most of steelwork. The fire burned at its highest intensity for approximately 2.5 hours (1:00-3:30 am) and lasted total of 4.5 hours until 5 am. Most of combustible materials in the hut were consumed during the fire and the temperature reached over 1000 °C.

(a) Elevation before fire (b) Fire fighting activity

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(e) Local buckling on column (f) Deformed truss end Figure 2-1 Photos of Broadgate fire (photo reference of SCI, 1991)

The first floor plan is rectangular in shape (approximately 80 m in length and 50 m in width) and size of the site hut in the floor is 40 m by 12 m. The floor was constructed using composite lattice trusses and composite beams. The maximum permanent deflection of the steel trusses, which spanned 13.5 m, was 552 mm and the deflections of the composite beams were between 82 mm and 270 mm. Local buckling was observed at the bottom flanges and webs of some of the beams, which implies high axial compression due to thermal expansion (Figure 2-1(d)). Because of the large floor area compared to the fire area, it is assumed that heated portion of floor framing was highly constrained by the surrounding non-heated floor structures. Local buckling was also observed at unprotected steel columns. These columns deformed and shortened by approximately 100 mm (Figure 2-1(e)). There were adjacent heavier columns which showed no signs of permanent deformation. It has been hypothesized that this shortening was a result of restrained thermal expansion, which was provided by transfer beams at an upper level of the building (SCI, 1991). Axial loads in the columns were redistributed to connecting structural members and alternative load carrying path was created. Although the building was under construction and the applied load on the structural members were much lower than design load, individual members would not have survived under the applied load and fire without help from the connecting structural members or components. This fire provided significant insight about potential strength and redundancy of steel structures against fires.

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2.1.3 One Meridian Plaza

On 23rd February 1991, a fire broke out at about 8 pm on the 22nd floor of the 38-story office building, One Meridian Plaza, in Philadelphia in Pennsylvania, USA. Initially the fire started on the 22nd floor spread vertically up to the 29th floor through an unprotected opening in floor, shaft assemblies, and broken windows on the outside of the building. The fully developed fire was not under control until 3 pm on the next day, lasting for 19 hours (Figure 2-2(a)). Construction of the building was completed in 1973. The floor plan is rectangular in shape, approximately 74 m in length by 28 m in width (2,080 m2 of floor area). The building has a steel frame with concrete slabs over steel metal decks. Columns and beams are covered by spray-on fire proofing material with required protection rates of 3 and 2-hour, respectively (Figure 2-2(b)). Typical structural damage is shown in the photos of the interior views after fire (Figure 2-2(c)- (d)). Temperatures of steel members were high enough to deform the beams up to 1 m of sag between columns. It can also be seen from the photos that combustible items were completely burn. The most notable point is that there was no structural collapse, even under such large deformation. The concrete slab cracked (Figure 2-2(e)) and the floor sagged; however, despite large deformations, its load carrying capacity was maintained throughout the severe fire, which lasted far beyond the standard required fire duration ratings.

(a) Exterior view (b) Typical spray-on fire protection

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(e) Interior view after fire (f) Crack in concrete slab on 28th floor Figure 2-2 Photos of One Meridian Plaza fire (photo reference of USFA TR049)

2.1.4 World Trade Center Building 7

The 47-story steel commercial building located in the north region of the WTC complex collapsed at 5:21 p.m. on 11th September 2001, about eight hours after the first aircraft struck WTC tower 1 (NIST, 2004). The construction of the WTC building 7 was an expansion project in 1987 using an existing structure of Con Ed Substation, which is a three-story steel framed building originally built in 1967. The overall dimensions were approximately 100 m (330 ft) long, 40 m (140 ft) width, and 190 m (610 ft) height. The column layout of the Con Ed Substation and the additionally built upper portion of WTC 7 did not align and a series of column transfer systems were constructed between Floors 5 and 7. The existing I-shaped Con Ed Substation’s columns were braced with welded thick plates to the tops (between the flange edges to make box sections) and strong diaphragm concrete slabs were built on Floors

16 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

5 and 7. Floors 8 through 45 had a typical framing plan with perimeter moment frames (Figure 2-3). It is reported that fires were observed on several floors (Floors 7, 8, 9, 11, 12 and 13) (Figure 2-4) after 2 pm on the day; however, the exact time of the fire break-out and details are uncertain. The building was damaged by falling debris from WTC tower 1 and 2 on the south façade, which may have been a potential contribution to the building collapse. However, the fires are more likely catalyst of this catastrophic event, given that collapse occurred about six hours after the second tower (WTC 1) collapsed. There were two fuel tanks located on Floor 5 for Con Ed’s emergency energy supply. It is uncertain if the fuel was burned before the building collapse, because visual observation was impossible due to the lack of windows on Floor 5. The scenario and mechanism of WTC 7 collapse is still under investigation; however, the NIST (2004) studies report the probable sequence, which are outlined and selectively quoted in the following description.

Figure 2-3 Floor plan and damages of WTC 7 (“June 2004 Progress Report on the Federal Building and Fire Safety Investigation of the World Trade Center Disaster, Appendix L- Interim Report on WTC 7,” NIST, 2004, Figure L-23c)

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(a) Fires on floors 11-12 on the east face (b) Fires on floors 7 and 12 on the north face Figure 2-4 Fires observed from the east and north face of WTC 7 (NIST, 2004, Figure L-24a, b)

The collapse of WTC 7 was recorded on several videos from the north of the building. It took about 8 seconds from first downward movement at the penthouse to initiation of the global collapse. The east penthouse fell followed by the west penthouse and the screen-wall drop, and then the entire building started collapsing. The probable sequence of events leading to the collapse of WTC 7 is illustrated in Figure 2-5. The collapse initiation most probably occurred at the column number 79, 80 and/or 81 possibly including other interior columns 69, 72, 75, 78 and 78A (see Figure 2-3) on Floor 13 or lower floors. Because the collapse initiated at eastern part of the building, traveled to the west in approximately 7 seconds and the global collapse started almost simultaneously, the column failures progressed horizontally around the transfer stories (between Floor 5 and 7) (see Figure 2-3(c)) and ultimately led to the global instability. Because WTC building 7 is the only steel building that is on record as having globally collapsed due to fire (although fire may not have been the only reason), further investigations including analytical simulations are desired.

18 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

(a) Collapse initiation by eastern column (b) Vertical collapse at eastern columns failure

(c) Horizontal collapse transition at (d) Progressed global collapse transfer system on Floors 5 and 7 Figure 2-5 Probable global collapse mechanism of WTC 7 (NIST, 2004, Figure L-33, 34)

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2.1.5 Windsor Building

On 12th February 2005, one of the most devastating fire disasters in the history of steel structures occurred in the Windsor Building in Madrid, Spain. The fire broke out at about 11 pm on the 21st floor of the 32-story office building and quickly developed up to the top floor by 1 am on the next day. The top ten floors were totally engulfed in flames and it gradually spread to the lower floors. The fire reached the 17th floor by 2 am and about that time a significant area of exterior cladding dropped. Upper floors partially collapsed at about 4 am and the fire spread downward to 4th floor by 9 am. The fire was not under control until 2 pm and the fire department declared the fire extinguished at 5 pm. The duration is between 18 to 20 hours. The building is 32-stories and 106 m in height, and was completed in 1979. The floor plan is rectangular in shape with approximately 40 m in length (7 bays with 5.6 m span) and 25 m in width (2 external 6.3 m bays and an internal 12.6 m core bay). The building is composite steel and reinforced concrete (RC) structure (i.e., RC core and waffle slabs supported by internal RC columns, internal steel beams, and perimeter steel columns). Mechanical floors are located between 3rd and 4th floors and 16th and 17th floors. RC wall girders (height 3750 mm, width 500 mm, and length 25 m) penetrate RC core in these mechanical floors and the axial load of perimeter steel columns are transferred to the core by the cantilevers of the wall girders. The perimeter steel columns are box shape in section and consist of two welded channels (C shape sections), located every 1.8 m. The building was constructed based on 1970’s Spanish design code, where the specifications on fire protection were minimal. Unfortunately, the building was under renovation to install new fire protection systems when the fire broke out. The installment included sprinklers, fire protection of perimeter steel columns and interior beams, fire walls, fire insulation of floors at perimeter cladding, and exterior stairs for evacuation. The renovation was carried out from lower to upper floors. Fire protection of steel work had been completed up to 17th floor, except for the 9th and part of 15th floor. No protection had been installed on the 18th floor and higher. It is considered that the fire quickly spread to upper floors through the uncompleted fire insulation of floors at the perimeter cladding. The fire also developed slowly to lower stories, in a similar way, through partially incompleted fire insulation of floors.

20 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

(a) Before fire (*1) (b) After fire (*2) Figure 2-6 Exterior view of Windsor Building before and after the fire (*1) : Pedro Gonzalez (EFE) in report NILIM, 2005 (*2) : NILIM, 2005

Structural damage is significant at the top 11 stories, where fire protection had not been installed to steelwork. Perimeter steel columns including exterior bays of waffle slabs almost completely collapsed. However, the RC core maintained the strength and total collapse was prevented (Figure 2-6). Lack of fire protection of the steel columns was critical to the partial collapse. The probable collapse mechanism reported in NILIM (2005) is that (1) the steel columns near the fire buckled due to material deterioration under elevated temperature, (2) the axial load of the buckled columns were redistributed to adjacent structures, (3) the number of deteriorated columns increased due to the developing fire, however, the waffle slab worked as cantilever and prevented structural collapse, (4) the fire further spread and waffle slabs reached their load carrying capacity as a cantilever for the extended supporting area and collapsed, and (5) the floor collapse induced failure of other floors and waffle slabs were ripped off at the connections to the core. It is certain that upper mechanical floor between 16th and 17th floors provided enough redundancy to prevent progressive collapse, resisting the impact of the partial collapse of upper floors and prevented further failure of lower floors.

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Figure 2-7 shows detailed photos during and after the fire. Figure 2-7(a) shows how the south side of upper floors collapsed sequentially. Figure 2-7(b) and (c) shows how the waffle slab tore off near RC columns. Figure 2-7(d) shows where fire protection is not installed perimeter steel columns on 9th floor, and the columns are buckled and highly deformed.

(a) Collapse of top south side (*1) (b) Tore waffle slab during fire (*2)

(c) Tore waffle slab after fire (*3) (d) Buckled perimeter columns (*4) Figure 2-7 Detailed photos of Windsor Building fire (*1) : Javier Lizon (EFE) in report NILIM, 2005 (*2) : EFE in report NILIM, 2005 (*3) : OTEP in report NILIM, 2005 (*4) : Miyamoto in report NILIM, 2005

22 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2.1.6 Cardington Fire Test

In addition to review of past fire disasters of steel buildings, a large full-scale fire test performed at Cardington in UK is briefly introduced in this section to further discuss the actual behavior of steel structures under fires. The eight-story full-scale building fire test (Figure 2-8) is unique compared to scales of other structural fire tests and provides interesting information regarding characteristic behavior of steel frames under fires such as redistribution of forces and thermally induced effects. The test building was designed in accordance with British Standard or Eurocode and targeted typical European steel buildings. The floor dimensions were 45 m in the length and 21 m in the width. Typical one-way steel decks were designed for composite floor structure and design load was applied by using sand bags during the test. Six compartment fire tests were performed in different floors and locations (Figure 2-9). Columns were covered with fire insulations but beams were not. Some of the beams experienced elevated temperatures greater than 1000 °C and large deformations (beam sagging); however, the building did not even partially collapse. As steel strength at 1000 °C retains only about 5 % of strength at ambient temperature, the composite effect played a significant role for the structural stability under fires. This finding raised questions about current fire insulation design practice and motivated steel composite floor design with only partial or even no fire insulation on composite beams, although the interactive effect with other building components such as compartment partitions must be carefully investigated for practical application. The ductile deformation capacity of floor structure is remarkable; however the continuity and integrity of the composite structures are to be further examined. This issue is especially important for US design, because the generally good performance was attributed to slab reinforcement, which is common in the UK but not usual in typical US construction practice. Despite the strength of composite beams at elevated temperatures, columns were vulnerable to fires by losing their load carrying capacity associated with local buckling. This was observed in tests of columns located near beam-column connections that were unprotected. The columns and connections were fully covered in the later tests. Further details about the Cardington Fire Test can be found in several publications such as SCI (2000), Kirby (1997, 1998), Kirby et al. (1996b, 1999) and Yang (2002).

23 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

(a) Frame overview (b) Deformed beams Figure 2-8 Photos of the Cardington Fire Test (Steel Construction Institute (SCI), (2000), “Fire Safety Design: A New Approach to Multi- Storey Steel-Framed Buildings,” SCI Publication P288, Figure A.1.1, B.3.18)

Figure 2-9 Floor framing and test locations (SCI, 2000, Figure B.3.1)

24 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2.1.7 Summary of Past Fire Disaster Review

Past fire disasters on steel buildings are reviewed in this section to learn from the observed behavior of actual steel buildings under fires. Among the listed fire disasters, four major events: Broadgate Phase 8, One Meridian Plaza, World Trade Center (WTC) building 7 and Windsor Building, as well as Cardington eight-story full-scale fire test are reviewed in detail. WTC tower 1 and 2 are not closely reviewed, because of the unique aspects of their design and the terrorism attack. The most important point from this review is that no steel building has totally collapsed by fire alone except perhaps WTC 7, which may have encountered some physical damage that contributed to its collapse and experienced the extremely unusual situation of not being attended to by fire fighters. This evidence illustrates the potential high resistance of steel buildings under current design practice. Also, the superior performance of steel beams observed in the cases of Broadgate Phase 8, One Meridian Plaza and Cardington Fire Test should be highlighted. Some of the beams experienced elevated temperature greater than 1000 °C without collapse, allowing large deformations with catenary actions. On the other hand, steel columns have proven to be quite vulnerable in past fire disasters. The Windsor Building partially collapsed in the upper stories, where fire insulation on the columns was missing due to renovation. Also, local buckling with large distortions occurred in the columns in the Cardington Fire Test, which must have significantly deteriorated the axial strength. These observations are very helpful in understanding of characteristic behavior of steel buildings under fires, although further careful investigations are necessary to generalize and use the findings for structural fire design.

2.2 MECHANICAL PROPERTIES OF STEEL UNDER ELEVATED TEMPERATURES

2.2.1 Experimental Results

Evaluation of the mechanical properties of steel at elevated temperatures is essential for analytical simulations of steel buildings exposed to fire. Large numbers of tests have been carried out to investigate these properties; however, it is difficult to review these experimental results comprehensively, given that some of the test results are contained in internal institutional reports and are not easily accessible. In this section, some of the available test results are reviewed and summarized to provide an overview of the basic characteristics of behavior of steel at elevated temperatures.

25 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Static material properties, specifically stress-strain curves, are reviewed in this section and will be used again later in the studies presented in Chapters 3 to 5. Transient properties such as rate dependence or creep strength are not specifically reviewed.

2.2.1.1 Experiments by Harmathy and Stanzak

Harmathy and Stanzak (1970) carried out tensile strength tests of structural steels at elevated temperatures and provided complete stress-strain curves up to 10 % strain. This study was among the first to examine large strain response of steel at high temperatures. In terms of the history of structural fire engineering, this research is significant in the sense that the primary focus is to provide useful information for design engineers who are concerned with assessing the fire endurance of building elements. Structural steels manufactured in the United States (ASTM A 36) were tested under 12 specified temperatures from 24 °C to 649 °C, and Canadian structural steels (CSA G40.12) were tested under 13 specified temperatures from 24 °C to 704 °C. The minimum specified yield strength of ASTM A 36 and CSA G40.12 are 250 MPa (36 ksi) and 300 MPa (44 ksi), respectively. The measured chemical composition from the specimens of these steels is shown in Table 2-2. Tensile tests of these steels under elevated temperatures were performed under average strain rates of 0.051 to 0.102 (min-1). The measured stress strain curves are shown in Figure 2-10. The relationships between the US and SI units are summarized in Appendix B. The strength at 300 °C (572 °F) is greater than that at ambient temperature above 1 % strain. At higher temperatures (≥ 400 °C or 572 °F), the strength is lower than that at ambient temperatures. A significant strength drop is observed for tests conducted between 500 °C (932 °F) and 600 °C (1112 °F). It is noteworthy that the explicit yield point and hardening plateau, which is one of the distinct characteristics structural steel at ambient temperature, is not clearly observed and nonlinear behavior is more significant at elevated temperatures.

Table 2-2 Chemical composition of steels by Harmathy and Stanzak (Wt%) Steel C Mn P S Si Ni Cr Al ASTM A 36 0.19 0.71 0.007 0.03 0.09 - - - CSA G40.12 0.195 1.40 0.015 0.019 0.022 0.03 0.01 < 0.01

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(a) ASTM A 36 steel

(b) CSA G40.12 steel Figure 2-10 Stress-strain curves by Harmathy and Stanzak (1970)

27 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2.2.1.2 Experiment by Skinner

Skinner (1972) performed comprehensive tests of Australian steels at elevated temperatures, specifically focusing on information for prediction of behavior of structural steel members in buildings. The tests covered not only mechanical properties such as modulus of elasticity, tensile stress-strain properties, creep and thermal expansion, but also thermal properties such as specific heat and thermal conductivity. 25.4mm (1 inch) thick plates were used for the tensile tests under specified constant temperatures ranging from ambient temperature to 650 °C. Stress was measured under controlled strain up to 5 % with six different strain rates from 1×10-5 (min-1) to 2×10-1 (min- 1). The measured composition of the material is shown in Table 2-3 (AS A149 or AS A186:250). Characteristic yield stress and tensile strength of the tested steel were 245 MPa (35.5 ksi) and 487 MPa (70.6 ksi), respectively.

Table 2-3 Chemical composition of steel by Skinner (Wt%) Steel C Mn P S Si Ni Cr Cu Al Mo AS A149 0.27 0.65 0.033 0.041 0.128 0.086 0.16 0.01 0.007 0.42

Figure 2-11 shows measured stress-strain curves with strain rate of 5×10-5 (min-1), which is the slowest strain rate available for various temperatures from ambient temperature to 650 °C. The stresses were measured under controlled displacement up to strains of 0.05 (every 0.0025 up to 0.01 and every 0.005 up to 0.05). Stress-strain curves at selected temperatures; ambient temperature, 300 °C, 400 °C, 500 °C and 600 °C, are plotted in Figure 2-11. The strength at 300 °C is greater than that at ambient temperature above 1 % strain, whereas the strength is lower than that at ambient temperatures at higher temperatures greater than 400 °C (as also observed in the test by Harmathy and Stanzak (1970)).

28 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

500

450

400

350

300

250

200 Stress (MPa) Stress 150 RT 300°C 100 400°C 500 C 50 ° 600°C 0 0 0.01 0.02 0.03 0.04 0.05 Strain Figure 2-11 Stress-strain curves by Skinner (1972)

2.2.1.3 Experiments by DeFalco

DeFalco (1974) focused on and examined compressive properties of US structural steels at elevated temperatures. Mechanical properties of three different types of structural steels ASTM A36, A441, and A588, were tested at ambient temperature 21.1 °C (70 °F) and elevated temperatures from 93.3 °C (200 °F) to 648.9 °C (1200 °F) at every 111.1 °C (200 °F). Tested specimens were round bars with 38.1 mm (1.5 inch) long and 12.7 mm (0.5 inch) diameter.

29 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

(a) A 36 steel

(b) A 441 steel

30 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Figure 2-12 Stress-strain curves by DeFalco (1974)

A total of 63 tests (21 for each type of steel) were performed and stress-strain curves were obtained up to 1.5 % strain under a constant strain rate of 5×10-3 (min-1). The characteristic yield strength of the steels varied depending on the shape and thickness. The yield strength of A36 plates with thickness of 19.1 mm (3/4 inch) or less was 250 MPa (36 ksi) and those of A441 and A588 were 345 MPa (50 ksi). A36 is a common carbon steel, whereas A441 and A588 are corrosion resistant high strength steels. The maximum permissible chemical compositions of these types of steel are shown in Table 2-4. The measured stress-strain curves of the three types of steel are shown in Figure 2-12. Three tests were performed under a specified temperature for each type of steel. Mean values were used to plot for the stress-strain curves in the figure.

31 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Table 2-4 Chemical composition of steel by DeFalco (Wt%) C P S V Steel Mn Si Ni Cr Cu max max max min ASTM A 36 (*1) 0.25 - 0.04 0.05 - - - (*2) - 0.85- 0.30 0.20 ASTM A 441 0.22 0.04 0.05 - - 0.02 1.25 max min 0.75- 0.15- 0.25- 0.40- 0.20- 0.01- ASTM A 588 0.20 0.04 0.05 1.25 0.30 0.50 0.70 0.40 0.10 (*1) : plate thickness of 19.1 mm (3/4 inch) or less (*2) : When copper is specified, the minimum copper is 0.20 percent.

2.2.1.4 Experiments by Fujimoto et al.

Fujimoto et al. (1980, 1981) carried out uniaxial tensile tests of two types of Japanese structural steel, SS 41 and SM 50A, at elevated temperatures up to 600 °C at every 100 °C. SS 41 is a hot rolled structural carbon steel, while SM 50A is a structural steel with especially high weldability. The minimum specified yield strength of SS 41 and SM 50A with plate thickness of 40 mm or less are 235 MPa and 325 MPa, respectively. The maximum permissible chemical composition of these steels is shown in Table 2-5 (AIJ, 1973). The uniaxial tensile tests were performed up to 2.5 % strain under specified constant temperatures. Measured stress-strain curves of SS 41 and SM 50A at ambient and elevated temperatures from 300 °C to 600 °C at every 100 °C are extracted and shown in Figure 2-13.

Table 2-5 Chemical composition of steel by Fujimoto et al. (Wt%) Steel C max Si max Mn max P max S max JIS SS 41 - - - 0.05 0.05 JIS SM 50A (*1) 0.20 0.55 1.50 0.04 0.04 (*1) : plate thickness of 50 mm or less

32 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

SS41 350

300

250

200

150

Stress (MPa) Stress RT 100 300°C 400°C 50 500°C 600°C 0 0 0.005 0.01 0.015 0.02 0.025 Strain (a) SS 41 steel

SM50 450

400

350

300

250

200 Stress (MPa) 150 RT 300°C 100 400°C 50 500°C 600°C 0 0 0.005 0.01 0.015 0.02 0.025 Strain (b) SM 50A steel Figure 2-13 Stress-strain curves by Fujimoto et al. (1980, 81)

2.2.1.5 Experiments by Kirby and Preston

Kirby and Preston (1988) performed tensile tests of British steel under transient heating conditions, where the mechanical load on a steel specimen was maintained at a constant and the temperature increased under specified rates. This transient temperature-controlled testing

33 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

approach attempts to simulate the realistic behavior of structural steel under fire conditions, given that the load on structures is almost always constant during fires and temperature increases. In this testing procedure, stress-strain curves are translated from the strain- temperature relationships. The steels tested were BS4360:1979 Grade 43A and 50B. The characteristic strength was 255 MPa for Grade 43A (275 MPa according to BS4360:1986) and 355 MPa for Grade 50B. The maximum permissible chemical compositions of these steels are shown in Table 2-6.

Table 2-6 Chemical composition of steel by Kirby and Preston (Wt%) Steel C max Si max Mn max P max S max BS Grade 43A 0.25 0.50 1.60 0.05 0.05 BS Grade 50B 0.23 0.50 1.50 0.05 0.05

Transient heating tests were carried out with the heating rate of 10 °C/min. The translated stress-strain curves up to 2 % strain at ambient and elevated temperatures (300 °C to 800 °C at every 100 °C) are shown in Figure 2-14. Comparing the results from their transient tests with past steady-state experiments (mechanical loading under constant elevated temperatures), they found that stress-strain relationships derived from these two types of experiments were not significantly different.

Grade 43A 300

250

200

150 RT 300°C Stress (MPa) Stress 100 400°C 500°C 600°C 50 700°C 800°C 0 0 0.005 0.01 0.015 0.02 Strain (a) Grade 43A steel

34 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Grade 50B 400

350

300

250

200 RT 300°C Stress (MPa) Stress 150 400°C 100 500°C 600°C 50 700°C 800°C 0 0 0.005 0.01 0.015 0.02 Strain (b) Grade 50B steel Figure 2-14 Stress-strain curves by Kirby and Preston

2.2.1.6 Comparison of the Experiments

Stress-strain curves shown in Figure 2-10 to Figure 2-14 were obtained in tests performed using different procedures with different types of steels. Comparing these results provides general understanding of structural steel at elevated temperatures. Nine stress-strain curves are normalized by measured 0.2 % offset proof strength and compared at ambient and elevated temperatures from 300 °C to 600 °C at every 100 °C (Temperature unit of the stress- strain curves by Harmathy and Stanzak (1970) and DeFalco (1974) is converted from Fahrenheit to Celsius and curves at elevated temperatures from 300 °C to 600 °C are obtained by linear interpolation). The measured proof strength, specified yield strength and other features of each test are summarized in Table 2-7.

35 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Table 2-7 Comparison of steel experiments at elevated temperatures

Measured Specified Testing (*1) (*2) Steel Researcher ε max Tmax Fy (MPa) Fy (MPa) procedure

ASTM A 36 300 250 Harmathy and Steady-state (*3) 650 10 CSA G40.12 350 300 Stanzak (1970) Tensile 700 Skinner Steady-state (*3) AS A 149 236 245 5 650 (1972) Tensile ASTM A 36 295 250 DeFalco Steady-state (*3) ASTM < 2 650 385 345 (1974) Compressive A 441 JIS SS 41 245 235 Fujimoto et al. Steady-state (*3) 2.5 600 JIS SM 50A 356 325 (1980, 1981) Tensile BS Gr. 43A 255 255 Kirby and Transient (*4) 2 800 BS Gr. 50B 355 355 Preston (1988) Tensile (*1) : upper limit of measured strain (%) (*2) : upper limit of measured temperature (°C) (*3) : fixed temperature (*4) : fixed load

These stress-strain curves up to 2.5 % strain are compared in Figure 2-15. Difference of these responses is relatively small up to 500 °C, while significant difference is observed at 600 °C. The two tests conducted by DeFalco agree well each other at any temperatures. Also the two tests by by Fujimoto et al., and Kirby and Preston agree. This fact is true even at 600 °C, where overall results are relatively scattered. On the other hand, the two tests by Harmathy and Stanzak are relatively different. This is probably because Harmathy and Stanzak tested steels manufactured in two different countries (the US and Canada), whereas the other studies tested steel from only one country. Skinner and Fujimoto et al. used similar testing approaches (i.e., tests under specified constant temperatures); however their results are not necessarily closer than others. Statistical information regarding the steel strength at elevated temperatures is investigated based on these data and is shown in Chapter 5.

36 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

1.2 ASTM A36 (Harmathy) 1 CSA G40 0.8 AS A149 ASTM A36 (DeFalco) 0.6 ASTM A441 JIS SS41 0.4 JIS SM50A Stress / Yield stress Stress 0.2 BS Gr43A BS Gr50B 0 0 0.005 0.01 0.015 0.02 0.025 Strain (a) Ambient temperature

300°C 400°C

1.5 1.2 1

1 0.8 0.6

0.5 0.4 Stress / Yield stress Yield / Stress stress Yield / Stress 0.2

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain Strain (b) 300°C (c) 400°C

500°C 600°C 1 0.6 0.8 0.5

0.6 0.4

0.3 0.4 0.2 Stress / Yield stress Stress 0.2 / Yield stress Stress 0.1

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain Strain (d) 500°C (e) 600°C Figure 2-15 Comparison of stress-strain curves in experiments

37 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

2.2.2 Equations of Stress-strain Curves

In order to use the experimentally measured stress-strain data for design analysis, stress- strain relationships need to be formulated in practical and continuous functions. The structural steel tests at elevated temperatures shown in the previous sections were carried out in five different countries; Canada, Australia, the US, Japan and the UK. Design standards for steel structures against fires are different in these countries with different levels of development. Design equations of the steel stress-strain curves at elevated temperatures in the design standards are compared and discussed.

2.2.2.1 Eurocode Stress-strain Curves

At present, the Eurocode is the most developed design standard for structural fire design. Each of following nine parts of the Eurocode contains significant coverage of structural fire engineering:

EN 1991 Eurocode 1 Basis of design and actions on structures EN 1992 Eurocode 2 Design of concrete structures EN 1993 Eurocode 3 Design of steel structures EN 1994 Eurocode 4 Design of composite steel and concrete structures EN 1995 Eurocode 5 Design of timber structures EN 1996 Eurocode 6 Design of masonry structures EN 1997 Eurocode 7 Geotechnical design EN 1998 Eurocode 8 Design provisions for earthquake resistance of structures EN 1999 Eurocode 9 Design of aluminum alloy structures

Guidance on fire simulations, and the design of concrete and steel structures for fire conditions are contained in Eurocode 1 (EC1, 2002), Eurocode 2 (EC2, 1993), and Eurocode 3 (EC3, 2003), respectively. Relationships describing time and gas temperatures in flashover fires are defined in EC1 as functions of influential factors, such as fuel load, geometry of fire compartment, openings and firefighting activities. The described relationships allow engineers to estimate temperatures of gas in fire compartments and consequently temperatures of structural steel depending on the type and thickness of fire insulation.

38 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Both conceptual and detailed descriptions regarding strength calculations of steel structures during fires are included in EC3. Design equations for stress-strain curves of structural steel at elevated temperatures are also defined in EC3. The idealized curves consist of three regions; elastic, transition, and perfectly-plastic. Figure 2-16 shows the shape and key parameters of the stress-strain curves, where E()T , FTp () and FTy () are modulus of elasticity, proportional-limit stress and yield stress, respectively. These key parameters are explicit functions of temperature of T . ε p ()T and ε y ()T are strains corresponding to

FTp () and FTy ().

(a) Stress-strain curves (b) Key parameters

39 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Reduction factors for the modulus of elasticity, stress at proportional-limit and yielding with respect to temperatures are shown in Figure 2-16(c). These are defined as proportions of values at elevated temperatures to those at ambient temperature as shown in Equation (2.1)

, where E0 , Fp0 and Fy0 are modulus of elasticity, proportional-limit stress and yield stress at ambient temperature, respectively.

ET() FTp () FTy () KTE ()= , KTp ()= , and KTy ()= (2.1) E0 Fp0 Fy0

Stress-strain curves at elevated temperatures are shown in Figure 2-16(a). These are obtained by substituting the reduction factors into the equations defined in EC3. The significant drop of strength between 400 °C and 700 °C, which was observed in experimental results, are represented in these curves.

2.2.2.2 AS4100 Stress-strain Curves

The Australian Standard for steel structures, AS4100 (ABCB, 1998), contains documentation about design of steel structures under fire conditions; however, it is less comprehensive than the Eurocode. Description in AS4100 about the mechanical properties of structural steel at elevated temperatures is limited to yield stress and modulus of elasticity (i.e., no proportional limit stress). Reduction factors of these values are defined as proportions of values at elevated temperatures to those at ambient temperature in similar way used in Eurocode. The defined reduction factors of the modulus of elasticity and yield strength are shown in Figure 2-17(a). Since there is no definition about stress-strain curves considering highly nonlinear behavior of steel at high temperatures, it is interpreted that characteristic bilinear stress-strain curves at ambient temperature is used at elevated temperatures. Simple structural analysis for fires can be performed by using these degraded modulus of elasticity and yield strength. Assuming perfect-plastic bilinear, stress-strain curves under elevated temperature by AS4100 are shown in Figure 2-17(b). These curves can be useful for simple member-based strength evaluation; however, they do not represent realistic behavior of structural steel and are not appropriate be used in advanced analytical simulations.

40 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

AS4100

K (T) 1 y K (T) E 0.8

0.6

0.4 Reduction factor 0.2

0 0 200 400 600 800 1000 1200 Temperature (°C) (a) Reduction factors (b) Stress-strain curves Figure 2-17 Stress-strain curves defined by AS4100

2.2.2.3 AIJ Stress-strain Curves

The Architectural Institute of Japan (AIJ) issued design recommendations for steel structures under fire conditions in 1999 (AIJ, 1999), which includes equations of stress-strain curves of structural steel at elevated temperatures. It is expected that these equations can be used in advanced structural fire design. The equations are based on test results by Fujimoto et al. (1980, 1981) and are prepared separately for SS400 (SS41) and SM490 (SM50A), which are common Japanese structural steels with characteristic yield strengths of 235 MPa and 325 MPa, respectively. Other properties and composition of these steels are described in Section 2.2.1.4 and the stress-strain curves up to 2 % and 10 % strain are shown in Figure 2-18.

(a) SS400 (up to 2% strain) (b) SS400 (up to 10% strain)

41 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

The stress-strain curves of these steels at elevated temperatures are composed of three portions; the elastic portion, the hardening plateau portion, and the strain hardening portion. The hardening plateau portion gradually reduces under elevated temperatures and completely disappears at 500 °C (see Figure 2-18). The equations, which define these curves, are as follows:

σ (,)TTTεσεσε= max{ 12 (,),(,)} (2.2)

where σ (,)T ε is defined stress under temperature T and strain ε , and:

σε1(,)TETT= min(),{ εσp ()} (2.3)

(ET()− EC () T )ε ETC ()ε σε2 (,)T =+ (2.4) 32 23 2 1()()()+−ET E Tεσ T 10.05+ ()ε {}()()CC

E()T and σ p ()T are modulus of elasticity and stress at proportional limit at T , respectively.

ET( )=×− 210,000 (1 10−62 T ) (MPa) (2.5)

−62 σ p (TT )=×−× 240 (1 4 10 ) (MPa) if T ≤ 500 °C and SS400 (2.6)

42 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

−62 σ p (TT )=×−× 330 (1 4 10 ) (MPa) if T ≤ 500 °C and SM490 (2.7)

σ p ()T = 0 (MPa) if T > 500 °C (2.8)

ETC () and σ C ()T are given in Table 2-8. Values can be linearly interpolated for each table.

Table 2-8 Coefficients in AIJ equations for stress-strain curves SS400 SS400 SM490 SM490

T ET() T σ ()T T ET() T σ C ()T C C C (°C) (MPa) (°C) (MPa) (°C) (MPa) (°C) (MPa) 0 4500 0 190 0 4500 0 270 300 5200 300 160 300 4500 320 270 600 750 500 90 600 600 600 110 1000 0 750 0 1000 0 750 0

The concept of reduction factors of steel properties at elevated temperatures is not explicitly adopted in AIJ equations, rather the precise stress-strain curves are defined instead. Reduction ratios of key parameters are derived from these stress-strain curves (see Figure 2-19). In order to compare stress-strain equations in AIJ with those in other design standards,

KTy () and KTp () are defined here as the ratios of stresses at 2 % strain and on the hardening plateau at temperature T , with respect to the characteristic yield strength.

Different values of KTy () are prepared for SS400 and SM490 steel, while the values of

KTp () are shared between them (see Eqs. (2.6) to (2.8)). KTE () is a proportion of modulus of elasticity at T to that at ambient temperature, as given in Eq. (2.5).

43 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

1.2 K (T) y SS400 K (T) 1 y SM490 K (T) p 0.8 K (T) E 0.6

0.4 Reduction factor

0.2

0 0 200 400 600 800 1000 1200 Temperature (°C) Figure 2-19 Reduction ratios in the stress-strain curves defined by AIJ

2.2.2.4 AISC Stress-strain Curves

Structural fire engineering toward performance-based design is still in a developing stage in the United States. Structural Design for Fire Conditions has been recently included in an appendix of the Specification for Structural Steel Buildings in the US design standard of steel structures by American Institute of Steel Construction (AISC-LRFD, 2005). The specification describes two types of analytical approaches, referred to as the Advanced Method and the Simple Method. Rigorous structural and thermal analyses are required for the advanced method, while design equations at ambient temperatures are used for the simple method. In these AISC methods, the degradation ratios of the material properties such as modulus of elasticity and yield stress under elevated temperatures are adopted from Eurocode 3 (EC3, 1995). The degraded material properties are substituted into the equations at ambient temperature and the strength of the structural members under fire conditions is individually evaluated in this simple method. Other than the material deterioration ratios, there is no stress-strain curve defined for structural steel under elevated temperatures. This approach is similar in concept to the Australian Standard, AS4100, described in Section 2.2.2.2. Therefore, it can be interpreted that the characteristic bilinear stress-strain curve of structural steel at ambient temperature is also assumed for those under elevated temperatures (Figure 2-20). Since the reduction factors are adopted directly from EC3, Figure 2-20(a) is the same as Figure 2-16(a) fitted to the elastic modulus and the yield strength, and Figure 2-20(b) is a bilinear approximation of the stress-strain curves previously shown in Figure 2-16(b).

44 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

K (T) 1 y K (T) E 0.8

0.6

0.4 Reduction factor 0.2

0 0 200 400 600 800 1000 1200 Temperature (°C) (a) Reduction factors (b) Stress-strain curves Figure 2-20 Stress-strain curves defined by AISC

2.2.2.5 Comparison of the Equations of Stress-strain Curves

The equations of the stress-strain curves of structural steel at elevated temperatures, defined in or interpreted from the four design standards, EC3, AS4100, AIJ (SM490), and AISC, are compared at ambient and elevated temperatures up to 1000 °C at every 100 °C in Figure 2-21. The stress-strain curves obtained through experiments and introduced in Section 2.2.1 are also superimposed in the figures at ambient temperature and 300 °C to 800 °C. These curves are normalized by the characteristic yield strength for the equations and the measured yield strength at the test results. Temperatures less than 400 °C are typically not important in structural fire engineering, because the overall strength degradation is fairly modest up to this temperature. Comparing the equations and test results between 400 °C and 600 °C, it is observed that the Eurocode equation is close to or slightly lower than the mean of the test results and is the best representative of actual steel properties at elevated temperatures among these four equations. It is unfortunate that test results at more than 600 °C are not obtained and comparison between the test data and equations is not possible. The perfectly plastic bilinear stress-strain curves from AS4100 and AISC do not represent highly nonlinear stress-strain response of steel at elevated temperatures. In addition, equations from AS4100 are conservative at 300 °C to 500 °C, while they are unconservative at 700 °C and 800 °C. The AIJ equations more precisely represent the shape of the stress-strain curves. The equations best fit the test results at ambient temperature, properly taking into account strain hardening. However, the strength of the AIJ equations is

45 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE very conservative at higher temperatures, such as 500 °C and higher. This is because the target stress-strain curves to define the AIJ equations are approximately mean minus three times the standard deviation of tested stress-strain curves. The safety margin of structural design for highly unknown building behavior under fire conditions is taken into account in the predictive equations. This may be an effective approach to assure the safety; however, it may not fit with the concept of Load and Resistance Factor Design (LRFD), where limit-state strength and nominal load are simulated as accurately as possible and structural safety is evaluated by taking into account the effect of uncertainties with the load and resistance factors.

20°C

1.2

1 EC3 0.8 AS 0.6 AIJ AISC 0.4 Test Stress / Yield / Yield stress Stress 0.2

0 0 0.01 0.02 0.03 0.04 Strain (a) Ambient temperature (20 °C)

100°C 200°C

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 Stress / Yield / Yield stress Stress / Yield stress Stress 0.2 0.2

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Strain Strain (b) 100 °C (c) 200 °C

46 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

300°C 400°C

1.2 1.2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 Stress / Yield / Yield stress Stress / Yield stress Stress 0.2 0.2

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Strain Strain (d) 300 °C (e) 400 °C

500°C 600°C 1

0.6 0.8 0.5

0.6 0.4

0.4 0.3 0.2 Stress / Yield / Yield stress Stress 0.2 / Yield stress Stress 0.1

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Strain Strain (f) 500 °C (g) 600 °C

700°C 800°C 0.4 0.2

0.3 0.15

0.2 0.1

0.1 0.05 Stress / Yield stress Yield / Stress stress Yield / Stress

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Strain Strain (h) 700 °C (i) 800 °C Figure 2-21 Comparison of stress-strain curves (equations and experiments)

47 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

Although the EC3 curve fits the test data well, the perfectly-plastic strength for strains greater than 2 % is observed to be conservative. Selecting 500 °C as representative of the elevated temperatures, the critical strength of columns at 500 °C is investigated with the EC3 curve and other stress-strain curves, which includes strain hardening (about 1 % of the initial stiffness) for strains greater than 2 % (see Figure 2-22). The simulation is performed in the way, which will be explained in Chapter 3. The column strengths at 500 °C with varied member length (the same study shown in Figure 3-6(d)) were identical to these two stress- strain curves. This limited study does not generalize the outcome; however, the strength at strains greater than 2 % seems not to be critical for structural fire simulations.

500°C

0.8

0.6

0.4 Stress / Yield / Yield stress Stress 0.2 EC3 EC3 study 0 0 0.01 0.02 0.03 0.04 Test Strain Figure 2-22 Eurocode stress-strain curve (500 °C) with hardening at large strains

Generally speaking, the actual values of yield strength at ambient temperature are greater than the characteristic values (i.e., the mean of actual yield strength is approximately 105 % of specified strength; Ellingwood, 1983). Taking this fact into consideration, it may be more meaningful to compare between the stress-strain curves with the design equations normalized by characteristic yield strength and the test data normalized by measured yield strength. The stress-strain curves at 500 °C and 600 °C, as well as ambient temperature, are shown in Figure 2-23. Comparing the Eurocode equations and test data, the stress-strain curves obtained by the Eurocode equation predicts lower strength than the strength obtained in the test data at 500 °C and 600 °C. At 2 % strain, the mean measured strength from tests is approximately 20 % higher than the strength predicted by Eurocode.

48 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE

20°C

1.5

EC3 1 AS AIJ AISC 0.5 Test Stress / Yield / Yield stress Stress

0 0 0.01 0.02 0.03 0.04 Strain (a) Ambient temperature (20 °C)

500°C 600°C

0.8 1

0.8 0.6

0.6 0.4 0.4

Stress / Yield / Yield stress Stress / Yield stress Stress 0.2 0.2

0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 0.04 Strain Strain (b) 500 °C (c) 600 °C Figure 2-23 Comparison of normalized stress-strain curves (equations and experiments)

2.2.3 Experiments by JISF

The Japan Iron and Steel Federation (JISF) experimentally investigated tensile stress-strain relationships of Japanese standard steels under ambient and elevated temperatures (20 °C and 300 °C to 800 °C). Six different types of steel were examined. Five sets (five different supplies) of test data are included for each type of steel and each set contains stress-strain curves under different temperatures (ambient and elevated temperatures from 300 °C to 800 °C at every 100 °C). The total number of tests is 280. Figure 2-24 shows comparison of stress-strain curves from the JISF test data and curves by EC3 and AIJ at 500 °C, 600 °C, and 700 °C. As is also observed in the comparative study discussed in the previous section (2.2.2.5), the EC3 curves agree better with the test data than

49 CHAPTER 2. OVERVIEW OF STEEL STRUCTURES EXPOSED TO FIRE the AIJ curves. The details of the JISF study are described in Appendix C, where statistical studies regarding the steel strength at elevated temperatures are included.

500 °C 600 °C 400 200

300 150

200 100 Stress (MPa) Stress (MPa) 100 50

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain Strain 500 °C 600 °C

700 °C 100

60 JISF EC3 40 AIJ

Stress (MPa) 20

0 0 0.005 0.01 0.015 0.02 0.025 Strain 700 °C Figure 2-24 Comparison of stress-strain curves by JISF, EC3, and AIJ

CHAPTER 3 ANALYSIS OF INDIVIDUAL MEMBERS

3 ANALYSIS OF INDIVIDUAL MEMBERS

Takagi, J., Deierlein, G.G. “Strength Design Criteria for Steel Members at Elevated Temperatures,” Journal of Constructional Steel Research, (in press) See Appendix A for further studies on behavior of individual members at elevated temperatures.

3.1 SUMMARY

Design equations for structural steel members at elevated (fire) temperatures are evaluated through comparisons to nonlinear finite element simulations. The study includes comparative analyses of the American Institute of Steel Construction (AISC) and European Committee for Standardization (CEN) design provisions for laterally unsupported I-shaped columns, beams, and beam-columns at temperatures between ambient to 800 oC. The Eurocode 3 provisions are shown to predict the simulated finite element results within about 10 % to 20 %. On the other hand, the AISC specification predicts strengths that are up to twice as large (unconservative) as the simulated results. The discrepancies are largest for members of intermediate slenderness and temperatures above 300 oC. Modifications to the AISC equations are proposed that provide improved accuracy with calculated strengths typically within 20 % to 30 % of the simulated results. Limitations of the member-based assessments and future research and development needs for structural fire engineering are discussed.

3.2 INTRODUCTION

While the basic concepts for structural fire engineering are well established, explicit assessment of structural response to fires is uncommon in engineering practice. Instead, building codes and design practice have traditionally relied on prescriptive requirements to provide adequate structural fire-resistance in building structures. In steel-framed structures, this is typically accomplished through thermal insulation requirements that are validated by

51 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS fire endurance tests. Such approaches may work well for routine design, however, the lack of alternative methods to establish structural performance by calculation impedes the design of structures where the prescriptive methods fall short of providing effective solutions. Recently, the situation is changing with the publication of standards to calculate structural fire resistance in a manner similar to how other strength limit states are evaluated. For example, the Eurocode 3 (EC3, 2003) standard includes detailed provisions to establish fire loads and evaluate their effects on steel structures. The latest edition of the Specification for Structural Steel Buildings in the American Institute of Steel Construction (AISC, 2005) includes a new appendix entitled, Structural Design for Fire Conditions, which provides guidance and criteria to evaluate the structural resistance of steel building components at elevated temperatures. The EC3 and AISC standards for steel structures both follow an approach whereby the structural resistance to gravity and other loads is calculated with the steel members at an assumed elevated temperature. Each standard permits this evaluation through either an “advanced” or “simple” method, where the former requires rigorous structural and thermal simulations and the latter method is accomplished through member-based strength limit state checks. The member-based approaches are similar to conventional checks made at ambient temperatures. In the AISC specification, for example, the member design strength equations are essentially the same as those for ambient temperatures, except that the input yield strength and elastic modulus of the steel are reduced based on the assumed elevated temperature. While the AISC method is straightforward and easy to implement, its accuracy has not been thoroughly verified. On the other hand, the structural fire provisions of EC3 have been published in codified form since the early 1990’s and have been reviewed and modified since their first publication. The objective of this study is to provide a critical assessment of the AISC and EC3 member strength equations through a comparison to results of detailed finite element simulations at elevated temperatures. The assessment includes laterally unsupported columns, beams, and beam-columns of bi-symmetric I-shaped steel sections with idealized loading and boundary conditions. The simulations employ three dimensional shell finite element models that capture inelastic yielding, overall and local buckling, and non-uniform torsion effects. The column strengths are evaluated in terms of critical axial load applied to a pin-ended column with initial geometric imperfections that represent fabrication tolerances. The beam strengths are evaluated in terms of critical bending moments, which are applied

52 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

about the major-axis at the ends of a simply supported beam that is laterally unsupported and susceptible to lateral-torsional buckling. Results are then compared for the combined effects of axial compression and bending in beam-columns. The assessment covers a range of design conditions by parametrically varying the elevated temperatures and member lengths.

Most of the analyses are for members of Grade 50 steel (yield strength of Fy = 345 MPa) with selected study of Grade 36 ( Fy = 250 MPa) steel. As will be demonstrated, the AISC strength equations at elevated temperatures are unconservative, and alternative strength equations are proposed. The proposed equations are similar in format to the AISC provisions, thus maintaining practicality for design. Beyond the immediate benefit of the improved design equations, this research provides fundamental information to improve the understanding of structural steel members under fire conditions.

3.3 BASIS OF MEMBER STRENGTH EVALUATIONS

Assessment of structural safety to fire hazards can generally be categorized into three stages. The first stage entails characterization of fire initiation and development, which can be done either through direct simulation or through parametric time-temperature models of compartment gas temperatures. The second stage involves heat transfer calculations to evaluate temperatures in structural members, considering insulation and other factors that affect heat transfer. The third stage is to assess structural behavior under elevated temperatures, including the effects of both thermal expansion and degradation of material properties. The assessments made within each of these three stages are generally treated as conditionally independent, where it is assumed that structural behavior does not impact heat transfer or fire development and heat transfer does not impact fire development. While there are situations where the assumption of conditional independence does not hold (e.g., where structural deformations may damage fire compartments, which in turn affects fire development), in most cases the conditional independence is a reasonable assumption. This assumption greatly simplifies the assessment since the analyses in each stage can be done separately and without interaction. This line of reasoning is implicit in the structural fire assessment presented herein (and in the AISC and EC3 design provisions), where the elevated steel temperatures are treated as input to the structural assessment and calculated independently.

53 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

In this study, it is assumed that the elevated temperature of the steel members is known (or can be determined) and used to evaluate the strength limit state of individual members under the combined effects of elevated temperature and applied loads. In concept, the strength limit state can then be evaluated either by (1) calculating the critical temperature (intensity and distribution) that the member can sustain under the given loads, or (2) calculating the strength (load resistance) of a member under a specified temperature. The former approach, referred to herein as the “temperature approach”, is more representative of the actual fire conditions, where the temperature increases as the while the applied gravity loads are constant. The latter approach (termed the “load approach”) is simpler to implement in nonlinear analysis and fits more naturally in existing formats for structural assessment, where member design equations or simulation tools (e.g., nonlinear analysis) are used to assess the critical loads based on the specified temperature-dependent material properties. Although material nonlinear analyses are, in concept, load path dependent, for monotonically increasing gravity loads and temperatures of individual members, it is reasonable to assume that the critical limit state calculated following a “temperature approach” and “load approach” should be similar.

Figure 3-1 Comparison of temperature and load control analyses

The authors have confirmed this assumption by conducting finite element analyses of individual members with fixed loads and variable temperature and vice versa. Shown in Figure 3-1 is an example of one such analysis, where the critical combinations of temperature and strength for a column are obtained by both approaches. This example is for a W14×90

54 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

Grade 50 column (W360×134; Fy = 345 MPa) with a length of 5.67m and weak axis slenderness ratio of 60. The column was modeled using shell finite elements, where its strength limit state is controlled by flexural buckling about weak axis. Further details of finite element analyses are described later. The AISC design provisions essentially adopt the load approach concept, where the member resistance under a specified temperature is obtained by substituting degraded modulus of elasticity and yield stress into strength design equations that are otherwise the same as those applied at ambient temperatures. This member-based check further assumes that the loads induced in the member by restraint to thermal expansion can be independently calculated and superimposed with other applied load effects. The extent to which this assumption is valid depends on the indeterminate nature of the structural system and loading - effects that are not represented in an isolated member analysis. Another assumption made in the AISC provisions and this study is that the member strength can be conservatively calculated with a uniform temperature distribution through the member.

3.3.1 Steel Properties under Elevated Temperatures

Shown in Figure 3-2(a) are idealized stress-strain curves for steel at elevated temperatures. These curves are based on parameters specified in EC3 and substantiated by test data collected by Wainman and Kirby (1988) and others. These stress-strain models are specified through reduction factors (see Figure 3-2(b)-(c)), which are defined for the proportional limit Fp , yield stress Fy , and modulus of elasticity E as follows:

FTp () FTy () ET() KTp ()= , KTy ()= and KTE ()= (3.1) Fp0 Fy0 E0

The terms in the denominator of Eq. (3.1), Fp0 , Fy0 , and E0 , correspond to properties at

ambient temperature (20 °C or 68 °F), and those in the numerator, FTp (), FTy (), and

ET(), are at the elevated temperature, T . Values of the reduction factors are summarized in Table 3-1 and plotted in Figure 3-2(c). Referring to Figure 3-2(c), at 600 °C (1112 °F) the yield strength decreases to about half its ambient temperature value, while the elastic

55 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS modulus and proportional limit decrease more rapidly to about 30 % and 20 %, respectively, of their ambient values.

(a) Stress-strain curves (b) Key parameters

Referring back to Figure 3-2(a), the bilinear elastic plastic relationship, which is commonly assumed in idealized stress-strain models at ambient temperature, disappears as the material becomes more inelastic under elevated temperatures. Finite element analyses that employ nonlinear stress-strain curves (such as in Figure 3-2(a)) model directly this behavior. As described later, the EC3 member design equations for elevated temperatures take this nonlinear stress-strain response into account through coefficients that vary nonlinearly with temperature. On the other hand, the AISC design equations only apply reduction factors to the modulus of elasticity and the yield stress, thereby implying that the

56 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS bilinear (elastic-plastic) properties are preserved at high temperatures. As described later, this assumption of bilinear behavior, which fails to take into account the graduate softening response, leads to unconservative results using the AISC member strength equations for elevated temperatures.

Table 3-1 Stress-strain reduction factors in EC3

Temperature °C (°F) KTy () KTp () KTE () 20 (68) 1.000 1.000 1.000 100 (212) 1.000 1.000 1.000 200 (392) 1.000 0.807 0.900 300 (572) 1.000 0.613 0.800 400 (752) 1.000 0.420 0.700 500 (932) 0.780 0.360 0.600 600 (1112) 0.470 0.180 0.310 700 (1292) 0.230 0.075 0.130 800 (1472) 0.110 0.050 0.090 900 (1652) 0.060 0.038 0.068 1000 (1832) 0.040 0.025 0.045 1100 (2012) 0.020 0.013 0.023 1200 (2192) 0 0 0

3.4 FINITE ELEMENT SIMULATION MODEL

Accuracy of the design models is judged against simulation data of detailed three- dimensional analyses of beam-columns using the finite element method (FEM). As shown in Figure 3-3, the steel members are simulated with shell finite element models created and run using the ABAQUS software (Hibbitt, Karlsson & Sorensen, 2002). The shell finite element models are well suited to simulating geometric and material nonlinearity, including global flexural and torsional-flexural buckling and local flange and web buckling. The analyses are conducted using the “load approach” where the critical strength is determined by incrementing the applied load on a model at various prescribed temperatures. The following are some features of the models:

57 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1. The member is subdivided into 32 shell elements along its length, and the flanges and web are each subdivided into eight elements across the cross section (Figure 3-3). Each element has eight nodes and four Gaussian integration points in the shell plane with three point Simpson’s rule integration through the shell thickness. 2. Nonlinear stress-strain curves of steel at the elevated temperatures are adopted from EC3, as shown in Figure 3-2(a). A uniform temperature distribution is assumed through the member cross section and along its length. Multiaxial yielding is modeled through the von-Mises yield criterion. The yield strengths are assumed to be equal to their nominal specified values, so as to provide consistent comparisons with the design models. 3. Linear kinematic constraints are applied to the flanges and web at the member end so as to enforce planar behavior within each flange and web but to allow cross-section warping (Figure 3-3). Displacements of the web along the Y- and Z-axis are restrained at both ends and longitudinal displacements along the X-axis are restrained at one end. Twisting rotation (about the X-axis) is restrained at both ends, and rotational displacements about Y- and Z-axes (weak and strong axes) are free at both ends. 4. For the column (axial) strength analyses, axial forces are applied along the kinematically restrained webs and flanges at one end of the member. In one set of analyses, the flange ends are free to rotate, thereby permitting flexural buckling about the Y-axis (weak axis); and in a second set of analyses, rotational displacements about the Y-axis are restrained in order to determine the flexural buckling strength about the Z-axis (strong axis). 5. For the beam (flexural) strength analyses, a concentrated force couple is applied at the center of upper and lower flanges at each end so as to induce a uniform strong axis moment along the beams. The kinematic constraint across the flanges ensures a uniform distribution of flexural stresses. 6. Initial geometric member “sweep” imperfections are modeled by introducing a single sinusoidal curve along the member length, with a maximum initial displacement of 1/1000 of the length at the mid-span.

58 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

Figure 3-3 Shell finite element mesh and boundary conditions

1 1 20°C 20°C 500°C 500°C 0.8 0.8 800°C 800°C p0 y0 0.6 0.6

0.4 0.4 P(T) / P / P(T) M(T) /M

0.2 0.2

0 0 0 0.002 0.004 0.006 0.008 0.01 0 0.005 0.01 0.015 0.02 δ / L θ (rad) (a) W14×90 column (L/ry=60 Gr.50) (b) W14×22 beam (L/ry = 60, Gr. 50) Figure 3-4 Load versus displacement response from FEM simulations under ambient and elevated temperatures

Shown in Figure 3-4 are example FEM simulation results for a laterally unsupported column and beam at ambient and elevated temperatures (20 oC, 500 oC, and 800 oC). Column results (Figure 3-4(a)) are shown in terms of the normalized axial load versus midspan

deflection for a W14×90 column with a slenderness of Lr/ y = 60, where L is the length and

o ry is radius of gyration about weak axis. The critical strength at 500 C is about 50 % of that at room temperature. This 50 % reduction is in contrast to the three material reduction factors

o o o of K y (500 C) = 0.78, K p (500 C) = 0.36 and KE (500 C) = 0.60, whose range of values suggests that all three parameters, including the change in proportional limit, play a role in the member strength reduction. At 800 oC the strength is about 8 % of that at room

o temperature, which is in contrast to three material reduction factors of K y (800 C) = 0.11,

o o o K p (800 C) = 0.05 and KE (800 C) = 0.09. The results for 800 C are intended as an upper

59 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

bound on the temperature response, since the large strength reduction at this temperature suggests that the practical value of calculating the strength at this temperature is limited. The

beam data (Figure 3-4(b)) are for a W14×22 beam with a lateral slenderness of Lr/ y = 60 subjected to a uniformly distributed strong-axis moment. Here, the strengths reductions at elevated temperatures are slightly less than for the column, suggesting that the beam behavior is more dependent on the reduction in yield strength and less on the reduction in proportional limit. In these two examples, the arc-length (Riks) solution method is used to track the post- peak response. For the parametric studies shown later, where only the peak strength is reported, the finite element analyses were run under load control up to the critical strength limit state. This was done as a practical measure to reduce the analysis run times. Analyses were also conducted to assess the effects of thermally-induced residual stresses and cross-section imperfections on critical loads. Residual stresses were introduced with the distribution shown in Figure 3-5(a), assuming a peak residual stress at ambient temperature

of Fr0 = 69 MPa (10 ksi). Under elevated temperatures the peak residual stresses are

assumed to reduce in proportional to the reduction in yield stress, i.e., FTryr()= K () TF0 . Shown in Figure 3-5(b) are critical column strengths obtained from simulations run with and without residual stresses at an elevated temperature of 500 oC. The largest difference occurs at slenderness of about Lr/ y = 100, where the residual stresses reduce the critical calculated load by less than 15 %. Residual stresses are modeled in this same way for the parametric subsequent parametric analyses presented later.

Without σ 1 r With σ r 0.8 (T) y 0.6 (T) / P / (T)

cr 0.4 P

0.2

0 0 50 100 150 200 L / r (a) Residual stress pattern (b) Critical weak-axis buckling strength Figure 3-5 Influence of residual stresses (W14×90 Gr. 50 column at 500 °C)

60 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

While the primary focus of this study is on compact or near-compact sections, the significance of local flange or web buckling and local geometric imperfections was considered. Previous research has reported that local buckling is not particularly critical at elevated temperatures for structural sections, for which local buckling is not critical at ambient temperature (Wang, 2002; Uy and Bradford, 1995; Ranby, 1998). This observation is corroborated by this study where local buckling was only observed as governing the strength limit state in a few of the FEM analyses of very short columns where the web width- thickness ratios exceed the AISC requirements for compact column sections. To help assess the significance of local geometric imperfections, a W14×90 (Grade 50) column of varying slenderness was analyzed with and without local imperfections at an elevated temperature of 500 oC. The local imperfections were defined by scaling the first-mode local buckling shape, obtained by a linear buckling analysis, to a maximum amplitude of 1/1000 of the local buckling length (equal to a peak flange and web imperfection of about 0.5 mm). Strength deterioration due to the local imperfection was only observed for the shorter members (lengths of 3.1 m for flexural buckling about the strong axis and 1.9 m for flexural buckling about the minor axis), where the maximum difference in critical strengths for analyses with and without the imperfection are 5 % and 3 % for flexural bucking about strong and weak axes, respectively. Thus, these analyses support the assumption that the member strengths are no more sensitive to local buckling at elevated temperatures as compared to ambient temperatures, and the response of compact (and near-compact) sections are fairly insensitive to local geometric imperfections.

3.5 COLUMN STRENGTH ASSESSMENT

Many numerical and experimental studies have been carried out on the behavior of steel columns under elevated temperatures (Burgess et al., 1992; Poh and Bennetts, 1995; Talamona et al., 1996; Toh et al., 2000; Baker et al., 1997). Franssen et al. (1995) used finite element techniques to numerically simulate column response under elevated temperatures and proposed new column design equations for EC3. Talamona et al. (1997) and Franssen et al. (1998) subsequently performed comprehensive analytical studies to investigate the critical temperatures for various I-shaped sections with varying slenderness ratios, yield stresses, member orientations, axial loads, and loading eccentricities. They used the critical axial column strengths from these analyses to confirm the proposed design equations by Franssen

61 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

et al. (1995). The EC3 column design equations (EC3, 2003) have since been modified to incorporate the proposed revisions. These prior studies provide the impetus for this current study to independently assess the nominal strength provisions of the latest EC3 (2003) standard and the new AISC (2005) specification.

3.5.1 AISC Column Strength Equations

The nominal column strength P of the AISC specification at ambient temperature is cr0, AISC calculated as follows:

⎡⎤Fy 0 Fe0 For FFye00≤ 2.25 PAcr0, AISC= ⎢⎥0.658 F y0 (3.2) ⎣⎦⎢⎥

For FFye00> 2.25 PAcr0, AISC = 0.877 Fe0 (3.3)

2 π E0 where Fe0 = 2 (3.4) ⎛⎞KL ⎜⎟ ⎝⎠r

where, F and E are the yield stress and elastic modulus; F is elastic buckling stress, y0 0 e0 given by (3.4); and A , r , and K are cross-sectional area, radius of gyration, and effective buckling length factor, respectively. The AISC equations for calculating the critical load at elevated temperatures,

PTcr, AISC (), are identical to Eqs. from (3.2) to (3.4), except that the elastic modulus and yield

strength terms are replaced by their temperature dependent counterparts, E()T and FTy (), which are determined using the EC3 reduction factors of Eq. (3.1) and Table 3-1.

3.5.2 EC3 Column Strength Equations

The EC3 column strength Pcr0, EC 3 at ambient temperature is calculated as follows:

(3.5) PPcr0, EC 3= χ 0y 0 1 χ =1.0≤ (3.6) 0 22 ϕϕλ000+−

62 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

⎡ 2 ⎤ ϕ =0.5 1+−+αλ00 0.2 λ 0 ⎣⎢ ()⎦⎥ (3.7)

F y0 (3.8) λ0 = Fe0

where α is an imperfection factor, which varies from 0.13 to 0.76 depending on the member properties, such as buckling orientation (i.e. about the weak axis or strong axis), web height to flange width ratio, flange thickness, and yield stress. is a slenderness ratio that is given λ0 by Eq. (3.8) for stocky sections (i.e. Class 1 , Class 2, or Class3 cross-sections, as defined in EC3) and the other parameters are as defined previously. One of the notable differences between the AISC and EC3 equations is that the critical load in EC3 depends on the slenderness ratio, buckling axis, and cross section properties, whereas the AISC strength only

varies with respect to the flexural slenderness ratio, KL r . Design equations at elevated temperatures in EC3 are similar to the ones at ambient

temperature, but with a few important differences. Equations for critical load PTcr,3 EC () are the same as Eqs. (3.5) and (3.6), except that the yield strength of Eq. (3.5) is replaced by its

temperature dependent, PTy (), as specified using the yield strength reduction factor of Table 3-1 and Figure 3-2(c). At elevated temperatures, χ T )( is calculated by Eq. (3.6) but with the following temperature dependent parameters that replace the expressions in Eqs. (3.7) and (3.8):

2 ϕα()TTT= 0.51⎡ ++λλ () ()⎤ ⎣⎢ ⎦⎥ (3.9)

KTy () λλ()T = 0 (3.10) KTE ()

(3.11) α = 0.65 235/ Fy0

where λ 0 is as specified in Eq. (3.8) and KTy () and KTE () are the reduction factors of Table 3-1 and Figure 3-2(c).

63 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.5.3 Assessment of Column Strengths

The AISC and EC3 column strength equations are compared to FEM simulations of two column sections under various temperatures and slenderness ratio. The columns consist of W14×22 and W14×90 sections with Gr. 50 and 36 steels (W360×32.9 and W360×134

sections with Fy =345 MPa and 250 MPa). Member section properties are summarized in

Table 3-2, where h , tw , bf , and t f are the height, web thickness, flange width and flange thickness, respectively. As is evident from the ratio of strong to weak axis moment of inertia,

I x / I y , the W14×22 represents a beam type geometry, whereas the W14×90 represents a column geometry. Per EC3, the imperfection factors for these cross sections are α x = 0.21 and α y = 0.34.

Table 3-2 Steel section data

h tw bf t f Section h /tw bf / 2t f I x / I y (mm) (mm) (mm) (mm) W14×90 356 11.2 369 18.0 25.9 10.3 2.8 W14×22 349 5.8 127 8.5 53.7 7.47 28.4 HEA100 96 5 100 8 19.2 5.0 2.5

The AISC characterizes column cross sections by the width to thickness ratios of the flanges and webs to denote the transition between sections that are expected to be controlled by local flange or web buckling prior to section yielding. Referring to Table 3-2, both of the

W14 sections satisfy the AISC criteria for compact flanges of btf 2 f < 13.5 and 15.8 for

Gr.50 and 36 steel, respectively. On the other hand, the web slenderness of the W14×22 section ( htw = 53.7) exceeds the limiting AISC compactness criteria of htw = 35.9 and 42.1 for Gr.50 and 36 steel, respectively. Therefore, these data indicate that the W14×22 is expected to be sensitive to local web buckling at high stresses, whereas other local buckling modes should not affect the results. These two W14 sections are intended to represent the range of behavior for rolled wide-flange members encountered in design practice.

64 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 ABAQUS-S 1 ABAQUS-S ABAQUS-W ABAQUS-W AISC AISC 0.8 0.8 EC3-S EC3 (T)

y0 EC3-W y Proposed 0.6 0.6 / P cr0 (T) / P (T) P 0.4 cr 0.4 P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) ambient temperature (b) 100 °C

1 ABAQUS-S 1 ABAQUS-S ABAQUS-W ABAQUS-W AISC AISC 0.8 0.8 EC3 EC3 (T) (T)

y Proposed y Proposed 0.6 0.6 (T) / P (T) / P

cr 0.4 cr 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (c) 200 °C (d) 300 °C

1 ABAQUS-S 1 ABAQUS-S ABAQUS-W ABAQUS-W AISC AISC 0.8 0.8 EC3 EC3 (T) (T)

y Proposed y Proposed 0.6 0.6 (T) / P (T) / P

cr 0.4 cr 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (e) 400 °C (f) 500 °C

65 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 ABAQUS-S 1 ABAQUS-S ABAQUS-W ABAQUS-W AISC AISC 0.8 0.8 EC3 EC3 (T) (T)

y Proposed y Proposed 0.6 0.6 (T) / P (T) / P (T)

cr 0.4 cr 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (g) 600 °C (h) 700 °C

1 ABAQUS-S ABAQUS-W AISC 0.8 EC3 (T)

y Proposed 0.6

(T) / P

cr 0.4 P

0.2

0 0 50 100 150 200 L / r (i) 800 °C Figure 3-6 Critical compressive strengths of W14×90 Gr.50 column

Superimposed in Figure 3-6(a) are FEM simulation results and nominal strengths calculated according to the AISC and EC3 provisions for the W14×90 (Gr. 50) column at ambient temperature. These results are shown as a benchmark against which to judge the differences in the models at elevated temperatures. Critical strengths of member were investigated for elevated temperature increments of 100 °C up to 800 °C. Representative results are shown in Figure 3-6(b)-(i) for temperatures from 200 °C to 800 °C. As noted previously, the results at 800 °C have limited practical impact but are included to show the bounds of response. Simulation results are shown for both strong and weak axis buckling for slenderness ratios between from 20 to 200. Compared to the ambient temperature case, the differences between simulated results for strong versus weak axes decrease at higher temperatures. Clearly evident in these figures is that the AISC strength equations are

66 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

unconservative at elevated temperatures, particularly for slenderness ratios between 40 and 100 and temperatures above 500 °C. For instance, referring to the strength ratio comparisons in Figure 3-7, at 500 °C the nominal strengths calculated by the AISC provisions are up to 60 % larger than the critical strengths as calculated by simulation. On the other hand, the EC3 column strength equations match the simulated results within about 20 %.

60 AISC EC3 Proposed 40

20 Error (%) 0

-20

0 50 100 150 200 L / r Figure 3-7 Percentage error in the calculated compression strength of W14×90 Gr.50 column at 500 °C

3.5.4 Proposed Column Strength Equations

Motivated by the large discrepancy between the AISC provisions and the simulated results, the authors developed an alternative column strength equation that is similar in format to the AISC equations but with greatly improved accuracy at high temperatures. The proposal is to use the following equation for elevated temperatures in lieu of Eqs. (3.2) and (3.3):

⎡⎤FTy () PT()= ⎢⎥ 0.42FTe () AFT () (3.12) cr, Prop⎢⎥ y ⎣⎦ π 2 ET() where FTe ()= 2 (3.13) ⎛⎞KL ⎜⎟ ⎝⎠r

This equation is similar to the AISC equation (2) but with a different coefficient and exponent. Like the AISC equations, this model utilizes the EC3 reduction factors to calculate

67 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

FTy () and ET(). As compared with other data in Figure 3-6(b)-(i) and Figure 3-7, the nominal strength by the proposed equations track the simulation data fairly well, closer in fact that than the EC3 equations at temperatures greater than 300 °C to 400 °C, where structural fire analyses are important.

3.5.5 Column Test Data

To further substantiate these analyses, results of the finite element simulations and nominal strengths are compared to test data reported by Franssen et al. (1998). The column tests were of a HEA100 section, whose sizes are shown in Table 3-2. Data for five column tests at varying temperatures and lengths are summarized in Table 3-3 along with results from finite element simulations and the three nominal equations (AISC, EC3, and the newly proposed equations). Measured steel yield strengths at ambient temperature (as reported by Franssen et al.) were used for Fy0 in the analytical simulations and strength equations. Flexural buckling about the weak axis was the dominant mode of failure in all cases. Referring to Table 3-3, four of the five finite element simulations predict strengths within 3 % of the measure strengths, thus confirming the validity of the simulations as a basis for evaluating the design models. Critical strengths calculated by EC3 are all within 30 % of the test data, whereas those by AISC equations are unconservative by up to 65 %. Strengths predicted using the proposed equations are within 10 % of the measured test data.

Table 3-3 Measured and calculated strengths of column tests

Test Pcr kN ( PPcr cr, Test ) Lry T °C name Test FEM AISC EC3 Proposed 107 142 123 112 CL1 20 694 110 (0.97) (1.29) (1.12) (1.02) 244 414 320 277 CL3 50 474 251 (0.97) (1.65) (1.27) (1.10) 143 250 177 159 SL40 79 525 170 (0.84) (1.47) (1.04) (0.94) 131 198 145 138 AL5 108 457 127 (1.03) (1.56) (1.14) (1.09) 103 125 104 100 BL6 137 446 105 (0.98) (1.19) (0.99) (0.95)

68 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.5.6 Influence of Yield Strength and Section Geometry

Results of analyses to examine the influence of yield strength and section properties are shown in Figure 3-8(a)-(b). Comparing Figure 3-8(a) to Figure 3-6(f), the trends in both the simulation and relative accuracy of the design equations is essentially the same for Gr. 36 as Gr. 50 steel. The influence of section proportions (W14×22 versus W14×90) is seen by comparing Figure 3-8(b) and Figure 3-6(f). For the W14×22 only minor axis flexural buckling is considered due to the large difference in strong versus weak axis properties. In Figure 3-8(b) the effect of web slenderness web in the W14×22 is apparent, where the simulated results drop off compared to the design equations at low slenderness ( L / r < 30)

where the critical stress exceeds about 0.6 FTy (). This occurs because at these stress levels local web buckling, which is not reflected in the column strength equations, begins to control the critical strength. It turns out that for this column, these discrepancies are not of much practical significance since the column length corresponding to L / r = 30 is only 0.80 m. Except for these cases where web buckling is critical (and is expected based on the fact that the ht/ w exceeds the AISC limit for compact webs), the critical strengths agree well with the proposed equation. Additional analyses of the W14×22 section at other temperatures confirm

that the critical web buckling stress of about 0.6 FTy () is fairly constant across various temperature ranges.

1 ABAQUS-S 1 ABAQUS-W ABAQUS-W AISC AISC EC3 0.8 0.8 EC3 Proposed (T) (T) y y Proposed 0.6 0.6 (T) / P (T) / P cr cr 0.4 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) W14×90 Gr.36 (b) W14×22 Gr.50 Figure 3-8 Comparative assessment of column compression strength at 500 °C

69 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.6 BEAM STRENGTH ASSESSMENT

Design equations for laterally unsupported beams require evaluation of torsional-flexural instability effects, which for I-shaped sections involves consideration of non-uniform torsion and warping restraint. Compared to columns and laterally supported beams, there are relatively few studies of laterally unsupported beams under fire conditions. Bailey et al. (1996) studied the critical temperatures for several beam sections with different loading patterns and commented that the then current EC3 provisions were unconservative for laterally unsupported beams. Piloto and Vila Real (2000) performed an experimental study of electronically heated beams and reported that the measured critical temperatures were scattered and generally higher than the theoretical or design temperatures. They attributed the variations to the complexity of the phenomena and the difficulty in conducting the experiments. Vila Real et al. (2000, 2004a, 2004b) numerically studied the critical temperatures and strength for various loading patterns, from which they proposed alternative design equations that were later incorporated in EC3 (2003). Building upon prior research, analytical results for laterally unsupported beams are compared with design equations of the AISC and EC3 specifications, similar to the column comparisons. As in the column study, the large discrepancy between the AISC strength equations and simulation results prompted the proposal of alternative equations for evaluating beams at elevated temperatures.

3.6.1 AISC Beam Strength Equations

The AISC equations for beam strength at ambient temperature are given by the following equations, where M p0 is the plastic moment and M r0 is the initial yield moment (reduced to account for residual stresses), E0 and G0 are the elastic moduli, J is the torsional constant,

Cw is the warping constant, and λ is the slenderness ratio (= Lr/ y ):

For λ ≤ λp0 M cr0, AISC= M p0 (3.14)

⎛⎞λλ− For λ <≤λλ MMMM=− − p0 (3.15) pr00 cr0, AISC p 0() p 0 r 0 ⎜⎟ ⎝⎠λλrp00−

70 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

2 π ⎛⎞π E For λ > λ MEIGJIC=+0 (3.16) r0 cr0, AISC 0 y 0 y w ⎜⎟ λλrryy⎝⎠

The slenderness ratios, λp0 and λr0 , correspond to the transitions between full plastic bending capacity, inelastic lateral-torsional buckling, and elastic lateral-torsional buckling, represented by Eqs. (3.14), (3.15) and (3.16), respectively. λp0 is determined from empirical data, while λr0 corresponds to the theoretical slenderness where the critical elastic buckling

moment, per Eq. (3.16), is equal to the initial yield moment M r0 . The critical moment for elastic lateral torsional buckling with Eq. (3.16) is theoretically derived (Timoshenko and Gere, 1961) and that for inelastic buckling with Eq. (3.15) is a linear interpolation between the transition points with Eqs. (3.14) and (3.16). These transition points are calculated by the following equations, where Sx is the elastic section modulus about strong axis and the other terms are as defined previously:

E0 λp0 =1.76 (3.17) Fy0

X10 2 λrL020=++11X F (3.18) FL0

M rxL00= SF (3.19)

π EGJA00 where X10 = (3.20) Sx 2

2 CSwx⎛⎞ X 2 = 4 ⎜⎟ (3.21) I y ⎝⎠GJ

FFLy00= 0.7 (3.22)

As specified in AISC (2005), the critical moment under elevated temperatures

M cr, AISC ()T is obtained from Eqs. (3.14) to (3.21) by modifying E, G and Fy using the reduction coefficients KTE () and KTy ().

71 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.6.2 EC3 Beam Strength Equations

The EC3 beam strength equations at ambient temperature have a similar format to the EC3 column equations:

(3.23) M cr0, EC 3= χ LT 0M p 0 1 χ =1.0≤ (3.24) LT 0 22 ϕϕλLT000+− LT LT where ⎡ 2 ⎤ ϕαλλ=+0.5 1 (LT 00 −+ 0.2) LT (3.25) LT 0 ⎣⎢ LT ⎦⎥

M p0 (3.26) λLT 0 = M cr0, e (3.27) M pxy00= ZF

and χLT 0 is the reduction factor for lateral torsional buckling, αLT is an imperfection

factor which depends on the section proportions (α LT = 0.21 is used for rolled sections with

the web height to flange width ratio hb/2f ≤ and αLT = 0.34 for hb/2f > ), M cr0, e is the

elastic critical moment for lateral torsional buckling, and Z x is plastic modulus about strong axis. Eq. (3.26) is specified for compact sections, which correspond to the Class 1 or Class 2 designations in EC3. Differences between the beam equations in EC3 under fire conditions and at ambient temperature are similar to those between the corresponding column equations. The primary

changes are in the definition of ϕLT , α LT , and λ LT , which are defined for elevated temperatures by the following equations that replace Eqs. (3.25) - (3.27):

2 (3.28) ϕαλλLT (TTT )=0.5[1+LT LT ( )+ LT ( )] (3.29) α LT =0.65 235/Fy0

KT() y (3.30) λλLT ()T = LT 0 KTE ()

72 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

Note that carries units of MPa in Eq. (3.29). Fy0

3.6.3 Proposed Beam Strength Equations

As an alternative to the AISC beam strength equations, the following equations are proposed to evaluate bending strengths at elevated temperatures using a similar format to the AISC design equations:

CT() ⎛⎞λ X ⎡⎤ For λ ≤ λr ()T Mcr, Prop() T=+ MT r ()⎣⎦ MT p () − MT r ()⎜⎟ 1 − (3.31) ⎝⎠λr ()T

2 ππ⎛⎞ET() For λ > λ ()T M() T=+ ETIGTJ () () IC (3.32) r cr, Prop y y w ⎜⎟ λλrryy⎝⎠

In contrast to the equations at ambient temperature where the design equations are distinguished into three regions of behavior, here only two equations are used to model inelastic and elastic lateral-torsional buckling. As will be shown later, these equations reflect that fact that at elevated temperatures the critical moment drops off quickly from the plastic moment at small slenderness values. The distinction between inelastic and elastic behavior is

indicated by the slenderness value λr ()T , which corresponds to the elastic moment at the onset of yielding, M r ()T . The governing equations for λr ()T and M r ()T are the same as the AISC values, Eqs. (3.18) to (3.22) with reduced yield stress and elastic modulus, except

that the initial yield stress FL is replaced by the following:

FTLpr()= FT ()− FT () (3.33)

FTppy()= KTF () 0 (3.34)

FTryr()= K () TF0 (3.35)

Compared to the original AISC equations, the major change is to base FTL () on the temperature dependent proportional limit FTp () rather than the yield stress. Fr0 is the

residual stress at the ambient temperature, which is specified in AISC as Fr0 = 69 MPa for

73 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

rolled shapes. Implied in Eq. (3.35) is the assumption that the residual stresses under elevated

temperatures are proportional to the reduction factor of yield strength, KTy (). The termCTX () in Eq. (3.31) is an exponent that is defined as a bilinear function with respect to the temperature according to the following equation for T > 100 oC,

T CTX ()=+ 0.6 ≤ 3.0 (3.36) 250 where T carries units of oC.

3.6.4 Assessment of Beam Strengths

Comparisons between the simulated results and design equations for bending strength are

shown in Figure 3-9 through Figure 3-11. Simulated results for a W14×22 Gr. 50 ( Fy =345 MPa) beam of varying lateral slenderness are compared to the AISC and EC3 equations at ambient temperatures in Figure 3-9(a). As in the column analyses, the simulated points correspond to the peak point in load versus deflection curves, such as shown previously in Figure 3-4(b). The comparison in Figure 3-9(a) demonstrates that even at ambient temperatures, the AISC flexure equations tend to be unconservative relative to the simulated results and EC3 equations. In this example, the maximum error occurs at an intermediate slenderness, λ = 100, where the AISC strength is about 30 % larger than the simulated results.

74 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 ABAQUS 1 ABAQUS AISC AISC 0.8 EC3 0.8 EC3 Proposed (T) p0 p 0.6 0.6 / M cr0 (T) / M

M 0.4 cr 0.4 M

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r b y b y (a) ambient temperature (b) 100 °C

1 ABAQUS 1 ABAQUS AISC AISC 0.8 EC3 0.8 EC3 Proposed Proposed (T) (T) p p 0.6 0.6 (T) / M (T) / M (T)

cr 0.4 cr 0.4 M M

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r b y b y (c) 200 °C (d) 300 °C

1 ABAQUS 1 ABAQUS AISC AISC 0.8 EC3 0.8 EC3 Proposed Proposed (T) (T) p p 0.6 0.6 (T) / M (T) / M (T)

cr 0.4 cr 0.4 M M

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r b y b y (e) 400 °C (f) 500 °C

75 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 ABAQUS 1 ABAQUS AISC AISC 0.8 EC3 0.8 EC3 Proposed Proposed (T) (T) p p 0.6 0.6 (T) / M (T) / M

cr 0.4 cr 0.4 M M

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r b y b y (g) 600 °C (h) 700 °C

1 ABAQUS AISC 0.8 EC3 Proposed (T) p 0.6

(T) / M (T)

cr 0.4 M

0.2

0 0 50 100 150 200 λ = L / r b y (i) 800 °C

Figure 3-9 Critical bending moment strengths of W14×22 Gr.50 beam

100 AISC EC3 80 Proposed

40 Error (%) 20

-20 0 50 100 150 200 L / r Figure 3-10 Percentage error in the calculated bending moment strength of W14×22 Gr.50 beam at 500 °C

76 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

W14x22 Fy 250MPa T500C W14x90 Fy 345MPa T500C

1 ABAQUS 1 ABAQUS AISC AISC 0.8 EC3 0.8 EC3 Proposed Proposed (T) (T) p p 0.6 0.6 (T) / M (T) /M cr cr 0.4 0.4 M M

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r b y b y (a) W14×22 Gr.36 (b) W14×90 Gr.50 Figure 3-11 Comparative assessment of beam bending moment strength at 500 °C

Strengths for the W14×22 beam at elevated temperatures, 200 °C, 500 °C and 800 °C, are compared in Figure 3-9(b)-(i). Included are the FEM simulation results and the nominal strengths calculated by the three design equations (AISC, EC3, and the proposed model). Immediately apparent from these comparisons are the large discrepancies between the AISC equations and the simulated results. Referring to Figure 3-10, at 500 °C the largest percentage differences occur for intermediate slenderness values of λ = 80 to 100, where the AISC strengths are about 80% higher than the simulated strengths at temperatures. In part, the discrepancies arise because at elevated temperatures the simulated results indicate that the moment strength drops off quickly with increasing slenderness, whereas the AISC equations

preserve the plastic moment, M p ()T , up to about λ = 40. In contrast to the AISC equations, the EC3 model tracks the simulated results fairly well. As indicated previously, the new equations proposed by the authors are intended to provide good accuracy while maintaining a similar concept and format of the AISC equations. One of the basic features of the proposed model is that it preserves use of the elastic critical load at high slenderness values. At lower temperatures (e.g., 200 °C) the transition between the inelastic and elastic response, at λ = 110, is quite abrupt. At higher temperature, as the proportional limit is reduced through KTp (), the inelastic curve controls over a larger range of slenderness. As evident from Figure 3-9(e)-(i) and Figure 3-10, at above about 400 °C results from the proposed model and the EC3 model are quite similar and agree well with the simulation data. The proposed model is less conservative than the EC3

77 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

equations at lower temperatures (e.g., 200 °C in Figure 3-9(b)-(d)), owing to the desire to maintain close conformance with the AISC relationships at ambient temperature. Results shown in Figure 3-11 illustrate that the trends observed in Figure 3-9 for the W14×22 Gr. 50 beam are generally representative of other yield strengths and section

properties. Results for Gr. 36 steel ( Fy = 250 MPa) at 500 °C are shown in Figure 3-11(a).

Here the effect of the residual stresses (assumed at Fr0 = 69 MPa at ambient temperature) are proportionally larger for the Gr. 36 steel, and the increased softening effect leads to closer agreement between the EC3 and proposed equations, as compared to the results shown in Figure 3-9(f) for Gr. 50 steel. Results for a W14×90 Gr. 50 beam at 500 °C are shown in Figure 3-11(b). In this case, the strengths calculated by the EC3 and proposed models agree very well with the simulated results up to about λ = 100. At higher slenderness these two models tend to underestimate the critical load, because the proportions of the W14×90

(smaller I x / I y ratio) are such that the in-plane pre-buckling deformations tend to increase the critical load. This beneficial effect of in-plane deflections is picked up in the simulation but not in the critical load equations. Overall, the results in Figure 3-11 confirm that the EC3 and the proposed models provide accurate results for the typical range of steel shapes and yield strengths used in practice.

3.7 BEAM-COLUMN STRENGTH ASSESSMENT

Laterally unsupported beam-columns subjected to combined axial compression and strong axis bending experience combined limit states of yielding, lateral buckling and lateral torsional buckling. As with laterally unsupported beams, there are relatively few studies of design equations for beam-columns under fire conditions. Lopes et al. (2004) compared numerical simulations to equations in the 1995 and 2003 editions of EC3 and confirmed that the 2003 provisions are more accurate and conservative than the 1995 provisions. Toh et al. (2000) proposed an approach to find combinations of the critical axial force and bending moment using Rankin’s method. In the following discussion, results of the present study of columns and beams is extended to evaluate the AISC and EC3 design equations for beam-columns subjected to axial load and major-axis bending.

78 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.7.1 AISC Beam-Column Strength Equations

The AISC beam-column strength equations employ a simple bilinear combination of the ratio of axial and bending effects. As given by the following, the equations for elevated temperatures are identical to those at ambient temperatures except that the nominal strengths are calculated at elevated temperatures:

P PM8 For u ≥ 0.2 uu+≤x1.0 (3.37) PTcry, AISC () PTMTcry,, AISC () 9crx AISC () P PM For u < 0.2 uu+≤x1.0 (3.38) PTcry, AISC () 2()()PTMTcry,, AISC crx AISC

where Pu and M ux are the factored axial load and bending moment about the strong axis and PTcry, AISC () and M crx, AISC ()T are the critical axial strength for flexural buckling and the critical bending moment for lateral torsional buckling, respectively. Assuming the member to be pin-ended about both axes, the column strength PTcry, AISC () is controlled by flexural

buckling about the weak axis. Per the AISC Specification, M ux should include second-order effects. For the pin-ended column subjected to uniform end moments, M x,end , the second- order moment at the mid-span is calculate as:

M xend, MTux ()= (3.39) 1()− PPucre, T where π 2 ET() A PT()= (3.40) cr, e λ 2

According to Eq. (3.39), the second-order amplification factor is calculated based on the critical load determined using ET() and per Eq. (3.1).

79 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

3.7.2 Proposed Beam-Column Strength Equations

The proposed equations employ the same interaction check and amplification factor as the

AISC equations, except that the nominal strength terms, PTcry,Prop () and M cry,Prop ()T , are calculated according to the newly proposed equations.

3.7.3 EC3 Beam-Column Strength Equations

The EC3 beam-column equations for combined axial load and bending moment are as follows:

PM()T uux (3.41) +≤kTLT () 1.0 PTcry,3 EC () MTcr,3 EC () where, (3.42) PTcry,3 EC ()= χ y () TPT y () P u (3.43) kTLT ()=− 1μLT () T PTcry,3 EC ()

μλLT (TT )=−≤ 0.165y ( ) 0.15 0.9 (3.44)

where Pu and M ux ()T are the factored axial load and bending moment about the strong axis and PTcr,3 EC () and M cr,3 EC ()T are the critical axial strength for flexural buckling and the critical bending moment for lateral torsional buckling, respectively, and other terms are as defined previously. Note that the Eq. (3.44) for μLT ()T is shown in simplified format for a pin-ended beam-column subjected to uniform end moments.

3.7.4 Assessment of Beam-Column Strengths

The same FEM analysis model used for the column and beam studies is used for beam- column study, including non-uniform torsion and warping restraint effects. The limit state combinations of axial load versus end moment are compared in Figure 3-12 for a W14×90

(Gr.50) member with λ = 60 at various elevated temperatures. The curve in the AISC and proposed equations is due to the second-order effects in M ux ()T per Eq. (3.39). In general both the EC3 and proposed equations show good agreement with the simulated results. It is

80 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS difficult to say whether the bi-linear or linear interaction equations are more appropriate, since much of the accuracy of the interaction check depends on the accuracy of the nominal axial load and moment strength. Following the previous discussion of the axial load and moment strengths, the AISC provisions at elevated temperatures are highly unconservative relative to the simulated results and other design equations. The errors are larger for bending dominated (as opposed to axial dominated) members, owing to the underlying errors in the

M cr, AISC ()T equations discussed previously.

1 1 ABAQUS ABAQUS AISC AISC 0.8 0.8 EC3 EC3 Proposed Proposed (T) (T)

y 0.6 y 0.6

(T) / P 0.4 (T) / P 0.4 u u P P

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 M (T) / M (T) M (T) / M (T) x,end p x,end p (a) 200 °C (b) 300 °C

1 1 ABAQUS ABAQUS AISC AISC 0.8 0.8 EC3 EC3 Proposed Proposed (T) (T)

y 0.6 y 0.6

(T) / P 0.4 (T) / P 0.4 u u P P

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 M (T) / M (T) M (T) / M (T) x,end p x,end p (c) 400 °C (d) 500 °C

81 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 1 ABAQUS ABAQUS AISC AISC 0.8 0.8 EC3 EC3 Proposed Proposed (T) (T)

y 0.6 y 0.6

(T) / P 0.4 (T) / P 0.4 u u P P

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 M (T) / M (T) M (T) / M (T) x,end p x,end p (e) 600 °C (f) 700 °C

1 ABAQUS AISC 0.8 EC3 Proposed (T)

y 0.6

(T) / P (T) 0.4 u P

0.2

0 0 0.2 0.4 0.6 0.8 1 M (T) / M (T) x,end p (g) 800 °C Figure 3-12 Critical axial load and moment strengths of W14×90 Gr.50 (λ=60) beam-column

Results in Figure 3-13 illustrate similar results for a lower yield steel strength and alternative steel section. Comparing Figure 3-12(d) and Figure 3-13(a), the differences between Gr. 50 and Gr. 36 steel at 500 °C indicate that results from the simulation, EC3 and proposed equations tend to converge for Gr. 36 steel, presumably because the residual stresses and non-proportional limit are closer together. Conversely, the AISC results, which do not take into account the reduced proportional limit loose accuracy for Gr. 36. Comparing Figure 3-12(d) and Figure 3-13(b), the differences between the W14×90 to W14×22 sections lead to minor changes that can be traced back to differences in the axial load and moment strengths.

82 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

1 1 ABAQUS ABAQUS AISC AISC 0.8 0.8 EC3 EC3 Proposed Proposed (T) (T)

y 0.6 y 0.6

(T) / P 0.4 (T) / P 0.4 u u P P

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 M (T) / M (T) M (T) / M (T) x,end p x,end p (a) W14×90 Gr.36 (b) W14×22 Gr.50 Figure 3-13 Comparative assessment of beam-column strengths at 500 °C

3.8 SUMMARY AND CONCLUSIONS

Codified design equations for steel members subjected to high temperatures are an important step towards facilitating consensus standards to evaluate the structural safety of buildings to fire. Incorporation of fire provisions in structural design standards also raises awareness of the issues and has encouraged research and development to validate and improve the provisions. Design equations first introduced in the Eurocode EC3 standard in 1992 were subsequently updated and improved through studies by a number of researchers. Similar design requirements have only recently been introduced in a new appendix to the 2005 AISC Specification. The results summarized herein provide an independent assessment of the EC3 and AISC provisions for columns, laterally unbraced beams, and laterally unbraced beam-columns, which are compared to data from detailed nonlinear finite element analyses. Utilizing three- dimensional shell finite elements, the numerical simulations incorporate the effects of local and overall buckling and instability, including non-uniform torsion and warping restraint effects. The simulation results are not entirely independent of the design standards, since the simulation models are based on stress-strain data at elevated temperatures that is specified in EC3 and referenced in AISC. However, through comparisons between the EC3 stress-strain models and test data of steel at elevated temperatures and column tests, the authors have reaffirmed the accuracy of the EC3 stress strain data. The simulation study included about

83 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

four hundred finite element analyses, including the effects of varying slenderness, steel temperature, steel yield strength, residual stresses, and section properties. Comparison between the AISC provisions and the simulation results indicate that the AISC provisions significantly over-estimate the nominal strength of columns, beams and beam-columns at elevated temperatures. The AISC column strengths were up to 60 % higher than the simulated results, and the beam and beam-column strengths were 80-100 % higher. These large discrepancies indicate that the approach used in the AISC provisions of simply

modifying the elastic moduli ( E and G ) and yield strength ( Fy ) in the otherwise standard (ambient temperature) design equations is inaccurate. Thus, the large variations are due primarily to the fact that the steel stress-strain curve loses its characteristic bi-linear elastic- plastic response at elevated temperatures. Comparisons between the EC3 provisions and simulation results indicate that the EC3 equations are within 20 % of the simulations. This good agreement reflects refinements made to the EC3 provisions since their first publication. While it would be tempting to simply recommend that the EC3 provisions be adopted into the AISC Specification, the format of the EC3 provisions is quite different from the AISC ambient strength design provisions. In the interest of maintaining similarity in format and style between ambient and elevated temperature provisions in the AISC Specification, new design equations are proposed as alternatives to those in the 2005 edition of the AISC specification. The proposed alternative equations for calculating the nominal column and beam strengths are validated against the simulation data and reduce the discrepancies to less than about 20-30 %, which is similar the accuracy of the EC3 provisions. While similar in format, the proposed equations are distinct from the AISC provisions for ambient temperatures, and thus there is a discontinuity in response between the two sets of equations. The proposed equations are only intended for use at elevated temperatures, which can be assumed as temperatures higher than 200 °C, a temperature that would rarely if ever be exceeded except under fire conditions.

3.9 LIMITATIONS AND FUTURE RESEARCH

While this study helps to both validate and improve current design provisions for steel members at elevated temperatures, the scope is limited to assessment of individual members comprised of bi-symmetric I-shaped members with compact (or near-compact) sections. The current study treats the problem deterministically, using nominal properties (material

84 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS strengths, initial imperfections, etc.) and nominal high-temperature material response parameters from EC3, whereas the actual conditions involve considerable uncertainty. Moreover, the current study assumes a uniform temperature distribution through the member cross section, which may not be conservative for slender members where non-uniform temperatures can induce member deformations that may accentuate destabilizing geometric nonlinear effects, i.e., moments induced by P-δ action (e.g., see Wang, 2002). Finally, the current study is limited to evaluating the structural response, conditioned on the induced fire temperature, which ignores the challenges and large uncertainties in predicting the elevated temperatures and other fire effects. Among the many areas that are ripe for future research and development, two areas most in need of further study are reliability analysis of uncertainties and the evaluation of indeterminate system response. The first of these should address a broader range of design and response parameters, including variations in cross-section types, thermal and mechanical loading intensities and distributions, material properties at ambient and elevated temperatures, and temperature induced distortions. The second topic would extend the assessment of individual components to assess collapse safety of indeterminate systems, considering the nonlinear redistribution of forces and the associated uncertainties in loading and response effects.

85 CHAPTER 3. ANALYSIS OF INDIVIDUAL MEMBERS

CHAPTER 4 ANALYSIS OF GRAVITY FRAMES

4 ANALYSIS OF GRAVITY FRAMES

4.1 GENERAL

4.1.1 Overview

This chapter is concerned with the behavior of indeterminate structures to fire-induced temperature effects. In particular, the chapter addresses issues associated with restraint to thermal expansion and inelastic redistribution of forces under large deformation. Failure of individual members could initiate global building collapse if an alternative load-carrying path does not exist. Evaluation of alternative load-carrying paths is often performed by eliminating the individual damaged members and checking stability of the rest of the structure. This approach works well for the limit state evaluation in the lower bound sense; however, it does not necessarily simulate structural behavior during fires including interactions between the heated members and surrounding structure. In order to understand structural building behavior under fires, such interactions need to be carefully simulated with indeterminate frame analysis models. Thermally induced longitudinal deformation is typically not a critical issue when stability of a single individual member is studied. However, this is more significant for members in indeterminate frame structures. If constraint of thermally induced longitudinal deformation of steel members is high, large compressive axial force can be induced at elevated temperatures. These two issues (redistribution of forces and thermally induced effects) can be simulated through global stability evaluation using indeterminate frame models. In this chapter, a typical steel-framed office building is used as a benchmark design to examine the behavior of an indeterminate structural system subjected to a fire. The primary focus is on simulating the initiation of building collapse using sub-assembly models to evaluate possible collapse mechanisms for localized fire scenarios. The sub-assembly models are useful for precise analytical simulations with a reasonable computational effort.

87 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Analysis results for several sub-assembly models are presented followed by general descriptions about the benchmark office-type building design.

4.1.2 Benchmark Office-type Building Design

Shown in Figure 4-1 is the structural floor plan of the benchmark office-type building design, which will be used throughout this chapter. The study focuses on the structural behavior of gravity frames (mainly the frame on the column line 3) in the fire compartment. The connections between the columns and beams are typical shear-tab bolted connections as shown in Figure 4-2. The lateral load resisting frames (shown as braced frames in Figure 4-1) are away from the fire compartment. Thus, vertical collapse is studied and the gravity frames are assumed to be laterally supported. The plan is rectangular measuring 25.6 m in the east-west direction and 32.0 m in the north-south direction with 3.2 m modules. The building has 10-stories with a 4 m floor height (i.e., 40 m of total height) and the plan is the same at each floor. One-way concrete slab/deck systems are supported by the secondary beams, which run north-south. There are three framing bays in the east-west direction, and five 6.4 m bays in the north-south direction. The core (including hallway, utilities, elevators, stairs, and other mechanical space) is located at the 6.4 m center bay, while side bays with 9.6 m span are office spaces.

Figure 4-1 Floor plan of benchmark building design

88 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

(a) Elevation on column line C (b) Elevation on column line 3 Figure 4-2 Details of column-beam shear tab connections

Design dead load and live load (i.e., unfactored loads) are 4310 N/m2 (90 psf) and 2870 N/m2 (60 psf), respectively. The dead load includes the weight of the concrete slab with an average thickness of 114 mm (4.5 in) and steel deck (2870 N/m2, 60 psf), steel weight (720 N/m2, 15 psf), and the weight of other items such as mechanical equipment, finishes, and partitions (720 N/m2, 15 psf). In addition, exterior cladding weight of 1200 N/m2 (25 psf) is assumed around the building perimeter. Member sections are selected from the American Institute of Steel Construction (AISC, 2005) W-shape rolled sections. The steel is ASTM A992 (Gr.50) with a characteristic (minimum specified) yield strength of 345 MPa (50 ksi) and ultimate tensile strength of 485 MPa (65 ksi).

Factored load ratios of the gravity columns, PPun/φ in accordance with AISC specifications are approximately 0.86 and 0.77 (accounting for 0.6 of the live load reduction factor due to the large supporting area) at the first and second floor level, respectively, where

Pu is the required axial strength, φ is the resistance factor for columns (0.9), and Pn is the nominal axial strength. The critical load combinations for Pu is 1.2DL+1.6LL, where DL is the dead load and LL is the live load. The orientations of the I-shaped columns are shown in the plan. Beams are conservatively designed for serviceability, and the deformation under unfactored live load is about 1/500 of the span. A shear-tab connection is used with three ASTM A325N bolts for beams a and b, and five bolts for beam c. This benchmark building design is simpler than actual designs in practice, but realistic enough to be used as a representative for simulations of building behavior under fires. Typical sections are shown in the plan and listed in Table 4-1.

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Table 4-1 Section sizes (mm)

Member Section h tw bf t f Interior column W14×90 356 11.2 369 18.0 Exterior column W12×72 311 10.9 306 17.0 Beam a W14×22 349 5.8 127 8.5 Beam b W14×38 358 7.9 172 13.1 Beam c W21×62 533 10.2 209 15.6

4.1.3 Failure Mechanisms and Sub-assembly Analysis Models

It is assumed that a fire breaks out in the bay between column lines 2-3 and C-D at the second floor. The fire compartment is shown in the plan (see Figure 4-1) and the compartment partitions are assumed to remain intact during the fire. Depending on the magnitude and duration of the fire as well as other factors such as the properties of the fire insulation, the localized fire may initiate building collapse. Shown in Figure 4-3 are the possible collapse initiation mechanisms: interior column buckling, exterior column buckling, beam yielding, and interactive failure. Buckling of a single column is critical for the interior and exterior column buckling mechanisms, and yielding/sagging of a beam is assumed critical for an isolated beam mechanism. Potential large deformations of Beam c causes lateral displacement of the exterior column at the third floor level and it leads to multi-story (second and third floor) exterior column buckling in the interactive failure mechanism. One of these mechanisms may induce other mechanisms in the subsequent events, which may lead to global building collapse; however, it is assumed that such interaction is limited and the combined failure mechanisms are not considered. This assumption allows use of sub- assembly analysis models, which will be described in the following sections, and drastically simplifies simulations of the complex structural behavior in each mechanism. Failure of bolted connections is a key issue and is considered in these simulations.

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Figure 4-3 Possible failure mechanisms (column line 3)

In order to simulate the failure mechanisms shown in Figure 4-3, three sub-assembly frame analysis models are introduced: interior column sub-assembly, exterior column sub- assembly and beam sub-assembly (Figure 4-4). The interior column sub-assembly is used for simulations of the interior column buckling failure mechanism; while the exterior column sub-assembly can be used for simulations of the three failure mechanisms: exterior column buckling, beam yielding for Beam c, and interactive failure. The beam sub-assembly is useful for simulations of isolated beam yielding failure, such as may occur for the secondary beam (Beam a) as well as Beam c. A notable point for these sub-assembly models is that boundary conditions of the longitudinal deformation constraint are modeled with inelastic translational springs. Therefore, the properties of these springs are carefully evaluated.

(a) Interior column (b) Exterior column (c) Beam sub-assembly sub-assembly sub-assembly Figure 4-4 Sub-assembly analysis models

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4.1.4 Time-temperature Relationships in Localized Fire

The compartment fire shown in Figure 4-1 is assumed to be a fully-developed and ventilation-controlled flashover fire. Relationships between gas temperature and time in the flashover fire are calculated by using a fire model described in Eurocode 1 (EC1, 2002). Details of this Eurocode “Parametric Fire Curve” are shown in Appendix B.4.1. The fire curve is a function of several factors including the geometry of the fire compartment, the fire load and firefighting activities. The assumptions used in the fire simulations are shown in Appendix B.4.1. The maximum gas temperature in the benchmark compartment fire is generally more than 1000 °C and the fire lasts less than one hour. Based on the obtained relationships between time and the gas temperature in the compartment flashover fire, the temperature of structural steel is calculated by using an iterative time-step heat-transfer method described by Buchanan (2002). The primary heat transfer mechanism to the insulated steel members from the gas is conduction, while those of the unprotected members are radiation and convection. Figure 4-5 shows temperatures of the gas and structural steel members with insulation and without insulation. The material of the insulation for the structural steel is gypsum board and the thickness is 9.5 mm. The material of the boundary enclosure of the fire compartment is also gypsum board. The other parameters and assumptions used for the temperature simulations are described in Table B-5 in Appendix B.4.1.

Figure 4-5 Time-temperature relationships in a fire simulation

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4.1.5 Organization of Chapter 4

In this chapter, three separate studies are performed in sections 4.2, 4.3, and 4.4 for the interior column, beam, and exterior sub-assemblies, respectively. These sections are designed be self-contained because they have been or are being planned to be published as individual journal papers. As a result, there may be some repetition of the material.

4.2 EVALUATION OF INTERIOR COLUMN SUB-ASSEMBLY

Takagi, J., Deierlein, G.G. “Collapse Assessment of Steel Gravity Columns under Localized Compartment Fire” (in preparation).

4.2.1 Summary

The stability of interior gravity columns in multi-story steel-framed buildings is evaluated under a localized compartment fire, considering the interaction of the heated column with steel framing in floors above the critical column. The study considers three building heights (five, ten and twenty stories) with structural configurations and details representative of standard construction. Finite element models are combined with analytic equations to evaluate the nonlinear interaction between the heated column and the surrounding floor framing, considering torsional-flexural instability and local flange and web buckling in the heated column and composite behavior and membrane action of the composite floor diaphragm. The simulations of all three building heights indicate that the restraint of the gravity framing above the affected column does not significantly affect the column collapse strength. The restraint is neither stiff enough to generate significant compressive stresses in the column due to thermal expansion nor strong enough to redistribute the gravity load after the heated column buckles. Alternative design options are suggested to improve the robustness of the framing through selective strengthening of gravity framing connection details.

4.2.2 Introduction

Interior gravity steel columns are one of the most vulnerable structural components against fires due to their high axial load ratio. Under deterioration of their vertical load-carrying capacity at elevated temperatures, the columns become unable to support the gravity load. Alternative load-carrying paths may be provided, depending on the strength of the rest of the structure. If an alternative load-carrying path is not provided, the column failure can initiate

93 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES building collapse. The purpose of this research is to investigate the behavior and limit-state of interior gravity columns under a compartment fire as initiation of possible overall building collapse. Research significance lies in realistic structural evaluation of the gravity columns and their interaction with the overall building frame. The gravity columns connect to composite beams with bolted shear-tab connections, which are often typically considered in design as pinned (rotationally free) connections. However, the actual rotational stiffness of this type of connection provides some restraint to longitudinal deformation of the gravity column during fire. The restraint induces additional axial compression to the column from the thermal expansion effect at elevated temperatures which may contribute to column buckling, conversely, the same connections and gravity framing may provide support after the column loses vertical load carrying capacity. The behavior of this mechanism is investigated in detailed analysis models and the effect of the restraint on building safety is discussed. Accurate analytical evaluation of the structural behavior of steel buildings under fires is difficult due to many complex and uncertain phenomena. Detailed analytical modeling of the overall structural system could provide the most reliable simulation results under current research development; however, this approach is generally computationally intensive and not practically applicable. In addition, the nonlinear behavior is complex and not fully understood. Therefore, analytical models of the overall structural system may make it difficult to capture local failure modes. The objectives of this research are to present a practical analytical approach to perform accurate analytical evaluation of steel structures under fires and to closely investigate its characteristic behavior. The approach taken is to limit the focus on localized compartment fires and investigate the case where the interior gravity column is the primary damaged structural component during the fire. By limiting the focus of the study, the analytical models can be simplified by following specifically appropriate sub-assembly models for analysis. Detailed simulations are practically performed with the extracted structural sub-assembly. Shown in Figure 4-6 is one of the simplest sub-assembly analysis models for columns subjected to localized fire. The model consists of a column and constraint springs, which are rotational springs at the top and bottom of the column and a longitudinal spring. While the material properties of the column deteriorate during fire, the properties of the springs, which contribute to simulating the interaction between the column and the surrounding building structure, are not subjected to temperature effects. This type of sub-assembly model has been

94 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES studied by several researchers such as Poh and Benenetts (1995), and Ali et al. (1998); however, the constraint springs are often modeled as elastic and the spring properties are not precisely developed to represent real steel building structures. In order to obtain stiffness of the elastic longitudinal and rotational constraint springs, Wang and Moore (1994) studied a simple two-dimensional two-bay framed model. They reported that the maximum stiffness of the longitudinal constraint spring is about 2 % of that of the column stiffness. Using the elastic spring properties from the simple model, parametric studies with varied length and load ratio of the column lead to their conclusion that enhancement of the critical temperature by the rotational springs and deterioration due to thermal restraint with the longitudinal spring can cancel out. In their study, slender and heavily loaded columns tend to have greater influence for both enhancement and deterioration of the critical temperatures. The research shows interesting results that warrant more realistic modeling of the constraint springs and further case studies to generalize the findings. Poh and Benenetts (1995) studied the same column model shown in Figure 4-6 and compared their analytical results to existing test data. The properties of the elastic constraint springs are obtained referring to experimental results. Ali et al. (1998) summarized a comparison of analytical simulations and test data for columns with a longitudinal spring and without rotational springs in Figure 4-6. Their analytical study, which was performed by parametrically varied member length, constraint stiffness, and load ratio was interesting; however, developing realistic constraint springs was not a primary interest in their studies. Suzuki et al. (2005) studied column behavior under fire in moment frames taking into account inelastic effects from the surrounding structure; however, the constraint spring is not fully modeled to include three-dimensional and large deformation effects.

Figure 4-6 Analysis model of a column with constraint springs

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Stability of these structural systems depends on the combination of several factors, including temperature, axial force, properties of the column, and stiffness and strength of the constraint springs. In simulations under constant axial load and increasing temperatures, axial force of the column first increases due to the thermal expansion effect and constraint by the spring, while the critical strength of the column against buckling decreases due to material degradation. At a certain temperature, the column buckles and drastically loses its axial load carrying capacity; however, this does not necessarily lead to the collapse of the system depending on the property of the constraint spring. The load may be supported by the spring (i.e., the surrounding structure) and an alternative load carrying path may be provided. During this process, the equilibrium points may be discontinuous with respect to simply increasing temperatures. This “snap-through” type behavior is difficult to analytically simulate due to this discontinuity. An additional analytical algorithm has been proposed by several researchers (Franssen, 2000; Suzuki et al., 2005) to switch the incremental analysis parameter from temperature to displacement control during this snap-through. In this study, developing an algorithm for this type of behavior is not a primary interest and the behavior is simulated in an iterative calculation, focusing on the vertical collapse of the system and limiting the number of free degrees of freedom. Realistic evaluation of the constraint springs in sub-assembly analysis model is important for accurate structural simulation under fires. The longitudinal inelastic constraint spring properties are obtained through evaluation of the rotational stiffness and strength of the bolted shear-tab connections considering three-dimensional and large deformation effects. The sub-assembly analysis model contains a continuous column representing realistic rotational boundary stiffness of the column at the fire floor. The limit-state is evaluated in terms of temperatures. Alternative design options are suggested to improve the frame response through selective strengthening of localized details to improve the robustness of the frames.

4.2.3 Analysis Model

4.2.3.1 Modeling of System

Under the localized compartment fires, it is assumed that lateral displacement of the gravity column at each floor is constrained by the surrounding structure. Introducing an inelastic vertical spring to represent the floor framing at each level, the analysis model can be

96 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES simplified as shown in Figure 4-7(a). The properties of the inelastic vertical springs shown in the figure are discussed later. The column at the fire floor (second floor) is heated, while the columns at adjacent floors are assumed to remain at ambient temperature (note the heat conduction effect in these columns is studied in Appendix A.4.3.). Assuming that longitudinal elongation of the gravity columns at the fourth and higher floors is negligible, the analysis model can be further condensed with respect to the load and vertical springs at the higher floors (Figure 4-7(b)). Columns at the first and third floor are not condensed in order to evaluate the effect of the continuous boundary condition. Since the floor plan is assumed to be identical at each floor, the inelastic vertical spring properties and axial load at each floor are also identical. The floor structure at the third floor includes heated beams in the fire compartment, such that the inelastic vertical spring properties differ from those at other floors. However, this effect is assumed to have limited impact on the critical temperatures of the system and the inelastic vertical spring properties at the third floor are assumed to be identical to those at other floors.

(a) Analysis model with nonlinear spring (b) Condensed load and spring at each floor model Figure 4-7 Analysis model for column buckling collapse mechanism

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4.2.3.2 Modeling of Column

The steel columns in the proposed model (Figure 4-7(b)) are simulated with shell finite element models created and run using the ABAQUS software (Hibbitt, Karlsson & Sorensen, 2002). The model assumptions are similar to those used in Chapter 3. Some additional assumptions and differences applied for this study are described as follows:

1. The member is subdivided into 20 shell elements along the length (4 m of the story height), and the flanges and web are each subdivided into eight elements across the cross section. 2. Nonlinear stress-strain curves of steel at the elevated temperatures are adopted from EC3, which are described in Chapter 2 and 3. 3. Linear constraints are applied to the column at each floor level. The three -story continuous column is modeled from the first to fourth floor level. Lateral support is provided at each floor level, while vertical support is only on the first floor level. Only the column on the fire floor (second floor) is heated and those on the other floors remain at ambient temperature. 4. Initial geometric imperfections are modeled by introducing a single sinusoidal curve along the member length, with a maximum initial displacement of 1/1000 of the floor height at the mid-span.

Figure 4-8(a) shows a preliminary model of the interior column sub-assembly without the vertical spring. This model is first studied in order to investigate the strength of the column alone at elevated temperatures. The column is W14×90 (Grade 50) and the section sizes are shown in Table 4-1. The height of the each floor is 4 m and the slenderness ratio about the weak axis is 42.3. Shown in Figure 4-9 is the preliminary study plotting relationships between vertical displacement on the fire floor (relative vertical displacement at the third floor to the second floor) and load carrying capacity of the column at ambient and elevated temperatures from 100 °C to 800 °C. In this analysis, the thermal expansion effect is taken into account in the displacement. Buckling about the weak axis is the critical mechanism. The column strength quickly deteriorates at temperatures between 500 °C and 800 °C, where the material strength and stiffness drastically drop. In addition to the continuous three-story shell element model,

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single story models with rotationally free and fixed boundary conditions are studied to investigate the effect of the rotational constraint provided by the continuous column below and above the fire floor.

(d) Continuous (b) Free-end single (c) Fixed-end single (d) Continuous model (Shell) model (Shell) model (Shell) model (Fiber) Figure 4-8 Preliminary model for interior column

Figure 4-9 Axial load carrying capacity of the interior column at elevated temperatures

Shown in Figure 4-10 are the vertical displacement and axial load carrying capacity of the columns at ambient (20 °C) and elevated temperatures (500 °C and 800 °C). The figure compares the strength of the column simulated with four different models: (1) continuous column (for three stories from below to above the fire floor) with shell elements, (2) single

99 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES column in the fire floor with shell elements and rotationally free end-supports, (3) single column with shell elements with rotationally fixed end-supports and (4) continuous column with fiber elements (assumptions of the fiber models are described in Appendix A.2.1). These four models are shown in Figure 4-8 (a), (b), (c), and (d), respectively. The strength with model (3) is close to that with model (1) at any temperature, while the strength with model (2) is lower at elevated temperatures (approximately 70 % of the maximum strength of the other models at 500 and 800 °C). These results indicate that high rotational fixity can be expected for continuous columns subjected to localized compartment fires.

7000 6000 Shell, cont. Shell, cont. 6000 Shell, pin 5000 Shell, pin Shell, fixed Shell, fixed 5000 Fiber, cont. 4000 Fiber, cont. 4000 3000

(T) (kN) 3000 (T) (kN) cr cr P P 2000 2000

1000 1000

0 0 -0.02 0 0.02 0.04 0.06 0.08 0.1 -0.04 -0.02 0 0.02 0.04 0.06 Vert. disp. (m) Vert. disp. (m) (a) Ambient temperature (20 °C) (b) 500 °C

800 Shell, cont. Shell, pin 600 Shell, fixed Fiber, cont.

400

(T) (kN) cr P 200

0 -0.06 -0.04 -0.02 0 0.02 0.04 Vert. disp. (m) (c) 800 °C Figure 4-10 Comparison of column strength with different models

The maximum strengths of these four models are close and within 10 % of each other at ambient temperature. The rotational constraint does not have a large effect on the maximum strengths, because plastic yielding is significant at the maximum strengths, which are close to

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the perfectly plastic axial force ( AFy0 = 6120 kN, where A is the cross-sectional area and

Fy0 is the yield strength at ambient temperature). Post buckling strength, however, significantly drops in model (3) compared with models (1) and (2). The remained rotational stiffness at the end hinges helps the strength at this stage. The deformed shape of the column at the post-buckling stage is shown in Figure 4-11. The maximum strength with model (4), which cannot simulate local buckling, is slightly higher than that with model (1); however the differences are limited (within 2 % at elevated temperatures). Post buckling strength is higher with model (4) without local buckling. Further comparison between shell and fiber models including flexural buckling about the strong axis and lateral torsional buckling are shown in Appendix A.2.

Figure 4-11 Post buckling deformation of shell element model

4.2.3.3 Modeling of Constraint Springs

Realistic evaluation of the constraint spring properties in Figure 4-7(b) is a key for the simulations using this subassembly. The spring represents the vertical stiffness and strength of the floor framing of levels above the critical column. Since the springs at every floor shown in Figure 4-7(a) are identical, the condensed spring stiffness and strength in Figure 4-7(b) is simply obtained by multiplying the spring properties for a single floor by the number of the floors that are supported by the gravity column at the fire floor. The vertical

101 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES stiffness and strength of a single floor is evaluated using beam models shown in Figure 4-12. As shown in the structural floor plan in Figure 4-1, four beams with three sections connect to the interior gravity column: Beam a, b and c as shown in Table 4-1. Details of the connections between the column and beams are shown in Figure 4-2. Since the span of Beam a (W14×22) and b (W14×38) is identical (6.4 m) and the difference in sectional properties is relatively small, the properties for the connection of Beam a are also used for Beam b. Therefore, two analytical models for Beam a/b and c are created. As shown in Figure 4-12, the model consists of three components: a steel composite beam, an inelastic rotational spring for the shear tab bolted connection, and an elastic longitudinal constraint spring by floor structure.

Figure 4-12 Analysis model of beams for vertical spring stiffness of floor structure

The steel beam is modeled with shell elements and inelastic longitudinal springs are used to model the concrete slab. The effective width of the concrete slab is assumed equal to one- quarter of the span length in accordance with AISC (1995). A bilinear stress-strain curve with 0.1 % strain hardening is used for the steel. Concrete properties are adopted from Eurocode 2 (EC2, 1993). No reinforcement and no tensile strength are assumed for the concrete slab, which is common in US practice (although reinforcement may be placed near columns depending on design). The strength of the steel deck is also ignored. Inelastic rotational springs are located at the ends of the composite beam representing the stiffness and strength of shear tab bolted connections. An experimental study for this

102 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES type of connection has been performed by Liu and Astaneh-Asl (2004). Their study focuses on the rotational and shear behavior of the connection, including the composite effect of the concrete slab and steel beam under cyclic seismic load. Although the loading in this study is not cyclic, the results are still applicable to evaluate the vertical stiffness and strength of the floor structure. According to the proposed modeling by Liu and Astaneh-Asl, the inelastic rotational properties of shear-tab connections for Beams a/b and c (Figure 4-1 and Figure 4-2) are calculated and shown in Figure 4-13.

Figure 4-13 Rotational properties of shear-tab connections

A large difference in positive and negative moment capacities are observed due to the different contribution of the concrete slab in each loading direction. The maximum positive moment capacity is controlled by the bolted connection failure in tension (Point 1 in Figure 4-13), where the corresponding compression force couple is carried by the concrete slab. Bolts located at lower part of the shear-tab connections fail under the maximum moment capacity (Point 1), while bolts higher in the connection retain some strength provide additional rotational capacity (Point 2-3). The maximum rotational deformation (Point 3) is controlled by binding of the beam flange and column. The contribution of the concrete slab is very limited and is ignored for the negative moment capacity (Point 4), which is obtained from the maximum moment resistance provided by the bolt group. The longitudinal constraint stiffness of each beam is calculated from the analysis model of the floor structure shown in Figure 4-14(a). Here, the floor system is modeled with a combination of axial struts, representing the steel beams, and shell finite elements, representing the concrete slab. Both beam strut and slab finite element models are elastic and

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the stiffness of the concrete, which work for tension as well as compression, is assumed to be 50 % of the initial stiffness for compression (8.0 GPa). A vertical displacement is imposed at the location of the interior gravity column and it induces, through membrane action, horizontal displacements at the far ends of the connecting beams. Shown in Figure 4-14(b) is the vertical section of beam a on column line C indicated in Figure 4-14(a).

(a) Floor model

(b) X-X section Figure 4-14 Analysis model of floor structure for in-plane stiffness calculation

The axial force of Beam a, pa , and corresponding horizontal displacements, δa , at the far ends of the beams are monitored, and the effective spring stiffness of the floor is obtained by comparing δa and paH , which is the horizontal component of axial force of Beam a.

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Shown in Figure 4-15 are the relationship between δa and paH for Beam a as well as those for Beam b and c. The constraint stiffness of Beam c is relatively small compared to beam a and b, because there is no continuous slab for Beam c. The lateral displacement and force under an imposed vertical displacement of 0.4 m are marked with circles in the figure. As it will be shown later (Figure 4-22), 0.4 m is almost the maximum vertical displacement studied in this paper. The equivalent secant elastic stiffness of the longitudinal constraint spring of the beams is defined and superimposed in the plot. The elastic stiffness is 125,000 kN/m for Beam a and b, and is 54,000 kN/m for Beam c. This simulation is approximate, because accurate calculation of this constraint stiffness is difficult and involves many issues such as nonlinear behavior of the concrete slab, the stud connections between the steel beams and concrete slab, the steel connections, and 3D effects from the stiffness of the vertical components including the lateral resisting system.

2500 Bm a,b model Bm c model 2000 Bm a analysis Bm b analysis Bm c analysis 1500

1000 Reaction (kN) Reaction

500

0 0 0.005 0.01 0.015 0.02 Horiz. disp. (m) Figure 4-15 Longitudinal constraint stiffness of beams

The properties of the vertical spring of each floor structure shown in Figure 4-7(a) are computed in the following steps: (1) the property of the longitudinal constraint springs at the end of the connecting beams to the interior gravity column are prepared by computing a model of the floor structure shown in Figure 4-14, (2) the vertical stiffness and strength of each composite beam are calculated under imposed vertical displacement (Figure 4-12), (3) the vertical spring for each floor is calculated by combining the contribution of each beam in the floor, (4) longitudinal elongation of the column at fourth and higher floors is assumed as negligible and the condensed vertical spring, shown in Figure 4-7(b), is obtained by simply multiplying the spring property of each floor.

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Shown in Figure 4-16 are the vertical spring properties of the floor structure and contribution of Beam a, b and c, where the vertical stiffness and strength of Beam a is practically identical to that of Beam b. The limit-state condition of the vertical resistance of the floor diaphragm (Figure 4-12) is connection failure for all beams. The critical vertical strengths are 37 kN for Beam a and b, and 100 kN for Beam c (indicated with circles in Figure 4-16). At the limit-state, the resultant of the axial and shear forces reaches the capacity of the connections, which are 393 kN (web plate bearing) for Beam a and b, and 955 kN (shear-tab plate bearing or bolt shear failure) for Beam c. The limit-state is simply evaluated based on this resultant and is not coupled with rotational behavior. The initial vertical stiffness of each floor structure is about 0.15 % of the longitudinal stiffness of the gravity column per floor, EAL0 , where A is the cross-sectional area and L is the floor height. This relatively low stiffness of the vertical spring with respect to the column supports the condensation of the upper floor columns to the simplified model shown in Figure 4-7(b).

200 Beam a, b Beam c 150 Total

100 P (kN)

0 0 0.1 0.2 0.3 0.4 0.5 0.6 Δ (m) Figure 4-16 Vertical resistance of floor structure

4.2.4 Evaluation of Critical Temperatures

In this section, the vertical displacement under varied temperatures of the heated interior gravity column shown in Figure 4-7(b) is numerically evaluated and the critical temperatures are consequently obtained. The equilibrium equation of this system with respect to the vertical displacement under a given temperature can be written as:

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PPLdTTPs ()Δ +Δ+cg (α ,) = (4.1)

,where Ps is the force resisted by the inelastic vertical spring, Pc is vertical load carrying

capacity of the column, and Pg is gravity load composed of dead and live load. Therefore, the left-hand-side of this equation is the resistance and the right-hand-side is the force. Δ is the vertical displacement of the heated column (relative displacement of third floor level to the second floor level). Δ is measured as positive in the downward direction. The origin is the top of the column under dead and live load without thermal expansion. L is the length of the heated column (4 m), α is the thermal expansion coefficient (1.4×10-5 m/°C), T is the

temperature, and dT is the increment of temperature (=T - 20 °C). Ps is a function of the

vertical displacement, Δ , and the resistance curves as shown in Figure 4-16. Pc is a function of Δ , T , and the column behavior. The effect of thermal expansion is included in this resistant capacity by adding LdTα to Δ . Relationships between Pc and Δ for a given T are shown in Figure 4-9. Because the thermal expansion effect is included in Pc , the applied

force Pg is a constant in this equation and calculated with mean dead and live loads (102.5 % and 25 % of the design loads, respectively (Ellingwood, 1983). Three conditions of the vertical constraint spring on column response are studied: (1) without a constraint spring, (2) with an elastic constraint spring, and (3) with an inelastic constraint spring. The stiffness of the elastic constraint spring is defined as the initial stiffness of the inelastic spring. In the case of the study without the constraint spring, there is no Ps term in Eq. (4.1). Therefore, for this case, equilibrium combinations of temperature

and the vertical displacement are obtained from the intersections of the resistant curves, Pc with respect to Δ under given T and the axial load level Pg shown in Figure 4-9. Similarly, results for the case with elastic and inelastic springs are shown in Figure 4-17. Plotted on the vertical axis of each figure is resistance of the system (the left-hand-side of Eq. (4.1)) at ambient and elevated (100-800 °C) temperatures resulting from the constraint spring properties, shown in Figure 4-16, and the column resistance. The response for the case with an elastic spring is shown in Figure 4-17(a) and with an inelastic spring in Figure 4-17(b).

107 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

(a) Elastic constraint spring

(b) Inelastic constraint spring Figure 4-17 Resistance of the system under elevated temperatures

The equilibrium combinations for these three vertical constraint spring conditions (no spring, elastic spring, and inelastic spring) under a fixed gravity load and changing temperature are numerically solved by an iterative calculation and the results are shown in Figure 4-18. The vertical displacement is 0 at ambient temperature with the gravity load (Point 1 in Figure 4-18). Under increasing temperature, the vertical displacement increases due to thermal expansion and reaches the critical temperature around 600 °C (Point 2). The

108 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES critical temperatures for the study cases with and without the constraint springs are 598 °C and 599 °C, respectively, and the constraint effect is very limited. The equilibrium points are further obtained under decreasing temperature (Point 3); however, the structure will not generally reach these points since the structure will become unstable at the peak points. However, in the case with an elastic spring, the equilibrium temperature decreases from the critical temperature at Point 2 and turns to increase again at around 500 °C. Eventually, it increases to equilibrium at a temperature higher than the initial critical temperature. Under simply increasing temperature, the displacement “snaps-though” from Point 2 to 4. Assuming the constraint spring stays elastic and continues to pick up load, the equilibrium temperature keeps increasing indefinitely. The gravity load is eventually all carried by the elastic spring. At this stage, the building may not function due to the large vertical displacement (about 0.2 m). However, the possibility of preventing the progressive collapse of the building is an important benefit of this behavior.

Figure 4-18 Vertical displacement of the interior column under elevated temperatures

This snap-through type behavior is difficult to numerically evaluate. In this research, it is possible by limiting degrees of freedom in structural systems (i.e., one degree of vertical freedom for the interior gravity column). This approach is not generally scalable to more complicated structural systems with more degrees of freedom. However, as the purpose of

109 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

this study is to examine the nature of gravity column behavior in frames, the simplified approach provides a practical way to track the problem. As is evident from Figure 4-18, the effect of the vertical inelastic constraint spring is not significant for the critical temperature in the ten story case study building. In order to examine further this issue, two other office buildings with different numbers of stories are studied. The floor plan is identical to the benchmark design shown in Figure 4-1; and the geometry of the fire compartment, fire scenario, beam sections, and location of heated interior column are assumed to be the same. The numbers of stories of these buildings are 5 and 20. The fire floor is the second floor in both cases. The heated interior gravity column sections at the fire floor are designed as W10×49 and W14×233 (Table 4-2) and factored load ratios of these columns are 0.76 and 0.77 for 5- and 20-story buildings, respectively. The corresponding factored load ratio in the original 10-story building design is 0.77. Thus this

ratio is comparable in all the three buildings. The critical temperatures Tcr with and without consideration of constraint of springs are similarly simulated with analysis model shown in Figure 4-7(b), and summarized in Table 4-3.

Table 4-2 Section sizes of columns in 5- and 20-story buildings (mm)

Number of story Section h tw bf t f 5 W10×49 253 8.6 254 14.2 20 W14×233 393 22.6 399 36.0

Table 4-3 Critical temperatures with different number of building stories

Number of Column Slenderness Tcr with Tcr without Difference story section ratio constraint (°C) constraint (°C) (°C) 5 W10×49 61.4 594 600 -6 10 W14×90 42.3 598 599 -1 20 W14×233 38.8 620 612 +12

The spring enhances the critical temperature for the 5-story building by about 6 %, and it deteriorates it by about 12 % for the 20-story building. The difference is not significant in either case. This is because the vertical displacements at the limit-state are small (longitudinal deformation of the column due to thermal expansion and material deterioration

110 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

cancels out as shown in Figure 4-18). As a result, reaction from the longitudinal constraint spring is limited.

4.2.5 Comparison between Design Equations and Sub-assembly Simulations

Compared in Figure 4-19 are column strengths determined from the limit points of finite element simulations and the proposed column design equations previously described in Chapter 3. Three sets of results are compared, with and without the restraining spring effect: (1) sub-assembly simulations, (2) single member simulations, and (3) the proposed design equations. In the design equation check, the thermally-induced force of the restraint spring is based on the initial elastic spring stiffness as it might be calculated in design.

Figure 4-19 Critical axial strength of W14×90 (4 m) based on member-based and sub-assembly approaches

As shown in Figure 4-19, by simple superposition of the thermally-induced force with the design equation strength, the effective column strength (i.e., strength available to resist gravity load) is reduced. At 800 °C the effective strength goes to zero since the thermally- induced force is equal to the calculated strength. Comparisons between the design strength

equation (without Ps term in Eq. (4.1)) and the ABAQUS finite element simulation for a single member agree well above 400 °C; at lower temperatures the member design strength check is conservative (about 20 % conservative at 200 °C), which follows since the member

111 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES design equation is specifically calibrated for elevated fire temperatures. Finally, comparing the ABAQUS simulation results between the single member and the sub-assembly with and without the restraining force ( Ps ) yields the following observations: (1) the rotational restraint provided by the columns above and below the heated floor have a significant beneficial effect at temperatures above 200 °C (up to a 40 % increase at 400 °C), (2) the restraining spring effect is negligible for temperatures below 700 °C, and (3) above 700 °C the restraining spring helps to preserve the system capacity, provided that the restraining elements have sufficient strength.

4.2.6 Improvement of Structural Robustness

It is observed from Figure 4-16 that resistance of the vertical constraint spring is controlled by the capacity of the bolted shear-tab connections. This observation suggests that strengthening of the connections may be an effective method for improving the collapse resistance of the structural system. Two alternative design options for improved connection details for Beam a and b are suggested in Figure 4-20. Option A entails increasing the number of bolts in the connections, while Option B entails providing continuous reinforcement in the concrete slab. The reinforcement is expected to contribute to supporting axial tensile force at the connection. Although special design details may be needed to ensure the connection between the steel beam and the added slab reinforcement, it is likely that the shear studs provided for composite beam action will be strong enough so as not to be critical to the failure mechanism.

(a) Option A (b) Option B (5-7/8 in bolts are assumed.) (8-#4 reinforcement bars are assumed.) Figure 4-20 Options for strengthened connections

112 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Assuming that the strength of the proposed connection options is adequate, then the critical failure mechanism for Beam a and b is likely to be steel beam yielding (without contribution of the introduced reinforcement for Option B). Note that other failure mechanisms, such as yielding of the shear-tab or fracture, must be considered to assume that beam yielding controls the critical strength. Accordingly, the connection strengthening could increase the calculated maximum vertical resistance of the floor structure from 162 kN to 377 kN.

Figure 4-21 Total vertical load carrying capacity with strengthened connection for Beam a and b

Figure 4-22 Vertical displacement of the buckled column with improved beam connection

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Shown in Figure 4-21 are the resistance of the system under elevated temperatures (100- 800 °C), assuming strengthened connections for Beam a and b (not for Beam c). The equilibrium combination of the temperature and vertical displacement is shown in Figure 4-22. The snap-through behavior is observed at approximately 600 °C, after which the equilibrium point is regained at around 0.5 m of the vertical displacement. At this point, the vertical load of the column is completely taken by the vertical spring, thus providing an alternative load carrying path. Although this study is limited to static analysis, it suggests that strengthening of connections could provide a more robust structural system, thereby preventing complete collapse of the system.

4.2.7 Conclusions

The fire resistance of a gravity column in a 10-story composite steel building with office occupancy is evaluated under a given localized fire scenario. An interior W-shape column frame is heated in the fire and buckling of the column is the critical collapse mechanism. Evaluation is carried out using a sub-assembly including the interior column with finite shell elements and an inelastic vertical spring, which is calibrated to represent the stiffness and strength of the surrounding floor framing. Conclusions through this limited case study using the interior sub-assembly analysis model for this archetypical steel-framed building are summarized in the following points:

1. The rotational restraint provided by the columns above and below the heated floor has a significant beneficial effect at elevated temperatures (up to a 40 % increase at 400 °C). 2. The additional axial compression induced by vertical restrain of the gravity framing to thermal elongation does not significantly reduce the critical load of the column (i.e., it is reasonable to neglect the possible negative effects of the vertical restraint). 3. The vertical strength and stiffness provided by typical gravity framing is not sufficient to provide an alternative load path once the critical column temperature is reached. 4. Details are suggested for how the gravity framing connections could be made more robust (stronger) to provide sufficient strength to provide an alternative load path after column buckling. However, this alternative load path would not engage until large (e.g., 0.5m) deformations occur.

114 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

4.3 EVALUATION OF BEAM SUB-ASSEMBLY

Takagi, J., Deierlein, G.G. “Behavior and Limit-state Assessment of Steel Composite Beams under Localized Compartment Fire” (in preparation).

4.3.1 Summary

The behavior and strength limit-state of composite steel beams in a steel-framed building are analytically evaluated under a localized compartment fire, considering the effects of the shear-tab type bolted connections and the interaction of the heated beam and the surrounding floor framing. The finite element models of the composite beams are composed of various elements including shell elements for the steel beam, longitudinal springs of the concrete slab, inelastic springs of the bolted connections, and a longitudinal constraint spring for the in-plane floor diaphragm stiffness. The properties of the bolted connections are carefully evaluated through comparison with existing experimental data, because simulations indicate that connections are subjected to thermally-induced force and bolt shear failure is the critical failure mode under the fire. Alternative design details for the connections are suggested and the suggested improvements are validated by simulations. It is shown that composite beams with the improved connections can survive to more than 800 °C (at the lower flange) if large mid-span sagging (more than 20 % of the beam span) is accepted in the effect of the fire.

4.3.2 Introduction

Beam failure including failure of bolted connections is one of the typical failure mechanisms of steel buildings under fires. Many researchers have studied the behavior and the strength limit-state of steel beams under fire conditions using both analytical and experimental approaches (Liu and Davies, 2001; Allam et al., 2002; Wainman and Kirby, 1988). In order to evaluate realistic behavior of composite beams in building structures, the interaction between the steel beams and other structural components (e.g., steel-concrete composite effect and longitudinal deformation constraint by floor framing) must be considered (Bailey et al., 1996b, Sanad et al., 2000, Baily, 2005). Composite steel beams with concrete slabs are a very popular structural system in steel buildings. In addition to enhancing the stiffness of the beams, concrete slabs contribute by controlling the steel temperature and by enabling the development of membrane action when slab reinforcement is present. Constraint of longitudinal deformation by floor framing is also influential to beam behavior. If the constraint is high, axial compressive force is induced in the beams, which can cause buckling

115 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES and connection failures. However, the high axial constraint can also enable the development of catenary action, which can provide an alternate load path. In this research, the behavior and limit-state of beams under a fire are evaluated considering the interaction with surrounding structural components in a benchmark steel- framed office building. Secondary floor beams are the primary focus along with behavior of shear-tab type bolted connections. The effect of the concrete slab is considered by introducing inelastic longitudinal springs and the steel beams are modeled with finite shell elements. The material properties of steel and concrete at elevated temperatures are adopted from Eurocode. It is found that failure of the bolted connection dominates the limit-state of typical beam design. Alternative design options to improve the critical strength are proposed and the performance is numerically verified.

4.3.3 Analysis Model

4.3.3.1 Modeling of System

Finite element models created for the composite beams are composed of various elements, including shell elements for the steel beam, longitudinal springs for the concrete slab, inelastic springs for the bolted connections, and a longitudinal constraint spring for the floor framing (Figure 4-23). This study focuses on the secondary beam (Beam a in the floor framing plan in Figure 4-1) and vertical deformation of the supporting beam (Beam c) is assumed as negligible (vertical displacements at the ends are fixed). The property of the longitudinal constraint spring is obtained through independent simulations using an elastic floor framing structural model. The properties of the longitudinal spring of the bolted connections are carefully evaluated through calibration with existing experimental data. Modeling of each structural element for the composite beam will be discussed in following sections.

116 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Figure 4-23 System of finite element composite beam model in floor framing

4.3.3.2 Modeling of Steel Beam

Three dimensional finite shell element models are created for the steel beams by using commercial finite element analysis software, ABAQUS (Hibbitt, Karlsson & Sorensen, 2002). The model assumptions are similar to those used in 4.2.3.2 for the interior column study and some additional assumptions particularly applied for the beam model are described in the following points: 1. Residual stress is not considered. 2. The temperature distribution in the composite section is defined based on reported experimental data (Wainman and Kirby, 1988) as shown in Figure 4-24. The temperature

of the lower flange and lower three-quarters of web is defined as TL . The temperature of

the upper one-quarter of web is 0.9TL , while that of the upper flange and concrete slab is

0.8TL and 0.4TL , respectively. The imposed temperatures change during analyses in proportion to this fixed distribution mode through the cross-section. The temperature distribution along the member length is assumed to be uniform. 3. The coefficient of thermal expansion is independent of temperature and assumed as 1.4×10-5 m/°C for both steel and concrete. 4. Out-of-plane (web plane) displacement and rotation about the longitudinal axis are fixed at the center of the upper flange along with the length. Vertical displacement is fixed at

117 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

the center of the upper flange at the ends, and longitudinal displacement at these points is free.

(Temperature at bolted connections is assumed as 0.9TL .) Figure 4-24 Temperature distribution of composite section

4.3.3.3 Modeling of Concrete Slab

Inelastic longitudinal springs are placed above the steel beam at the center of the concrete slab. The length of each spring is 0.2 m and the end displacement of each spring is linked to the beam section with a kinematic linear constraint as shown in Figure 4-23. The average thickness of the concrete slab is 112.5 mm and the effective width described in AISC (2005) is used for the cross-sectional area to calculate the spring properties. The stress-strain curves of concrete at the elevated temperatures are adopted from Eurocode 2 (1993) (Figure 4-25). No tensile strength is considered in the slab assuming that no reinforcement is placed in the concrete slab, following typical construction practice in the US. The property of the longitudinal spring of the concrete slab above the bolted connection (two springs at the ends) is different from that of the other springs along the beam length. Evaluation of the effective concrete slab length at the ends is not straightforward; however, it is not highly influential to overall beam behavior and strict evaluation is not needed. This is because these springs are normally subjected to stretching deformation due to end rotation, such that no force is induced in the springs. The effective length of these springs is assumed as 1.6 m, which is one-quarter of the beam span and eight times longer than the length of the other slab springs. An alternative design option with reinforcement in the concrete slab is studied later and 1.6 m is chosen to represent the embedment length of reinforcement.

118 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

1 20°C 100°C 0.8 200°C 300°C 0.6 400°C 500°C 0.4

0.2 Stress /Compressive strength 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Strain Figure 4-25 Compressive stress-strain curve of concrete

4.3.3.4 Modeling of Bolted Connection

Figure 4-26 shows a conceptual relationship between the longitudinal displacement of a beam and temperature. As temperature increases, the beam elongates. At this stage, the beam does not significantly deform (sag) and is subjected to compressive axial force. Further increasing temperature induces significant deterioration of steel strength and stiffness, which leads to loss of flexural stiffness of beams. This induces large deformation and the beams may act like cables to resist load through catenary action. The effectiveness of catenary action depends on the boundary conditions (e.g., longitudinal support at beam ends and strength of the connections). At this stage, the beams are subjected to tensile axial force and the longitudinal displacement may change direction (i.e., The beams pull inward).

Figure 4-26 Gravity load supporting systems of beams at elevated temperatures

119 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Figure 4-27 shows a detail of a typical shear-tab bolted connection used for the secondary beam (Beam a) in the benchmark building design (Figure 4-1). This type of connection is usually designed for the shear force associated with gravity dead and live load at ambient temperature. Under elevated temperatures, however, this type connection can be critical to beam failure when subjected to high compressive or tensile force as shown in Figure 4-26.

Figure 4-27 Detail of beam connection

The behavior and strength of high-strength bolted connections at elevated temperatures have been evaluated by Yu (2006). Shown in Figure 4-29 are force-displacement relationships of a single shear bolted connection with an ASTM A325 high strength bolt and 9.5 mm (3/8 inch) thick Grade 50 plates (Figure 4-28). The mechanical tests were carried out under constant elevated temperatures up to 800 °C (with increments at every 100 °C).

Figure 4-28 Single shear bolt test by Yu (2006)

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150 25°C 100°C 200°C 300°C 400°C 100 500°C 600°C 700°C

Load (kN) 800°C 50

0 0 5 10 15 20 25 30 35 40 Displacement (mm)

(a) Edge distance 1.0DB

25°C 200 100°C 200°C 300°C 150 400°C 500°C 600°C 100 700°C

Load (kN) 800°C

0 0 5 10 15 20 25 30 35 40 Displacement (mm)

(a) Edge distance 1.5 DB Figure 4-29 Load-displacement relationships of single shear connections by Yu (2006)

The distance between the center of the bolts and the edge of the plates is 22.2 and 33.3 mm (Figure 4-28), which are equivalent to 1.0 and 1.5 times the bolt diameter DB . The failure mechanisms in this test are bolt shear and plate bearing. A rapid drop in post-capping strength is observed in the force-displacement relationships at relatively low temperatures, while post capping strength remains relatively high with ductile deformation at higher temperatures. Pre-loading initial displacements up to 6 mm are observed. The reason for the

121 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

initial displacements is mainly thermal expansion of the specimen, which were set in the furnace at ambient temperature and loaded at the specified elevated temperatures. Yu (2006) also investigated the shear strength of bolts at elevated temperatures. Doubly sheared ASTM A325 and A490 high strength bolts were tested with thick plates such that deformation of plate bearing is negligible. Figure 4-30 shows the shear strength of ASTM A325 bolts (strength per single shear) superimposed with the maximum strength of single shear connections obtained from Figure 4-29. The strength of the single shear connections

with 1.0 DB and 1.5 DB edge distance at ambient temperature are approximately 50 % and 80 % of the bolt shear strength, respectively. This is because plate bearing is the dominating failure mode for the single shear connections. However, the difference decreases at elevated temperatures and there is almost no difference more than 500 °C. This means that bolt shear failure is more critical at elevated temperatures than plate bearing. This is consistent with the observation in the experiments by Yu.

conn. 1.0D 250 conn. 1.5D bolt 200

150

100

Max. shear strength (kN) strength shear Max. 50

0 0 200 400 600 800 Temperature (°C) Figure 4-30 Maximum single shear strength of A325 bolts by Yu (2006)

Three ASTM A325N bolts with 22.2 mm (7/8 inch) diameter are used for the connection as shown in Figure 4-27. The force-displacement relationship of the longitudinal springs of the bolted connections in Figure 4-23 is assumed as bilinear with post capping softening (Figure 4-31). One longitudinal spring is used in the analysis model (Figure 4-23) instead of multiple springs for each bolt in order to simplify the model system and avoid difficulty in analytical convergence dealing with the strength softening. Three components of beam deformation can cause shear deformation of each bolt at the shear-tab connections: (1) longitudinal deformation of the beam, (2) rotational deformation the connection, and (3)

122 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

vertical shear deformation. Displacement of each bolt is varied considering possible rotational deformation at the connection, while longitudinal deformation of the beam induces the same horizontal relative displacement to each bolt.

Figure 4-31 Force-displacement relationship model of bolted connection

Parameters to represent this force-displacement relationship include peak strength

PTB (), displacement at the peak strength ΔBp , and displacement at vanishing strength ΔBe

(Figure 4-31). ΔBp and ΔBe are assumed as independent of temperature, while PB is a function of temperature as shown in Eq. (4.2).

PTBy()= KBBBRB () Tζ NCRn0 (4.2)

where KTyB () is the reduction ratio of the bolt strength at elevated temperatures, which is

adopted from ECCS (2001). The values of KTyB () are compared with the reduction factors

of steel by EC3 in Figure 4-32 and Table 4-4. ζ B is an adjustment factor to account for the

non-uniform distribution of bolt forces due to rotation. NB is the number of the bolts (three).

CR is a ratio between the mean bolt shear strength and RBn0 , which is the nominal single shear strength of bolt, and is calculated to be 160 kN according to AISC (2005). The value of

CR is calibrated with experimental data by Yu (2006) and is defined as 1.5. The detail of the calibration is described later in 5.2.4.

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Figure 4-32 Reduction factor of bolt strength by ECCS

Table 4-4 Values of reduction factor of bolt strength

Temperature °C (°F) KTyB () KTy () KTp () KTE () 20 (68) 1.000 1.000 1.000 1.000 100 (212) 0.968 1.000 1.000 1.000 200 (392) 0.935 1.000 0.807 0.900 300 (572) 0.903 1.000 0.613 0.800 400 (752) 0.775 1.000 0.420 0.700 500 (932) 0.550 0.780 0.360 0.600 600 (1112) 0.220 0.470 0.180 0.310 700 (1292) 0.100 0.230 0.075 0.130 800 (1472) 0.067 0.110 0.050 0.090 900 (1652) 0.033 0.060 0.038 0.068 1000 (1832) 0 0.040 0.025 0.045 1100 (2012) 0 0.020 0.013 0.023 1200 (2192) 0 0 0 0

The rotational displacement at the connection varies under the flexural deformation of the beam. This rotation makes it difficult to evaluate the properties of the single longitudinal spring for the bolted connection. In order to simplify the evaluation, the rotational displacement at the bolted connection is assumed to be a constant and 0.05 radian. This rotational displacement is observed at the limit-states in the typical beam simulations. This

124 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES rotation results in a difference of 3.75 mm in longitudinal displacements between two adjacent bolts (spaced 75 mm apart). The connection strength is shown in Figure 4-33, which is obtained by adding strength of three single bolts, measured by Yu (2006) with this difference in the longitudinal displacement. The measured shear strength by Yu with edge distances of 1.0 DB and 1.5 DB are shown in the figure. The calibrated adjustment factor accounts for the non-uniform distribution of bolt forces due to rotation, ζ B , is 0.8. ΔBp and

ΔBe are also calibrated as 10 mm and 25 mm, respectively. These parameters are defined such that the force-displacement relationships of the model agree at elevated temperatures between 500 °C and 700 °C, where bolted connections typically lose the load carrying capacity under fires.

25°C 600

500

400 1.0D Conn. 1.0D Bolt 300 1.5D Conn.

Force (kN) 1.5D Bolt 200 Model

100

0 0 5 10 15 20 25 30 Displacement (mm) (a) Ambient temperature

200°C 400°C 600 600

500 500

400 400

300 300

Force (kN) 200 Force (kN) 200

100 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Displacement (mm) Displacement (mm) (b) 200 °C (c) 400 °C

125 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

500°C 600°C 600 600

500 500

400 400

300 300

Force (kN) 200 Force (kN) 200

100 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Displacement (mm) Displacement (mm) (d) 500 °C (e) 600 °C

700°C 800°C 600 600

500 500

400 400

300 300

Force (kN) 200 Force (kN) 200

100 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Displacement (mm) Displacement (mm) (f) 700 °C (g) 800 °C

Figure 4-33 Comparison of force-displacement relationships of bolted connection between analysis model and test data by Yu (2006)

4.3.3.5 Modeling of Longitudinal Constraint by Floor Framing

Evaluation of the longitudinal constraint of steel beams in floor framings is difficult due to the complex behavior of composite floor structures, such as inelastic response of the concrete in the slab, stud connections between the concrete slab and steel beams, steel connections, and three-dimensional (3D) interaction with vertical members. Also, the in-plain diaphragm constraint stiffness depends on the location of the heated beams, geometry of the fire compartment and fire scenarios. Consequently, the constraint stiffness is highly uncertain and an approximate evaluation is performed in this section. Figure 4-34 shows an analysis model of the floor structure composed of truss elements for the steel beams and shell elements for the concrete slab. These elements are elastic and the elastic modulus of the shell

126 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES elements is set to 50 % of the initial compressive modulus of the concrete. Vertical load is applied at the center of the heated secondary beams in the fire compartment and the longitudinal constraint stiffness of the beams is calculated from the horizontal component of tensile force of the beams pa (Figure 4-34(b)) and corresponding horizontal displacements

δa . The calculated constraint stiffness of the four beams in the fire compartment is very different for each beam. The average linear longitudinal constraint stiffness is approximately 1.0×107 N/m. This is about 8 % of the elastic axial stiffness of a W14×22 beam with a length of 6.4 m at ambient temperature. Although investigating the constraint stiffness in more rigorous approaches is an interesting study subject, the approximated linear stiffness is used in this research primarily focusing on understanding fundamental behavior of the composite beams in frames under fire conditions. The sensitivity of this constraint stiffness to the behavior of the composite beam will be investigated later.

Floor model

(b) Y-Y section Figure 4-34 Analysis model for constraint stiffness

127 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

4.3.4 Evaluation of Behavior and Limit-state

4.3.4.1 Performance of Typical Design

Figure 4-35(a) shows relationships between lower flange temperature of the composite beam and vertical displacement at the mid-span with and without consideration of longitudinal constraint by the surrounding floor system. The critical temperatures are 634 °C and 660 °C with and without the constraint, respectively. Loss of flexural stiffness due to steel yielding is the failure mechanism of the beam without the constraint, while shear failure of the bolted connection is critical for the beam with the constraint. The connection fails during beam elongation (i.e., in the compression phase in Figure 4-26), after which the connection can no longer supported the vertical shear force. The temperatures at the capping strength and failure of the connection are marked in the Figure 4-35(a). Although the failure mechanisms are different between these cases, the critical temperatures are about the same.

0 0

-0.1 -0.05 -0.2

-0.3 -0.1 -0.4

With constraint -0.5 With slab -0.15 Without constraint Without slab Capping Capping

Mid-span displacement (m) -0.6 Mid-span displacement. (m) Conn. failure Conn. failure -0.2 -0.7 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (a) Longitudinal constraint effect (b) Concrete slab effect Figure 4-35 Mid-span displacement and modeling comparison

Another comparison is carried out investigating the effect of the concrete slab on the behavior of the composite beam. Figure 4-35(b) shows mid-span displacement of the beams at elevated temperatures with and without the longitudinal springs for the concrete floor slab. It is observed that the mid-span displacement is smaller with the concrete slab but the critical temperature is lower. This is because shear failure of the connection in the compression phase (Figure 4-26) is critical for the beam with the concrete slab (longitudinal elongation due to thermal expansion is larger with small sagging deformation with higher stiffness),

128 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES while connection failure in the tension phase is critical without the slab. Assessment of the critical mechanism has a fair degree of uncertainty because of the underlying and uncertain assumptions regarding the force-displacement relationship of the bolted connections and the spring stiffness for the constraint of the floor structure. However, this study is useful for analytically deriving possible failure mechanisms, which have not been fully examined in past research, and the critical temperatures in typical composite beam design. Analytical difficulties were found during these simulations due to the modeling of the longitudinal springs of the bolted connections, which have bilinear force-displacement relationships including post peak-strength softening (Figure 4-31). Analyses were often aborted at the peak-strength and the post peak-strength behavior was not properly simulated. In order to stabilize the simulation, an alternative analytical technique is introduced. As shown in Figure 4-36, simulations were run with springs having perfect-plastic force- displacement relationships instead of post peak-strength softening. Multiple perfect-plastic models were prepared and run with different yielding strengths of the connection springs. A post peak-strength equilibrium point was found in each simulation, and temperatures and displacements at the equilibrium points were traced. Combinations of the temperatures and displacements at the equilibrium points traced in this approach are not necessarily ordered by simply increasing temperature, i.e., a critical temperature at connection failure can be lower than that at the peak-strength as shown in Figure 4-35(b). This approach is computationally intensive; however it provides the ability to simulate the post peak-strength behavior with reasonable analytical stability under current modeling development. The simulations were performed to investigate the behavior for research purposes; however, further modeling and analytical development is desired for practical application.

Figure 4-36 Post peak-strength evaluation of bolted connection

129 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

4.3.4.2 Performance of Alternative Design

Longitudinal deformation of composite beam floor framing is constrained in most cases, although the stiffness of the constraint varies. It is observed from Figure 4-35 that shear failure of bolted connections is critical and improving the connections may be an effective means to enhance the performance of composite beams against fires. Two alternative design options for the connections are proposed: (1) using slotted bolt holes (Figure 4-37(a)) and (2) placing continuous reinforcement in the concrete slab in addition to the slotted bolt holes (Figure 4-37(a) and (b)). As noted previously, critical failure mechanism for standard connection is shear failure of bolted connection in the compression phase (Figure 4-35). The proposed design option (1) is, therefore, to use slotted bolt holes, which allow beam elongation (Figure 4-37(a)). This detail is designed such that the axial force is released at the bolted connections in the compressive phase (Figure 4-26) but provides support in the tension phase. No horizontal response is expected in the compressive phase and the size of the slot should be large enough so as not to come into bearing under the maximum deformation. In the option (2), reinforcement in the concrete slab increases the tension strength of the connections. The temperature of the concrete slab is relatively low compared with the steel beam (Figure 4-24), such that the tension strength of the reinforcement remains high. This strength can enhance the performance of the beam connections near the tension phase (Figure 4-26) However, without combining with option (1), reinforcement alone would not improve the critical temperature, because the failure mechanism remains shear failure of the connections.

2-#4 reinforcing bars are assumed. (a) Slotted bolt hole (b) Continuous reinforcement Figure 4-37 Proposed design options for bolted connections

130 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Figure 4-38 compares analysis results of the typical composite beam and two proposed alternative designs. Vertical displacement at mid-span and longitudinal relative displacement at the spring of bolted connections (compression positive) are plotted in Figure 4-38(a) and (b), respectively. The critical temperatures and failure mechanisms are summarized in Table 4-5. The critical temperature of the typical design is 634 °C with bolt shear failure due to thermal elongation of the beam. The relative displacement of the bolted connection significantly increases around 600 °C, once the relative displacement reaches the capping point (0.01 m) (Figure 4-38(b)). Increasing the number of bolts at the connection would not significantly improve the critical temperature, because thermal elongation of the beam without insulation imposes shear forces on the bolts and leads to bolt failure. On the other hand, slotted bolt holes (alternative design option (1)) prevent this bolt shear failure and improve the critical temperature by 95 °C to 729 °C. Mid-span displacement increases dramatically near the critical temperature, as flexural stiffness and strength of the beam is lost. The maximum relative displacement of the bolted connection in this case is approximately 0.03 m at 695 °C. The relative displacement decreases at the higher temperature. The current analytical algorithm cannot further trace this behavior; however, there may still be equilibrium points at higher temperatures under a fully developed tension system (catenary system with large deformation).

0 Typical 0.04 Slotted hole -0.1 Slotted hole + rebar 0.03 Capping -0.2 Conn. failure

0.02 -0.3 Typical Slotted hole -0.4 Slotted hole + rebar 0.01 Capping Mid-span displacement (m)

Conn. failure Displacement (m) at connection -0.5 0 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (b) Longitudinal relative displacement at (a) Mid-span displacement the connection Figure 4-38 Performance of composite beams with alternative design options for the connections

131 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

In this research, the critical temperature for this alternative design option is defined as the maximum temperature in Figure 4-38. Placing reinforcement in the concrete slab (alternative design option (2)) prevents the significant drop of the mid-span displacement at more than 700 °C. The tension strength of the longitudinal spring enables catenary action; however, the shear strength of the bolts deteriorate at high temperature at the connection so that the gravity load is no longer supported at elevated temperatures above 812 °C. Thus 812 °C is the critical temperature of this design option. If this shear failure of the bolted connection due to the gravity is prevented by some design improvement, such as a fail-safe type mechanical support (e.g., Figure 4-39), the critical temperature can increase to more than 1000 °C. This implies a possibility to reduce or partially eliminate fire insulation of steel beams.

Table 4-5 Comparison of the critical temperatures

Design options Tcr (°C) Failure mechanism Typical 634 Bolt shear by thermal elongation Slotted holes 729 Beam yielding Slotted holes + Reinforcement 812 Bolt shear by vertical load

Figure 4-39 Alternative connection detail of secondary beams to prevent shear failure at elevated temperatures

132 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

4.3.4.3 Effect of Longitudinal Constraint

Stiffness of the longitudinal constraint of beams can significantly vary depending on multiple factors including fire scenario, location of heated beam, and analysis modeling. The stiffness is not precisely calculated in this research; rather the primary focus is on overall structural fire simulations of beams in steel-framed buildings. Lacking accurate data to validate the constraint spring stiffness, the influence of the spring stiffness on beam behavior is investigated. Figure 4-40(a) compares mid-span displacements by changing the constraint

7 6 8 stiffness (denoted as Ks ) from 10 N/m to 10 N/m and 10 N/m, in addition to no constraint. The constraint stiffness of 107 N/m is approximately 8 % of the elastic axial stiffness of W14×22 beam (without concrete slab) with 6.4 m at ambient temperature. Therefore, 106 N/m and 108 N/m are equivalent to 0.8 % and 80 % of the beam stiffness, respectively. The behavior with the longitudinal constraint stiffness of 106 N/m is almost identical to that without the constraint (Figure 4-40(a)) and the failure mechanism is beam yielding. Shear failure of bolts due to thermal elongation of the beam is the failure mechanism for the beam with the constraint stiffness of 107 N/m and 108 N/m. The critical temperatures for these cases are 634 °C and 586 °C, respectively, which are lower than that with the constraint stiffness of 106 N/m (622 °C). The critical temperature is decreased under high longitudinal constraint with the failure mechanism of bolt shear in beam elongation. The critical temperatures are summarized in Table 4-6.

0 0

-0.1 -0.05 -0.2

-0.1 w/o constraint -0.3 Ks = 106 N/m w/o constraint 6 Ks = 107 N/m -0.4 Ks = 10 N/m 7 -0.15 Ks = 108 N/m Ks = 10 N/m Capping -0.5 Ks = 108 N/m Mid-span displacement (m) Mid-span displacement (m) displacement Mid-span Conn. failure Conn. failure -0.2 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (a) Typical design (b) Slotted holes + reinforcement Figure 4-40 Influence of the longitudinal constraint stiffness

133 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Table 4-6 Effect of the constraint stiffness to the critical temperatures

Design options Ks (N/m) Tcr (°C) Failure mechanism - 660 Beam yielding 106 662 Typical 7 10 634 Bolt shear by 108 586 thermal elongation - 735 Beam yielding Option (2) 106 812 Slotted holes + Bolt shear by 107 812 Reinforcement vertical load 108 812

Similarly, Figure 4-40(b) compares mid-span displacements of alternative design option

6 7 (2) (slotted bolt holes and reinforcement in slab) with the constraint stiffness Ks of 10 , 10 , 108 N/m and no constraint. The critical temperature of this design without the constraint is 735 °C, while that with the constraint is 812 °C (as limited by shear failure of the bolts for gravity loads). As described earlier, preventing this shear failure of the bolts significantly improves the critical temperatures. Such improvement cannot be simulated without modeling of this constraint spring, although the behavior is not highly sensitive to the constraint stiffness.

4.3.5 Conclusions

Behavior of composite beams in a benchmark steel building with office occupancy is simulated under elevated temperatures. Finite element models are created for a composite steel beam, which consists of shell elements for the steel beam, inelastic longitudinal springs for the concrete slab, bolted shear-tab connections and constraint by the floor framing structure. Stress-strain curves of steel and concrete under elevated temperatures are adopted from Eurocode 3. The critical temperatures are simulated in incremental analyses with respect to temperatures taking into account geometric and material nonlinearity. Notable findings and conclusions about composite steel beams (secondary beams) in this archetypical steel-framed building under the compartment fire are summarized in following points:

134 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

1. Failure of bolted shear-tab connections is critical. Alternative design options for the connections are suggested using slotted bolt holes and installing continuous steel reinforcement in the concrete slab. The improvement is analytically confirmed and the critical temperatures can significantly increase from 634 °C to 812 °C at the lower flange temperature. The failure mechanism changes from beam yielding to shear failure of the bolts due to the gravity in this design improvement. Furthermore, preventing this bolt failure against gravity can increase the critical temperature to more than 1000 °C. This fact implies potential strength of composite steel beams in floor framings and possibilities for new types of structural fire design.

2. The longitudinal constraint stiffness Ks is approximately calculated using a relatively simple elastic floor framing FEM model, which should pick up the most significant aspects of the floor diaphragm behavior. The calculated lateral constraint stiffness is about 8 % of the longitudinal stiffness of the steel beam (without composite effect). The sensitivity of this constraint stiffness to the beam behavior is investigated for the typical

and improved alternative design with 0.1× Ks and 10× Ks . Overall, the behavior is not very sensitive to the longitudinal stiffness; however, the critical temperature for the

alternative design is not improved without this constraint (i.e., Tcr = 735 °C without

constraint and 812 °C with constraint between 0.1× Ks and 10× Ks )

4.4 EVALUATION OF EXTERIOR COLUMN SUB-ASSEMBLY

4.4.1 Overview

The behavior and limit-states of the exterior column sub-assembly model are investigated in this section. The model is composed of steel column and composite beam. The column is modeled using shell finite elements, and the composite beam is modeled in the same way as the beams in the beam study (Figure 4-23). The elevated temperatures of these members vary depending on the assumed fire scenario, and structural simulation is carried out incorporating the time-temperature relationships of the compartment fire based on Eurocode 1 (2002). For the typical design, the critical failure mechanism during the fire is exterior column buckling. Insulating the column changes the failure mechanism to a connection failure, caused by large deformations and the associated tension forces in the beam. Alternative

135 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES structural fire resisting designs are examined without fire insulation of the beam. Although large deformation of the beam (1.5 m at the mid-span) is expected during the fires, the analyses show that structural failure can be avoided by adopting the proposed design improvements.

4.4.2 Analysis Model

4.4.2.1 Modeling of System

The exterior column sub-assembly is composed of the exterior column on column line 3 and Beam c on the third floor in the fire compartment (Figure 4-1). The exterior column section on the second floor is W12×72 (AISC) and Beam c is W21×62 (Table 4-1). The orientation of the exterior column is also shown in the floor plan (Figure 4-1), where the column flange is parallel to Beam c. This sub-assembly model can simulate three possible failure mechanisms: exterior column buckling, beam yielding (Beam c), and interactive failure between the beam yielding and multi-story (the second and third floor) column buckling (Figure 4-41). The dominating failure mechanism, which is identified in the structural fire simulations, is dependent on combinations of various factors such as strength, stiffness and temperatures of members and connections.

Figure 4-41 Failure mechanisms simulated with exterior column sub-assembly

The column is modeled with finite shell elements for four stories from the first to fifth floor level. Although the column on the second floor is the only portion subjected to the compartment fire, the columns on the first, third and fourth floor are continuously modeled with the shell elements to provide accurate boundary conditions (rotational constraints to the heated portion). The column on the fourth floor is necessary to evaluate the interactive failure mechanism.

136 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Shown in Figure 4-42 is the finite shell element model of the exterior column sub- assembly. The modeling of the steel column is essentially the same as that for the interior column sub-assembly described in 4.2.3.2. The longitudinal constraint spring at the top of the column is not included in this model, because it does not have a significant effect on the critical strength, as found in simulations for the interior column sub-assembly. The modeling of the steel beam and concrete slab is similar to that used for the simulations of the beam sub- assembly, as described in 4.3.3.2 (also Appendix A.4.1). The modeling assumptions for the concrete slab are discussed in 4.3.3.3 including the properties of concrete at elevated temperature shown in Figure 4-25. The longitudinal constraint at the interior end of the beams is assumed to be rigid. This is because the stiffness of the longitudinal constraint developed in the beam is controlled by the exterior column, which is much more flexible than the floor framing (e.g., the longitudinal constraint of the beam by the exterior column is approximately 1.5×106 N/m without strength deterioration of the column in the fire compartment, while that by the floor framing calculated for the beam sub-assembly in 4.3.3.5. is on the order of 1.0×107 N/m). The constraint at the exterior end is based on only the bending stiffness of the exterior column. As shown in Figure 4-43, the adjacent floor structure may significantly constrain the lateral displacement at the exterior end through membrane action. However, in this case the lateral stiffness would be coupled with the vertical displacement at the exterior column, and including this effect is beyond the scope of the present analysis model.

137 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Figure 4-42 System of exterior column sub-assembly model

Figure 4-43 Lateral constraint by floor slab with membrane action

138 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

To avoid the modeling complexity and uncertainty, the constraint membrane effect of the floor structure is not considered in this exterior column sub-assembly model. This means that interaction between frames in the east-west direction (e.g., column lines 3 and 4) is neglected. If the fire compartment were larger (Figure 4-44(b)) and the frames on column lines 1 to 6 behave similarly during the fire, neglecting the membrane effect is a more convincing assumption. In this research, it is assumed that the membrane effect in the one- bay fire is not significant and study on the effect is left for future work.

(a) One-bay compartment fire (b) Large compartment fire Figure 4-44 Comparison of compartment fire for exterior column sub-assembly simulations

4.4.2.2 Modeling of Bolted Connection

In beam sub-assembly simulations, it is found that bolted connections are critical for beams at elevated temperatures. Nonlinear longitudinal springs for the shear-tab bolted connections are developed and validated using experimental data obtained by Yu (2006). A similar approach is taken by introducing a longitudinal spring for the shear-tab bolted connections

139 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

between the exterior column and Beam c. Shown in Figure 4-45 is the detail of the shear-tab connection. Five high-strength ASTM A325 bolts with 9.5 mm (3/8 inch) shear-tab plate are used for the connection. As shown in the figure of the sub-assembly model (Figure 4-42), the bolted connection is modeled as a combination of a slot and nonlinear spring. This connection prevents longitudinal displacement of the beam and restraints the transverse displacement. Therefore, shear failure of the connection due to the gravity load on the beam is separately monitored from the longitudinal displacement and connection force.

Figure 4-45 Detail of exterior column connection

The properties of the nonlinear longitudinal spring are obtained in similar fashion to those for the beam sub-assembly models. Due to analytical difficulty in including the peak- strength softening, the bolted connection is modeled using a single longitudinal spring in the simulations. Assuming bi-linear force-displacement relationships including post peak- strength softening for the longitudinal spring (Figure 4-31), the temperature-dependent relationships are calibrated with the experimental data obtained by Yu (2006) (single shear connection test with 1.0 DB and 1.5 DB edge distance, where DB is the bolt diameter). The rotational displacement at the connection, which leads to different displacements for each bolt, makes it difficult to evaluate the properties of the single longitudinal spring for the connection. For simplicity, a constant rotational displacement of 0.05 is assumed for the evaluation of the spring property. The value of the constant rotational displacement is same

140 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES as that used for the beam sub-assembly. The capping strength of the longitudinal strength is defined in Eq. (4.2). The calibrated parameters are consistent with those obtained in the beam study. The adjustment factor to account for the non-uniform distribution of bolt forces due to rotation, ζ B , is 0.8, and the displacement at the peak and vanishing strength ΔBp and

ΔBe are 10 mm and 25 mm, respectively. The force-displacement relationships of the longitudinal spring model are shown in Figure 4-46. The model agrees at elevated temperatures especially between 600 °C and 700 °C, where bolted connections are critical under fires. Although the model agrees with the test data, this depends on the assumed rotational displacement of 0.05, which leads to the difference of the longitudinal shear deformation at each bolt.

500°C 600°C 600 600 1.0D Conn. 1.0D Conn. 1.0D Bolt 1.0D Bolt 500 500 1.5D Conn. 1.5D Conn. 1.5D Bolt 1.5D Bolt 400 400 Model Model 300 300

Force (kN) 200 Force (kN) 200

100 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Displacement (mm) Displacement (mm) (a) 500 °C (b) 600 °C

700°C 800°C 600 600 1.0D Conn. 1.0D Conn. 1.0D Bolt 1.0D Bolt 500 500 1.5D Conn. 1.5D Conn. 1.5D Bolt 1.5D Bolt 400 400 Model Model 300 300 Force (kN) 200 Force (kN) 200

100 100

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Displacement (mm) Displacement (mm) (c) 700 °C (d) 800 °C Figure 4-46 Comparison between analysis model and test data by Yu (2006) of longitudinal force-displacement relationships of beam-column bolted connection

141 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

4.4.3 Evaluation of Behavior and Limit-state

4.4.3.1 Basis of Simulations

Simulations of the exterior column sub-assembly are carried out in the time domain (i.e., time is the primary parameter in simulations), while the simulations of the interior column and beam sub-assemblies are performed in the temperature domain. The beam-column sub- assembly response is described in terms of time since the beam and exterior column members in this sub-assembly are subjected to different temperatures, depending on the type of insulation and the gradient in the beam. Temperatures of the beam and exterior column are obtained using the parametric fire (time-gas temperature) curve of Eurocode 1 (2002) and a one-dimensional heat transfer analysis (by Buchanan 2002). The following input parameters are assumed for the time-temperature curve: fuel load of 430 MJ/m2, fire compartment area of 61 m2, total surface area of compartment equal to 235 m2, and opening factor of 0.05, and a fire-fighting factor of 0.292. Where used, thermal insulation is assumed to be (or equivalent to) 9.5 mm of gypsum wall board. Details of the parametric fire simulations and numerical heat transfer approach are described in Appendix B.4.

Figure 4-47 Time-temperature relationships in a fire simulation

4.4.3.2 Simulation Results

Shown in Figure 4-48 are relationships between time, steel temperature, and displacements in the sub-assembly simulation. The beam displacement is measured at mid-span (Figure 4-48(a)), and the horizontal column displacement is measured at the exterior column at the

142 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES third floor (Figure 4-48(b)). Assuming no fire insulation on any elements, the critical failure mechanism is exterior column buckling at 5.8 minutes and at a column temperature of 578 °C. Failure is detected as the point when the mid-span displacement of the exterior column in the fire compartment diverges as shown in Figure 4-48(b). Providing insulation on the column changes the critical mechanism to failure of bolted connection at 7.8 minutes and 136 °C in the column and 767 °C in the beam at the lower flange. The horizontal displacement at the third floor first increases (pushes outwards) due to thermal expansion of the beam and then decreases (pull inwards) to catenary action developing in the beam. The bolted connection fails then due to the tension force in the beam. Insulating the connection improves the duration to 8.2 minutes (812 °C in the beam), changing the failure mechanism from a pure axial load failure to an interactive failure. If, in addition to insulation on the column, the column is oriented with its web parallel to the beam and the connection is strengthened against fires, the structure could survive the entire fire. This survival would require large displacements, up to 1.5 m (or 16 % of the beam length) at mid-span, to develop catenary action. Insulating the beam as well would dramatically reduce these beam deflections. The temperatures at the limit-states and the duration in the fire simulations are summarized in Table 4-7.

0 0.1

-0.2 0.05

-0.4 0 -0.6 -0.05 No insulation -0.8 No insulation No ins. at mid-floor

Mid-span disp. (m) Ins. col. Ins. col.

Column horiz. disp. (m) disp. horiz. Column -0.1 -1 Ins. col. + improved Ins. col. + improved Conn. failure Conn. failure 0 2 4 6 8 10 0 2 4 6 8 10 Time (min) Time (min) (a) Mid-span displacement (b) Horizontal displacement at third floor

Time (min) 0 2 4 6 8 10 With insulation (°C) 0 32 67 104 141 178 Without insulation (°C) 0 168 381 602 791 921 Figure 4-48 Displacement for the exterior column sub-assembly model

143 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

Table 4-7 Limit-states for exterior column sub-assembly model T (°C) tcr Tcr (°C) cr Failure Design options (min) (Gas) Column Beam mechanism Column Typical 5.8 920 578 578 buckling Connection Insulation on column 7.8 965 136 767 failure Insulation on column + Interactive 8.2 973 145 812 Insulation on connection failure Insulation on column + Strengthened connection + - 1114 (*1) 413 (*1) 1055 (*1) Survive fire Rotated column orientation (*1) Experienced highest temperature during fire

4.4.3.3 Alternative Connection Design

In order to strengthen the connection against fires, two alternative design options are considered: (1) providing insulation on the connection and reinforcement in the slab, (2) using alternative flange-supported connection detail as shown in Figure 4-49. The slab reinforcement in option (1) provides the longitudinal tensile strength at the connections to maintain catenary action (insulating the connection does not provide enough strength). The proposed flange-supported connection in option (2) directly supports the beam without the bolts, such that the risk of shear failure at the connection during fires is significantly reduced. It is also advantageous that temperature of the upper flange at the connection is relatively low and the connection strength remains high. Thermally induced axial force in the beams, which could lead to failure of the connections, is also less significant in option (2). Another benefit of option (2) is that there will be less deformation of the bolts since rotational displacement at beam ends is permitted. The supporting plate of option (2) needs to be thick enough to transfer shear force from the beam to the column. An additional cover plate on the beam flange at the connection may be needed to prevent localized yielding at the flange. Ways to improve on this detail should be further studied. In terms of constructional aspects, the benefit of the drop-off assembly of the beams and easier installation of fire insulation on columns may compensate the extra fabrication required for option (2).

144 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

(a) Typical design (b) Proposed design Figure 4-49 Connection details between external column and beam

4.4.4 Conclusions

The exterior column sub-assembly model is composed of the finite shell element column and composite beam. The temperatures of the beam and column are generally different during fires and structural simulation should be carried out in the time domain, where time is the primary parameter. The behavior and limit-state of the created sub-assembly model are simulated in this time domain based on the time-temperature relationships of the compartment fire based on Eurocode 1 (2002). Conclusions through study about the exterior column sub-assembly for this archetypical steel-framed building design are summarized in following points:

1. For the case without insulation, the critical failure mechanism is buckling of the exterior column; the column buckles after a duration of 5.8 min and at a column temperature of 578 °C. Insulating the column changes the mechanism to the connection failure, slightly extending the duration to 7.8 min. The temperatures of the exterior column and beam at the limit-state are 136 °C and 767 °C, respectively. Adding insulation to the connection extends the duration to 8.2 min.

145 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

2. To fully survive the fire requires insulation on the exterior column, the connection, and the beam. However, as an alternative to insulating the beam, other design options are considered. The proposed design improvement includes rotating the column to provide more longitudinal constraint to the beam and reinforcement in slab or alternative flange- support connection detail as shown in Figure 4-49. Although large deformation of the beam (1.5 m at the mid-span) is expected, the failure may be avoided without the fire insulation on the beam by adopting the design improvement.

4.5 OVERALL LIMIT-STATE EVALUATION

Based on the limit-state analyses using the three sub-assembly models, the critical mechanisms with different design conditions are summarized in Table 4-8. The critical time is obtained from the time-temperature relationships of the steel members shown in Figure 4-47.

Table 4-8 Critical time, steel temperatures and failure mechanisms of sub-assemblies (time, min), [steel temperature, °C] ([Column temperature, Beam temperature] (°C) for exterior column sub-assembly) Insulation Insulation Sub-assembly No insulation on columns (*1) + on columns (*1) Connection design (*2) Column buckling No failure No failure Interior column (6.0), [599] (-), [-] (-), [-] Exterior column Column buckling Connection failure Interactive failure and beam (5.8), [578, 578] (7.6), [136, 767] (8.2), [145, 812] Connection failure Connection failure Connection failure Beam (6.3), [634] (6.3), [634] (7.3), [729] (*1) 9.5 mm gypsum board (*2) Insulation on connections for the exterior column sub-assembly and slotted bolt holes for the beam sub-assembly

Without fire insulation, column buckling occurs early in the fire. For the interior column, insulating the column avoids failure, whereas for the exterior assembly, insulating the column (but not the beam or connection) only changes the critical mechanism to bolted connection failure with a slightly longer endurance time (7.6 minutes versus 5.8 minutes). Insulating the bolted connection allows slightly longer endurance (8.2 minutes versus 7.6

146 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES minutes) changing the failure mechanism from the connection failure to the interactive failure. Insulating the beam (in addition to the column) avoids the failure of the exterior sub- assembly. The sub-assembly failure may also be avoided without this fire insulation on the beam by adopting design improvements, which are discussed in 4.4.3.3 (i.e., column rotation and reinforcement in slab as shown in Figure 4-49). Regarding the beam sub-assembly, insulating the connections does not prevent connection failure, if there is no insulation on beams. This is because connection forces developed by resistance to thermal expansion of the beams are often cause failure of the shear tab connections in compression. To prevent the connection failure, alternative design options are discussed, such as using slotted bolt holes and steel reinforcement in the concrete slab. A key finding from this study is that there is some possibility of avoiding failure without fire insulation on beams, while insulation on columns is essential. This finding is consistent to the observations from the Cardington fire test (SCI, 2000; Kirby, 1997, 1998) and the Broadgate Phase 8 fire (SCI, 1991) as described in Chapter 2. It is also found that failure of the bolted connections is a critical failure mechanism, as well as the column buckling. However, beam-drop failure due to the connection failure has not been reported in the past fires and the Cardington fire test. In the Cardington fire test, shear failure of the bolts were observed, but the beams did not fall, possibly due to the shear strength of the slabs. Several possible reasons for the difference between this study and observations from past fires are considered: (1) the uniform temperature of beams along the length assumed in simulation may be a highly conservative assumption (non-uniform temperature distribution in the beam length is expected in real fires and it would induce less significant longitudinal deformation due to the thermal expansion), (2) bolted connections in the beams may have lost the load transfer capacity in real fires; however, shear capacity of the concrete slab, which is neglected in this study, may have been high enough to prevent the beam-drop, and (3) the rotational deformation and longitudinal translation at the connections are not fully coupled in the analyses and this may cause the difference between the simulation and reality.

4.6 CONCLUSIONS OF GRAVITY FRAME ANALYSIS

The sub-assembly simulations for a representative steel building support some of the following observations and conclusions:

147 CHAPTER 4. ANALYSIS OF GRAVITY FRAMES

1. The restraint to vertical thermal column elongation provided by floors above the fire floor does not significantly impact the strength limit state of the columns at the fire floor. 2. Column buckling is a critical mechanism and insulation on the columns is essential to avoid collapse during building fires. 3. When the interior gravity column buckles during the fires, the surrounding structure of the column is not strong enough to provide an alternative load-carrying path. However, bracing the bolted connections between the column and connecting beams can provide an alternative path and prevent the initiation of the global building collapse. 4. Failure of the bolted connections may occur due to thermal elongation of the beams. Short of providing full beam insulation, beam failures can be controlled through enhanced connection details that employ slotted bolt holes to permit the thermal elongation and incorporate thermally protected reinforcement in the slab. 5. Protecting the connections with enhanced details and accepting large deformation (sagging) during fires, the beams could survive more than 1000 °C by catenary action. 6. Insulating the exterior columns changes the failure mechanism from the single-story buckling to the multi-story buckling associated with large deformation of the connecting beam during the fires. Increasing the column stiffness for the frame by rotating the column orientation (web is parallel to the frame) prevents this multi-story buckling and the connection failure is then critical. 7. The connection between the exterior column and beam fails under tension force associated with the catenary action. The slotted bolt holes, which are effective to prevent the failure due to thermal elongation of the beam, are not helpful against this failure mechanism. Therefore, other improved design options (e.g., reinforcement in the slab and insulation on the connection) should be adopted for the connection.

These specific conclusions are based on a limited case study example and require substantiation by further study. The proposed sub-assembly models are suggested as a means to conduct such studies.

148

CHAPTER 5 PROBABILISTIC ASSESSMENT

5 PROBABILISTIC ASSESSMENT

5.1 OVERVIEW

In this chapter, probabilistic studies of steel structures under fires are performed for individual members (columns) and the gravity frames of the benchmark building discussed in Chapter 4. Statistical properties (mean and coefficient of variation) of characteristic assessment parameters are defined based on previously published information and by applying engineering judgment to estimate factors whose statistical information is unknown. Sensitivity of the critical temperatures to these probabilistic parameters is evaluated for the three sub-assembly structures introduced in Chapter 4. The uncertainty associated with fire behavior is briefly addressed, but the primary focus of the assessment is on the structural aspects. Probability of collapse of the sub-assemblies under given intensity (temperatures) is assessed by the mean-value first-order second-moment (FOSM) method.

5.2 STRUCTURAL UNCERTAINTIES IN FIRE ENGINEERING

5.2.1 Summary of Statistical Data

Summarized in Table 5-1 is statistical information (mean and coefficient of variation, c.o.v.) of input parameters used for structural simulations. This information will be used later in this chapter for the probabilistic risk assessment. The parameters include the dead load, live load,

yield strength of steel at ambient and elevated temperatures ( Fy0 and FTy ()), elastic modulus of steel at ambient and elevated temperatures ( E0 and E()T ), strength (force- displacement relationship) of the vertical constraint spring for the interior column ( Ps ), longitudinal constraint stiffness of the beams ( Ks ), and the shear strength and deformation capacity of the bolted connections ( PB and ΔB ).

149 CHAPTER 5. PROBABILISTIC ASSESSMENT

Table 5-1 Statistical data for uncertainties μ (mean) δ (c.o.v) Type (*1) Reference Ellingwood, Dead load 4420 Pa (*2) 0.10 Normal 1983 Ellingwood, Live load 718 Pa (*2) 0.60 (*3) Gamma 1983 (*4) Ellingwood, F 1.05 F 0.10 Lognormal y0 ychar, 1983

FTy () KTFyy() 0 0.22 - Section 5.2.2 Ellingwood, E 200 GPa 0.06 Normal 0 1983 (*5) E()T KTEE ()0 0.22 - P , vertical spring s Figure 5-2 Figure 5-2 - Section 5.2.3 for the interior column

PB , strength (*6) 1.2 KTNRyB() B Bn0 0.29 - Section 5.2.4 of bolted connections Δ , deformation capacity B Figure 5-5 0.25 - Section 5.2.6 of bolted connections K , longitudinal constraint s 1.0×107 N/m (*7) Lognormal stiffness of the beams (*1) Information about distribution types is reported for completeness but is not used in the probabilistic assessment by first-order second moment (FOSM) method. (*2) Mean values of the dead and live loads are 102.5 % and 25 % of the unfactored design loads (4310 Pa (90 psf) for dead load and 2870 Pa (60 psf) for live load), respectively (Ellingwood, 1983). (*3) c.o.v. of live load is defined as 0.40 to 0.80 (area-dependent) in Ellingwood (1983). The average value is taken. (*4) Fychar, is the nominal specified value of the yield strength (345 MPa for Grade 50 steel). (*5) Sufficient information has not been found. The c.o.v. was assumed to be the same as that for the yield strength, 0.22. This may be a conservative assumption considering that the yield strength and elastic modulus at ambient temperature have c.o.v.’s of 0.1 and 0.06, respectively. (*6) NB is the number of bolts in a connection, and RBn0 the is nominal shear strength (bolt

shear failure) of the bolts from AISC (2005). RBn0 is 160 kN for ASTM A325 bolts. (*7) The mean stiffness is derived in Section 4.3.3.5. The 16 % and 84 % percentiles are assumed to be 400 % and 25 % of the median, respectively.

The statistical information about the loads (dead and live loads) and steel properties at ambient temperature is obtained from Ellingwood (1983). The load properties under fire

150 CHAPTER 5. PROBABILISTIC ASSESSMENT

conditions are assumed as the same as those under non-fire conditions. Variations of the yield strength of steel and the shear strength of the bolted connections at elevated

temperatures are derived from past test data (c.o.v of FTy () is 0.22 and c.o.v of PB is 0.29).

The c.o.v. of ET() is assumed as the same as the c.o.v. of FTy () (0.22), because no test data were obtained for this property and the value is likely conservative (since c.o.v. of Fy0 (0.10) is greater than c.o.v. of E0 (0.06)). As will be shown later, the critical temperatures are not sensitive to E()T . This will be shown using this conservatively assumed variation of E()T .

The statistical information of the deformation capacity of bolted connections ( ΔB ) is approximately defined by calibration to the test data.

5.2.2 Variability of Yield Strength of Steel

Statistical information on the yield strength of structural steel at elevated temperatures is necessary for the probabilistic assessments using mean-value FOSM. In particular, the mean strength and the coefficient of variation (c.o.v.) at 2 % strain at elevated temperatures, denoted as μ ()T and δ , respectively, are needed. δ can be a function of Fy FTy FTy temperature; however, it is assumed to be a constant for all elevated temperatures. Using the

reduction factor of steel strength at elevated temperature KTy (), described in Chapter 3 (Eq.

(3.1)), the mean of the 2 % strength, μ ()T , is defined as follows: Fy

μ ()TKT= ()μ (5.1) FyFyy0 μ = μ (5.2) FFPSyy00, μ =1.05F (5.3) Fy 0, PS y0, char

where μ and μ are the means of the 2 % strength and 0.2 % off-set strength at Fy 0 FPSy 0, ambient temperature, respectively. Fy0, char is the characteristic 0.2 % off-set strength. The symbols used in this section are summarized in Table 5-2. It is assumed in Eq. (5.2) that the 2 % strength at ambient temperature is the same as the 0.2 % off-set strength, although strain hardening may cause μ to be greater than μ . The Fy 0 FPSy 0,

151 CHAPTER 5. PROBABILISTIC ASSESSMENT

1.05 coefficient in Eq. (5.3) is the ratio between the actual and the nominal 0.2 % off-set yield strength (Ellingwood, 1983). Given F and KT(), μ ()T is obtained through ychar0, y Fy Eqs. (5.1)-(5.3) as a function of elevated temperature, T .

Table 5-2 Symbols regarding statistical properties of steel strength Property Strength mean c.o.v. FT() μ ()T δ 2 % strength at elevated temperatures y Fy FTy F μ 2 % strength at ambient temperature y0 Fy 0 - F μ δ 0.2 % off-set strength yPS0, FPSy 0, FPSy 0,

Characteristic strength Fychar0, Fychar0, - Ratio between 2 % strength at elevated - μ ()T δ temperatures and 0.2 % off-set strength FPSy | FTPSy |

It is assumed that the mean of the 2 % strength at elevated temperatures, μ ()T , is a Fy product of the means of the 0.2 % off-set strength, μ , and the ratio between the 2 % FPSy 0,

strength at elevated temperatures and the 0.2 % off-set strength, μ ()T . FPSy |

μ ()TT= μμ() (5.4) FFPyyy0,SF |PS

Taking a first order approximation, the coefficient of variation (c.o.v.) of the 2 % strength at elevated temperatures, δ , is defined as follows: FTy

22 δδ=+ δ (5.5) FTyy()() F0, PS FTPS y |

where δ and δ are the c.o.v. of the 0.2 % off-set strength and the c.o.v. of the ratio FPSy 0, FTPSy | between the 2 % strength at elevated temperatures and the 0.2 % off-set strength, respectively. In Eq. (5.5), it is assumed that δ and δ are not correlated. δ FPSy 0, FTPSy | FPSy 0, has been derived as 0.1 in a past study (Ellingwood, 1983), while δ is evaluated by FTPSy | using the test data reviewed in Chapter 2 (Harmathy and Stanzak, 1970; Skinner, 1972;

152 CHAPTER 5. PROBABILISTIC ASSESSMENT

DeFalco, 1974; Fujimoto et al.,1980, 1981; Kirby and Preston, 1988). Shown in Figure 5-1(a) is the measured strength at 1.5 % and 2.0 % strain at ambient and elevated temperatures from 300 °C to 600 °C. The 1.5 % strength is investigated, because some of the test data do not include strength at 2.0 % strain. The 1.5 and 2.0 % strengths are both normalized with respect to the measured 0.2 % off-set yield strength in each test. Shown in Figure 5-1(b) are the c.o.v.s of the 1.5 % and the 2.0 % strength. There are approximately 10 data points at each temperature. The effect of increasing temperature on the c.o.v. is somewhat negligible until the temperature range between 500 °C and 600 °C, at which point c.o.v. drastically increases.

0.25 1.5 ε=1.5% 0.2 ε=2.0%

1 0.15

c.o.v 0.1 0.5 Mean at ε=1.5% ε Stress / Yield stress Stress Mean at =2.0% 0.05 Data at ε=1.5% Data at ε=2.0% 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Temperature (°C) Temperature (°C) (a) Mean (b) Coefficient of variation Figure 5-1 Variation of tested steel strength under elevated temperatures

Based on Figure 5-1(b), δ is assumed to be 0.2, which is close to the c.o.v. at 600 FTPSy | °C. This assumption is based on the fact that the critical temperatures of steel structures are generally greater than 500 °C and, therefore, data at 600 °C are more important than data at lower temperatures. δ is defined to be a constant value for elevated temperatures, since FTPSy | this assumption greatly simplifies the probabilistic analyses given the limited data set. Substituting 0.1 and 0.2 into δ and δ in Eq. (5.5) yields 0.22 for δ . FPSy 0, FTPSy | FTy

5.2.3 Variability of Longitudinal Spring Stiffness for Interior Column

The properties of the vertical spring connected to the interior column in the benchmark building were discussed in Section 4.2.3.3, and the relationship between the vertical

153 CHAPTER 5. PROBABILISTIC ASSESSMENT displacement and reacting force of the spring was shown in Figure 4-16. The relationship was evaluated approximately by introducing several assumptions and simplifications. Therefore, precise statistical evaluation of this relationship is difficult. Possible variations of this relationship are conservatively studied by considering the upper and lower bounds. In Section 4.2.3.3, the vertical force-displacement relationship of the spring was calculated using the analysis model shown in Figure 4-14. The model is composed of a composite beam, bolted connection and the longitudinal constraint of the beam by the surrounding floor structure. The variation of the properties of these components affects the calculated properties of the vertical spring. Three factors of these components are considered to influence the properties of the vertical spring: (1) the strength and (2) stiffness of the bolted connection and (3) the stiffness of the longitudinal constraint spring for the beams. Possible upper and lower bounds of these factors are discussed and the variation of the vertical spring properties is evaluated. The strength of the connections strongly influences the strength of the vertical spring, because steel plate bearing (shear-tab or web plate) at the connections is the critical failure mechanism of the vertical springs. Considering that the c.o.v. of the steel yield strength is 0.1 (Ellingwood, 1983) and the uncertainties introduced by other factors such as accuracy of fabrication, influence from other failure mechanisms (e.g., bolt shear failure), and interactive behavior with axial and rotational deformation, the c.o.v. of the connection strength is assumed to be 0.2. Using this information, the upper and lower bounds of the connection strength are assumed to be 120 % and 80 % of the typical (mean) strength. The rotational spring stiffness of the bolted shear-tab connection strength influences the stiffness of the vertical spring. Liu and Astaneh-Asl (2004) investigated the rotational stiffness and strength of this type of connection, but they did not specifically address statistical information about the rotational stiffness. They did, however, present data on key rotational deformations and estimated their c.o.v. to be 0.2. Assuming this value is closely related to the vertical spring stiffness and considering other uncertainties including interaction with the axial force, the c.o.v. of the rotational stiffness of the connections is assumed to be 0.3. There is no significant statistical data available to characterize the uncertainty of the horizontal elastic restraint stiffness that the floor structure provide to the beams. Past studies have not focused on this property, which depends on many factors including building shape and column location. Since generalizing statistical information is difficult, the upper and

154 CHAPTER 5. PROBABILISTIC ASSESSMENT

lower bounds of the longitudinal spring stiffness are assumed to be 400 % and 25 % of the originally defined stiffness (1.25×105 N/m for Beam a and b, and 5.4×104 N/m for Beam c). Other factors in the analysis model used to calculate the vertical spring property (shown in Figure 4-14) are less influential and the effect from the uncertainties of these factors are considered to be included in the three primary factors: connection (1) strength and (2) stiffness, and (3) constraint stiffness for the beams. Ratios of the assumed upper and lower bounds of these three factors with respect to characteristic (or nominal) values are summarized in Table 5-3.

Table 5-3 Ratios of upper and lower bounds of factors for the vertical spring Lower bound Upper bound Connection strength 0.8 1.2 Connection stiffness 0.7 1.3 Constraint stiffness for beams 0.25 4.0

Variations of the force-displacement relationships of the vertical constraint spring of the interior column are investigated, in conjunction with the variation of the three factors in Table 5-3. The combinations of these factors that result in the highest and lowest vertical spring strength are shown in Table 5-4 and the corresponding force-displacement relationships are plotted in Figure 5-2. The maximum strength of the lowest spring does not differ significantly from that of the typical (mean) spring (4 % lower), while the strength of the highest spring is significantly larger (28 % higher than mean). The effect of the longitudinal floor constraint stiffness on this spring property is further investigated with two additional combinations of the factors (high and low constraint stiffness for the beams, see last two rows of Table 5-4). Typical (mean) values are used for other two factors (rotational strength and stiffness of the connections). Although the upper and lower bounds at the constraint stiffness have a large range (400 % and 25 % of the mean), the difference compared with the force-displacement relationship of the vertical spring is not significant (the greatest difference at the maximum strength is 11 %). Therefore, the relationship is not sensitive to the longitudinal floor constraint stiffness. Because the constraint stiffness is highly uncertain and derived with a relatively simple elastic model as shown in Figure 4-14, this study result substantiates the use of the simplified model for the longitudinal constraint stiffness.

155 CHAPTER 5. PROBABILISTIC ASSESSMENT

Table 5-4 Combinations of factors for vertical spring of interior column Connection Connection Constraint stiffness

strength stiffness for beams Typical Typ Typ Typ Highest UB LB LB Lowest LB UB UB Higher beam constraint Typ Typ UB Lower beam constraint Typ Typ LB (*) Typ: typical value (mean), UB: upper bound, and LB: lower bound

Figure 5-2 Variation of vertical spring properties

5.2.4 Variability of Shear Strength of Bolts

Yu (2006) experimentally investigated the performance of single shear bolted connections with 22.2 mm (7/8 inch) diameter bolts (ASTM A325) and 9.5 mm (3/8 inch) thick Grade 50 plates. Plate bearing was observed as the failure mode at ambient temperature, while at elevated temperatures the failure mode becomes shear failure of the bolts. This is due to the more rapid strength deterioration of bolt capacity versus steel plate capacity (Figure 4-32 and Table 4-4). Presuming that the bolt shear failure is the dominating failure mode for the bolted shear-tab connections of the benchmark steel building, statistical properties of the bolt shear strength at elevated temperatures are investigated in this section. Yu (2006) also tested the shear strength of ASTM A325 and A490 bolts using thicker steel plates so that bolt shear failure is the critical failure mode at all temperatures. Kirby

156 CHAPTER 5. PROBABILISTIC ASSESSMENT

(1995) tested the shear strength of British M20 Grade 8.8 high strength bolts in a similar way, where two sets of tests (set A and C) were performed for different bolt steel compositions. The shear strengths of the bolts from these tests (Yu 2006 and Kirby, 1995) are plotted in Figure 5-3. Figure 5-3(a) shows the measured critical shear strength (double shear) of the bolts at elevated temperatures up to 800 °C, and Figure 5-3(b) shows the strength normalized with respect to the shear strength at ambient temperature and superimposed with the ECCS (2001) reduction factor of bolt strength at elevated temperatures (the rule for the normalization will be described later). Notice that the measured strengths around 300 °C are slightly higher than that at ambient temperature, whereas the ECCS strength prediction continuously decreases under elevated temperatures. Overall, the normalized test data agree to ECCS within 20-30 % up to 700 °C.

600 Yu A490 Yu A490 Yu A325 Yu A325 1 500 Kirby A Kirby A Kirby B Kirby C 0.8 ECCS 400

0.6 300

200 0.4 Shear strength (kN) strength Shear

100 strength shear Normalized 0.2

0 0 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (a) Measured shear strength (b) Normalized shear strength Figure 5-3 Shear strength of bolts at elevated temperatures

The nominal shear strength of high-strength bolted connections at ambient temperature

RBn0 (bolt shear dominating strength) is defined in AISC (2005) as:

RBn0 = FA Bv B (5.6)

where FBv and AB are the ultimate shearing stress and the cross-sectional area of the unthreaded part of the bolts, respectively. The British Grade 8.8 high strength bolts tested by Kirby are equivalent to ASTM A325 and the nominal strength is evaluated in the same way

157 CHAPTER 5. PROBABILISTIC ASSESSMENT

for ASTM A325 based on AISC. The calculated FBv is 415 MPa (60 ksi) for both ASTM A325 and British Grade 8.8, and 520 MPa (75 ksi) for ASTM A490. The tested bolts do not

2 2 include threads in the shear section, and AB is 388 mm (0.60 in ) for both ASTM A325 and A490 with 9.5 mm (7/8 inch) diameter and 288 mm2 (0.45 in2) for British grade 8.8 M20

bolts. The RBn0 values based on Eq. (5.6) are 160 kN, 200 kN, and 120 kN for ASTM A325, A490 22.2 mm diameter, and British grade 8.8 M20, respectively. The measured shear

strength of the bolts at ambient temperature is denoted as RBtest0 and the mean value is μ yB0 .

The ratio between μ yB0 and RBn0 is termed CR , as in the following equation.

μ yB00= CR R Bn (5.7)

where CR is found to be equal to 1.5 from the test data. The tested shear strength of bolts

shown in Figure 5-3(b) is normalized with respect to μ yB0 . The statistical properties of the shear strength of bolted connections, which will be

needed later for the FOSM analysis, are the mean, μ yB , and c.o.v., δ yB . The mean at

ambient temperature μ yB0 is CRRBn0 , while c.o.v. at ambient temperature δ yB0 is obtained from the test data by Yu (2006) and Kirby (1995) as 0.05. The mean at elevated temperatures μ yB ()T is defined in the following equation using the reduction factor of bolted connection strength KTyB () by ECCS (2001):

μ yB ()TKT= yB ()μ yB0 (5.8)

The values of KTyB () are shown in Figure 4-32 and Table 4-4 in Chapter 4. Under this definition, μ yB ()T is not exactly equal to the mean of the normalized test data as shown in Figure 5-3(b); however, it is close to the data and the formulated strength at elevated temperatures consistent with ECCS prediction. Using ECCS is more advantageous than using the test data for analytical manipulation.

158 CHAPTER 5. PROBABILISTIC ASSESSMENT

The c.o.v. of the shear strength of the bolts at elevated temperatures is assumed to be a

constant at any elevated temperature and denoted as δ yBT . The c.o.v. is defined in the following equation as the variation of the tested bolt shear strength with respect to μ yB ()T at temperatures between 400 °C and 700 °C.

n 2 ∑{}RTBiμ yB()1 i − δ = i=1 for 400 °C ≤ T ≤ 700 °C (5.9) yBT n −1 i

where RBi is the shear strength of i th test data point, Ti is the temperature of the i th test data point, and n is the number of tests performed at elevated temperatures between 400 °C

and 700 °C. The calculated value of δ yBT using Eq. (5.9) is 0.29. Shown in Figure 5-4 are the test data of Figure 5-3, normalized by the mean strength defined in Eq. (5.8). The assumed mean strength is lower than the measured mean strength at high temperature. This is clearly observed at temperatures greater than 600 °C, although by this point the mean value is only about 10 % to 15 % of the ambient strength values. The strength of the assumed mean plus and minus the derived standard deviation is superimposed in the figure. The assumed variation seems reasonable with the test data at elevated temperatures between 400 °C and 700 °C. The mean and c.o.v. of the bolt strength in this section are summarized in Table 5-5.

2.2 Yu A490 Yu A325 2 Kirby A 1.8 Kirby C ECCS 1.6 μ ± σ 1.4 1.2 1 0.8

Normalized shear strength / ECCS / strength shear Normalized 0.6 0 200 400 600 800 Temperature (°C) Figure 5-4 Shear strength of bolts normalized with ECCS strength

159 CHAPTER 5. PROBABILISTIC ASSESSMENT

Table 5-5 Mean and c.o.v. of shear strength of bolts

μ yB (mean) δ yB (c.o.v.)

Ambient temperature CRRBn0 0.05

Elevated temperatures KTyB()μ yB0 0.29

5.2.5 Variability of Longitudinal Strength of Springs for Bolted Connections

Bolted beam connections are modeled as inelastic longitudinal (axial) springs (see Figure 4- 23). As shown previously in Figure 4-31, the force-displacement relationships of the springs are modeled using a bi-linear curve with a post peak-strength softening branch. The peak-

strength, PTB (), is defined by the following equation:

PTBy()= KBBBy () Tζ Nμ B0 (5.10)

where ζ B is an adjustment factor to account for the non-uniform distribution of bolt forces due to rotation. NB is the number of the bolts. Assuming 0.05 for the rotational

displacement of the connection at the limit-state, the calibrated value of ζ B is 0.8, as described in 4.3.3.4 and 4.4.2.2. Substituting Eq. (5.7) (CR = 1.5) leads to the following relationship between the mean strength of the longitudinal (axial) spring strength for the connection at elevated temperatures and nominal shear strength of bolts at ambient temperature:

PTBy()= 1.2 KBB () TNRBn0 (5.11)

The c.o.v. of the peak strength of the connections is assumed to be the same as the c.o.v. of the shear strength of bolts (0.29).

5.2.6 Variability of Deformation Capacity of Bolted Connections

While bolt shear failure is the critical mechanism for shear-tab connections, data from bolts alone is not sufficient to evaluate the deformation characteristics of the connections, since the beam web or bracket deformation is not negligible. Yu (2006) investigated force-

160 CHAPTER 5. PROBABILISTIC ASSESSMENT displacement relationships of single shear bolted connections at elevated temperatures. The tests provide useful information for the evaluation of the actual behavior of shear-tab connections. As Yu’s tests were the only data available in this study, the variation of the deformation capacity of the bolted connections is based on Yu’s study. The force-displacement relationships of the longitudinal (axial) spring for the bolted connections in the beam sub-assembly are obtained in Section 4.3.3.4. (Figure 4-33) Although this model reasonably agrees with the test data, a range of variation is observed in Figure 5-5, where the test data shown in the figure is identical to those in Figure 4-33. The upper and lower bounds of the possible variation of the deformation capacity are assumed to be 125 % and 75 % of the original. These upper and lower bound models are superimposed on the data in Figure 5-5, assuming that the peak strength and proportion of ΔBp and ΔBe are maintained. These upper and lower bounds compare well with the test data by Yu (2006)

(the bolt edge distance of 1.0 DB and 1.5 DB , where DB is bolt diameter), especially at 600 °C and 700 °C. Due to insufficient statistical data, the c.o.v. of the deformation capacity of the longitudinal spring of the connection is not obtained in a formal manner. Rather, the c.o.v. is assumed as equal to 0.25.

161 CHAPTER 5. PROBABILISTIC ASSESSMENT

500°C 600°C 350 350 1.0D Conn. 1.0D Conn. 300 1.5D Conn. 300 1.5D Conn. Model μ Model μ 250 250 Model μ ± δ Model μ ± δ 200 200

150 150 Force (kN) Force (kN) 100 100

50 50

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Displacement (mm) Displacement (mm) (a) 500 °C (b) 600 °C

700°C 800°C 350 350 1.0D Conn. 1.0D Conn. 300 1.5D Conn. 300 1.5D Conn. Model μ Model μ 250 250 Model μ ± δ Model μ ± δ 200 200

150 150 Force (kN) Force (kN) 100 100

50 50

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Displacement (mm) Displacement (mm) (c) 700 °C (d) 800 °C Figure 5-5 Uncertainty of deformation capacity of bolted connection

5.2.7 Variability of Time-temperature Relationships in Compartment Fire

The behavior in buildings under fires is highly uncertain. The uncertainty is due to various factors such as the geometry of buildings and fire compartments, the fire location, fuel load, the materials in the fire compartments, and the firefighting activities. Assessing such uncertain behavior is not a primary goal of this research. Nevertheless, to provide a more complete perspective on the probabilistic issues, the variability of the time-temperature relationships (gas and resulting steel temperatures) in the compartment fire is briefly discussed in this section. In Section 4.4.3.1 (Figure 4-47), the compartment fire behavior of the benchmark building was deterministically simulated and discussed using the parametric fire curve obtained following Eurocode 1 (2002) with the following input parameters: fuel load, 430 MJ/m2; fire compartment area, 61 m2; compartment total surface area, 235 m2;

162 CHAPTER 5. PROBABILISTIC ASSESSMENT opening factor, 0.05; and fire fighting factor, 0.282. The thermal heat transfer follows an approach described by Buchanan (2002), assuming 9.5 mm gypsum board for the insulated members. Details of this parametric fire simulation and the time-step heat transfer approach are described in Appendix B.4.1. The fuel load, firefighting factor, and opening factor are influential factors for this fire simulation. Consequently, they are they are the variables used to investigate the variation in the fire simulation. Values of these factors are parametrically varied between the upper and lower bounds as shown in Table 5-6. The resulting variations to the time versus temperature curves are shown in Figure 5-6. The lower and upper bounds for the fuel load are defined as mean plus and minus one standard deviation (the c.o.v. is 0.3 by EC1, 2002). Those for the firefighting factor are the minimum and maximum value in EC1; and those for the opening factors are calculated from possible window sizes for the office-type benchmark building. Two cases are studied with different values for the upper bound of firefighting factor: 0.360 for poor conditions and 1.215 for extremely poor conditions (no safe access routes, no fire fighting devices, and no smoke exhaust systems in staircases, as described in EC1).

Table 5-6 Band of influential factors for fire simulation Typical Lower bound Upper bound Fuel load (MJ/m2) 420 301 559 0.360 (Poor condition) Firefighting factor 0.282 0.149 1.215 (Extremely poor condition) Opening factor 0.05 0.03 0.07

163 CHAPTER 5. PROBABILISTIC ASSESSMENT

(a) Normal firefighting condition

(b) Poor firefighting condition (different time scale from (a)) Figure 5-6 Variations of time-temperature relationships

The extremely poor firefighting condition delays the time to achieve the peak gas temperature to more than 100 min and the total fire duration is more than 180 min. This long fire duration increases the maximum steel temperature, especially for the insulated members. The maximum temperatures of gas and steel are investigated under the three firefighting conditions (good, poor, and extremely poor) with varied fuel load and opening factor. The

164 CHAPTER 5. PROBABILISTIC ASSESSMENT results are summarized in Table 5-7. Large variation of the maximum temperatures is clearly seen in this table, especially for insulated steel, where the lowest maximum temperature is 344 °C, while the highest is 1009 °C.

Table 5-7 Maximum temperatures in variation of fire simulation (°C) Steel Steel Fire fighting condition Gas with insulation without insulation Good 913 344 777 Poor 1255 557 1244 Extremely poor 1311 1009 1307

Although the opening factor may be more certain once the building is designed, a wide range between the upper and lower bounds is considered to account for variations of different building designs. There are many other factors that influence fire simulations such as the material of the fire compartment boundary and the extent of fire insulation, which greatly affects steel temperatures. However, these factors are treated deterministically in this introductory simulation. Full consideration of these probabilistic factors will provide even greater variations; however simulating the variation is not a goal of this research. Rather the probabilistic assessment in this study focuses on structural aspects (e.g., the probability of structural failure under given temperatures).

5.3 PROBABILISTIC STUDIES

5.3.1 Sensitivity of Critical Temperatures to Uncertainties

Structural stability of the benchmark steel building under the localized compartment fire was deterministically evaluated in Chapter 4. Three types of sub-assembly analysis models (interior column, beam, and exterior column beam sub-assemblies) were introduced in order to precisely simulate possible failure mechanisms. Using the statistical information about structural parameters obtained in the previous section, sensitivities of the critical temperatures for the three sub-assemblies to the uncertain parameters are evaluated in this section.

165 CHAPTER 5. PROBABILISTIC ASSESSMENT

5.3.1.1 Sensitivities in Interior Column Sub-assembly Study

The critical temperature for the buckling at the interior column is 598 °C by the deterministic evaluation. Structural uncertainties introduced to the probabilistic assessment for the interior column are dead load, live load, elastic modulus of steel, yield stress of steel, and the vertical spring properties of floor framing. The statistical properties of these variables are summarized in Table 5-1. Variation of the critical temperature with the upper and lower bounds (mean plus and minus standard deviation) of each variable is summarized in Figure 5-7.

Uncertainty LL

-60 -40 -20 0 20 40 60 Sensitivity (°C) Figure 5-7 Sensitivity of the critical temperature of interior column sub-assembly

Referring to Figure 5-7, the yield strength of steel is the most influential factor among the uncertain variables. Although the upper and lower bounds of the vertical spring

properties, Ps are liberally evaluated due to their high uncertainty, they have little effect on the results. The crucial temperature is insensitive to the vertical spring properties. This observation is consistent with the study performed in Section 4.2.4 showing that the critical temperature of the system without the vertical spring 599 °C is essentially the same as that with the spring (598 °C). Modulus of elasticity also does not strongly influence the critical temperature. This is because the slenderness ratio of the column is relatively small ( Lry = 42.3; assuming L = 4 m as the story height) and the limit-state mechanism is inelastic buckling, which is more controlled by the yield strength than the elastic modulus. Sensitivity of the limit-state to the yield strength and elastic stiffness with varied member length is studied additionally in Appendix A.3.5.

166 CHAPTER 5. PROBABILISTIC ASSESSMENT

5.3.1.2 Sensitivities in Beam Sub-assembly Study

The critical mechanism of the beam sub-assembly at elevated temperatures is failure of the bolted connections due to thermal expansion of the beam. Uncertain variables for this system

are the dead load (DL), live load (LL), deformation capacity ( ΔB ) and strength ( PB ) of bolted

connections, yield strength of steel ( Fy ), and longitudinal constraint stiffness by the

surrounding floor framing ( Ks ). Sensitivity of the critical temperature (634 °C) to these variables is shown in Figure 5-8. The most influential factor is the deformation capacity of the bolted connections ( ΔB ), which has about four times the influence of the connection strength ( PB ). This implies that the axial force induced by the thermal expansion overwhelms the strength of the bolted connections. As such, strengthening the connection is a less effective method of preventing the connection failure than is providing greater deformation capacity. Whereas the yield strength of steel at elevated temperatures is the most influential factor for the interior column study, this is not the case for beams due to their different failure mechanism. The elastic modulus of steel is not included in Figure 5-8; however, it has been confirmed that the influence is limited.

Δ B P B DL Uncertainty LL

-60 -40 -20 0 20 40 60 Sensitivity (°C) Figure 5-8 Sensitivity of the critical temperature of beam sub-assembly

The sensitivity of the longitudinal spring stiffness from the floor framing, Ks , is relatively large (the second largest next to the deformation capacity of the bolted connections). This result makes sense, because the longitudinal spring stiffness influences the induced force at the connections and the critical mechanism is connection failure due to

167 CHAPTER 5. PROBABILISTIC ASSESSMENT

the thermal expansion of the beam. However, it should be noted that the upper and lower bounds (mean plus and minus one standard deviation) of the spring stiffness were assumed to

7 be 400 % and 25 % of the standard (mean) stiffness μKs (=10 N/m). The higher spring

stiffness (4.0 μKs ) induces greater axial force at the connections and consequently decreases the critical temperature. The lower spring stiffness increases the critical temperature and, in fact, changes the failure mechanism from connection failure to beam yielding. Section 4.3.4.3. in Chapter 4 describes details of the behavior and variation of the critical

temperatures with different longitudinal spring stiffness (10 μKs and 0.1 μKs ). Combining the results of this section and Section 4.3.4.3., the critical temperature, Tcr , is compared with Ks in Table 5-8. This table shows that Tcr does not change in proportion to Ks . Considering the uncertainty of Ks and the wide range of the variation, Tcr is not strongly sensitive to Ks .

Table 5-8 Critical temperature with various constraint stiffness

Ks / μKs 0.1 0.25 1.0 4 10

Tcr (°C) 662 (+28) 642 (+8) 634 (±0) 605 (-29) 586 (-48)

5.3.1.3 Sensitivities in Exterior Column Sub-assembly Study

Behavior of the exterior column sub-assembly at elevated temperatures is deterministically evaluated in Section 4.4.3. The critical mechanism is exterior column buckling, but insulating the column changes the mechanism to connection failure. Temperatures of the exterior column and the connecting beam (Beam c) are different and the limit-state is evaluated in the time domain as shown in Chapter 4. Although sensitivity of the limit-state to uncertain variables can also be evaluated in the time domain, the temperature domain is used in this section in order to make the evaluation less dependent on the fire simulation and more convenient to compare with other studies for the interior column and beam sub-assemblies. The temperature of the exterior column and beam in the fire compartment at the limit- state is defined as the critical temperature. The sensitivity of the critical temperature for the buckling of exterior column (without fire insulation on the column) to the uncertain variables is similar to that studied for the interior column sub-assembly. The sensitivity of the critical temperature of the beam with fire insulation on the column is investigated in this section to

168 CHAPTER 5. PROBABILISTIC ASSESSMENT determine the failure mechanism for the exterior column sub-assembly. The column insulation prevents column buckling and the connection failure occurs at 5.8 min during the particular compartment fire studied in Section 4.4.3.1. The critical temperatures at the limit- state of the exterior column and beam are 136 °C and 767 °C, respectively. The uncertain variables in this study are yield strength of steel ( Fy ), deformation capacity ( ΔB ) and strength ( PB ) of the bolted connection, dead load (DL), live load (LL). The sensitivity of the critical temperature of the beam to these variables is shown in Figure 5-9. The most influential variable is the yield strength of steel, Fy . The strength of the bolted connection,

PB , is not as influential, although the connection failure is critical. This is because the flexural stiffness loss of the beam changes the vertical load carrying mechanism from bending to tension with catenary action (Figure 4-26), and the axial force overcomes the connection strength. Deformation capacity of the bolted connection does not influence the critical temperature, which is in contrast to its significant effect for the beam sub-assembly. This contrast is determined from the connection failure in different phases: the connection in the beam sub-assembly fails in the compression phase (Figure 4-26), while that in the exterior column sub-assembly fails in the tension phase. Connection failure in the compression phase is deformation controlled because of thermal expansion, while that in the tension phase is strength controlled because of the catenary action.

Δ B

P B Uncertainty DL

-60 -40 -20 0 20 40 60 Sensitivity (°C) Figure 5-9 Sensitivity of the critical beam temperature of exterior column sub- assembly

169 CHAPTER 5. PROBABILISTIC ASSESSMENT

5.3.2 Collapse Probabilities of Sub-assemblies given Temperatures

Using the statistical information shown in Table 5-1, the probability of failure is calculated for the three sub-assemblies: interior column, beam, and exterior column. Deterministic relationships between the gas and steel temperatures in the fire compartment are linearly scaled for the temperature and the probability of failure under given elevated gas temperatures is assessed. The deterministic temperature-time relationships are shown in Figure 5-10, which is previously shown as Figure 4-47 based on the fire simulation conditions described in 4.4.3.1. The mean-value first-order second-moment method (FOSM, equations are shown in Appendix B.4) is used for the assessment, and all random variables are assumed to be uncorrelated. A calculated cumulative distribution function (CDF) for failure with respect to the gas temperature is shown in Figure 5-11. The CDFs are evaluated for the three sub-assemblies (interior column sub-assembly with and without fire insulation, beam sub-assembly without insulation, and exterior column sub-assembly with insulation on the column and without insulation on the beam). Of the three sub-assemblies, the critical collapse mechanism is buckling of the un- insulated interior column, assuming that the exterior column is insulated. Figure 5-11 shows that the failure of the beam sub-assembly (connection failure) can be critical (probability of interior column buckling at 620 °C is about 50 %, while that of the connection failure at 620 °C is about 25 %). At 620 °C, the probability for the failure of the exterior column sub- assembly is very low. The maximum gas temperature in the fire simulation shown in Figure 5-10 is 1114 °C. Therefore, the sub-assemblies would fail in most cases except the interior column sub- assembly with fire insulation. Once performance of the proposed connection details is precisely examined, additional study on the probability of failure for the improved beam and exterior column sub-assembly design would provide interesting information.

170 CHAPTER 5. PROBABILISTIC ASSESSMENT

Figure 5-10 Time-temperature relationships in a fire simulation

1 Int. col. w/o ins. 0.9 Int. col. w/ ins. 0.8 Beam Exterior col. 0.7

0.6

0.5

0.4

0.3 Probability of collapse of Probability

0.2

0.1

0 400 600 800 1000 1200 1400 1600 Max gas temperature (°C) Figure 5-11 Collapse probability of sub-assemblies

The variabilities of the critical collapse probabilities are summarized in Table 5-9. This table shows the gas temperatures associated with 16 %, 50 %, and 84 % chances of collapse of the sub-assemblies. The c.o.v.s in the critical temperatures are simply calculated by normalizing the difference in temperatures at the 16 % and 84 % probability levels with respect to the mean (50 %) temperatures. The c.o.v. of the beam sub-assembly is relatively high (0.074), which indicates greater uncertainty for the failure of the sub-assembly. The steel temperatures corresponding to the mean gas temperatures are 598 °C in the interior

171 CHAPTER 5. PROBABILISTIC ASSESSMENT

column, 634 °C in the beam, 136 °C in the exterior column, and 767 °C in the beam in the exterior column sub-assembly.

Table 5-9 Variability of the collapse probability with respect to gas temperature 16 percentile mean 84 percentile Sub-assembly c.o.v. (°C) (°C) (°C) Interior column without 577 616 655 0.064 insulation (-39) (±0) (+39) Interior column with 1310 1399 1488 0.064 insulation (-89) (±0) (+89) 604 652 700 Beam without insulation 0.074 (-48) (±0) (+48) Exterior column with 763 792 820 insulation on column and 0.036 (-28) (±0) (+28) without on beam

5.3.3 Reliability of AISC-LRFD Fire Equation

The rationale of the simple design method in the AISC (2005) specification was discussed and some alternative member-based design equations are proposed in Chapter 3. The AISC method and proposed design equations are further evaluated in this section by introducing probabilistic assessment concept. Shown in Figure 5-12 is the fundamental concept of load and resistance in the load and resistance factor design (LRFD) in AISC. The load and resistance are functions of random variables and the expected load and resistance are their means. Nominal load and resistance are separately and deterministically defined by the expected (mean) values for design. Multiplying nominal values by factors gives the factored load and resistance, which are used for the AISC strength evaluation according to the following equation:

φRnu≥ Q (5.12)

where, φ , Rn , and Qu are the resistance factor, nominal resistance, and factored load, respectively. If Eq. (5.12) is satisfied, the probability of structural failure is deemed as meeting the specified criteria.

172 CHAPTER 5. PROBABILISTIC ASSESSMENT

Figure 5-12 Load and resistance relationships in AISC-LRFD

Using the statistical information obtained in Table 5-1, the probability of failure of columns, which satisfy Eq. (5.12) under fire conditions (the factored resistance equals to the factored load) is investigated. The column considered is a AISC W14×90 section with Grade 50 steel. The nominal strength is evaluated by the AISC simple approach and proposed approach as described in Chapter 3. The resistance factor φ for columns is 0.9 (AISC, 2005), and the relationships between the expected, nominal, and factored loads are summarized in Table 5-10. According to Eq. (5.12), the design check would be to evaluate the required strength ( Pu = 1.2DL+0.5LL) to the design strength, φPTn (), where PTn () is calculated according to the design checking provisions of Chapter 3.

Table 5-10 Comparison of expected, nominal, and factored load (N/m2) Expected Nominal Factored (1.2DL+0.5LL) DL 4420 4310 (= 90 psf) 5712 LL 718 2870 (= 60 psf) 1435 DL+LL 5138 (1.00) 7180 (1.40) 7147 (1.39)

173 CHAPTER 5. PROBABILISTIC ASSESSMENT

The relationships between the expected and nominal load are based on a past study (Ellingwood, 1983) and the load factors for the dead and live load are defined in AISC (2005). The ratio of the factored load (combined dead and live load) to the expected (mean) load is 1.39. This ratio is assumed to be constant in this study. The probability of failure of the columns under a specified constant elevated temperature of 500 °C is assessed by using the mean-value first-order second-moment (FOSM) method by the following procedure: (1) evaluate the nominal strength of the column by the AISC simple approach, using both the AISC nominal strength equations and the proposed strength equations (Chapter 3), (2) calculate the factored load such that Eq. (5.12) is satisfied, (3) define the expected load as 1.00/1.39 of the factored load, and (4) evaluate the probability that the load is greater than the resistance by FOSM. The purpose of this study is twofold: (1) to confirm the accuracy of the design equations (AISC and proposed) and (2) to investigate the effect of uncertainties. Dead and live loads are the random variables for the load distribution, and the yield strength and elastic modulus of steel at elevated temperatures

( FTy () and ET()) are considered as the random variables for the resistance distribution. The statistical information (mean and c.o.v.) of these random variables is shown in Table 5-1. FEM simulations are performed for the evaluation of the gradient of the resistance with respect to the random variables. Details of the modeling are described in Chapter 3. The probability of failure of W14×90 columns with varied member length is shown in Figure 5-13. Two sets of data are shown in the figure. One is for cases where the nominal strength is calculated by the AISC approach, and the other is for cases where the proposed strength equations are used. Figure 5-13(a) shows the beta function (the definition is shown in Appendix B.4) and Figure 5-13(b) shows the probability of failure (these figures have a complementary relationship). As described in Chapter 3, the nominal strength equations of AISC are known to be unconservative, especially for columns with intermediate length (slenderness ratio of 40-100). The results from this study are consistent with those from Chapter 3, where referring to Figure 5-13(b), the probability of failure with the slenderness ratio of 60 is about 37 %. In terms of accuracy, deterministic studied are performed for different temperatures, section sizes, and steel strength in Chapter 3 and the accuracy of the proposed equations are confirmed. In order to investigate the effect of uncertainty, this case study represents other cases with different section sizes and steel strength, because the statistical properties of the

174 CHAPTER 5. PROBABILISTIC ASSESSMENT

uncertainties are not functions of temperatures and constantly defined regardless section sizes and steel strength.

3 0.4 AISC 2.5 Proposed 0.3 2

β 1.5 0.2

1 P (failure) 0.1 0.5 AISC Proposed 0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r y y (a) β (b) Probability of failure Figure 5-13 Probability of failure of W14×90 column at 500 °C with varied length

The probability of failure shown in Figure 5-13 is calculated assuming that the resistance factor φ = 0.9. Alternatively, required φ values for the target probabilities of failure at 500

°C can be evaluated. Assuming a target probability of 0.47 % (β = 2.6) as the AISC commentary indicates for members at ambient temperature, the required φ values are shown in Figure 5-14. The required φ factors would need to be about 0.5 to 0.7 for the AISC equations and 0.7 to 0.9 using the proposed member strength provisions.

0.8

φ 0.6

0.4

0.2 AISC Proposed 0 0 50 100 150 200 L / r y Figure 5-14 φ factors for 0.47 % (β = 2.6) probability of failure of W14×90 column at 500 °C with varied length

175 CHAPTER 5. PROBABILISTIC ASSESSMENT

A similar study is performed for the column with a fixed length of 4 m (this gives a slenderness ratio of 42.3 about the weak axis) with varied elevated temperatures up to 800 °C. The probability of failure is shown in Figure 5-15. A large probability of failure (about 30 %) is observed for the column design based on the AISC approach at elevated temperatures greater than 400 °C, whereas the probability of failure based on the proposed approach is less than 3 % and varies less with respect to temperature.

3.5 0.4 AISC AISC 3 Proposed Proposed 0.3 2.5

β 0.2 1.5 P (failure) 1 0.1 0.5

0 0 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (a) β (b) Probability of failure Figure 5-15 Probability of failure of W14×90 (L = 4 m) column with varied temperatures

Required φ values for the 0.47 % (β = 2.6) target probability of failure are also investigated with the fixed length (4 m) column at elevated temperatures. The values for the column based on the AISC approach are about 0.5 at elevated temperatures greater than 400 °C, while those based on the proposed approach are about 0.7.

176 CHAPTER 5. PROBABILISTIC ASSESSMENT

0.8

φ 0.6

0.4

0.2 AISC Proposed 0 0 200 400 600 800 Temperature (°C) Figure 5-16 φ factors for 0.47 % (β = 2.6) probability of failure of W14×90 (L = 4 m) column with varied temperatures

Considering that the φ value for the column design at ambient temperature is 0.9 and

the value of the β function is 2.6 (0.47 % for the probability of failure) according to the AISC commentary, a higher probability of failure is expected for the columns designed based on the AISC equations under fire conditions. This is mainly because the AISC equations are unconservative as is shown in Chapter 3. In addition to the unconservative nature of the equations, material properties at elevated temperatures are more uncertain than those at ambient temperature (e.g., c.o.v. of the yield strength is defined as 0.22 at elevated temperatures, while that at ambient temperature is 0.1 as shown in Table 5-1). The proposed equations provide β ≅ 2 to 2.5 (with φ = 0.9). Conversely, the required φ value for β = 2.6 is found to be 0.7-0.8. These values are not too far off from the target of β = 2.6 with φ = 0.9. This slight unconservativeness of the proposed equations may be acceptable, because evaluation of the overall risks of structures need to consider the probability of the occurrence of the temperature, T , in the steel members and such probability is quite small. The probability of failure is obtained by the total probability theorem as shown in the following equation:

P()(|)() failure= ∫ P failure T P T dT (5.13) T

177 CHAPTER 5. PROBABILISTIC ASSESSMENT

where PfailureT(|) is the probability of failure at a given elevated temperature and PT() is the probability of the occurrence of T in the steel members. PT() is also evaluated considering the probability of the temperature of the steel members reaching T under flashover fires, P(| T flashover fire), and the probability of flashover fires, P() flashover fire , during a certain time, i.e., mean annual probability of flashover fires. Similar to Eq. (5.13), PT() can be obtained by using the total probability theorem, integrating P(| T flashover fire) and P() flashover fire over time t .

P() T= ∫ P ( T | flashover fire)( P flashover fire) dt (5.14) t

This brief overview of the overall risk assessment of steel structures against fires provides some ideas about the issues of structural risk assessment for fires. This research focuses on the structural aspects. Future coordination with other aspects of risk assessment against fires is still needed.

5.3.4 Conclusions

Conclusions obtained in this chapter are summarized in the following points: 1. Statistical properties (mean and coefficient of variation) of uncertain structural variables are defined by reviewing past test data and studies, as well as by using engineering judgment for those factors whose statistical information is unknown. The uncertain variables are dead and live loads, yield strength and elastic modulus of steel at elevated temperatures, strength and deformation capacity of bolted connections, and the longitudinal constraint spring properties. The coefficient of variation of the yield strength of the steel and the shear strength of bolts are defined as 0.22 and 0.29, respectively, based on existing test data. Statistical properties of the loads are assumed to be the same as those at ambient temperature. The properties of the elastic modulus of steel and other variables are approximately defined using engineering judgment. 2. The variation of the time-temperature relationships is investigated by applying various values to fuel load, firefighting factor, and opening factor. These are used in the parametric fire curve defined in Eurocode 1 (2002). The maximum temperature of insulated steel members can vary from 340 °C to 1000 °C and those of unprotected

178 CHAPTER 5. PROBABILISTIC ASSESSMENT

members can be 780-1300 °C. These highly varied maximum steel temperatures are the consequence of the uncertain fire behavior that controls structural stability. Further research and development is needed to integrate the fire and structural simulations in a unified probabilistic assessment. 3. The sensitivity of the critical temperatures of three sub-assembly structures (interior column, beam, and exterior column) is evaluated for the uncertain variables. Yield strength of steel is the most influential factor for the interior column, because the slenderness ratio of the column is relatively small and the limit-state mechanism is inelastic buckling, which is controlled by material strength. For the beam sub-assembly, the deformation capacity of the bolted connections is the most influential factor. This implies that axial force induced by the thermal expansion overwhelms the strength of the bolted connections, and strengthening connections is less effective at preventing the connection failure than enhancing the deformation capacity of the connection. For the exterior beam-column sub-assembly, the strength of the bolted connection is the most influential parameter. This is because the connection fails due to the tensile axial force induced in the beam by catenary action. 4. The magnitude of the gas temperature during the compartment fire is scaled and the probability of collapse with respect to the given gas temperature is assessed for the three sub-assemblies, using the mean-value first-order second-moment (FOSM) method. The interior column is studied with and without fire insulation. The mean gas temperature for the limit-states significantly increases from 616 °C to 1399 °C with the fire insulation. The mean gas temperature for the un-insulated beam sub-assembly is 604 °C due to the connection failure. Furthermore, the variability of the failure point of the beam sub-assembly is greater than that of the interior column sub-assembly (i.e., greater uncertainty in the beam sub-assembly). Conversely, relatively smaller variability is observed in the exterior column sub-assembly (with insulated column and un-insulated beam), where the mean gas temperature for the limit-state is 763 °C. Of the three sub- assemblies, the interior column sub-assembly without insulation is the most critical. This assessment shows that the failure of the beam sub-assembly can be the dominating mechanism (probability of interior column buckling at 620 °C is about 50 %, while that of the connection failure of the beam at the temperature is about 25 %); however failure of the exterior column sub-assembly would almost never control.

179 CHAPTER 5. PROBABILISTIC ASSESSMENT

5. The probability of failure of the columns under a given elevated temperature is evaluated for columns designed using the AISC-LRFD simple method (2005 specification) and the proposed equations in Chapter 3. The uncertain variables are dead and live load, and yield strength and elastic modulus of steel at elevated temperatures. The evaluation is performed using the mean-value FOSM method for a W14×90 column (Grade 50) with varied member length or temperature. The maximum probability of failure at elevated temperature is about 40 % for the columns based on the AISC design strength equations, while it is 3-4 % for the columns based on the proposed design strength equations. Consistent results (high unconservativeness for columns with intermediate length) are observed for the AISC method as discussed in Chapter 3. Required values for the resistance factor to satisfy 0.47 % (β = 2.6) probability of failure given elevated temperatures are evaluated. The values are 0.5-0.7 and 0.7-0.9 for the design based on the AISC and proposed equations, respectively. This slight unconservativeness of the proposed equations may be acceptable, because evaluation of the overall risks of structures need to consider the probability of the occurrence of the temperature in the steel members and such probability is quite small.

180

CHAPTER 6 CONCLUSIONS

6 CONCLUSIONS

6.1 GENERAL

The main objective of this research is to investigate the collapse performance of steel-framed buildings under fully developed (flashover) fires. This research approach employs detailed finite element simulations to assess the strength of individual members (beams and columns) and indeterminate structural sub-assemblies (beams, columns, connections and floor diaphragms). One specific focus of the investigation is to assess the accuracy of beam and column strength design equations of the Eurocode 3 (2003) and the American Institute of Steel Construction (AISC) Specification (2005). An outcome of the structural member study is a proposal for alternate equations to improve the accuracy of the AISC Specification provisions. To examine the collapse limit-state response of building systems, structural sub- assemblies are devised to examine indeterminate effects of gravity-framing systems, including forces induced by restraint to thermal expansion and nonlinear force redistribution due to yielding and large deformations. The sub-assemblies are based on gravity framing for an archetypical mid-rise office building, with structural steel framing details that are representative of design and construction practice in the United States. These studies identify the governing factors in structural system collapse, including the effectiveness of fire insulation or alternate techniques (e.g., strengthening of connections) to increase the collapse resistance. The significance of uncertainties in gravity loading and structural parameters on collapse due to fire are investigated through a probabilistic assessment that includes statistical characterization of key parameters and the mean-value first-order second-moment (FOSM) analyses to integrate their effects. At the individual component level, the reliability of the current AISC Specification and proposed alternative design equations are evaluated.

181 CHAPTER 6. CONCLUSIONS

Reliability studies of the sub-assembly models are conducted to assess conditional probabilities of collapse in gravity framing systems for a specified elevated temperature. In this chapter, the notable findings from this research are highlighted and summarized.

6.2 SUMMARY

6.2.1 Steel Properties at Elevated Temperatures

Existing test data of structural steel properties (stress-strain curves) at elevated temperatures, reported by five research groups (Harmathy and Stanzak, 1970; Skinner, 1972; DeFalco, 1974; Fujimoto et al.,1980, 1981; Kirby and Preston, 1988), are reviewed and compared with design equations in four specifications: Eurocode (European countries), AS4100 (Australia), AISC (the US) and AIJ (Japan) in Chapter 2. The measured stress-strain curves are normalized with respect to the measured 0.2 % off-set strength at ambient temperature, and reduction ratios of the strength at elevated temperatures are evaluated. The test data clearly show that the stress-strain curves at elevated temperatures experience early non-linearity (deterioration of the tangent stiffness) below the yield point, which is distinctly different from the characteristic elastic-plastic behavior of steel at ambient temperatures. This behavior is in contrast to the elevated temperature provisions of the AS4100 and AISC standards, where the specified degradation parameters imply use of the same characteristic elastic-plastic stress- strain response at both ambient and elevated temperatures. Consequently, the implied elevated temperature stress-strain curves do not fit the experimental data well. The AIJ provisions define non-linear curves incorporating deterioration of the tangent stiffness; however, the curves are conservatively defined by including large safety margins in the material properties. The design equations by Eurocode are the most representative of the experimental data. Based on comparisons of measured data and proposed stress-strain models, the stress- strain equations of steel response at elevated temperatures defined by Eurocode are adopted in this research for analytical simulations. Statistical data to describe the variability in steel strength deterioration ratios at elevated temperatures are also obtained and used in and the probabilistic studies, reported in Chapter 5.

182 CHAPTER 6. CONCLUSIONS

6.2.2 Past Fire Disasters

Past fire disasters on steel buildings are reviewed in Chapter 2 to investigate the observed collapse behavior of actual steel buildings subjected to fires. Four major fire events: Broadgate Phase 8, One Meridian Plaza, the World Trade Center (WTC) building 7 and the Windsor Building, as well as the Cardington eight-story full-scale fire test are reviewed in detail. The most important finding from this review is that no steel building has totally collapsed from fire except the WTC 7 building, which had some unique circumstances (a non-redundant transfer truss system and possibly damaged caused by falling debris from the WTC 1 tower). These observations demonstrate the potential strength of steel buildings designed as per current practice against fires. The superior fire resisting ability of steel beams was clearly observed in the Broadgate Phase 8, One Meridian Plaza and the Cardington Fire Test, where insulation was not present on the beams. Some of the beams in these structures experienced temperatures greater than 1000 °C and deformed considerably; however, they did not collapse. In contrast, the vulnerability of steel columns to fire was clearly observed. The Windsor Building partially collapsed in the upper stories where fire insulation on the columns was missing due to renovation. Also, local buckling on columns with significant distortion, which greatly deteriorated the vertical load carrying capacity of the columns, was observed in the Cardington Fire Test. These observations are useful for understanding the characteristic behavior of steel buildings under fires, although further careful investigations are necessary to generalize the findings for structural fire design.

6.2.3 Member-based Strength Study

The individual member strengths of I-shaped steel columns and beams are studied using detailed three-dimensional finite shell element models, where the strengths under specified elevated temperatures are parametrically investigated. Residual stress and geometric imperfections are taken into account in the simulations; and the analytical simulations are validated by comparison to existing test data for columns. In column models subjected to axial force, flexural buckling is found to be the critical mechanism. Beam models capable of simulating non-uniform torsion are subjected to uniformly-distributed moment about the strong-axis, and lateral torsional buckling as well as yielding are the critical mechanisms. Accuracy of the member-based design criteria defined in Eurocode is confirmed through comparison with the simulation data, while the AISC criteria is found to be highly

183 CHAPTER 6. CONCLUSIONS unconservative. Alternative design equations for AISC are proposed for I-shaped steel columns, unbraced beams, and beam-columns at elevated temperatures.

6.2.4 Benchmark Building Study

The performance of gravity framing in steel-framed office buildings under localized fires is assessed through study of an archetypical building design in Chapter 4. The plan of the ten story building is 32.0 m by 25.6 m and a localized fire is assumed to break out on the second floor. Four possible failure mechanisms (i.e., interior column buckling, exterior column buckling, beam yielding and interactive failure, where beam deformation in the fire compartment causes multi-story exterior column buckling) are simulated with sub-assembly analysis models. Beam and columns in the sub-assembly models are composed of shell finite elements; and nonlinear springs are used to model connections and boundary conditions imposed by the surrounding structure. Non-linear spring properties are based on supplemental models that take into account the effects of the concrete floor slab and the behavior of the bolted connections. The critical failure mechanisms and temperatures are investigated, and design recommendations to improve the collapse capacity are suggested.

6.2.5 Probabilistic Studies

Assessment of structural collapse due to fire involves large uncertainties in fire development, heat transfer, and structural behavior. Fire development is highly uncertain and depends on various factors such as fuel load, geometry of the fire compartments, and firefighting activities. Additionally, heat transfer into the structural steel involves uncertainties in compartment gas temperatures, radiation effects, and effectiveness of thermal insulation. While the importance of these two factors (fire development and heat transfer) is recognized, detailed study of them is outside the scope of this study. Rather, the probabilistic aspect of this research focuses on uncertainties in the structural gravity loading and response aspects under elevated temperatures. Statistical information (i.e., mean and coefficient of variation) of characteristic structural factors (e.g., steel properties at elevated temperatures) are investigated. Using the statistical information, the sensitivities of the collapse limit-state of both individual steel members and structural sub-assemblies under the localized fire with respect to the uncertain factors are evaluated. Specifically, the collapse probabilities of the structural sub-assemblies are evaluated using the mean-value FOSM approach for given gas

184 CHAPTER 6. CONCLUSIONS temperatures in the fire compartment. Reliability aspects of the AISC design equations for columns are included in the study.

6.3 MAJOR FINDINGS AND CONCLUSIONS

The main findings and general conclusions obtained from this research are summarized in the following sections.

6.3.1 AISC Member-based Design Criteria

The AISC Specification’s (2005) new provisions for assessment of the design strength of members at elevated temperatures are based on the standard design equations at ambient temperatures, where the input material properties (elastic stiffness and yield strength) are reduced using the material degradation factors defined in the Eurocode (2003). Therefore, it can be interpreted that the AISC method assumes use of the same elastic-plastic stress-strain relationships for steel at elevated and ambient temperatures. This is in contrast to the actual nonlinear stress-strain relationships, as shown in Chapter 2. For stability sensitive members, errors arise because of the nonlinear deterioration of tangent stiffness below the nominal yield point. Finite element simulations are performed for columns, unbraced beams and beam-columns under parametrically changed elevated temperatures and member length. The simulation results show that the AISC provisions are unconservative by up to 60 % for the columns and 80-100 % for the beams and beam-columns. This unconservative behavior is typically observed for members with intermediate length (slenderness ratio of 60-80), where inelastic buckling is the critical failure mechanism. Member-based critical strengths investigated by applying the tangent modulus theory (Appendix A), with the reduced stiffness, agree well with the simulation data. The study shows explicit relationships between material deterioration of the tangent stiffness and member strengths at elevated temperatures.

6.3.2 Effect of Residual Stress and Local Buckling

Three-dimensional finite shell element models, used in the simulations, are well-suited for investigation of residual stress and local buckling. The maximum residual stress is assumed as 20 % of the yield strength for members with Grade 50 steel, such that it deteriorates in proportion to the degraded yield strength at elevated temperatures. The maximum difference of the critical strength with and without consideration of the residual stress is less than 15 %,

185 CHAPTER 6. CONCLUSIONS which is found for members with intermediate length (slenderness ratio of 100) at around 500 °C. Overall, the order of magnitude of the effect of residual stresses is comparable at ambient and elevated temperatures. Local buckling is not critical for compact or near-compact sections at ambient temperature. It is shown in this research that local buckling is similarly not critical for compact and near-compact members at elevated temperatures. Local buckling is observed in the post-buckling stage, but it does not control the critical strength. The only exception to this was observed for a very short segment of a non-compact W14×22 column, where the strength was limited by local buckling at a stress approximately equal to 60 % of the yield strength at elevated temperatures. This extreme case study is purposely carried out to verify that local buckling is not a major concern for compact sections at elevated temperatures relative to ambient temperature.

6.3.3 Proposed Design Criteria for AISC

Motivated by the unconservativeness of the AISC member design equations, alternative design equations for steel columns and beams at elevated temperatures are proposed. These equations are functions of temperature and member length, and follow a similar format to the existing AISC design equations at ambient temperature. The accuracy of these equations is validated with the analytical simulations under various specified elevated temperatures and member lengths. The strengths calculated by the proposed equations are within 20-30 % of the strengths calculated by finite element simulations. This agreement is comparable to that observed between Eurocode equations and simulations (10-20 %). More than eight hundred simulations are performed in total to verify the proposed equations, accompanied by studies on members with different steel strengths and section sizes.

6.3.4 Steel-framed Building under Localized Fire

In simulations of the steel-framed benchmark building under localized fires, the members are initially assumed to be bare (un-insulated). In this case, the governing failure mechanism is interior column buckling at a critical column temperature of 578 °C (gas temperature of 920 °C). Assuming insulation on the columns (with no insulation on the beams) increases the critical gas temperature to 965 °C. The second critical failure mode occurs for the interior (secondary) beams, which experience failure of the bolted shear-tab connection at a critical

186 CHAPTER 6. CONCLUSIONS beam temperature of 634 °C in the lower flange. This failure is caused by thermal expansion which induces large axial forces in the beams and the bolted connections. Typical shear-tab connections are designed for beam shear force due to gravity and failure of this type of connections is critical for beams under fires. Design improvements of these bolted shear-tab connections for interior beams are proposed (discussed later at 6.4.1 Design Recommendations), which can increase the critical temperature for interior beams. The third critical failure mode occurs through an interactive mechanism between the exterior column and the beam which provided out of plane support to the column. For this limit-state, the temperatures at the exterior column and lower flange of the beam are 145 °C and 812 °C, respectively (low column temperature in the column because of the insulation). Using fire insulation is the only practical solution to prevent column buckling, because using a larger steel section as an alternative design approach is economically infeasible; however, bolted connection failure for beams can be prevented without fire insulation. Using slotted bolt holes or placing steel reinforcement in the composite concrete slab can significantly improve the performance of the connections under fires by releasing additional shear force or bracing longitudinal strength of the connections. Where connections are appropriately reinforced, steel beams have potentially high load carrying capacity with catenary action that can develop under large sagging deformations. Assuming that this large deformation is allowed, fire insulation for beams can be reduced or even eliminated by improving the connections.

6.3.5 Longitudinal Constraint of Interior Column

Building columns that expand during a fire may be subjected to high axial compression loads, depending on the constraint stiffness provided by framing levels above the heated column. Typical bolted shear-tab connections, which are used at connections between columns and gravity floor beams, are often considered as having negligible rotational stiffness. This assumption significantly simplifies design calculations and is usually conservative; however, this assumption is not necessarily conservative when the rotational stiffness provides vertical constraint that can induce axial compression forces in a heated column. To investigate this effect, the rotational stiffness of typical shear-tab connections is evaluated based on tests by Liu and Astaneh-Asl (2004). Results calculated for three building heights (5, 10 and 20 stories) with a fire at the second floor, indicate that the axial forces

187 CHAPTER 6. CONCLUSIONS induced by vertical restraint have a negligible effect (less than 1 % change in critical column temperature) on the column axial strength. The reason for this is that the column loses both stiffness and yield strength as it elongates with increasing temperature. Therefore, while constraint to the thermal elongation alone may induce large axial compressions, when combined with the associated material degradation the induced forces are negligible.

6.3.6 Longitudinal Constraint of Beam

Floor framing that surrounds the beams subjected to a compartment fire can provide significant constraint to longitudinal deformation of the heated floor beams. This constraint has beneficial and negative effects. The negative effect is that the restraint to beam elongation during the heating phase of response can lead to premature failure, such as bolt shear (as described previously). Provided steps are taken to avoid premature failure modes, the constraint can enable catenary action that can greatly increase the gravity load resistance at elevated temperatures. Studies of interior beams showed that where constraint is provided by the surrounding framing, the critical temperature increased from 634 °C to 812 °C. However, typical beam connections are not strong enough to support the catenary action and connection design enhancements are needed to achieve this improved performance. Evaluation of the longitudinal constraint of beams is difficult, however, and involves many complex issues such as non-linear behavior of the concrete slab, the stud connections between steel beams and concrete slab, steel connections, and 3D effects from the stiffness of vertical components including the lateral resisting system. Relatively simple analyses are carried out for this evaluation using elastic FEM models for the floor structure. Although evaluation of the longitudinal constraint stiffness of beams is approximated, a sensitivity study shows that accuracy of the stiffness is not critical to overall beam behavior under fire conditions.

6.3.7 Properties of Bolted Connections

Since failure of bolted connections is critical in some failure mechanisms, realistic evaluation and modeling of the bolted connections are essential for this research. The strength reduction ratio of bolts defined in ECCS (2001) is compared with existing test data by Kirby (1995) and Yu (2006). The ECCS reduction ratio agrees well with the test data within 20-30 % up to 700 °C. Bilinear force-displacement relationships including post capping softening is assumed for longitudinal springs of shear-tab type bolted connections for beam sub-assembly

188 CHAPTER 6. CONCLUSIONS analysis models. The simulation results show that compressive axial force of the beams due to thermal elongation causes the bolted connection failure. In order to prevent this failure, slotted bolt holes are suggested as a design improvement and its effectiveness is validated in simulations.

6.3.8 Evaluation of Structural Uncertainties

The characteristic parameters needed for the structural fire simulations are reviewed for their statistical properties. These factors are the dead load, live load, yield strength and elastic modulus of steel at elevated temperatures, strength (force-displacement relationship) of the vertical constraint spring for the interior column, longitudinal constraint stiffness for the beams, and shear strength and deformation capacity of bolted connections. In particular, the statistical properties of the steel strength at elevated temperatures are carefully investigated by reviewing past experimental data, and the coefficient of variation is determined to be 0.22. Investigations of the effect of these uncertainties indicate that the strength limit-states are most sensitive to the degraded yield strength of steel at elevated temperatures. Also, the limit-state of the beams that fail due to thermal elongation is sensitive to the deformation capacity of bolted connections.

6.3.9 Probabilistic Studies

Using the statistical information on the variability in structural parameters, the collapse probabilities of the sub-assemblies in the benchmark building are evaluated as a function of gas temperatures in the localized compartment fire. The evaluation is performed by the mean-value (FOSM) approach. Assuming deterministic relationships between the steel and gas temperatures in the fire compartment (which are obtained through the incremental time step simulation), the probability of failure of the three sub-assembly models is evaluated with respect to the gas temperature. The magnitude of the gas temperature during the compartment fire is scaled and the probability of collapse with respect to the given gas temperature is assessed for the three sub- assemblies. No fire insulation is assumed for the steel members except for the exterior column. Among the three sub-assemblies, the largest variability is observed for the beam sub-assembly. The critical collapse mechanism is buckling of the interior column. This assessment shows that the connection failure of the beam sub-assembly is also a significant mechanism, since the probability of interior column buckling at 620 °C is about 50 %, while

189 CHAPTER 6. CONCLUSIONS

that of the connection failure of the beam at the temperature is about 25 %. Failure of the exterior column sub-assembly would almost never control over the interior column and beam mechanisms. The probability of failure of columns given elevated temperatures is evaluated for those designed based on the AISC-LRFD simple method (2005 specification) and the proposed equations. The uncertain variables are dead and live load, and yield strength and elastic modulus of steel at elevated temperatures. The evaluation is performed by using the mean- value FOSM method for AISC W14×90 I-shaped columns (Grade 50) with varying member length or temperatures. The probability of column failure ranges from 4 % to 38 % (β = 0.3- 1.8) for designs based on the AISC strength provisions (with φ = 0.9). These probabilities reduce to 0.5 % to 3 % (β = 1.9-2.6) based on the proposed equations (with φ = 0.9).

6.4 DESIGN AND ANALYTICAL MODELING RECOMMENDATIONS

6.4.1 Design Recommendations

In the review of the past fire disasters and the Cardington fire test in Chapter 2, it is reported that some of the steel buildings did not collapse under fires without insulations. This fact shows the potential strength of steel buildings against fires and questions the rationale of current structural fire engineering practice. Three interesting points are found through the simulations of the benchmark building in this research: (1) buckling of un-insulated gravity columns are inevitable under flashover fires, making fire insulation on columns necessary to prevent collapse, (2) beams can maintain their gravity load carrying ability at elevated temperatures associated with large deformation, (3) failure of typical shear-tab bolted connections is critical for the beams. One design recommendation derived from these findings is to brace or protect connections for axial strength of beams at high temperatures. Consequently, fire insulation on beams can be reduced or eliminated. The benefit of the reduction or elimination of the beam insulation can have both economical and environmental advantages. In terms of economics, conventional fire insulation has been reported to contribute up to 30 % of bare steel costs for steel buildings (Lawson, 2001). Therefore, the resulting savings by using less insulation could be applied to improve other fire protection systems (e.g., egress). From an environmental (indoor air quality standpoint), spray type fire insulation, which is commonly employed for beam insulation, is considered to be

190 CHAPTER 6. CONCLUSIONS environmentally unfriendly. Therefore, removal or minimization of beam insulation (which is more prone to dislodging and getting into the air) is desirable. By bracing the connections against fires, steel structures will be more redundant and thus preventing progressive collapse. On the other hand, large deformation of the beams under fires may not be acceptable in some cases such as possible break of the fire partitions, prevention of smooth evacuation, and difficulty in post-fire renovation. The recommendation must be further studied with discussion on quantitative evaluation on improvement of structural performance as well as risk, environmental, and economical assessment.

6.4.2 Analytical Modeling Recommendations

The advantage to using shell element models for I-shaped steel sections is its capability to simulate local buckling. It is observed in this research that local buckling is significant to the post-buckling strength and limit-state strength of non-compact sections. In order to study these issues, shell element models are used in this research; however, using fiber element models would lead to the same results if the research focuses on only the member-based limit-state strength study for compact sections. Post-buckling strength of columns is an important factor for study of indeterminate systems related to progressive collapse under fires and shell element models are recommended for this purpose. Beam behavior is controlled less by local buckling and fiber models can work well for beams. Using hybrid shell and fiber models for columns and beams is a possible option for frame studies to reduce computational time. Although the proposed sub-assembly models are concise and efficient to simulate accurately the characteristic behavior of building frames under fires, even simpler models are preferred for practical use. For this purpose, two modeling improvements that are worth investigation are modeling of bolted connections and the concrete floor slab. Development of lumped spring models for bolted connections, which can simulate behavior under combined axial, shear and moment at elevated temperatures, could drastically simplify simulations. This effort should be carried out through calibration with test data. Modeling of the concrete slab is a difficult aspect in structural fire simulations. However, approximate modeling of the concrete slab is acceptable for practical purpose, since structural behavior is shown in this research not to be sensitive to the stiffness. Using membrane elements instead of shell elements for the concrete slab is another possible alternative. In both modeling methods, analytical convergence would be a critical problem due to physical softening

191 CHAPTER 6. CONCLUSIONS

behavior of bolted connections and concrete slabs. An efficient analytical algorithm needs to be developed for this goal.

6.5 FUTURE WORK

6.5.1 Member-based Strength Evaluation

Although this research on the critical strength of individual steel members provides useful information for improvement of current structural fire design specification, there are many areas that need to be developed in future research. Current research focuses on bi-symmetric I-shaped members with compact (or near-compact) sections assuming a uniform temperature distribution through the member cross section. In order to provide thoroughly a useful simple individual member design approach under fire conditions, studies on non-symmetric, non-compact sections are needed. A non-uniform temperature distribution along the member length generally increases the critical strength; however, a non-uniform temperature distribution in the section can be unconservative for slender members where induced deformation deteriorates the member stability due to P-δ effects. The critical strength can deteriorate by 50 % with a linear temperature distribution (ambient temperature at one edge and elevated temperature at the other) and a slenderness ratio of 140 or greater. Also, the assumption of a uniformly-distributed bending moment for beams and beam-columns is useful for a prototype study; however, it must be extended to non-uniform moment for generalization. Reliability analysis of uncertainties, which is carried out in the performance evaluation of frames, is in need of further study for individual member strength evaluation. It should address a broader range of design and response parameters, including variations in cross- section types, thermal and mechanical loading intensities and distributions, material properties at ambient and elevated temperatures, and temperature induced distortions.

6.5.2 Performance Evaluation of Steel Buildings under Fires

A limit-state study of a benchmark steel building under a given localized fire scenario is performed by finite element analyses introducing sub-assembly simulation models. Although characteristic behaviors for an indeterminate structure such as post-buckling stability of columns and catenary action of beams is evaluated, more case studies with different types of steel buildings and various fire scenarios are needed for generalizing the findings. More

192 CHAPTER 6. CONCLUSIONS simulations of alternative designs are also needed with different conditions in order to confirm the effects and allow for practical application.

193 CHAPTER 6. CONCLUSIONS

194

APPENDIX A SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

A SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

A.1 TANGENT MODULUS THEORY

A.1.1 Flexural Buckling

Inelastic critical strength of ideally straight columns can be theoretically derived by tangent modulus theory (TMT), which has been primarily studied in 1960’s, and these efforts are described in previous literature (e.g., Galambos, 1998). According to the theory, the critical strength is simply given by substituting tangent stiffness into elastic stiffness in the elastic buckling equation.

π 2 EA()ε P = t0 (A.1) cr0, tan (/)Lr2

where Pcr0, tan is the critical strength by tangent modulus theory. ε is a strain and uniform in

sections. Et0 ()ε is tangent stiffness under the stain ε . A, r and L are the cross-sectional area, radius of gyration, and length, respectively. Although the theory provides useful insight for column strength against inelastic buckling, derived critical strengths do not necessarily agree well with test results due to lack of consideration of imperfections. Geometric imperfections and deformed bent form deteriorate the critical strength because of the P-δ effect. Also, residual stress causes non- uniform strain and tangent stiffness in sections. Using this theory and accepting some degree of inaccuracy, the tangent modulus theory is applied to analyze column strength at elevated temperatures. Ideally straight steel columns without residual stress are first considered. The critical strength under elevated temperatures against flexural buckling is obtained by the tangent modulus theory as Eq. (A.2).

195 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

πε2 ETA(, ) PT()= t (A.2) cr, tan (/)Lr2

where PTcr, tan () is the critical strength by the tangent modulus theory and ETt (,ε ) is tangent stiffness under stain ε at temperature of T . Considering effect of residual stress, the critical strength is obtained as Eq. (A.3) and (A.4).

πε2 ETA(, ) PT()= tave, (A.3) cr, tan ()Lr2

Exyt (, ,,ε TdA ) ET(,ε )= ∫ (A.4) tave, A

Since distribution of residual stress is not uniform in sections, tangent stiffness in a

section is not constant either. Exyt (, ,,ε T ) is a function of the location in a section, which is given as coordinate of (,x y ). ε is redefined as a strain specifically generated by the applied axial load. Therefore, the strain is uniform in a section and independent of the

residual stress distribution. Etave, (,ε T ) is average tangent stiffness. Stress in a section can be calculated as summation of generated stress and residual stress.

σ (,x yT ,,εσεσ )= ar (, T )+ (, xyT ,, ε ) (A.5)

where σ a (,ε T )is the generated stress by applied load and σ r (,x yT ,,ε ) is the residual stress. Equilibrium to the applied force PT() in a section is shown in Eq. (A.6).

PT()= ∫σε (,,,) x y T dA (A.6)

It is assumed that the initial residual stress (i.e., σ r (,x yT ,0,): residual stress without applied axial force) under elevated temperature is proportional to the yield strength with respect to temperatures.

196 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

σ ry(,xy ,0,) T= K () Tσ r (, xy ,0,20°C) (A.7)

where KTy () is reduction factor of yield strength in EC3. The initial residual stress satisfies equilibrium in a section as shown in Eq. (A.8).

σ (,xy ,0,) TdA= 0 (A.8) ∫ r

Eq. (A.8) is generally not true under applied axial force, because the magnitude of residual stress is dependent of the generated strain by the axial force (Figure A-1).

Figure A-1 Strain level and residual stress

Relationship between the slenderness ratio and the critical strength by tangent modulus

theory is obtained by substituting PT() in Eq. (A.6) into PTcr, tan () in Eq. (A.3), where the slenderness ratio is defined as Lr, a ratio of the length with respect to radius of gyration about buckling axis. Since ETtave, () and PT() are uniquely defined with respect to ε , relationship between slenderness ratio and the critical strength can be iteratively calculated with a variable of ε . Adopting the stress-strain curves of steel at elevated temperature from EC3 (2003), the calculated critical strength under elevated temperatures with and without consideration of residual stress is shown in Figure A-2. The critical strength in the figure is

normalized with plastic axial strength PTyy()= Aσ () T, where σ y ()T is yield strength of steel at elevated temperatures.

197 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

Observing these relationships without residual stress, the curves consist of two regions: the inelastic buckling controlled region and the elastic buckling controlled region. The critical strength is lower than the elastic buckling strength in the inelastic region, while it agrees in the elastic region. Stress at the boundary of these regions equals the proportional limit and the transition is made with a discontinuity of the gradient. This discontinuity is not clearly observed in the curves with residual stress. This is because the average tangent stiffness defined in Eq. (A.4) gradually transitions from elastic to inelastic with residual stress. Figure A-3 shows stress-strain curves obtained from the average tangent stiffness in the section at ambient temperature and 500 °C. The elastic limit of stress is smoothed by effect of the residual stress.

W14x90 (Gr.50), T=100°C W14x90 (Gr.50), T=200°C

P without σ P without σ 1 cr,tan r 1 cr,tan r P with σ P with σ cr,tan r cr,tan r 0.8 min ( P , P ) 0.8 min ( P , P ) cr,e y cr,e y (T) (T) y y 0.6 0.6 (T) / P (T) (T) / P (T) cr cr 0.4 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r (a) Ambient temperature, 100 °C (b) 200 °C

W14x90 (Gr.50), T=500°C W14x90 (Gr.50), T=700°C

P without σ P without σ 1 cr,tan r 1 cr,tan r P with σ P with σ cr,tan r cr,tan r 0.8 min ( P , P ) 0.8 min ( P , P ) cr,e y cr,e y (T) (T) y y 0.6 0.6 (T) / P (T) (T) / P (T) cr cr 0.4 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r (c) 500 °C (d) 700 °C Figure A-2 Critical strength of columns by tangent modulus theory

198 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

The difference of the critical strength with and without residual stress is most significant around the discontinuous point. The location of the point increases in slenderness ratio with higher temperature (e.g., Lr = 75 is the location of the discontinuous point at ambient temperature, while Lr = 97 at 500 °C). Considering that the range of slenderness ratios commonly used in practice is between 30 and 60 in building structures, it can be said that the effect of residual stress for columns is less influential under elevated temperatures. It is also noteworthy that the difference of the critical strength with and without residual stress

decreases under elevated temperatures (e.g., the difference is 14 % of FTy () at ambient temperature, while it is 8 % at 500 °C).

Figure A-3 Stress-strain curves with the average tangent stiffness in section

Numerical analysis is carried out for a W14×90, which is a standard W-series section issued by the American Institute of Steel Construction (AISC, 2005). This I-shaped section is used as a prototype in this Appendix in both flexural and lateral torsional buckling studies. The section is commonly used in practice for columns and its width-thickness ratio is greater than most of the compact W-series sections in AISC. Therefore, the section is relatively subjected to local buckling and a good model case to study strength against local buckling at elevated temperatures. The critical strength of columns against buckling about weak and strong axes at 500 °C is calculated by FEM analysis with parametrically changed slenderness ratio (Figure A-4).

199 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

The figure superimposes the critical strength by the tangent modulus theory. The steel grade

is 50 ( Fy0 = 345 MPa (50 ksi)) and the maximum residual stress is 69 MPa (10 ksi). Other modeling assumptions are described in Chapter 3. The greatest differences of the critical strength calculated with and without residual

stress are 1.8 % of PTy () for buckling about the strong axis and 4.3 % about the weak axis. These maximum differences are observed when the slenderness ratio equals 100 for both buckling strength about the strong and weak axes. This observation agrees with study results of the tangent modulus theory. The critical strength computed by FEM analyses is lower than that produced by the tangent modulus theory where the effect of residual stress is significant. This is because additional stress generated by deformation at the limit state is influential in this range of the slenderness ratio.

W14x90 Fy=345MPa T=500°C Strong A x is W14x90 Fy=345MPa T=500°C Weak A x is

Shell without σ Shell without σ 1 r 1 r Shell with σ Shell with σ r r 0.8 TMT without σ 0.8 TMT without σ r r (T) (T) y TMT with σ y TMT with σ 0.6 r 0.6 r (T) / P (T) / P

cr 0.4 cr 0.4 P P

0.2 0.2

0 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Buckling about strong axis (b) Buckling about weak axis Figure A-4 The critical strength of W14×90 column

A.1.2 Lateral Torsional Buckling

The elastic critical bending moment of beams against lateral torsional buckling (LTB) under constant bending moment about the strong axis is theoretically derived as Eq. (A.9) (Timoshenko and Gere, 1961).

2 ππ⎛⎞E MEIGJICcr, e=+ y y w ⎜⎟ (A.9) LL⎝⎠

200 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

where I y , G , J , and Cw are moment of inertia about weak axis, shear modulus, torsional constant and warping constant, respectively. The first term in the square root in Eq. (A.9) correspond to St. Venant torsion and the second term corresponds to warping torsion. Studying a 4 m long W14×90 section, for instance, the second term is 440 % of the first term and warping torsion dominates the overall elastic strength for LTB. In order to understand behavior of lateral torsional buckling under elevated temperatures, tangent modulus theory is again introduced to predict the critical moment for LTB, similar to the flexural buckling study. Average tangent stiffness is approximately calculated by using simplified models with two lumped sections as shown in Figure A-5. The area of each lumped section is equals to half of the I-shaped section and, the distance between the lumped sections hnet is defined such that it satisfies the equation given in Eq. (A.10).

Figure A-5 Lumped fiber model

M py()TPTh= f () (A.10) where

M pxy()TZ= σ () T (A.11)

PTyf ()= Afσ y () T (A.12)

where Z x is the plastic section modulus about the strong axis and Af is the area of a lumped section. Without consideration of axial stress, strain of each lumped section is the same magnitude and opposite sign. The absolute value of strain in the lumped section is defined as

ε and the corresponding curvature φ is given as 2/ε hnet under the plane-section-remains- plane assumption. The moment under this curvature is calculated in Eq. (A.13) and (A.14).

201 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

M f ()TPTh= fnet () (A.13)

PT()= σε (,,,) xy TdA (A.14) f ∫ f

Stress in each lumped section is calculated in Eq. (A.5) taking into account distribution of residual stress. Axial force of each lumped section is calculated in Eq. (A.14) by integrating stress over half of the section. Average tangent stiffness is similarly defined as

Eq. (A.4) with respect to Af and the critical bending moment is obtained in Eq. (A.15) by simply substituting tangent stiffness and corresponding tangent shear stiffness into Eq. (A.9).

2 π ⎛⎞π ET() MT()=+ ETIGTJIC () () tave, (A.15) cr,,, tan t ave y t ave y w ⎜⎟ λλrryy⎝⎠

The relationship between the critical moment and slenderness ratio is obtained by

finding a slenderness ratio such that M f ()T in Eq. (A.13) equals M cr, tan ()T in Eq. (A.15). The calculated critical bending moment in various temperatures with respect to slenderness ratio for W14×90 (Gr.50) is shown in Figure A-6. The critical moment by this approach is significantly lower than the plastic moment or the elastic critical moment in the range of smaller slenderness ratio, while it agrees with the elastic critical moment in the range of greater slenderness ratios. The effect of residual stress is most influential with intermediate slenderness ratios around 90 at ambient temperature, while the influenced slenderness ratios becomes greater under elevated temperatures (e.g., around 130 at 500 °C). It is also noteworthy that the difference of the critical moment with and without residual stress decreases under elevated temperatures as is seen in the flexural buckling study.

202 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

W14x90 (Gr.50), T=100°C W14x90 (Gr.50), T=200°C

1 1

0.8 0.8 (T) (T) p p 0.6 0.6 (T) /(T) M (T) / M cr cr 0.4 0.4 M M M without σ M without σ cr,tan r cr,tan r M with σ M with σ 0.2 cr,tan r 0.2 cr,tan r min ( M , M ) min ( M , M ) cr,e p cr,e p 0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r (a) Ambient temperature, 100 °C (b) 200 °C

W14x90 (Gr.50), T=500°C W14x90 (Gr.50), T=700°C

M without σ 1 1 cr,tan r M with σ cr,tan r 0.8 0.8 min ( M , M ) cr,e p (T) (T) p p 0.6 0.6 (T) /M (T) /M

cr 0.4 cr 0.4

M M without σ M cr,tan r M with σ 0.2 cr,tan r 0.2 min ( M , M ) cr,e p 0 0 0 50 100 150 200 0 50 100 150 200 λ = L / r λ = L / r (c) 500 °C (d) 700 °C Figure A-6 Critical moment by tangent modulus theory

The tangent modulus theory provides the correct critical strength for flexural buckling, if the columns are perfectly straight at the limit state. However, this is not the case for LTB, because the calculated average tangent stiffness by this approach is already approximated by introducing the lumped section, and bending deformation increases the critical moment. Despite these factors, the calculated critical moment is a good prediction of the critical moment calculated by the analyses as is seen in Figure A-7. The analyses are carried out with shell element models with and without consideration of residual stress. The effect of residual stress is observed in the range of slenderness ratio between 120 and 180. As expected, the analytical results agree well with those produced by the tangent modulus theory.

203 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

W14x90 Fy=345MPa T=500°C

1.2 Shell w/o σ r Shell w/ σ 1 r TMT w/o σ r (T)

p 0.8 TMT w/ σ r 0.6 (T) / M cr

M 0.4

0.2

0 0 50 100 150 200 λ = L / r y Figure A-7 Comparison of the critical moment by analyses and tangent modulus theory

A.2 MODELING COMPARISON OF INDIVIDUAL MEMBERS

A.2.1 Fiber Model

The influence of local buckling to the critical strength is investigated in this section by comparing the strength simulated with shell element models and fiber models. The beam theory (i.e., plane-section-remains-plane assumption) is adopted for the fiber models so that local buckling is not simulated. The modeling hypotheses for the fiber models are similar to those for the shell element models. Additional hypotheses specifically applied to the fiber models are summarized in the following points:

1 Residual stress is not considered. 2 A member is subdivided into 32 elements along its length and each element has three sections at the ends and center. Displacements between the sections are interpolated by quadratic functions. 3 The element stiffness matrices are calculated by Gauss’s integration method with two integration points along with the length and Simpson’s rule is used for calculation of the section properties. The integration points in a section are shown in Figure A-8.

The difference of the critical strength calculated with the shell element models and fiber models is not necessarily due to the local buckling, but may be derived from other factors such as numerical error or modeling assumptions. Therefore, the effect of the local buckling

204 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

is also studied by introducing an imperfection for local buckling to the shell element models. If local buckling is the dominating mechanism to the critical strength, it must be sensitive to

the imperfection. The critical imperfection mode for local buckling (denoted as bLB ) is defined as the lowest local buckling mode obtained by linear buckling analysis. The imperfection mode is scaled such that the maximum initial displacement is equivalent to 1/1000 of the single local buckling length and the critical imperfection is provided as translational initial displacements.

Figure A-8 Integration points in fiber model section

A.2.2 Effect of Local Buckling

In this section, the critical axial strengths of columns at an elevated temperature (500 °C) are investigated under varied member lengths. The column section is a W14×90 (AISC, 2005) and flexural buckling about strong and weak axes is simulated with four different analytical

models: (1) shell element models with residual stress σ r and without a critical imperfection for local buckling bLB , (2) shell elements model with σ r and bLB , (3) shell elements model

without σ r and with bLB , and (4) fiber models (no σ r or bLB ). The simulated critical strengths with the models (2)-(4) are normalized with the strength of model (1) and shown in Figure A-9. Residual stress is not considered in models (3) and (4). Consequently, the evaluated critical strengths are unconservative compared to those with model (1) (more than 1.0 in Figure A-9). This trend is more clearly observed in the strengths against buckling about the weak axis and the unconservativeness is 15.9 % at maximum. The strengths of these two models agree well with each other except when the slenderness ratio equals 20, where local buckling dominates in model (3). The effect of the local buckling is also observed at the slenderness ratio of 20 by comparing the critical strengths of models (1) and (2). The

205 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

strengths of these models are close (the values in Figure A-9 are close to 1.0). Strength deterioration due to the local buckling at this slenderness ratio is approximately 10 % for both the strong and weak axes. Corresponding lengths with this slenderness ratio are 3.12m and 1.89 m for columns subjected to buckling about strong axis and weak axis, respectively. Considering that buckling about the weak axis is more critical for columns and 1.89 m is much shorter than a typical column length, limited practical impact of local buckling is observed in this study.

W14x90 Fy=345MPa T=500°C Strong A x is W14x90 Fy=345MPa T=500°C Weak A x is 1.2 1.2

1.1 1.1 r r σ σ

cr | shell, shell, | cr 1 shell, | cr 1 / P / P cr cr

P Shell, σ , b P Shell, σ , b 0.9 r LB 0.9 r LB Shell, b Shell, b LB LB Fiber Fiber 0.8 0.8 0 50 100 150 200 0 50 100 150 200 L / r L / r Strong axis Weak axis Figure A-9 Effect of imperfection for local buckling

A.2.3 Post Buckling Strength

Depending on the existence of an alternative load carrying path, failure of individual members in framed structures may not lead to the global structural collapse. Accurate simulation of this global building collapse is generally difficult due to the complex behavior. In this research, introductive studies are performed for global building collapse using the interior column sub-assembly; however further effort is needed for this research area. Failure of individual structural members or elements can initiate a global collapse and evaluation of post critical strength of the individual members may be needed. In this section, post-buckling strengths of columns are investigated with different analysis modeling assumptions for future development of research on global collapse simulations. Figure A-10 shows axial strengths of W14×90 columns at elevated temperature (500 °C) against flexural buckling about the strong and weak axes. The length of the column is 4 m

206 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

and the corresponding slenderness ratios are 25.6 and 42.3 for buckling about the strong axis and weak axis, respectively. The simulations are performed with four different models: (1)

fiber model, (2) shell element model without σ r or bLB , (3) shell element model with σ r and without bLB , and (4) shell element model with σ r and bLB .

W14x90 L=4m 500°C Strong A x is W14x90 L=4m 500°C Weak A x is 0.8 0.8 Fiber Shell 0.6 0.6 Shell, σ r

(T) (T) Shell, σ , b

y y r LB 0.4 0.4 (T) / P (T) / P

cr Fiber cr

P Shell P 0.2 0.2 Shell, σ r Shell, σ , b r LB 0 0 0 0.005 0.01 0.015 0 0.005 0.01 0.015 δ / L δ / L top top (a) Buckling about strong axis (L/ry = 25.6) (b) Buckling about weak axis (L/ry = 42.3) Figure A-10 Post buckling strength (W14×90, Gr.50, L=4m)

The critical strength with residual stress (model 3) is slightly lower than that without residual stress (model 2) especially for the weak axis; however, the difference is limited (less than 5 % difference). The effect of local buckling is influential to post-buckling behavior (model 3 and 4). This is more clearly observed in the strength against buckling about the strong axis. Strength calculated with fiber model (model 1), where local buckling is not simulated, is much higher than the strength with shell element models. Similar studies are performed for the strength against lateral torsional buckling (LTB) using the same models. Relationships between uniformly distributed bending moment and rotation at the end are plotted in Figure A-11(a). The post-buckling moment calculated with shell element models is significantly lower than the moment of the fiber model due to the effect of local buckling. Comparing shell element models with and without residual stress and an imperfection for local buckling, the difference of the bending moment is not significant. The critical moment of this column is approximately 70 % of the plastic moment at 500 °C, while it is almost 100 % at ambient temperature with this slenderness ratio. Vulnerability against LTB is observed at the elevated temperature.

207 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

Shell, σ , b 1 r LB 1 Shell 0.8 Fiber 0.8 (T) (T) p p 0.6 0.6 (T) / M (T) / M

cr cr Shell, σ , b 0.4 0.4 r LB M M Shell, σ r 0.2 0.2 Shell Fiber 0 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 θ (rad) θ (rad) (a) Uniformly distributed moment (b) Anti-symmetrically distributed moment Figure A-11 Post-buckling behavior for LTB

The strength against LTB under anti-symmetric bending moment (linear distribution along member with minimum and maximum moment at the ends) is also investigated (Figure A-11(b)). The absolute value of the minimum and maximum moment is equal and the inflection point is located at the mid-span. LTB will least likely occur under this type of moment distribution. Therefore, studying these two extreme cases with uniformly and anti- symmetrically distributed bending moment covers any other cases for risks of LTB under fire conditions. The assumptions of the study is the same as previous ones (i.e., W14×90 Gr.50 L = 4 m, 500 °C). An imperfection for LTB is given in the direction of the weak axis, and takes the form of the sinusoidal curve shown in Eq. (A.16).

⎧ Lz⎛⎞2π ⎫ bxyzLB (, ,)= ⎨ sin⎜⎟ ,0,0⎬ (A.16) ⎩⎭2000 ⎝⎠L

The critical bending moments, obtained with the four analysis models, are almost the same and equal to the plastic moment at the temperature 500 °C. There is no deterioration of strength due to LTB. The effect of local buckling is again significant to the post buckling strength. Strength rapidly reduces for the shell element models, while it remains as plastic moment for the fiber model.

208 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

A.3 EFFECT OF UNCERTAIN CONDITIONS

A.3.1 Overview

Member-based strength studies are deterministically performed in Chapter 3 to evaluate the simple design approach described in AISC (2005). Because the approach can give unconservative member strengths, alternative design equations are proposed. Furthermore, the AISC method and the proposed equations are probabilistically evaluated in Section 5.3.3 using the defined statistical data (with Table 5-1 in Section 5.2) for the characteristic factors (loads and steel properties at elevated temperatures). Although this study provides interesting information regarding the probability of failure considering uncertainties, there are several aspects, where deterministic conditions are assumed (e.g., uniformly distributed temperatures, boundary conditions, and imperfections). Evaluation of such conditions is typically difficult due to the high uncertainty in reality and is not fully performed in this research. Instead, sensitivities of the critical strength to these conditions are investigated to understand variation of the actual strength in this section. Temperature distribution, boundary conditions (rotational constraint stiffness at the ends), imperfections (residual stress and geometric imperfection), and steel properties at elevated temperatures are the uncertain conditions whose sensitivities are investigated. The study is carried out for AISC W14×90 I-shaped steel columns (Grade 50) at 500 °C with varied member length. The critical strength of the columns is assessed with different values for each condition and the sensitivities of the strength are evaluated.

A.3.2 Non-uniform Temperature Distribution

Wang (2002) studied the effect of non-uniform temperature distribution within cross-sections of I-shaped steel columns to the critical strength against buckling about the strong axis. It is reported that a non-uniform temperature distribution may deteriorate the strength, despite the average temperature in the sections being lower. Uneven temperature distributions in steel cross-sections induce bending deformation due to thermal expansion and create P-δ effects. Consequently, the critical strength may be decreased. In this research, bucking about the weak axis is studied as a typical failure mechanism of columns during fires. Figure A-12 and Table A-1 show combinations of non-uniform temperature distributions in I-shaped steel columns. Temperature is uniformly (500 °C) or

209 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS linearly (0-500 °C) distributed in a section for the strong direction (S-a and S-b in Figure A-2) and weak axis direction (W-a, W-b, and W-c). Two symmetric, linearly distributed temperature distributions for the weak axis are studied (W-b and W-c) in order to study the combined effect on a the geometric imperfection. The geometric imperfection is represented by initial distributions that follow a single sinusoidal curve with a maximum displacement of 1/1000 of the column length at the middle. In addition to these non-uniform temperature distributions within the cross-sections, three types of temperature distributions along the member length are investigated: uniform (A-a), linear (A-b), and sinusoidal (A-c) distributions.

Figure A-12 Non-uniform temperature distribution modes

Table A-1 Combinations of non-uniform temperature distributions Temperature distribution mode Name strong axis in weak axis in Note along length section section Typical A-a S-a W-a Uniform temperature W-1 A-a S-a W-b Linear for weak axis 1 W-2 A-a S-a W-c Linear for weak axis 2 S A-a S-b W-a Linear for strong axis A-linear A-b S-a W-a Linear in length A-sign A-c S-a W-a Sinusoidal in length

210 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

The critical strength of the columns with the six combinations of the temperature distribution modes (Table A-1) are evaluated with the shell element FEM models described in Chapter 3. The highest temperature is 500 °C for all combinations. The analyses are performed in two steps: temperature is increased in the first step and the axial load is applied in the second step. P-δ effects with thermally induced bending deformation is simulated in this analysis approach. Shown in Figure A-13 is the critical strength of the column with varied lengths for five combinations of the temperature distribution modes in Table A-1: Typical, W-1, S, A-linear, and A-sign. Figure A-13(a) shows the critical strength normalized with plastic strength at

500 °C ( PTyyy()= AKTF () 0 , where A , Fy0 , and KTy () are cross-sectional area, yield strength of steel at elevated temperatures, and reduction factor of the yield strength, respectively), while Figure A-13(b) shows ratios of the critical strength with respect to the strength of the typical case.

1.6 Typical

1 Linear in sec. W-1 C) 1.4 ° C)

° Linear in sec. S 0.8 Linear in length 1.2 Sine in length (T= 500 (T= 500 (T= (T)

(T) 0.6 1 y cr,Typ 0.4 0.8 (T) / P cr (T) / P P 0.2 cr 0.6 P

0 0.4 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-13 Sensitivity of critical strength to non-uniform temperature distribution

The critical strength is greater than the typical strength in most cases; therefore, assuming a uniform temperature distribution is conservative in most cases. However, the critical strength in case W-1 (linear temperature distribution in the weak axis) is lower than the typical strength when slenderness ratio is greater than 60. The strength is less than 50 % of the typical strength when the slenderness ratio is greater than 140. The critical strength in case S (linear temperature distribution for the strong axis in the section) is also lower than the typical strength for slenderness ratios greater than 160 (Figure A-13(b)); however, the

211 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

relative strength does not go lower than 90 % of the typical strength, which is not a significant deterioration in strength. This non-uniform temperature distribution mode for the strong axis would have a greater impact, if the critical mechanism under uniform temperature is buckling about the strong axis. Slender columns are more vulnerable to non-uniform temperatures within the cross-section. This result is consistent with the study by Wang (2002). Non-uniform temperature along the member length (A-linear and A-sign in Table A-1) constantly yields greater critical strengths than the typical strength. Shown in Figure A-14 are the critical strengths of non-uniform temperature combination cases W-1 and W-2 in Table A-1. The combined effect of non-uniform temperature distribution about the weak axis and geometric imperfections are investigated. The geometric imperfection increases and decreases the bending deformation due to the non- uniform temperature for W-1 and W-2, respectively. The geometric imperfection does not impact the strength as much as non-uniform temperature distribution does, because the critical strength in these two cases is not much different. The effect of the non-uniform temperature distribution dominates that of the geometric imperfection.

1.6 Typical

1 Linear in sec. W-1 C) 1.4 ° C)

° Linear in sec. W-2 0.8 1.2 (T= 500 (T= 500 (T= (T)

(T) 0.6 1 y cr,Typ 0.4 0.8 (T) / P cr (T) / P P 0.2 cr 0.6 P

0 0.4 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-14 Sensitivity of critical strength to non-uniform temperature distribution for the weak axis

A.3.3 Imperfections

Sensitivity of the critical strength of the columns at an elevated temperature (500 °C) to imperfections (geometric imperfection and residual stress) is investigated by changing the

212 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

magnitude of the imperfection by 200 % and 50 %. The geometric imperfection is modeled by initial displacements that form a sinusoidal curve along the weak axis. The maximum initial displacement at the center is 1/1000 of the length for the typical setting. The displacement is varied to 1/500 and 1/2000 of the length for this sensitivity study. The residual stress is given by the linear distribution mode shown in Figure 3-5(a) and the maximum residual stress is 20 % of the yield stress both for tension and compression (for Grade 50 steel). The proportion of the yield stress and residual stress is assumed as constant at ambient and elevated temperatures. Shown in Figure A-15 is the critical strength with different magnitudes of the imperfections (200 % and 50 % of the original magnitude both for the geometric imperfection and residual stress). The influence of the geometric imperfection and residual stress is about the same order and is relatively significant to the members with slenderness ratio of 100-160. However, since the maximum influence is about 10 % of the critical strength, the strength can be considered not significantly sensitive to the imperfections.

1.6 Typical

1 L / 500 C) 1.4 ° C)

° L / 2000 2.0 σ 0.8 r 1.2 (T= 500 (T=

(T= 500 (T= 0.5 σ r (T)

(T) 0.6 1 y cr,Typ 0.4 0.8 (T) / P (T) cr (T) / P P 0.2 cr 0.6 P

0 0.4 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-15 Sensitivity of critical strength to imperfections

A.3.4 Boundary Conditions

The rotational constraint at the boundaries of the members is assumed to be free for the member-based strength assessment performed in Chapter 3 and Section 5.3.3. This assumption simplifies the assessment and derives conservative results. The effect of the boundary conditions (rotational constraint of the column ends) is investigated in this section.

213 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

The AISC W14×90 I-shaped steel columns (Grade 50) at 500 °C are investigated under four boundary conditions: (1) free top and bottom (Typical), (2) fixed top and bottom, (3) fixed, bottom and free, top, and (4) springs top and bottom. The rotational springs are elastic and given a constraint stiffness that is from the adjacent column (continuing column) and the beams. The temperature of the continuing column is assumed to remain ambient and the rotational constraint stiffness from the column is obtained from Eq. (A.17).

3EI0 y ⎛⎞P krot, col =−⎜⎟1 (A.17) LP⎝⎠cr

where krot, col is the rotational constraint stiffness from the continuing column. E0 , I y , and L are the elastic modulus at ambient temperature, moment of inertia about the weak axis, and the member length, respectively. The member length of the continuing column is assumed to be the same as the length of the heated column. P and Pcr are the applied and critical axial forces of the continuing column, respectively. The ratio of P and Pcr is approximately evaluated from the pre-calculated critical axial strength of the column (i.e., P is the critical strength at 500 °C and Pcr is the critical strength at ambient temperature with rotationally free boundary conditions). The contribution of the beams to the rotational boundary constraint stiffness of the

columns krot, bm is calculated by considering the effect of the bolted shear-tab connections. The initial stiffness of the connection (shown in Figure 4-13) is used for the calculation. The

total constraint stiffness at the column boundary is therefore given as kkrot,, col+ rot bm . Shown in Figure A-16 is the strength of the columns under the four boundary conditions with varied member length. The slender columns are more influenced by the boundary conditions than stocky columns. The strength with the fixed boundary conditions is about 300 % of the strength with the free conditions at a slenderness ratio of 200. The strength increase with the bottom end fixed condition is about 50 % of the increase with the both ends fixed condition, while the increase with the springs is about 75 % (Figure A-16(b)).

The rotational constraint by the connecting beams krot, bm is taken into account in this

study in addition to the constraint by the continuing column krot, col ; however, krot, bm is not

214 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

included in the interior and exterior sub-assembly models used in chapter 4. The critical

strength with and without krot, bm is investigated for the column (W14×90, Grade 50 at 500

°C, length equals 4 m, slenderness ratio equals 42.3). The critical strength with both krot, col and krot, bm , with only krot, col , and without neither krot, col and krot, bm are 3660 kN, 3620 kN (99 %), and 2770 kN (76 %), respectively. Therefore, the effect of the beam constraint is only considered for the critical strength for the sub-assembly study.

Typical 3

1 Fixed ends C) ° C)

° Fixed bottom 2.5 0.8 Spring ends (T= 500 (T= 500 (T)

(T) 0.6 2 y cr,Typ 0.4

(T) / P (T) 1.5 cr (T) / P (T) P 0.2 cr P 1 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-16 Sensitivity of critical strength to boundary conditions at 500 °C

In a similar manner, the critical strength of the columns at ambient temperature is investigated with the four boundary conditions as shown in Figure A-17. The constraint effect is more significant for the slender columns, which is considered with the results from the 500 °C columns. The strength increase with the spring boundary is about 50 % of the increase with the fixed ends condition. This increase ratio is lower than the ratio studied for the columns at 500 °C. This is because the rotational constraint from the continuous columns

krot, col , based on Eq. (A.17), nearly vanishes. Therefore, the increase of the critical strength by the rotational constraint from the connecting members is more significant for the columns at elevated temperatures (because the stiffness of the connecting members is relatively higher).

215 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

3.5 1 3 0.8

y0 2.5 cr0,Typ

/ P 0.6 / P cr0 2 P cr0

0.4 Typical P Fixed ends 1.5 0.2 Fixed bottom Spring ends 1 0 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-17 Sensitivity of critical strength to boundary conditions at 20 °C

A.3.5 Steel Properties

The sensitivity of the critical strength of the column to the characteristic properties of steel at elevated temperatures is investigated in this section. The characteristic properties are the elastic modulus, the yield strength, and the strength at the proportional limit (proportional strength), as defined in Eurocode 3 (EC3, 1995). Reduction factors of these properties with respect to temperatures are defined as the proportion of values at elevated temperatures to those at ambient temperature as shown in Eq.(A.18).

ET() FTp () FTy () KTE ()= , KTp ()= , and KTy ()= (A.18) E0 Fp0 Fy0

where E0 , Fp0 and Fy0 are modulus of elasticity, proportional-limit stress and yield stress

at ambient temperature, respectively, and ET(), FTp () and FTy () are those at elevated temperature T . Other details about the stress-strain curves defined by EC3 are described in Section 2.2.2.1.

Statistical properties (mean and coefficient of variation, c.o.v.) of FTy () and E()T are defined in Table 5-1 in Section 5.2.1. In addition, the mean and c.o.v. of FTp () are assumed

to be KTFyy() 0 and 0.22 (same as c.o.v. of FTy ()), respectively, for the study in this

216 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

section. The critical strength of W14×90 (Grade 50) columns at 500 °C with varied length

are simulated with mean plus and minus one standard deviation of FTy (), FTp (), and E()T (Figure A-18). The yield strength controls the critical strength of shorter columns (slenderness ratio of 80 or smaller), while the elastic modulus controls for longer columns (slenderness ratio of 100 or greater). This is because yielding or plastic buckling is the critical mechanism for shorter columns and elastic buckling is that for longer columns. The maximum difference of the critical strength with mean plus and minus one standard deviation of FTp () is approximately 10 % for the columns with intermediate length (around the slenderness ratio of 80); therefore, the sensitivity of the critical strength to FTp () is less than

the sensitivity of the critical strength to FTy () and E()T .

1.6 Typical

1 + K (T) C) 1.4

y ° C) ° - K (T) y 0.8 1.2

+ K (T) 500 (T= p (T= 500 (T=

- K (T) (T)

(T) 0.6 p 1 y

+ K (T) cr,Typ E 0.4 - K (T) 0.8 (T) / P (T) E cr (T) / P P 0.2 cr 0.6 P

0 0.4 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-18 Sensitivity of critical strength to steel properties at 500 °C

The c.o.v.s of FTy (), FTp (), and ET() at elevated temperatures are defined as 0.22,

while those of Fy0 and E0 at ambient temperature are 0.1 and 0.06, respectively (Ellingwood, 1983). The variation of the critical strength due to the uncertainty of the steel material properties at elevated and ambient temperatures is compared. Shown in Figure A-19 is the sensitivity of the critical strength of the column to Fy0 and E0 at ambient temperature. Due to the difference of the material uncertainties (c.o.v.s), the critical strength varies less at ambient temperature.

217 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

1.6 Typical + K (T) 1 y 1.4 - K (T) y 0.8 + K (T) 1.2

y0 E

- K (T) cr0,Typ / P 0.6 E 1 / P cr0 P cr0

0.4 P 0.8

0.2 0.6

0 0.4 0 50 100 150 200 0 50 100 150 200 L / r L / r (a) Critical strength (b) Ratio to the typical case Figure A-19 Sensitivity of critical strength to steel properties at 20 °C

A.4 OTHER MISCELLANEOUS STUDIES

A.4.1 Temperature Distribution of Composite Beams

A non-uniform temperature distribution mode for composite beam sections is assumed as shown in Figure A-20 referring to experimental data obtained by Wainman and Kirby (1988).

The temperature between the lower flange and 3/4 of web is defined as TL , while the temperatures at the upper 1/4 of web and the upper flange are 0.9TL and 0.8TL , respectively.

Figure A-20 Temperature distribution of composite section

Wainman and Kirby (1988) carried out multiple tests for simply supported beams at elevated temperatures. Temperatures and displacements of the beams were recorded under constant mechanical load with increasing temperature. One of their typical tests is selected (data #89) and discussed in this section. Figure A-21 shows the longitudinal section of the test assembly. The beam is British BS4360, 356×171×67UB Grade 43A ( Fy = 292 MPa) and

218 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS the nominal section properties are shown in Table A-2. The span of the simply supported beam was 4.5 m. Concrete slab blocks were placed at the top of the steel beam; however, composite effect was not expected without stud connections between the slab and beam. Although the tested beam was non-composite, it is a good reference for the temperature distribution of composite steel beams with concrete slabs.

Table A-2 Section sizes of beam tested by Wainman and Kirby (mm)

Section Web height, h Web thickness, tw Flange width, bf Flange thickness, t f 356×171 364.0 9.1 173.2 15.7

Figure A-21 Beam experiment by Wainman and Kirby (1988) (Wainman, D. E., Kirby, B. R. (1988), “Compendium of UK Standard Fire Test Data, Unprotected Structural Steel - 2,” Ref. No. RS/RSC/S1199/8/88/B, British Steel Corporation (now Corus), Swinden Laboratories, Rotherham, pp 14 Figure 1(a))

The beam is heated in a furnace and temperatures of the beam at several sections and the mid-span displacement were recorded. The recorded temperatures at three sections are shown in Figure A-22. The locations of these three sections were 0.62 m left from the center (section 1), the center (section 2), and 0.31 m right from the center (section 3) as shown in

219 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

Figure A-21. The assumed temperature distribution mode is superimposed on the recorded temperatures. The mode is defined such that it agrees better at higher temperatures with the test results considering the limit-states of the beams. The temperature of the concrete slab is assumed to be 0.4TL based on previous research (Lamont et al., 2000).

Left Center 1 1

0.8 0.8

0.6 0.6 2 min 2 min

Height 0.4 10 min Height 0.4 10 min 20 min 20 min 30 min 30 min 0.2 0.2 35 min 35 min model model 0 0 0 200 400 600 800 0 200 400 600 800 Temperature (°C) Temperature (°C) (a) Section 1 (left) (b) Section 2 (center)

Right 1

0.8

0.6 2 min Height 0.4 10 min 20 min 30 min 0.2 35 min model 0 0 200 400 600 800 Temperature (°C) (c) Section 3 (right) Figure A-22 Recorded temperatures in three sections by Wainman and Kirby (1988)

A.4.2 Modeling Comparison of Composite Beam

The mid-span displacement of the simply supported beam tested by Wainman and Kirby (1998) is used to verify the accuracy of the simulations using ABAQUS. The details of the beam test are explained in the previous section (A.4.1). Three types of temperature distribution modes are studied in the ABAQUS simulations: (1) uniform distribution, (2)

220 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS bilinear distribution (uniform at lower half and linearly interpolated at the upper half with 60 % of the temperature at the top) and (3) detailed distribution calibrated with the recorded temperatures. These three temperature modes along the height of the section are compared with the test data as shown in Figure A-23(a). The temperature distributions from the test were recorded at the three sections and the lower flange temperatures are around 350-400 °C. The detailed distribution mode agrees with the test data fairly well, while the uniform mode has a large temperature gap at the top. Relationships between the mid-span displacement and the elevated lower flange temperatures are compared in Figure A-23(b) for the test data and simulations with these three distribution modes. Although the beam had no stud connections to engages composite behavior, the concrete slab on top of the steel beam significantly influenced the non-uniform temperature distribution from the test (accordingly, the simulation were modeled as non-composite). The mid-span displacement from the test is -30 mm at 400 °C of the lower flange temperature. The mid-span displacement calculated by the simulation with the detailed temperature distribution at this temperature is -24 mm, while those with bilinear and uniform distribution are -19 mm and -5 mm, respectively. The non- uniform temperature distribution induced greater mid-span displacement and it is accurately simulated in the analysis.

1 0

0.8 -50

0.6 -100

Height 0.4 Test (at 10 min) Test -150

Uniform Mid-span disp. (mm) Uniform 0.2 Linear Linear Detailed Detailed 0 -200 0 100 200 300 400 500 0 200 400 600 Temperature (°C) Temperature (°C) (a) Temperature distribution (b) Mid-span displacement Figure A-23 Comparison between analysis and test by Wainman and Kirby (1988)

221 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

A.4.3 Effect of Heat Conduction

In the structural fire simulations for the frames in the benchmark building, it is assumed that the temperatures of the members are constant along the member length. Therefore, the columns above and below the fire compartment remain cool with perfect heat insulation assumed at the floor level. However, the temperature of these columns near the floors must elevate due to heat conduction. In order to investigate the temperature of the columns above and below the fire compartment and evaluate the rationale of the constant temperature assumption along the member length, the effect of heat convection is studied in this section. Figure A-24 shows a study model for a one dimensional transient heat conduction problem. The column in the fire compartment is heated and the temperature is assumed uniformly constant at 500 °C. The temperature of the column above the fire compartment is originally 20 °C at time t = 0 along its length. Fire insulation at the floor level is assumed to be perfect; therefore, the heat transfers from the heated column to the column above only by conduction (i.e., no radiation or convection). The temperature at the floor level is maintained at 500 °C and the heat supply to the above column is infinite.

Figure A-24 Study model for heat conduction

The diffusion equation of the column above the fire compartment is stated in Eq. (A.19).

∂TT∂2 ρck= (A.19) p ∂tx∂ 2

222 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

where t and x are time and the distance from the floor level, respectively. T is the temperature of the column and is a function of t and x . Other thermal properties of the steel column are shown in Table A-3.

Table A-3 Thermal properties of steel column Density, ρ (kg/m3) 7850

Specific heat, cp (J/kg°C) 600 Thermal conductivity, k (W/m°C) 45.8

The time to reach steady-state behavior (denoted as t0 ) is approximately calculated for reference using thermal diffusivity α and diffusion length L0 (Incropera and DeWitt, 2002).

-6 2 The thermal diffusivity is defined as α = kcρ p , which is 9.72×10 m/sec using the

properties in Table A-3. The diffusion length is given as Lt00= α , and assuming L0 = 2

5 m, which is the half of the column length, leads to t0 = 4.1×10 sec (114 hr). Knowing this approximate time to reach the steady-state, the diffusion equation is solved numerically.

TTpp−− TT pp T p+1 − T p kkmm−+11+=Δ mmρ cx m m (A.20) ΔΔxxp Δ t

where m and p are the member of evenly discretized elements (along the member length) and the time step number, respectively. Each element length is Δx and time length is Δt . Solving Eq. (A.20) derives relationships between time, temperature and the distance from the floor level. Two cases are studied with different boundary conditions at the top of the column: (1) the column continues with infinite length and (2) the column ends after 4 m (no heat transfer at the top to the above). Shown in Figure A-25 are temperature-distance from floor level relationships at t = 1, 10 and 100 hour. A slightly different temperature distribution is observed (at the top of the column) at t = 100 hours, while there is almost no difference at t = 1 and 10 hours.

223 APPENDIX A. SUPPLEMENTAL STUDIES ON INDIVIDUAL MEMBERS

4 1 hr 10 hr 3 100 hr 100 hr 4m

2 Length (m) 1

0 0 100 200 300 400 500 Temperature (°C)

Figure A-25 Temperature increase by heat conduction

This study shows a very limited temperature increase on the column above the fire compartment (e.g., 100 °C up at 1 m from the floor level after 10 hours). The result is consistent with the previously shown preliminary check using the thermal diffusivity. It does not mean, however, that the column remains cool and is almost intact after a 10-hour fire in reality. Partitions of the fire compartment including the insulation at the floor level may break under such a long-lasting fire and the spreading fire would heat the column. This study simply investigates the effect of heat conduction and shows that the rationale not to take into account conduction for relatively short-lasting (less than one hour) compartment fire studied in this research. The derived limited effect of heat conduction may be counter-intuitive; however, simulations using ABAQUS agree with this result. Moreover, Dr. Lamont and Dr. Lane with Arup have verified it with an independent study using SAFIR, a structural fire analysis program. Their help and advice based on proficient knowledge regarding structural fire engineering are greatly appreciated.

224

APPENDIX B REFERENCE EQUATIONS

B REFERENCE EQUATIONS

B.1 CONVERSION OF UNITS

In this thesis, SI units are adopted as the primary unit system. In some cases, converted values in the US units are also shown for easier understanding for some readers. However, values are shown only in the US units in some quoted plots (e.g., Figure 2-10 and Figure 2- 12: plots for measured stress-strain curves of steel at elevated temperatures in past studies). Conversion of the SI and US units is briefly introduced below. The SI and US units regarding temperatures are Celsius (°C) and Fahrenheit (°F). The relationships between these units are given in Eq. (B.1)

5 TT=−( 32) (B.1) CF9

where TC and TF are temperatures in Celsius and Fahrenheit, respectively. Some converted values are shown in Table B-1

Table B-1 Conversion of temperature units

TF (°F) 70 100 200 300 400 500 600 700 800 900 1000 1100 1200

TC (°C) 21 38 93 149 204 260 316 371 427 482 538 593 649

Table B-2 shows the relationships between the SI and US units for length and force.

Table B-2 Conversion of length and force units 1 inch (in) 1 feet (ft) 1 pound (lb) 1 kilopound (kip) 25.4 mm 0.3048 m 4.448 N 4.448 kN

225 APPENDIX B. REFERENCE EQUATIONS

In addition, relationships of the units for pressure are shown in Table B-3.

Table B-3 Conversion of pressure units 1 kilopound per square inch (ksi) 1 pound per square feet (psf) 6.895 MPa (= 6.895×106 N/m2) 47.88 Pa (= 47.88 N/m2)

B.2 SYMBOLS

Expressions of the symbols used in this research are primarily based on AISC (2005). However, some of the symbols are differently defined in order to clarify the properties at elevated temperatures from those at ambient temperature. Some expressions used in the Eurocode (EC1, 2002 and EC3, 2003) are also introduced. Special care is needed in introducing the Eurocode symbols because they are fairly different from those in AISC. Also, the definition of sectional axes is different between AISC and Eurocode (Figure B-1). Consequently, sectional properties are differently expressed. The symbols used in this research are compared in Table B-4 with those in AISC and Eurocode for reference.

Figure B-1 Section axes in AISC and Eurocode

Table B-4 Symbols in AISC and Eurocode

Symbol AISC Eurocode Unit Property (mm6) Warping constant Cw Cw Iw (MPa) Elastic modulus (ambient and elevated temperature) E0 , ET() E E

Fy0 , FTy () Fy f y (MPa) Yield strength (ambient and elevated temperature) Shear modulus of elasticity (ambient and elevated , (MPa) G0 GT() G G temperature) (mm4) Moment of inertia about STRONG axis I x I x I y

226 APPENDIX B. REFERENCE EQUATIONS

Symbol AISC Eurocode Unit Property

(mm4) Moment of inertia about WEAK axis I y I y I z (mm4) Torsional constant J J IT

- Reduction factor for the elastic modulus KTE () kTE () kE,θ

KTp () - k p,θ - Reduction factor for the proportional limit

KTy () kTy () ky,θ - Reduction factor for the yield stress

(N) Elastic critical force for flexural buckling (2D) Pcr0, e Pcr Ncr Elastic critical force for flexural buckling about strong (N) Pcrx0, e - Ncrit, y axis Elastic critical force for flexural buckling about weak (N) Pcry0, e - Ncrit, z axis (mm3) Section modulus about STRONG axis Sx Sx Wy (mm3) Section modulus about WEAK axis S y S y Wz (mm3) Plastic section modulus about STRONG axis Z x Z x Wpl, y 3 Z y Z y Wpl, z (mm ) Plastic section modulus about WEAK axis (mm) Radius of gyration about STRONG axis rx rx iy (mm) Radius of gyration about WEAK axis ry ry iz Slenderness ratio in Eurocode at AMBIENT - - λ0 λ temperature for flexural buckling (2D) Slenderness ratio in Eurocode at AMBIENT - - λx0 λy temperature for flexural buckling about STRONG axis Slenderness ratio in Eurocode at AMBIENT - - λy0 λz temperature for flexural buckling about WEAK axis Slenderness ratio in Eurocode at AMBIENT - - λLT 0 λLT temperature for lateral torsional buckling Slenderness ratio in Eurocode at ELEVATED - - λ ()T λθ temperature for flexural buckling (2D) Slenderness ratio in Eurocode at ELEVATED - - λx ()T λy,θ temperature for flexural buckling about STRONG axis Slenderness ratio in Eurocode at ELEVATED - - λy ()T λz,θ temperature for flexural buckling about WEAK axis Slenderness ratio in Eurocode at ELEVATED - - λLT ()T λLT,,θ com temperature for lateral torsional buckling Reduction factor in Eurocode at AMBIENT - χ - χ0 temperature for flexural buckling (2D) Reduction factor in Eurocode at AMBIENT - χ - χ x0 y temperature for flexural buckling about STRONG axis Reduction factor in Eurocode at AMBIENT χ - - y0 χ z temperature for flexural buckling about WEAK axis

227 APPENDIX B. REFERENCE EQUATIONS

Symbol AISC Eurocode Unit Property Reduction factor in Eurocode at AMBIENT - - χLT 0 χLT temperature for lateral torsional buckling Reduction factor in Eurocode at ELEVATED - - χ()T χ fi temperature for flexural buckling (2D) Reduction factor in Eurocode at ELEVATED - χ - χ x ()T yfi, temperature for flexural buckling about STRONG axis Reduction factor in Eurocode at ELEVATED - - χ y ()T χ zfi, temperature for flexural buckling about WEAK axis Reduction factor in Eurocode at ELEVATED - χ - χLT ()T LT, fi temperature for lateral torsional buckling

B.3 DESIGN EQUATIONS OF STEEL AT ELEVATED TEMPERATURES

B.3.1 Eurocode 3

Design equations of steel at elevated temperatures are defined as follows in Eurocode 3 (Eurocode 3, Design of Steel Structures – part 1-2. General rules – Structural fire design, Draft prEN 1993-1-2, Stage 49 Draft, Brussels, Belgium, 2003, pp 17-20)

228 APPENDIX B. REFERENCE EQUATIONS

229 APPENDIX B. REFERENCE EQUATIONS

230 APPENDIX B. REFERENCE EQUATIONS

B.3.2 AS4100

AS4100 defines bilinear stress-strain relationships for steel at elevated temperatures. The stress-strain relationships are perfect-plastic, and the reduction factors of elastic modulus

KTE () and yield strength KTy () at elevated temperatures are defined by the following Eqs. (B.2)-(B.5).

T KTE () = 1.0 + when 0 °C < T ≤ 600 °C (B.2) ⎡⎤⎛⎞T 2000⎢⎥ ln ⎜⎟ ⎣⎦⎝⎠1100 ⎛⎞T 690 1− ⎜⎟1000 = ⎝⎠ when 600 °C < T ≤ 1000 °C (B.3) T − 53.5

KTy () = 1.0 when 0 °C < T ≤ 215 °C (B.4) 905 −T = when 215 °C < T ≤ 905 °C (B.5) 690

B.4 TIME-TEMPERATURE RELATIONSHIPS

B.4.1 Parametric Fire Curve

Parametric fire curves (time and gas temperature relationships during fire) defined in Eurocode 1 (2002) are used in this research for the fire simulations. The equations for the curves are quoted below for the reference. (Eurocode 1, Actions on structures – part 1-2. General actions – Actions on structures exposed to fire, Final Draft prEN 1991-1-2, Brussels, Belgium, 2002, pp 31-33)

231 APPENDIX B. REFERENCE EQUATIONS

232 APPENDIX B. REFERENCE EQUATIONS

233 APPENDIX B. REFERENCE EQUATIONS

Table B-5 summarizes the parameters and conditions, which are used for the parametric fire curve equations in this research. These parameters and conditions are set for the simulations of the compartment fire in the benchmark office building shown in Figure 4-1.

Table B-5 Parameters and conditions for parametric fire curves

Parameter Value or condition Boundary enclosure material Gypsum Opening factor, O 0.05

2 Fire compartment floor area, Af 61 m

2 Compartment total surface area, At 235 m

Compartment factor, δq1 1.5

Occupancy factor, δq2 1.0

Fire fighting factor, δn 0.282 Combustion factor, m 0.8

Some of the parameters in Table B-5 are defined in Eurocode 1 (2002). Part of the descriptions about the parameters is quoted below for reference. Values used in the fire simulations in this research are indicated in the original document. (Eurocode 1, Actions on structures – part 1-2. General actions – Actions on structures exposed to fire, Final Draft prEN 1991-1-2, Brussels, Belgium, 2002, pp 47, 48 and 51)

234 APPENDIX B. REFERENCE EQUATIONS

235 APPENDIX B. REFERENCE EQUATIONS

B.4.2 Step-by-step Steel Temperature Simulation

Simulations of steel temperatures are carried out by using an iterative time-step method described by Buchanan (2002). The temperatures of the unprotected and insulated steel members are calculated based on the compartment gas temperatures, which are obtained by the parametric fire curves defined in Eurocode 1 (2002). The time-step method uses two equations for unprotected and insulated steel members. The temperature increment ΔTsteel of unprotected steel members during the time increment step Δt seconds is calculated by using

236 APPENDIX B. REFERENCE EQUATIONS

Eq. (B.6). The steel temperature increases by heat transfer from the gas in the fire compartment. Convection and radiation are the mechanism of the heat transfer for unprotected steel members under fires. The first and second terms in the outer brackets in Eq. (B.6) correspond to the convection and radiation, respectively.

Fsteel 1 44 Δ=Tsteel{( h c, steel TT gas− steel ) +σε steel ( TT gas − steel )} Δ t (B.6) Vcsteelρ steel p, steel

where, Fsteel and Fsteel are the surface area and volume of unit length of the steel member, respectively. ρ is the density, cp is the specific heat, hc is the convection coefficient, ε is the emissivity, and σ is the Stefan-Boltzmann constant (5.67×10-8 W/m2°C4). The values of these constants are shown in Table B-6 (Buchanan, 2002). On the other hand, the primary mechanism of the heat transfer for the insulated steel members under fires is conduction. The temperature increment of insulated steel members is calculated by using Eq. (B.7), where, k is the thermal conductivity, dinsul is the thickness of the fire insulation.

⎡⎤ Fk⎢⎥ρ c Δ=TTTtsteel insul ⎢⎥steel p, steel () − Δ (B.7) steel Vdρ c F dcρ gas steel steel insul steel p, steel ⎢⎥(ρ c + insul insul p, insul ⎣⎦⎢⎥steel p, steel V 2

Table B-6 Thermal properties of steel and fire insulation Steel Gypsum Density, ρ (kg/m3) 7850 800

Specific heat, cp (J/kg°C) 600 1700 Thermal conductivity, k (W/m°C) 45.8 0.2 2 Convection coefficient, hc (W/m °C) 25 - Emissivity, ε 0.5 -

237 APPENDIX B. REFERENCE EQUATIONS

B.5 FOSM

The mean-value first-order second-moment (FOSM) method is used for the probabilistic assessment of the structural failure. The considered random variables are assumed as uncorrelated in this research due to insufficient statistical information. This assumption of the uncorrelated random variables is unconservative; however, assuming perfect correlation for all combinations of the random variables is unrealistic (although it is a conservative assumption). Focusing on proposing a probabilistic assessment approach, the assumption of the uncorrelated random variables is considered as reasonable under current research development of statistical data about the random variables. Equations used in the FOSM analyses are shown below in Eq. (B.8)-(B.14) for reference (Benjamin and Cornell, 1970). Considering future research development, these equations are shown in general format so that they can be used for analyses with correlated random variables. The probability of failure Pfailure() can be evaluated in following equations.

Pf ≅ Φ−()βMVFOSM (B.8) where

2 u 1 ⎛⎞z Φ=()udz exp − (B.9) ∫−∞ ⎜⎟ 2π ⎝⎠2 g()M βMVFOSM = (B.10) ∇∑∇gT ()M g ()M

g is the limit-state function defined as difference between the critical temperature of the structural members Tcr and the maximum steel temperature Tsmax obtained in the fire simulation. Negative g implies loss of the structural stability during the fires. M is a mean vector of random variables, and ∇g()M is the gradient of g (x) at the mean and Σ is a covariance matrix.

gT= cr− T smax (B.11)

T M = {}μ12μμ... n (B.12)

238 APPENDIX B. REFERENCE EQUATIONS

T ⎧ ∂∂gg ∂ g⎫ ∇=gM() ⎨ ... ⎬ (B.13) ⎩⎭∂∂xx12 ∂ xn

2 ⎡⎤σ11212ρσσ" ρσσ 11nn ⎢⎥ σ 2 ρσσ Σ=⎢⎥222nn (B.14) ⎢⎥%# ⎢⎥2 ⎣⎦⎢⎥sym σ n

where x is a random variable, μ is a mean, σ is a standard deviation, ρ is a covariance and n is the number of the random variables. ∇g()M is approximately calculated in this research in Eq. (B.15).

T ⎧⎫gg()()μσ+− μσ − gg()()μσ+− μσ − ∇=gM()⎨⎬11 11" nn nn (B.15) ⎩⎭22σσ1 n

The distribution of some random variables (e.g., steel strength at elevated temperatures and longitudinal constraint stiffness of the beams) is assumed to be lognormal, which is asymmetric distribution. The lognormal distributions are converted to normal distributions by taking logarithm and the converted values of mean and standard deviation are used for the FOSM equations shown above.

239 APPENDIX B. REFERENCE EQUATIONS

240

APPENDIX C JISF EXPERIMENT

C APPENDIX C JISF EXPERIMENT

C.1 SUMMARY

A comparative assessment is performed on experimental data on steel materials at elevated temperatures provided by the Japan Iron and Steel Federation (JISF). The stress-strain curves are investigated and compared with the design equations in the Architectural Institute of Japan (AIJ) and Eurocode 3 (EC3). Statistical information of strength at 1 % and 2 % strain is evaluated and summarized.

C.2 DATA CONDITIONS

Conditions of the provided test data by JISF are as follows:

(1) Steels with other specifications and grades than those of the steels that JISF provided may have different characteristics from those of the provided data. (2) The provided data are all based on basic oxygen furnace steels and electric furnace steels are not included. When electric furnace steels are included, the data characteristics may be different due to the differences in production process, chemical compositions, etc. (3) The steel specimens from which the provided data were obtained are limited in thickness up to 40 mm. The steels with a thickness of over 40 mm may have different characteristics. (4) The characteristics of steel under elevated temperatures may change when welding or any type of forming is given to the steel.

C.3 GENERAL

This report summarizes a comparative assessment of experimental data on steel materials at elevated temperatures provided by the Japan Iron and Steel Federation (JISF) dated on 23 Dec 2003. The data include tensile stress-strain relationships of Japanese standard steel under ambient and elevated temperatures (20 °C and 300 °C to 800 °C). It is expected that

241 APPENDIX C. JISF EXPERIMENT

the stress-strain relationships obtained in the test will contribute to a better understanding of behavior of steel structures under fire conditions and fire engineering design, specifically in establishing reasonably simplified stress-strain curves which can be used for analytical simulations. The following six different types of steel are examined:

SM490A rolled plate SM490A wide flange SM490AW rolled plate SN490C rolled plate STKN400B structural tube STKN490B structural tube

These types of steel are roughly equivalent to Grade 50 mild steel in the US standard except STKN400B, which has a strength equivalent to Grade 36. Simple data analyses were performed as part of this assessment to look at two aspects: (1) investigation of the stress- strain relationships and comparison to those described in Recommendation for Fire Resistant Design of Steel Structures by Architectural Institute of Japan (AIJ, 1999) and Eurocode 3 (EC3) (CEN, 1995), and (2) statistical study of the strength of the steel at 1 % and 2 % strain under elevated temperatures.

C.4 JISF STRESS-STRAIN CURVES

Plots of the stress-strain data provided by JISF are shown in Figure C-1 to Figure C-6. Five sets (five different supplies) of test data are included for each type of steel and each set contains stress-strain curves under different temperatures (ambient and elevated temperatures from 300 °C to 800 °C at every 100 °C). Two stress-strain curves are included in each set of the test data at 400 °C, 500 °C and 600 °C. The test at 800 °C is performed only with SM490A(PL) and STKN490B steel. As summarized in Table C-1, the total number of tests is 280. It is observed from the stress-strain curves at elevated temperature shown in Figure C-1 to Figure C-6 that (1) the yield points and yield plateaus disappear, (2) the stress-strain relationships become more nonlinear, and (3) strain hardening reduces and disappears.

242 APPENDIX C. JISF EXPERIMENT

Table C-1 Number of tests for each steel type and temperatures Temperature (°C) Steel Type Total 20 300 400 500 600 700 800 SM490A (PL) 5 5 10 10 10 5 5 50 SM490A (WF) 5 5 10 10 10 5 - 45 SM490AW (PL) 5 5 10 10 10 5 - 45 SN490C (PL) 5 5 10 10 10 5 - 45 STKN400B 5 5 10 10 10 5 - 45 STKN490B 5 5 10 10 10 5 5 50 Total 30 30 60 60 60 30 10 280

243 APPENDIX C. JISF EXPERIMENT

SM490A Plate (1) SM490A Plate (2)

600 600

500 500

400 400

300 300

Stress (MPa) Stress 200 (MPa) Stress 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490A Plate (3) SM490A Plate (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490A Plate (5)

600 20°C 500 300°C 400°C 400 500°C 300 600°C 700°C Stress (MPa) Stress 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-1 JISF stress-strain curves (SM490A Plate)

244 APPENDIX C. JISF EXPERIMENT

SM490A Wide Flange (1) SM490A Wide Flange (2)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490A Wide Flange (3) SM490A Wide Flange (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490A Wide Flange (5)

600 20°C 500 300°C 400°C 400 500°C 300 600°C 700°C

Stress (MPa) 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-2 JISF stress-strain curves (SM490A Wide Flange)

245 APPENDIX C. JISF EXPERIMENT

SM490AW Plate (1) SM490AW Plate (2)

600 600

500 500

400 400

300 300

Stress (MPa) Stress 200 (MPa) Stress 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490AW Plate (3) SM490AW Plate (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SM490AW Plate (5)

600 20°C 500 300°C 400°C 400 500°C 300 600°C 700°C

Stress (MPa) Stress 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-3 JISF stress-strain curves (SM490AW Plate)

246 APPENDIX C. JISF EXPERIMENT

SN490C Plate (1) SN490C Plate (2)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SN490C Plate (3) SN490C Plate (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

SN490C Plate (5)

600 20°C 500 300°C 400°C 400 500°C 300 600°C 700°C Stress (MPa) 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-4 JISF stress-strain curves (SN490C Plate)

247 APPENDIX C. JISF EXPERIMENT

STKN400B (1) STKN400B (2)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

STKN400B (3) STKN400B (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

STKN400B (5)

600 20°C 500 300°C 400°C 400 500°C 300 600°C 700°C

Stress (MPa) Stress 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-5 JISF stress-strain curves (STKN400B)

248 APPENDIX C. JISF EXPERIMENT

STKN490B (1) STKN490B (2)

600 600

500 500

400 400

300 300

Stress (MPa) Stress 200 (MPa) Stress 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

STKN490B (3) STKN490B (4)

600 600

500 500

400 400

300 300

Stress (MPa) 200 Stress (MPa) 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain

STKN490B (5)

600 20°C 300°C 500 400°C 400 500°C 600 C 300 ° 700°C Stress (MPa) Stress 200 800°C 100

0 0 0.02 0.04 0.06 0.08 0.1 Strain

Figure C-6 JISF stress-strain curves (STKN490B)

249 APPENDIX C. JISF EXPERIMENT

C.5 COMPARISON OF THE TEST DATA WITH AIJ AND EC3

Under current development of structural fire engineering, there are not many design standards that specify stress-strain curves of steel under elevated temperatures. The Architectural Institute of Japan (AIJ, 1999) defines stress-strain curves in Recommendation for Fire Resistant Design of Steel Structures. The AIJ equations are developed to provide conservative lower strengths that correspond to the mean minus three standard deviations of strength test data. Additionally, the AIJ curves are configured to provide a more realistic response for strains larger than 2%. Eurocode 3 (EC3) (CEN, 1995) is one of the most developed design standards for fire engineering and defines stress-strain curves that can be used in advanced fire engineering analysis. The stress-strain curves are established based on data from extensive testing carried out by Corus (British Steel). Kirby and Preston (1988) summarized the large amount of test data from Corus and provided useful information of steel properties under elevated temperatures. The Eurocode curves are the basis of the high temperature provisions of the American Institute of Steel Construction (AISC) and Australian design standards (AZ4100). The JISF test data are compared with the AIJ and EC3 equations. Four types of steel (SM490A, SM490A, SM490AW, and SN490C) are selected for this study as Grade 50 class rolled steel. The design yield strength of steel and elastic stiffness at ambient temperature are defined slightly differently by the various design standards, and these properties are the basis for the design equations for higher temperatures. The AIJ and EC3 equations are calculated based on steel properties defined in their respective design standards at ambient temperature. These properties are shown in Table C-2, along with those of AISC.

Table C-2 Elastic modulus and yield strength (Gr.50) defined in AIJ, EC3, and AISC AIJ EC3 AISC E (GPa) 210 210 200

Fy (MPa) 325 355 345

Figures C-7 and C-8 are the plots of the stress-strain relationships of the JISF test data, and the AIJ and EC3 equations at 300 °C to 700 °C. Figure C-7 shows curves up to 2.5 % strain and FigurecC-8 shows curves up to 10 % strain. JISF data for 800 °C are limited and not studied here. The primary range of focus in the EC3 data is on strains less than 2 %,

250 APPENDIX C. JISF EXPERIMENT because the tangent stiffness vanishes at 2 % strain in EC3 curves. This follows from the fact that Kirby’s data, upon which EC3 is based, are limited to less than 2 % strain. The AIJ equations are defined such that they can be used for strains greater than 2 %. Notable observations from Figure C-7 are: (1) The JISF data have higher variation at higher temperatures, and (2) The AIJ equation is conservative.

251 APPENDIX C. JISF EXPERIMENT

300 °C 400 °C 500 500

400 400

300 300

200 200 Stress (MPa) Stress (MPa)

100 100

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain Strain 300 °C 400 °C

500 °C 600 °C 400 200

300 150

200 100 Stress (MPa) Stress (MPa) 100 50

0 0 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain Strain 500 °C 600 °C

700 °C 100

60 JISF EC3 40 AIJ Stress (MPa) 20

0 0 0.005 0.01 0.015 0.02 0.025 Strain 700 °C Figure C-7 Comparison of stress-strain curves (up to 2.5 % strain)

252 APPENDIX C. JISF EXPERIMENT

300 °C 400 °C

600 600

500 500

400 400

300 300

Stress (MPa) Stress 200 (MPa) Stress 200

100 100

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain 300 °C 400 °C

500 °C 600 °C 250 400 200

300 150

200 100 Stress (MPa) Stress (MPa)

100 50

0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Strain Strain 500 °C 600 °C

700 °C 120

100

80 JISF 60 EC3 AIJ Stress (MPa) 40 20

0 0 0.02 0.04 0.06 0.08 0.1 Strain 700 °C Figure C-8 Comparison of stress-strain curves (up to 10 % strain)

253 APPENDIX C. JISF EXPERIMENT

C.6 STATISTICAL STUDY

At elevated temperatures, the yield plateau, which is characteristic of structural steel, is no longer clearly observed and the 0.2 % offset strength is not appropriate to define the yield strength. Alternatively, the AIJ defines the yield strength as the stress at 1 % strain (1 % strength), while EC3 defines it as the stress at 2 % strain (2 % strength), where the tangent stiffness vanishes in the EC3 stress-strain curves. Although either of the definitions can be used to evaluate the degradation of steel strength, this difference is confusing and makes it difficult to understand and share experimental data performed in various countries. A large number of tests for steel under elevated temperatures have been carried out; however, the results are unexpectedly scattered despite the well-controlled steel quality. The different definitions of the material properties or different testing procedures may have significantly influenced these results. Figures C-9 and C-10 show the stresses of the JISF test data from 300 °C to 700 °C at 1 % and 2 % strain. Four types of steel (SM490A, SM490A, SM490AW, and SN490C) are selected for this study as Grade 50 class rolled steel. The numbers of the tests are 20 for 300 °C and 700 °C, and 40 for 400 °C to 600 °C. The mean, mean plus and minus one standard deviation, mean minus three standard deviation, and coefficient of variation (c.o.v.) are plotted. The mean strength and coefficient of variation are summarized in Table 3. It is observed that the 1 % and 2 % strength is less closer at higher temperatures (i.e., the 1 % strength at 300 °C is 84 % of the 2 % strength, while the 1 % strength at 700 °C is 102 % of the 2 % strength). The c.o.v. has similar relationship with respect to the temperatures (i.e., 0.06 and 0.05 at 300 °C for 1 % and 2 % strength, respectively; and 0.18 and 0.17 at 700 °C for 1 % and 2 % strength, respectively). The AIJ and EC3 equations are superimposed in Figures C-9 and C-10 from 300 °C to 700 °C. The JISF test data agree with EC3 strengths. The 2 % strengths at 100 °C to 400 °C are generally greater than the strength at ambient temperature; however, EC3 conservatively defines constant yield strength from ambient temperature up to 400 °C. It is observed, therefore, that the 2 % strength by EC3 is lower than the test data.

254 APPENDIX C. JISF EXPERIMENT

Figure C-9 Comparison of stresses at 1 % strain

Figure C-10 Comparison of stresses at 2 % strain

255 APPENDIX C. JISF EXPERIMENT

Table C-3 Mean and coefficient of variation of 1 % and 2 % strength 1 % strength 2 % strength Temperature (°C) mean (MPa) c.o.v. mean (MPa) c.o.v. 20 365 0.061 390 0.042 300 337 0.063 399 0.052 400 310 0.062 363 0.049 500 252 0.078 276 0.077 600 146 0.097 148 0.099 700 67 0.176 66 0.172

ACKNOWLEDGEMENTS Experimental data provided by Japan Iron and Steel Federation are greatly appreciated.

256 APPENDIX C. JISF EXPERIMENT

The following Japanese paper is requested to be attached to this appendix by JISF.

Figure C-11 JISF paper (page 1)

257 APPENDIX C. JISF EXPERIMENT

Figure C-12 JISF paper (page 2)

258

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266 SYMBOLS

SYMBOLS

General Notation

X 0 and X ()T indicate that X is a function of temperature.

X 0 is a property at ambient temperature and X ()T is that at elevated temperature T .

XT is also used, if the value of X at elevated temperatures is different from that at ambient temperature and X is constant at any elevated temperature.

A = Cross-sectional area (m2) 2 AB = Cross-sectional area of unthreaded part of bolt (m )

Af = Cross-sectional area of lumped beam section (half of cross- sectional area of beam) (m2) 6 Cw = Warping constant (m )

CX = Exponent in equation for proposed critical moment DL = Dead load

DB = Diameter of bolt (m) E = Modulus of elasticity (N/m2) 2 Et = Tangent stiffness (N/m ) 2 Etave, = Average tangent stiffness in a section (N/m )

2 FBv = Ultimate shearing stress of bolt (N/m ) 2 FL = Initial yield stress (N/m ) 2 Fe = Elastic buckling stress (N/m ) 2 Fp = Stress at the proportional limit (N/m )

2 Fr = Residual stress (N/m ) 2 Fy = Yield stress (N/m )

2 Fychar, = Characteristic 0.2 % off-set yield strength (N/m ) G = Shear modulus of elasticity (N/m2) 4 I x , I y = Moment of inertia about strong and weak axis (m ) J = Torsional constant (m4)

267 SYMBOLS

K = Effective buckling length factor

K p , K y , KE = Reduction factors for the proportional limit, yield stress, and modulus of elasticity respectively

Ks = Longitudinal constraint spring stiffness for beams (N/m)

K yB = Reduction factor for bolt shear strength L = Length (m)

Lb = Unbraced length for beams (m) LL = Live load

M p = Plastic moment (Nm)

M r = Initial yield moment (Nm)

M cr, AISC , ( M crx, AISC ) = Nominal moment (about strong axis) in AISC (Nm)

M cr,3 EC , ( M crx,3 EC ) = Nominal moment (about strong axis) in EC3 (Nm)

M cr, Prop , ( M crx, Prop ) = Proposed nominal moment (about strong axis) (Nm)

M cr, e = Elastic critical moment (Nm)

M cr,tan = Nominal moment by tangent modulus theory (Nm)

M ux = Factored bending moment about strong axis (Nm)

M x,end = Bending moment about strong axis at the ends (Nm)

NB = Number of bolt at connection

PB = Peak strength of longitudinal spring for bolted connections (N)

Pc = Vertical load carrying capacity of column (N)

Pcr, AISC , ( Pcry, AISC ) = Nominal axial strength of column (for flexural buckling about weak axis) in AISC (N)

Pcr,3 EC , ( Pcry,3 EC ) = Nominal axial strength of columns (for flexural buckling about weak axis) in EC3 (N)

Pcr, Prop , ( Pcry, Prop ) = Proposed nominal axial strength of column (for flexural buckling about weak axis) (N)

Pcr,tan = Nominal strength by tangent modulus theory (N)

Pg = Gravity load (N)

Pn = Nominal axial strength of column in AISC (N)

Ps = Resistance force of vertical spring for interior column (N)

268 SYMBOLS

Pu = Factored axial load (N)

Qu = Factored load

RBn = Nominal shear strength of bolt (N)

Rn = Nominal resistance in AISC

RBtest = Experimental shear strength of bolt (N) S = Elastic section modulus (m3) 3 Sx = Elastic section modulus about strong axis (m ) Z = Plastic section modulus (m3) 3 Z x = Plastic section modulus about strong axis (m ) T = Temperature (°C)

TL = Temperature at lower flange of beam (°C)

Tcr = Critical temperature (°C)

Tsmax = Maximum temperature of steel in fire simulation (°C) dT = Increment of temperature (= T - 20 °C) (°C) bLB = Critical imperfection mode for local buckling (m) bf = Flange width of section (m) cp = Specific heat (J/kg°C) g = Limit-state function h = Height of section (m) 2 hc = Convection coefficient (W/m °C) hnet = Height of lumped beam section (m) k = Thermal conductivity (W/m°C) r = Governing radius of gyration (m) rx , ry = Radius of gyration about strong and weak axis (m) t = Time (sec)

t f , tw = Flange and web thickness of section respectively (m) ( x , y ) = Coordinate in sections (m) Δ = Vertical displacement of column (m)

ΔB = Deformation capacity of longitudinal spring for bolted connections (m)

269 SYMBOLS

ΔBp , ΔBe = Relative displacement at longitudinal spring for bolted connections at peak strength, and vanishing strength (m) α = Imperfection factor for flexural buckling in EC3, Thermal expansion coefficient

α LT = Imperfection factor for lateral-torsional buckling in EC3

α x , α y = Imperfection factor for flexural buckling about strong and weak axis in EC3 δ = Displacement (m), Coefficient of variation δ = Coefficient of variation of steel strength at 2 % strain Fy δ = Coefficient of variation of 0.2 % off-set steel yield strength FPSy , δ = Coefficient of variation of steel strength at 2 % strain at elevated FTPSy | temperatures with respect to 0.2 % off-set yield strength

δ yB = Coefficient of variation of shear strength of bolt ε = Strain, Emissivity φ = Curvature in beam section (1/m), Resistance factor in AISC λ = Slenderness ratio in AISC λ = Slenderness ratio for flexural buckling in EC3

λ LT = Slenderness ratio for lateral torsional buckling in EC3

λp = Slenderness ratio for transition between full plastic bending and inelastic lateral-torsional buckling in AISC

λr = Slenderness ratio for transition between inelastic and elastic lateral-torsional buckling in AISC

λrf , λrw = Limiting width-thickness ratio for local buckling of flange and web in AISC ρ = Density (kg/m3) σ = Stress (N/m2), Standard deviation, the Stefan-Boltzmann constant (5.67×10-8 W/m2°C4) 2 σ a = Generated stress by applied load (N/m ) 2 σ r = Residual stress (N/m ) 2 σ y = Yield stress (N/m )

270 SYMBOLS

μ = Mean μ = Mean of steel strength at 2 % strain (N/m2) Fy μ = Mean of 0.2 % off-set steel yield strength (N/m2) FPSy , μ ()T = Mean of Ratio between 2 % strength at elevated temperatures and FPSy | 0.2 % off-set strength

μKs = Mean of longitudinal spring stiffness for beams by surrounding floor framing (N/m)

μ yB = Mean of shear strength of bolt (N)

μ yBtest = Mean of tested shear strength of bolt (N) χ = Reduction factor for flexural buckling in EC3

χLT = Reduction factor for lateral torsional buckling in EC3

271