. . . . ... . .. ‘. .. ‘ . - u c u o This is to certify that the dissertation entitled Response of Restrained Steel Beams Subjected to Fire Induced Thermal Gradients presented by Mahmoud M. S. Dwaikat has been accepted towards fulfillment of the requirements for the Doctoral degree in Civil Engineering W ' ' Major Professor’s Signature Alhd' 2010 Date MSU is an Affirmative Action/Equal Oppom'Jnify Employer LIBRARY Michigan State Universrty PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5108 KIProj/Aod-PreleIRCIDateDueJndd RESPONSE OF RESTRAINED STEEL BEAMS SUBJECTED TO FIRE INDUCED THERMAL GRADIENTS By Mahmoud M. S. Dwaikat A DISSERTATION Submitted to Michigan State University in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY Civil Engineering 2010 ABSTRACT Response of Restrained Steel Beams Subjected to Fire Induced Thermal Gradients By Mahmoud M.S. Dwaikat Structural steel is frequently used in high rise building construction due to its low weight-to- strength ratio and high ductility characteristics. However, steel suffers fast degradation of its strength and stiffness properties at elevated temperature, and hence, steel structural members are to be provided with a certain level of fire protection to achieve required fire resistance. The current approaches for evaluating fire resistance of steel members have many limitations and do not take into consideration critical factors governing fire response. For instance, a restrained steel beam develops significant fire induced restraint forces when exposed to fire and these forces transform the behavior from beam to that of a beam-column. However, current design provisions do not take into consideration the effect of fire induced restraint forces, and thus beams are continued to be designed for flexural capacity only under fire conditions. To obtain test data on the influence of thermal gradients on steel beam-columns, fire resistance tests on four beam-columns were conducted as part of this study. The test variables included magnitude and direction of thermal gradients, load level, insulation scheme, and fire scenarios. Results from the fire tests show that beam-columns develop significant thermal gradients under uneven fire exposure and these gradients alter the failure mode from that predicted using conventional P—M curves. Test data were utilized to validate finite element models created using ANSYS for tracing thermal and structural response of beam-columns under fire conditions. The finite element models account for various critical factors, namely high temperature material properties, fire induced restraints and thermal gradients, and the different strain components (including high temperature creep), that have significant influence on the fire response of steel beam-columns. Once validated, the models were used to cany out detailed parametric studies to quantify the influence of critical parameters, such as load, end restraints, and fire scenario, on the fire response of beam-columns. Results from the parametric study showed that fire resistance of restrained steel beams is adversely affected by increasing load ratio and fire severity, while fire resistance gets enhanced by increasing axial and rotational restraint stiffnesses. Results from the parametric studies and from fire experiments were utilized to develop two simplified calculation methodologies at sectional and global levels for tracing the response of steel beam-columns under fire. At sectional level, a simple methodology for adjusting the conventional plastic P-M diagrams to account for the effect of fire induced thermal gradients is derived At global level, an engineering methodology is developed for evaluating the fire response of restrained steel beams utilizing equilibrium and compatibility principles. The proposed approach accounts for load, boundary conditions, fire scenarios, and beam geometry, and can be applied for evaluating fire resistance of restrained steel beams utilizing either deflection or strength limit state. A comparison of results show that the proposed approaches provide better fire resistance estimates of restrained beams than that predicted by current provisions in codes and standards The simplicity of the proposed approaches makes them attractive for undertaking a rational fire design under a performance-based environment. Application of these rational approaches in the design process of beam-columns will contribute to reliable and cost effective fire response predictions. iv DEDICATION To the grace and pleasure of Gd, the soul of my mother, and my father. ACKNOWLEDGMENT I would like to sincerely thank my advisor Prof. Venkatesh Kodur for the encouragement, guidance, and support he provided me throughout the 4.5 years of research I spent with him. Also, special thanks to the distinguished faculty members who served on my committee: Professors Ronald Harichandran, Rigoberto Burguefio, and Alejandro Diaz. Thanks to all my committee members for their support, patience, encouragement, and useful suggestions and guidance. Sincere and warm thanks to my friends who supported me during the course of my research, particularly Monther Dwaikat and Saskia Polder—Verkiel, and the rest of my friends and colleagues in the CEE department. Thanks to the faculty and personnel in the CEE department for their support and help, especially the manager of the Civil Infrastructure Laboratory Mr. Siavosh Ravanbakhsh, and the rest of the personnel. 1 and my advisor also wish to acknowledge the support of National Institute of Standards and Technology (through Building and Fire Research Laboratory Fire Grant # 60NANB7D6120), National Science Foundation (through Award# CMMI 0652292} TABLE OF CONTENT LIST OF TABLES x LIST or FIGURES Xi CHAPTER ONE INTRODUCTION 1 1.1 General 1 1.2 Fire Response of Beams and Columns 4 1.3 Current Approaches of Fire Resistance Evaluation 6 1.4 Research Objectives 9 1.5 Scope 10 CHAPTER Two STATE-OF-THE-ART REVIEW 14 2.1 General , 14 2.2 Fire Response of Restrained Steel Beams 15 2.2.1 Beam-Column Response 17 2.2.2 Thermal Gradient 18 2.2.3 Fire Scenario 20 2.3 Current Approaches for Evaluation of Fire Resistance 22 2.3.1 Prescriptive Approach 22 2.3.2 Performance Based Approach 23 2.3.2.1 Fire Scenarios 24 2.3.2.2 Loading Conditions 26 2.3.2.3 Failure Criteria 27 2.3.3 Provisions in Codes and Standards 28 2.3.3.1 Prescriptive-Based Fire Design 29 2.3.3.1 Performance-Based Fire Design 31 2.3.4 Summary 33 2.4 Previous Studies on Steel Beam-Columns under Fire 33 2.4.1 Experimental Studies 33 2.4.2 Analytical Studies 37 2.5 High Temperature Steel Properties 40 2.5.1 Thermal Properties 41 2.5.2 Mechanical Properties 42 2.5.2.1 Yield Strength and Elastic Modulus 44 2.5.2.2 Temperature-Stress-Strain Relationships 46 2.5.3 Deformation Properties 47 2.5.3.1 Thermal Strain 47 2.5.3.2 High Temperature Creep 48 2.6 Summary 49 CHAPTER THREE FIRE RESISTANCE EXPERIMENTS 60 3.1 General 60 3.2 Test Apparatus 60 3.3 Test Specimens 61 3.4 Instrumentation 63 3.5 Test Conditions and Procedure 63 3.6 Experimental Results 65 3.6.1 Thermal Response 65 3.6.2 Structural Response 68 3.6.3 Failure Pattern and Fire Resistance 71 3.7 Summary 72 CHAPTER FOUR NUMERICAL MODELING AND VALIDATION 82 4.1 General 82 4.2 Selection of Finite Element Program 82 4.3 Finite Element Modeling 83 4.3.1 General Approach 83 4.3.1 Thermal Analysis 85 4.3.2 Structural Analysis 89 4.3.3 Modeling End Restraints 91 4.4 Selection of Material Models 93 4.4.1 Thermal Properties 94 4.4.2 Mechanical Properties 94 4.4.3 High Temperature Creep 96 4.4.3.1 ANSYS Creep Model 96 4.4.3.2 Calibration of ANSYS Creep Model 99 4.4.3.3 Comparison with Other Creep Models 100 4.5 Model Validation 103 4.5.1 Validation of Thermal Response 103 4.5.1.1 Unprotected Steel 104 4.5.1.2 Protected Steel 105 4.5.2. Validation of Structural Response 107 4.5.2.2 Li and Guo Tests 107 4.5.2.1 MSU Fire Tests 110 4.6 Effect of Variation of Constitutive Material Models 114 4.7 Summary 1 17 CHAPTER FIVE PARAMETRIC STUDIES 137 5.1 General 137 5.2 Factors Influencing Fire Resistance 138 5.3 Definition and Range of Parameters 139 5.3.1 Load Ratio 139 vii 5.3.2 End Restraints 139 5.3.3 Location of Axial Restraint 141 5.3.4 Thermal Gradient 142 5.3.5 Fire Scenario 144 5.4 Results of Parametric Study 145 5.4.1 Effect of Load Ratio 146 5.4.2 Effect of Rotational Restraint 147 5.4.3 Effect of Axial Restraint 149 5.4.4 Effect of Location of Axial Restraint 151 5.4.5 Effect of Thermal Gradient 152 5.4.6 Effect of Fire Scenario 153 5.5 Summary 157 CHAPTER SIx METHODOLOGY FOR EVALUATING PLASTIC P-M CURVES UNDER THERMAL GRADIENTS 173 6.1 General 173 6.2 Evaluation of Steel Temperature 174 6.2.1 General 174 6.2.2 Exposure to Standard Fire 176 6.2.3 Exposure to Design Fire 178 6.2.4 Comparison to Finite Element Analysis 181 6.3 Plastic P—M Diagrams 182 6.3.1 General 182 6.3.2 Generation of P-M Diagrams 183 6.3.2.1 Strong Axis Orientation 185 6.3.2.2 Weak Axis Orientation 188 6.3.3 Comparison to Fire Tests 190 6.3.4 Design Application 192 6.3.4.1 Temperature Calculation 193 6.3.4.2 AISC Specifications 194 6.3.4.3 Accounting for Thermal Gradient Using the Proposed Approach ............ 194 6.5 Summary 195 CHAPTER SEVEN METHODOLOGY FOR PREDICTING FIRE RESPONSE OF RESTRAINED STEEL BEAMS . 213 7.1 General 213 7.2 Characterizing the Fire Response 213 7.2.1 Fire Induced Forces 215 7.2.2 Fire Induced Deflection 224 7.2.3 Influence of Fire Scenario 226 7.3 Comparison to Test Results 228 7.4 Comparison to Finite Element Analysis 230 7.4.1 Fire-Induced Forces 230 7.4.2 Mid-span Deflection 231 7.4.3 Deflection Limit State 234 viii 7.5 Design Applicability 235 7.6 Summary 241 CHAPTER EIGHT CONCLUSIONS AND RECOMMENDATIONS 258 8.1 Summary 258 8.2 Key Findings 259 8.3 Recommendations for Future Research 261 8.4 Research Impact 263 APPENDIX ....................................................................................... 264 REFERENCES 267 ix LIST OF TABLES Table 3.1: Matrix of fire test parameters. 73 Table 4.1: Comparison of measured fire resistance with that obtained from numerical simulations using ASCE, Eurocode and Poh constitutive relationships. ....... 118 Table 5.1: Characteristics of the beams used in parametric study of finite element analysis 158 Table 6.1: The range of parameters used in the Simulations 196 Table 6.2: Temperatures in Wl4xl76 as predicted by Eq. 6.5 and by detailed finite element analysis 196 Table 6.3: Plastic axial and moment capacities calculated as per AISC design manual.196 Table 6.4: Calculation of eccentricity of center of Stiffness and the gradient-induced shift MTG as per proposed equations 197 Table 7.1: Characteristics of the beams used in finite element analysis for comparison with proposed approach 242 Table 7.2: Fire resistance predicted using proposed approach and provisions in codes and standards under different limit states 242 LIST OF FIGURES Fig. 1.1: Types of fire exposure and fire scenario commonly encountered in buildings.. 12 Fig. 1.2: Transformation of the P-M interaction diagram under thermal gradients ........... 13 Fig. 2.1: Development of fire induced restraint forces in restrained steel beams and columns. 51 Fig. 2.2: Typical fire response of a restrained steel beam that fails by yielding ................. 52 Fig. 2.3: Types of (a) fire exposure and (b) fire scenario 53 Fig. 2.4: Changes in the Shape of the P—M interaction diagram under the influence of thermal gradients 53 Fig. 2.5: Various Stages of Fire in a Typical Compartment with common idealization... 54 Fig. 2.6: Thermal properties of steel as predicted by different models and as measured in different test programs 55 Fig. 2.7: Yield strength and elastic modulus of steel as predicted by different models and as measured in different tests 56 Fig. 2.8: Temperature-stress-strain relationships for structural Steel as per ASCE, Eurocode and Poh models 57 Fig. 2.9: Thermal strain of steel as predicted by different models and as measured in different tests 58 Fig. 2.10: Comparison of measured and predicted (using ANSYS and Harmathy models) creep strains 59 Fig. 3.1: Structural-fire test furnace at MSU’S Civil and Infrastructure Laboratory (dimensions shown in mm) 73 xi Fig. 3.2: Insulation schemes for the beam-column specimens 74 Fig. 3.3: Instrumentation mapping for the tested specimens (dimensions shown in mm) 75 Fig. 3.4: Photos of the boundary conditions for the tested beam-columns 77 Fig. 3.5: Temperatures recorded at section B-B for beam-columns 78 Fig. 3.6: Sign convention adopted for the fire induced restraint forces in beam-columns79 Fig. 3.7: Failed shape of each beam-column, shown with failure locations ......................... 80 Fig. 3.7: Continued 81 Fig. 4.1: Standard and design fire scenarios 119 Fig. 4.2: Discretization of steel sections for thermal analysis 120 Fig. 4.3: Elements and meshes that are used for discretization structural analysis .......... 121 Fig. 4.4: Restrained beam with symmetric restraint conditions 122 Fig. 4.5: Structural discretization and modeling of end restraint 122 Fig 4.6: Kinematics for simulating end restraint in a beam - 122 Fig. 4.7: Stress-strain relationships adopted for structural fire analysis 123 Fig. 4.8: Stress-strain relationships adopted for structural fire analysis 124 Fig. 4.9: Calibration of ANSYS high-temperature creep model for structural steel ........ 125 Fig. 4.10: Creep strains predicted using ANSYS and Harmathy models compared to test data 126 Fig. 4.11: Comparing predicted steel temperatures with measured values from fire tests 127 Fig. 4.11: Continued 128 Fig. 4.12: Predicted and measured temperatures in the steel beam tested by Li and Guo (2007) 129 xii Fig. 4.13: Temperature zones for the validation and thermocouple locations .................. 129 Fig. 4.14: Recorded and predicted temperatures of tested columns (section B-B) .......... 130 Fig. 4.15: Simulation of fire resistance test on restrained beam 131 Fig. 4.16: Recorded and predicted deflection and restraint force as a function of fire exposure time 132 Fig. 4.16: Continued 133 Fig. 4.17: Predicted and measured axial deformation and moment for beam-column Cl-S 134 Fig. 4.18: Predicted and measured axial deformation and moment for beam-column Cl- W 134 Fig. 4.19: Predicted and measured axial deformation and moment for beam-column C2-S 135 Fig. 4.20: Predicted and measured axial deformation and moment for beam-column C2- W 135 Fig. 4.21: Effect of using different material constitutive models on fire resistance ......... 136 Fig. 4.22: Effect of using different constitutive models on fire response of Beam # 13 136 Fig. 5.1: Assumed and real behavior of restraint stiffnesses in restrained beams ............ 159 Fig. 5.2: Connection types and location of axial restraint in steel beams 160 Fig. 5.4: Comparison between nonlinear and linearized thermal gradients in restrained steel beams 161 Fig. 5.5: Time-temperature curves for the selected fire scenarios 161 Fig. 5.6: Effect of load ratio on response of simply supported steel beam exposed to uniform 162 xiii Fig. 5.7: Effect of load ratio on response of restrained steel beam exposed to uniform heating 162 Fig. 5.7: Continued 163 Fig. 5.8: Effect of rotational restraint on response of axially unrestrained beam under uniform 163 Fig. 5.9: Effect of rotational restraint on response of axially restrained beam under uniform 164 Fig. 5.10: Effect of axial restraint on response of rotationally unrestrained steel beam under 165 Fig. 5.11: Effect of axial restraint on response of rotationally restrained steel beam under uniform heating 166 Fig. 5.12: Effect of location of axial restraint on the response of uniformly heated steel beam 167 Fig. 5.13: Effect of location of axial restraint on the response of steel beam subjected to thermal gradient 168 Fig. 5.14: Development of fire induced restraint force in a restrained steel beam ........... 169 Fig. 5.15: Effect of thermal gradient on fire response of restrained steel beams .............. 170 Fig. 5.16: Steel temperature in a restrained steel beam under different fire scenarios ....171 Fig. 5.17: Effect of fire scenario on the response of restrained beams 172 Fig. 6.1 Characteristics of typical parametric fire exposure 198 Fig. 6.2: Comparison of steel temperature predicted by Eq. 6.5 with that obtained from finite element analysis 199 xiv Fig. 6.3: Comparing predictions from proposed equations and from finite element analysis for Ts,max and ts,ma.x 200 Fig. 6.4: Variation of Es and F y as a function of steel temperature 201 Fig. 6.5: Characterizing P-M diagram for a WP section with thermal gradient in the strong direction 202 Fig. 6.6: Comparing the Shift in moment (MTG) as estimated from simplified approach with that obtained from analysis 203 Fig. 6.7: Linearized P-M diagram for WF sections with linear thermal gradient along the strong direction 204 Fig. 6.8: P-M diagrams Obtained from analysis (solid curves) and as predicted by Eq. 14 (dashed curves) for W24x76 section with different linear thermal gradients along its strong axis (AT=Ts,HF—Ts, CF) 205 Fig. 6.9: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 14 (dashed curves) for W14x311 section with different linear thermal gradients along its strong axis (AT=Ts,HF—TS,CF) 206 Fig. 6.10: Linearized P-M diagram for a WF section with thermal gradient in weak direction 207 Fig. 6.11: P—M diagrams obtained from analysis (solid curves) and as predicted by Eq. 15 (dashed curves) for W24x76 section with different linear thermal gradients along its weak axis (AT=Ts,HF—TS, CF) 207 Fig. 6.12: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 15 (dashed curves) for Wl4x311 section with different linear thermal gradients along its weak axis (AT=Ts,HF—Ts, CF) 208 XV Fig. 6.13: Measured fire induced axial force and bending moment in the tested beam- columns (Refer to Fig. 3.3 for details of tested beam-columns) 209 Fig. 6.14: Predicted P-M capacity envelopes for the tested beam-column Cl-S at various fire exposure times 210 Fig. 6.15: Predicted capacity envelop for the tested beam-column Cl-W 2 1 1 Fig. 6.16: P-M diagrams for W14x176 using uniform temperature and thermal gradient 212 Fig. 7.1: Typical fire response of a restrained steel beam that fails by yielding ............... 243 Fig. 7.2: Variation of the restraint factors (FA and FR) as a function of end restraint ratio (x) and temperature using ASCE and Eurocode steel high temperature properties ......... 244 Fig. 7.3: Development of tensile catenary force in restrained beam 245 Fig. 7.4: Approximated deflection profile at the catenary point 245 Fig. 7.5: Interpolating deflection between catenary point and point of initiation of yielding 246 Fig. 7.6: Standard and design (realistic) fire scenarios 246 Fig. 7.7: Approximated deflection profile after cooling of Steel 247 Fig. 7.8: Comparison of fire response predictions from the proposed approach with results from finite element analysis and from fire test 248 Fig. 7.9: Comparison of maximum compressive fire-induced axial force as obtained from the proposed approach and finite element analysis 249 Fig. 7.10: Comparison of predictions from propOsed approach with results from finite element analysis for Ty and Tc 250 Fig. 7.11: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different axial restraints (Kr = 0) ..... 251 Fig. 7.12: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different axial restraints (Kr = 100% EsI/L) ( symbols for lines are like previous figure) 252 Fig. 7.13: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different thermal gradients (Kr = 100% EsI/L, Ka = 10%EsA/L) 253 Fig. 7.14: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 braced beam with different locations of axial restraint (Kr = 100% EsI/L, Ka = 10%EsA/L, AT: 200°C) 254 Fig. 7.15: Comparison of predicted temperatures at deflection limit state as predicted by proposed Eq. 7.26 for LF = [/20 and [/30 with results from finite element analysis ..... 255 Fig. 7.16: Properties of the continuous beam used in the design example ......................... 256 Fig. 7.17: Steel temperature and rrridspan deflection as a function of fire exposure time for restrained beam 257 xvii CHAPTER ONE INTRODUCTION 1.1 GENERAL In high-rise buildings, steel is often the material of choice for structural framing consisting of beams and columns. Steel structures are to be designed to withstand various loads including effects arising due to fire accidents. Fire incidents and laboratory experiments have shown that fire is one of the most severe hazards to which structures are subjected during their life time. Thus, the provision of appropriate fire safety measures is an important aspect of building design in order to enhance the safety of the occupants and to control the spread of fire (NIST 2005). Fire safety can be achieved through active and passive fire safety systems. Active systems are generally activated once the fire has started and they include fire detectors, smoke control systems and sprinklers. Passive fire protection systems, which include fire resistance, are those inherent in the structure and do not require specific operation to get activated in the event of a fire. The rationale for passive fire safety systems is that structural integrity is the last line of defense when other measures for resisting fire fail. Fire resistance is defined as the duration during which a structural member (system) exhibits resistance with respect to structural integrity, stability, and temperature transmission under fire conditions (Buchanan 2002). Typical fire resistance requirements for specific building members are specified in building codes. Fire resistance issues play a crucial role in the performance of buildings and infrastructure in the event of a fire as seen in the collapse of the WTC twin towers and the damage to the Eurotunnel. Beams and columns are primary load carrying members in a framed structure. When exposed to fire, not only does steel loses its strength and stiffness at accelerated rates, but beams and columns also develop significant internal forces that transform their behavior from beams or columns into beam-columns. Limited guidance is provided in codes and standards for the design of restrained steel members to resist fire conditions. There are two main approaches through which such design is carried out, namely; prescriptive- and performance-based approaches. While the prescriptive approach is based on empirical relations that are derived from fire tests carried out under standard conditions, the performance- based approach focuses on the performance of the structural system under given load, fire, and boundary conditions. The implementation of a performance-based approach in fire design of structural members is a recent development, and therefore, most provisions provided by performance-based codes (such as the Eurocode (EC3 2005)) are an extension of room-temperature design equations. In the Eurocode for instance, the ambient temperature design equations are used to check for load-bearing capacity of the structural members under fire, but with the replacement of the material properties by those of high temperature properties. Based on this methodology, beams under fire are still structurally treated as beams under ambient temperature. The influence of fire and boundary conditions on the fire response of such structural members is not explicitly treated. Three critical issues are neglected when fire design is based on ambient temperature design equations with the material properties reduced as a function of temperature. These critical issues are the change of fire response, the change of failure mode, and the change of failure criterion. The fire response of a beam within a framed structure is different from its ambient temperature response. For instance, a beam at room temperature generally carries the applied load through flexure, however, under later stages of fire exposure; the same beam within a framed structure carries the load through a cable (catenary) mechanism. Further, as the beam at ambient temperature may fail either by flexure, shear, or buckling; under fire temperature the same beam is more likely to fail by tension or by developing excessive deflection. Also, while failure at ambient temperature is generally based on strength limit state for steel beams, deflection limit state becomes more realistic for assessing the failure of steel beams under fire conditions (Skowronski 1988 and 1990) When exposed to fire, strength and stiffness properties of steel degrade rapidly with the rise of temperature, leading to deterioration in the behavior of steel structural members and systems. Also, due to the effect of end restraints, significant forces can develop in a steel beam or column. In current design practice, these fire induced restraint forces are usually not taken into consideration because they do not exist at room temperature. Nevertheless, these fire-induced forces transform the behavior of a beam or column into that of a beam-column and alter the fire response of a structural system (Kodur and Dwaikat 2009). 1.2 FIRE RESPONSE OF BEAMS AND COLUMNS Under fire conditions, temperatures attained in steel lead to both degradation in material properties and generation of additional internal forces that do not exist at room temperature. These fire induced restraint forces (P & M), which can often reach 30% of the room-temperature capacity, transfer the behavior of the beam (or column) to that of a beam-column (Kodur and Dwaikat 2009). Generally, beam- columns are defined as members that are subjected to an axial load (P) and a bending moment (M). At room temperature, the applied forces (P and M) are assumed to remain constant and can be computed directly from the applied loads (gravity, snow, wind, or seismic loads). However, under fire conditions, the fire induced restraint forces (P and M) are a function of a number of factors including the characteristics of the beam (or column), and type of fire exposure, and the magnitude and duration of these forces vary with time. The response of a beam-column primarily depends on the interaction between bending moment (M) and axial force (P). The combination of P & M affects the load carrying capacity of beam-columns which is generally expressed, in codes and standards, using the following relationship at ambient conditions: P + (:2 —M— S 1.0 QDP QM “'11 n n C1 where c1 and c2 are interaction coefficients, and (DMn and ¢Pn are the moment and axial capacities, respectively. Since there are no validated expressions or methodologies for computing the fire—induced P & M acting on restrained beams (or columns), Eq. [1.1] cannot be applied under fire conditions. Therefore, one of the main objectives in this research is to quantify the critical values for the P & M that emerge as a result of fire exposure. Another aspect is the development of fire-induced thermal gradients across the cross-section of a beam-column due to 1-, 2-, or 3-Sided fire exposure, as shown in Fig. 1.1(a). Generally, 4-sided fire exposure leads to nearly uniform temperature within the cross section, while 1-, 2-, or 3- sided fire exposure tends to generate thermal gradients in the cross section. For example, a restrained column subjected to a thermal gradient (as in perimeter columns exposed to fire from one side) develops a bending moment in addition to the axial force. Similarly, a restrained beam subjected to a thermal gradient (since beams generally support a concrete slab and thus fire exposure is from 3 sides) develops an axial force in addition to the bending moment. These fire induced thermal gradients can severely distort the plastic P-M diagram (Pn and Mn in Eq. [1.1]) due to varying strength and stiffness properties of steel across the depth of the section. This “nonhomogeneity” within the steel section produces a distorted P-M interaction diagram that is different from that of a “homogenous” steel section at room temperature (Garlock and Quiel 2008). Further, the type of fire scenario has a Significant influence on the development of restraint forces in beam-columns. Much of the current knowledge is based on fire resistance tests on restrained columns or beams where the exposure is typically assumed to be that of a standard fire. However, in buildings the fire exposure is likely to be different and dependent on fuel load and ventilation characteristics in the fire compartment (Buchanan 2002). Fig. 1.1(b), illustrates the difference between standard fire and realistic (design) fire scenarios. The main difference between the two is that in standard fire exposure, the temperature of the fire is assumed to increase without any decaying (cooling). However, fires in buildings die down after reaching a peak temperature, and the compartment enters a cooling phase as a result of exhaustion of fuel and/ or ventilation (oxygen). This decay phase results in cooling of steel and thus development of tensile stresses due to the shrinkage of steel. This aspect of the cooling phase has significant influence on the fire response of structural members, especially on restrained members such as beam-columns. There is a very limited number of studies on the effect of this cooling phase of fire on the structural response of steel members in general (Li and Guo 2006 and 2008), and beam-columns in particular. 1.3 CURRENT APPROACHES OF FIRE RESISTANCE EVALUATION There are many drawbacks in the current approach for evaluating fire resistance of beam-columns. The fire resistance of a restrained beam is evaluated by a mere reduction in strength properties and without any consideration to restraint forces that develop under fire conditions. This is mainly due to the lack of methodologies for quantifying the internal forces that develop in restrained beams (or columns). In addition, the current fire resistance approaches for steel beams and columns are based on fire tests which were carried out on specimens of standard sizes under service loads, simply supported end (unrestrained) conditions, and standard fire exposure which does not reflect reality. Furthermore, in the current approaches, the failure is often based on a thermal (prescriptive) criterion, which assumes a member to fail onCe steel reaches a “critical" temperature, regardless of loading, restraint conditions, or fire scenario. Further, the effect of fire induced restraints on steel members is not fully accounted for in current code provisions. Therefore, the current prescriptive approaches for evaluating fire resistance do not represent realistic fire, loading, and restraint scenarios as encountered in practice. This makes the current design approaches inapplicable for use under the recently introduced performance-based codes that seek to facilitate innovative, cost-effective and rational designs. In a performance-based environment, the response of beam-columns is based on realistic performance criteria (Wang and Kodur 2000). The failure criteria may vary depending on the function of the steel member. For example, for a restrained beam under fire, a deflection limit state may be more critical than achieving full yield capacity, while for a column, a strength (or buckling) limit state can be more critical. Excessive deflection can not only severely affect the structural integrity; it can also accelerate the spread of fire to other compartments. ArlOther limitation of the current design approaches lies in the evaluation of the cap acity of beam-columns. The ultimate capacity of beam-columns is assessed using P'M interaction diagrams. These interaction diagrams are generated based on the aS'sumption of uniform temperature. This means that a constant yield stress, Fy, is assumed through the depth of the section when predicting the capacity of beam- columns (e.g., AISC 2005, EC3 2005). However, under real fire exposure, thermal gradients are likely to emerge and they can alter the shape of the P-M interaction diagram. These gradients are significant in the case of beams and columns under 1-, 2-, or 3-side exposure (See Fig. 1.1(a)). Figure 1.2 illustrates the plastic P-M interaction curve for a W24x76 beam section under different thermal gradients. The average temperature of the section (across the depth) is maintained at 500°C, while the thermal gradient was increased from zero to 150°C. The interaction curves are computed through direct integration of ultimate stresses across the cross section for each case of thermal gradient. The values of the moment and axial force are normalized to the plastic moment and axial capacities of the section at each case of thermal gradient. It can be seen from Fig. 1.2 that: 1) Using constant minimum Fy (constant maximum T of 650°C) may be overly conservative as seen by comparing this yield surface to the true yield surfaces at different thermal gradients; and 2) the true plastic P-M interaction curve (that considers variable Fy through the section as a result of thermal gradient) shifts and departs from the shape of the curve considering constant average Fy. This shift and distortion of the PM diagram depends on the level of thermal gradient in the Se<=tion. Fig. 1.2 also shows that the assumption of constant (i.e. uniform) average Fy [AT = 0) through the depth of the section can be conservative in some cases and nOrlconservative in other cases. The response of beam-columns was closely examined in WTC twin towers collapse Studies. Based on these investigations it was recommended that the behavior of steel beam-columns is an urgent research need since they are critical components of structural frames in high rise buildings and other‘built infrastructures (NIST 2005). The limited studies undertaken in literature on fire response of beam-columns did not properly account for many important factors, such as the fire scenario, thermal gradient, restraint configuration, and high-temperature properties. Also, there is a lack of understanding of the interaction between fire induced internal forces and capacities, such as between bending moment and axial force and the respective capacities. Knowledge of such interactions is critical for the proper design of beam- columns under fire conditions. 1.4 RESEARCH OBIECTIVES From the above discussion, it is Clear that there is a considerable lack of knowledge of the fire response of steel beam-columns under realistic fire, loading, and restraint scenarios. To address this knowledge gap, it is proposed to undertake a comprehensive study of the structural response of restrained steel beams under fire conditions with the ultimate objective of developing a rational methodology for evaluating the fire resistance of restrained steel beams. The specific objectives of the re 8 earch are: - Conduct a detailed state-of-the-art review on the fire response of restrained steel beams. The comprehensive review will cover experimental and numerical studies, provisions in codes and standards, and high temperature material properties. - Model the thermal and structural response of restrained steel beams under realistic fire, loading, and failure conditions. The models for thermal and structural analysis will account for nonlinear high temperature material properties, various strain components and fire induced restraint effects. - Conduct fire resistance tests on steel beam-columns under realistic fire, loading and restraint scenarios to generate data for the validation of the models. I Verify the computational models using data from fire resistance tests on steel beam-columns. - Carry out parametric studies to quantify the influence of various factors on the fire resistance of restrained steel beams. . Develop a rational design methodology, using data from fire tests and parametric studies, for evaluating the response of restrained steel beams ' under realistic fire, load, restraint, and failure limit states. 1 - 5 SCOPE The research, undertaken to address the above objectives, is presented in eight Chapters. Chapter 1 gives a general background to the fire response of steel beam- COIUmnS and presents the objectives of this study. Chapter 2 provides a state-of-the- art review on the behavior of steel beam-columns exposed to fire. The review inchrdes a summary of experimental and analytical studies, as well as the fire design Provisions for restrained steel beams in current codes of practice. A review of the 10 high temperature material properties and associated constitutive relationships for structural steel is also presented in Chapter 2. Chapter 3 deals with fire resistance experiments on four steel beam-columns under realistic fire, loading and restraint scenarios. Results from these fire tests are used to discuss the response of steel beam-columns under these realistic conditions. Chapter 4 presents details of finite element modeling of the fire response of steel beam-column beams. The validation of the finite element models (thermal and structural) is also presented in Chapter 4, where predictions from the model are compared with available test data from literature, and also with results from the fire tests presented in Chapter 3. Parametric studies are presented in Chapter 5. The definitions and ranges of parameters governing the fire resistance of restrained steel beams are described in Chapter 5, followed by a discussion of the results from the parametric studies. In Chapter 6, a methodology for evaluating the P-M diagrams of steel beam-column under fire induced thermal gradients is presented. In Chapter 7, simplified equations for predicting the response of restrained steel beams under fire are developed. The proposed equations in Chapter 6 and 7 are verified using results from fire resistance tests and from the parametric studies. Finally, Chapter 8 SuITZImarizes the main findings arising from the current study and lays out reCommendations for further research. 11 4-side exposure 3"?» ’3'? r it P It ;? 3'. v gr I» 3-side exposure a t E i 2-side exposure E at 3’ E £52 “an. A)“. I 1-srde exposure E I E AAA (0) Fire exposure types ISO 834 standard fir 1000 _ .U 800 - f 2“ 600 _ ASTM E119 standard fire :3 \ H Ex 1 34°01 .15."? 32‘? a 200 g r3 O I I I m r I I I 0 20 40 60 80100120140 Time, min (b) Fire scenarios Fig. 1.1: Types of fire exposure and fire scenario commonly encountered in buildings 12 1.2 of 1 3 AT = O C AT = 25°C 03 AT = 50°C AT = 100°C 0-6 AT = 125°C 0.4 AT = 150 c 1 2 Constant T = 650°C -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 Fig. 1.2: Transformation of the P-M interaction diagram under thermal gradients 13 iii: is s ,. .aJ J CHAPTER Two STATE-OF-THE-ART REVIEW 2.1 GENERAL A restrained beam in a framed steel structure, when exposed to fire, can develop significant internal forces that transform its behavior to that of a beam-column. This Change in behavior and resulting failure patterns of restrained beams is often neglected in the design and analysis. Current provisions in codes and standards do not address this change in behavior under fire conditions, and therefore, restrained beams in a framed structure are often treated as simply supported beams (AISC 2005, EC3 2005, ASCE 1992). Since 1960’s, few experimental and numerical studies have shown that the fire resistance of steel structures is critically influenced by a number of important factors, including load, fire, and restraint conditions. While mOSt of these studies focused on individual beam or column responses, very limited Studies addressed the more likely beam-column behavior. In this chapter, the typical fire behavior of steel beam-columns, with emphasis on restrained steel beams, is first discussed, followed by a critical review of design provisions in codes and Sta‘ll‘ldards. for evaluating their fire resistance. Also, a summary of previous 9x13 erimental and analytical studies on the fire performance of restrained beams or Collens is presented. Further, since high temperature properties are critical for understanding fire response of steel structures, the variation of high temperature 14 material properties of structural steel in material property tests and in constitutive relationships are also discussed. 2.2 FIRE RESPONSE or RESTRAINED STEEL BEAMS The behavior of steel structures under tire is quite different from that at room temperature (Kodur and Dwaikat 2009). This may be attributed to many factors that influence the response under fire conditions. Beams are primary load-bearing members typically used in a steel framed structure, and are often restrained from free expansion. This restraint condition of the steel beams under fire leads to generation of significant internal forces that are nonexistent at ambient temperature. Combined with the degradation in steel strength and stiffness as temperature increases, the effect of restraint under fire generally transforms the structural response of a steel beam to that of a beam—column. To illustrate the development of fire-induced forces in a restrained beam (or column), a steel frame as shown in Fig. 2.1(a) is exposed to fire. In current fire design practice, this type of structure is modeled as a simply supported beam (or column) (Fig. 2.1(b)). However, due to the restraint offered by the adjacent members, the beams and columns are likely to behave as restrained members (Fig. 2-1 (c, (1)). When the beam (or column) expands under increasing temperatures, the adj acent members impose both axial and rotational restraints (Ka and Kr) on the beam (or column) that prevent it from free expansion. This generates axial force (P) and bending moment (M) as seen in Fig. 2.1 (c, d). Also, with the increase of temperature, steel loses its strength and stiffness properties. Therefore, both the 15 midspan deflection (A) and the end-rotation (6) of the beam (or column) gradually increase with the increase in steel temperature. Further, the fire induced axial force (P) causes additional bending moment on the beam (or column) due to P-A effect, as seen in Fig. 2.1(c, d). The increase in the bending moment (M) produces additional deflections and hence further deterioration in the response of the beam (or column) under fire. Furthermore, the increasing temperature and stress in steel accelerate the development of high temperature creep deformations leading to rapid increase in deflection and ultimately failure of the member. The general discussion presented above can be illustrated specifically for a restrained steel beam exposed to fire. The fire response of a restrained steel beam generally undergoes three distinct stages, as shown in Fig. 2.2. In Stage I, elastic response dominates the behavior wherein the beam expands as a result of continuous heating, and compressive axial force and bending moment develop due to the effect of end-restraints. Fire-induced internal forces and deflections continue to increase until yielding or buckling occurs in the beam. For the purpose of illustration, the behavior shown in Fig. 2.2 is for the case where no buckling occurs in the beam, and therefore, elasto-plastic response dominates the behavior in Stage "- as shown in Fig. 2.2(b). As steel temperature continues to increase with fire e"‘DOsure time, softening of steel causes larger deflections and rotations until the first plastic hinge develop in the beam. The plastic hinge, which forms at the location of maximum bending moment in the beam, causes further increase in deflection as ShOWn in Fig. 2.2(c), which leads to a gradual transformation of the fire induced akia] force from compressive to tensile force. The beam enters a catenary phase in 16 ———— Stage 111 when the fire-induced compressive axial force vanishes as shown in Fig. 2.2(b). In the catenary phase, tensile force develops in the beam and the load bearing mechanism gradually Changes form flexural to cable (tensile) until failure occurs by rupture of the beam (or in the connections). As the beam undergoes the above three stages, it is assumed that the connections continue to perform elastically. Therefore, fire-induced forces and rotations in the beam must not exceed the connections capacity. During the above process, the magnitude of response of the beam (fire induced deflection and restraint forces) is affected by many factors and is quite different than the response at room temperature. The beam-column response of either restrained steel beam or restrained steel column under fire is affected by three main aspects, namely; a) the development of fire induced restrained forces (P & M), b) the influence of thermal gradient, and c) the influence of fire scenario. The three aspects are discussed in the following pa ra graphs. 2.2. 1 Beam-Column Response The fire induced restraint forces (P & M) in a beam-column, can often reach 30% of the room-temperature capacity of either the beam or the column, and these forces transfer the behavior of the beam (or column) to that of a beam-column (Kodur and DWaikat 2009). At room temperature, the applied forces (P and M) are assumed to rerl'lain constant and can be computed directly from the applied loads (gravity, Snow, wind, or seismic loads). However, under fire conditions, the fire induced restraint forces (P and M) vary as a function of a number of factors including the 17 ‘- geometry of the beam (or column), the boundary conditions, applied load and fire characteristics. The response of the beam-column primarily depends on the interaction between bending moment (M) and axial force (P). The combination of P & M affects the load carrying capacity of beam-columns which is generally expressed using the following relationship at ambient conditions: P M CIE'tCzW—Sll) [2.1] or or where c1 and c2 are interaction coefficients, and Mcr and Pcr are the moment and axial critical capacities, respectively. The applied forces (P & M) in Eq. [2.1] are generally the second order values that result due to geometric nonlinearities. However, Eq. [2.1] cannot be applied as is under fire conditions since there are no validated expressions or methodologies for computing the fire-induced P & M acting on restrained beams (or columns). Also, Eq. [2.1] is valid for uniform temperature (uniform strength stresses) distribution in the beam-column. As it will be explained in the following section, Eq. [2.1) does not properly account for the influence of thernial gradient across the cross section of the beam-column. 2.2.2 Thermal Gradient when a beam-column is exposed to fire, significant thermal gradients can develop ac31‘055 the cross-section due to 1-, 2-, or 3-sided fire exposure, as Shown in Fig. 2‘3 (0). Generally, 4-sided fire exposure produces nearly uniform temperature, while 1" 2-, or 3- sides of fire exposures can generate thermal gradients in the section. For 18 example, a restrained column subjected to a thermal gradient (as in perimeter columns exposed to fire from one side) develops a bending moment in addition to the axial force. Similarly, a restrained beam subjected to a thermal gradient (since beams generally support a concrete slab and thus fire exposure is from 3 sides) develops an axial force in addition to the bending moment. The influence of thermal gradients is partially considered in some codes of practice by carrying out cross-sectional analysis. The Eurocode (EC3 2005) accounts for thermal gradient by applying numerical integration (sectional analysis) for the axial and moment plastic capacities of the section only. However, the influence of thermal gradient on the shape of the interaction equation (2.1] is not addressed at all. Due to uneven heat distribution in the section, strength and stiffness properties of steel vary across the depth of the section. This variation of strength and stiffness of steel leads to an eccentricity between the center of stiffness and center of geometry of the cross section. Because of this gradient-induced shift in the center of stiffness, the axial force (P) will act eccentrically on the section, and thus generates bending moment. This bending moment can cause a shift in the plastic P-M diagram. Therefore, this migration of center of stiffness causes a distortion in the plastic P-M interactive diagram. Figure 2.4 illustrates the plastic P-M interaction curve for a W24x76 beam section unCler different thermal gradients. The average temperature of the section (across the depth) is maintained at 500°C, while the thermal gradient was increased from Zero to 150°C. The interaction curves are computed through direct integration of ulti mate stresses across the cross section for each case of thermal gradient. The 19 values of the moment and axial force are normalized to the plastic moment and axial capacities of the section at each case of thermal gradient. It can be seen from Fig. 2.4 that: 1) Using constant minimum Fy (constant maximum T of 650°C) may be overly conservative as seen by comparing this yield surface to the true yield surfaces at different thermal gradients; and 2) the true plastic P-M interaction curve (that considers variable Fy through the section as a result of thermal gradient) shifts and departs from the shape of the curve considering constant average Fy. This shift and distortiOn of the P-M diagram depends on the level of thermal gradient in the section. Figure 2.4 also shows that the assumption of constant (i.e. uniform) average Fy (AT = 0) through the depth of the section can be conservative in some cases and unconservative in other cases. In a similar fashion, thermal gradient can also cause a shift in the shear center of the section, and this shift influences the lateral torsional buckling capacity of steel beams exposed to realistic fire exposures. This issue of shifting center of stiffness, which can have a significant influence on beam-column equation, is not treated in almost all design codes or standards. 2.2-3 Fire Scenario The type of fire scenario has a significant influence on the evolution of steel temperature and on the development of restraint forces in beam-columns. Much of the current knowledge is based on fire resistance tests on restrained columns or beams where the exposure is typically assumed to be that of a standard fire. Ho‘Never, in buildings the fire exposure is likely to be different and dependent on 20 :1 fuel load and ventilation Characteristics in the fire compartment (Buchanan 2002). Fig. 2.3(b), illustrates the difference between standard fire and realistic (design) fire scenarios. The main difference between the two is that in standard fire exposure the temperature of the fire is assumed to increase without any decaying (cooling). However, fires in buildings die down and the compartment enters a cooling phase as a result of exhaustion of fuel and/or ventilation (oxygen). This aspect of cooling phase has significant influence on the fire response of structural members, specially restrained members such as beam-columns. While under standard fire steel properties continue to deteriorate due to increasing temperature, the decay phase of a design fire leads to a recovery of strength and stiffness in steel as a result of decreasing temperature in steel. Also, the decay phase of a design fire leads to cooling and contracting of steel and this contraction causes an increase in tensile stresses in the beam and thus leads to recovery of some of the deformations in the beam. The total recovered deformation in the beam depends on the stress state of steel prior to the start of cooling of the beam. There is a lack of information on the behavior of steel in the cooling phase under realistic fires. There is a very limited number of studies on the effect of this cooling phase of fire on the StrLlCtural response of steel members in general (Li and Guo 2006 and 2008), and bearn-columns in particular. 21 2.3 CURRENT APPROACHES FOR EVALUATION or FIRE RESISTANCE Generally, there are two broad approaches through which the fire resistance of steel members is evaluated, namely; prescriptive- and performance-based approaches. A brief discussion on these two broad approaches is presented below. 2.3.1 Prescriptive Approach In this approach, the fire resistance is evaluated based on standard fire tests. The purpose of the standard fire test is to determine the failure time (fire resistance) at which the structure/assembly loses its load bearing and/or compartmentation ability to withstand fire exposure. Generally, in standard fire tests, the failure is based on a thermal (prescriptive) criterion, which assumes a member to fail once steel reaches a “critical” temperature, regardless of loading, restraint conditions, or fire scenario. The critical temperature as per ASTM E119 is defined as the temperature at which steel loses 50% of its room-temperature yield strength (ASTM E1 19a 2008). Therefore, standard fire tests are a comparative tests and do not reflect actual performance of the member. Standard fire resistance tests are generally carried out on building elements such as Walls, floors, beams, or columns in accordance with national standards such as ASTM E119a (2008) and ISO 834 (1975). Test specimens are constructed in a SirI'lilar manner as the building elements in practice. ASTM E119 specifies the dimensions of the test specimen and the size of the furnace being used for the Sta hdard fire test. Further, the furnace chamber is heated by liquid fuel or gas such 22 that the average temperature in the furnace follows the Specified standard temperature-time curve shown in Fig. 2.3(b). During the fire test, load bearing members, such as a steel beam, are loaded with service loads (dead load + live load). A common practice in standard fire tests is to apply a service load level (about 50% of the room temperature capacity) on the tested member. The test continues until failure occurs by exceeding the critical temperature or when the structural member is no longer able to sustain the applied load. Based on these standardized tests, tabulated data or empirical relations are derived for fire resistance. Therefore, the current prescriptive approaches that are based on standard tests do not account for realistic fire, loading, and restraint scenarios as encountered in practice. This makes the prescriptive design approaches in conflict with the recently introduced performance-based codes as it will be explained in the following section. 2.3.2 Performance Based Approach In recent years, there has been an increased focus on moving towards performance- based fire safety design from the current prescriptive-based approaches (Meacham and Custer 1992, Kodur 1999, and Wang and Kodur 2000). This is mainly due to cost-effective and rational fire safety solutions that can be arrived at through performance-based design approach. One of the key aspects in any performance- based design is the fire resistance design of structural members. At present, there is limited information or tools (design equations) that can be applied for performance- 23. based fire safety design of restrained steel beams. This is due to the lack of both experimental and analytical studies on the fire response of restrained steel beams. In the development of the performance-based approach for evaluating the fire resistance of steel beams, three main factors must be considered, namely, (1) fire scenario, (2) loading conditions and (3) failure criteria (Kodur 1999). These three factors are discussed in the following sections. 2.3.2.1 Fire Scenarios The development of a fire scenario in a building compartment, which is commonly referred to as “a room fire" involves three stages; namely, growth, burning, and decay stages as shown in Fig. 2.5. In the growth stage, the temperature of the fire gradually increases and the fire spreads Slowly over combustible surfaces within the room. Generally, in this stage, the structural integrity is not significantly influenced because the fire temperature does not reach high levels. However, once the fire temperature reaches about 600°C, flashover occurs and the fire enters the intense burning stage, where the temperature may exceed 1000°C. In the burning stage, the temperatures and the radiant heat flux are so high within the room that all exposed surfaces will ignite and burn. The structural integrity may be severely damaged at this stage. Fires that pass the flashover threshold are commonly referred to as “post flashover" fires. Eventually, the fire enters the decay stage where the fire temperature drops once the fuel burns out or there is a lack of ventilation (oxygen) (Buchanan 2002). Often, for structural analysis, the fire growth is idealized through time-temperature relationships, as shown in Fig. 2.3(b) and Fig. 2.5. 24 The development of a fire scenario is quite complex since it is dependent on a number of factors. For ease of calculation, a standard time temperature curve is often used to represent a room fire scenario in various standards (ASTM 8119a 2008, EC1 2002, ISO 834 1975). Such standard fires are used to define the temperature profile that is generally used in standard fire tests as discussed earlier. While standard fire resistance tests are useful benchmarks to establish the relative performance of different steel members under the standard fire condition, they Should not be relied upon to determine the survival time of steel members under realistic fire scenarios. Nor does the standard heating condition bear any relation to the often less severe heating environments encountered in real fires. Figure 2.3(b) illustrates the time-temperature curves for a standard and two realistic design fire scenarios. In the standard fire (ASTM E119a 2008), the fire Size is the same (irrespective of compartment characteristics), temperature increases with time throughout the fire duration, and there is no decay phase. This standard fire does not represent an actual fire scenario in a building. While the temperature in a standard fire continues to increase indefinitely with time, a natural room-fire starts to gradually cool down as Shown in Fig. 2.5. Room- fires are generally expressed in terms of “design or realistic” fires. A typical design fire can be mathematically represented by a parametric time-temperature curve that is dependent on the fuel and ventilation characteristics of fire compartment Examples of these parametric fires are those Specified in Eurocode (BC1 2002) and one of them is shown in Fig. 2.5. In the Eurocode, the heating phase of a design fire is Specified parametrically in terms of the factor I‘ which is dependent on the 25 ventilation characteristics and thermal inertia of fire compartment For I‘ = 1.0 the heating phase of a parametric fire represents the ISO 834 standard fire. Higher the value of I‘ represents more severe fire conditions with faster rise of temperature in fire compartment. In these idealized parametric fires, the decay phase is typically linear with a constant decay rate that is also dependent on the ventilation characteristics, as shown in Fig. 2.3 (b) and Fig. 2.5. 2.3.2.2 Loading Conditions The current provisions in codes of practice for evaluating fire resistance are generally based on a load ratio of about 50%. Load ratio is defined as the ratio of the applied load on the beam under fire conditions to the strength capacity of the beam at room temperature. The load ratio depends on many factors including the type of occupancy of the building, dead-load-to-Iive-load ratio, safety factors (load and capacity factors) used for design under both room temperature and fire conditions. The loads that are to be applied on a steel beam, in the event of fire, can be estimated based on the guidance given in design codes. The ASCE-07 (2005) and Eurocode (EC1 2004) have different specifications for the critical load combination under fire conditions (wj), and these are: {ASCE-07 2005}: wf = 1.2 dead load + 0.5 live load [2.2a] {EC1 2004 }: wf = 1.0 dead load + 0.5 live load [2.2b] Different combinations are Specified for different cases of load types. Based on these guidelines for load combinations, and for typical dead-load-to-live-Ioad ratios (in 26 the range of 2 to 3), the actual load ratio typically found under fire conditions ranges between 50% and 70%. 2.3.2.3 Failure Criteria The conventional prescriptive approach of evaluating fire resistance (defining failure under fire conditions) is based on thermal and/or strength failure criteria as Specified in ASTM E119a (2008). Accordingly, the thermal failure of a steel beam is said to occur when at least one of the following criteria is met: - The average temperature in steel exceeds the critical temperature, which is 538° C (1000° F) for ASTM A36 structural steel. - The beam is unable to resist the applied service load (50% of ambient temperature capacity). The deflection and rate of deflection play a crucial role on the response of steel beams exposed to fire. In fact, a deflection limit is often applied to check the status of steel beams (serviceability limit state) at ambient conditions (AISC 2005 and EC3 2005). This criterion Should be considered to determine failure under fire conditions. This is because generally, strength failure in a steel beam exposed to fire is reached after undergoing large deflections. The resulting deflections in fire scenarios are generally higher than those at room temperature due to deterioration of member stiffness and also due to temperature-induced creep. The British Standard (BS 476 1987) contains deflection and rate of deflection criteria for defining failure of steel beams tested in a furnace. Although these deflection limit States might have been set to limit damage to the furnace during fire tests, deflection 27 and rate of deflection can be important under certain fire conditions. This is because the integrity of the structural member cannot be guaranteed with excessive deformations. Moreover, fire resistance based on limiting deflection will help to facilitate the safety of fire fighters and also to safely evacuate occupants prior to structural collapse. According to BS 476, failure is assumed to occur when: 0 The maximum deflection (A) of the beam exceeds L/20 (or L/30 in some cases) at any fire exposure time, or o The rate of deflection (€15?) exceeds the limit given by the following expression: dA 1.2 dt = 900061 (mm/min) [2.3] where L = span length of the beam (mm), and d = effective depth of the beam (mm). 2.3.3 Provisions in Codes and Standards Provisions for evaluating the fire resistance of steel members are generally specified in codes and standards such as American Institute for Steel Construction (AISC 2005) or American Society of Civil Engineers (ASCE 1992), or the Eurocode Standards (EC3 2005). Based on the two broad approaches described earlier, the current provisions in codes and standards can be split into two main categories, namely; prescriptive- and performance-based provisions. While the prescriptive approach is predominantly used in the American standards, the European standards 28 use a performance-based procedure for evaluating fire resistance. There is still extensive research taking place for transforming the current design provisions into purely performance-based approach. 2.3.3.1 Prescriptive-Based Fire Design Most of the provisions in codes and standards share a common strategy of computing a critical temperature of steel member, and use that critical temperature for evaluating fire resistance. The critical temperature is defined as the temperature at which steel loses 50% of its room-temperature yield strength. According to ASTM E119, critical temperature of 538°C (1000°F) is generally used in for structural steel. Based on this definition of critical temperature, empirical relations are derived from standard fire tests for computing fire resistance of various steel sections with various insulation schemes. These prescriptive-based relations are derived from standardized fire tests and sectional seizes and generally depend on the sectional geometry and insulation scheme. An example of such relations is the following formula provided by the International Building Code (IBC 2000) section 720513 for evaluating fire resistance of I-shaped steel columns insulated with spray applied fire proofing material: A 4.86 =[c,[v_r’]+c,]x.p [2.41 p where ,II‘PC is the fire resistance (in minutes) according to IBC equation, Ap/Vp is the ratio of the heated perimeter (Ap) to the cross sectional area (Vp) of the steel section, and tp is the insulation thickness. C1 and C2 are regression coefficients. 29 Equation [2.4] is used to either evaluate fire resistance, or the required insulation thickness for achieving a specific fire resistance rating. Such equations are too simplistic and are based on critical temperature obtained from standard fire tests with standard test conditions. Due to its Simplicity, the prescriptive design approach continues to be used in codes adopting performance-based approach such as the Eurocode (EC3 2005), New - Zealand Standards (SNZ 1997), and japanese Building Code (Harada et al. 2004). In these codes and standards, the concept of critical temperature has been Slightly modified to be the average temperature at which steel member (rather than steel material) loses 50% of its room-temperature capacity. Empirical or semi-empirical formulas are provided in these codes to evaluate the critical temperature. For instance, Eurocode 3, New Zealand and japanese steel design codes provide the following relations for computing the critical temperature (Tcr) of steel beams: 1 TEC3 = 39.191r{ —1:|+ 482 6’ 0.967x r3333 [253] T3” =905 —690x r W] chrBC = 700— 375x r me] where the load ratio (r) is defined as the ratio between the bending moment (Mo) resulting from reduced load during fire to the room-temperature plastic moment capacity of the steel beam (Mp). Obviously, in these equations and the like, the influence of restraint conditions and fire is not properly accounted for. 30 2.3.3.1 Performance-Based Fire Design The advanced calculation methods in codes adopting performance-based approach evaluate the fire resistance based on strength criteria. The implementation of performance-based approach in fire design of structural members is recent, and therefore, most fire design provisions are based on the room-temperature design equations. In the Eurocode for instance, the ambient temperature design equations are used to check for load-bearing capacity of the structural members under fire but by updating steel properties as a function of steel temperature. For example, for fire design of a steel beam according to EC3, the designer must ensure that the MEC3 temperature-dependent bending capacity ( cr,T ) of the steel section is greater than the imposed action (M) on the beam due to loading, i.e.: EC 3 M cr,T Z M [2.6] EC3 In the relation above, M cr,T is generally evaluated by using the same room- temperature design equations but with reduced steel properties. The imposed action (M) is also evaluated based on reduced load combination at the event of fire. Based on this methodology, beams at ambient temperature are treated structurally as beams under fire. The influence of fire and boundary conditions on the fire response of such structural members is not considered. Generally, three critical issues are neglected when the fire design is based on using ambient temperature design equations, but with updated reduced steel properties. 31 These critical issues are the change of Structural response, the change of failure mode, and the change of failure criterion. Change in structural response occurs due to the development of fire induced forces in the restrained steel beams or columns within framed structure. For instance, a beam at room temperature generally carries the applied load through flexure, however, under fire conditions; the same beam in a framed structure (where it is very likely to be restrained) carries the load through either arching or tensile cate nary mechanisms at later stages of fire exposure. Change in failure mode can occur due to the effect of restraint conditions. For instance, a beam at ambient temperature may fail either by flexure, shear, or buckling (global or local), while under fire conditions the same beam is more likely to fail by tension or by developing excessive deflection. Change in failure criterion results as a direct consequence of the change in structural response and in failure mode mentioned above. While at ambient temperature, strength failure is considered as an ultimate limit state, however, under fire conditions, and due to high ductility of steel, strength failure is an impractical ultimate limit state criterion since strength failure is reached after undergoing very large deflection. Thus, an ultimate limit state based on deflection might be a better assessment for restrained beams under fire. Most fire tests and aCtual fire incidents have Shown that fire resistance of restrained beams is generally governed by deflection criteria rather than strength criteria (Skowronsky 1990 and 1988, Kirby 1997, Newman et al. 2000, Wang and Kodur 2000, Wang 2000). Large deflection in steel beams can cause severe damage to structural integrity of the 32 —=r framed building and thus may hinder evacuation of people and facilitate the spread of fire. 2.3.4 Summary The fire resistance of steel members is currently evaluated through prescriptive- based approaches that are derived based on standard fire tests and thus have limited application. Realistic fire resistance assessment can be obtained through a performance-based approach in which fire scenario, load level, restraint effects, and failure criteria, can be accounted for based on actual (rather than standard) scenarios. Presently, there is lack of design approaches for undertaking performance-based fire design of restrained steel beams. The aim of the current study is to develop rational guidelines for fire design of restrained steel beam under realistic load, fire and end conditions. 2.4- PRvaus STUDIES ON STEEL BEAM-COLUMNS UNDER FIRE A review of the literature indicates that limited experimental and numerical studies have been undertaken on the fire response of beam-columns. This section provides an overview of experimental and analytical studies that were undertaken to study the fire performance of restrained steel beams or columns. 14.1 Experimental Studies There have been few experimental studies that have been performed on steel beam- COlumns. In these experiments, attempts have been made to simulate fire-induced 33 thermal gradients and axial and rotational restraint effects. The following is a summary of reported relevant fire tests. The effect of fire induced restraint in beams was studied by Bletzacker at The Ohio State University (OSU) for the American Iron and Steel Institute (AISl) (Bletzacker, 1966). This research was initiated to establish criteria termed as “restrained rating criteria" by which steel beams in slab-beam assemblies can be identified as restrained or unrestrained under fire. The experimental program consisted of twelve slab-beam assemblies consisting of 4-in- thick structural concrete slab on 22-gauge steel floor units supported by a W12 ><27 steel beam. The beam and floor deck were protected with a spray-applied cementitious fire protection material. A representative floor construction of 3 ft wide and the full length (16.75 ft) of the beam was assembled and loaded uniformly. The parameters considered in the test program included connection types, composite action between beam and concrete slab (noncomposite, partially composite, and fully composite action), axial and rotational restraints through the application of various levels of axial thrust and end moment, and the applied vertical 108d on the critical temperature of the beams. ReSults from OSU fire tests showed that any increase of the stiffnesses of end n‘3Straints beyond a certain limit does not improve fire resistance of beams (BIEtZacker, 1966). In the tests, as the magnitude of restraining stiffness was increased the flanges and/or the web of beams became more susceptible to local buckling. This is because the larger the restraining stiffness, the larger the fire induced axial force in the beam. Also, test results showed that composite action can 34 enhance fire resistance of beams by providing additional restraint (especially rotational restraint near the supports) in conjunction with the restraint provided by the beam-to-column connections. It was also found that the higher the load levels the lower will be the fire resistance, regardless of whether the beam is restrained or not. The generated test data from the OSU fire tests were studied extensively (lding et al. 1988) and many numerical models were created (llTRl, FASBUS l 8: ll) to simulate the effect of restraint on the fire response of beam-slab assemblies. However, these studies were of a commercial nature and were not used to develop an understanding of the fire response of restrained beams. The effects of critical factors such as thermal gradient, fire scenario, or failure criteria on fire response of restrained beams were not considered. The main purpose of these studies was to highlight that ASTM fire ratings (which are prescriptive in nature) are generally conservative for use in actual practice, since these ratings are based on unrestraint fire tests, employ thermal failure criteria, and are conducted under service load conditions. The OSU fire tests and related numerical studies culminated in the development of deScriptive criteria for classification of restrained steel assemblies. These qualitative and simple criteria were later adopted by the ASTM E119 standard, Table X3.1 (ASTM 2008) and are used for assigning fire ratings to assemblies. This table cliilssifies the construction types into "restrained” or "unrestrained" beam-slab assemblies based on some “rules of thumb". There were no detailed or specific 35 guidelines as how to rationally design such restrained steel members (which become beam-columns due to interaction with fire). During 1995 and 1996, large-scale fire tests were conducted on an eight-story, steel- framed office building at the Cardington Large Building Test Facility (LBTF) by the Building Research Establishment in the United Kingdom (Newman 1999). The purpose of these tests was to study the behavior of full scale structural systems (frames) under realistic fire and loading conditions, and to generate data for developing computer models that are capable of analyzing framed structures under fire- 'The steel frame building was five bays long (148 ft) by 3 bays wide (69 ft) and 8 stories (108 ft) high. The beams in most of the tests were designed as simply supported members acting compositely with a concrete slab cast on metal deck. Columns were protected (insulated) up to the underside of the floor slab, but the beams, deck and floor slab in this unsprinklered building were unprotected. A total of eight fire tests were undertaken on this steel framed building, including tW0 fire tests on restrained beams that were unprotected and directly exposed to a deSign fire through a special furnace built around the beams. Results from these fire teSts confirmed the OSU observations, and it was noted that for restrained beams, catenary tensile membrane action can significantly improve fire resistance. Also, it Was observed that the weakest link in a steel construction under fire were the COlllmns, because of the severe local buckling that occurred in the column near the Connection. Thermal gradients induced significant bending moments during these fun-scale structural-fire tests (Bailey et al. 1999) but the effect of these thermal gradients on the load carrying capacity of beam-columns was not investigated. 36 SO me of the recent experimental studies focused on the influence of end restraints on the fire response of beam-columns. Li et al. (2000) conducted a fire test on axially restrained steel beams and found noticeable axial forces building up in the beam. Ali et al. (1998), Rodrigues et al. (2000), Ali and O’Connor (2001), Yang et al. (2006), and Tan et al. (2007) conducted a series of fire tests on axially and rotationally restrained steel columns under monotonic uniform heating. The main findings from these experiments were that increasing the axial restraint lead to earlier failure of the column and this corresponds to a lower critical temperature in steel. Wang and Davies (2003) tested columns that developed thermal gradients across the sections. They inferred that the thermal gradients produced additional bending moments in the columns. There were no attempts to quantify these fire induced bending moments. Axial ly restrained steel beams, with fire induced thermal gradients, were tested by Liu et al. (2002) and Li and Guo (2008). Based on these fire tests, the authors concluded that the restraint effect and catenary action significantly influence the bohavior of a beam under fire conditions. However, in their experimental studies, the effect of thermal gradient on the fire response was not explicitly measured or 9Ven discussed. 2.4.2 Analytical Studies There have been a limited number of analytical studies on the fire behavior of beam- Columns. Most of these studies presented results from finite element analysis, and in the analysis a number of assumptions were made to reduce the complexity of the 37 problem. These studies did not yield simple design tools that can aid the designer to properly and efficiently account for fire effects U smani et al. (2001) presented a discussion of the basic principles that govern fire response of restrained steel beams. The analytical study focused on axially and rotationally restrained beams subjected to linear thermal gradient over the cross section. The study was based on simplified assumptions for the strains arising due to fire exposure. Further, the study did not include material nonlinearities or transient material characteristics such as strain hardening, high-temperature creep and fire scenarios. Moreover, geometric nonlinearities, such as large deflections were not addressed in this study. Tan and Huang (2005) investigated the development of fire-induced restraint forces in steel beams using one dimensional single span beams. The effect of slendemess ratio, load utilization factor, thermal gradient across the steel section and axial and rotational restraints on the fire response of steel beams was evaluated. In the analysis, the failure of the beam was said to occur when the steel section reaches the Critical temperature. The critical temperature was redefined as the temperature at Which steel member loses 50% of its room temperature strength. Results from the analysis indicated that the axial restraint reduces the critical temperature, while the rotational restraint increases it. The effects of local buckling of the steel beam and the different fire scenarios were not considered in this study. Yin and Wang (2003) applied a finite-element method to study the behavior of restrained steel beams under fire conditions. Results generated from these numerical studies were used by Wang and Yin (2006) to develop a method for 38 predicting the fire behavior of restrained steel beams. However, the method is applicable for a narrow range of situations due to the nature of the assumptions the method is based on. For example, the method requires the user to assume a deflected profile of the entire beam, which makes the method case-specific since the deflected profile is a function of the load shape. Local buckling, high-temperature creep and fire scenarios, which influence the fire response of restrained steel beams, were not included in these studies A closer examination of the previous studies reveals that the location of the fire induced axial restraint was assumed to remain at the geometrical centroid of the beam. In actual practice, the location of the axial restraint can vary depending on the type and configuration of the connection between the beam and the column. The literature review reveals that the effect of varying the location of the axial restraint on the fire response of restrained beams has not been investigated (Dwaikat and Kodur 2009). For a simply supported beam, assuming the restraint location to be at the geometric centroid results in the fire induced axial force not generating any moment around the beam center. However, if the location of the axial restraint is not at the centroid of the section, then a significant bending moment might develop at SUpports due to the eccentricity of the fire induced restraint force. This generated bending moment can affect the fire response of a beam-column (Dwaikat and Kodur 2009). 39 2 - 5 HIGH TEMPERATURE STEEL PROPERTIES Knowledge of high-temperature properties of steel is critical for evaluating fire resistance using numerical models. In most previous studies, researchers have either used material models specified by codes and standards such as the Eurocode or ASCE manual of practice (EC3 2005, ASCE 1992), or they devised their own material models (Poh 2001, Anderberg 1988, William-Leir 1983) for evaluating fire resistance of steel structures. In this section, a review of the high temperature properties of steel is presented and the sources for the variations between the diffe rent steel material models will be highlighted. Steel has excellent strength properties at ambient temperature, however, like other materials; steel loses its strength and stiffness with rise of temperature. The temperature dependent properties that are important for modeling the fire response of steel structures include thermal, mechanical, and deformation Properties. These properties vary with temperature and are also dependent on a number of other parameters. Much of the current knowledge on the high- temperature properties of steel is based on limited material property tests. Also, moSt of the current information on high temperature material properties of Structural steel is suitable for the heating phase of fires. This is because most (if not all) material tests were conducted under either transient or steady state tests with increasing temperature. However, there is lack of data on how properties of steel Vary during the cooling phase of fire. 40 The response of a steel structure exposed to fire is governed by: (a) thermal (b) mechanical and (c) deformation properties. Thermal properties determine the temperature profile in the steel sections resulting from exposure to fire, while the mechanical properties govern the loss of strength and stiffness as a function of temperature. Deformation and mechanical properties determine the extent of deformation of the steel member under fire conditions. Until recently there were no standard test methods for evaluating high temperature mechanical properties. This has led to researchers using their own test methods to measure the high temperature properties of steel. Due to differences in test methods, heating conditions, and data collection techniques, there is a noticeable variation in the different sets of data available in the literature. Thus, there are considerable variations in the different constitutive relationships presented in different codes and standards for many of the high-temperature properties of steel (Kodur and Harmathy 2002). 2. 5.1 Thermal Properties The main thermal properties that influence the temperature rise in steel are thermal conductivity and specific heat (often expressed in terms of heat capacity). Figs. 2-6(a) and (b) plot the available data on thermal conductivity, and specific heat of Steel as a function of temperature, respectively. Relationships from codes and Starldards (EC3 2005, ASCE 1992), as well as published test data were used to compile Figs. 2.6(a) and (b) (Rempe and Knudson 2008, Bentz and Prasad 2007, Toul Oukian 1972, Powel and Tye 1960, and Yawata 1969). 41 It can be seen in Fig. 2.6(a) that thermal conductivity decreases with temperature in an almost linear fashion, and there is little variation between the models presented in ASCE manual and Eurocode. On the contrary, specific heat models vary considerably between 700°C and 800°C, as can be seen in Fig. 2.6(b). In general, the specific heat of steel increases with an increase in temperature with a large spike occurring around 750°C. The spike in the specific heat at around 750°C is due to the phase change that occurs in steel in which the atoms transition from a face centered cubic “FCC" to a body centered cubic “BCC” structure. In overall assessment, for high temperature thermal properties of steel, minor variations exist in the specified models in design codes and standards. This process absorbs considerable energy (heat), thus accounting for the spike around 750°C seen in Fig. 2.6(b). The variation between the test data and the models shown in Fig. 2.6 is partly due to the fact that the majority of the existing data on specific heat originates from studies carried out on iron and non-structural steel alloys. Additionally, the maximum temperature reached in these studies on iron and non-structural steel was below 75 0°C, thus not capturing the full range of temperatures observed in fire conditions. 2. 5.2 Mechanical Properties A review of the literature indicated that there have been more studies on high- tenlperature mechanical properties of steel than that on thermal properties. Tests for high temperature strength properties are conducted mainly in two ways: transient- and steady-state tests. In transient-state tests, the test specimen is SUbj ected to a constant load and then exposed to uniformly increasing temperature. 42 Temperature and strain are continuously recorded under constant stress. Thermal strain (evaluated from a separate test) is then subtracted from the total measured strain (Outinen 2007). In the transient-state tests, the heating rate has a great influence on the strain rate and thus different heating rates produce different strain rates. The heating rate of steel depends on the nature of the fire as well as on the thermal protection (insulation) and geometry of the cross section. Generally, for a typical beam with 2-hour fire rated protection (i.e. at least 2 hours to reach critical temperature of the steel), the heating rate of steel can be in the range between 3- 7°C/min. However, for unprotected steel sections, the heating rate can be in the range between 25-40°C/min. In the literature, (Outinen 2007) transient mechanical property tests were conducted at heating rates ranging between 10 to 50°C/min. Th is heating rate can be suitable for unprotected steel members, but not for protected members with slow heating rates. Steady-state tests are generally faster and easier to conduct than the transient-state tests. In steady-state tests, the test specimen is heated to a specific temperature and after that a tensile test is carried out. Stress and strain values are recorded continuously under constant temperature. The test can be either load-controlled (loading rate is constant) or strain-controlled (strain rate is constant) (Outinen 2007, Anderberg 1988). Despite the fact that strain rate has a significant effect on the test results, a large amount of test data on conventional steel is published Without the information on strain rates (Outinen 2007, Anderberg 1988, and Cooke 19881 43 2.5.2.1 Yield Strength and Elastic Modulus As mentioned earlier, different test regimes were used to obtain yield strength and elastic modulus of steel at elevated temperatures. The variations in test parameters resulted in different test measurements, thereby leading to differences in constitutive relationships presented in different codes and standards. Generally, tensile tests are conducted to obtain elastic modulus and yield strength of steel. There is lack of experiments for steel modulus under compression. This is because in tensile tests, complications that may arise due to geometric instabilities and confinement of specimens are eliminated. However, it is generally assumed that the modulus of elasticity for steel derived based on tensile tests is the same under compression. Figs. 2.7(a) and 2.7(b) show the yield strength and modulus of elasticity of steel as a function of temperature respectively. The test data plotted in the figures are compiled from various high temperature property tests as shown on the figures. Both the yield strength and elastic modulus decrease as temperature increases. This decrease can be attributed to the nucleus of the iron atoms in steel moving farther apart due to rising temperature in steel, leading to decreased bond strength, which in turn reduces the yield strength and elastic modulus. It Can be seen in the figures that there is significant variation in test data on yield Strength and modulus of elasticity at temperatures above 300°C. This variation can be attributed to many factors, primarily variable heating and strain/load rates during the test. For example, when the heating rate of the stressed specimen is Small’ the specimen will be subjected to stress at elevated temperatures for longer 44 time duration, therefore, other factors, such as high-temperature creep (time- dependent plastic strain under constant stress and temperature), can influence the resulting temperature-stress-strain curves of the tested specimen. The yield strength and elastic modulus constitutive relationships from ASCE manual, Eurocode, and those proposed by Poh (2001) are also shown in Figs. 2.7(a) and 2.7(b) respectively. As is the case with test data, there is also a considerable variation in the constitutive models for yield strength and modulus of elasticity. These variations in constitutive models are due to the large variation in the test data used to compile the respective constitutive models. A review of the models shows that the Eurocode model predicts less reduction in yield strength of steel with temperature as compared to the ASCE or Poh models. However, the Eurocode model provides a higher reduction in elastic modulus of steel with temperature as compared to the ASCE and Poh constitutive models as shown in Fig. 2.7(b). Also, up to 400°C, the Eurocode model assumes no reduction in steel yield strength; however, ASCE and Poh models assume a loss of 30% and 40%, respectively, at 400°C, as shown in Fig. 2.7(b). This major difference between Eurocode and other constitutive models can be attributed to the fact that Eurocode distinguishes betWeen two limits on the temperature-stress-strain curves, namely, the proportionality limit (Fp), and the yield limit (Fy). The proportionality limit in the EurOcode is the end of the linear portion of the stress-strain curve, after which point the Stress-strain relation becomes plastic and nonlinear (ellipsoidal shape). The yield limit in Eurocode is the defined as the end point of the inelastic portion of the 45 stress-strain curve. This feature of the stress-strain in Eurocode does not exist in either ASCE or Poh models which both assume a sharp point as a limit between linear-elastic and inelastic material response. The theory behind introducing proportionality limit in the Eurocode stress-strain curves at high temperatures is to capture visco-elastic behavior that is partly due to creep effect. The nonlinearity after the proportionality limit indicates that stress causes more strain after this point than in the linear elastic range. This simplification enables the stress-strain curves of the Eurocode to partly account for creep strains at elevated temperature. 2 . 5.2.2 Temperature-Stress-Strain Relationships Figures 2.8(a) and 2.8(b) show the temperature-stress-strain curves from Eurocode 3, ASCE manual, and as proposed by Poh (2001) at room temperature and at 600°C, respectively. The y-axis represents normalized stress and the x-axis represents normalized strain. Originally, the Eurocode temperature-stress-strain curves were derived based on transient-tests under slow heating rates (Twilt 1991 and Anderberg 1988). Therefore, when used in the fire resistance analysis, the Eurocode constitutive relationships generally produce slightly more flexible response as compared to ASCE or Poh constitutive models. This is attributed to the fact that in the transient-state tests, adopted by the Eurocode, part of high temperature creep is inClllded in the resulting temperature-stress-strain curves (Anderberg 1988, Bu Chanan 2002). However, less information is known about the test data that were used to derive ASCE temperature-stress-strai-n curves of steel. On the other hand, Poh (2001) developed generalized temperature-stress-strain relations for structural 46 steel based on a large set of experimental data. These relations account for specific features, such as the yield plateau and the effect of strain hardening. These temperature-stress-strain relations have been validated against the specified relationships in codes and standards, and they have been shown to give more realistic predictions of fire resistance when used in conjunction with high temperature creep model. 2. 5.3 Deformation Properties The deformation properties that influence the fire response of steel structures are thermal strain and high-temperature creep. The following sections discuss the va riations in deformation properties of steel at elevated temperatures. 2 - 5.3.1 Thermal Strain There have been many tests to characterize thermal strain of steel at elevated temperatures, results from some of which are compiled in Fig. 2.9 (Outinen 2000, Cooke 1988, Anderberg 1988, and Stirland 1980,). As seen in Fig. 2.9, thermal strain of steel increases with temperature up to nearly 750°C, at which point a phase Change takes place (as discussed previously) and the thermal strain becomes nearly constant up to 800°C, after which point thermal strain starts to increase again. Variation of thermal strain models as specified in ASCE and EC3 are also plotted in Fig- 2.9. Minimal differences exist between the Eurocode and ASCE models for theI‘Inal strain of steel up to 700°C. However, in the temperature range of 700°C- 850°C the ASCE model assumes a continuously increasing thermal strain while the Eur'ocode model accounts for the phase change that occurs in steel in this 47 foil: 2.5 Cre KETT C03 j. LE? 3531 temperature range by assuming a constant thermal strain from 750°C to 850°C, followed by an increasing thermal strain up to 1000°C (See Fig. 2.9). 2.5.3.2 High Temperature Creep Creep is defined as the time-dependent plastic strain under constant stress and temperature. The influence of high temperature creep on the response of structures is much more pronounced at elevated temperatures than at room temperature. This is because at elevated temperatures steel becomes less viscous and thus the continuous deformations under constant stress and temperature, referred to as high temperature creep, become more significant. The codes and standards are not specific about how to incorporate creep into fire analysis, and the choice of creep models is often left to the user (Kodur et al. 2009). At present, most fire resistance analyses are carried out using Harmathy’s high temperature creep model that is mainly based on Dorn’s theory (Dorn 1954; Harmathy 1967). Dorn's creep theory assumes constant stress (dO's / dt = 0). However, in the case of beam-columns 6Xposed to fire, the fire induced stresses vary considerably and rapidly with time and temperature, and these conditions are not captured by Dorn's theory. Therefore, an alternate high temperature creep model that can capture variable Stress scenarios is required to account for these extreme conditions. Generally, creep deformations in steel become noticeable at temperatures above 400°C. However, it was found experimentally that when the stress level is high, the effeCt of creep becomes significant in steel members even at temperatures of 300°C (“Hang et al. 2006, and Huang and Tan 2003). Creep tests (Kirby and Preston 1988, 48 Harmathy and Stanzak 1970, and Born 1954) were generally conducted for steel and metal alloys with variable chemical compositions, and very little information is available on the effect of high temperature creep on the structural response. Contradictions also exist in codes of practice when it comes to high temperature creep. Eurocode states that "the effects of transient thermal creep need not be given explicit consideration" (Clause 433(4) of EC3 2005). However, the ASCE manual of practice states that high temperature creep should be accounted for in fire resistance analysis through one of two options. The first option is to use “effective” temperature-stress-strain curves derived from transient-state tests at relevant heating and strain rates (ASCE 1992 and Buchanan 2002), while the second option is to use specific high temperature creep models developed for structural steel. As such, it is generally left to the user to choose which creep model to use. 2.6 SUMMARY The state-of-the art review clearly indicates that there is a lack of knowledge with respect to a number of key areas that are critical for developing a holistic approach for the fire safety design of beam-columns. The following are some of the key factors that need to be considered for developing such design guidelines: ' No fire resistance experiments have been performed on beam-columns that develop thermal gradients. Data from fire experiments are needed to validate finite element models. " The effect of fire scenario on the response of beam-columns was not investigated in previous studies. Fire scenario can vary from a standard fire 49 curve (with no decay phase) to a design fire (with decay phase). The decay phase can result in cooling of steel in a restrained beam and thus leads to generation of tensile forces due to the shrinkage of steel. The development of these tensile forces was not investigated in previous work. Evaluating and quantifying fire induced forces in beam-columns. The fire induced axial force and bending moment have a significant influence on the performance of the beam-columns themselves and on beam-to-column connections. For example, simple shear connections are not designed to resist tensile forces that develop due to fire induced catenary action or due to shrinkage of steel in cooling phase of a design fire. Generally, these fire induced forces are not accounted for in fire design due to lack of tools for evaluating them. Simple expressions for evaluating these fire induced forces become a necessity for proper fire design of beam-columns. There is a large variation in high temperature properties of steel presented in different codes. For example, the Eurocode states that high temperature creep can be neglected, while ASCE manual of practice states that high temperature creep must be included in the analysis. This led the researchers to either use the discrepant code-specified material models or devise their own material models. Accounting for the response of beam-columns is quite complex and has to be carried out using sophisticated finite element analysis. Such analysis requires a large amount of input data and also produces a huge amount of numerical results that are not easy to analyze. To avoid all these complexities, 50 simplified design tools need to be developed based on intensive numerical studies that take into account all of the previously mentioned factors. w Ka, ' W KO, :<——:. a L ——>; K" Simply supported beam Restrained beam Perimeter column M = wL2/8 + PXA Bending moment and axial force P+ (b)Simply supported (c) Restrained beam (d) Restrained column-beam Fig. 2.1: Development of fire induced restraint forces in restrained steel beams and columns. 51 Ka, Kr w Ka, Kr ?\ ’1 a ) Layout of restrained steel beam Stage I Stage II I I Stage III Elastic \IElasto-plastic L \ I I Initiation of yielding I T I I I I I l I I l I I I Catenary action I I I l I I I l v I v 3 Steel temperature / I I I I l l I I l I l I I Fire-induced axial force _--- ———V i \l I . b) Fire-induced axial force Steel temperature / Tc I I I I I I :1, I I L p--————-————_- -—————-——q—-—--—————__—- Initiation of catenary action Rupture of steel I F, PH ”U H CD ya I I I I I I I I l I I I I I I I I I I I I I I I l l c) Midspan deflection Midsnan deflection b p D 9: -_--—.—_-——.——_-—-—- Fig. 2.2: Typical fire response of a restrained steel beam that fails by yielding 52 "an."unuvnnuunn ....... 5AM a?» 0.8; AA; LA); ska AAA (a) Fire exposure types ISO 834 standard fzr 1000 r P 800 4 if 600 ) ASTM E] 19 standard fire :3; 400 _ V Examples of E 200 deszgn fire E" 0 . I I r e 1 I j 0 20 40 60 80 100 120 140 Time, min ( b ) Fire scenario Fig. 2.3: Types of (a) fire exposure and (b) fire scenario 1.2 — a 1 E AT = we 90,3; AT=25°C 0.0 ' AT=50°C AT: 100°C 0.6 7 \\ 07-.- 125°C 0.4 - $\ AT=150°C 02 “Q‘ '0 M/Mp \\ -0.2 - I -o.4 4.6 W24x7' 0.3 I 35% _1 Ease . ” Constant T = 650°C -102 AT .. -1.5 4.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 Fig. 2.4: Changes in the shape of the P-M interaction diagram under the influence of thermal gradients 53 A Growth phase Q) L- :3 H CO L- O) o (15) Natural fire F Ignition Intense burning Flashover Decay phase \ Standard fire Design, realistic, or \\ parametric fire Time Fig. 2.5: Various Stages of Fire in a Typical Compartment with common idealization 54 Thermal Conductivity, Wlm.°C (a) O O Rempe & Knudson 2008 El Dale 8: Prasad 2007 20 _ A Touloukain 1972 0 Yawata 1969 I Powel1956 1° ‘ — E03 model —ASCE model 0 T l l l ‘I 0 200 400 600 800 1000 Temperature, °C (0) Thermal conductivity 5000 ~ Flempe & Knudson 2008 q 4000 ) Flempe & Knudson 2008 Specific Heat, .ng.°c N 0) § § I? 0 n O Bentz & Prasad 2007 I Touloukian 1972 E23 model —ASCE m odel O 200 400 600 800 1000 Temperature, ‘C ( b ) Specific heat Fig. 2.6: Thermal properties of steel as predicted by different models and as measured in different test programs 55 Outlnen & Makelalnen 2004 _ A 1'2 o Outlnen et al. 1997 A Makelainen et al.1998 0 Chen et al. 2006 0 LI et al. 2003 323 model ASCE model Poh model ”>3 . ‘ o \ I ..~ . I i— 0.4 ~ ._, >2 e “- Yield point "E03" o 0.2 - .. Proportionality limit "EC3" A " ~-- ’_ o I I I T ‘ T. 0 200 400 600 800 1000 Temperature, ‘C (a) Yield strength 1.2 1 1 a . . - . 0.8 — o ‘ 0 £ . 0.6 ~ ° 9 ‘A \ ‘x. E} 0 Chen 8: Young 2008 0 ° ‘41 0,4 _ a Outlnen & Makelfilnen 2004 o Outlnen et al. 1997 I Cooke 1988 A Dever 1972 0.2 — A Clark 1953 EC3 model ASCE model 0 ------- Poh model 0 200 400 600 800 1000 Temperature, 'C (b) Elastic modulus Fig. 2.7: Yield strength and elastic modulus of steel as predicted by different models and as measured in different tests 56 F, T Q75 . Steel temperature: 20 °C Fy 05 ‘ ASCE model 0.25 — E3 madel - - - - Poh model 0 U U I I fl I 0 5 10 15 20 25 30 85,T (Fy / ES) 0.6 - 0.5 q """ 0.4 - F Steel temperature: 600 °C s,T 0 3 ‘ F 0 2 ASCE model y EC3 model 0'1 - - - - Poh model 0 I I U I I 0 5 10 15 20 25 30 85f (F), / E) Fig. 2.8: Temperature-stress-strain relationships for structural steel as per ASCE, Eurocode and Poh models 57 1.5 I Outlnen et al. 2004 I 1.25 , O Anderberg 1988 o Cooke 1988 n Stirland 1980 1 I —EC3 model —ASCE model 0.75 — 0.5 J Thermal Strain, % 0.25 a m 0 200 400 600 800 1000 Temperature, °C Fig. 2.9: Thermal strain of steel as predicted by different models and as measured in different tests 58 Applied variable stress 5 ~ - ”-160 ”j (do, Mr )Avemge = 10.22 MPa/mm 140 4 ‘ I"ir‘1nl °\o 3.5 _ I I - 1* 120 5: 3 _ — 1003 2 E 1;; 25 — - 80 a; El 12“ . .c 60 gl 3: .’ t: 01.5 4 l Test data _ 40 ‘D 1 " ° — ANSYS model 20 0.5 — ‘— 0 --- Harmathymodel 0 0 100 200 300 400 500 600 Time,min dos/dt ¢ 0, Ts=550°C Fig. 2.10: Comparison of measured and predicted (using ANSYS and Harmathy models) creep strains 59 CHAPTER THREE FIRE RESISTANCE EXPERIMENTS 3.1 GENERAL The state-of-the-art review indicates that there is a lack of fire tests on beam- columns subjected to realistic conditions, such as thermal gradients and design fire scenarios. There have been few fire tests conducted on restrained steel beams subjected to thermal gradients and realistic fires. However, almost all of the reported fire tests on steel columns were carried out under standard fire exposure and under uniform heating. Because of the lack of fire tests on steel columns subjected to thermal gradients, four steel columns subjected to thermal gradients were tested under realistic conditions. The main purpose of these tests is to investigate the effect of thermal gradient on the beam-column response of the restrained columns and to generate test data for validation of finite element models. Full details of the fire experiments, including specimens' instrumentation, fire tests procedure and measured parameters are presented in this chapter. 3.2 TEST APPARATUS The fire resistance tests on steel columns were carried out using the structural fire testing furnace recently commissioned at MSU (Dwaikat et al. 2009). The test 60 furnace, shown in Fig. 3.1, has been specially designed to produce conditions, such as temperature, loads and heat transfer, to which a structural member might be exposed during a fire. The furnace consists of a steel framework supported by four steel columns, with a fire chamber that is 2.44 m wide, 3.05 m long, and 1.78 m high. The maximum heat power the furnace can generate is 2.5 MW. Six natural gas burners located within the furnace provide thermal energy, while six thermocouples, distributed throughout the test chamber, monitor the furnace temperature during a fire test. Two small view ports on either side of the furnace wall are provided for visual monitoring of the fire-exposed specimens during a test. Vertical pressure actuators with individual capacities of 2400 kN were used to apply axial load to the top of each column. The furnace can accommodate two columns to be tested at a time, and different load levels can be applied for each column with the two separate vertical actuators. 3.3 TEST SPECIMENS The experimental program consisted of fire resistance tests on four steel columns designated C1-S, C1-W, C2-S, and C2-W. All specimens used a W8x48 cross-section with A992 Grade 50 steel (Fy = 345 MPa) and were fabricated to a length of 3.3 m. The classification of each column was based on the direction of thermal gradient in the section. Columns C1-S and C2-S experienced a thermal gradient along the strong axis, while columns C1-W and C2-W experienced a thermal gradient along the weak axis. The C1 columns were tested simultaneously during the same fire test, and likewise for the C2 columns. Since the beam-columns are exposed to fire from the 61 four sides, insulation was removed in certain location so as to generate thermal gradient in the required direction, as shown in Fig. 3.2. The average measured dimensions of each specimen are shown with their corresponding insulation scheme in Fig. 3.2. Initially, each specimen was insulated with spray-on fire resistive material (SFRM) on all four sides along the majority of its length. As shown in Fig. 3.2(a), the top and bottom ends of each specimen, which were not to be exposed to fire, were left unprotected. The sections dimensions and average insulation thickness for each column is reported in Fig. 3.2(b). The insulation material that was used is CAFCO 300 with a specified thermal conductivity of 0.078 W/m-K and a density of 240 kg/m3 at room temperature (lsolatek International 2008). The insulation was applied by a contractor so as to simulate the conditions actually found on a construction site. During the application of the insulation, special care was taken to measure the SFRM thickness at various locations on the specimens to ensure reasonable agreement with nominal Specifications. The insulation was left for three weeks to allow the insulation to adequately dry prior to testing and develop thermal resistance properties realistic to actual building construction. Since the columns were exposed to fire from four sides, the insulation was removed in specific locations, as Shown in Fig. 3.2(b), to allow a thermal gradient in the desired direction. These insulation schemes simulated the thermal gradients that would realistically emerge in perimeter columns and floor beams due to three-sided heating. 62 3.4 INSTRUMENTATION The test specimens were instrumented with thermocouples, strain gauges, and displacement transducers to monitor their thermal and mechanical response during the fire test. Steel temperatures were measured using Type-K Chromel-alumel thermocouples, 0.91 mm thick, installed at four cross sections along the length of each column, as shown in Fig. 3.3. The furnace temperature was monitored using six thermocouples distributed Spatially inside the furnace. Five high-temperature strain gages were attached closer to the base of every column at section D-D to directly measure the total strains. Vertically and horizontally oriented linear variable displacement transducers (LVDT’S) were attached at three distinct locations at the top of each member in order to calculate axial and lateral displacements and top-end rotations. The pressure in the vertical actuators was also recorded as a measure of applied axial load. 3.5 TEST CONDITIONS AND PROCEDURE Two fire tests were performed, each with two steel columns placed vertically inside the furnace as shown in Fig. 3.1(b). A length of 1775-mm over the middle of the 3300-mm long column was directly exposed to fire. The columns were fixed to a very thick steel base plate at the bottom by the use of four 100x150x12.5-mm steel angles and 25.4-mm diameter high strength bolts (see Fig. 3.4(a)). Heavy steel plate (300x300x50 mm) was placed at the top of each column to ensure a uniform distribution of the applied axial load (see Fig. 3.4(b)). There was no bolted or welded connection between the heavy steel plate and the vertical actuator’s loading 63 piston (i.e. only bearing), and horizontal sway at the top end was not fully restrained. Restraint to axial elongation was provided by the loading frame. These boundary conditions simulated realistic restraint experienced by a fire-exposed perimeter column and allowed for the development of bending moments and increased axial load due to restraint of thermal effects. The fire exposure and loading details for each of the four specimens are shown in Table 3.1. As mentioned previously, the columns were exposed to fire on four sides. Columns Cl-S and C1-W were tested under ASTM 8119 standard fire exposure (ASTM, 2008), while columns CZ-S and C2-W were tested under a design fire scenario. The design fire included a growth phase simulating the ASTM E119 standard fire for about 90 minutes and then underwent a rapid cooling phase. The fire scenarios used in the two tests are shown in Fig. 3.5. All columns were tested with axial loads applied at their top end by the vertical actuators. Columns Cl-S and C1-W were subjected to an initial axial load of 650 kN, which is equal to 25% of the columns’ yield capacity at room temperature. Columns C2-S and C2-W were subjected to an initial axial load of 1430 kN, 45% of the columns' yield capacity at room temperature. The load was applied in average increments of 200 kN for 10 minutes until full load was applied. The total applied load was maintained for approximately 30 minutes before the start of the fire test and was maintained until a condition was reached at which no further increase of the axial deformation of the columns could be measured. This was selected as the initial condition for measuring the axial deformation of the columns. During the test, the columns were exposed to heat controlled in such a way that the average 64 temperature in the furnace followed, as closely as possible, the targeted time temperature curve. The load was kept constant throughout the test. The columns were considered to have failed and the tests were terminated when the columns suffered strength failure and could no longer carry the load. 3.6 EXPERIMENTAL RESULTS Data generated from the fire tests were used to study the overall performance of steel beam-columns under fire induced thermal gradients. Thermal response, structural response, and failure patterns were compared to evaluate the effect of thermal gradient orientation, load level, and type of fire exposure. 3. 6.1 Thermal Response The temperatures recorded during the fire tests at location B-B in each column are shown in Fig. 3.5. Section B-B consistently recorded the highest temperatures because it had the most direct exposure to the burners due to its position (center of column) in the furnace (see Fig. 3.3). Thermocouples across the section at D-D recorded nearly uniform temperatures that did not exceed 100°C, and therefore temperature effects at this location were negligible. Each plot in Fig. 3.5 shows a steel temperature plateau at about 100°C that can be primarily attributed to the evaporation of residual water in the insulation that remained following curing. This reaction consumed a significant amount of energy and initially slowed the transfer of heat from the hot gases through the insulation to the steel. Once the residual water was dispelled, steel temperature increased with 65 fire exposure and a thermal gradient developed due to the strategic removal of the insulation. This effect is more pronounced for the C1 columns as shown in Figs. 3.5(a) and (b). Columns C1-S and C1—W, exposed to ASTM E119 standard fire as shown in Figs. 3.5(a) and (b), experienced increasing temperature until the end of the test at t = 240 minutes. The average rate of temperature increase for these columns was approximately 3.5°C/min. Columns C2-S and C2-W, exposed to the design fire shown in Figs. 3.5(c) and (d), experienced increasing temperature until t = 90 minutes when a rapid decay phase was invoked. During the decay phase, the temperature in these columns decreased as heat was exchanged with the now cooler ambient environment. The average rate of steel temperature increase prior to the decay phase for these columns was approximately 7°C/min, and the average rate of temperature decrease during the decay phase was approximately 3°C/min. The difference in heating rate between the C1 and C2 specimens is primarily attributed to the difference in insulation thickness (44 mm for the C1 columns, and 38 mm for the C2 columns). In addition, the C2 columns had bare steel exposed to the fire at the locations of insulation removal shown in Fig. 3.2, as opposed to the C1 columns which had a trace (~2.5 mm) of remaining SFRM on its stripped areas. Column C2-S also had a wider insulation gap than C1-S, thus allowing more heat transfer directly to the steel. The plots in Fig. 3.5 show that each section experienced the largest temperature increase at the location nearest the gap in insulation with temperatures decreasing with distance from the gap. 66 Since the insulation was applied by a contractor, the SFRM thickness had variations that would be realistically seen on Site, and these variations affected the thermal response. Pre-test measurements of insulation thickness taken at section B-B of column Cl-W are shown in Fig. 3.6(a), and the location of WTC6 had 20% less insulation thickness than that of WTCS. Therefore, this location of WTC6 developed higher temperatures than that of WTCS for column C1-W as shown in Fig. 3.5(b). Column Cl-S experienced a similar effect, as seen in Fig. 3.5(a), and the temperature of its web and bottom flange are almost always equal. Fig. 3.6(b) shows that the insulation on this specimen's web was nearly 30% thicker than on the rest of the section (due to pooling when the insulation was spray-applied), causing a Slower growth of web temperature relative to that of the flanges. Fig. 3.5 shows the development of thermal gradients as represented by the difference in temperature at the various locations through the depth of the section. The magnitude of these gradients undergo a rapid initial increase followed by a slight and gradual decrease as the section begins to approach thermal equilibrium during the later stages of the fire growth. As shown in Figs. 3.5(a) and (c), the strong axis gradients are significant because heat transfer from the hotter flange (with the insulation gap) to the cooler flange is slowed due to their separation by the thin web plate. Columns C1-W and CZ-W developed less severe thermal gradients about the weak axis (Figs. 3.5(b) and (d)) because the gradient now forms along the width of the flanges (which represent the majority of the sections’ thermal mass) and heat can be transferred more readily across the thicker flange plate. 67 In the case of columns exposed to design fire, Figs. 3.5(c) and (d) show that the thermal gradients began to reverse direction once the decay phase initiated. During the decay phase of the fire, the portions of the section with less insulation (i.e. at the insulation gaps) cooled at a faster rate while fully insulated regions lost heat more slowly to the cooler ambient environment. The plots in Figs. 3.5(c) and (d) also reveal that during the decay phase, column C2-W experienced faster reversal in the direction of its thermal gradient as compared to that of C2-S. This phenomenon can be attributed to the fact that in the case of column C2-W, the uninsulated tip surfaces of the flanges have a higher emissivity factor when compared to the insulation gap on the flange of column C2-S. The adopted insulation schemes for C2- 3 with the insulation removed from the middle of the flange (as shown in Fig. 3.2) makes the exposed flange “less visible" to the surrounding thermal environment, and therefore it will have less heat loss compared to C2-W. 3. 6.2 Structural Response The structural performance of the column specimens was monitored using the deformations and strains recorded during the fire tests, and that data is reflected in Fig. 3.7. The axial displacement and top-end rotation curves shown in Figs. 37(3) and 3.7(b) were calculated using vertical displacement data recorded by the three vertical LVDT'S attached to the top of each member. Axial displacement was calculated as an average of these three data points, and rotations were calculated by analyzing the orientation of the plane between the three LVDT displacements. The experimental axial loads shown in Fig. 3.7(c) were calculated by multiplying the 68 recorded changes in loading pressure in the vertical actuator by the cross-sectional area of the actuator’s piston. Bending moments shown in Fig. 3.7(d) were calculated using strain data recorded using strain gauges at section D-D. Because the temperature at D-D did not exceed 50°C, the experimental moment can therefore be calculated by multiplying the strain gradient, (D, through the depth of the section by the bending stiffness at room temperature (M = El (D ). Sign conventions for deflection and rotation are shown in Fig. 3.8. Positive axial force denotes compression, and positive bending moment corresponds to added compression on the hotter face of the column. During the fire tests, each column experienced thermal expansion upward against the restraint of the loading frame due to increasing temperature, which in turn generated a corresponding increase in axial load. Figs. 3.7(a) and 3.7(c) show that reduction in strength and stiffness eventually lead to a rapid decrease in both axial displacement and axial load as the column approached failure. This decrease initiated once the average temperature in the column's hottest regions begin to exceed about 600-700°C, as shown in Fig. 3.5. The C1 columns take nearly three times longer to reach this point than the C2 columns because they have 25% more insulation thickness, less significant insulation gaps, and less than half the initially applied axial load. As the column cross-sections heated unevenly and developed a thermal gradient, the columns initially developed a positive bending moment due to the rotational restraint of the end connections (i.e. the hotter face of the section became more compressed in response to the gradient). At the same time, the effective centroid 69 (i.e. the center of stiffness) of each section began to migrate toward the cooler face of the section as the gradient generated non-uniform stiffness (i.e. a non-uniform distribution of B) through the section. Axial loads applied about the section’s geometric centroid via the vertical actuator now generated a bending moment because the axial force resultant was eccentric to the center of stiffness. This moment is opposite in direction to that caused by restraint of thermal bowing even though the direction of the thermal gradient has not changed. Fig. 3.7(d) shows that the eccentric moments in each column eventually became larger than the thermal bowing moments, reversing the moment-time curve from positive to negative until the specimen reached its plastic capacity. The strong axis specimens experienced larger bending moments than the weak axis specimens due to their larger moment of inertia in the direction of bending (i.e. in the direction of the gradient). Fig. 3.7(b) shows that rotations of the top end of the beam-columns increased with the increase of fire exposure time, but then the rotations decrease at later stages of fire exposure. The initial increase of top-end rotations can be attributed to the development of thermal gradients, which produce thermal curvatures in the beam-columns. However, as the center of stiffness of the cross section is moving towards the cooler side, and with spread of plasticity in the hotter regions of the sections at later stages of fire exposure, the top end rotations start to reduce and switch direction, as shown in Fig. 3.7(b). In the later stages of fire exposure, differences. in behavior between C2 and C1 columns were caused by the presence of decay phase in the C2 fire test. The C2 columns continued to expand under exposure to the growth phase of the design fire, 70 and then the columns yielded through their full depth a few minutes before entering decay phase of the design fire. At this point, the loaded columns underwent severe deformation similar to that of the C1 columns until the decay phase initiated. During the decay phase, steel temperature in the C2 columns started to decrease, allowing the steel to regain some of its strength and stiffness and therefore reducing their rate of deformation as seen in Fig. 3.7(a). 3. 6.3 Failure Pattern and Fire Resistance The failure times for the four tested columns are Shown in Table 3.1. Failure is defined as the time when the columns cannot maintain the applied load (i.e. when the sudden decrease in axial load and moment shown in Figs. 3.7(c) and 3.7(d) initiates). There were no incidents of sudden global or local instability observed during the fire test. Validation studies using computer models Show that failure is reached via full section yielding as discussed in Chapter 4. Fig. 3.9 shows the failed shape of each column, with the location of plastic hinge marked as a distance from the bottom of the member. The time to reach failure from the start of fire exposure is defined as fire resistance. This definition of failure contrasts with the traditional prescriptive-based approach in which failure is often assumed to occur when steel attains its critical temperature regardless of loading level or thermal gradients. The beam-columns failed by developing full plastic response. It can be seen from Fig. 3.9 that the location of the plastic failure for every specimen is nearly in the hottest region (at the center) of the beam-columns far from the base. This failure trend can 71 be explained by referring to the distorted P-M diagrams of the beam-columns. This failure pattern will be explained with full details in Chapter 6. 3.7 SUMMARY Fire tests were conducted to study the behavior of axially loaded steel columns that develop thermal gradients through their depth. Thermal gradients were created by removing part of insulation on one side of the specimen thus simulating the thermal response of a three-sided heating scenario such as a column on the perimeter of a building frame. The column specimens developed significant bending moments in response to these gradients and therefore they behaved as beam-columns. Thermal gradients also produced a change in plastic capacity to combinations of axial load (P) and moment (M). All four specimens failed by plastic yielding under the effect of combined axial force and bending moment. The location at which failure occurred was where the temperature was the largest and the moment was the smallest (the latter due to the effects of the thermal gradients on the combined P and M demand and capacity). Results of the tests showed that load level, fire scenario, and the magnitude and the direction of the thermal gradient (i.e. along the strong or weak axis) had a significant influence on the fire response of these beam columns. 72 Table 3.1: Matrix of fire test parameters. , Initial Fire Axis Insulation P Recorded Specimen . . Thickness time of failure Exposure Orientation (mm) P (min ) y . Cl-S [$8111.13 strong 44 0.25 220 c1-w gig weak 44 0.25 215 C2-S diff" strong 38 0.45 90 cz-w d‘gsr‘eg“ weak 38 0.45 93 _ ;-_',_.,_. -,_., - ~' . ~_ -7:- ;‘.— - - .-——-:’..éi Vertical fl Actuator 5f -_.: T305 T WSPGCIMGH __ i... I ska the w (a) Furnace and loading frame (b) Schematic for furnace front view Fig. 3.1: Structural-fire test furnace at MSU's Civil and Infrastructure Laboratory (dimensions shown in mm) 73 305 2385 Column C1-S and C2-S Column Cl-W and C2-W Fig. 3.2: Insulation schemes for the beam-column Specimens 74 Cl: 460 C2: 430 C1: 460 C2: 430 C1: 740 C2: 990 610DA ”if; L .I Fig. 3.3: Instrumentation mapping for the tested specimens (dimensions shown in mm) 75 —- -—50 3% WTC3 j N \ STCl I— "x \STCZ 1 8 \WTC4 ll yt< STC3 +103 ‘— WTC2 WTC] (a) Thermocouple locations (b) Thermocouple locations for Cl-S and C2-S at A-A, for Cl-W and C2-W at A-A, B-B, C-C and D-D B-B, CC and D-D 90 m *‘50 | T7 1 //SSG1 25 “I R :33? W591 W863 I 5H?" N1 3364 WSG2 I WSG4 wsgs I ‘ SSGS 4 103 ‘— (c) Strain gauges at D-D for ((1) Strain gauges at D-D for Cl-S and C2-S Cl-W and CZ-W Fig. 3.4: Instrumentation mapping for the tested specimens (dimensions shown in mm) 76 -- ~a—-.--- . . ..- .....- a) Base fixation b) Top end Fig. 3.5: Photos of the boundary conditions for the tested beam-columns 77 120° ASTM E119 \ __ 7 §1000 — —-’ I 800 j avg. furnace 5 temp. g 600 — 3 E 400 ‘ +STCI .2 200 a +STC2 +STC3 o l I I 0 60 . 120 . 180 Time (mm) (a) Column CI-S 1200 240 i avg. furnace temp. Temperature (°C_) 0) 8 ASTM E119 \ ___ 400 200 0 .~ 0 60 120 Time (min) (c) Column CI-W Fig. 3.6: Temperatures recorded at section B-B for beam-columns 78 1200 s design fire _._ STCI @000 ‘ \ + srcz E 300 4 + STCS : E 600 r 3 E 400 o r— 200 d L o . . . . I 0 30 60 90 120 150 Time (min) (b) Column CZ-S 1200 design fire +WTCI £31000 ~ \ +WTC2 :1: 800 - +WTCS 3 —WTC4 e 600‘ g I 2 40° Q I .2..- L‘ 0 I I l I I o 30 60 90 Time (min) (d) Column CZ-W 120 150 sow LIT l a: o: E L2: L11 E e r: o o O O m U :1: m w \\\\ \\ “— Fig. 3.7: Sign convention adopted for the fire induced restraint forces in beam- columns 79 a) as b) c1-w Fig. 3.8: Failed shape of each beam-column, shown with failure locations 80 .. :0. i 3.0.: Lava-Pa . .1. m m 0 1 2 2 18301nn1 c) c2-s :Confinued Fig. 3.8 81 CHAPTER FOUR NUMERICAL MODELING AND VALIDATION _ 4.1 GENERAL Undertaking fire tests is quite expensive, complex, and time consuming. An alternative is to use numerical models to investigate the influence of critical parameters on the fire response of beam-columns. For this purpose, the general purpose finite element program ANSYS is used. The fire response is evaluated through two stages of analysis, namely, thermal and structural analysis. In this chapter, the development of finite element model for thermal and structural analyses is presented. Also, the material models used in the analysis are described. Validation of the finite element models for thermal and structural analysis is carried out by comparing predictions from the analysis with results from fire tests. In the validation process, the influence Of variation Of constitutive models Of structural steel on the predictions of fire response is carefully examined. 4.2 SELECTION OF FINITE ELEMENT PROGRAM Undertaking detailed parametric studies requires the use Of a numerical model. The general-purpose finite element program ANSYS (ANSYS 2007) was chosen to carry 82 out the numerical studies because Of its diverse capabilities. ANSYS has the ability to efficiently model highly sophisticated material and geometrical nonlinearities, and allows the user to specify temperature-dependent thermal and mechanical properties Of almost any material. Also, ANSYS provides a wide range of high temperature creep models in addition to any user-specified high temperature creep model. ANSYS contains an Optimization tool that can be used effectively for automating parametric studies. Further, ANSYS contains a rich library of diverse categories of elements that are well suited for simulating various structural systems. ANSYS has been used by various researchers for undertaking fire resistance analysis of steel structures (NIST 2005). The analysis was carried out by incorporating all significant parameters, such as material and geometrical nonlinearities, end restraints, fire scenarios, and thermal gradients. Details on the finite element analysis are provided in the following sections. 4.3 FINITE ELEMENT MODELING 4.3.1 General Approach The fire resistance analysis is generally carried out through two stages, namely; thermal and structural analyses. The purpose of the thermal analysis is to Obtain the evolution of temperature in a structural member as a function of fire exposure time. The purpose Of the structural analysis is to Obtain the deformations and stresses resulting due to fire exposure. For practical cases of analysis, the following assumptions are adopted while carrying out structural fire analysis: 83 0 Fire characteristics are independent Of both thermal and structural analysis results. 0 Thermal analysis is independent of the structural analysis. 0 Temperature distribution is uniform along the beam span length. The fire scenarios that are typically used in structural fire engineering can be grouped under two categories. Standard fire scenarios; where fire continues to increase indefinitely, and realistic, natural, or design fire scenarios where fire temperature undergoes a decay phase. Examples of standard fire scenarios are ASTM E119 and ISO 834 standard fire exposures. The fire temperature (Tf in °C) as a function of time (t in minutes) for the standard fire curves are expressed as: ASTM E119: Tf = 20+ 750(1—exp(— 3.79553Jt / 60))+170.41Jt / 60 [4,1] 150 334; Tf = 20 + 34510g(8t +1) [4.2] In case of a design or “realistic" fire, a decay phase follows after reaching a maximum fire temperature. Unlike standard fire curves that are function Of time only, the growth and decay phases Of design fires are dependent on the compartment geometry and fuel and ventilation characteristics. Examples of design fires are the Swedish fire curves and the parametric fire curves specified in the Eurocode (EC1 2002). Generally, the rise Of fire temperature is dependent on geometric, fuel and ventilation characteristics of the compartment. Computational fluid dynamics (CFD) models are required to compute the evolution of fire in a certain enclosure. Since this kind Of analysis is beyond the scope of this study, fire characteristics are 84 reduced tO time-temperature relationships and are taken from the specified design fire curves in codes. The fire resistance analysis, with its thermal and structural phases, is carried out via ANSYS finite element program using the following general procedure: 0 Thermal analysis: Fire exposure time is incremented and temperature distribution in the beam cross section is Obtained at every time step and for the entire fire exposure history. 0 Room-temperature structural response: Static structural analysis is performed on the entire structural model Of the beam to Obtain the room- temperature structural response (deformations and stresses). 0 Structural fire analysis: Fire exposure time is incremented and the temperature distribution obtained from the thermal analysis is applied, at each time step, on the room-temperature deformed model of the beam. In the following sections, more details are given about the thermal and structural analyses. 4.3.1 Thermal Analysis Since the temperature along the beam span length is assumed to be uniform, heat transfer in the beam is reduced to a two-dimensional problem. Thus, the governing partial differential heat transfer equation within the beam cross section can be written as (Purkiss 2007): 85 dT ——=v- kVT PC dt ( ) [4.3] where k = conductivity matrix, pc = heat capacity, T = temperature, t = time, and V = is the spatial gradient operator. At the fire-beam interface, heat is transferred through radiation and convection. The heat flux on the boundary due to convection and radiation can be given by the following equation: qb : (hcon + h rad )(T — Tf ) [4.4] where hrad and hcon are the radiative and convective heat transfer coefficients, and are defined as: __ 2 2 km _ 4073(T + T, )(T + T,) [,5] Tf = temperature Of the atmosphere surrounding the boundary (in this case it is the fire temperature), 0' = Stefan-Boltzman constant = 5.67x10-8 (W /m2.°K4), and e: emissivity factor and it is related to the “visibility" of the surface to the fire. The heat flux and temperature gradient are related through Fourier's Law Of conduction: q = "kVT [4.6] Using Fourier’s Law, the governing heat transfer equation on the boundary Of the beam can be expressed as: 86 3T 3T k —n +—nz =—qb By y dz [4.7] where ny and nz = components of the vector normal to the boundary in the plane of the cross-section. The right hand side Of Eq. [4.7] is dependent on the type of boundary condition. Since the beam is exposed to fire from three sides, there are two types Of boundaries: 0 Fire exposed boundaries where the heat flux is governed by: qb = —hf (T — Ti) [4.81 o Unexposed boundary where the heat flux equation is given by: qb : _h0 (T _ T0) [4.9] where hf and h0 are heat transfer coefficient Of the fire side and the cold side, respectively. Tf and T0 = fire and cold side temperatures, respectively. By relating nodal temperatures through proper shape functions matrix {N} such that: T={N}T 'Te [4.10] the heat transfer equation (Eq.[4.3]) subjected to the boundary conditions (Eq. [4.7]) can be discretized as (ANSYS 2007): 87 CéTe + KéTe : Qe [411] t Kt e is the specific heat matrix. e is the thermal “stiffness" matrix and is the sum of conductivity and convection matrices. Qe is the applied nodal thermal load and is composed of the convective and radiative heat fluxes. Te is the nodal temperatures. The cross section Of the beam is meshed using two types Of elements, namely PLANESS and SURF151. Figure 4.2 shows the two elements and the typical discretization Of steel cross section for thermal analysis for different cases Of exposures. PLANESS is used as a solid plane element with two-dimensional thermal conduction capability. The element has four nodes with a single degree of freedom, temperature, at each node. This element is applicable to a two-dimensional, steady- state or transient thermal analysis. The temperature within PLANESS element is interpolated from the degrees Of freedom (Te = Ti,',k,l) using the following isoparametric function (ANSYS 2007): _(1— SXI —t ( )_ Ta.- n 4 (1:41;; ) _1—s)(1+t [4.12] _I where s and t are the isoparametric locus Of a point in the element domain. To carry out the numerical integrations, 2x2 integration points are used in PLANESS. 88 The SURF 151 element is generally used for various load and surface effect applications. The one-dimensional SURF 151 is overlaid onto the face Of PLANESS 2D thermal solid element to simulate the effect Of both thermal radiation and heat convection from the surrounding environment to the boundaries of the steel section (see Eq. [44]). Heat convection load was applied around the section, and a convection coefficient of hcon = 25 W/ (m2 .°C) was assumed in the analysis. The ambient (bulk) temperature on the node K Of SUR151 element was assumed to be equal to either the fire temperature (Tf) in case the boundary is exposed to fire, or tO room-temperature, in case the boundary is not exposed to fire. SURF151 uses linear interpolation function and two integration points for numerical integration. 4.3.2 Structural Analysis According to the principle Of virtual work, any virtual change Of the internal strain energy must be balanced by a change in the external work due tO the applied loads. OH = 6V [4.13] where U is the strain energy and V is the external work. Variation Of strain energy (6U) can be expressed as: 6U = I {dekdvol [4.14] vol Under fire conditions, the strain vector (5 ) is the sum Of mechanical (em), thermal (eth), and creep (scr) strains in steel, i.e.: 89 8 = 8m + 81h + gcr [4.15] n Variation of the external work (6V) due to the applied nodal forces ( F e ) can be computed by assuming a variation of nodal displacement {(54} as: 9V = {‘i‘IT IFenI [4.16] By relating nodal displacements (ue) Of the finite elements to the displacement field (u) through shape functions matrix {N} as follows: “e = {NIT °“ [4.17] the virtual work equation (Eq. [4.13]) can be rewritten in matrix form as (ANSYS 2007) K6 ue — F? = F," [4.18] th where K6 is the element stiffness matrix, and Fe is the element thermal load vector The finite element analysis is carried out using 8-noded SHELL93 element in ANSYS. Figure 4.3(a) shows the main features Of the SHELL93 element while Fig. 4.3(b) Shows the discretization of a beam-column for structural analysis. SHELL93 element is suitable for analyzing slender to moderately thick beam-column members, and it accounts for material and geometric nonlinearities, non-uniform temperature distribution over either the length Of the element and/or the cross-section of the element Material nonlinearities include nonlinear temperature-stress-strain curves, 90 m. high temperature creep, and temperature-dependent thermal strain. Geometric nonlinearities include large deformations and large rotations with bending capabilities in and out Of the plane of the element. 4.3.3 Modeling End Restraints Restraint conditions on the boundary Of the steel beams arise due to adjacent structural members. Rotational restraint (generally in the plane Of bending) is a function Of many factors including the continuity of the beam (to adjacent spans), rotational stiffness offered by the supporting columns, rotational stiffness of the beam-tO-column connection, and the extent of composite action that develops between concrete slab and steel beam. The axial restraint is also a fimction Of a number of factors including the continuity Of the beam, lateral stiffness Of the supporting columns, and the composite interaction between concrete slab and steel beam. The computation of these restraint stiffnesses (axial and rotational) at room temperature can be carried out using conventional methods Of analysis for indeterminate structures that can be found in structural analysis textbooks (Ghali et al. 2003). In this study, we assume that the axial and rotational restraints on the ends Of the beam are elastic, constant, and symmetric as shown in the illustration in Fig. 4.4. Further, the location Of the axial restraint force can vary depending on the type and configuration Of the connection between the beam and the column as shown in Fig. 4.5(b). The axial restraint stiffness may not pass through the centroidal axis Of the cross section, but may be eccentric by a distance y from the center Of geometry of 91 the section. In almost all previous studies, the location of axial restraint stiffness was always assumed to be in the centroid of the section. In order to account for the above factors, axial and rotational restraints are modeled using a set Of three axial springs on each end Of the beam. The three springs are arranged such that both axial and rotational restraint can be imposed on the beam as shown in Fig. 4.6. The top and bottom springs have an axial stiffness of ks,1 while the middle spring has an axial stiffness Of ks,2.The linear springs were modeled using COMBIN39 spring element available in ANSYS. It is assumed that these three springs has constant elastic stiffness properties Using the configuration shown in Fig. 4.6, the axial restraint stiffness (Ka) and rotational restraint stiffness (Kr) are Obtained as (Dwaikat and Kodur 2010): Ka = ks,1 +2st,2 [4.19] _ M _ F, ><(d/2+ y)+FB x(d/2— y) r 6’ lRad = 2k” tan(1Rad)[(d / 2)2 + yzl K [4.20] where ks,1 and ks,2 are the stiffnesses Of the spring elements, y is the eccentricity of the axial restraint stiffness from the geometric centroid Of the section. FT and F8 are the forces in the top and bottom axial springs as illustrated in Fig. 4.6. By changing the values of ks,1 and ks,2, different values of axial and rotational restraint can be imposed on the beam. In order tO use the 3-axial Spring approach for modeling rotational and axial restraints, the three spring elements must act on a rigid diaphragm. Therefore, to 92 simulate this effect, the following kinematic constraints equations are imposed on the restrained ends of the beam (Dwaikat and Kodur 2010): (44.), =(uxlm + (2n - zm lay) m lay) = lay) [4.22] n m [4.21] where ux represents the translational degree of freedom in the X axis, and By is the rotational degree of freedom about the Y axis. Z is the coordinate of the node. Figure 4.7 shows a schematic of this kinematic constraint condition where the subscript n refers to the nodes that are located on the end of the beam, and the subscript m refers to the master node. These constraint equations (Eq. [4.21] and [4.22]) force all the nodes on the restrained edge of the beam to follow the master node that is represented by the roller in Fig. 4.7, and hence results in a rigid diaphragm condition on that end. 4.4 SELECTION OF MATERIAL MODELS It was shown in Chapter 2 that significant variations exist among different constitutive models for high temperature properties of structural steel. These variations can lead to differences in predicted response of the restrained beams. Therefore, it is important to decide on the material models for use in finite element models. In the following sections, the selected constitutive models, for thermal and structural analyses, will be presented. The selection Of the constitutive models that will be used in the analysis is based on comparing the experimental results to 93 numerical results obtained from using different constitutive models for thermal and structural analysis. The results of the comparison are presented in the validation section (Section 4.5). 4.4.1 Thermal Properties The main thermal properties that influence the temperature rise in steel are thermal conductivity and specific heat. As previously illustrated in the literature review on high temperature material properties, there is not significant variation in the thermal properties as Specified in different codes and standards. Therefore, high- temperature thermal properties and thermal expansion Of steel specified in the Eurocode (EC3 2005) are used as input parameters to both thermal and structural analysis in ANSYS. 4.4.2 Mechanical Properties The literature review (Chapter 2) showed that there is a high variation in the high temperature mechanical properties Of structural steel as specified in different codes and standards. Also, the mechanical properties specified in codes and standards are mainly derived from “Old" test data and there is lack of information on the methodologies through which these tests were conducted. For example, transient effect of high temperature creep is not treated properly in most design codes and standards, despite the fact that it influences the response Of steel members under elevated stresses and temperatures (Kodur and Dwaikat 2009, Kodur et al. 2009). It is not clear how and tO what extent high temperature creep is included in the temperature-stress-strain relations for structural steel as specified in codes and 94 SIB? I‘ECI ana use fig IEY Ba Eli: ex] Ira standards (e.g. ASCE 1992, AISC 2005, EC3 2005). For example, while ASCE manual recommends explicit consideration Of high temperature creep in the structural analysis, the Eurocode states that "the eflects of transient thermal creep need not be given explicit consideration" (Clause 4.3.3(4) Of EC3 2005). Therefore, in an attempt to isolate the contribution of high-temperature creep to the response of restrained steel beams, temperature-stress-strain relations recommended by Poh (2001) were used in the analysis. Figure 4.8 shows the stress-strain model Of the structural steel at elevated temperatures as proposed by Poh. The generalized temperature-stress-strain curves developed by Poh were based on an intensive experimental program and on statistical studies. These curves were used to predict fire response of tested steel columns and found to be sufficiently accurate (Poh 1995 and 2001). Also, contrary to the confusion regarding the treatment Of creep in codes and standards, Poh clearly states that his stress-strain curves do not include the effect Of creep (Poh 2001) Based on a comparison between the constitutive models that will ensue in this chapter, temperature-stress-strain curves of Poh are used (in conjunction with an explicit high temperature creep model) in the structural analysis in case of a transient structural analysis is carried out, such as exposure to a design or standard fire time-temperature curve. However, when the analysis is not concerned with heating rate (i.e. when temperature is applied monotonically), the temperature- stress-strain curves of Eurocode 3 (EC3 2005) will be used in the analysis. This is 95 because Eurocode curves partially account for the transient effect of creep as will be demonstrated later in this chapter. 4.4.3 High Temperature Creep At present, most fire resistance analyses are carried out using Harmathy's high temperature creep model that is mainly based on Dorn's theory (Dorn 1954; Harmathy 1967). Dorn's creep theory assumes constant stress (6110's / dt = O ). However, in the case of restrained beams exposed to fire, the fire induced stresses vary considerably and rapidly with time and temperature, and these conditions are not captured by Dorn's theory. Therefore, an alternate high temperature creep model that can capture variable stress scenarios is required to account for these extreme conditions. 4.4.3.1 ANSYS Creep Model ANSYS provides 13 creep models that can be fitted empirically. Also, ANSYS provides nonlinear regression tools for fitting creep models to experimental creep test data. Implicit creep model number 11 is selected because it can model both the primary and secondary creep, and is capable of predicting creep strain regardless Of any coupling between time and either stress or temperature of steel. The fundamental theory for this creep model is based on the work of Zienkiewicz and Cormeau ( 1974) on unified theory on plasticity and creep strains. The model is presented in simplified format in ANSYS theoretical manual. 96 The selected ANSYS high-temperature creep model is given in a simplified format, as (ANSYS 2007, Kodur and Dwaikat 2009b): c c +1 —c /T [clas2t3 e 4 s AScreep____€ . + e = ‘eq Pmmw secondaw c3 +1 c —c /T {@036} 7 S [4.23] where c1 through c7 are coefficients that are dependent on type Of steel and are Obtained through nonlinear regression analysis to high temperature creep test data; with c1 = 6x10-6 per minute, c2 = 6.95, c3 = — 0.4, c4 =16500°C, c5 = 6.0x10—6 per minute, c6 = 6x10-5, c7 = 5x10-3°C, t = time at end of sub step (min), as = steel 0 creep _ . stress (MPa), TS = steel temperature (C), and A5“, = mOdIfied creep stram increment ANSYS calculates creep strain in incremental form in which creep strain increment is dependent on the creep strain at a given time step (t). First, the modified total strain vector at time step (t) is computed using the creep strain vector of the previous time step (t - 1) as: mod _ total t "' 81‘ plastic t thermal _ t creep 8 t—l [4.24] 8 “-8 8 The modified total strain (aim) is then used to compute the equivalent modified total strain as: l 2 2 2 ‘5 (6m0d_8mod) +(8mod_6mod) +(Emod_£mod) + €mod,total ___ l t,x t, y t, y t.z t.z 1.x [4.25] 8(1 flow) 3 Od 2 3 mod 2 3 mod 2 +§(y:flxy) 4.209042) +50”! 22:) 97 mod,total The equivalent modified total strain (Seq ) is used to compute the equivalent stress ( Ueq ) as _ mod,total 0.4, —- E8“! [4.26] where E is the temperature-dependent young modulus Of steel, and v is Poisson's ratio. This equivalent stress is used tO compute the equivalent creep strain increment creep (AEeq ) as a scalar quantity using ANSYS model for creep (Eq. [4.23]) and employing temperature at current time step (t). The scalar equivalent creep strain _ A creep . . Increment ( Seq ) 1S then converted to a full creep straIn tensor as follows: "A468" ’2 —1 —1 0 0 02;?“— Aej'e‘r 42-10008];d Aej’eep _ Age” —1 —1 2 0 0 0 eff“ Aegee” ' 2(1+v)ejq‘°°‘°‘“’ ° 0 0 0 3 0 0 fig“ [427] Aeff” 0 0 0 0 3 0 24'2" _Aeffepd _0 0 0 0 0 3__,,),_ creep creep creep creep creep creep where A8,, 2 A5), ,AEZ ,Aé'xy 9A5“ 9 and Agyz are the final creep strain increments in x, y, 2, xy, xz, and yz, respectively. elastic creep Next, the elastic strains ( XJ ) and the total creep strains (ex; ) are calculated as follows, using the example of the x-component: 98 x,t JCJ elastic _ mod _ creep 8 — 8 A8): [4.28] creep _ creep creep 8 _ 8 + A8x [4.29] x,t x,t-1 Comparison between ANSYS creep model and existing Dom-based creep models will be carried out in the following section. As a representative for Dom-based models, Dom-Harmathy modified model will be used to demonstrate the effect of non-trivial stress rate development ( das / dt 7"- O) on high temperature creep strain Of steel. 4.4.3.2 Calibration of ANSYS Creep Model High temperature creep test data is required to generate the seven coefficients (c1,...,c7) Of ANSYS creep model. Nonlinear regression fitting tool is available in ANSYS finite element software, and this fitting tool is dedicated for fitting creep test data in order to obtain the coefficients of the creep model. Performing creep tests at high temperature is very complex and time consuming. Therefore, there is a lack of creep test data in literature. However, two independent sets Of high-temperature creep test data were used to fit the ANSYS creep model (Kodur and Dwaikat 2009b). The first set of high temperature creep test data, shown in Fig. 4.9(a), is published by Nippon Steel Corporation (NKK 1989) for structural steel SMSOA (equivalent to A992 in American designation, or S355 in European norms). The second set Of high-temperature creep test data, shown in Fig. 4.9(b), is from tests carried out by Kirby and Preston (1988) for A36 (8275) steel. 99 The first set Of high temperature creep test data is from transient tests wherein temperature is increased at a rate of 10°C/min, while stress is kept constant. The second set is from steady state tests wherein temperature is kept constant at 550°C, while stress is kept constant. For both sets of test data, different values Of applied stress were used as shown in Fig. 4.9. The results of the calibration is plotted in Figure 4.9, which shows the comparison of high-temperature creep strain computed using Equation [4.23] with that of test data for various stress levels. The model shows good agreement with the test data for both types of A36 and Grade 50 steel. It can be seen from Figure 4.9(b) that Creep strain can become very Significant at high temperatures and increased stress levels. The calibrated creep model of ANSYS was integrated into the finite element structural model. 4.4.3.3 Comparison with Other Creep Models Calibrated ANSYS creep model is compared against Dorn-based creep models. dos/dt=0 Dorn’s creep theory assumes constant stress ( ), and states that creep strain is a fimction of temperature (TR), stress (as), and time (t). Creep strain in Dorn is derived using the following creep strain rate equation (Dorn 1954): =0 dc d0" cr = Sq) . ‘Q/TR , s dt [ (00] 6 dt [4.30] where the compound function S(as)=1 e as S"0'" Bo >1.4 [4'31] S S where B, n and Q are constants that depend on the type of steel. It should be noted that the above equations, as proposed by Dorn, were essentially devised from dos/dt=0 material tests Of metal alloys under constant stress ( ) and constant temperature increments (dTR ldt = O ). Harmathy extended Dorn's approach and proposed a revised creep model for structural steel (Harmathy 1967). Harmathy-Dorm modified creep theory is given as: de — Q/ T ——d:—r—=Z-coth2(bH/£O)-e R fordos/dt=0 [4,32] ' —Q/T 3 tZ . e R (time hardening rule) b H :4 [4.33] Zldecr /dt' - At (strain hardening rule) where (Z) is the Zener-Hollomon experimental parameter and is given for A36 steel as: 0.02602” as s 15000 psi Z =1 16 0.00030 [4,34] 1.23x10 Xe S 15000 x zm JLé<—Roller ..... 71.; 1“,: (9)9”? Fig 4.6: Kinematics for simulating end restraint in a beam 122 Fig.4 Fs, T/ F y,20°C Fs, r/ F y,20°C 1 Es,T \ le,20° C/Es,20°C l Eurocode 3 model Temperature-stress-strain curves Fig. 4.7: Stress-strain relationships adopted for structural fire analysis 123 1 _ 0.8 ~ 0.6 — fl \ E; l 0.4 7 \ \ L“ 503 model 0.2 - - - - - Poh model 0 V l l fl 0 250 500 750 1000 Temperature, °C Young elastic modulus - - - - Poh model 0.8 ~ > Yield point "E03" E 0.6 — ['1 > L“ 0.4 — 02 _ Proportionality limit "EC3" 0.0 f l l l l O 200 400 600 800 1000 Temperature, °C Yield stress Temperature-dependent reduction factors Fig. 4.8: Stress-strain relationships adopted for structural fire analysis 124 l 2.0 _ _ j 122.6 MPa * 93 MPa 3 . . 4/ 1.5 4— . * ' w . \o g . ANSYS creep model ' O E. 1 0 , o Creep test data 8 . 21": §- 73.6 MPa 2 63 (t5 .Af”’ , ‘ » 4" 49 MPa ‘ ‘WIF"""t:' ‘;‘ 0.0 v + ‘. . . .‘ 0 50 100 150 200 250 300 350 Time, min ( a ) Calibrating using Nippon creep test data (T=550°C) 1.0 ' I _ - o - — Creep test data (10°C/min) . . . ANSYS creep model ; 5 : 0.8 fl ’ i : ‘l 9 i = o\° 0:50 MPa :' . I: ch 0 6 “ :\ : ' E 0=100 MPa . ‘65 0 4 ‘ / CL . a - I p g 0'150 MP3 \ . 1‘ o ' I, 0.2 ~ » , .... .. . . .... . .. ......... 0.0 « f. . , I 0 150 300 45° 500 750 Steel temperature, °C (b) Calibrating using Kirby and Preston creep test data Fig. 4.9: Calibration of ANSYS high-temperature creep model for structural steel 125 Applied variable stress 5 a ~ - a— 160 4.5 _ (dos/dt)Avemge £0.22 MPa/nun $140 4 _ $13.5 1 “ 12° ‘3 .E 3 _ 1000. g E '5', 2.5 — 80 a; g 2 __60 g g b 01.5 I Test data g 40 (0 1 — ANSYS model 2 0.5 — “‘ ° 0 --- Harmathymodel 0 0 100 200 300 400 500 600 Time,min das/dt #5 0, Ts=550°C Fig. 4.10: Creep strains predicted using ANSYS and Harmathy models compared to test data 126 Measured Predicted Cool flange: - - - 900 __ Web: —I— —D— l-lot flange: + —A— o 750 - ° . I 254x146x7.5x12 9 600 4 3-sided : ‘5 450 a a g 300 ‘ l— 150 - o l T l o 19 . 20 30 Time, min. (a) Section 254x146x7.5x12 Measured Predicted Cool flange: —— — — - 900 J Web: _I._ —-D— l-lot flan e: _ . Q 9 750 ~ g a; 600 a 356x171 x9.5x15 ' . S I 3-sided E 450 4 8. g 300 5 l- 150 a O l i l 0 10 20 30 Time, min. ( b ] Section 356x1 71x9.5x15 Fig. 4.11: Comparing predicted steel temperatures with measured values from fire tests 127 Measured Predicted 1050 .. Cool flange: _ - - Web: —|_ —l:l— 900 ~ Hot flange: + ——A— 750 ‘ I 406x178x8x12.8 9 di‘ 5 600 4 3-sided fl 3 450 — a. E, 300 — '— 150 - ' fl 0 l j j 0 1O 20 30 Time, min. (c) Section 406x1 78x8x12.8 MeaSured Predicted 1050 1 COO' flange: — - - I Web: _I_ _n_ 9 900 r Hot flange: + —A— 2" 75° 5 305x165x8x12.8 a 600 — cu if» 450 — o. g 300 — " 150 ~ 0 1 r n 0 1O 20 30 Time, min. [d] Section 305x165x8x12.8 Fig. 4.11: Continued 128 l Furnace (fire) temp. l l l 1 Temperature, °C §§§§§§§§ l ‘ h‘\ '. ‘ ' . I ..”I ' I ', I 1' '- 5 \ .’ Temp. in bottom flange 0‘ \ .37\ Q ‘ \ "H s ' w. ‘ \ ......... “ I" ~ ~ . " . ' I‘-.-'."n .I-u _.. - ~ . Q ‘ . . ’ /.' ' . - d I / Test data 100 - I .-" TemP- l" tOP —-— 503 model . " . v" , 1;. " flange —o—ASCE model 0 fir T l T l T I j 16 20 24 Time, min 28 32 36 40 Fig. 4.12: Predicted and measured temperatures in the steel beam tested by Li and Guo (2007) CH Mill \I 305 305 1 ‘ x \ ‘ fi 1 \ Transition 460 B-B 355 A-A C’,‘ y / I7, '1 ’/ ,4 V ; l’ g . x —v _ ‘ V ~\ \ V \‘W \fi fl ‘/ A IV CI 460 C-C ;, \ ’l‘ \ A-A 655 l \'\ “ “\\ x \.x \\_\> w x ’1‘ ’i‘ B-B 460 v f \ \\ L.‘ ‘ 3 ,1. C-C 660 >02 l1 305 x. ‘C \ \ \ \ \\ \\ \ \ \ \ \\ ‘\ \ \ . ~x \ t < ‘__.‘ \‘ ~‘\ ,‘x _ \ x ,\ l \‘\ \ \ \ \ \“\ x ‘ x ~ _ .. . . ~ ~ \ ~ . \ \ \ ‘ \ \ \ \ ‘ \ \ x ‘ ~ x: \ \ ‘1 O . ‘ \ t \ \ \ \ x , \ ‘ \ \ \ \ \‘ ‘ ~ ~ 3 \ _ g \ x \\ \ 1 \ .y x‘ \\ ‘ \‘ ‘ ~ . ‘ ~ - \ ‘\ “ \ L\ \ ‘ ‘\ ‘ t t . x x‘ ‘A x t \ \ \ . V 1 \ ‘ L x‘ ‘ xx; \ \\\ \ ~ ‘ \ \\ i \ . \ x H e‘ x L \ X ‘ \ , \ / I . r / I ' ’ / ’/ Transition 610 7:<———> D-D v 380\ L—__l Fig. 4.13: Temperature zones for the validation and thermocouple locations 129 120° . — Experiment 1000 - --- Model § 300 ‘ 0 5 600 - ‘2’ 3. 400 é *- 200 o I V I I 0 60 120 180 240 Time (min) (a) C1-S column 1200 - , — Experiment 1000 - "- Model AWTC1ASTM E119 . 9 800 - E / 3 600 - wrcz ,5, fl '0' g 400 // t :27 0 , 1 l- 200 I, WTC3 /, ‘ ** \ WTC4 0 l fi I j 0 60 120 180 240 Time (min) (b) C1-Wcolumn Fig. 4.14: Recorded and predicted temperatures of tested columns (section 8-8) 130 a J Discretization of the structural model ":- t :r c) Deformed shaped of the model ( First principal stress] Fig. 4.15: Simulation of fire resistance test on restrained beam 131 Fire exposure time, min 0 51015202530354045 o l l l l I l T l 5 '5° ‘ —ASCEmodel E. 400 ~ ------- EC3model ,5 "on Test data a -150 — 8 ~. 3 -200 - ~ ~. ....... -2so — """" ''''' knenaucua -300 . 550 — __ _ _ ,J’ 4“ z 400 '5 x 3 250 ~ ,' .53 .’ 3,3: 100 5 9 Fire exposure time, min 3 -50 w l '1' l l l l l l 3 o 5 10 15 20 25 30 35 4o 45 a E '200 “ . —Poh mwe| 2 "t; —ASCE model 33'350“ ----E03model ———o---Testdata -500 — (a,b) Predictions using Poh (2001), ASCE and Eurocode temperature-stress-strain models Fig. 4.16: Recorded and predicted deflection and restraint force as a function of fire exposure time 132 Fire exposure time, min 0 51015 20 25 3O 35 4O 45 o l l T l T T —l l l '50 _ a Test data —ANSYS creep model g .100 - \. Harmathy creep model i t - - - - Nocreep .2 -150 ~ ‘ ‘5 2 \ “53-200 ~ .—-.-_......_ ..... ___ a #2 .250 a -300 , 600 — 0 450 ~ 2300 ~ :1 :3 E150 T -v-~" -' Flre ex ure time, min .2 o m 1 l l l 1 ms 1 l l ‘05 202530354045 .150 A 0 Test data _300 q —ANSYS creep model l-hrmatlly creep model -450 _ - - - -Nocreep (c,d) Effect of high-temperature creep on predictions ( using Poh constitutive model ) Fig. 4.16: Continued 133 Axial Displ. (mm) Axial Dlspl. (mm) E 2 £5 ‘E -5, 4* Test data _ g + Partial Simulation ‘2? -10 - + Full Simulation , -15 l l I 1 Time (min) “”9 (mm) a ] Axial deformation b ) Bending moment [at D-D) Fig. 4.17: Predicted and measured axial deformation and moment for beam-column Cl-S 15 , 4o 30 _ 10 1 A _ E 20 5~ 2' 0 _ _ ._ _g 5 0 : l ' ‘ ' E _10 J ,5 _ + Test data . 'E’ _20 l _._ Partial Simulation 0 _10 - Full Simulation 2 '30 i + _40 _l -15 . T . - -50 1 . . o 60 120 180 240 o 60 120 180 240 Time (min) Time (min) a) Axial deformation b ) Bending moment (at D-D) Fig. 4.18: Predicted and measured axial deformation and moment for beam-column Cl-W 134 a , - O _ - 4 _ - F--— Moment (kNom) Axial Displ. (mm) 0 _5 ' _ -e Test data _ _. Partial Simulation ‘ -10 _ Full Simulation -15 a l i T T l -150 T l j T T 1 o 20 40 60 so 100 120 0 2° 49l’lmealmlntlo 10° ‘20 Time (min) a ) Axial deformation b) Bending moment (at D-D) Fig. 4.19: Predicted and measured axial deformation and moment for beam-column C2-S 15 ~~—— 22km“ A 55 5 10 - 5 -1 0- i \Q -5 4 -0- Test data Partial simulation Axial Dlspl. (mm) Moment (kN-m) -10 1 " Full simulation '15 l l l l l‘ 0 20 4o 60 80 100 120 0 20 40 60 80 100 120 Time (min) Time (min) a) Axial deformation b) Bending moment ( at D-D) Fig. 4.20: Predicted and measured axial deformation and moment for beam-column CZ-W 135 Predicted fire resistance, nin. h o l O A Unconservative a 30 1 . I a ‘ 1': I Conservative A 20 4 J A EC3 model A ASCE model I Poh model no creep) 10 I Poh model with creep) 10 20 30 40 Measured fire resistance, min. Fig. 4.21: Effect of using different material constitutive models on fire resistance Deflection, mm is 2; § -120 .. -140 - Time, min 20 40 60 80 100 120 140 ) l IIIIIII Test data -—e— EC3 model —.—ASCE model _.__ Poh model —o— Poh model +ANSYS creep model - —o—— EC3 model + ANSYS creep model - '. ' ------- ASCE model + ANSYS creep model l i Fig. 4.22: Effect of using different constitutive models on fire response of Beam # 13 136 CHAPTER FIVE PARAMETRIC STUDIES 5.1 GENERAL The numerical model presented in Chapter 4 is applied to study the effect of various parameters on the fire response of restrained steel beams. The varying parameters included: load level, axial and rotational restraint stiffnesses, location of axial restraint force, thermal gradient, and fire scenario. The fire resistance analysis is generally coupled in one direction, i.e.: the results of thermal analysis affect the structural response, but the effect of structural response on the thermal analysis is neglected. Based on this, in the parametric study, the temperature will be applied on the steel beams without reference to time, i.e. rate- independent monotonic heating. The only exception is when the influence of fire scenario is studied, and thus heating is rate-dependent. In the analysis where heating is rate-independent, the Eurocode high temperature mechanical properties of structural steel are used in the structural analysis. This is because the temperature-stress-strain relationships of steel in the Eurocode partially account for high temperature creep in a form of additional stress-dependent plastic strains. However, in the case where heating is rate-dependent (i.e. in case of exposure to a 137 fire scenario), high temperature mechanical properties as per Poh constitutive model are used in conjunction with ANSYS creep model. The reason for using two different stress-strain curves is to avoid convergence complexities that arise due to the use of high-temperature creep models in the finite element analysis. In all cases of analysis, high temperature thermal properties of structural steel specified in Eurocode are used in the analysis. Results from the parametric studies on the structural fire response of restrained beams are presented in this chapter. 5.2 FACTORS INFLUENCING FIRE RESISTANCE A state-of—the-art review Clearly indicates that several factors govern the fire response of restrained steel beams. It was shown through qualitative parametric studies (Kodur and Dwaikat 2009, Dwaikat and Kodur 2009) that the main factors influencing the fire response of steel beam-columns beams are: 0 Load ratio 0 Rotational restraint stiffness 0 Axial restraint stiffness 0 Location of axial restraint 0 Thermal gradient 0 Fire scenario Detailed parametric studies were conducted to quantify the influence of each of the above factors on the fire resistance of restrained steel beams and the results are presented in Section 5.4. 138 5.3 DEFINITION AND RANGE or PARAMETERS In this section, the parameters influencing fire response of restrained steel beams are introduced and defined. Realistic range of these parameters is also discussed. 5.3.1 Load Ratio The load ratio (LR) is a measure of stress level in the beam just prior to fire exposure. LR is defined as the ratio between the maximum bending moment (Mmax) induced in the beam due to gravity load to the unfactored plastic bending capacity of the beam at room temperature, i.e. LR =Ex100% F Z y x [5.1] Since fire is a rare incident, the critical load combination is less than the ultimate load used for the room-temperature design of the beams. For a live-to-dead load ratio of 2, the critical load combination at fire conditions reaches 70% of the room- temperature capacity. Therefore, the LR was varied from 30% to 70% as shown in Table 5.1. 5.3.2 End Restraints The end restraint conditions are modeled using spring combinations as was described in Section 4.3.3. The resultant axial and rotational spring stiffnesses of the spring configurations (Ka and Kr) were varied independently so as to study the influence of both axial and rotational restraint conditions on the fire response. The 139 axial restraint ratio (AR) is defined as the ratio between the resultant axial restraint stiffness applied on the beam to the beam axial stiffness, i.e.: KG AR = ESAS /L [5.2] In the same fashion, the rotational restraint ratio (RR) is defined as: _ Kr E51,, /L [5.31 RR It will be shown through the parametric study that as the value of the applied AR and RR increases, the Change in the fire response of the beam diminishes. For practical cases of steel framed buildings, the value of AR and RR acting on the beam range between 10% and 20% (Purkiss 2007, Wang 2002). Therefore, the values of AR and RR that are used in the parametric study were varied according to Table 5.1. The axial and rotational restraints are assumed in this study to be elastic and constant. Also, the restraint stiffnesses are assumed to be invariable with respect to time of fire exposure. This assumption is justified for the cases where the beam-to- column connection and the columns are well insulated against fire exposure, and/ or the adjacent frames (to which the beam extends) are not exposed to fire. The real and assumed behavior for the case of rotational restraint for a typical connection is compared in Fig. 5.1. The restrained beam within a framed structure (Fig. S.1(a)) is idealized to a beam with end restraints as shown in Fig. 5.1(b). As observed in special tests on different connections under fire (Al-jabri et al. 2005), the rotational stiffness of the connection remains elastic until yielding (or local buckling) occurs in 140 the connection, which in turn causes a reduction in the rotational stiffness of the connection, as Shown in Fig. 5.1(d). In case of extended plate shear connections (Fig. S.1(c)), the rotation increases until the bottom flange of the beam hits the flange of the supporting column and this leads to further increase in the rotational stiffness of the connection, as shown in Fig. 5.1(d). However, in this study, the connection is assumed to be isolated from fire exposure and thus the variation in connection stiffness is not considered. Similarly, the axial stiffness mainly depends on the adjacent frames and supporting columns. If the adjacent frame is under fire, its lateral resistance will deteriorate and the lateral stiffness will decrease, as Shown in Fig. 5.1(e and f). However, if the adjacent frames are not under fire, the lateral stiffness will experience minimum variation and thus it can be assumed to be constant with time, as shown in Fig. 5.1(d and f). 5.3.3 Location of Axial Restraint The literature review revealed that in all previous studies the location of the fire induced axial restraint was assumed to be at the geometrical centroid of the beam. Location of axial restraint force can vary depending on the type and configuration of the connection between the beam and the column. For a simply supported beam, assuming the restraint location to be at the geometric centroid results in the fire induced axial force not generating any moment around the beam center. However, if the location of the axial restraint is not at the centroid of the section, then a significant bending moment might develop at supports due to the eccentricity of the 141 fire induced restraint force. This generated bending moment can affect the response of the beam-column under fire. The assumption that the line of thrust is along the centroidal axis is valid if the beam-to-column connection is similar to the type shown in Fig. 5.2(a) and the rotation of the connection is not large. If the rotation of the connection is large enough, the bottom flange of the beam will touch the flange of the column and this will generate higher compressive stresses in the bottom flange of the beam. In such a situation, the resultant line of thrust of the axial force will be shifted towards the bottom flange, where stress concentration is large. Other types of connections can cause a shift in the line of thrust of the fire induced axial force. An example is shown in Fig. 5.2(b), where “welded angle seat” connections are used to support the beam. Therefore, the lint of action of the fire induced axial force (at the support section) can vary depending on the connection configuration. For this purpose of study, the location of axial restraint force (y) was varied from zero (centroidal axis) to d/2 (parallel to the bottom flange). 5.3.4 Thermal Gradient Generally fire induced thermal gradients in steel beams (due to 1, 2, or 3-Sided exposure) are nonlinear and vary depending on the geometric properties of the cross section and insulation schemes. However, thermal gradients can be approximated linearly without significant loss of their actual influence on the fire response of steel beams (Dwaikat and Kodur 2009). Figure 5.3 shows the linearization of the thermal gradient in Steel sections exposed to fire. The average 142 temperature of the top and bottom steel flanges of the section in the linearized thermal gradient profile are computed as follows: Tav,s + Tt0p,NL 2 [5.4] Ttop,L = 3Tav,s — Trop,NL 2 [5.5] Tbot,L : where the average temperature in the entire section is computed as: T =i—_TS dA _ZT av, IA ”’5 jdA _ AS [5.6] where Tav and A5 are the average temperature and the cross sectional area of the entire steel section, respectively. Tav,i and Ai are the average temperature and cross sectional area of the i-th steel plate of the cross section. The subscripts NL and L refer to nonlinear and linear temperature profile in the section, respectively. The notations in the above equations are illustrated in Fig. 5.3. The linearization of thermal gradient is applied on a typical steel section (W24x76) exposed to a standard fire from three sides. The nonlinear and linear thermal profiles are compared in Fig. 5.4(a and c) for case of no insulation applied, and in Fig. 5.4 (b and (1) when the section is protected by 1- inch spray applied fire proof material. The results in Fig. 5.4 show that the maximum thermal gradient can be as high as 150°C between the top and bottom flanges of the section. Also, as seen in Fig. 5.4(d), 143 the linearization of thermal gradient is better approximation in the case when fire protection is applied as compared to unprotected section. It will be shown in the parametric study that using the linearized thermal profile will result in a conservation prediction of fire response. 5.3.5 Fire Scenario Fire scenarios used for structural fire design are generally grouped under two categories, namely; standard and design fire scenarios. The first type assumes a nearly constant or increasing heat flux from the fire to the exposed structural elements throughout the duration of fire exposure, while the second type represents an increasing heat flux first and then a decreasing heat flux from the fire to the structural members, and thus gradual reduction of fire temperature after achieving a maximum value. The first type of fires requires continuous supply of fuel and oxygen, while the second type can be either fuel or ventilation controlled or both. In this study, the first type of fire is represented by ASTM E119 standard fire curve, while for the second type of fires scenarios; the parametric fire curves Specified in Eurocode 1 (EC1 2005) are used. ASTM E119 scenario, which does not have a decay phase, represents a baseline case for comparing the response of the steel beam under different fire scenarios. Two parametric fires will be used; fire scenario 1, representing a severe fire condition compared to ASTM [3119, and fire scenario ll, representing a mild fire severity. The time-temperature curves for the three fire scenarios are shown in Figure 5.5. The time-temperature curves of fire scenarios are assumed to include the effect of heat radiation and convection from the ambient air 144 and surrounding walls of the compartment. It is beyond the scope of this study to investigate the development of these fire scenarios. The design fires, 1 and H, were selected to represent different cases of fire severity. Fire 1 represents a typical fire scenario that can occur in a compartment with the following characteristics: Total area of compartment = 320m2, opening area = 10 m2, floor area =100m2, average opening height = 1.5m, thermal inertia “b factor" = 500 ]/ m2$0.5K, fire load density = 500 M]/ m2. For the mild fire scenario (fire ll), the same compartment characteristic (of fire I) were assumed with the only change is in the value of thermal inertia “b" factor, changed to 1500 ]/ m250.5K. The thermal inertia factor “b” is a measure of how much heat is absorbed by the compartment boundaries, and this is dependent on the thermal properties of the material of walls, ground, and ceiling (Buchanan 2002). Smaller values of “b" indicate that smaller amount of fire generated heat flux is absorbed by the boundaries, and thus, greater amount of heat flux goes to increasing the temperature of the atmosphere of the compartment, and hence increasing the fire severity. 5.4 RESULTS or PARAMETRIC STUDY Results from the parametric studies are presented in Figures 5.6 to 5.16 where time-deflection (or temperature-deflection) curves for the analyzed beams are plotted in addition to the fire-induced axial force. The effect of the studied parameters on the thermal response of the beam is not presented here because many of these parameters such as load ratio and restraint conditions do not 145 “IE! influence the thermal response of the beams. The effect of each of the parameters on the fire response is discussed bellow. 5.4.1 Effect of Load Ratio To investigate the effect of load on fire resistance, the beams were analyzed under three load ratios, namely, LR = 30%, 50% and 70%. Figure 5.6 shows the effect of load ratio on the response of a simply supported beam exposed to uniform temperature. Generally, higher load ratio results in higher deflection, and this is due to higher stresses developed the beam prior to fire exposure, and this causes earlier spread of plasticity in the beam compared to the cases with lower load ratio. The influence of load ratio on the response of restrained steel beam is presented in Fig. 5.7. The fire induced axial force, due to restraint effect (shown in Fig. 5.7(b)) has a direct influence on the midspan deflection as Shown in Fig. 5.7(a). Since the beam is restrained form expanding, compressive axial force develops in the beam, and this causes a reduction in the overall geometric stiffness of the beam. Also, as the axial force increases, the stresses in the beam increase until plasticity is initiated in the beam and this occurs at the moment of the peak value of the compressive fire restraint force. Plasticity causes a severe reduction in the beam stiffness which leads to higher deflection that in turn leads to increased midspan bending moment due to increased P-delta effect on the beam. This continues until the deflection in the beam is so large that the fire induced force switches from compression to tensile force. The peak value of the fire induced force is smaller for higher load ratios. Since this peak value of the restraint force corresponds to the initiation of plasticity in the 146 beam (no local buckling in elastic range), the beam with higher load ratio (higher initial stress level prior to fire) will experience plasticity at earlier stages of fire exposure. As expected, higher load levels produce larger downward deflections in the restrained steel beam. On the other hand, the detrimental compressive axial force, which develops due to thermal expansion of the steel beam, reduces with the increased load level. However, the increasing load level produces larger deflection, which makes the beam go into catenary action at earlier stages of fire exposure. Hence higher tensile forces develop in the restrained steel beam under catenary action as the load level is increased. 5.4.2 Effect of Rotational Restraint To investigate the effect of rotational restraint on the fire resistance, the beams were analyzed under four rotational restraint ratios, namely, RR = 0, 10%, 30% and infinity. Two sets of results are presented here, first set is for beams with no axial restraint (AR = 0), while the second set of results is for beams with AR = 10%. Results from two cases are plotted in Fig. 5.8 and 5.9, respectively, as function of steel temperature. The increase in rotational restraint leads to a Change in the shape of room- temperature bending moment diagram. To compensate for this change in bending moment along the span of the beam, the value of the actual applied load (w) was varied such that load ratio (LR) remains constant under different rotational 147 W restraint ratios (RR). For the two sets of results shown in Fig. 5.8 and 5.9, the load ratio was maintained at 50%. The results Shown in Fig. 5.8 indicate that higher rotational restraint ratio leads to improved performance under fire. This can be attributed to the fact that higher rotational restraint stiffness leads to greater overall stiffness of the beam. The results also Show that it requires small value of rotational restraint (RR = 30%) to cause a significant improvement in fire response. The effect of increasing rotational restraint on the fire response of axially restrained beam is shown in Fig. 5.9. Similar to the trend of results in Fig. 5.8, the improvement of fire response is also observed in the results plotted in Fig. 5.9. The occurrence of local buckling in the post-yield stage in the bottom flange of the support section slightly affects the response of the rotationally restrained beams as shown in Fig. 5.9(a). The increase in RR does not affect the fire induced axial force in the elastic range, however, higher values of RR lead to lower tensile catenary forces in the beam, as seen in Fig. 5.9(b). For small values of RR, the failure mode in the beam results from a combination of tensile and flexural failures. However, as the value of RR increases, the failure mode becomes more related to flexure and the beam fails by developing simultaneous plastic hinges at midspan and support sections, rather than by reaching full tensile capacity as in the case of a beam with axial restraint only, as it will be described in the following section. 148 5.4.3 Effect of Axial Restraint Figure 5.10 shows the mid-span deflection of a restrained beam as a function of fire exposure time for different values of axial restraint (AR) stiffness, i.e. AR = 0, 10%, 30%, and infinity. The AR was varied for two cases, first with no rotational restraint (RR = O), and second with RR = 25%. It can be seen in Fig. 5.10 in the elastic range of fire response, increasing AR leads to insignificant increase in midspan deflection as seen in Fig. 5.10(a). However, with the buildup of compressive force, as shown in Fig. 5.10(b), plasticity starts to spread in the section and larger deflection is induced in the beam by higher values of AR. The increased deflection in this stage is mainly attributed to softening of steel, P- delta effect, and the negative influence the compressive fire induced axial force on the flexural stiffness of the beam. Despite the relatively larger midspan deflection in the beam that is caused by increasing AR, the ultimate bearing capacity of the restrained beam is greatly enhanced due to restraint effect as seen in Fig. 5.10(a). This enhanced response is mainly due to the development of tensile catenary action which leads to increased overall stiffness of the beam. Similar pattern of the influence of AR on the rotationally restrained beams is observed in the results plotted in Fig. 5.11. The plastic tensile capacity profile of the beam (ky(T) x Fu x As) is also shown on figures that plot the fire induced axial force as a function of fire exposure time. This profile represents the maximum tensile axial force that can possibly be attained by the beam. As shown in Fig. 5.10(b), the beams failed when the fire induced axial force almost reached the maximum plastic axial capacity of the beam (ky(T) x Fu x 149 As). In the case of rotationally unrestrained beams (RR = 0, Fig. 5.10), the development of plastic hinge in the mid span of the beam did not cause failure and the load bearing mechanism changed from flexural to cable mechanism. And therefore, failure occurred in the beams when the catenary tensile force reached the maximum plastic axial capacity of the beam (ky(T) x Fu x As). However, in the case of rotational restrained (RR 5‘ 0, Fig. 5.11), the failure occurred before developing full plastic tensile capacity of the beam in the catenary phase. This is because for rotationally unrestrained beam, strength failure requires the development of three plastic hinges simultaneously. The strength failure of these rotationally restrained beams occurred when the combined effect of tension and bending (on the support section) exceeded the combined plastic axial and moment capacity of the section. It is worth noting that in both cases (Fig. 5.10 and 5.11) the occurrence of local buckling has minor influence on fire response of the restrained beams. The occurrence of local buckling leads to either a slight increase in the midspan deflection, or a change in the rate of deflection due to localized reduction of flexural rigidity of the beam near support. Also, occurrence of local buckling leads to a drop in the fire induced axial force due to relaxation effect. As repeatedly observed in fire tests and actual fire incidents (discussed in Chapters 2 and 3), occurrence of local buckling at the bottom flange of the beam near support does not lead to failure in the restrained beams. 150 5.4.4 Effect of Location of Axial Restraint To investigate the influence of varying the position of fire induced axial restraint force on the response of beams; the location of axial restraint force was varied from the mid-height of the web (y = 0) to the mid-height of the bottom flange (y = d/Z). Two sets of results are presented in Fig. 5.12 and 5.13; first set is for uniform heating, while the second is for a linear thermal gradient of 200°C. Trends of results in Fig. 5.12(a) indicate that the fire response of a restraint beam improves when the axial restraint is located at the bottom flange (case y = d/Z). The improvement in fire response can be attributed to the development of additional hogging moment at the support of the beam (P x y). The hogging moment leads to a reduction in the total moment in the mid-span of the beam, and thus, enhances the response of the beam. This is illustrated in Figs. 5.14 which show the development of thermal gradient and restraint force for mid-span and support sections in the beam exposed to fire. When the beam is exposed to fire from three sides (top side is unexposed), the bottom-half of the steel section will experience higher temperatures as compared to the top-half of the section. This leads to higher expansion in the bottom—half as compared to the top-half of the section, leading to the development of large axial restraining force (P). This fire induced axial force generates a counter-moment at the support (PXd/Z) that opposes the sagging moment (MGravity+ MP- 6) at the mid-span of the beam as shown in Fig. 5.14. This leads to a reduction in the total moment in the mid-span, and therefore, enhances the fire resistance of the beam. However, when the deflection of the beam becomes large, the mid-span moment (MGravity+ MP- 6') becomes very large (due to 151 increased P-6 effect) as compared to the moment due to fire induced force (PXd/Z), therefore, the improvement due to the shift of the axial restraint is diminished. Local buckling occurs in the bottom flange for the case y = d/Z, as indicated in Fig. 5.12(b), and this is due to the concentration of stresses in the bottom flange due to restraint effect. The occurrence of local buckling leads to a reduction in fire induced compressive force, but the overall fire response is marginally affected by the occurrence of local buckling. In the case of the beam with thermal gradients, better improvement in fire response is observed as shown in Fig. 5.13(a). Thermal gradient leads to higher expansion in the bottom-half of the section, as compared to the top-half. Therefore, larger fire induced axial restraining force (P) is generated in the beam for the case of y = d/Z as compared to the case of y = 0. While thermal gradient caused relatively larger elastic deflection in the beam with y = 0, thermal gradient caused an upward “bowing effect" for the beam with y = d/Z, as seen in Fig. 5.13(a). However, due to the development of high compressive stress in the bottom flange (case of y = d/Z), local buckling occurs at post-yield (Fig. 5.13(b)) and this causes a sudden increase in the midspan deflection with a corresponding drop in the fire induced axial force. 5.4.5 Effect of Thermal Gradient In order to study the influence of thermal gradient on the fire response of restrained steel beams, thermal gradients are applied on the beam. Two values of thermal gradient were used: zero and 200°C and the average temperature of the beam was 152 maintained constant throughout the analysis The structural parameters of the restrained beam and the results of the analysis are Shown in Fig. 5.15. Results show that that increasing thermal gradient leads to increased elastic deflection of the beam, as shown in Fig. 5.15(a). This can be attributed to the fact that thermal gradient leads to the generation of additional thermal curvature in the beam, and this additional curvature increases the midspan deflection of the beam. Since the average temperature in steel was maintained constant during all the three cases of analysis, small variations in fire induced compressive force were observed, as seen in Fig. 5.15(b). However, the influence of thermal gradient on the elasto-plastic deflection of the beam is less pronounced. This can be attributed to the fact that once yielding spreads in the beam, the rotational stiffness of the beam will reduce and thus thermal curvature will leads to smaller deflection in the beam. 5.4.6 Effect of Fire Scenario To study the influence of fire scenario on restrained beam response, three types of fire exposures were selected for analysis, namely: ASTM E119 standard fire, design fire l and design fire ll. Typical steel beam, with characteristics shown on Fig. S.16(b), is subjected to the three fire scenarios whose time-temperature curves are shown in Figure 5.5. The beam has 15 mm spray-applied fire proof insulation and is exposed to fire from 3 Sides, with concrete slab on top of the beam. The thermal properties of steel and concrete are assumed to follow Eurocode and the physical and thermal properties of insulation are as presented in Chapter 4. For the 153 structural analysis, high temperature mechanical properties of structural steel as per Poh constitutive model (2001) and ANSYS high temperature creep model are used in the analysis. Figure 5.16 shows the temperature in steel section due to exposure to the three fire scenarios. The results Show that a maximum of 200°C thermal gradient between the top and bottom flanges of the beam had developed in the beam due to the three side fire exposure. Factors affecting the development of such thermal gradients include the depth of the steel beam and the presence of concrete slab which acts as a heat sink for the beam. In the case of a standard fire exposure (Fig. S.16(a)), the thermal gradient in the beam reduces as fire progresses. This can be attributed to the fact that the steel beam tends towards thermal equilibrium as fire progresses. The plateau that occurs at steel temperature between 750-800°C (Fig. S.16(a)) is attributed to the phase change that occurs to the Structural steel and this process absorbs a considerable amount of heat. In the case of design fires (Fig. 5.16 b and c), the maximum average temperature in steel generally occurs during the cooling phase of the fire. This is mainly attributed to the thermal lag effect and redistribution of heat inside the steel section in addition to the influence of heat stored in the insulation. Thermal gradient reduces after achieving maximum temperature in steel and then it reverses direction. This reversal of thermal gradient can be attributed to many factors including the redistribution of heat due to two dimensional heat conduction in the steel section, the effect of concrete slab, which will supply the top flange with heat as it cools down (reverse of heat flux direction), and the effect of insulation in slowing down the heat dissipation. 154 The asn anal the par: TESl SCE] but def iO' Th sta pr hi fii The results of thermal analysis (from each fire scenario individually) were applied as nodal temperatures on the structural mesh of the restrained beam. Structural analysis is carried out to investigate the influence of fire scenario on the response of the restrained. The results of the structural analysis as well as the structural parameters of the analyzed beam are shown in Fig. 5.17. It can be seen that the resulting mid-span deflection and the axial restraint force are dependent on the fire scenario. The restrained beam experiences run away failure under ASTM E119 fire, but no failure occurs in the beam under design fire scenarios. The midspan deflection of the beam is larger under the growth phase of design fire I as compared to the midspan deflection under ASTM E119 or design fire 11 as seen in Fig. 5.17(a). This is because the growth phase of design fire I is more severe than the ASTM standard fire or design fire II, and this leads to rapid increase in temperature and larger thermal gradients, and thus faster degradation of steel strength and stiffness properties. For a similar reason, the fire induced compressive force in the beam is higher when it is exposed to design fire 1 as compared to when exposed to design fire 11, as shown in Fig. 5.17(b). Local buckling occurred in all fire exposures, with earlier occurrence of local buckling under design fire I, and this is reasonable because of higher thermal gradient and temperatures under design fire I which leads to earlier spread of plasticity in the beam as compared to exposure to other fire scenarios. While the beam failed under ASTM [3119 fire exposure, the beam survived the two design fire scenarios, but with the maximum attained deflection in the beam occurring under exposure to design fire II, as shown in Fig. S.17(a). This can be attributed to the fact 155 that the beam under design fire 11 experienced slow cooling as compared to design fire I. By referring to Fig. 5.16(b and c), beam under design fire ll sustained an average temperature above 500°C for more 70 minutes (between 40-110 min.), while the beam under design fire 1] sustained temperature above 500°C for almost 40 minutes (between 30-70 min.). Therefore, in case of exposure to slow heating process (such as design fire 11), the spread of plasticity and the influence of high temperature creep have a major effect on the fire response of the restrained steel beams. The effect of cooling phase in design fire is also shown in Fig. S.17(b), where the tensile force (due to catenary action) increases when the beam enter the decay phase of design fire, as seen in Figure S.17(b). This is because steel regains much of its strength upon cooling (as a result of decay phase in design fire I & II), and starts to contract in cooling phase. In the case of ASTM E119 standard fire scenario, deflection continues to increase with temperature due to the absence of decay phase until the beam enters a catenary action phase at around 60 min. into fire exposure (see Fig. S.17(b)). The tensile force that develops in the beam due to the catenary action results in an improvement in response. However, due to the continuous deterioration of the strength and stiffness properties of steel and due to the increasing influence of high temperature creep, deflection of the beam continues to increase until the full tensile capacity of the beam is exhausted at around 93 min. into fire exposure. In case of a design fire, the tensile force that results from catenary action (that results due to large deflection) leads to improvement of fire response as seen in Fig. S.17(a and b). Also, fire response of restrained beam is 156 further enhanced due to the tensile force that develops due to the shrinkage of steel in the cooling phase. 5.5 SUMMARY This chapter presents the influence of various factors on the fire response of restrained steel beams. The studied parameters are: load level, axial and rotational restraint stiffnesses, location of axial restraint force, thermal gradient, and fire scenario. Data from parametric studies indicates that fire scenario, load ratio, and end restraints have significant influence on the fire response of restrained steel beams. Increasing load level leads to earlier deterioration of fire response of steel beams. Axial restraint leads to increased deflection at early stages of fire exposure; however, at later stages of fire exposure, the response of steel beams is significantly improved due to the effect of axial restraint. Increasing rotational restraint leads to improved fire response of a steel beam since rotational restraint helps redistribute the bending moment in the beam. While restrained beams always fail under standard fire exposures, design fires result in recovery of strength, stiffness, and deflection of the restrained beam as a result of the cooling phase, and thus improved fire response. Data from parametric studies also showed that thermal gradient, location of restraint force have a minor effect on the fire response of restrained steel beams. While thermal gradient leads to a slight increase in midspan deflection, the location of axial restraint force leads to a counter bending moment at the support which leads to a slight improvement in fire response. Results from the parametric 157 studies are utilized in Chapters 6 and 7 to develop guidelines for evaluating fire resistance of restrained steel beams. Table 5.1: Characteristics of the beams used in parametric study of finite element analysis Parameter Values Comments Load ratio (LR) 30%, 50%,70% LR = M max / (F y Z x ) Thermal gradient (A7) 0°C, 200°C AT = 77‘ F - TBF ASTM E119, Severe Fire scenario (Fire 1), Moderate -- (Fire II) Measured downward Location of restraint O, d/Z form the centroid of the section Axial restraint ratio 0, 10%, 30%, and . . . Relative to room- (AR) infimty temperature beam Rotational restraint 0, 10%, 30%, and stiffnesses ratio (R19 infinity 158 (a) Layout of typical Steel frame Ka, Kr w Ka, Kr K L ' ......V. (b) Beam with idealized restraints : A U: E ---------- ‘5 GE) T = 20°C g T = 300°C Real behavior T = 500°C Rotation ”0" . _ , (d) Variation of M-6 curves with (c) Idealized rotation of connection temperature of conn ection , Assumed ff“, behavior 0 cc up» ’1 ............... ‘ Displacement :8“ """" Real behavior T = 300 °C T = 20°C V ' tio P-6 'th [e] Idealized lateral displacement 0) arm n of curves W1 temperature of adjacent frame Fig. 5 - 1 : Assumed and real behavior of restraint stiffnesses in restrained beams 159 .-¢-t""15'::. .‘Qs l . . l J Connection configuration[ I Conceptual model :2?- 1:- 7:" ‘12:: ..IT“~-’ 7’ [are 1 l 300; ‘ ._. -._.._............ {fa} [ Centroidal axis ““‘T‘-—--L-—/"' I I l (a) Single plate shear connection. Connection configuration Conceptual model Centroidal axis \N t I; ,‘fl-‘LJ h:::) ) l — ( b ) Welded angle seat connection. Fig. 5.2: Connection types and location of axial restraint in steel beams 160 (=5min 10,20 30 ...,t=60mln. t:50mln,100,150,..., t=400mln. “Xx. 1 \ xx : 0.5 ~ 0.5 - Sea- Sax— “ R g 0 1 g 0 '1 5.0.25 “ 5’0-25 "‘ -0.5 - -O.5 — 0 200 400 600 800 1000 0 150 300 450 600 750 900 Temperature, °C Temperature, °C a ) Unprotected steel section ( b ] 1-inch insulation thickness with concrete slab on top Nonlinear thermal gradient in the steel section t=5min 10,20 30 ...,t=fl7mln. i=50mln,100,150,.,., i=400mln. new \. w. 1 '\. '0‘ '5 ‘. K ‘ 0.5 1 ‘ Ease Xxx \\::3 3a%— ‘ R . . , . . R , g o — g 0 ~ '. .g-O 25 - .g-O 25 - - o ' , , a l o5- f f "" eel O 200 400 600 800 1000 0 150 300 450 600 750 900 Temperature, °C Temperature, °C c) Unprotected steel section (b) 1-inch insulation thickness with concrete slab on top Linearized thermal gradients for unprotected section Fig. 5.3: Comparison between nonlinear and linearized thermal gradients in restrained steel beams ASTM E119 fire Design fire I Design flre II Temperature, °C o l l l I l l l l l 0 20 40 60 80 100120140160 Time, min. Fig. 5.4: Time-temperature curves for the selected fire scenarios 161 Steel temperature, °C 0 200 400 600 800 —LR = 30% .400 4 —-—LR=50% —— LR = 70% -600 - _ .1—1 Sect:W24x76, L = 9 m (RR = AR = 0) Midspan deflection, mm. do 8 Fig. 5.5: Effect of load ratio on response of simply supported steel beam exposed to uniform heating Steel temperature, °C 0 150 300 450 600 750 900 o —T‘—_—x 1 T m e- 1‘...““ oca E. -200 — Buckling 5:3 —LR = 30% g '40“ ‘ —-—LR = 50% g — LR = 70% C -600 a — 3. H fl Section W24x76, L = 9 m 2-800— RR=AR=10°/o a) Midspan deflection Fig. 5.6: Effect of load ratio on response of restrained steel beam exposed to uniform heating 162 1 300 _ Start of 1500 _ yielding 1200 ~ 900 a 600 a 300 ~ Local Buckling l l l '300 ‘ FuxAsxkyfl') -eoo — \ -900 ~ 0 200 400 600 800 1000 Steel temperature, °C Fire-induced force, kN b] Fire-induced axial force Fig. 5.6: Continued Steel temperature, °C 0 200 400 600 800 1000 Local :- '200 _ buckling o -400 - _ RR = 0 g '600 RR =10% 'a .300 - —I—RR = 30% ------- RR: °° H 2-1200 « E - Section W24x76, L = 9 m, AB = 0, LR = 50% Fig. 5.7: Effect of rotational restraint on response of axially unrestrained beam under uniform heating 163 ['0 8:: egg Midspan deflection, mm. 'a; s 1250 — Steel temperature, °C 0 200 400 600 800 1000 ~ Local 1 7 buckling L l l I Section W24x76, L = 9 m, ‘ AR =10%, LR = 50% a) Midspan deflection Local buckling FuxA,xky('l)~ 200 400 600 800 1000 Steel temperature, °C b) Fire induced axial force Fig. 5.8: Effect of rotational restraint on response of axially restrained beam under uniform heating 164 g. an i . Midspan deflection, mm. Steel temperature, °C 200 400 600 800 1000 xx» Local “ \\ buckling AR =10% (web) +AR = 30% ------- AR: °° l H # Section W24x76, L = 9 m, RR = 0, LR = 50% a) Midspan deflection buckling 0 200 400 600 800 Steel temperature, °C b) Fire induced axial force Fig. 5.9: Effect of axial restraint on response of rotationally unrestrained steel beam under uniform heating 16S Steel temperature, °C 0 200 400 600 800 1000 ' 0 l l 2...... t“:- " uc In 5 -400 . ~ .‘ 9 AH=0 ‘ 3 6°“ 2213:? ~~ .0 '800 _ -—I— = o c 1000 ....... AR: .. - _ it... H“ 2 1 Section W24x76, L=9m, ' 40° ‘ RR: 10%, LR=50°/o a) Midspan deflection 2500 —- , — - ~. L l ' 3. oca E 2000 - .' ‘. , . ; buckhng 8 1500 — - h .2 1000 — g 500 - :3 1, 0 'E 500 2 LL-1000 7 0 200 400 600 800 1000 Steel temperature, °C b) Fire induced axial force Fig. 5.10: Effect of axial restraint on response of rotationally restrained steel beam under uniform heating 166 0 Steel temperature, °C 200 400 600 800 1000 -550 e -700 ~ -850 — Midspan deflection, mm. -1000 — l I l l _ yzdfl _ y=0 i ll:llllllllllllllllllllllllllli|;.l Section W24x76 , L = 9 m, LR = 50% RR = AB = 10% yflZ'I'a” .4-0- Uniform temperature Fig. 5.11: Effect of location of axial restraint on the response of uniformly heated steel beam 167 Average steel temperature, °C 0 200 400 600 800 1000 r -1oo — y = d /2 -250 — ill—Ill Local -400 g buckling Section W24x76 , ~55O — L = 9 M, LR = 50%, RR = AB = 10% -700 — yiL'I'k -1000 — AT: 200°C Midspan deflection, mm. Fig. 5.12: Effect of location of axial restraint on the response of steel beam subjected to thermal gradient 168 Load “w “in; :-:-' £055 :~..-:-:-:.:-:- (0) Restrained beam exposed to fire [1 C..<— DMGravity + MP-6 Temperature Stress / Force (1)) (bSection at mid-span Location of restraint y y=d/2 I (c) Section at support Fig. 5.13: Development of fire induced restraint force in a restrained steel beam 169 Average steel temperature, °C 0 200 400 600 800 1000 l l I — AT= 0°C -100 — 4 E. — AT = 200° C E '25" ‘ H Local g“ _400 _ buckling “3 Section W24x76 , g .550 . L=9m,LR=50%, ‘5 RR = AR = 10% g -700 ~ ' .8 ........ ' .- E -850 — I ‘ t .0. 4000 - AT a) Midspan deflection Fig. 5.14: Effect of thermal gradient on fire response of restrained steel beams 170 1200 "i 15 mm insulation w24x76 1000 l —Flre curve Bottom flange —c— Middle of web ------- Top flange 0 r l T j T Temperature, °C a e e g Temperature, °C ._§§§§ l l 0306090120150180 0305090 Time,min. 120 150 180 Time,min. (a) AS TM E1 1 9 standard fire (b) Design fire 1 9m _ § l Temperature, °C § 5 § .5 0| 0 0 30 60 90 120 150 180 Time, min. (c) Design fire 11 Fig. 5.15: Steel temperature in a restrained steel beam under different fire scenarios 171 Fire exposure time, min. 0 20 40 60 80100120140160180 o I I l l l— j fl 1 . — H 2450 — .. W24x76, L = 9m, LR = 50% § 000 - .. AB = RR =10% *6 g . g _450 - , Design fire I 3 DeSign fire II 7 t: -600 ~ - ' " a S 2450 ~ ASTM E119 fire -900 — a ) Midspan deflection Fire exposure time, min. 0 20 40 60 80100120140160180 1200 — Local . . . wI bucklin ------- DeSIgn fire I I/\ g o l 300 ‘ ' . —o——DeSign fire ll .'/' 400 4 ; ASTM E119 ,kN 60$ 88:: -1200- g ' -1600o ' Fire Induced axial force '3 ux A sxky('l7 b) Fire induced axial force Fig. 5.16: Effect of fire scenario on the response of restrained beams 172 CHAPTER SIX METHODOLOGY FOR EVALUATING PLASTIC P-M CURVES UNDER THERMAL GRADIENTS 6.1 GENERAL As illustrated in Chapter 2, there are major drawbacks in the current approaches and code provisions for evaluating fire resistance of steel beam-columns. A rational methodology is proposed in this chapter to overcome some of these drawbacks. The methodology, derived based on heat transfer and structural mechanics principles, takes into consideration the influence Of thermal gradients on the plastic P-M diagrams of steel beam-columns. First, the thermal gradient in steel section is quantified and then the plastic P-M diagram of the section is adjusted accordingly. The validity of the proposed approach is established by comparing the fire response predictions from the prOposed approach with values obtained from rigorous numerical studies as well as from fire resistance tests (on beam-columns). Also, predictions from the proposed methodology are compared with fire resistance estimates from current codes of practice. The applicability Of the proposed approaches to design situations is illustrated through numerical examples. 173 6.2 EVALUATION OF STEEL TEMPERATURE In order to quantify fire induced thermal gradient along the depth of the cross section, simplified method for computing temperature in steel section as a function of exposure time is presented. Once the thermal gradient is quantified, it will be used for modifying the shape of the plastic P-M diagrams Of the steel cross section. Therefore, this section will present the development of simple equation for predicting temperature in steel. 6.2.1 General The generalized equation for heat transfer from fire to the structural member can be expressed as (Purkiss 2007): 2 0T kV T+Q=pC—a't— [6.1] in which Q is the total heat flux vector which is comprised of two parts; namely; convective heat flux (Qcon) and radiative heat flux (Qr) as follows: _ _ 4 4 Q _ Qcon + Qrad _ hcon (T - Tf ) + 404T — Tf) [6'2] The terms in Eqs. [6.1 and 6.2] are defined in Chapter 5. Generally, Eq. [6.1] is simplified into one dimensional problem and then finite difference method is applied for solving the simplified equation. Eurocode 3 (EC3 2005) presents incremental finite difference relations for solving Eq. [6.1] with adjustments made in order to compensate for the simplification. However, since the Eurocode relations are based on finite difference solution, the accuracy of the Eurocode relations is 174 highly dependent on the size of time increment, in addition to them being complex and require intensive use Of spreadsheets calculations. Other simpler methods for evaluating steel temperature are totally empirical, such as the “best fit method", and these methods have numerous limitations and also may not yield accurate temperature predictions under wide range of fire scenarios (Buchanan 2002). These limitations can be ascribed to the fact that the best fit method was derived for calculating average steel temperature in proximity to the critical temperature of steel, which corresponds to 538°C for carbon steel. Thus; the temperature predicted by best fit method may not be reliable at temperatures beyond roughly 125% Of critical temperature of steel which is 538°C (Buchanan 2002). Based on the previous discussion, a simple equation for predicting temperature in steel sections is required. In order to simplify the partial differential equation [6.1], the following assumptions are made (Dwaikat and Kodur 2010a): 2 Steel temperature has a uniform distribution, leading to V T = 0 , - the radiation problem can be approximately described as an equivalent convection problem, - temperature in insulation material is assumed to be equal to an average of steel and fire temperatures, i.e. (Ts + Tf)/m, where m = 2 for linear variation of temperature across the insulation material, - insulation is thin and thus the volume Of the insulation is computed as its thickness (tp) multiplied by the heated perimeter (Fp) of the section, and - the thermal parameters of steel and insulation materials are assumed to be constant over the temperature range. 175 Based on the assumptions listed above, Eq. [6.1] can be reduced to the following one dimensional differential equation: F /A )(T —T) dT ldt de : ( p s f S __ f [6.3] dt C ,0 F t C A csps(l/hwn+tplkp)[l+ p Pili] 1+_5”_OS__I?S__’Z'_ C ,0 m c I S S S PpP P P where hcon is the convective heat transfer coefficient, CSpS and Cppp are the heat capacity of steel and insulation, respectively, kp is conductivity Of the insulation material, and As is the cross sectional area of steel section. Equation [6.3] can be rewritten as dT dT f __S_= _ _ _ dt FlhTf TS) F2 d: [6'4] with F1 and F2 corresponding to the terms in Eq. [6.3]. 6.2.2 Exposure to Standard Fire In order to Obtain a simplified solution for Eq. [6.4], the fire time-temperature curve of a standard fire (Tf) is fitted by a power function of the form: Tf= atn. For ISO 834 standard fire a = 469.9 and n = 0.1677 (R2 = 0.995), and for ASTM E119 standard fire a = 496.5 and n = 0.1478 (R2 = 0.989). Figure 6.1 shows the relative error due to this fitting for both standard fires. It can be seen that the error is less than 2% after the first 10 minutes of fire exposure. If we assume that the temperature in steel has the following form: 176 I799 TS(t) = Tf(1—e_ 5’ j [65] where s is a correlation coefficient, then by substituting Eq. [6.5] into Eq. [6.4], de /dt n dividing by Tf, substituting—T— = 7 , and multiplying by est we obtain: f n st _ _ _ 2 St I (e I) + s — F1 F2 t e [6.6] Based on comparison to results from thermal analysis, the value of 3 must be very small (st < 1), and thus the exponential function can be approximated as: (stlz (stli 2! 3i e"=1+st+ + +...El+St, f0r5t<1 [6.7] by substituting Eq. [6.7] into Eq. [6.6] and solving for s we Obtain: =a—an) n(1 + F2 )+1 [6-8] Since F2 << 1 and the factor n/t becomes significantly smaller than 1.0 as t increases, we can neglect the second term Of the nominator, leading to the following simplified expression for s: * F /A S: F1 = h p S) n+1 c ,0 F t [6'9] c ,0 (1/h+t /k )1+ p p—pi (n+1) S S P P Cs’os As m Using 5 from Eq. [6.9], Eq. [6.5] can be used to predict steel temperature at any time step, without the need for incrementing time steps. Equation [6.5] can be applied in 177 one step (no time-steps needed) and this is an advantage over one-dimensional finite different equations (such as those specified in the Eurocode), whose accuracy is highly dependent on the size Of time increment used. 6.2.3 Exposure to Design Fire While the temperature in a standard fire continues to increase indefinitely, the design fire enters a decay phase after reaching a maximum value (Tfimax). A typical design fire can be represented by a parametric fire curve as specified in Eurocode 1. Figure 6.2 (a) illustrates the characteristics of a typical parametric fire scenario. The heating phase of a design fire is specified parametrically in Eurocode 1 in terms of the factor I‘ which is dependent on the ventilation characteristics and thermal inertia of fire compartment. For I‘ = 1.0 the heating phase Of a parametric fire represents the ISO 834 standard fire. Heating phase of a design fire can also be fitted using a power function (i.e. Tf= atn). For instance, for a parametric fire with l‘ = 0.5, a = 312 and n = 0.229 (R2 = 0.95), while for parametric fire with l‘ = 1.5, a = 474 and n = 0.176 (R2 = 0.97). Steel temperature during the heating phase of a design fire can then be computed using Eq. [6.5]. The decay phase of a design fire starts at t = t1, as shown in Fig. 6.2. In the parametric fires specified by the Eurocode the decay phase is always linear with a decay rate “r" as shown in Fig. 6.2 (b). The fire temperature reduces to room temperature at t = t2, however, steel requires more time to cool down to room temperature and this happens at t = t3. The maximum temperature in steel occurs at 178 t2 > ts,max > t1 because in that time interval fire temperature is still greater than average steel temperature, however fire is decaying. If the rise in steel temperature steel after t1 (the segment AB in Fig. 6.2(b)) is represented by a quadratic function, and if we assume that the maximum steel temperature occurs on the same decay curve of the fire (i.e. point B is on the fire curve) then the equation for the average steel temperature between t1 and ts,max can be written as: Ts=aa2+flt+y [6.10] where a, B, and y are orientation coefficients. In order to estimate the maximum temperature in steel (Ts,max) the following boundary conditions for Eq. [6.10] are utilized: - at pointA (t: t1): Ts = T51 (using Eq. [6.5] at t = t1), and de/dt = slope from Eq. [6.5] at t = t1, - and at point B (t = ts,max): T5 = Tfi and de/dt = 0. By applying the above four boundary conditions on Eq. [6.10] and solving for ts,max and Ts,max we Obtain: f,max _Ts,l t =t1 + 3’1““ 1 de [6.11] —— +r 2 dt _ t—q 179 dT 1 s -— — + x 2Tf,max dt r Ts,1 t = t T = 1 s,max dT [6.12] 1__s_ + r 2 dt t_t _ 1 dz, The term dt t—t can be assumed to be equal to the secant slope Of the steel — 1 dTS _ T5,] temperature curve, Le. (11‘ — t1 . Substitution back into Eqs. [6.11 and (=11 6.12] gives: —T -t 1+ f,max s,1 s,max _ 1 lT +rt [6.13] 2 3,1 1 T + 2t r _ f ,max 1 name “ 1+ 2. HT [6441 1 s,1 As it will be shown through comparison to rigorous finite element analysis, the last two simple equations give a good approximation for both the maximum temperature attained in steel beam during exposure to a design fire scenario and for the corresponding time of attaining that maximum average steel temperature. 180 6.2.4 Comparison to Finite Element Analysis In order to verify the proposed equations, results from finite element analysis are compared to the results from the proposed equations. The finite element model developed using ANSYS in Chapter 4 for thermal analysis is used for generating the finite element results. Table 6.1 shows the range of parameters that were used to in the simulations. A comparison between finite element predictions of steel temperature and those predicted using Eq. [6.5] is shown in Fig. 6.3. The comparison shows that Eq. 6.5 can reasonably predict steel temperatures below 750°C; while steel temperatures are overestimated about 10-12% by Eq. [6.5] for Ts > 750°C. This overestimation is because Of the phase change that occurs in steel at temperatures between 750°C and 800°C, which is not captured by Eq. [6.5]. Any attempt to make adjustments to Eq. [6.5] in the range Of Ts > 750°C will result in a complex equation for predicting steel temperature which departs from the purpose of this study. Thus, Eq. 6.5 is deemed to be acceptable for generating steel temperature conservatively below 750°C and this is the range Of temperatures encountered in the failure of beam-columns in most practical situations. The results of the finite element analysis for the maximum temperature in steel (Ts,max) and the time to reach maximum temperature in steel (ts,max) are compared in Fig. 6.4 and against the predictions using Eqs. [6.13 and 6.14]. The results in Fig. 6.4 indicate that the proposed equations are capable of predicting the maximum temperature attained in steel section when exposed to design fire scenarios. 181 6.3 PLASTIC P-M DIAGRAMS 6.3.1 General Current provisions in codes and standards recommend the use Of uniform average temperature for establishing the plastic P-M curves at elevated temperatures. This assumption, though adequate for cases where temperature in steel is uniform, such as a column exposed to fire from four sides, may not be valid for columns or beams exposed to fire from 1, 2, or 3 sides since significant thermal gradients develop across the section. These thermal gradients can cause severe distortion in the P-M curves and render the capacity curves based on uniform temperature inadequate for evaluating the strength of such beam-columns. In this section, a simplified approach is proposed for adjusting the uniform temperature plastic P-M curves to account for the shape distortion resulting from fire-induced thermal gradients. The focus Of this study is on the shape of the plastic P-M interaction equations that is influenced by thermal gradients. Therefore, in the generation Of the P-M diagrams, the main assumption is that the beam-columns are braced against lateral or torsional buckling. Further, the influence of local buckling is neglected since the beam-column is assumed to be made of a compact section, as required in a design situation. Stability factors (e.g. KL for columns) can be achieved independently from the P-M diagrams. This can be done through adjusting both axial and bending capacities (using Pcr and Mcr instead of plastic capacities FyAs, and Fny, respectively), and by amplifying applied moment due to lateral bowing of the member. Pcr and Mcr, under the influence of thermal gradient, can be computed 182 based on Eurocode recommendations (EC3 2005), or the AISC specifications manual (AISC 2005). In terms Of global stability, Eurocode recommends an effective length Of 0.5Lo for columns in intermediate storey and 0. 7L0 for columns in top storey for fire resistance design. 6.3.2 Generation OfP-M Diagrams To generate the plastic beam-column capacity for a section with thermal gradient, each fiber Of steel section is subjected to full ultimate stress that corresponds to the temperature in that fiber, i.e. FuXky(Ti). Fig. 6.5 shows a wide flanged (WF) section with thermal gradient and the corresponding ultimate stress Fu><(yl. —YGC)xFu xky(Tl.)> 0 and -1 for x < 0) and the positive sign indicates compression. The terms in Eqs. [6.15 and 6.16] are illustrated in Fig. 6.5. The bending moment MGC is calculated around the center Of geometry (YGC) of the section, and the positive sign 183 4‘" indicates tension at the top (cooler) fiber. The center Of geometry is defined as the geometrical centroid Of the section, while the neutral axis (YNA) is defined as the axis at which the strain is zero. Because the section must be in equilibrium, the sum in Eq. [6.15] must equal the externally applied axial force (P) acting on the section, while the sum in Eq. [6.16] must equal the externally applied bending moment (M). To Obtain the entire P-M diagram for the same temperature profile of the section, the location Of the neutral axis (YNA) is varied along the depth of the section. For each location of the neutral axis, a pair Of P and M values is computed using Eqs. [6.15 and 6.16], respectively. The location of the neutral axis (YNA) is varied from zero to din increments of Ayi = 0.001d. The computations of Eqs. [6.15 and 6.16] are carried out using a MATLAB code for all the cases of sectional geometries and thermal gradients. Figure 6.6 shows the degradation in strength and stiffness of steel as a function of temperature that is adopted in the calculation of F-M diagrams. These strength- temperature relationships are adopted from the ASCE manual of practice (ASCE 1992) Preliminary research by Garlock and Quiel (2007) showed that the distortion in the shape of the normalized P-M diagrams does not depend on the relative sizes Of the sections; however, the shape critically depends on the axis orientation (i.e. on the direction of thermal gradient). Their study highlighted that the distortion in the PM diagrams is more severe for the case of thermal gradient in strong direction as compared to the case of weak direction. Therefore, it is proposed here to adjust the P-M diagrams for the case of severe distortion due to thermal gradient in the strong 184 direction, while the shape of P-M diagrams will be maintained for the case Of weak direction. The following subsections will present treatment of P-M diagrams for both strong and weak axis orientation. 6.3.2.1 Strong Axis Orientation The basic features of the distorted plastic P-M diagrams for a WF section with thermal gradient in the strong direction are compared to the case of a uniform temperature in Fig. 6.7. The figure shows that the value Of moment capacity under peak axial capacity (point A in Fig. 6.7) moves back and forth (to point A’ in Fig. 6.7) depending on the eccentricity (e) between center Of stiffness “CS" and center of geometry “GC” that is caused by the thermal gradient in a WF section. In order to Obtain a simple method for deriving the distorted P-M diagrams the following assumptions are made (Dwaikat and Kodur 2010a,b): - The P-M diagram represents a parallelogram with its sides being parallel to the P-M segments derived based on the average temperature of the section. Specifically, we assume that the segments AB and AC are parallel to the segments A 'B' and A 'C' respectively, as shown in Fig. 6.7. - The magnitude of the shift (MTG) in the P-M capacity envelope (Fig. 6.7) is ’ assumed to be numerically equal to the ultimate axial capacity (Pu,Tave) of the section multiplied by the eccentricity (e) between the center of geometry (GC) and of the center of stiffness (GS) of the section as shown in Fig. 6.10. The ultimate capacity is computed based on the average temperature of the section, i.e: 185 MTG = eXPu,Tave = e x ky(Ts,Ave)> ‘% [6'19] + E s,CF E s,HF BF’FkE(Ts,CF)+[ 2 ]’wd+BF’FkE(Ts,HF) k T —k T d ZBFtF E( s,CF) E( s,Ave) 3:... X [620] 2 23 t +t d k (T ) - FF w E s,Ave where BF, tF, tw, d are dimensions of the section as shown in Fig. 6.8, and kE(T) is the reduction factor for elastic modulus at steel temperature T. For verifying the validity of Eq. 6.20, the computed value Of MTG (using Eq. [6.17]) is compared to the actual value Obtained from the true P-M interaction curves (via MATLAB code) for 186 E... [I sections with different thermal gradients. Fig. 6.9 shows the result of this comparison for different cases of thermal gradients. The figure shows that the eccentricity (e) increases more rapidly as the temperature increases. This is because stiffness (Es) of steel suffers faster degradation with the increase in temperature. It can be seen in the figure that MTG can be reasonably predicted for realistic cases Of linear thermal gradients using the simple methodology described above. With the calculation of MTG using Eq. [6.17], an adjustment for the P-M interaction curves can now be proposed. The adjustment Of the P-M interaction [curves is based on using the average temperature Of steel section with a shift MTG that occurs as a result Of thermal gradient in the section. Fig. 6.10 shows the proposed adjustment for the P-M diagrams to account for the effect Of thermal gradients. The adjustment of P-M diagram is aimed at preserving the room-temperature shape of the P-M diagram and only introducing the shift MTG to account for the thermal gradient effect. The adjusted equations of the plastic P-M diagrams for wide flange section with linearized thermal gradient in the strong direction can be written as (Dwaikat and Kodur 2010b): |M+M I lM—M | TG + P 31.0 and TG — P 31.0 [6.21] Z F A F A F x u,Ts,ave s u,Ts,ave x u,Tave s u,Ts,ave In order to verify the proposed simple approach for predicting P-M diagrams, two sections with different cases of thermal gradients are analyzed. Figs. 6.11 and 6.12 show the comparison between the true P-M diagrams (solid lines) for these two sections are Obtained via MATLAB code that carries out the complex computations using Eqs. [6.15 and 6.16], and the approximated linearized P-M diagrams (dashed 187 lines) that are Obtained using Eq.[6.21]. The two sections are W24x76 with relatively thin steel plates and W14x311 with thicker steel plates. These sections were selected to compare the effect Of the different steel plate thicknesses on the P- M diagrams. The selected sections (W24x76 and W14x311) were subjected to different thermal gradients along their strong axis, and the resulting P—M diagrams are normalized to the axial (Pu) and moment (Mu) plastic capacities using average temperatures. It can be seen in Figs. 6.11 and 6.12 that the distortion in the P-M diagrams is well predicted using the proposed simple Eq. [6.21] for both sections. Also, it can be seen in the figures that the distortion is larger for higher temperatures and higher thermal gradients. This is because at higher temperatures (Ts > 400°C) the reduction in steel elastic modulus (Es) becomes steeper and thus causes larger eccentricity (e) between center Of stiffness and center Of geometry Of the non homogenous section. Further, the comparisons in Figs. 6.11 and 6.12 indicate that increasing plate thicknesses has a very small influence on the shape Of the P-M diagrams under extreme cases of thermal gradients (case AT = 225°C). It can be seen that Eq. [6.21] conservatively predicts the P-M capacity envelopes for realistic cases of thermal gradients. 6.3.2.2 Weak Axis Orientation In the case of thermal gradient along the weak axis, the distortion in plastic P-M diagrams is smaller as compared to the case of thermal gradient along strong axis. The ensuing analysis shows that for realistic cases Of thermal gradients, the effect of thermal gradient on the P-M diagrams can be neglected for weak axis orientation. 188 “I 'L I h— _0‘" m I . . Therefore, as shown in Fig. 6.13, the P-M diagram. for WF sections with thermal gradients in the weak direction can be constructed by assuming the moment capacity ratio (M/Mu) to increase linearly as the axial capacity ratio (P/Pu) decreases until the moment capacity ratio (M/Mu) reaches its maximum value Of 1.0 at P/Pu = 0.5, after which the moment capacity remains constant. Based on this assumption, the P-M diagrams for WF sections subjected to thermal gradients in their weak direction can be expressed as (Dwaikat and Kodur 2010a): H FF 25 APP Ly FM Ln s u,Ts,ave s u,Ts,ave| I y u,Ts,ave l [6.22] If P Si, M 51.0 A F 2 Z F s u,Ts,ave y u,Ts,ave Figs. 6.14 and 6.15 show P-M diagrams for two different crOss sections with different thermal gradients. The two sections are a W24x76 with relatively thin steel plates and a W14x311 with thicker steel plates. These sections were selected to compare the effect Of the different steel plate thicknesses on the P-M diagrams. As seen in Figs. 6.14 and 6.15, the P-M diagrams experience smaller shifts due to the effect of thermal gradient in the weak direction as compared to larger shifts for the case of thermal gradient in strong direction (as seen in Figs. 6.11 and 6.12). The shift in P-M diagrams becomes more noticeable when both the average temperature of the section and the thermal gradient are high (e.g. AT = 225°C and T s,CF = 700°C in Figs. 6.14 and 6.15). Such cases of extreme temperatures and thermal gradients are rarely encountered in reality because steel sections tend towards thermal equilibrium as temperature in steel rises with time of exposure. Also, the shape of 189 the P-M diagram seems tO be smoother and more rounded for sections with thicker plates. For both cases Of sections (thick or thin plates) and the different thermal gradients they are exposed to, Eq. [6.22] seems to provide sufficiently reasonable prediction of the P-M diagrams for WF sections subjected to thermal gradient in the weak direction. 6.3.3 Comparison to Fire Tests The proposed approach for generating the distorted P-M diagrams induced by thermal gradients is validated by comparing the predictions with measured data in fire resistance tests on beam-columns. For this validation, the same beam-columns (Dwaikat et al. 2009) that were used in the validation of the thermal analysis are used. Beam-columns designated C1-S and C1-W were selected to demonstrate the ability of the simplified approach to predict the distorted P-M envelopes that are induced by non-uniform thermal gradients along strong and weak axes, respectively. As shown in Fig. 3.3, the tested beam-columns were instrumented with thermocouples to measure steel temperatures, strain gauges to measure bending moment, and LVDT’s to measure displacement and rotations. The measured fire induced axial forces and bending moments for the two cases of axis orientations (weak and strong axes) are plotted in Fig.6.16. The generated bending moments were directly computed using the readings of the strain gauges at the base section D-D (see Fig. 3.3 for location Of section D-D) which remained cool (below 40°C) during the fire test (Dwaikat et al. 2009). Since the top end Of the beam-column was 190 lightly rotationally restrained, it is assumed that the bending moment at section D-D is the maximum and values Of the moment at the critical section where plastic hinge developed (at 1930 mm from the based) was interpolated between MD-D and zero at the top end (Dwaikat et al. 2009). The pressure recorded in the vertical actuators was also used to directly measure the fire induced axial forces. Fig. 6.16 shows that both the axial force and bending moment increase first and then decrease with fire exposure time. This is because the beam-columns expand non-uniformly under the influence of temperature and thermal gradients until the spread Of plasticity in the beam-columns causes a reduction in the axial force. The recorded temperatures in the two beam-columns in fire resistance experiments are shown in Fig. 3.5. In order to use the simplified approach, the temperature profiles are linearized such that the top flange and average temperatures were assumed to equal the lowest and the average temperatures of the section, respectively. Fig. 6.17 plots the development of the fire-induced P and M in the tested beam-column that had thermal gradient along its strong axis. Also, the PM envelope as predicted using the simplified approach (Eq. 6.19) for different fire exposure times is also plotted in Fig. 6.17 for the same beam-column at specific times of fire exposure. It is seen in the figure that at different time steps, the beam- columns experience different thermal gradients which cause different shifts in the P-M diagrams. Based on the results in Fig. 6.17, the capacity failure of the beam- column is conservatively predicted at t = 215 min. using Eq. [6.21], compared to the Observed failure Of the beam-column at t = 220 min. in fire tests. Actual failure in tests is said to occur at the time after which the beam-column is no longer capable of 191 carrying the applied axial load and thus starts to deflect at an accelerated rate. While, the predicted failure is assumed to be the time at which the fire induced P and M exceed the capacity envelope given in Eq. [6.21]. Figure 6.18 plots the development Of fire induced axial force and bending moment for the tested beam-column C1-W, subjected to thermal gradient along its weak axis. The figure also shows the predicted P-M capacity envelope as predicted using Eq. [6.22]. In this case, since the shape of the P-M diagrams is not adjusted, only the P-M diagram at failure is shown in Fig. 6.18. It can be seen that Eq. [6.22] predicts the failure to occur few minutes (t = 212 min.) prior to the Observed (actual) failure (at t = 215 min.) Of the beam-column in fire resistance experiments. 6.3.4 Design Application To demonstrate the applicability Of the simplified approach in design situations, a numerical example is presented for a beam-column exposed to ASTM E119 standard fire (ASTM 2008). The beam-column is of section W14x176 (Fy = 345 MPa.) and is insulated with 0.5 in. (12.7 mm.) spray applied fire proofing material. Section dimensions are: d = 386 mm, BF = 399 mm, tF = 33.3 mm, tw = 21 mm. The thermal conductivity of insulation is 0.1 W/m.°K The beam-column is exposed to fire from three sides such that thermal gradient is assumed to develop in the strong direction. Thermal gradients induce thermal bowing which causes additional bending moments to develop in the beam-column. These bending moments include moments resulting from restraining thermal curvature, and from lateral deflection caused by thermal curvature. In both cases, the resulting bending moments need to be 192 W: accounted for in structural analysis under fire conditions. In this example, the beam-column is braced in both directions in order to ensure fully plastic failure mode. The plastic P-M capacity envelope for the beam-column will be evaluated for two cases. Case one is for a uniform temperature and case two is with thermal gradient in the section of the beam-column. For the case of uniform temperature, the P-M diagram is evaluated using AISC design specifications (AISC 2005) for the maximum and average temperatures in the section. For the case of thermal gradient, the P-M diagrams will be generated using the simplified approach. For either case, the P-M capacity envelopes will be established at 2 and 3 hours of fire exposure. The temperature in the section will be evaluated using Eq. [6.5]. 6.3.4.1 Temperature Calculation Using the geometric properties of W14x176, section factors (Ap/V) were computed for the entire section, top flange, and bottom flange as 54.16, 33.5, and 63.5 m-1, respectively. Table 6.2 shows the calculated temperatures at 2 and 3 hours of fire exposure and compares them to the results of finite element analysis. Equations [5.4] and [5.5] are used to estimate the linearized thermal gradient in the cross section after 2 and 3 hours of fire exposure. As shown in Table 6.2, linearized thermal gradient (according to proposed approach) increased from 139°C at 2 hours of fire exposure to 180°C at 3 hours Of fire exposure. This compares well with finite element predictions, wherein the thermal gradient increased from 193°C to 219°C, at 2 and 3 hours of fire exposure, respectively, as shown in Table 6.2. The 193 reduction factors for E5 and Fy are computed using equations specified in the ASCE manual Of practice (ASCE 1992). 6.3.4.2 AISC Specifications The plastic axial and moment capacities are calculated as per AISC design specifications (AISC 2005) for two cases Of uniform temperatures, namely average and maximum section temperatures at 2 and 3 hours of fire exposure. The results Of the calculations are presented in Table 6.3. The plastic P-M diagrams are plotted in Fig. 6.19 for 2 and 3 hours of fire exposure and using maximum and average uniform temperatures in the section. The figure shows that the beam-column experiences significant reduction in capacity between 2 and 3 hours Of fire exposure. However, the shape of the P-M diagram does not change despite the change in thermal gradient between the 2-hour and 3-hour fire exposure. This is simply because the AISC procedure does not account for the effect of thermal gradient on the P-M diagrams. 6.3.4.3 Accounting for Thermal Gradient Using the Proposed Approach The effect of thermal gradient on the P-M diagrams can be included by computing the eccentricity (e) between the center of stiffness and the center of geometry at time Of fire exposure. Both the eccentricity (e) and the shift (MTG) in the plastic P-M diagrams are computed at 2-hour and 3-hour fire exposures and are presented in Table 6.4. 194 The MTG is used to adjust the plastic P-M diagrams as per Eq. [6.21] for strong axis direction. The distorted P-M diagrams are plotted in Fig. [6.21] for 2-hour and 3- hour fire exposure. It can be seen that at 2-hour fire exposure, the distortion in P-M diagrams is small, however, at 3-hour of fire exposure, the distortion in PM diagrams becomes significant. This is because at temperatures above 400°C the reduction in steel young modulus (Es) becomes steeper and thus thermal gradients at such elevated temperatures will have higher effect on the plastic capacity Of the section. Clearly, the plastic P-M diagrams as per AISC design manual do not account for the shift in the center of stiffness of the section due to thermal gradient, and therefore can be unconservative as seen in Fig. 6.19. 6.5 SUMMARY The development of simplified approach for evaluating plastic P-M diagrams of steel beam-columns under realistic fire exposure is presented in this chapter. The guidelines are derived based on basic heat transfer and structural mechanics principles and were supported by detailed parametric studies presented in Chapter 5. The proposed guidelines include a method for computing steel temperature and fire induced thermal gradients, and a simplified methodology for adjusting the PM diagram to account for the influence of fire induced thermal gradient The validity Of the proposed approaches is established by comparing predictions with results from detailed finite element analysis and fire tests. The proposed methodology provides simple and rational approach for generating the plastic P-M interaction diagrams for steel beam-column subjected tO thermal gradient under standard or design fire 195 exposures. The applicability Of the proposed methodology to a design simulation is illustrated through numerical example. Table 6.1: The range Of parameters used in the simulations Parameter Min. Max. Step d (mm) 200 600 200 b (mmL 200 600 200 tf(m@ 10 30 20 tw (mm) 7 14 7 tp (mm) 0 30 10 kp (W/m.°K) @20°C 0.07 0.21 0.14 pxcp ()/m3.°K)@ 20°C 250x900 350x900 150x900 Gamma factor (I‘) 0.5 1.5 0.5 t1 (min.) 30 120 30 t2 (min.) t1 + 30 t1 + 150 60 Table 6.2: Temperatures in W14x176 as predicted by Eq. 6.5 and by detailed finite element analysis After 2 hours After 3 hours 5 Ts(°C) Ts(°C) Ts(°C) Ts(°C) Eq. 6.5 F.E. "y “3 Eq. 6.5 F.E. "y “5 Cool flange 0.00277 297 314 0.81 0.89 435 429 0.65 0.77 Average 0.00448 436 428 0.65 0.76 615 582 0.35 0.47 Hot flange 0.00526 491 507 0.57 0.69 679 648 0.25 0.35 Table 6.3: Plastic axial and moment capacities calculated as per AISC design manual After 2 hours After 3 hours (DPn qDMn CDPn (DMn kN kN.m kN kN.m Average temp. 6759 1060 3622 568 Max. temp. 5922 929 2580 405 196 Table 6.4: Calculation Of eccentricity Of center Of stiffness and the gradient-induced shift MTG as per proposed equations AT (°C) AT (°C) e (mm) MTG (kN.m) Eq. 6.5 F.E. Eq. 6.20 Eq. 6.17 After 2 hours 194 193 23.7 160.2 After 3 hours 244 219 90.6 328.35 197 Temperature Temperature T ---- Design fire curve f,max A verage steel temp. Ts,max 75,1 - E Decay rate r : m 20 °C ' -- i """" ; ; t 1 t 2 Time 1' 3 (a) Design fire and steel temperatures T ........ Design fire curve f,max : T g Average steel temp. Ts,max i B ’l t2 .-1 t s,max , (b) Rise of steel temperature during decay phase Fig. 6.1 Characteristics Of typical parametric fire exposure 198 E g 400_ . , M .1 g 200 1 0o 200 400 600 800 1000 1200 Ts Obtained from F.E.A. , °C Fig. 6.2: Comparison Of steel temperature predicted by Eq. 6.5 with that Obtained from finite element analysis 199 CD '7, 900W. ““““““““““““““““ _:‘ + 10% margin : ° : \llr/z 00”)! 2 . . x '0 ”0” n g 700 ------------- : ————————————— 1 ------- ------ '— —————— : E : I .”o,’. l ’ I g 1 I I, ’0”. ’1’ i I I, I I 3’. E , 5:3. I E 3: 500l “““““““““ T"'”":};?’“ :“:’ “““““ i ““““““““ f .- | . ’/ I I 'h I "9.. xi’ 0 . l E : ,4 o ; -10A)marg1n ; ‘0- 4” 1': 300r “““““ , 3” .17", """""" 1'“ “ l o , ’ 4' : : : 2., l i l l g x ’ E E i E I I I l I a; 10? 1 l I E , tr '— 00 300 500 700 900 °C) From proposed approach Ts,max I (a) Maximum steel temperature +10% margin d N 3 8 _L O O 4 I Is m" (m in.) From F.E.A. so 100 150 260 230 ts,max (min. ) From proposed approach (b) Time for reaching maximum steel temperature Fig. 6.3: Comparing predictions from proposed equations and from finite element analysis for Ts,max and ts,max 200 1'0 . Es,r/Es,20°c 0.8 d / 0.6 ‘ 0.4 ‘ FLT/FY,” °C 0.2 4 F u,T/F u,2a °c 0.0 r u u u - O 200 400 600 800 1000 Temperature, °C Fig. 6.4: Variation of Es and Ify as a function of steel temperature 201 Fig. 6.5: Characterizing P-M diagram for a WP section with thermal gradient in the strong direction 202 Cool flange temperature (T CF), °C 0 1 50 300 450 600 750 900 o o ‘ f :— ‘ ' _U 1 I I no - .N AT=SOOC -0.2 - ‘ S‘ ‘ ~ ~ ’g ‘ ‘. ~ ‘ ~ AT=100°C :- -o.4 - x u. s s X ‘ ‘ ~ 5.: ~ ‘ AT: 150°C \ -o.6 I ‘ g s E s ‘ O '0'8 ' — M m obtained from analysis \ AT: 200 C -." MTG = e X ASXFuJ‘ave ‘ -1.0 . Fig. 6.6: Comparing the shift in moment (MTG) as estimated from simplified approach with that obtained from analysis 203 |M + M ml + P Z x F y (Tave ) AS FY (TOW ) MTG =eXB‘ (72m) Actual P-M diagram s“~ / ~\ M/M. \ |M—MTG|_ p <1 szy(T ) ASFy(T ) ave ave Linearized P-M diagram Fig. 6.7: Linearized P-M diagram for WF sections with linear thermal gradient along the strong direction 204 m ' 1 1 -1 AT= 150C T = 400C M/M AT = 150C x T = 550C $3 0. i AT = 150C T = 700C 1 -1 1 \ ‘V .\ ’ -1 A J 1 AT= 225C T = 400C -1 AT = 225C T = 550C AT = 225C T = 700C Fig. 6.8: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 14 (dashed curves) for W24x76 section with different linear thermal gradients along its strong axis (AT=Ts,HF—Ts,CF) 205 AT: 150C AT= 225C T = 400C “ T = 400C M/MU AT = 150C AT = 225C T=SSOC /,/‘~\ T=ssoc NVMU ”:15“: AT 225C 1 T = 700(: x = /‘~\ T: 7ooc LL” 0 e” o a. \ 0. -1>. A -1 . A L .1 0 1 -1 0 1 WM M/M M/M U U U Fig. 6.9: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 14 (dashed curves) for W14x311 section with different linear thermal gradients along its strong axis (AT: Ts,HF—Ts,CF) 206 Wt Linearized P-M diagram § 1 M p fl H 'L ) (1.0 m 521:}: 5 2,15; (T ) T AsFy (Tm ave .------------.‘ M/Mu Actual P-M diagram Fig. 6.10: Linearized P-M diagram for a WF section with thermal gradient in weak direction AT = 75C AT: 150C AT: 225C 1 T=400C 1* , \ T=400C ' T=400C AT=150C 1 /\ T=550C 1i ’///—:\,\\ T=550C i AT=225c T=ssoc A MU AT = 225C “ ‘ T = 7ooc AT = 150C T = 700C Fig. 6.11: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 15 (dashed curves) for W24x76 section with different linear thermal gradients along its weak axis (AT=Ts,HF—Ts,CF) 207 AT: 150C T = 400C AT = 150C T = 550C AT=150C , \ T=700C -1 AT: 225C \ T = 400(: AT = 225C _ T = ssoc AT = 225C T = 700C Fig. 6.12: P-M diagrams obtained from analysis (solid curves) and as predicted by Eq. 15 (dashed curves) for W14x311 section with different linear thermal gradients along its weak axis (AT=Ts,HF—Ts,CF) 208 Mmema‘ D'D ”bl Force 1” mam at DD ”'81 meg _ 15 .1000 i --------- E r 10 . fig 800 a . E 36°05 1 . '. c' . .isfg u. A - , . :- ' _ o .‘9'. 40° .— ' ' . , ._ 2a x ' * 2°° ‘4' ~-5 ‘ Momentatcrltlcalsectlon ‘ . < Momentatcrltlcal section 0 T I I i" '18 0 I I T— T '10 0 60 120 180 240 O 60 120 180 240 Time, nin. Time, ruin. (a) Strong axis ( C1 -5) (b) Weak axis (CI-W) Fig. 6.13: Measured fire induced axial force and bending moment in the tested beam- columns (Refer to Fig. 3.3 for details of tested beam-columns) 209 t = 216 min. 1-4 — Tave = 655°C 1.2 AT = 150°C Eq. 6.21 0.4 Predicted failure (1‘ = 216 min.) Finite element AISC PM 0.2 envelope 1 l l 0 1 ‘U I -O.75 -0.5 -0.25 0 0.25 0.5 0.75 1 MM u, Ta ve (a) Plastic P-M diagrams at 21 6 minutes of fire exposure 1.4 1.2 Measured failure (t = 220 min.) 0 Eq. 6.21 .'. f“ O § § t = 220 min. Tave = 670°C AT = 200°C Finite element 1 i P/Pu.Tave 0.8 .6 0.4 AISC P-M envelope envelope O l l l l l T l l -‘| -0.75 -0.5 -0.25 0.25 0.5 0.75 1 u, Ta ve 0 (b) Plastic P-M diagrams at 220 minutes of fire exposure Fig- various fire exposure times 6.14: Predicted P-M capacity envelopes for the tested beam-column Cl-S at 210 Measured failure 1 6 a t = 215 min. I; Tave = 655°C Finite element % AT = 130°C envelope AISC P-M Eq. 6.22 0.2 k t = l—fi lifi 0 l 0 min. I l 4 1 -1 -O.8 -O.5 -0.3 0 0.25 0.5 0.75 1 MMuJave Fig. 6.15: Predicted capacity envelop for the tested beam-column Cl-W 211 6800 AISC specifications /Average temp. ‘ Maximum temp. 5440 4080 2720 1 360 0 I -1 360 -2720 -4080 .5440 Simplified approach.“~..:“~.‘u (Eq.14) ,. -6800 ‘3 -1200 -900 -600 -300 0 300 600 900 1200 (a) At 2 hours of ISO 834fire exposure 4000 AISC . specifications Average temp. Maximum temp. 2000 / O I -2000 Simplified approaeh"-.v,.o"‘. _4000 (E q. 14) J -600 -300 O 300 600 (b) At 3 hours of ISO 834 fire exposure Fig. 6.16: P-M diagrams for W14x176 using uniform temperature and thermal gradient 212 l CHAPTER SEVEN METHODOLOGY FOR PREDICTING FIRE RESPONSE OF RESTRAINED STEEL BEAMS 7.1 GENERAL This chapter presents the development of a simple approach for evaluating fire induced forces and deflections in restrained steel beams at any given fire exposure time. The approach utilizes basic equilibrium and compatibility principles to compute fire-induced forces and deflections in the beam. The validity of the proposed approach is established by comparing fire response predictions with results obtained from rigorous finite element analysis as well as from measured data in fire resistance tests. In addition, predictions from the proposed equations are compared with fire resistance estimates from current codes of practice. The applicability of the proposed approaches to design situations is illustrated through numerical example. 7 .2 CHARACTERIZING THE FIRE RESPONSE A restrained steel beam, shown in Fig. 7.1(a), when exposed to fire, develops significant internal forces (axial force and bending moment) and large deflections due to the effect of restraints and deterioration in steel properties with increasing temperatures. If the beam (shown in Fig. 7.1(a)) is laterally braced, and is made of a 213 compact section (i.e. no local buckling), then the beam undergoes three distinct stages, as shown in Fig. 7.1(b and c) when exposed to fire. In Stage I, elastic response dominates the behavior wherein the beam expands as a result of continuous heating, and compressive axial force and bending moment develop in the beam due to the effect of end-restraints. Fire-induced internal forces and deflections continue to increase until yielding occurs in the beam. Elasto-plastic response dominates the behavior in Stage II, as shown in Fig. 7.1(b). As steel temperature continues to increase with fire exposure time, softening of steel causes larger deflections and rotations until the first plastic hinge develop in the beam. The plastic hinge, which forms at the location of maximum bending moment in the beam, causes sudden increase in deflection as shown in Fig.7.1(c), which leads to reduction and then reversal of the axial force in the beam from compressive to tensile force. The beam enters a catenary phase in Stage 111 when the fire-induced compressive axial force vanishes as shown in Fig. 7.1(b). In the catenary phase, tensile force develops in the beam and the load bearing mechanism gradually changes form flexural to cable (tensile) until failure occurs by rupture of the beam (or in the connections). When the beam undergoes the above three stages, it is assumed that the connections continue to perform elastically. Therefore, fire-induced forces and rotations in the beam must not exceed the connections capacity. In the following sections, the response of a restrained steel beam will be traced in temperature domain. The steel beam is assumed to be braced against lateral or torsional buckling. Also, the effect of local buckling is ignored in the development of the proposed approach since in most building applications compact sections are 214 used. lt will be shown through comparison with results from finite element analysis (that takes local buckling into consideration) that the occurrence of local buckling has a minor effect on the fire response of restrained steel beams. Under realistic simplifications, expressions for evaluating critical internal forces and deflections are derived based on equilibrium and compatibility principles. These expressions will be derived for monotonic heating with linear thermal gradient through the cross section. The monotonic heating represents temperature in steel that results due to exposure to a standard fire wherein temperature always increases with time. Influence of realistic fires, with cooling phase, will then be introduced. The developed expressions are aimed to serve as a simple, rational, and engineering approach for fire design of restrained steel beams, over a wide range of scenarios. 7.2.1 Fire Induced Forces As the beam expands with increasing temperature in Stage I, the beam develops significant compressive axial force due to the presence of axial restraints (see Fig. 7.1(b)). At this stage, the response of the beam is in the elastic range but with reduced properties of steel due to increasing temperature. Since the midspan deflection under service load at this stage is very small compared to the beam length (Ae << L) and can be neglected, the fire-induced axial force can be obtained by applying compatibility principles as: PL P 56 __6th —6k 3 m—a(T—20)L-2 K— [7.1] S E a 215 where 6e and 6th, are the elastic axial deformation and thermal expansion in the beam, respectively, and 6k is the elastic deformation in the restraining elements modeled as axial springs with stiffness Ka (as shown in Fig. 7.1(a)). P is the fire- induced axial force in the beam at steel temperature T, Es and a are the elastic modulus and coefficient of thermal expansion of steel, respectively, RE is the temperature-dependent reduction factor for Es based on average steel temperature, and As is the cross sectional area of the beam. Solving Eq.[7.1] for P, with algebraic manipulation, produces: where Fy is yield strength of steel at ambient temperature, and FA is the axial restraint factor defined as: aES kEKaL/(EISAS) F = A F 2k +K L/[E A [73] y E a s 5 Since the beam is most likely to be exposed to fire from three sides, it will develop thermal gradient that causes a thermal bowing in the beam. If we assume that the thermal gradient is linear across the section and uniform along the entire span of the beam, the rotational restraints will restrain the thermal bowing of the beam and thus cause an additional bending moment (M6) in the beam as shown in Fig. 7.2(a). If the thermal gradient is linearly approximated as ((TTF - TBF)/d), where d is the depth of the section, TT F and TBF are the temperatures of top and bottom flanges, respectively, then the elastic MG can be computed using superposition and compatibility principles as: 216 L/2M M o =oh—ok :> j—de=£ G e I O 1 2k E] s E s _ 7.3 = TBF TTF 5_ MG [ 1 d 2 Kr in which He and 6th, are the elastic and free thermal rotations in the beam, respectively, and GR is the elastic rotation in the restraining elements modeled as rotational springs with rotational stiffness Kr (as shown in Fig. 7.1(a)). The reduction in steel elastic modulus (kl?) is computed based on average steel temperature. MG is the bending moment induced by thermal gradient, and I is the second moment of area of the section in the direction of thermal gradient. Solving Eq [7.3] for MG, with algebraic manipulation, produces the following: AT MG =[—2—FR)Fny [7.4] where Sx is the elastic section modulus, AT = TTF — TBF, and the factor FR is the rotational restraint coefficient and is defined as follows: _ aES kEKrL/(ESI) Fy 2k E + KrL/(ESI) [7-51 R Also, thermal gradient will cause an increase in the curvature of the beam, at midspan. 1f the beam is not rotationally restrained then the “ ree” thermal curvature that results from a linear thermal gradient will be (zpth = a(TTF - TBF)/d). However, since the beam is rotationally restrained, the “resultant" thermal curvature 111th; can be obtained by superposition as: 217 £____MG — A]: 1.. FYFR :a _ _— W’h” d kEESI d aESkE "-61 Fire-induced forces (P and MG) build up in the beam (as shown in Fig. 7.1(b)) as steel temperature increases until yielding takes place. The increase in elastic stress in the beam, in Stage I, can be used as an indicator for the occurrence of yielding. If the thermal gradient is linear then the mechanical strain is also linear, and thus the build-up of elastic stress in the beam can be evaluated using the following equation: S x where Fs,max is the maximum stress in the steel section, Mmax is the maximum bending moment in the beam. Initially and before yielding, the increase in elastic deflection is small, and thus the slight increase in bending moments due to P-A effect in Stage I can be neglected, and Eq. [7.7] can be used until yielding takes place in the section. Because Eq. [7.7] checks for yielding of steel, Mmax is the maximum bending moment at either the support or midspan. Under thermal gradient, additional bending moment (MG) is generated if the beam is rotationally restrained. The gradient-induced bending moment (MG) will cause compression in the hotter (bottom) fibers of the section. Therefore, as shown in Fig. 7.2(c and d), the midspan bending moment due to gravity load (Mm) will be reduced by a magnitude of MG, while the bending moment at support (Ms) will be increased by a magnitude of MG. Also, if the beam-to-column connection is built such that the axial restraint stiffness 218 acts eccentrically to the centroid of the beam, then an additional bending moment (M = ny) is generated. This is shown in Fig. 7.2(b) where the topology of the beam- to-column connection causes the axial restraint force to be eccentric by a distance y measured from the centroidal axis of the section. Therefore, Mmax can be generally computed by accounting for all the previously described bending moments using the following relationship: M =maX(Mm_MG+PX(Ae-y),MS+MG+ny) [7.8] max By substituting Eqs [7.2], [7.4], and [7.8] into Eq. [7.7] we obtain: _ YAS _ AT Mo FAFy[1+ s )(T—2O)+7(FRFy)=kyFy — s [7.9] x x where the positive/negative sign depends on whether M0 is either Ms or Mm, respectively. Y = Ae - y, where y is the eccentricity of axial restraint force. The elastic deflection (Ae) can be neglected in case of uniform heating, however, in case of thermal gradient; the elastic deflection is significant and must be accounted for. The computation of elastic deflection will be explained in subsequent sections. Equation [7.9] is nonlinear since all the terms are dependent on steel temperature. In order to solve Eq. [7.9] for the temperature at which yielding occurs in steel (Ty), the temperature-dependent factors FA, FR and IQ! in Eq. [7.9] are approximated as follows. The variation of FA and FR with temperature, using ASCE and EC3 high temperature properties of steel, is plotted in Fig 7.3. The trends in the figure show that for temperatures below 550°C, FA and FR are almost constant for restraint condition (x) less than 100%. Based on this, the temperature-dependent factors FA and FR can be 219 assumed constant for a given restraint condition based on the temperature reduction factors for the elastic modulus of steel (kE') as specified in Eurocode and ASCE manual. This approximation for FA and FR is reasonable since for typical steel structures exposed to fire, steel yields at temperatures below 550°C. For T < 550°C and x < 1, where x is either KaL/(EsAs) or KrL/(Esl), the following approximation is used: k Ex alx Fx = z ZkE + x 2a1 + x [7.10] The coefficient a1 in Eq. [7.10] is dependent on the high temperature reduction factors for elastic modulus of structural steel which vary depending on the provisions for steel properties specified in different codes. The coefficient a1 is obtained by regression through minimizing the “sum of absolute errors” for Fx in the range T< 550°C: - In case of steel properties as specified in Eurocode 3: a1 = 0.6. - In case of steel properties as specified in ASCE manual: a1 = 0.6829. The temperature-dependent reduction factor for steel yield strength is approximated as: ky zl—aZT [7.11] where the coefficient a2 is also dependent on the material properties as specified in the codes and it is obtained through regression analysis for ky in the range of T < 600°C as: 220 - in case of Eurocode 3 steel properties: a2 = 0.0013. - In case of steel properties as specified in ASCE manual: 02 = 0.0008. By substituting Eqs. [7.10] and [7.11] into Eq. [7.9] the following relationship is obtained for the critical value of 7)! evaluated at both critical sections (midspan and support sections): T _ 1—M0 /My —0.5FRAT y " FA(1+ YAS /Sx)+ a2 , Ty < 600°C [7.12] In Eq. [7.12], M0 is the maximum of Mm and Ms, Y = Ae — y, where y is the eccentricity of the axial restraint force, FR and FA are the rotational and axial restraint factors, respectively, and are defined as: _ aEs alKrL/(ESI) R Fy Zal + KrL/(ESI) [“31 F : aES alKaL/(ESAS) A Fy 2a1 + KaL/(ESAS) [7441 The maximum fire-induced compressive force generated in the beam can be computed by substituting Ty into Eq. [7.2]: PC,max = FyAS >< FA(Ty - 20) . [7.15] As plasticity spreads throughout the section, the axial compressive force reduces until it reaches zero at temperature Tc as shown in Fig 7.1(b). The catenary action is initiated at Tc upon the deveIOpment of a first plastic hinge in the restrained beam. Since P = 0 at Tc, the condition for developing a plastic hinge can be written as: 221 Mo-‘FMG =ky(Tc)FuZx [7.16] where Fu is the ultimate tensile strength of steel at ambient temperature, and Zx is the plastic sectional modulus. Due to lack of information, the degradation of the ultimate tensile strength of steel (Fu) in Eq. [7.16] as a function of steel temperature is implicitly assumed to follow the same degradation (ky) as that of yield strength (Fy). Using the approximation for ky as described earlier, the temperature at catenary action can be computed as: T _1 1_M0_My FRAT C_a M M 2 ””1 2 u u where Mo is the maximum of Mm and Ms, Mu = FuZx and My = Fny. Assuming no local buckling, and knowing that P = 0 at T = Tc, then using the room temperature values of midspan and support moments (Mo = max(Mm and Ms)) in Eq. [7.17] can provide a reasonable approximation for the steel temperature at catenary point (Tc). It will further be shown through comparison to results from rigorous finite element analysis that the error in computing Tc does not severely affect the evaluation of deflection. At Stage III, the tensile force in the beam increases until it reaches a maximum value at which full plasticity is achieved at the two sections (at midspan and at support). After that, the tensile force drops in the beam following the same trend as that of the degradation of the steel yield strength with temperature. This is illustrated in Fig. 7.4 which shows the development of tensile force in the restrained beam during catenary phase. If the connections can sustain the tensile force, then we can obtain a 222 conservative approximation for the maximum tensile force in the beam through extending a line between the point of maximum compressive force (Ty, Pc,max) and the point of catenary (Tc, P = 0), as shown in Fig. 7.4. Using this approach, the temperature at which maximum catenary force occurs in the beam is the temperature Tthat satisfies the following relation: P T—Ty cmax — cmax 9 9 Tc_Ty = _k y (“F y As [7.18] Pc,max is the maximum compressive force in the beam computed using Eq.[7.15] at steel temperature Ty. Since the maximum tensile catenary action force is expected to occur at temperatures beyond 500°C, the yield reduction factor (lg!) can be approximated as: ky 2 a3 — a4T, 500°C S T 5 900°C [7.19] where a3 and a4 are regression coefficients that are dependent on steel properties: - For steel properties as specified in Eurocode 3: a3 = 1.139; a4 = 0.0013 - For steel properties as specified in ASCE manual; a3 = 1.329; a4 = 0.0014 By direct substitution, an expression for estimating the temperature at which the catenary force is maximum is obtained as: _ TCTyFA + a1a3(Tc —Ty) en,max Ty FA + (1104 (Tc _ Ty ) [7.20] Tt The temperature obtained from Eq. [7.20] is used to compute the maximum tensile catenary force as: 223 ml Pten,max = Fy As (a3 — a4Tten,max)’ 500°C S Ttenmax S 9000 C [721} 7.2.2 Fire Induced Deflection Elastic deflection at T = Ty can be obtained using conventional room-temperature deflection expressions found in structural analysis textbooks [19], but the revised steel elastic properties, reduced as a function of temperature, are to be substituted in place of room-temperature steel properties. Influence of thermal gradient on elastic deflection can be quite significant, and can be incorporated by direct integration of the “resultant" thermal curvature in Eq. [7.6]. For restrained beam with uniformly distributed load, subjected to linear thermal gradient along the depth of the cross section, the total elastic midspan deflection is: M“ wig. 1_F_R Fy e 384kEESI 8 d all aES [7.22] where the reduction in elastic modulus (kE) is computed based on the average steel temperature. The second order effect on the elastic deflection can be neglected since the total elastic deflection is generally small. EIasto-plastic temperature-deflection history of the restrained beam can be conservatively evaluated based on linear interpolation between the deflection at the end of elastic phase (at T: Ty) and at the point of catenary action of the beam (at T = Tc), as shown in Fig. 7.6. For evaluating the deflection at point of catenary action (Ac), the condition P = 0 at T = Tc is utilized. Fig. 7.5 shows a schematic of approximating the deflected shape of the beam, only half of the beam is shown due to symmetry. When the beam expands 224 under the influence of temperature, it compresses the axial spring by a magnitude of 6 = P/Ka. However, when the axial force P = 0, the value of 6 is zero. Using this information, a compatibility equation can be written (see Fig. 7.5) as: A2_ L,2 L62_L L 52 L2 723 c- 3 —3+ _ —2—+a§(Tc—20)- p, —3 [- l where (L/2) is half the deformed length of the beam. The plastic permanent shortening (6p!) in the beam length (due to compression) is small and can be neglected as compared to the thermal expansion of the beam. Solving for Ac and neglecting second order terms we obtain: L AC = EJZoch — 20) [7.24] Using linear interpolation, as shown in Fig 6, the deflection between Ty and Tc can be obtained as: A=A +—A:C;A—y(T—Ty) y Tc _Ty , Ty < T< TC [7.25] Using Eq. [7.25], a limiting temperature based on a deflection limiting criterion A = LF can be obtained as: (LF —A )(T —T ) TDLS =Ty + (Ac: Acy) y [7.26] where TDLS is steel temperature at deflection limit state, LP is the deflection limit state and is usually taken as (310 — 513%. It will be shown through a design example 225 that Eq. [7.26] can effectively be used as a guideline for fire design of restrained beam based on deflection criteria. 7.2.3 Influence of Fire Scenario The temperature rise in steel, as a function of fire exposure time, depends on section geometry, insulation, and fire scenario. Generally, from a structural fire design point of view, fire scenarios are classified into two types, namely; standard and design or “realistic” fire scenarios. Examples of standard and typical design fires with resulting average steel temperatures are compared in Fig. 7.7. In case of a standard fire exposure, the fire and steel temperatures increase continually with time, as shown in Fig. 7.7. However, because of depletion of fuel and/or oxygen, a realistic fire (also known as design fire) in buildings undergoes a decay (cooling) phase after achieving a maximum value of Tfimax at t = t1 as shown in Fig. 7.7. In the early stages of the cooling phase, the temperature of the fire is still higher than the average temperature of the steel in the beam (due to lag effect) and therefore steel temperature continues to increase for a little while during the cooling phase of fire until steel temperature reaches a maximum value of Ts,max at t = ts,max, as shown in Fig. 7.7, after which, the steel temperature starts to decline gradually. . The computation of steel temperature as a function of fire exposure time is beyond the scope of this study. Eurocode 3 provides simple expressions for calculating steel temperature at small time increments [2] both under standard and design fires. These incremental expressions are the finite difference solutions of the 226 approximated one-dimensional heat transfer differential equation (assuming constant steel temperature) for protected and unprotected steel sections. The methods specified in the Eurocode can be used to evaluate the time-temperature history in fire exposed steel sections. While under standard fire steel properties continue to deteriorate due to increasing temperature, the decay phase of a design fire leads to a recovery of strength and stiffness in steel as a result of decreasing temperature in steel. Also, the decay phase of a design fire leads to cooling and contracting of steel and this contraction causes an increase in tensile stresses in the beam and thus leads to recovery of some deformations in the beam. The total recovered deformation in the beam depends on the stress state of steel prior to the start of cooling of the beam. There is a lack of information on the behavior of steel in the cooling phase under realistic fires. Generally, it is conservative to assume that the elastic deflection of steel (Ay) is recovered by the end of the decay phase of a design fire. In order to evaluate the tensile force that results due to the shrinkage of steel, Fig. 7.8 compares the deflected shape of the beam prior to fire exposure and after cooling of steel. In Fig. 7.8, the half length of the beam prior to start of the cooling of steel (at t = ts,max) is denoted as (L/Z) and is equal to the original half length plus the change in length due to the effects of thermal expansion and plastic shortening. For typical cases of design, plastic shortening 6p] can be neglected as compared to the thermal expansion experienced by the beam. For the approximation depicted in Fig. 7.8, the following compatibility relationship can be written after the complete cooling of steel: 227 2 2 L L 2 L P h [3 + a2- (Ts,max _ 20) " 6P1] — (A max _ Arec) = [E _ 21:0 ] [7°27] The maximum tensile force that results due to the shrinkage of steel (Psh) at the end of the decay phase can be obtained by solving Eq. [7.27] and neglecting squared terms as they tend to be very small: 2 2 Psh = (aL(Ts,max _ 20) _ _I: (A max — Arec) )Ka [7.28] Amax is the maximum deflection achieved in the beam and can be computed using Eq. [7.25] at Ts = Ts,max, and Arec is the recovered deflection due to the cooling of steel, and can be assumed to be equal to the total elastic deflection Ay. 7.3 COMPARISON T0 TEST RESULTS The proposed approach for evaluating fire induced restraint force and deflection in restrained beams is compared against the results of a fire test. Unprotected restrained beam, tested under a design fire by Li and Guo (2008), was selected for the validation of the proposed approach. The tested beam, made of mild steel (Fy = 275 MPa), section H250x250x8x12, and length 4500mm, was loaded by two-point load with 130kN each (70% of its plastic bending capacity), and then was subjected to a design fire from four sides with the top flange protected using ceramic insulation to simulate 3-side exposure. Details of the fire test are presented in Chapter 4. The axial and rotational restraint stiffnesses are estimated as Ka = 39.5kN/mm and Kr = 54 kN.m/milirad, respectively. The recorded fire induced deflection and axial 228 force are plotted in Fig. 9(a and b). Average steel temperature and thermal gradient obtained from the fire test are used in the analysis. Equation [7.25] was used to predict midspan deflection, while Eqs [7.15] and [7.28] are used to predict the fire induced forces during growth phase and decay phase of the fire test. The fire induced deflection and axial force predicted using the proposed approach is also plotted in Fig. 7.9(a and b). The elasto-plastic deflection is conservatively captured by the proposed approach as shown in Fig. 7.9(a). Also, the proposed approach predicts catenary action to occur at 16 min. of fire exposure, and this compares very well with the recorded time of 15 min for catenary action in the test. However, because the proposed approach uses linear interpolation of the deflection, the maximum midspan deflection (Amax) is conservatively overestimated by 20% as shown in Fig. 7.9(a). In Fig. 7.9(c), the proposed approach predicts the fire-induced axial force fairly well before the catenary point (at 15 min.). However, in the decay phase, the proposed approach slightly underestimates the tensile force that is due to the shrinkage of steel. Slight differences between the predictions from the proposed approach and results of the fire test can be attributed to the differences in material properties, uncertainties in the stiffness values of end restraints (Ka and Kr), and to the simplifying assumptions of the proposed approach. The occurrence of local buckling in the actual test had a minor influence on the fire response. This is due to the fact that in restrained beams, tensile force develops in the beam and this tensile force alleviates the effect of local buckling. Based on the comparison shown in Fig. 7.9(a), Eq. [7.25] can be used to conservatively predict deflection of the restrained beam, and thus can be used to design the beam based on a deflection limit state. 229 " 7.4 COMPARISON TO FINITE ELEMENT ANALYSIS Since there is a lack Of fire test data on restrained steel beams under fire, predictions from the proposed approach are compared with results from finite element analysis. The finite element model developed using ANSYS in Chapter 4 for fire resistance analysis is used for generating the finite element results. The characteristics Of the beam used in the finite element analysis are shown in Table 7.1. In the following sections the proposed equations for predicting fire-induced forces and deflection are verified by comparing its predictions against finite element analysis results. 7.4.1 Fire-Induced Forces Figure 7.10 compares maximum compressive axial force Obtained from finite element analysis to that Obtained via Eq. [7.15]. The axial force is normalized to the plastic axial capacity at the temperature Of yielding (Py = ky(7:y)>< 384x0.79x2><10 x8.7><10 8 0.61 [ 0.0032 355 j 1— x _6 5 =15mm 0.6829 14x10 x2><10 Deflection at catenary point (as per Eq. [724]) A, = gjzxa(rc —20) = E299J2x14x10'6w788— 20) = 542 mm - Temperature at deflection limit state (as per Eq. [726]) (LPTAyxTcTTy):411+(7000/30-15X878.8-411) = 605°C (A, —Ay) (542-15) TDLS :T)’ + Temperature at ultimate strength failure (tensile capacity) T _ TcTyFA +a1a3 (Tc -Ty) f _ TyFA + a1a4 (Tc - Ty) _ 878.8x411x0.00037 +0.6829x1.329><(878.8—41 1) 411x0.00037+0.6829x0.0014><(878.8—411) =931°C Design for standard fire exposure After computing the limiting temperature (TDLS) based on deflection limit state, the fire resistance duration can be computed Eq. [6.5]. The insulation material is assumed tO have thermal conductivity Of 0.1 W/m.OC and heat capacity of 375 k]/m3/°C. The insulation thickness is increased and the average steel temperature 237 is calculated using Eq. [6.5] at t = 120 min. Of fire exposure until the Ts z TDLS. This occurred when an insulation thickness Of 25 mm is used. For the same insulation thickness, the limiting temperature based on strength failure (Tf) is reached at 218 min. Of standard fire exposure Design for design fire exposure If the beam is exposed to a design fire scenario Shown in Fig. 7.18(a), then the maximum temperature attained in steel (Ts,max) should not exceed the limiting temperature based on deflection limit state (TDLS = 605°C). Using Equations 6.13 and 6.14, the required insulation thickness at which Ts,max z TDLS is 15 mm, for which the maximum steel temperature under fire is Ts,max = 597°C < TDLS = 605°C (at 90 min. of fire exposure). The average steel temperature as a function Of exposure time is plotted in Fig. 7.18(a) with the applied insulation thickness Of 15 mm. Fire resistance using code provisions In the current code provisions, the critical temperature iS computed based on Specified semi-empirical equations. First, the load ratio under fire conditions (r) is computed as: r = Mo/Mp = 285.8/1308.9 = 0.21. Second, the critical temperature is computed as per Eqs. [2.5a — 2.5c]: T563 =717°C, T53” =760°C and chrBC =621°C By using Eq. [6.5] for computing steel temperature and assuming insulation thickness Of 25 mm, the following fire resistances are Obtained for each critical temperatures computed above: 238 tEC3 =160min.' tSNZ =170min. and NBC =132min. The computed values above illustrate the discrepancies between the critical temperature relationships specified in codes and standards. A comparison for the fire resistances computed using proposed approach and using equations specified in codes and standards is shown in Table 7.2. The relationship specified in the lapanese Building Code for evaluating critical temperature gave 132 min. of fire resistance for the designed beam. The New Zealand and Eurocode relationships overestimated the fire resistance by 50 min and 40 min., respectively, and thus, a design according tO these relationships would have resulted in an unconservative prediction Of insulation thickness (less than 25 mm.). Finite element analysis The solution developed above using the proposed Simplified approach is verified against results Of finite element analysis on the beam with the computed insulation thicknesses under both standard and design fire scenarios. The temperature attained in steel section is computed through heat transfer analysis. The details Of finite element modeling for heat transfer analysis are presented in Chapter 4. The analysis is carried out by subjecting the beam, with the Specified boundary conditions above, tO the uniformly distributed load until room—temperature response is obtained. TO apply the fire scenario, steel nodal temperatures obtained from the heat transfer analysis are applied, at each time step, at the corresponding nodal locations Of the structural model. The midspan deflection Of the beam under standard and design fires is plotted in Fig. 7.18(b). 239 While the midspan deflection continues to increase in the case Of standard fire exposure, as shown in Fig. 7.18(b), a recovery Of deflection occurs in beam in the case of design fire exposure, and this is attributed tO the cooling phase Of design fire, as shown in Fig. 7.18(a). The sudden increase in midspan deflection at the128th minute Of standard fire exposure (Fig. 7.18(b)) is attributed to the occurrence Of local buckling in the bottom flange Of the beam near the support. The stiffened response Of the beam after the 150th minute into standard fire exposure iS attributed to the catenary action wherein tensile force develops in the beam until complete ultimate rupture occurs after 202 minutes into standard fire exposure. It is worth mentioning that the fire design in this example is based on deflection limit state, which is more realistic than strength failure criteria. Applying the proposed approach resulted in a conservative fire design Of the restrained beam, as shown in Fig. 7.18. Under standard fire exposure, ultimate strength failure is achieved at 200 minutes as seen in Fig 7.18(b), and the fire induced deflection prior to ultimate failure was very large. However, under design fire exposure, nO strength failure occurs in the beam, although deflection in the beam did reach large values, as Shown in Fig. 7.18(b). Therefore, the proposed approach for fire design based on limiting deflection provides a realistic assessment Of fire resistance Of beams as compared to strength design criteria. Table 7.2 compares the fire resistances computed using proposed approach, relations specified in codes and standards, and Obtained form finite element analysis under different limit states. The fire resistances computed using the proposed approach are close to the fire resistances predicted using finite element analysis for 240 deflection and strength limit states. However, the fire resistances computed using relations Specified in codes and standards are scattered and do not distinguish between deflection or strength limit states. This is because the relations specified in codes and standard for computing fire resistance are mainly based on evaluating critical temperature, which is does not reflect the realistic performance Of the beam. 7 .6 SUMMARY A Simplified methodology for evaluating fire response Of restrained steel beams is developed in this chapter. The methodology is derived based on utilizing compatibility and equilibrium principles. The approach developed in this chapter accounts for various parameters that have a significant influence on the fire resistance Of restrained steel beams such as fire scenario, load ratio, thermal gradient, and restrained conditions. The validity Of the proposed approaches is established by comparing predictions with results from detailed finite element analysis and fire tests. The proposed methods provides Simple, engineering approach for computing the fire resistance Of restrained steel beams exposed tO standard or design fire scenarios. The approach facilitates a rational fire design Of restrained beams under various parameters such as sectional size, length, load, insulation thickness and material properties. The applicability Of the proposed approach in a design Simulation is illustrated through a numerical example. The proposed methodology is compared against relations Specified in codes and standards, and it is shown that the approach provides better assessment Of fire resistance based on different limit states. 241 Table 7.1: Characteristics of the beams used in finite element analysis for comparison with proposed approach Parameter Values Comments Span length (L) 6m, 9m, 12m Total length Load ratio (LR) 30%, 50%,70% LR = M m /(FyZ,) Thermal gradient 0°C, 100°C, 200°C AT: TTF — TBF (AT) Measured downward Location of 0, d/Z form the centroid of the restraint section Axial stiffness d . f . (Ka) 0, 0.1, 0.3EsAS/L an In Imty Relative to room- Rotational temperature beam 0, 0.1, 0.3EsI/L and infinity stiffnesses stiffness (Kr) Table 7.2: Fire resistance predicted using proposed approach and provisions in codes and standards under different limit states Fire resistance (minutes) Deflection limit state Strength limit state (L/ 3 0) Proposed approach 120 218 japanese BuildingCOde 132 132 Eurocode 3 160 160 New Zealand Standard 170 170 Finite element analysis 128 202 242 Ka, Kr W Ka, Kr a) Layout of restrained steel beam Stage I a Stage II E Stage III "“59“,?”“K Initiation Ofyielding Elastic >ElastO-plastic ! Fire-induced axial force Steel temperature Catenary action / g 5 . Ty: \: I V l b) Fire- induced axzal force Steel temperature 13' 3 Tc .:_-<../VV w 5 0' A Ae : ' A - E c: 5 1°: Initiation of 1 8 catenary action E Rupture 0f steel E LP :: 01 1 l “U _‘ __________________________ l I I: Ac ! l g i '3 c) Midspan deflection E E i v E Fig. 7.1: Typical fire response Of a restrained steel beam that fails by yielding 243 x = °° I x = 10 0.8 — _ Ka r K, x .. 1.0 ESAS/L ESI/L [E 0'6 ‘ x = 0.6 a x = 0.3 II. 0.4 - x = 0.1 0.2 — \_ O I— l I I I 0 200 400 600 800 1000 Steel temperature, °C a) ASCE high temp. steel properties 1 -— x = °° 1 I x = 10 0.8 " _ Ka or Kr x = 1.0 ESAS/L ESI/L [E 0.6 r x = 0.6 2 x = 0.3 "- 0.4 — = 0.1 0.2 — \ o I l I l - ‘l 0 200 400 600 800 1000 Steel temperature, °C b ) E urocode 3 high temp. steel properties Fig. 7.2: Variation of the restraint factors (FA and FR) as a function Of end restraint ratio (x) and temperature using ASCE and Eurocode steel high temperature properties 244 \ Pc,max Tc T ten,max Steel Temperature kymeyXAS Fire-induced axial force Fig. 7.3: Development of tensile catenary force in I)! \3/ !\ I l l 1 I l l l l l l\ l\\ I l I \ I \ l \l \l l I I Pten, max V" 'l restrained beam Ac Fig. 7 .4: Approximated deflection profile at the catenary point 245 Steel temperature V ............................................ ‘5‘ . Linear interpolation Midspan deflection V Elastic Elasto-plastlcg Ca tenary phase - ~ Fig. 7.5: Interpolating deflection between catenary point and point Of initiation Of yielding T Standard fire 1000 Mix Design "realistic" fire \ D 800 _ I T ”a"’ o ' ’ .szmax"‘,o' 45‘ 600 r : I” I "~ \ ‘5 L" 4’ \ Average steel temp. 2 400 {E I”: : x“ / E I l" I Q 200 ’1”. : ~‘~~ ~ 8 " : t] . ts,max .--.- 0 V VI *I I I I T l 0 20 40 60 80 100 120 140 Time , min Fig. 7.6: Standard and design (realistic) fire scenarios 246 Kr Ka IIIIIIIIJIIIIIIIIIII' (L/ZJ' (a) Equilibrium position before fire exposure _T_ N I | N 5 K_ N 7: a T Amax — Arec Y1 /\ Final position (L/ZJ \/ of the support [b] Equilibrium position after cooling -—\IL-—- Fig. 7.7: Approximated deflection profile after cooling of steel 247 Fire exposure time, min 0 51015202530354045 o F I l l I I I E ANSYS g" -100 — 0 Test data g ------- Proposed appr. «3 -200 — '0 g ‘90... o o o o. -300 I In E _______________________ E -400 J (a) Predicted and measured deflection 550 - . o 400 ~ E g 250 —« ' . o _ '. g 100 4 . J Time, min _50 N1 1 .r/ 1 T 0 5 1015 20 25 30 354045 -200 — ._ V ANSYS g '350 " 0 Test data I.I. ------- Proposed appr. -500 - (c) Predicted and measured fire-induced axial force Fig. 7.8: Comparison of fire response predictions from the proposed approach with results from finite element analysis and from fire test 248 ’ O O 9 0 6* ,4 o o <. I!” :0 o. u'!’ .1, . ' u. IA O 9 A" o ’9 t 4'? E 04 ’9 o 0 g ’0‘. E o a. O V 0.2 00 0.2 0.4 0.6 0.8 l:,maxIPy)Proposed approach Fig. 7.9: Comparison of maximum compressive fire-induced axial force as obtained from the proposed approach and finite element analysis 249 600- .9 +10% margin 5,“ I g , E 500‘ 9 Q, ’i’ .3, o y” ’94” '2’ 400] 09,” 919’ % 90,3 ’3’ :93 300]. O ”’0 x’“ g ”89’ , ’ -10% margin “ .3’. ' g 200i no” «6’ I: “O ’ 0’ A» - xi 6‘: 100r 0 Q ’ 8 “5 GO 1 00 200 300 400 500 600 Ty (°C) From simplified approach a) Temperature at yield (Ty) 1000— x .2 +10% margin 4, z’ s. x :5 900* 0 .’ u E ,o’ 0} GE) 800- O x .2 ,g’ Q x’i 3 700- ’4" 9g ’4 *0/40 "F g , ,p -10% margin g 600- ,x’o 10* It ‘ $8: a ’/ §: 500. x" Q,“ ” O u , I, h I / 40200 ‘ ’ 560 600 760 800 960 1600 To (°C) From simplified approach b) Temperature at catenary point (Tc) Fig. 7.10: Comparison of predictions from proposed approach with results from finite element analysis for Ty and Tc 250 LFi:30% L: 6m LR:30% L: 9m “‘ F.E.A. AR = 10% 360 600 Temperature, °C 0 LR: 70% 9 /° L: 12m 360 600 Temperature, °C ' ' Proposed: AR = 10% 900 o 300 600 900 Temperature, °C F.E.A. "' AR = 0 “'“ Proposed: AR = 0 Fig. 7.11: Midspan deflections as Obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different axial restraints (Kr = 0) 251 v c o. 2°/ol ‘ 2°/o’ 33 4% - 4% 6%: LR:30% - 6%: L: 6m 0 . O 8 /°o 300 9008 /°0 C W Orr 2% 4 2%» 33 4% . 4% 6%] LFl:30% ‘ 6% L: 9m 0 I . O L . 8 /°o 300 600 900° :0 300 600 900 2°/o \‘ [ -\ I 6% _ LH:30% *4 6%] LFl:50% 3%] L: 12m L: 12m 0 . . O A . 8 /°o 300 600 9008 /°o 300 600 900 Temperature, °C Temperature, °C 6% _ LFi:70% .. . L: 9m ““5. O A . 8:0 300 600 900 2% 4% * 6% 8% LFI:70% ‘ L: 12m “x 0 300 600 900 Temperature, °C Fig. 7.12: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different axial restraints (Kr = 100% EsI/L) (symbols for lines are like previous figure) 252 . 1% «2% 0% ‘ LR:30% LR:70% 3. I 3% L: 6m ‘ 3%? L: 6’" a. 4% g . . “40/0 I I 4% 1 I ‘9 o 200 400 600 800 0 200 400 600 300 o 200 400 600 800 . . . 0%, a . 0° A»! . . q 1% - 1%» a ‘x 4 2%» 2%, LH:30% *. “550% LR:70% . 3/.- ngm X,\« 3/0» L_,9m 4 3%.» Log”, (Q I. \ t o O . n o A \ 4° 3004 /°0 200 400 600 8004 /°0 300 o 200 400 600 0% “1% ‘ 20/0‘ ‘ 370* f LFI:30% 0 5“" L: 12m 4% . o 200 400 600 800 o 4% Temperature, °C — F.E.A (AT=0°C) ....--.. Proposed app. (AT=0°C) -- F.E.A (AT=200°C) - - - Proposed app. (AT=200°C) Fig. 7.13: 200 400 600‘ 800 0 Temperature, °C 0% .. 1%, ‘ 2°/o * ‘ 30/0. LR:70% L: 12m 4% 200 400 600 Temperature, °C 800 Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 laterally-braced beam under different thermal gradients (Kr = 100% EsI/L, Ka = 10%EsA/L) 253 070 T 0%] 1% \th I 1%» g 2% . 2% LR:30% g LFI:50% ‘. '3 30/0’ L: 6’" ‘.\ ‘ 30/0' L: 6"" .i 3%) ’ ‘i 0 4 . O O 4 A’0 200 400 600 8004 /°0 13004 A’0 0%. - a 0% 0% 10/0‘ W. x ‘ 1°/o 1%’ g 2% ‘1, 2% . 2% LR:30% .‘a . Lia-70% .3, 3%“ L: 9m ‘21 3% L: 9m 4: ‘ 3% L: 9m “(‘3 40A, . . . ‘ 4% I A I R 40% I I L ‘ o 200 400 600 800 0 200 400 600 800 O 200 400 600 800 0%. . . . 0% - - . 0% . . - . _ 1% s _ 1% - 1%' b g 2% ‘. 4 2%» - 2%» LFl':30% ‘4 LR:70% b 1 30/6. L.’ 12m ‘. ‘1, 313/0. ~ 30/o~ L.‘ 12m \. 4°/ - . 1 (‘13 42 "\ 4% \‘ °0 200 400 600 800 °0 200 400 600 800 Temperature, °C + F.E.A (y = 0) - 1 - Proposed app. (y = 0) — F.E.A (y = d12)--~- Proposed app. (y = dIZ) 200 460 600an0 Temperature, °C Temperature, °C Fig. 7.14: Midspan deflections as obtained from proposed approach and finite element analysis for W24x76 braced beam with different locations of axial restraint (Kr = 100% EsI/L, Ka = 10%EsA/L, AT = 200°C) 254 —L J \ 1 +10%margin ”x m, 800 - ’ ’0. ll. / .. / g 7 fl / g 2", l: l/ a ’ .‘ °~ 600 — / i”. .2 ’ 500 ~ / V -10%margin 400 ,'l/ I I I I I l 400 500 600 700 800 900 1000 TDLS (°C) From simplified approach a) Deflection limit state (LF) = L/20 1000 - ~ I X,’ 900 a q +10%margin .,O’ Ill A ’ / IL. 800 Q.. I,’ 5 x’. 9' It 700 a .0 , ’ a a" I” ...- 2" .2 a, 500 4 / / -10%margin 4m f// I T I I I l 400 500 600 700 800 900 1000 TDLS (°C) From simplified approach b) Deflection limit state (LF) = L/30 Fig. 7 .15: Comparison Of predicted temperatures at deflection limit state as predicted by proposed Eq. 7.26 for LF = L/ZO and L/30 with results from finite element analysis 255 w]. = 70 kN/m, wD = 35 kN/m g é W24- x76 I :36 ‘5 a) Beam loading and prOperties cs /é>\/é>\ss Mm = 285.8 kN.m b) Bending moment diagram under fire load Ms = 143 kN.m Fig. 7.16: Properties of the continuous beam used in the design example 256 1200 ~ Design fire 1000 - Standard fire 9 800 — 2 3 600'— g Tm = 605°C E 400 :33 x 200 I Steel temperature 0 l I ' I'----- ----l I O 50 100 150 200 250 Time, min. ( a ) Steel temperature Fire exposure time, min. 0 50 100 150 200 250 I Deflection limit 400 — state "L,.-" Midspan deflection, mm do 8 '500 “ Strength limi Standard fire (F.E.A.) Standard flre (Proposed app.) - - - Design fire (F.E.A.) ------- Design fire (Proposed app.) (b) Midspan deflection Fig. 7.17: Steel temperature and midspan deflection as a function of fire exposure time for restrained beam 257 CHAPTER EIGHT CONCLUSIONS AND RECOMMENDATIONS 8.1 SUMMARY The thermal and structural response of steel beam-columns exposed to fire conditions was investigated as part Of this study. Both the plastic axial force (P) — moment (M) behavior and the global response of steel beams under fire induced thermal gradient and restraint conditions were studied. As part of experimental studies, fire resistance tests were carried out on four steel beam-columns to obtain data on the plastic P-M behavior Of beam-columns under realistic fire conditions. The parameters that were varied in the fire tests included load ratio, fire scenario, insulation thickness and magnitude and orientation of thermal gradient. Data from the fire resistance tests, in addition to available test data from literature, were used tO validate finite element models that were created using ANSYS. In the analysis, all the critical factors, namely high temperature material properties, fire induced restraints and thermal gradients, and the different strain components (including high temperature creep) that have significant influence on the fire response of steel beam-columns were incorporated in the models. The finite element models were applied to conduct a set of parametric studies to quantify the influence Of various factors on the fire response Of steel beam-columns. Results from fire tests and parametric studies were utilized to develop two methodologies for 258 evaluating response Of beam-columns under fire conditions at sectional and global levels. At the sectional level, a simple methodology for generating plastic P-M diagrams was derived by accounting for the effect of thermal gradients. At global level, a simplified approach was developed for evaluating fire induced axial force and deflection in restraint steel beams at any given time. The proposed approaches take into consideration the influence Of various factors governing the fire response, including fire scenario, end restraints, connection configuration (location Of axial restraint force), thermal gradient, load level, beam geometry, and failure criteria. The applicability Of the proposed methods to design situations is illustrated through numerical examples. it is shown that the proposed design guidelines require minimal computational effort, and thus are attractive for incorporation in codes and standards. 8.2 KEY FINDINGS Based on the information presented in this study, the following key conclusions are drawn: I There is very limited information on the fire performance of steel beam- columns, especially under realistic fire scenarios and restraint conditions. The current provisions, developed based on limited fire tests under "standard fire scenarios", are prescriptive and do not facilitate a rational approach for evaluating the fire resistance. Also, to date, there are no engineering calculation methodologies for evaluating the response of restrained steel beams under thermal gradients. 259 The fire resistance tests on four steel beam-columns provided unique data on the behavior Of steel beam-columns under realistic fire, load, and restraint scenarios. Data from fire resistance tests show that fire induced thermal gradient causes a shift in the plastic P-M diagrams Of steel beam-columns made Of wide flanged sections. This shift in plastic P-M diagrams is more significant when thermal gradient is in the strong direction Of the cross section. Results from parametric studies, as well as data from fire resistance experiments, indicate that fire scenario, load ratio, thermal gradient, magnitude and location of axial restraint and rotational restraint have a significant influence on the fire response of steel beam-columns, specifically: - A higher load level has an adverse effect on the fire response Of steel beam-columns. - A higher axial restraint ratio leads to higher fire induced restraint force and larger mid-span deflection. - Fire resistance is higher when the axial restraint is shifted downward below, the centroidal axis Of the beam. - Fire resistance decreases with increased axial restraint stiffness under deflection limit state; however, fire resistance increases with increased axial restraint stiffness under strength failure criteria. - Increased rotational restraint stiffness leads to improved fire response in restrained steel beams. 260 - Higher fire induced thermal gradients lead to larger elastic midspan deflection in steel beams. - The type of fire exposure has a significant effect on the fire resistance of restrained steel beams and these beams can survive bum-out conditions under most design fire scenarios. I The proposed methodology for adjusting the plastic P-M diagrams provides a simple and rational approach for generating P-M diagrams for steel beam- columns. The approach accounts for to thermal gradients generated under 1-, 2-, or 3-side exposure to either standard or design fire scenarios. I The proposed methodology for tracing the fire response of restrained steel beams, derived based on equilibrium and compatibility principles, accounts for realistic load, boundary conditions, thermal gradient, and fire scenarios. The approach can be applied for evaluating fire resistance of restrained beams under standard or design fire exposures and utilizing strength or deflection limit states. I The proposed approach provides, for the first time, simple design tools for fire design Of restrained steel beams. The approach requires minimal computational effort, and provides better fire resistance predictions than those Obtained from current codes provisions. Therefore, it can be integrated into structural design processes and substantially improve the fire safety Of the structures designed. 8.3 RECOMMENDATIONS FOR FUTURE RESEARCH While this study has developed fundamental understanding Of the fire response Of steel beam-columns, further research is required to extend the principles to other 261 practical situations that are present in typical steel framed buildings. The following are some of the key recommendations for further research in this area: 0 The influence of thermal gradients on plastic P-M diagrams was presented here for wide flanged (WF) sections. Further research is required to quantify the effect of thermal gradient on other types Of steel sections such as boxed sections and concrete-filled steel tube sections. 0 The research presented here can be extended to include the effect Of composite action between steel beams and concrete slabs. Current proviSions Of fire design of composite beams do not take into consideration the effect of end restraint and fire scenario. 0 In the current study, the end restraints were idealized with constant and elastic axial and rotational stiffnesses. Also, tO-date, there is no information on how to quantify the temperature-dependent variation Of axial and rotational restraint stiffnesses (Ka and Kr) acting on the boundaries of restrained steel beams. Further research is required to account for the more realistic nonlinear behavior of end restraints (Ka and Kr). o In this study, the steel beam-columns were assumed to have a zero effective length [KLO), i.e., fully braced beam-column with plastic failure mode. This research can be extended to account for the variation of the effective length Of restrained steel columns under fire conditions. 0 The current constitutive models for high temperature material properties of steel (including high temperature creep strain), particularly in the cooling phase, are not sufficiently accurate. Improvements in such constitutive 262 models will help to enhance the accuracy Of the model predictions under design fire scenarios. 8.4- RESEARCH IMPACT The current approaches for evaluating fire resistance through standard fire tests on full-scale steel members are expensive, time consuming and have a number Of drawbacks. An alternative is to use calculation methods for predicting fire resistance. However, such calculation methods are not widely available at present Further, the current fire resistance provisions in codes and standards (ASCE 1992, EC3 2005) are prescriptive and do not account for realistic conditions such as end restraint, loading and fire scenarios. Thus, the current design approaches may not be fully applicable for undertaking performance-based design which provides rational and cost-effective fire safety solutions. The proposed design approach provides a convenient way Of Obtaining fire response and fire resistance of restrained steel beams, and thus can be used for estimating fire resistance in lieu Of full-scale standard fire resistance tests. The proposed design approach can be applied to evaluate fire resistance under realistic fire, loading and restraint scenarios. The proposed equations express fire resistance in terms of structural parameters, and thus the approach isattractive for incorporation in codes and standards. in summary, the use Of the proposed approach will facilitate a rational fire design under a performance-based code environment. Such a rational design approach will contribute to reduced loss of life and property damage in fire incidents. 263 APPENDIX This research has produced the following papers: Peer-Reviewed Journal Papers 1. Dwaikat, M.M.S, and Kodur V.K.R, (2010) “Effect Of Location Of Restraint on Fire Response Of Steel Beams", Journal of Fire Technology, (46)1, pp. 109- 128. 2. Dwaikat, M.M.S and Kodur, V.K.R. (2009) “A Simplified Approach for Evaluating Plastic Axial and Moment Capacity Curves for Beam-Columns with Non-uniform Thermal Gradients”, Engineering Structures, 32, pp.1423-1436 3. Kodur V.K.R. and Dwaikat M.M.S. (2009), “Response of Steel Beam-Columns Exposed to Fire”, Engineering Structures, (31), pp. 369-379. 4. Kodur V.K.R, Dwaikat M. M. S., and Dwaikat M. B. (2008), “High-Temperature Properties Of Concrete for Fire Resistance Modelling of Structures", ACl Material Journal, 105(5), pp. 517-527. 5. Kodur, V.K.R., Dwaikat, M.M.S, and Pike R, (2009) “High-Temperature Properties Of Steel for Fire Resistance Modelling Of Structures", In Press, Journal of Materials in Civil Engineering- ASCE. 6. Kodur, V.KR., and Dwaikat, M.M.S, (2009) “Effect Of High Temperature Creep on the Fire Response Of Restrained Steel Beams". in Press, Materials & Structures Journal. 264 Conference Proceedings 1. Dwaikat, M.M.S. and Kodur, V.K.R. (2010) "Performance based design approach for evaluating fire resistance Of restrained steel beams”, Proceedings Of the Sixth International Conference on Structures in Fire (SiF’lO), East Lansing. 2. Dwaikat, M.M.S. and Kodur, V.K.R. (2010) “Lateral-Torsional Buckling Of Steel Beam-Columns under Fire Exposure", SSRC Annual Stability Conference, Orlando, Florida. 3. Dwaikat, M.M.S., and Kodur, V.R., (2009) “Effect Of Fire Induced Restraint on the Fire Response of Steel Beams", 5th International Symposium on Steel Structures: 1555 '09, Seoul, Korea. 4. Dwaikat, M.M.S., and Kodur, V.K.R, (2009) “Effect Of Restraint Force Location on the Response of Steel Beams Exposed to Fire”, Structures Congress Conference '09, ACSE, Austin, TX 5. Dwaikat, M.M.S., and Kodur, V.R., (2008) “Fire Performance of Steel Beam- COIumns under Design Fire Exposures”, Proceedings of the Fifth International Conference on Structures in Fire (SiF’08), Singapore. 6. Dwaikat, M.M.S., and Kodur, V.R., (2008) "Effect Of High Temperature Creep and Fire Scenario on the Response Of Steel Beam-Columns", BFRL Fire Conference, NIST/BFRL, Gaithersburg, MD. 7. Kodur, V.R, Garlock, M.E, Dwaikat, M.S, Quiel, S.,(2009) “Collaborative Research: Fire Engineering Guidelines for the Design Of Steel Beam- 265 Columns”, Proceedings Of 2009 NSF Engineering Research and Innovation Conference, Honolulu, Hawaii. . Iqbal, S., Dwaikat, M.M.S., and Harichandran R S., (2009) “Simple Approach for Calculating Inelastic Deflections Of Simply Support Steel Beams under Fire." Structures Congress Conference ’09, ACSE, Austin, TX. 266 REFERENCES . AISC (2005), "Steel Construction Manual 3rd Edition", American Institute Of Steel Construction, Chicago, Illinois. . Ali, F. and O'Connor, D., (2001) “Structural performance Of rotationally restrained steel columns in fire.” Fire Safety Journal, Vol. 36, No. 7, pp. 679- 691. . Ali, F., Shepherd, P., Randall, M., Simms, I.W., O’Connor, D.J., Burgess, I.W. (1988) “The effect of axial restraint on the fire resistance of steel columns." Journal Of Constructional Steel Research, Vol. 46, NO. 1-3, pp. 305-306(2). . 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