Copyright By Chung Thi Thu Ho 2010 The Thesis committee for Chung Thi Thu Ho Certifies that this is the approved version of the following thesis ANALYSIS OF THERMALLY INDUCED FORCES IN STEEL COLUMNS SUBJECTED TO FIRE APPROVED BY SUPERVISING COMMITTEE: Supervisor: ____________________________________ Michael D. Engelhardt ____________________________________ Todd A. Helwig ANALYSIS OF THERMALLY INDUCED FORCES IN STEEL COLUMNS SUBJECTED TO FIRE By Chung Thi Thu Ho, B.E, B.E THESIS Presented to the Faculty of the Graduate School of the University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN ENGINEERING The University of Texas at Austin August 2010 To my father and mother v ACKOWLEDGEMENT I would like to thank Vietnam Education Foundation (VEF), who sponsored my graduate study at UT Austin in the international education exchange program to improve Vietnamese Science and Technology capacities. I wish to express my gratitude to my advisor, Dr. Michael D. Engelhardt. His support, guidance, patience and profound knowledge have motivated me a lot from the beginning of my master’s program and through the study in this thesis. Without him, this thesis could not be completed. Finally, I would like to thank my parents, Ho Thu Quang and Nguyen Thi Hue, whose unconditional love and encouragement always enlighten the road that I go and provide me the strength to overcome many difficulties in the life. August 13, 2010 vi ANALYSIS OF THERMALLY INDUCED FORCES IN STEEL COLUMNS SUBJECTED TO FIRE by Chung Thi Thu Ho, M.S.E. The University of Texas at Austin, 2010 Supervisor: Michael D. Engelhardt The effects that thermally induced forces and deformations have on the performance and safety of steel columns subjected to fire are not well understood and are not clearly treated in building codes and standards. This thesis investigates the behavior of steel columns subjected to fire, with an emphasis on studying the significance of thermally induced forces and deformations. The approach used in this research is to conduct a series of analyses of steel columns using the finite element computer program ABAQUS. Columns are modeled in ABAQUS using beam elements that include nonlinear geometry, nonlinear temperature dependent material properties, and initial geometric imperfections. Using the ABAQUS model, a series of analyses are conducted on the behavior of columns under axial compression for temperatures varying from room temperature up to 2400° F. A series of individual columns are analyzed with and without restraint to thermal expansion. A column that is part of a truss is also analyzed to study a simple case of a flexible restraint to thermal expansion. Finally, the behavior of columns that are part of multi-story steel moment frames are investigated. All of the analyses conducted in this research indicate that forces generated by restraint to thermal expansion can have a very large impact on the performance of a steel column in fire. When evaluating the safety of a column in a fire, it is important to recognize that the vii total axial force in the column is the sum of the force generated by external gravity load on the frame and the force generated by restraint to thermal expansion. The force generated by restrained thermal expansion can be very large, and neglecting this force can lead to unsafe designs. viii TABLE OF CONTENTS TABLE OF CONTENTS ............................................................................................. viii LIST OF TABLES......................................................................................................... xi LIST OF FIGURES ..................................................................................................... xii CHAPTER 1 INTRODUCTION .................................................................................... 1 1.1 Overview .............................................................................................................. 1 1.2 Approaches to Structural Fire Safety Design ..................................................... 1 1.3 Thesis Objective and Organization ..................................................................... 3 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW ..................................... 4 2.1 Overview .............................................................................................................. 4 2.2 Steel Material Properties at Elevated Temperature ........................................... 4 2.2.1 Mechanical Properties ..................................................................................... 5 2.2.2 Stress-Strain Curves ........................................................................................ 9 2.2.2 Coefficient of Thermal Expansion ................................................................. 11 2.3 Previous Research on Column Strength at Elevated Temperatures and Column Strength According to AISC and Eurocode 3 .......................................... 13 2.3.1 Previous Research on Column Strength at Elevated Temperatures ................ 14 2.3.2 Buckling Strength of Steel Columns at Elevated Temperature in AISC Specification and in Eurocode 3 ............................................................................. 15 2.4 Previous Research on Thermally Induced Forces and Deformation in Columns ................................................................................................................... 20 2.5 General Assumptions Applied in This Thesis ................................................... 23 2.6 Summary ............................................................................................................ 24 CHAPTER 3 MODELING TECHNIQUES ................................................................. 25 3.1 Overview ............................................................................................................ 25 3.2 ABAQUS Program ............................................................................................ 25 3.3 Structural Modeling in ABAQUS ..................................................................... 26 3.4 General Issues in Creating Frame Models in ABAQUS .................................. 27 3.4.1 Element Types and General ABAQUS Assumptions ..................................... 27 ix 3.4.2 Material Stress – Strain Curve Model ............................................................ 28 3.4.3 Mesh Size ..................................................................................................... 32 3.4.4 Number of Load Increments .......................................................................... 35 3.4.5 Linear Geometry and Non-Linear Geometry Analyses .................................. 36 3.4.6 Load Control and Deflection Control ............................................................ 39 3.5 Summary ............................................................................................................ 42 CHAPTER 4 ANALYSIS OF INDIVIDUAL COLUMNS ........................................... 43 4.1 Overview ............................................................................................................ 43 4.2 Buckling of Axially Unrestrained Elastic Column Subjected to Axial Load at Room Temperature ................................................................................................. 43 4.2.1 Theoretical Methods for Elastic Buckling Analysis ....................................... 43 4.2.2 ABAQUS Models for Buckling Analysis ...................................................... 47 4.2.3 Analysis of Axially Unrestrained Elastic Columns Subjected to Compressive Load ...................................................................................................................... 48 4.3 Axially Restrained Elastic Column Subjected to Thermal Gradient ............... 50 4.4 Buckling of Axially Restrained Elastic Column Subjected to Thermal Expansion................................................................................................................. 53 4.4.1 Theoretical Methods for Buckling of Axially Restrained Elastic Column Subjected to Thermal Expansion ............................................................................ 53 4.4.2 ABAQUS Analysis of Axially Restrained Elastic Column Subjected to Thermal Expansion ................................................................................................ 55 4.5 Buckling of Axially Unrestrained Inelastic Columns Subjected to Axial Load at Elevated Temperatures ....................................................................................... 58 4.6 Behavior of Interior Column in a Truss Subjected to Gravity Load and Thermal Expansion ................................................................................................. 63 4.7 Summary ............................................................................................................ 72 CHAPTER 5 ANALYSIS OF COLUMNS IN FRAMES ............................................. 74 5.1 Overview ............................................................................................................ 74 5.2 Behavior of Interior Columns in a Ten-Story Steel Moment Frame ............... 76 x 5.2.1 Description of Frame .................................................................................... 76 5.2.2 Analysis of Ten Story Frame Using Load Control ......................................... 78 5.2.3 Analysis of Ten Story Frame Using Temperature Control ............................. 86 5.3 Behavior of Interior Columns in the SAC Buildings ........................................ 88 5.3.1 General Information about SAC Buildings .................................................... 88 5.3.2 Load and Load Combinations for Structural Fire Design ............................... 92 5.3.3 ABAQUS Analyses for SAC Buildings Subjected to Fire ............................. 93 5.4 Summary ............................................................................................................ 97 CHAPTER 6 SUMMARY AND CONCLUSIONS ....................................................... 99 6.1 Summary ............................................................................................................ 99 6.2 Conclusions ...................................................................................................... 100 6.3 Recommendations for Further Study ............................................................. 101 REFERENCES .......................................................................................................... 102 VITA ........................................................................................................................... 105 xi LIST OF TABLES Table 2.1 Reduction Factors for Mechanical Properties per Eurocode 3 ........................... 6 Table 2.2 Reduction Factors for Mechanical Properties per AISC .................................... 8 Table 2.3 Stress-Strain Relationships for Carbon Steel per Eurocode 3 .......................... 10 Table 5.1 Beam and Colum Sections of Moment Frame in 3-Story SAC Building (FEMA 2000) ............................................................................................................................. 90 Table 5.2 Beam and Colum Sections in Moment Frame of 9-Story SAC Building (FEMA 2000) ............................................................................................................................. 90 Table 5.3 Beam and Colum Sections in Moment Frame of 20-Story SAC Building (FEMA 2000) ................................................................................................................ 91 Table 5.4 Dead and Live Loads and Load Combinations ............................................... 93 xii LIST OF FIGURES Figure 2.1 Reduction Factors for Mechanical Properties per Eurocode 3 ......................... 7 Figure 2.2 Reduction Factors for Mechanical Properties per AISC .................................. 9 Figure 2.3 Stress-strain relationships for carbon steel at elevated temperature per Eurocode 3 .................................................................................................................... 11 Figure 2.4 Thermal Strain of Steel at Elevated Temperature per Eurocode 3 .................. 12 Figure 2.5 Thermal Expansion of Steel at Elevated Temperature ................................... 13 Figure 2.6 Strength Curves for Steel Columns at Elevated Temperatures per AISC Appendix 4 .................................................................................................................... 18 Figure 2.7 Strength Curves for Steel Columns at Elevated Temperatures per Eurocode 3 ...................................................................................................................................... 19 Figure 2.8 Strength Curves for Steel Columns at 600 oF per AISC Appendix 4 and Eurocode 3 .................................................................................................................... 19 Figure 2.9 Strength Curves for Steel Columns at 1000 oF per AISC Appendix 4 and Eurocode 3 .................................................................................................................... 20 Figure 3.1 ABAQUS Stages of a Complete Simulation (ABAQUS, 2008a) ................... 26 Figure 3.2 Typical Models of Stress-Strain Curves ........................................................ 29 Figure 3.3 Deflections at Midspan of a Fixed-Fixed Beam Subjected to Distributed Load ...................................................................................................................................... 31 Figure 3.4 Load-Deflection Response of a Fixed-Fixed Beam Subjected to a Distributed Load with Different Mesh Sizes .................................................................................... 34 Figure 3.5 Load-Deflection Response for an Initially Imperfect Column Subjected to Axial Load with Different Mesh Sizes ........................................................................... 35 Figure 3.6 Load-Deflection Response of a Fixed-Fixed Beam Subjected to Distributed Load Using Linear and Nonlinear Geometry Analyses .................................................. 38 Figure 3.7 Load-Deflection Response for an Initially Imperfect Column Subjected to an Axial Load Using Linear and Nonlinear Geometry Analyses ......................................... 39 xiii Figure 3.8 Load- Deflection response of a Column Subjected to an Axial Load Using Load Control and Deflection Control ............................................................................. 41 Figure 4.1 Deflection Behavior of Perfectly Straight and Imperfect Elastic Columns ..... 46 Figure 4.2 Load-Deflection Response for an Initially Imperfect Elastic Column Subjected to Axial Load ................................................................................................................ 49 Figure 4.3 Axially Restrained Elastic Column Subjected to a Thermal Gradient ............ 52 Figure 4.4 Analyses of Axially Restrained Elastic Column Subjected to Temperature Increase ......................................................................................................................... 56 Figure 4.5 Axial Force – Deflection at Midspan of Axially Unrestrained Column Subjected to Compressive Load at Elevated Temperatures per ABAQUS Analyses ....... 60 Figure 4.6 Compressive Strength of Axially Unrestrained Column at Elevated Temperatures................................................................................................................. 61 Figure 4.7 Truss with Interior Column Subjected to a Fire ............................................. 63 Figure 4.8 Compressive Strength of Interior Column in a Truss Subjected to Axial Load at Elevated Temperatures .............................................................................................. 68 Figure 4.9 Column Axial Force versus Applied Load P at 200 oF .................................. 69 Figure 4.10 Column Axial Force versus Applied Load P at 400 oF ................................ 70 Figure 4.11 Column Axial Force versus Applied Load P at 800 oF ................................ 70 Figure 4.12 Column Axial Force versus Applied Load P at 1200 oF .............................. 71 Figure 5.1 Ten Story Moment Frame with an Interior Column Subjected to Fire ........... 77 Figure 5.2 Compressive Strength or Induced Axial Force for Column C in Ten Story Moment Frame .............................................................................................................. 80 Figure 5.3 Interior Column Axial Force versus Equivalent Applied Load P at 200 oF .... 82 Figure 5.4 Interior Column Axial Force versus Equivalent Applied Load P at 400 oF .... 82 Figure 5.5 Interior Column Axial Force versus Equivalent Applied Load P at 800 oF .... 83 Figure 5.6 Interior Column Axial Force versus Equivalent Applied Load P at 1200 oF .. 83 Figure 5.7 Column Axial Force versus Distributed Load w at 1000 oF ........................... 85 Figure 5.8 Axial Force in the Interior Column of Ten Story Moment Frame for Temperature Control Analysis ....................................................................................... 87 xiv Figure 5.9 Elevation of External Moment Frames of 3-Story, 9-Story and 20-Story SAC Buildings (FEMA 2000) ................................................................................................ 89 Figure 5.10 Column Axial Force in 3 – Story SAC Frame from ABAQUS Temperature Control Analysis............................................................................................................ 94 Figure 5.11 Column Axial Force in 9 – Story SAC Frame from ABAQUS Temperature Control Analysis............................................................................................................ 95 Figure 5.12 Column Axial Force in 20 – Story SAC Frame from ABAQUS Temperature Control Analysis............................................................................................................ 96 1 CHAPTER 1 INTRODUCTION 1.1 Overview With the increasing interest in developing performance-based approaches for structural fire safety design, the ability to reasonably predict the response of a structure under fire effects is of great importance. Of particular interest is the ability to predict the response of columns in buildings subjected to fire, as columns are critical to the safety of a structure. This thesis presents the results of a study on the behavior of columns in steel buildings subjected to fire. More specifically, this study focused on the influence that restraints to thermal expansion have on the performance and safety of columns in steel buildings. Fire can be critical to the structural safety of the building. When a fire occurs in a building, the increase in temperature due to the fire can lead to a large reduction in strength and stiffness of the structural members. It can also result in large thermally induced forces and deformations in structural members. These effects can lead to collapse of buildings in fire. While there have been significant advances in the understanding of structural response to fire in recent years, there are still many aspects of structure-fire behavior that are not well understood and require further research. The investigation reported in this thesis is intended to contribute to an improved understanding of the response of steel structures to fire. 1.2 Approaches to Structural Fire Safety Design In general, there are two fundamental approaches used in structural fire safety design. One is a prescriptive approach, and the other is an engineered or performance-based approach. In the prescriptive approach, simple rules for structural fire safety prescribed in 2 building codes are followed. Following these rules generally requires no engineering calculations. The prescriptive rules in building codes are generally based on furnace testing of individual structural components (beams, columns, walls, etc) subjected to a standard fire to develop an hourly fire resistance rating. The prescriptive approach has been widely used for many years, and still dominates structural fire safety design practice in the U.S. In contrast, with the engineered approach to structural fire safety, the response of the structure to fire is computed and appropriate design measures are taken to provide acceptable structural performance. With this approach, fire is considered a loading condition in the structural design process, similar to other loading conditions such as gravity load, wind load, earthquake load, etc. The engineered approach to structural fire safety is also frequently referred to as a performance based approach. An engineered, or performance based approach is recognized as a more rational approach that can lead to more cost effective structural fire safety design (Buchanan 2002). In the final report on the collapse of the towers of the World Trade Center and the collapse of the 47-story WTC-7 Building on September 11, 2001, NIST (NIST 2005a, NIST 2005b) emphasized the need for an engineered approach as an alternative to the prescriptive approach for structural fire safety design. Some building standards, most notably the Eurocodes, already provide guidance for engineered structural fire safety design for buildings. In the U.S., the American Institute of Steel Construction (AISC) has begun to develop standards for structural fire safety design of steel buildings. These standards provide clear guidance on how to calculate the reduction in strength of structural elements that results from the reduction in material strength and stiffness at elevated temperatures. These standards, however, still are unclear on the importance and the treatment of thermally-induced forces and deformations in the structural fire safety design process. The Eurocodes suggest that the effect of thermally induced forces and deformation can be neglected, while AISC suggests that the effect of thermally-induced forces and deformations should be included, but does not provide guidance on how this should be done. 3 1.3 Thesis Objective and Organization A number of researchers have demonstrated that during a fire, thermally-induced forces and deformations can have a significant effect on structure’s response (Usmani et al 2002, Quiel and Garlock 2008, Buchanan 2002, NIST 2005b). However, the role that thermally-induced forces play in the response of steel columns in fire has not been clearly defined. As such, the objective of this thesis is to better understand the importance of thermally-induced forces in steel columns in fire. More specifically, the objective is to determine if it is safe to neglect these forces when assessing column performance in fire. This study will be carried through the use of finite element analysis, first on individual columns and then on columns that are part of multi-story steel frames. Chapter 2 of this thesis provides a discussion about steel properties at elevated temperature, a review of the literature review on past studies on the response of steel columns subjected to fire, and the key assumptions that will be used for the analyses in this thesis. Chapter 3 presents a brief overview of the ABAQUS program and its structural models which will be used for the analyses in the thesis. Included is a discussion of several modeling issues in related to the use of ABAQUS the analysis conducted in this thesis. Chapter 4 presents the results of analysis on the behavior of isolated individual columns at elevated temperature. Several analyses on axially restrained and unrestrained columns at room and elevated temperature are conducted in this chapter using both theoretical solutions and ABAQUS simulations. Also included is an ABAQUS analysis of a simple truss with an interior column subjected to temperature increase to provide preliminary insights on the influence of flexible axial restraint to thermal expansion. Chapter 5 presents results of a series of ABAQUS analyses for multi- story steel moment frames with interior columns subjected to temperature increase. Included are 3, 9, 10 and 20 story moment frames. Finally, Chapter 6 provides a summary, conclusions and recommendations for further study on this subject. 4 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW 2.1 Overview As described in Chapter 1, the focus of this thesis is the behavior of steel columns subjected to fire, with an emphasis on column axial forces induced from restrained thermal expansion. This chapter provides a brief review of the effects of elevated temperature on steel material properties and discusses the properties that are used for the analysis conducted later in this thesis. This is followed by a review of previous studies on the response of columns subjected to fire, including research on column strength at elevated temperature and on thermally induced forces and strain in structural members subjected to temperature increase. The chapter concludes with a discussion of the key assumptions that are used for the analysis of columns in this thesis. 2.2 Steel Material Properties at Elevated Temperature An analysis of steel columns subjected to fire requires information on the elevated temperature properties of steel. One particularly important aspect is the uniaxial stress- strain properties of steel. At elevated temperatures, both the modulus of elasticity and the yield strength are reduced from their normal room temperature values. These reductions in modulus and yield strength affect the thermally induced forces and as well as the column buckling capacity. In addition to the reduction in yield strength and modulus, the shape of the stress-strain curve for steel at elevated temperatures is fundamentally different than at normal temperatures. At elevated temperatures, the stress-strain curve does not exhibit a well defined yield plateau and becomes highly nonlinear at low levels of stress. This early nonlinearity will also affect thermally induced forces and buckling capacity. Also of interest is the coefficient of thermal expansion. This is needed to predict thermally induced strains and deformations, and the resulting thermally induced forces. 5 Elevated temperature properties of steel are widely reported in the literature, including Buchanan (2002), Wang (2002), Purkiss (2007), the SFPE Handbook of Fire Protection Engineering (SFPE 2002), and others. A number of building standards also provide information on elevated temperature properties of steel. Foremost among these is Eurocode 3 (CEN 2003). Eurocode 3 covers the design of steel structures, and Part 1-2 of Eurocode 3 provides rules for structural fire design of steel structures. In the U.S., the AISC Specification for Structural Steel Buildings (AISC 2010) provides rules for structural fire design in Appendix 4 on “Structural Design for Fire Conditions.” These two standards will be referred to herein as Eurocode 3 and as AISC. For the purposes of this thesis, elevated temperature properties of steel provided in Eurocode 3 and in AISC will be used for the majority of the analyses. These proprieties generally agree well with data reported in other literature, and represent a consensus opinion of the building standards community on the elevated temperature properties of steel for use in structural-fire engineering analysis and design. Provided below is basic information on elevated temperature properties of steel provided in Eurocode 3 and in AISC. 2.2.1 Mechanical Properties Increases in the temperature result in reductions in the the strength and stiffness of steel. Both Eurocode 3 (Eurocode 3, 2003) and AISC (AISC, 2010) provide reduction factors for modulus of elasticity, yield strength and proportional limit at elevated temperature. These reduction factors are applied to the corresponding room temperature (20 oC or 68 oF) properties. Table 2.1 lists reduction factors for the elastic modulus and yield strength for carbon steel according to Eurocode 3. Also listed is a reduction factor for the proportional limit stress. 6 At room temperature, the proportional limit is assumed to be equal to the yield strength. At higher temperatures, the proportional limit occurs at a stress less than the yield strength. This reflects the fact that the stress strain curve for steel at elevated temperature becomes highly nonlinear at stresses well below the yield stress. Note that from Table 2.1, Eurocode 3 shows no reduction in yield strength until the temperature exceeds 400 °C. However, the elastic modulus and proportional limit are reduced for temperatures in excess of only 100°C. Table 1.1 Reduction Factors for Mechanical Properties per Eurocode 3 Steel Temperature T (oC) Reduction factor at temperature T to the values of fy and Ea at 20oC Reduction factor for yield strength ky,T = fy,T /fy Reduction factor for proportional limit kp,T = fp,T /fy Reduction factor for elastic modulus kE,T = Ea,T /Ea 20 1.0000 1.0000 1.0000 100 1.0000 1.0000 1.0000 200 1.0000 0.8070 0.9000 300 1.0000 0.6130 0.8000 400 1.0000 0.4200 0.7000 500 0.7800 0.3600 0.6000 600 0.4700 0.1800 0.3100 700 0.2300 0.0750 0.1300 800 0.1100 0.0500 0.0900 900 0.0600 0.0375 0.0675 1000 0.0400 0.0250 0.0450 1100 0.0200 0.0125 0.0225 1200 0.0000 0.0000 0.0000 The reduction factors in Table 2.1 are plotted in Figure 2.1 7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 200 400 600 800 1000 1200 1400 Temperature ( oC ) R ed uc tio n Fa ct or Reduction Factors For Yield Strength Reduction Factors For Proportional Limit Reduction Factors For Elastic Modulus Temperature ( oF ) 0 392 752 1112 1472 1832 2192 2552 Figure 1.1 Reduction Factors for Mechanical Properties per Eurocode 3 Table 2.2 lists reduction factors for the elastic modulus, yield strength and proportional limit stress for carbon steel according to AISC. AISC lists reduction factors for temperatures expressed in degrees Fahrenheit. Also, shown in Table 2.2 are the corresponding temperatures in degrees Centigrade. Note that other than small differences due to rounding, AISC uses essentially the same reduction factors as Eurocode 3. 8 Table 2.2 Reduction Factors for Mechanical Properties per AISC Steel Temperature Reduction factor at temperature T to the values of fy and Ea at 68oF T (oC) T (oF) Reduction factor for yield strength ky,T = fy,T /fy Reduction factor for proportional limit kp,T = fp,T /fy Reduction factor for elastic modulus kE,T = Ea,T /Ea 20.0 68 1 1 1 93.3 200 1 1 1 204.4 400 1 0.8 0.9 315.6 600 1 0.58 0.78 398.9 750 1 0.42 0.7 426.7 800 0.94 0.4 0.67 537.8 1000 0.66 0.29 0.49 648.9 1200 0.35 0.13 0.22 760.0 1400 0.16 0.06 0.11 871.1 1600 0.07 0.04 0.07 982.2 1800 0.04 0.03 0.05 1093.3 2000 0.02 0.01 0.02 1204.4 2200 0 0 0 The reduction factors in Table 2.2 are plotted in Figure 2.2. It can be seen clearly from this graph that the reduction factors for mechanical properties of steel at elevated temperature provided by AISC agree well with those provided by Eurocode 3. 9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 500 1000 1500 2000 2500 Temperature ( oF ) R ed uc tio n Fa ct or Reduction Factors For Yield Strength Reduction Factors For Proportional Limit Reduction Factors For Elastic Modulus Figure 2.2 Reduction Factors for Mechanical Properties per AISC 2.2.2 Stress-Strain Curves When temperature increases, steel not only undergoes a reduction in strength and elastic modulus but undergoes a change in the basic shape of the stress strain curve. At room temperature, the initial portion of the stress-strain curve for structural steel can normally be accurately idealized as bilinear. However, as temperature increases, the stress-strain curve becomes highly nonlinear, and a bilinear approximation becomes increasingly inaccurate. This nonlinearity is particularly significant in stability problems, where the buckling strength is fundamentally related to the tangent modulus. Eurocode 3 provides equations to model the stress-strain curve for carbon steel at elevated temperatures. These relationships are given in Table 2.3. Note that AISC does not provide such relationships. 10 Table 3.3 Stress-Strain Relationships for Carbon Steel per Eurocode 3 Strain Range Stress σ Tangent Modulus Tp,εε ≤ TaE ,ε TaE , TyTp ,, εεε << ( )[ ] 5.02,2, )/( εε −−+− TyTp aabcf ( ) ( )[ ] 5.02,2 , εε εε −− − Ty Ty aa b TtTy ,, εεε ≤≤ fy,T 0 TuTt ,, εεε << ( ) ( )[ ]TttuTtTyf ,,,, /1 εεεε −−− - Tu ,εε = 0 - Parameters TaTpTp Ef ,,, /=ε ; 02.0, =Tyε ; 15.0, =Ttε ; 2.0, =Tuε Functions ( )( )TaTpTyTpTy Eca ,,,,,2 /+−−= εεεε ( ) 2,,,2 cEcb TaTpTy +−= εε ( ) ( ) ( )TpTyTaTpTy TpTy ffE ff c ,,,,, 2 ,, 2 −−− − = εε Where: Ea,T is elastic modulus. fy,T is yield strength. fp,T is proportional limit. εy,T is strain at yield. εp,T is strain at proportional limit. εt,T is limit strain for yield strength. εu,T is ultimate strain. 11 The stress-strain curves for the equations given in Table 2.3 are plotted in US units in the Figure 2.3. These plots clearly illustrate the early nonlinearity in the stress-strain curves at elevated temperatures. 0 10 20 30 40 50 60 70 0.000 0.050 0.100 0.150 0.200 Strain St re ss (k si ) 68 oF 400 oF 600 oF 750 oF 800 oF 1000 oF 1200 oF 1400 oF 1600 oF 1800 oF 2200 oF oF oF oF oF o oF oF oF o oF o Figure 3.3 Stress-strain relationships for carbon steel at elevated temperature per Eurocode 3 2.2.2 Coefficient of Thermal Expansion According to Eurocode 3, thermal strains for carbon steel can be computed as follows: + 20oC ≤ T < 750oC: 4285 10416.2104.0102.1 −−− −+=∆ xTxTx L L (2.1) + 750oC ≤ T ≤ 860oC: 2101.1 −=∆ x L L (2.2) + 860oC < T ≤ 1200oC: 35 102.6102 −− −=∆ xTx L L (2.3) 12 Thermal strains for carbon steel at elevated temperature according to Eurocode 3 are plotted in Figure 2.4. Note that the thermal strain increases in an approximately linear manner with temperature, except for temperatures from 750 oC to 860 oC. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 200 400 600 800 1000 1200 Temperature ( oC ) T he rm al S tr ai n D L /L (% ) Temperature ( oF ) 0 392 752 1112 1472 1832 2192 2552 Figure 4.4 Thermal Strain of Steel at Elevated Temperature per Eurocode 3 Eurocode 3 does not provide the coefficient of thermal expansion, α. However, it can be computed as the first derivative of the equations above, with the following result: + 20oC ≤ T < 750oC: Txx 85 108.0102.1 −− +=α 1/oC (2.4) + 750oC ≤ T ≤ 860oC: 0=α (2.5) + 860oC < T ≤ 1200oC: 5102 −= xα 1/oC (2.6) 13 Thermal expansion of steel at elevated temperature is plotted in Figure 2.5. Within the range of 750oC to 860 oC, the coefficient of thermal expansion is equal to zero, 0.0E+00 5.0E-06 1.0E-05 1.5E-05 2.0E-05 2.5E-05 0 200 400 600 800 1000 1200 Temperature ( oC ) T he rm al E xp an si on (1 /o C ) Temperature ( oF ) 0 392 752 1112 1472 1832 2192 2552 Figure 5.5 Thermal Expansion of Steel at Elevated Temperature AISC does not provide equations for the thermal elongation or thermal expansion of steel. Nonetheless, it suggests a coefficient of thermal expansion of steel equal to 7.8x10- 6/oF for temperature above 150oF (or 1.4x10-5/oC for temperature above 65oC) and is assumed constant with temperature. 2.3 Previous Research on Column Strength at Elevated Temperatures and Column Strength According to AISC and Eurocode 3 Several studies have investigated the behavior of steel columns subjected to fire. A brief review of past studies on steel column strength at elevated temperature as well as a 14 discussion of formulas for column strength at elevated temperature adopted by AISC and Eurocode 3 is provided in the following subsections. 2.3.1 Previous Research on Column Strength at Elevated Temperatures Previous research on column strength at elevated temperature has included experimental studies, analytical studies, and computational studies. A number of experiments on the response of steel columns at elevated temperatures have been reported including the works of Ali (1998), Talamona et al (1996), Franssen et al (1998), Ali (2003-2004), Yang (2006), and Tan K. H. (2007). The general approach used in the majority of these studies was to heat a steel column up to a predetermined temperature with no load applied to the column. Once the column reached the desired temperature, axial load was applied to the column and increased to failure of the column. An alternative approach used by some researchers was to apply a specified axial load on the column prior to heating. The temperature of the column is then increased while maintaining a constant load until the column buckles, thereby identifying a critical temperature for the column. In another approach, a specified axial force is applied to the column prior to heating. The temperature of the column is then increased following a specified time temperature curve while maintaining a constant load on the column. The test is continued until the column buckles, and the time to failure is identified. This last approach provides a fire resistance rating, in hours, that can be used to show compliance with hourly ratings required by building codes. Research has also considered the influence of various factors, including slenderness ratios, load eccentricity, heating-rate, and end support conditions. Overall, however, the number of experimental investigations on column strength at elevated temperature is relatively limited. Beside experimental studies, a number of analytical studies have been conducted with the objective of developing formulas for buckling strength of steel columns at elevated temperatures, such as the work of Skowronski (1993), Toh (2000), and Zeng (2003). The 15 formulas developed in these studies were based on assumed material properties at elevated temperature, assumed residual stresses at elevated temperature, and assumed a uniform temperature distribution in the column. Analytical studies were usually calibrated by experimental data or finite element analysis for the same columns. Finally, a number of researchers have conducted computational studies of column strength at elevated temperature using finite element analysis. Examples of past computational studies are the works of Poh and Bennetts (1995), Huang and Tan (2007), and Takagi and Deierlein (2007). Like analytical studies, computational studies are also based on assumptions on material properties at elevated temperature, assumptions on residual stresses, assumptions on initial imperfections, etc. These studies, typically validated against limited experimental data, have resulted in recommendations for formulas to predict column strength at elevated temperature intended for use in design. 2.3.2 Buckling Strength of Steel Columns at Elevated Temperature in AISC Specification and in Eurocode 3 The experimental, analytical and computational studies described above have resulted in a number of expressions for predicting column strength at elevated temperature (Buchanan 2002, Wang 2002, Purkiss 2007, Franssen 1995, Takagi and Deierlein 2007). This, in turn, has resulted in equations for column strength at elevated temperature adopted by building codes, including Appendix 4 of the 2010 AISC Specification and Eurocode 3. These equations are described below. AISC Equations for Column Strength at Elevated Temperature For columns at elevated temperature, AISC 2010 Appendix 4 provides the following equation for nominal compressive strength: 16 ( ) gTyFFTn AFP TeTy )(/)( )()(42.0= (2.7) Where: Ag is gross section of member. Fy(T) is yield strength at elevated temperature. Fe(T) is critical elastic buckling stress at elevated temperature 2 )( 2 )(       = r kL E F TTe π (2.8) With E(T) is elastic modulus at elevated temperature. r is radius of gyration. kL is effective length of the column. Both Fy(T) and E(T) are specified in Appendix 4, and are also listed in Table 2.2. This equation was the result of studies conducted by Takagi and Dierlein (2007) and is based on finite element analysis of columns at elevated temperature. However, as the authors commented, this equation is only suitable for temperature greater than 300 oC to 400 oC (or 572 oF to 752 oF) when the effect of material degradation becomes significant. Eurocode 3 Equations for Column Strength at Elevated Temperature At elevated temperature, the compressive strength of a column is computed according to Eurocode 3 as follows: gTyTTcr AFP )()()( χ= (2.9) 2 )( 2 )()( )( 1 TTT T λϕϕ χ −+ = (2.10) ( )2 )()()( 15.0 TTT λλαϕ ++= (2.11) 17 )0( 23565.0 yF =α (2.12) )( )( )0()( TE Ty T k k λλ = )0( )0( )0( e y F F =λ (2.13) Where: Fy(T) is yield strength at elevated temperature. Fy(0) is yield strength at room temperature. Fe(0) is critical elastic buckling stress at elevated temperature 2 )0( )0(       = r kL E Fe π (2.14) With E(0) is elastic modulus at room temperature. ky(T) & kE(T) are reduction factors for yield strength and elastic modulus at elevated temperature specified in Eurocode 3. These factors are also listed in Table 2.1. Note that units for yield strength and elastic modulus in the above equations is MPa These equations are based on the work conducted by Franssen (1995). Studies by Takagi and Dierlein (2007) show that column strength at elevated temperature determined from Eurocode 3 matches well with those from finite element analyses within approximately 20%. Figures 2.6 and 2.7 are the relationships between critical stress Fcr (equal to Pcr/Ag) and effective slenderness KL/r for steel columns at elevated temperatures according to AISC Appendix 4 and Eurocode 3 respectively. Figures 2.8 and 2.9 plot column strength according to both AISC and Eurocode 3 for temperatures of 600 oF and 1000 oF. These plots were developed for columns made of steel with a yield strength of 50 ksi and an 18 elastic modulus of 29000 ksi. For column strength values according to Eurocode 3, the material reduction factors were determined by interpolation from Table 2.1 for the corresponding temperatures, and the units for critical stress has been converted from MPa to ksi. As can be seen from these figures, the relationships between Fcr and KL/r for steel columns at elevated temperatures according to AISC Appendix 4 and Eurocode 3 are similar. AISC equations result in lower strengths in comparison with Eurocode 3, however, the differences are relatively small. 0.0 10.0 20.0 30.0 40.0 50.0 60.0 0 20 40 60 80 100 120 140 160 180 200 KL/r Fc r (k si ) T = 200 oF T = 400 oF T = 600 oF T = 750 oF T = 800 oF T = 1000 oF T = 1200 oF T = 1400 oFoFo oo o o oF oF Figure 6.6 Strength Curves for Steel Columns at Elevated Temperatures per AISC Appendix 4 19 0.0 10.0 20.0 30.0 40.0 50.0 60.0 0 20 40 60 80 100 120 140 160 180 200 KL/r Fc r (k si ) T = 200 oF T = 400 oF T = 600 oF T = 750 oF T = 800 oF T = 1000 oF T = 1200 oF T = 1400 oF oF oFo oF ooF oFoF Figure 7.7 Strength Curves for Steel Columns at Elevated Temperatures per Eurocode 3 0.0 10.0 20.0 30.0 40.0 50.0 60.0 0 20 40 60 80 100 120 140 160 180 200 KL/r Fc r (k si ) AISC: T = 600 oF Eurocode 3: T = 600 oFoFoF Figure 8.8 Strength Curves for Steel Columns at 600 oF per AISC Appendix 4 and Eurocode 3 20 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 0 20 40 60 80 100 120 140 160 180 200 KL/r Fc r (k si ) AISC: T = 1000 oF Eurocode 3: T = 1000 oF oo Figure 9.9 Strength Curves for Steel Columns at 1000 oF per AISC Appendix 4 and Eurocode 3 2.4 Previous Research on Thermally Induced Forces and Deformation in Columns When a fire occurs, it will cause an increase in the temperature in structural members. It is this temperature increase that affects the response and performance of the structure subjected to fire. In general, the effects of temperature increase can be characterized in the broad terms of thermal degradation and thermal restraint. The term thermal degradation, as used herein, refers to the reduction in member strength and stiffness that results from the reduction in material strength and stiffness at elevated temperature, as described in the previous section. For example, the buckling strength of a column will be reduced at elevated temperature because of the reduction in steel material stiffness and strength at elevated temperature. Thermal restraint, on the other hand, refers to forces and deformations generated in a member due to restrained thermal expansion. As a steel member is heated, it expands. If this expansion is restrained, very large forces can be 21 developed in a structure. Note that even if there was no degradation in material properties at elevated temperature, fires can still cause a great deal of structural damage or collapse due to the effects of thermal restraint. In general, however, the effects of both thermal restraint and thermal degradation combine to endanger structures subjected to fire. The effect of thermal degradation on column strength has been widely reported in literature and adopted in AISC and Eurocodes 3 as equations for column strength at elevated temperatures as discussed in previous section. However, there are a relatively small number of studies reported in the literature investigating the effect of thermal restraint on the behavior of columns at elevated temperature, including Usmani et al (2001), and Quiel & Garlock (2008). Usmani et al (2001) provides a useful discussion on methods to compute thermally induced strains, deformations and forces in structural members that are subject to uniform and non-uniform temperature increases. At the most basic level, the strain at any point in a member can be expressed as follows: mechanicalthermaltotal εεε += (2.15) In this equation, thermalε is the thermal strain resulting from a change in temperature and is computed from the coefficient of thermal expansion, α. That is, εthermal = α ∆T, where ∆T is the change in temperature. Note that the thermal strain from an increase in temperature that is uniform over the cross-section will cause elongation of the member. However, in the presence of a thermal gradient over the cross section, the thermal strain will also cause bowing of the member. The term mechanicalε refers to the mechanical strain, i.e. the strain that produces stress in the member. Mechanical strains result from external loads and also from restrained thermal deformations. If a member is unrestrained and there are no external loads and is subjected only to a temperature change thermaltotal εε = (2.16) 22 For this case, there is no mechanical strain, and therefore no stress develops in the member as a result of the temperature change. One the other hand, if a member is fully restrained without external load and is subjected to a temperature change, the total strain is zero, and: thermalmechanicalmechanicalthermal εεεε −=→+=0 (2.17) Thus, if a member in a structure has significant restraint to thermal expansion, significant mechanical strains, and therefore significant internal forces, can develop in the member due to temperature increase in a fire. Usmani et al (2001) note thermally induced forces and deformations, as opposed to thermal degradation of material properties, can be the dominant factor in the response of a structure to fire. Analytical expressions for thermally induced forces in columns developed by Usmani et all are described in greater detail in Chapter 4 of this thesis. The equations provided by Eurocode 3 and AISC for column strength at elevated temperatures described in the previous section only explicitly consider the effects of thermal degradation. That is, these equations predict the buckling capacity of a column based on the reduction in material stiffness and strength at elevated temperature. Methods to account for the effects of thermal restraint, on the other hand, are not as clearly described in these design standards. Nonetheless, according to the study of Usmani et al (2001), the effects of thermal restraint may be potentially significant to the behavior of columns in a fire. Restrained thermal expansion of a heated column can potentially generate significant force in the column, thereby reducing its ability to resist externally applied loads such as dead and live loads. Appendix 4 of AISC 2010 states the forces resulting from thermal restraint should be included in the analysis of members subjected to fire, but provides no guidance on how this should be done, while Eurocode 3 allows neglecting the effect of thermal restraint for member analysis. In the case of columns 23 subjected to fire, both of AISC and Eurocde 3 are unclear in how significant the effects of thermal restraint are in affecting column performance. The main objective of this thesis is to investigate this issue. 2.5 General Assumptions Applied in This Thesis Factors that affect the strength of columns at normal temperature are well understood, and include material strength and stiffness, effective slenderness, residual stresses, initial crookedness. The response of steel columns at elevated temperatures is more complex and depends on additional factors such as the temperature dependent material strength and stiffness, temperature dependent nonlinearity in stress-strain curves, the change in residual stress patterns at elevated temperatures, creep effects, and nonuniform temperature distributions over the cross-section and length of the column. To facilitate the analyses conducted for this thesis, several simplifying assumptions will be made. These are described below. When a steel structure is subjected to a fire, the fire will cause a temperature increase in the structural members. The temperature increase in the members will depend on the surrounding gas temperatures and on the thermal properties of the steel and any protective insulation. With this information, temperatures in the structural members can be computed from heat transfer analysis. Buchanan (2002) discusses methods of heat transfer analysis as applied to structural fire engineering problems. In general, temperatures will vary throughout a structural member and will also vary as a function of time during a fire event. However, due to the high thermal conductivity of steel, temperatures are often assumed to be uniform throughout a steel member at any given time during a fire (Buchanan 2002). For the analysis of columns at elevated temperature in this thesis, the assumption is made that temperatures are uniform over the cross-section of the column and along its full length. This assumption simplifies the analysis but may not always be realistic. In cases where the assumption of uniform temperature is violated, 24 the conclusions from the analyses conducted herein may no longer be applicable. Nonetheless, even in these cases, it is believed that analysis of columns subjected to uniform temperature increase still provides useful insights into column behavior in fire. Structural steel shapes always exhibit an initial residual stress mainly caused from uneven cooling after rolling. Residual stress has little effect on buckling strength of very slender columns, but does reduce inelastic buckling strength of intermediate slenderness columns (Galambos, 1998). The study in this thesis does not include the effect of residual stress on the response of interior columns in steel buildings under a fire. A review of the literature found no data on the distribution or magnitude of residual stresses in steel members at elevated temperatures. However, a number of researchers, including Vila Real (2004) and Takagi and Dierlien (2007) suggest that the magnitude of residual stresses are significantly reduced at elevated temperature, and their influence on column strength is significantly less than at normal temperatures. Further, the influence of material creep is explicitly considered in the analyses in this thesis. Lastly, the effects of local buckling are not included in this study. The analyses conducted herein only consider the overall flexural buckling of columns. 2.6 Summary This chapter has provided a brief discussion of steel material properties at elevated temperatures, and a brief review of past studies on the effect of thermal degradation and thermal induced force and deformation in columns at a fire General simplifying assumptions that will be used in this thesis were also discussed. The next chapter provides a more significant discussion of the finite element modeling techniques that are used in this study. 25 CHAPTER 3 MODELING TECHNIQUES 3.1 Overview As discussed in Chapter 1, the objective of the research described in this thesis is to investigate the forces developed in columns in steel buildings subjected to fire. The overall approach that was used in this research was to conduct computational simulations using the general purpose finite element analysis program ABAQUS. More specifically, analyses were conducted using ABAQUS Version 6.8-3, following the instructions in ABAQUS Version 6.8 Documentations. This chapter provides a brief overview of ABAQUS program and its structural models. This is followed by a discussion about several key modeling issues in the use of ABAQUS for conducting the analyses in this thesis. 3.2 ABAQUS Program ABAQUS is a set of finite element analytical programs originally developed by Hibbitt, Karlsson & Sorensen, Inc. and currently maintained by SIMULIA Corp. ABAQUS is a general-purpose simulation tool, and can solve a wide range of engineering problems, including structural analysis and heat transfer problems. ABAQUS has extensive element and material libraries capable of modeling a variety of geometries and material constitutive laws. ABAQUS consists of three main products: ABAQUS/Standard, ABAQUS/Explicit and ABAQUS/CAE. While ABAQUS/Standard and ABAQUS/Explicit perform analysis, ABAQUS/CAE provides a graphical environment for pre and post-processing. ABAQUS/Standard is a general-purpose analysis program for solving linear, nonlinear, static and dynamic problems. ABAQUS/Explicit is a special-purpose analysis program 26 that uses an explicit dynamic finite element formulation. It is suitable for modeling brief, transient dynamic events, such as impact and blast problems (ABAQUS, 2008a). ABAQUS/Standard and ABAQUS/CAE are used in this thesis for structural analysis. In general, a complete ABAQUS simulation consists of 3 distinct stages: preprocessing, simulation and postprocessing as shown in Figure 3.1 (ABAQUS, 2008a) Input files: job.odb, job.dat, job.res, job.fil Postprocessing Abaqus/CAE or other software Preprocessing Abaqus/CAE or other software Input file job.in Simulation Abaqus/Standard (or) Abaqus/Explicit Figure 10.1 ABAQUS Stages of a Complete Simulation (ABAQUS, 2008a) 3.3 Structural Modeling in ABAQUS Each analytical model in ABAQUS includes 10 modules: Part, Property, Assembly, Step, Interaction, Load, Mesh, Job, Visualization, and Sketch. To create a complete analysis model, it is usually necessary to go through most of these modules, as described below: 27 + Build up the geometry of the structure under a set of parts. (Part module, Sketch module, Mesh module) + Create element sections (Property module) + Introduce material data (Property module) + Assign section and material properties to the members (Property module) + Assemble parts to create the entire structure (Assembly module, Mesh module and Interaction module) + Create steps and choose analysis method (Step module) + Introduce load and boundary conditions (Load module) + Create jobs and submit for analysis (Job module) + Visualize the result. (Visualization module) 3.4 General Issues in Creating Frame Models in ABAQUS Many elements and options are available in ABAQUS that permit analysis of complex structural problems. However, there are many modeling issues that must be considered by the user when formulating a problem in ABAQUS. The following sections discuss some general issues in creating frame models in ABAQUS which are used in the subsequent analysis in this thesis, including element type, mesh size, load increment, material stress – strain curve model, treatment of geometric nonlinearity, load control and deflection control. Additional ABAQUS modeling issues relating to specific problems are discussed in greater detail in later chapters. 3.4.1 Element Types and General ABAQUS Assumptions The element types that are used in this thesis include beams in two and three dimensional (3D) spaces and truss elements in two dimensional (2D) space. Each element has two nodes at its ends. The ABAQUS names for these beams elements are B21 and B31, and 28 for truss element is T21. The number 2 and 3 indicate two and three dimensional elements, and the number 1 indicates that these elements use linear interpolation to calculate the displacement between the nodes. The beam element is described as a Timoshenko shear flexible beam which allows transverse shear strain. In this thesis, 3D beam elements are used when considering buckling in the weak axis of individual columns. 2D beam elements are used in almost analyses including the investigation of column in high rise frames, and the ABAQUS verification in beam members described later. 2D truss elements are used to study the behavior of a column with flexible restraints. Each 3D beam element has six degrees of freedom at each node, including 3 translational degree of freedom and 3 rotational degrees of freedom. Open-section beam types in ABAQUS have an additional degree of freedom for warping of the cross section. This thesis, however, does not use the ABAQUS open-section beam and does not include the effect of torsion in the 3D beam element. Each 2D beam element has three degrees of freedom at each node, including 2 translational degrees of freedom and 1 rotational degree of freedom. The truss element in 2D space has two translational degrees of freedom at each node (ABAQUS, 2008b). For the beam elements, the user inputs the cross-sectional geometry and ABAQUS calculates cross-sectional properties such as area and moments of inertia. For elastic – plastic analyses, ABAQUS also generates the yield moment, plastic moment, and compression and tensile forces at yield. The change in cross sectional area due to strain has been included in every analysis in this thesis by specifying the Poisson’s ratio of 0.3. More information about ABAQUS model for nonlinear material and nonlinear geometry will be discussed below. 3.4.2 Material Stress – Strain Curve Model For the beam and truss elements in ABAQUS, the user inputs a uniaxial stress-strain curve to model the material for that element. At room temperature, a number of 29 idealizations of the stress-strain curve for steel can be employed. Depending on the purpose of the analysis, the stress-strain curve can be simplified a number of different models, including an elastic model, elastic perfectly plastic model, bilinear model, multi- linear models, Ramberg-Osgood models and many other. Figure 3.2 illustrates some typical simple models for stress-strain curves, in which fs is stress, fy is yield strength, ε is strain and E is elastic modulus. f s ε0 E f s ε0 E f y f s ε0 E f y f s ε0 E a) Elastic Model b) Elastic Perfecly Plastic Model c) Bilinear Model d) Multi-Linear Model Figure 11.2 Typical Models of Stress-Strain Curves In ABAQUS, the stress-strain curve is introduced in the Property Module by two separate properties: elastic properties and plastic properties. For the elastic properties, the main required data are the values of elastic modulus and Poisson’s ratio. In the plastic 30 properties, the main required data are the values of yield stress and plastic strain. To perform an inelastic analysis, ABAQUS requires both elastic and plastic properties. ABAQUS permits the user to model any desired stress-strain curve by defining points on the curve. For example, to create an elastic perfectly plastic stress-strain curve as shown in Figure 3.2, the elastic properties are defined first, with the values of elastic modulus and Poisson’s ratio. Then the plastic properties are created with two rows of data. The first row includes the values of stress at the beginning of the plastic region, which is equal to stress at the end of elastic region or yield strength in this case, and the plastic strain at the beginning of plastic region, which is zero. The second row includes the value of the strength at the fracture, which is same as the yield strength in this case, and the value of plastic strain at fracture, which is equal to total strain at fracture minus the elastic strain. The elastic strain is computed as the stress divided by the elastic modulus. For stress-strain curves more complex than elastic perfectly plastic, such as the Eurocode 3 models for steel at elevated temperature, the stress-strain curve will be divided into several points, with an assumption that the curves between two adjacent points are linear. The material stress-strain data for both elastic and plastic properties can be defined to correspond to a specific temperature thereby permitting temperature-dependent material properties. ABAQUS also automatically calculates the values of stress-strain data between specified temperatures by interpolation. To demonstrate the behavior of the ABAQUS beam element with nonlinear material properties, a simple analysis was conducted on a 30-ft long W18x35 beam that is fixed against rotation at each end. The beam is constructed of material with a yield strength of 50 ksi, and is subjected to a uniformly distributed load. Results of the analysis are plotted in Figure 3.3. The results for two different ABAQUS models are included. One model used an elastic perfectly plastic material. The second model used a bilinear stress-strain curve. Also plotted is a theoretical solution using a simple plastic hinge idealization to represent flexural yielding. This theoretical solution can be found in standard textbooks on plastic analysis, such as Horne and Morris (1982). 31 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Vertical Defection at Midspan (in) D is tr ib ut ed L oa d (k ip /ft ) Theoretical - Elastic Perfectly Plastic Model ABAQUS - Elastic Perfectly Plastic Model ABAQUS - Bi-Linear Model 30 ft W18x35 fy=50 ksi Figure 12.3 Deflections at Midspan of a Fixed-Fixed Beam Subjected to Distributed Load As seen in Figure 3.3, both of the ABAQUS analyses agree well with the theoretical calculation in the elastic region. ABAQUS predicts a slightly lower elastic stiffness than the theoretical solution. This can be attributed to the fact that the ABAQUS model includes shear deformations, whereas the theoretical solution considers only flexure. In the inelastic range, the curve from the ABAQUS analysis with an elastic perfectly plastic material model follows the theoretical solution well. Note that the ABAQUS beam element considers yielding distributed along the length of the member, whereas the theoretical solution assumes that yielding occurs at a plastic hinge of zero length. Consequently, the theoretical solution shows an instantaneous change in stiffness when yielding occurs at the member ends and at midspan. The ABAQUS model shows a more gradual and a more realistic change in stiffness when flexural yielding occurs in the member. The ABAQUS analysis with a bi-linear stress-strain model results in a higher capacity for the member, as expected. Overall, the results shown in Figure 3.3 indicate 32 that the ABAQUS beam element can reasonably model the response of beams with flexural yielding. 3.4.3 Mesh Size One of the important factors that influence the accuracy of a finite element analysis is mesh size. A finer mesh generally leads to more accurate analysis, but also requires larger computational resources and time. For the problems considered in this thesis, mesh size relates to the number of beam elements used to model a single member. The effect of the mesh size on the accuracy of the predicted load-deflection response of the fixed-fixed beam previously shown in Figure 3.3 was investigated. Results are shown in Figure 3.4. This figure shows the load-deflection response computed by ABAQUS using four different mesh refinements, wherein the length of the beam elements were taken as 25, 10, 5 and 1-percent of the length of the member. This corresponds to the use of 4, 10, 20 and 100 elements, respectively, to model the beam. For all four cases, the ABAQUS model used an elastic perfectly plastic material model. Also shown in Figure 3.4 is the theoretical load deflection response using a simple plastic hinge model. The plots in Figure 3.4 show that the accuracy of the ABAQUS model was highly dependent on mesh size for this problem, and that a very fine mesh was needed to achieve good agreement with the theoretical solution. Note that 100 elements were also used to model the beam for the ABAQUS results previously shown in Figure 3.3. An additional evaluation of the effects of mesh size is shown in Figure 3.5. For this case, the elastic response of a pinned-pinned column subjected to axial compression is investigated. The ABAQUS analysis used an elastic material model but included nonlinear geometry. The member modeled in ABAQUS included an initial geometric imperfection with a shape that followed a half-sine wave and a magnitude at mid-height equal to 1/1000 times the length of the column. The initial geometric imperfection was introduced in the ABAQUS model by dividing the 10-ft long column into 10 elements. 33 The initial nodal coordinates at the ends of each element were chosen to correspond to the initially imperfect geometry described above. Consequently, the half-sine curve was approximated with a series of 10 straight segments. (More detailed descriptions of analysis of columns with initial geometric imperfections are provided in Chapter 4). ABAQUS analyses were then conducted by further subdividing each of the 10 elements into subelements with a length equal to 25, 10, 5 and 1-percent of the element. Results are shown in Figure 3.5. The load plotted in this figure is normalized by the Euler buckling load, Pcr which is equal to π2EI/L2. Also plotted in Figure 3.5 is a theoretical solution for a column reported by Gerard (1962) with the same geometric imperfection used in the ABAQUS model. Note that the theoretical solution assumes a linear relationship between curvature and deflection, and is therefore approximate at large deformations. The ABAQUS solution, on the other hand, provides a more accurate modeling of nonlinear geometry. Consequently, the ABAQUS solution and the theoretical solution cannot be directly compared at large deformations. Nonetheless, at smaller deformations, there is reasonable agreement between the ABAQUS and the theoretical solutions. The key observation for this figure, however, is that the ABAQUS solutions for the four different mesh sizes were essentially identical, indicating much less sensitivity to mesh size than the previous beam problem. It is clear that the mesh size refinement needed to achieve an accurate solution with the ABAQUS beam element is quite problem dependent. For the ABAQUS analyses conducted later in this thesis, mesh size was determined on a trial and error basis by using progressively finer meshes until key structural response quantities no longer changes increasing mesh refinement. 34 0.0 2.0 4.0 6.0 8.0 10.0 12.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Vertical Defection at Midspan (in) D is tr ib ut ed L oa d (k ip /ft ) Theoretical ABAQUS-Mesh Size = 25% ABAQUS-Mesh Size = 10% ABAQUS-Mesh Size = 5% ABAQUS-Mesh Size = 1% 30 ft W18x35 fy=50 ksi Figure 13.4 Load-Deflection Response of a Fixed-Fixed Beam Subjected to a Distributed Load with Different Mesh Sizes 35 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 70 Displacement at Midspan (in) P/ P c r Theoretical - Gerard ABAQUS - Mesh Size = 100% ABAQUS - Mesh Size = 50% ABAQUS - Mesh Size = 25% ABAQUS - Mesh Size = 1% W14x90 10 ft P ∆0 Figure 14.5 Load-Deflection Response for an Initially Imperfect Column Subjected to Axial Load with Different Mesh Sizes 3.4.4 Number of Load Increments Similar to mesh size, the number of load increment used in an ABAQUS analysis can have a significant effect on the accuracy of the results. To compute the response of a structure under a load with magnitude that varies from zero to a specified value or to the formation of plastic mechanism, ABAQUS divides the applied load into smaller increments and calculates structural response at each increment. The maximum number of increments and the increment size are specified in Step Module. A large increment size results in a small number of load increments, which leads to fewer points in the load- deformation response of the structure. If the increment is too large, the ABAQUS analysis may miss key events in the response, such as the onset of yielding in a member. This, in turn, can lead to an inaccurate load-deformation response history. A smaller load 36 increment size can result is a more accurate load-deformation response prediction, but can substantially increase the time required to complete the analysis. For the ABAQUS analyses conducted for this thesis, load increment size was chosen on a trial and error basis in an attempt to provide an accurate solution with reasonable computation time. 3.4.5 Linear Geometry and Non-Linear Geometry Analyses ABAQUS permits the solution of structural analysis problems using either linear or nonlinear geometry. When linear geometry is specified, the solution is based on the initial geometry of the structure, which is assumed to remain unchanged for any level of load or deformation on the structure. This is comparable to classical methods of structural analysis that are based on an assumption of small deformations. ABAQUS also permits the user to conduct an analysis based on nonlinear geometry. With this option, the geometry of the structure is updated throughout the analysis to account for deformations of the structure and the resulting change in geometry. Analyses using the nonlinear geometry option will provide more accurate results, since it accounts for changes in geometry under load. In classical methods of structural analysis for frame structures, approximate nonlinear geometry analysis is often conducted by accounting for the effect of changing geometry on the equations of equilibrium. This type of analysis is frequently referred to as a “P-∆” analysis. However, a more exact analysis for the effects of nonlinear geometry can be conducted by including the exact, nonlinear relationship between curvature and deflection and by accounting for changes of cross-section shape under load and the resulting changes in cross-sectional properties. This more exact approach for a nonlinear geometric analyses is difficult to implement in closed-form solutions, but is included in ABAQUS when nonlinear geometry analysis is specified. Consequently, ABAQUS provides a more exact formulation of nonlinear geometry effects than the typical P-∆ analysis included in closed form solutions or in simpler structural analysis computer programs. 37 The effect of using linear versus nonlinear geometry analysis on a simple beam problem is illustrated in Figure 3.6. This figure shows the load-deflection response of the same fixed-fixed beam used for previous analyses in this chapter. The beam is subjected to a uniformly distributed load, and the load is increased up to the formation of a plastic mechanism. The theoretical load-deflection curve shown in this figure is based on a simple plastic analysis and an assumption of linear geometry. Also shown are the results of two ABAQUS analyses; one using linear geometry analysis and another using nonlinear geometry analysis. Both ABAQUS analyses included inelasticity using an elastic perfectly plastic material model. Note that the two ABAQUS analyses give very similar results and both agree well with the theoretical solution. The ABAQUS nonlinear geometry analysis gives a load capacity slightly higher than the plastic mechanism load, even though no strain hardening is included in the material model. This increase in load capacity is the result of the effects of nonlinear geometry which models the development of tensile action in the beam at large deformations. For many structural analysis problems, such as that shown in Figure 3.6, an analysis based on linear geometry gives very accurate results, and there is little benefit in conducting a nonlinear geometry analysis. In stability problems, such as those considered in this thesis, it is essential to consider nonlinear geometry effects in the solution. This is illustrated in the simple column analysis shown in Figure 3.7. This figure shows the relationship between axial load and transverse deflection at mid-height for a pin ended column, based on elastic analysis. This is the same column previously described in the section on Mesh Size. The theoretical solution plotted in this figure is based on the solution reported by Gerard (1962). The theoretical solution uses an approximate nonlinear geometry analysis that considers equilibrium in the deformed geometry of the member, but does not include the nonlinear geometric relationship between curvature and deflection. Also shown in Figure 3.7 are the results of two ABAQUS analyses; one using linear geometry analysis and 38 another using nonlinear geometry analysis. The ABAQUS analysis using linear geometry completely misses the development of instability in this column, and only accounts for axial shortening of the column under load. The ABAQUS analysis using nonlinear geometry, on the other hand, captures the development of instability in the column. The ABAQUS nonlinear geometry analysis agrees reasonably well with the theoretical solution at small levels of deformation, but deviates considerably for the theoretical solution at larger deformations. The theoretical solution is inaccurate at large deformations since it neglects the nonlinear relationship between deflection and curvature. Consequently, ABAQUS provides a more exact nonlinear geometry analysis than typical closed form solutions reported in the literature. All ABAQUS analyses conducted for this thesis included nonlinear geometrical analysis. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Vertical Defection at Midspan (in) D is tr ib ut ed L oa d (k ip /ft ) Theoretical - Linear Geometry ABAQUS - Linear Geometry ABAQUS - Nonlinear Geometry 30 ft W18x35 fy=50 ksi Figure 15.6 Load-Deflection Response of a Fixed-Fixed Beam Subjected to Distributed Load Using Linear and Nonlinear Geometry Analyses 39 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 10 20 30 40 50 60 70 Displacement at Midspan (in) P/ P c r Theoretical - Gerard ABAQUS - Linear Geometry ABAQUS - Nonlinear Geometry W14x90 10 ft P ∆0 Figure 16.7 Load-Deflection Response for an Initially Imperfect Column Subjected to an Axial Load Using Linear and Nonlinear Geometry Analyses 3.4.6 Load Control and Deflection Control Two methods for imposing actions on structures available in ABAQUS are load control and deflection control. In the load control method, an external load is applied, and then internal forces, moment and deflection are calculated. In the deflection control method, a deflection or rotation is specified at a degree of freedom, and then the internal forces induced by the deflection are calculated. The external load required to achieve the specified deflection can be calculated by equilibrium with internal forces. Both external load and deflection are introduced in the ABAQUS Load Module. External load is specified in Load Creation, and deflection is specified in Boundary Condition Creation. The load control method is simple and straightforward in determining the response of the structure, while the deflection control method is little more complex to apply because it 40 requires calculation of the equivalent external load. However, the main disadvantage of the load control method in ABAQUS is that it can not track the response of the structure after the resistance of the structure begins to degrade. That is, it is not possible to track the degrading portion of the load-deflection response of a structure using load control. Deflection control, on the other hand, permits analysis of the structures once the resistance begins to decrease. Figure 3.8 shows the axial force – deflection relationship for an initially imperfect column from ABAQUS analyses with load control and with deflection control. The analysis includes both geometric nonlinearity and material inelasticity using the room temperature stress strain curve from Eurocode 3, as described in Chapter 2. In the load control method, an axial load P is applied at the roller end of the column. In the deflection control method, a downward deflection is specified at the roller end instead of the external load P. The axial force – deflection response predicted by ABAQUS in the ascending portion of the curve is identical for load control and deflection control. However, once the strength of the column begins to degrade, the load control analysis stops. The deflection control method, on the other hand, can track the descending portion of the load-deflection response. 41 0 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 7 8 9 Displacement at Midspan (in) A xi al F or ce (k ip s) ABAQUS - Deflection Control Method ABAQUS - Load Control Method 10 ft W14x90 fy=50 ksi ∆ 0 P Figure 17.8 Load- Deflection response of a Column Subjected to an Axial Load Using Load Control and Deflection Control The ABAQUS analyses conducted for this thesis used the deflection control method to predict the behavior of individual columns with material inelasticity, since the inclusion of inelasticity in the analysis results in a descending portion of the load-deflection response. (Gerard 1962). For the case of columns with elastic material model, it is not necessary to use deflection control because the load resistance does not degrade. It is also not necessary to use deflection control in studying the full structural response of interior columns in high-rise buildings with inelastic materials. High-rise buildings have a high level of redundancy, so that after the yielding or buckling of the investigated column, the building still has other load paths to resist higher loads, and this in turn, allows the ABAQUS analysis to continue. 42 3.5 Summary This chapter has provided an overview of the modeling techniques that are used to analyze the response of columns subjected to elevated temperatures. The analyses were conducted using the finite element computer program ABAQUS. Columns were analyzed using the ABAQUS beam element, which can model the effects of both material and geometric nonlinearities. A brief discussion was provided of issues related to the selection of mesh size and load increment size that affect the accuracy of the solution. Solution strategies using both load control and deflection control were described. In the following chapter, results using the ABAQUS modeling techniques described in this chapter to analyze the response of individual columns subjected to elevated temperatures are discussed. ABAQUS solutions are compared to theoretical solutions for selected problems to provide further validation of the ABAQUS solutions. ABAQUS is then used to further investigate the forces induced in columns due to restraint to thermal deformations. 43 CHAPTER 4 ANALYSIS OF INDIVIDUAL COLUMNS 4.1 Overview In this chapter, a series of analyses are conducted on individual columns that are not part of a building frame. Columns are analyzed under axial compression, both at room temperature and elevated temperature. Columns with restrained ends are also analyzed when subjected to temperature changes. Columns will be analyzed with ABAQUS, using the modeling techniques described in Chapter 3. Where available, the ABAQUS solutions are compared with theoretical solutions reported in the literature. Finally, results from ABAQUS analyses conducted for a column that is part of a simple truss are discussed to evaluate the effect of flexible boundary conditions. The purpose of the analyses described in this chapter is to provide further validation of the ABAQUS solutions, and to provide insights into the elevated temperature behavior of individual columns with simple boundary conditions. 4.2 Buckling of Axially Unrestrained Elastic Column Subjected to Axial Load at Room Temperature 4.2.1 Theoretical Methods for Elastic Buckling Analysis Before investigating column behavior in more complex problems involving temperature effects, restraint and inelasticity, this section briefly reviews classical solutions for the analysis of elastic columns. These solutions are widely reported in the literature including Timoshenko and Gere (1961), Gerard (1962), Galambos (1998), and Chen and Lui (1987). Available solutions include those for columns that are initially perfectly straight and columns with an initial geometric imperfection. Furthermore, solutions can be developed using linear or nonlinear kinematic relationships between displacement and 44 curvature. The theoretical solutions discussed below are based on linear strain- displacement relationships. Buckling Analysis for Perfect Straight Columns For a perfectly straight elastic column, the buckling load is calculated by the Euler equation with an effective unbraced length: ( )2kL EIPcr π = (4.1) Where: E is elastic modulus. I is moment of inertia. L is the length of the column. k is the effective length factor, k =1 for a pin-ended column. This equation is constructed by considering equilibrium of the column in its deflected shape when being subjected to an axial load. As long as the load is smaller than Pcr, the column remains perfectly straight. When the load attains Pcr, the column becomes unstable, and will deflect rapidly without additional load. The load deflection relationship for a perfectly straight member has a bifurcation point at the load Pcr. (Gerard 1962). Buckling Analysis for Columns with Initial Geometric Imperfection In reality, columns are not perfectly straight and always have initial imperfections. The solutions for columns with initial curvature are widely reported in the literature such as Timoshenko (1961) and Gerard (1962). If the shape of the initial deflection follows a sine curve, simple solutions are possible and are discussed below. 45 If the initial deflection of a column follows a sine function, an equation for the initial deflection can be written as follows: 2 sin xy o π ∆= (4.2) In this equation, ∆o is the initial deflection at midspan, and x is distance from the end of column. Then the deflection at midspan can be calculated by the following equation: cr o P P − ∆ =∆ 1 (4.3) Where: P is applied load Pcr is critical buckling load calculated from the Euler equation (4.1). Note that ∆ includes initial deflection ∆o and additional deflection at midspan caused by applied load P. Relationships between deflection at midspan and the applied load for columns with a variety of initial imperfections are shown in Figure 4.1. In this Figure, L is the column’s length, P, Pcr, ∆ and ∆o are as defined above. 46 0 0 0 1 1 1 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ∆/L P/ P c r Perfect Straight Column: ∆o/L = 0 ∆o/L = 10 ∆o/L = 1/50∆o/L = 1/100 ∆o/L = 1/500 ∆o/L = 1/1000 Figure 18.1 Deflection Behavior of Perfectly Straight and Imperfect Elastic Columns As seen from Figure 4.1, perfectly straight columns show a point of bifurcation in the load-deflection response. Columns with initial imperfections, on the other hand, do not exhibit a point of bifurcation. Rather, these columns show a continuous nonlinear load- deflection path that is asymptotic to the elastic critical load. The load-deflection response for columns with fairly small initial imperfections is similar to the perfectly straight column, exhibiting a rather sudden and well-defined buckling load. Columns with large imperfections exhibit a more gradual increase in deflection as load increases. For a steel column, ASTM limits the maximum initial imperfection to 0.1% of the column length. That is, the maximum initial mispan deflection is L/1000. For analyses of columns with initial imperfections in this thesis, it is assumed that the initial deflection follows the sine curve in Equation 4.2, with a maximum initial deflection at midspan equal to 0.1% of the column length. 47 4.2.2 ABAQUS Models for Buckling Analysis Several approaches are possible in ABAQUS for obtaining the load-deflection response of a column to characterize buckling behavior. One approach is to conduct an eigenvalue analysis followed by a Riks analysis. Another approach is to include nonlinear geometrical effects in a general static analysis of a column modeled with an initial imperfection (ABAQUS, 2008c). The first approach requires performing two analyses: an eigenvalue analysis and a Riks analysis. The eigenvalue analysis is conducted first on a perfectly straight column in to get the buckling loads (eigenvalues) and the corresponding buckled shapes (eigenvectors). The eigenvalue analysis only provides buckling loads and buckled shapes, but does not provide the full loaf-deflection response of the column. The load-deflection response can be computed by conducting an additional ABAQUS analysis called as Static-Riks Analysis. An initial deflection that follows the shape of the first buckling mode from the eigenvalue analysis, scaled to a specified value, is introduced in the column for the Static-Riks Analysis. ABAQUS then performs the Riks analysis for this column to obtain the full load-deflection response. Even if the strength of the column deteriorates, as can be the case when material inelasticity is included in the model, the Riks analysis can still compute the load-deflection response. The Static-Riks analysis can be very useful in analyzing the buckling behavior for simple structures such as an individual column. For more complex problems, where the response of the structure involves a sequence of yielding and buckling of several members or where loading conditions are complex, the Static-Riks analysis can be cumbersome or impossible to use. For such cases, solutions can be obtained by conducting an ABAQUS General Static analysis, including nonlinear geometry, on a structure modeled with initial imperfections. However, difficulties with this approach can occur when the strength of the structure begins to deteriorate, i.e., when the stiffness becomes negative. In some cases, the degrading portion of the load-deflection response can still be obtained by using 48 displacement-control solution strategies rather than load control, as described in Chapter 3. 4.2.3 Analysis of Axially Unrestrained Elastic Columns Subjected to Compressive Load This section presents results of buckling analyses for axially unrestrained elastic columns subjected to an axial compressive load. ABAQUS results are compared with a theoretical solution calculated from Equation 4.3 for a column with an initial imperfection. ABAQUS solutions are developed using the two approaches described above: an eigenvalue and Riks analyses, and a General Static analysis, both of which are for columns with an initial imperfection. The relationships between deflection at midspan and P/Pcr from ABAQUS are then plotted in comparison with the theoretical solution. For this analysis, the material is assumed to be elastic with a Young’s modulus of 29000 ksi. The section profile is a W14x90 as specified in the AISC Manual. The length of the column is 10 ft. An initial imperfection is introduced in the columns’ minor axes following the sine function as in Equation 4.2 with a maximum initial imperfection at midspan equal to 0.1% of column’s length or 0.12 in. Note that in every analysis conducted in this thesis, the root fillet is not included when using section profile as specified in AISC. The relationship between P/Pcr and deflection at midspan from these analyses are plotted in Figure 4.2. Pcr is computed using Equation 4.1 with k=1.0. Note that the theoretical solution is based on a linear strain-displacement kinematic relationship. The ABAQUS solutions, on the other hand, are based on a more exact formulation of nonlinear geometry that includes the nonlinear relationship between strain and displacement. 49 0 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 70 Displacement at Midspan (in) P/ P c r Theoretical - Linear Kinematics ABAQUS - Riks Analysis ABAQUS - General Static Analysis W14x90 10 ft P ∆0 Figure 19.2 Load-Deflection Response for an Initially Imperfect Elastic Column Subjected to Axial Load The ABAQUS solutions match well with the theoretical solution for the initial portion of the load-deflection response, up to a midspan deflection of about 20 inches. Beyond this, the two solutions differ significantly, as the nonlinearity in the strain-displacement relationships become significant. Theoretical solutions that include nonlinear strain- displacement relationships for perfectly straight elastic columns show an increase in load capacity beyond Pcr, similar to that shown by the ABAQUS results in Figure 4.2. (Timoshenko and Gere 1961, Chen and Lui 1987). These theoretical solutions show that as the column continues to deform, the two ends of the column pass each other causing the applied load to change from a compressive load to a tensile load, which results in a stiffening effect and decrease in midspan deflection. This behavior is purely mathematical and not physically possible. This behavior is also seen in the ABAQUS solutions in Figure 4.2. No theoretical solution was found in the literature that includes 50 nonlinear strain-displacement relationships for an imperfect column. Consequently, it is not possible to provide a direct comparison between the ABAQUS solutions and the theoretical solution. Nonetheless, the comparisons provided in Figure 4.2 and the general trends expected for solutions incorporating nonlinear strain-displacement relationships suggest that the ABAQUS solutions are accurate for elastic buckling analysis. Another observation that can be made from Figure 4.2 is that the two different ABAQUS analysis techniques matched very closely. The majority of the ABAQUS analytical solutions presented in the remainder of this thesis made use of the General Static analysis approach. 4.3 Axially Restrained Elastic Column Subjected to Thermal Gradient As noted before, due to the high thermal conductivity of steel, temperatures are often assumed to be uniform throughout a steel member at any given time during a fire. However, it may also be possible to obtain variations in temperature along the length of a column and also over the cross-section of a column. In this section, an elastic column subjected to a thermal gradient over the depth of the cross-section will be analyzed. A theoretical solution for this problem is reported by Usmani et al (2001). The purpose of the analysis reported in this section is to provide some validation of ABAQUS solutions involving thermally induced deformations and forces, as well as to provide some insight into the effects of thermal gradients on columns. Thermal gradient causes a variation in thermal strain over the cross-section, which induces a curvature along the length of the column. This curvature then results in a contraction along the column’s length. If the column is simply supported, the contraction occurs freely without inducing force. If the column is pinned at both ends, however, this contraction is prevented and induces tensile force in the column. Usmani et al (2001) provides an equation for contraction strain as following: 51 ( ) 2/ 2/sin1 φ φεφ L L −= (4.4) Where: L is column length φ is curvature induced by the effect of thermal gradient. yTαφ = , where α is coefficient of thermal expansion. Ty is temperature gradient over cross section. Then induced tensile force can be calculated as below: EAN φε= (4.5) Where: E is elastic modulus of material A is area of cross section This theoretical method was used to analyze the behavior of an axially restrained elastic column subjected to a thermal gradient over the cross-section. An ABAQUS analysis was performed for the same column. For both the theoretical calculation and the ABAQUS analysis, the material is assumed to be elastic with a Young’s modulus of 29000 ksi and a coefficient of thermal expansion of 7.8*10-6 (1/ oF). The section profile is a W14x90 and the length of the column is 10 ft. The column cross-section will be subjected to a linear thermal gradient with a temperature change of zero at the centroid of the cross-section. The relationships between temperature gradient and induced axial force are presented in Figure 4.3. 52 0.0 50.0 100.0 150.0 200.0 250.0 300.0 0.0 20.0 40.0 60.0 80.0 100.0 120.0 Thermal Gradient (oF/in) A xi al F or ce (k ip s) Theoretical - Usmani ABAQUS 10 ftT2 T1 W14x90 Figure 20.3 Axially Restrained Elastic Column Subjected to a Thermal Gradient It can be seen from Figure 4.3 that the ABAQUS simulation for thermal gradient agrees well with the theoretical solution provided by Usmani et al (2002). The small differences between these relationships at high thermal gradients may be due to the differences in computing the contraction strain for two methods. Usmani used a linear strain- displacement relationship in deriving the contraction strain equation, while ABAQUS uses a more exact formulation for nonlinear geometry. Nonetheless, the ABAQUS solution and the theoretical solution agree well. The results plotted in Figure 4.3 suggest that significant thermally induced forces can be developed in a restrained column subjected to thermal gradient. However, as discussed earlier, significant thermal gradients are unlikely in steel columns. For example, a thermal gradient of 80°C/inch requires a temperature change of about 1100°F over the depth of a W14x90. Due to the high thermal conductivity of steel, such a large 53 temperature difference is unlikely (Buchanan 2002). Thus, the results plotted in Figure 4.3 have little practical impact. However, these plots indicate that ABAQUS provides accurate results for thermally induced forces. 4.4 Buckling of Axially Restrained Elastic Column Subjected to Thermal Expansion 4.4.1 Theoretical Methods for Buckling of Axially Restrained Elastic Column Subjected to Thermal Expansion As discussed in Chapter 2, due to the high thermal conductivity of steel, temperatures are typically assumed to be uniform throughout a steel member during a fire. Application of a uniform increase in temperature over the cross-section and along the length of the member during a fire will result in thermal expansion. If a member is unrestrained in the axial direction, thermal expansion will occur freely and no force will be induced. However, when a member is axially restrained to prevent thermal expansion, an axial force will be induced. If the stiffness of the axial restraint is small, the induced force may be small. However, when the stiffness of the restraint is large, the induced axial force will be significant, and may be large enough to cause buckling. A number of studies have investigated the effect of thermal restraint on induced forces and deflections in columns, such as Usmani et al (2001) and Quiel and Garlock(2008). Usmani el al (2001) derived an equation to calculate a critical increase of temperature at which an initially straight column begins to bend: 2 2 αλ π =∆ crT (4.6) Where: α is the coefficient of thermal expansion λ is the slenderness ratio of the column. 54 According to Usmani, for axially restrained columns subjected to temperature increase only (i.e., no external load), the axial force induced in these columns is due only to thermal restraint. However, these columns remain straight as long as the increase in temperature is smaller than the critical temperature increase defined by Equation 4.6. When the increase in temperature reaches the critical increase, the column will experience significant out of plane deflection and the induced axial force is equal to the critical load at buckling of the same column subjected to external axial force only. For temperature increase in excess of the critical increase from Equation 4.6, Usmani provides the following equation to predict the midspan deflection. 2 2 2T T L εε π δ += (4.7) Where: L is column’s length εT is thermal expansion strain TT ∆= *αε , with ∆T is the increase of temperature. Another approach that can be used to predict the response of a fully axially restrained column to a temperature increase is to modify Equation 4.3 to provide deflection in terms of temperature change rather than in terms of load. Note that because Equation 4.3 is only for loads smaller than critical load at buckling, this method is only applicable for temperatures smaller than the critical temperature. According to Usmani et al (2001), the compressive force induced by a temperature change for a column with no bending will be: TEAEAEAP Tm ∆−=−== αεε (4.8) (εm and εT are mechanical strain and thermal strain as mentioned in Chapter 2) 55 By inserting Pcr from Equation 4.1 and the absolute value of P from Equation 4.8 into Equation 4.3, and taking k = 1 for a column with no rotational restraint at either end, (but with full axial restraint), the deflection at midpan of the column subjected to a temperature increase ∆T will be: 1 / 11 2 2 0 22 00 I TAL LEI TEA P P cr π α π α ∆ − ∆ = ∆ − ∆ = − ∆ =∆ (4.9) 4.4.2 ABAQUS Analysis of Axially Restrained Elastic Column Subjected to Thermal Expansion This section presents the results of ABAQUS analyses of an axially restrained elastic column subjected to a temperature increase. The ABAQUS results are compared with the theoretical solutions described above. ABAQUS solutions are generated by the two methods described earlier in this chapter: 1) an eigenvalue analysis followed by a Riks analysis, and 2) a General Static analysis. There is a small difference between these analyses with previous analyses with external load. That is, instead of an external load, a temperature increase is applied in the column, and ABAQUS performs the analysis to calculate the response of the column following the increase of temperature. In the eigenvalue analyses, a temperature increase of 1 oF is applied in the column. Then the value of the eigenvalue at the first buckling mode results from ABAQUS analysis is the value of the temperature increase that causes the significant bend out in the column. The critical temperature therefore is equal to this temperature increase adding up to 68 oF to refer to room temperature. As with previous examples, the material is assumed to be elastic with a Young’s modulus of 29000 ksi. The section profile is a W14x90 and the length of the column is 10 ft. For the theoretical solution by Usmani (Equations 4.6 and 4.7), the column is assumed to be initially perfectly straight. For other cases, including the ABAQUS solutions and the 56 theoretical solution in Equation 4.9, an initial imperfection is introduced in the column’s minor axes following a sine function in Equation 4.2 with a maximum initial imperfection at midspan is equal to 0.1% of column’s length or 0.12 in. The columns are subjected to uniform temperature along their length, and there is no thermal gradient over the cross-section. The coefficient of thermal expansion is taken as 7.8*10-6 (1/ oF). These analyses are conducted considering bending on the column’s minor axis. Relationships between temperature and deflection at midspan resulting from these analyses are presented in Figure 4.4. Note that in this plot, the temperature increase has been added to 68 oF to refer to room temperature. That is, at a temperature of 68° F, there is no temperature change in the column. 68 568 1068 1568 2068 2568 0 1 2 3 4 5 6 7 8 Displacement at Midspan (in) T em pe ra tu re (o F) Critical Temperature - Usmani (Eq. 4.6) Critical Temperature - ABAQUS - Eigenvalue Usmani (Eq. 4.7) Equation 4.9 ABAQUS - Riks Analysis ABAQUS - General Static Analysis W14x90 10 ft∆ T 0 Figure 21.4 Analyses of Axially Restrained Elastic Column Subjected to Temperature Increase 57 It can be seen from Figure 4.4 that the ABAQUS analyses agree reasonably well with the theoretical calculations for the value of critical temperature and for the load-deflection response of the column. The critical temperature calculated by ABAQUS with an eigenvalue analysis is 1210 oF, which is rather close to the critical temperature of 1273 oF calculated by Usmani’s method. For temperatures well below the critical temperature (pre buckling), the relationship between temperature and displacement at midspan from ABAQUS analysis is very similar to that predicted by Eq. 4.9. For temperatures in excess of the critical temperature (post buckling), the relationship between temperature and midpsan displacement from ABAQUS reasonably follows that calculated from Usmani’s equation. The difference between the curves at the beginning of the post buckling region is because Usmani’s method is developed for perfectly straight column, and that column only starts to bend when the temperature exceeds the critical temperature. The ABAQUS analyses, on the other hand, is for a column with an initial imperfection, where bending starts from the beginning of the temperature increase. Nonetheless, the ABAQUS solution and Usmani’s solution are very close at higher temperatures. Another observation from Figure 4.4 is that the ABAQUS General Static analysis matches with the ABAQUS solutions with eigenvalue and Riks analysis in predicting the response of columns with an initial imperfection subjected to a temperature increase. Subsequent ABAQUS analyses in this thesis will be based on the General Static analysis approach. The analyses performed in this section suggest that restraint to thermal expansion can have a significant impact on behavior of columns. For the column analyzed in Figure 4.4, large out of plane displacements due to restrained thermal expansion occur at a temperature on the order of 1200°F. Beyond this temperature, the column likely has little or no ability to carry external load. Note that temperatures of about 1200°F can be developed in typical building fires (Buchanan 2002). This analysis, however, only 58 considered elastic response. In reality, material inelasticity must also be considered in the response of column to elevated temperatures. This is considered in the next section. 4.5 Buckling of Axially Unrestrained Inelastic Columns Subjected to Axial Load at Elevated Temperatures In this section, the effects of material nonlinearity and inelasticity on the buckling capacity of columns at elevated temperature are considered. Methods to evaluate inelastic buckling of columns at room temperature are widely reported in literatures, including Galambos (1998), Chen and Lui (1985), Shanley (1947), etc. Building standards, such as the AISC Specification and Eurocode 3, also provide methods to calculate the capacity of real columns, including the effects of material inelasticity Moreover, as indicated in Chapter 2, as temperature increases, the strength and stiffness of steel are reduced, and the stress-strain curve becomes highly nonlinear. Buckling no longer can be predicted by the methods used for columns at room temperature (Takagi and Deierlein 2007). The 2010 AISC Specification and Eurocode 3 provide equations to calculate the nominal buckling strength of a column at elevated temperature. These equations account for the reduction in strength and stiffness and the change in the stress- strain relationship of steel at elevated temperature. The behavior of axially unrestrained columns under axial compression load at elevated temperatures will be investigated in this section using ABAQUS and compared to strength predictions by the 2010 AISC Specification 2010, as discussed in Chapter 2. The objective of this analysis is to provide some validation of ABAQUS predictions for inelastic buckling at elevated temperature. Column strength predictions from ABAQUS are compared to two calculations of column strength using AISC equations. One comparison is with column strength equations provided in Appendix 4 of the 2010 AISC Specification. These equations, based on the work of Tagaki and Deirlein (2007), are intended specifically to compute column 59 strength at elevated temperature. A second comparison is made by modifying the room temperature column strength equations in Chapter E of the AISC Specification. The Chapter E equations is modified by using elevated temperature values for elastic modulus and yield stress, as specified in Appendix 4 of the AISC Specification and as listed in Table 2.2 of this thesis. These elevated temperature values of elastic and modulus are used in lieu of the normal room temperature values in the Chapter E equations. In the analysis using ABAQUS, the nonlinear stress-strain curves for steel at elevated temperatures specified in Eurocode 3 (Figure 2.3) are used. The relationship between axial load on the column and midspan displacement was derived for various temperatures. Solutions were developed using a displacement control strategy in order to capture the descending branch of the load-deflection response. As with the previous example, the column that was analyzed was a 10 ft. long W14x90, with a pin at one end and a roller at the other end. The steel material used in the analysis had a room temperature yield stress of 50 ksi. As noted above, the elevated temperature stress-strain curves specified in Eurocode 3 were used for the ABAQUS analysis. Also, for the ABAQUS analysis, an initial imperfection was introduced in the column’’ minor axis following the sine function as in Equation 4.2 with maximum initial imperfection at midspan is equal to 0.1% of column’s length or 0.12 in. The coefficient of thermal expansion was taken as 7.8*10-6 (1/ oF) as specified in Appendix 4 of the AISC Specification. The analyses considered only minor axes bending of the column. The axial force – deflection curves of the columns at elevated temperatures resulting from ABAQUS analyses are plotted in Figure 4.5. 60 0 200 400 600 800 1000 1200 1400 0 2 4 6 8 10 12 14 16 18 20 Displacement at Midspan (in) A xi al F or ce (k ip s) T = 68 oF & T = 200 oF T = 400 oF T = 600 oF T = 750 oF T = 800 oF T = 1600 oF T = 1400 oF T = 1200 oF T = 1000 oF W14x90 fy=50 ksi 10 ft∆ o T Figure 22.5 Axial Force – Deflection at Midspan of Axially Unrestrained Column Subjected to Compressive Load at Elevated Temperatures per ABAQUS Analyses The curves in Figure 4.5 illustrate that column strength decreases rapidly with increase in temperature. This decrease is due to the reduction in strength and stiffness and the nonlinearity of the stress-strain curve of steel at elevated temperature. The Eurocode 3 stress-strain curves for steel show no degradation in material properties and no change in the shape of the stress-strain curve when the temperature is lower then 200 oF. Consequently, the axial force – deflection relationship of the column at 200 oF is identical to at 68 oF. When the temperature is higher than 200 oF, column strength begins to reduce. In comparison with the strength at room temperature, the column loses about 25% of its strength at 600 oF, loses more than 50% of its strength at 1000 oF and loses more than 90% of its strength at 1600 oF. Temperatures in typical building fires can exceed 1600°F (Richardson 2003). Consequently, the reductions in column strength at elevated temperatures shown in Figure 4.5 are very significant. 61 Figure 4.6 plots the maximum axial force resisted by the column at various temperatures based on the ABAQUS results in Figure 4.5. Also plotted is the strength of this 10 ft. long W14x90 column calculated based on the elevated temperature column strength equations in Appendix 4 of AISC. In addition, the strength of the column is computed using the column strength equations in Chapter E of–AISC modified by replacing the value of yield strength and elastic modulus from room temperature to those at elevated temperatures. Finally, the room temperature column strength based on Chapter E, which is 1179 kips, is also plotted for reference. 0 200 400 600 800 1000 1200 1400 0 500 1000 1500 2000 2500 Temprature (oF) C om pr es si ve S tr en gt h (k ip s) ABAQUS AISC - Appendix 4 for Elevated Temperature Modified from AISC - Chapter E for Room Temperature AISC Chapter E Room Temperature Strength W14x90 fy=50 ksi 10 ft∆ o T Figure 23.6 Compressive Strength of Axially Unrestrained Column at Elevated Temperatures The curves in Figure 4.6 show that the column strength computed from ABAQUS matches well with values predicted by AISC, both at room temperature and at elevated 62 temperature. At room temperature, the compressive strength computed from ABAQUS matches closely with that calculated from Chapter E of AISC. ABAQUS results in little higher compressive strength than the solution from Chapter E of AISC since residual stress is not included in this ABAQUS’s analysis. At elevated temperature, the compressive strengths computed from ABAQUS match well with those calculated from Appendix 4 of AISC for temperatures greater than 750 oF. For the lower temperatures, ABAQUS analyses do not agree with equation in Appendix 4 of AISC. As mentioned in Takagi and Deierlein (2007), which is the basis of the Appendix 4 column strength equations, these equations are intended for temperatures greater than about 570 oF to 750 oF. Thus, the Appendix 4 equations in AISC are not intended for temperatures less than about 750°F, although this limitation is not clearly stated in Appendix 4. However, it is clear that the Appendix 4 equations are inaccurate for lower temperatures, since they give significantly lower column strength at room temperature than that calculated by Chapter E of AISC. Figure 4.6 also shows a large difference between column strength predicted by ABAQUS and that calculated from the Chapter E equations modified for elevated temperature values for elastic modulus and yield stress. The modified Chapter E equations significantly overestimate column strength. This indicates that using the normal room temperature column strength equations in Chapter E, and simply replacing E and Fy with their evaluated temperature values, results in an unconservative prediction of column strength at elevate temperature. This is because simply replacing E and Fy does not account for the basic change of the stress-strain curve of steel at elevated temperature. The highly nonlinear stress-strain curve at elevate temperature leads to a significant reduction in tangent modulus, and therefore leads to a significant reduction in column strength. A similar observation was made by Tagaki and Deierlein (2007). 63 4.6 Behavior of Interior Column in a Truss Subjected to Gravity Load and Thermal Expansion The previous sections investigated the response of isolated individual columns subjected either to external load with no axial restraint or to temperature increase with complete axial restraint. However, in reality, during a fire columns are exposed to both external load and to temperature increase and interact with the surrounding structural members. Response of columns subjected to a fire are therefore influenced by a number of factors, including material degradation due to temperature increase, restraint to thermal expansion from surrounding members and the resulting development of additional force in the column, and the redistribution of force from the column to the surrounding structure resulting from the column’s loss of strength and stiffness. In the next chapter, these effects and interactions are investigated for columns subjected to temperature increase in multi-story moment frames. However, in this section, the interactions between a column and the surrounding structure are first examined for a more simple case. This section describes the analysis of a column that is part of a very simple truss. The structure is shown in Figure 4.7. W10x45 fy=50 ksi 10ft T A=100in 2∆ o P 10ft A=100in 2 Figure 24.7 Truss with Interior Column Subjected to a Fire 64 The investigated truss includes an interior column connected to two exterior struts. All connections are modeled as pins, and so there is no rotational restraint at the column ends. The struts are chosen to provide a finite axial restraint to the column. In the analysis, the struts are modeled to remain elastic, and are not subjected to temperature changes. The struts, in effect, provide for a flexible axial restraint. The interior column is subjected to a temperature increase, and is modeled to include initial geometric imperfection, material inelasticity, and temperature dependent material strength and stiffness degradation. An external load P is applied to the truss. The truss is statically indeterminate, so the proportion of P resisted by the column depends on the relative stiffness of the column compared to the struts. This relative stiffness will change as a function of load and temperature. In this problem, the axial force in the column will come from two sources. First, the columns will resist a portion of the external load P. Second, the column will be subject to additional axial force due to restrained thermal expansion. The column in the truss is a W10x45 section that is 10 ft. in length. The material is modeled with a room temperature yield strength of 50 ksi. Stress-strain curves at elevated temperature follow Eurocode 3 (see Figure 2.3). The coefficient of thermal expansion for steel is taken as 7.8*10-6 (1/ oF). An initial imperfection is introduced in the column’s minor axis following the sine function per Equation 4.2, with a maximum initial imperfection at midspan equal to 0.1% of column’s length or 0.12 in. Only minor axis bending of the column is considered in the analysis. The behavior of the interior column in the truss is investigated using ABAQUS. For comparison, the strength of the column using the elevated temperature column strength equations in Appendix 4 of the 2010 AISC Specification is also computed. Four ABAQUS analyses were carried out including three analyses of the truss and one analysis of the individual axially unrestrained column. The ABAQUS analysis for the individual column was conducted using the deflection control method. However, load control was 65 used for the ABAQUS analyses of the truss to predict the full response of the column at both pre and post buckling stages since the truss was still able to carry higher loads after the failure of the interior column. Detailed descriptions of the four ABAQUS analysis cases are described below. Case 1: An ABAQUS analysis was conducted of the truss to investigate the effect of thermal restraint and thermal degradation on the axial force induced in the interior column. In this analysis, no external load P is applied to the truss. The temperature in the column was increased from room temperature up to 2400 oF, the temperature at which steel loses virtually all of its strength and stiffness. The analysis considered the effects of both the thermal expansion of the column as well as the temperature dependent degradation of material strength and stiffness. As the temperature of the column was increased, the axial force induced in the column was computed. Case 2: ABAQUS analyses were conducted to investigate the effect of temperature dependent material degradation on compressive strength of the interior column, when an axial external load P is applied to the truss. For these analyses, thermal expansion of the column was not modeled, and so there is no thermally induced force in the column. Thus, only the external load P produced axial force in the column. For each analysis, the interior column in the truss is assumed to be exposed to a specified elevated temperature and this temperature was kept constant during the analysis. Temperature dependent material properties for the specified temperature were used in the analysis, but as noted above, thermal expansion was not modeled. Thirteen ABAQUS analyses were carried out for temperatures of 68 oF, 200 oF, 400 oF, 600 oF, 750 oF, 800 oF, 1000 oF, 1200 oF, 1400 oF, 1600 oF, 1800 oF, 2000 oF, and 2200 oF. For each temperature, the external load P was applied to the truss and increased until ABAQUS stops running or a mechanism formed. For each analysis, the relationship between the axial force in the column and the deflection of the column at its midspan were determined, along with the maximum axial 66 force sustained by the column at that temperature. As noted earlier, the axial force in the column for this case resulted only from applied load. Case 3: ABAQUS analyses were conducted to investigate the effect of both temperature dependent material degradation and thermal restraint on the behavior of the column. For these analyses, an external load P is applied to the truss, and the thermal expansion of the column was modeled. Consequently, for these analyses, axial force in the column was generated both from the external load and from the restraint to thermal expansion. For each analysis, the temperature of the column was increased from room temperature (68° F) to a specified maximum temperature, and then maintained at that specified temperature. After that, the external load P was applied to the truss and increased until ABAQUS stoped running or a mechanism formed. Thirteen ABAQUS analyses were carried out for maximum temperatures of 68 oF, 200 oF, 400 oF, 600 oF, 750 oF, 800 oF, 1000 oF, 1200 oF, 1400 oF, 1600 oF, 1800 oF, 2000 oF, 2200 oF. For each analysis, the relationship between the axial force in the column and the deflection of the column at its midspan were determined, and along with the maximum axial force sustained by the column at that maximum temperature. As noted above, the axial force in the column for this case resulted both from restraint to thermal expansion during the temperature increase and from the subsequently applied external load P. Case 4: ABAQUS analyses were conducted to determine the compressive strength of an axially unrestrained column at elevated temperature. This isolated column had the same material, thermal and geometrical properties as the column in the truss. The column was modeled with a pin at the bottom and a roller at the top. Consequently, there was no restraint to thermal expansion, and 100-percent of the applied load is resisted by the column. For each analysis, this column was assumed to be exposed to an elevated temperature and this temperature was kept constant during the analysis. Thirteen ABAQUS analyses were carried out corresponding to temperatures of 68 oF, 200 oF, 400 oF, 600 oF, 750 oF, 800 oF, 1000 oF, 1200 oF, 1400 oF, 1600 oF, 1800 oF, 2000 oF, 2200 67 oF. These columns were analyzed using deflection control. At each case of temperature, a deflection in the vertical direction was specified at the roller end of the column and was increased until ABAQUS stoped running or a mechanism formed. After finishing the analysis corresponding to each temperature, the relationship between the induced axial force in the column and deflection of the column at its midspan was determined, as well as the maximum induced axial force at that temperature. Note that for this case, there was no axial force in the column due to restraint to thermal expansion. Figure 4.8 shows the relationship between compressive strength (or maximum induced axial force) and temperature of the investigated columns for Case 2, 3 and 4 of the ABAQUS analyses. These relationships are plotted in comparison with the curve representing column strength at elevated temperatures computed from the equations in AISC – Appendix 4. Also included in Figure 4.8 is the relationship between induced axial force in the interior column and temperature increase for Case 1 of the ABAQUS analysis. 68 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 2000 2500 3000 Temperature (oF) C ol um n C om pr es si ve S tr en gt hs o r In du ce d A xi al F or ce s (k ip s) ABAQUS - Case 1 - Temperature Increase with Thermal Restraint - No Applied Load P ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P ABAQUS - Case 4 - Axially Unrestrained Column AISC - Appendix 4 - Axially Unrestrained Column W10x45 fy=50 ksi A=100in 2 10ft 10ft 10 ft P T A=100in 2 Figure 25.8 Compressive Strength of Interior Column in a Truss Subjected to Axial Load at Elevated Temperatures The plots in Figure 4.8 show that the strength versus temperature curves for the column are the same for Cases 2, 3 and 4. This indicates that the axial strength of the column is the same whether the axial force is generated by external load alone, or by a combination of thermal restraint and external load. Further, as seen previously from the analysis described in Section 4.5, that the column strength values predicted by ABAQUS analyses Case 2, 3 and 4 match well with the column strength calculated by AISC Appendix 4 for temperatures higher than 600 oF. As described earlier, this indicates that AISC Appendix 4 provides inaccurate predictions of column strength for temperatures below 600° F. Figure 4.8 also plots the results from ABAQUS – Case 1 for which the column was restrained from thermal expansion by the struts of the truss, but no external load was applied. As indicated in the plot, the axial force in the column initially increases with 69 temperature due to the thermal restraint. When the temperature reaches about 1100 oF to 1200 oF, the axial force in the column begins to decrease due to the decrease in material strength and stiffness and due to buckling of the column. The induced axial force is zero as temperature reaches about 2200 oF since the material has lost essentially all of its strength and stiffness. ABAQUS Cases 2 and 3 are examined in greater detail to provide further insight into the behavior of the column at elevated temperature. As seen in Figure 4.8, the compressive strength of the column for both cases is identical. However, the relationship between applied external load P and the axial force induced in the column is different for these two cases. Figures 4.9 to 4.12 show graphs of this relationship for temperatures of 200° F, 400° F, 800° F and 1200°F. 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 700 800 900 1000 Load P (kips) A xi al F or ce In du ce d in In te ri or C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P W10x45 fy=50 ksi A=100in 2 10ft 10ft 10 ft P T A=100in 2 Figure 26.9 Column Axial Force versus Applied Load P at 200 oF 70 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 800 900 1000 Load P (kips) A xi al F or ce In du ce d in In te ri or C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P W10x45 fy=50 ksi A=100in 2 10ft 10ft 10 ft P T A=100in 2 Figure 27.10 Column Axial Force versus Applied Load P at 400 oF 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 800 900 1000 Load P (kips) A xi al F or ce In du ce d in In te ri or C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P W10x45 fy=50 ksi A=100in 2 10ft 10ft 10 ft P T A=100in 2 Figure 28.11 Column Axial Force versus Applied Load P at 800 oF 71 0 50 100 150 200 250 300 350 400 0 100 200 300 400 500 600 700 800 900 1000 Load P (kips) A xi al F or ce In du ce d in In te ri or C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P W10x45 fy=50 ksi A=100in 2 10ft 10ft 10 ft P T A=100in 2 Figure 29.12 Column Axial Force versus Applied Load P at 1200 oF These figures illustrate the effect of thermal restraint on the axial force in interior column. At zero external load (P = 0), the column in Case 3 already has an axial force. This axial force is a result of the restrained thermal expansion of the column with temperature increase. Because of this thermally induced force, the column in Case 3 always sees a higher total axial force than in Case 2, although the truss is subjected to the same external load P. Since the compressive strength of the columns is the same for both cases the column in Case 3 reaches its compressive strength at a lower external load P than the column in Case 2. Therefore, the presence of thermal restraint decreases the column’s ability to resist the externally applied load P. For example, at 1200° F the compressive strength, i.e. the buckling capacity of the column is about 160 kips, as shown in Figure 4.8. In the absence of thermal restraint (Case 2), the external load P on the truss at the time that the column buckles is approximately 300 kips. However, in the presence of thermal restraint (Case 3), the external load P on the truss at the time that the column buckles is only about 100 kips (Figure 4.12). Thus, the axial force generated in 72 the column from restrained thermal expansion significantly reduced the magnitude of external load that could be resisted by the column. This indicates that thermally induced forces can have a very large effect on the performance of columns subjected to fire. There are some other observations that can be made from the figures above. One is the changing patterns of these curves that reflects load redistribution from the columns to the exterior struts. When the temperature is relatively low as shown in Figure 4.9 and 4.10, before approaching the critical load of the column, the relationship between applied load P and column axial force is linear. However, when the temperature is larger, as shown in Figure 4.11 and 4.12, these relationships become nonlinear at relatively low values of P. This reflects a redistribution of internal force from the column to the strut as the column loses stiffness from material strength and stiffness degradation at elevated temperature. Another observation from Figures 4.8 to 4.12 relates to the contribution of thermally induced force in the total force in the column. At each temperature considered in the analysis, with the same applied load P, the axial force in interior column in Case 3 is always larger than that in Case 2 by an amount equal to or less than the thermally induced force. So the total axial force in a column subjected to both thermal degradation and thermal restraint can be conservatively estimated by adding the thermally induced force and the force caused by applied external load. 4.7 Summary This chapter has presented the results of a series of analyses on individual columns under axial compression, both at room temperature and elevated temperature, and with and without restraint to thermal expansion. In addition, analysis results were presented for a column that is part of a simple indeterminate truss. 73 ABAQUS results were compared to theoretical solutions, where available, and were also compared with AISC Appendix 4 predictions for the buckling strength of columns at elevated temperature. These comparisons showed that the ABAQUS analysis was able to accurately predict thermally induced forces in columns, and was also able to accurately predict the buckling behavior of columns at elevated temperature. The analyses presented in this chapter indicate that at elevated temperatures, very large forces can be generated in columns as a result of restraint to thermal expansion. These thermally induced forces reduce the ability of the column to resist external load. Consequently, thermally induced forces can have a significant effect on the performance of columns subjected to fire. The next chapter presents ABAQUS solutions that were used to investigate the behavior of interior columns in multi-story moment frames subjected to fire. The analyses focused on the significance of thermally induced forces on the behavior of the columns and the entire frame. 74 CHAPTER 5 ANALYSIS OF COLUMNS IN FRAMES 5.1 Overview As described earlier, the purpose of this thesis is to investigate the influence of thermally induced forces on the behavior of columns in steel buildings subjected to a fire. So far this thesis has examined the behavior of individual columns subjected to either external load and/or temperature increase to validate the techniques used to model this problem on ABAQUS, and to provide some initial insights into the problem. In this Chapter, the response of columns in steel buildings subjected to fire are further investigated by modeling multi-story steel building frames on ABAQUS. A ten story steel moment frame with arbitrarily chosen member sizes are analyzed first. To then study column response in more realistically designed frames, analyses were conducted on a series of 3, 9 and 20- story model buildings known as the “SAC buildings.” The SAC buildings were used to study earthquake response of steel moment frame buildings as part of the SAC investigation (FEMA 2000), and are used herein because they represent realistic building designs. For the building frames investigated in this chapter, two analysis approached were considered. The first type of analysis is referred to as “load control.” With load control analysis, selected columns in the frame were heated to a selected temperature with no dead or live load on the frame. The external load on the frame was then increased while the temperature remained constant. Load control analysis does not represent a realistic condition during a building fire, but provides some insights into basic behavior. The second type of analysis presented in this chapter is referred to as “temperature control.” With temperature control analysis, dead and live load are first applied to frame and then held constant. The temperature of selected members is then increased to represent the 75 effects of a fire in a localized portion of the building. Temperature control analysis is a more realistic representation of the conditions present during a building fire. The ten story steel moment frame is analyzed using both load control and temperature control. The 3, 9 and 20-story SAC moment frames are analyzed using temperature control only. For all analyses conducted in this chapter, only a single 2-dimensional moment-resisting frame is analyzed. That is, a three-dimensional analysis of the complete building was not conducted. Further, only the bare steel framing was analyzed. The influence of the concrete floor system was not included in the analysis. Consequently, restraint to thermal expansion of columns provided by out-of-plane framing and by the concrete floor system was not considered in this analysis. It is anticipated that columns that are part of a moment frame will see greater restraint to thermal expansion than columns that are part of a gravity framing system. From this point of view, the analyses conducted in this chapter likely overestimate the effects of thermal restraint on columns that are part of a building’s gravity frames. On the other hand, neglecting the influence of out-of-plane framing and neglecting the influence of the floor slab likely underestimates the effects of thermal restraint on columns in actual building frames. Ultimately, to provide the most realistic assessment of the influence of restraint to thermal expansion for columns in steel buildings subjected, a more complete three- dimensional analysis, including the floor system, is needed. Such an analysis is beyond the scope of this thesis. Nonetheless, the two-dimensional analyses presented in this chapter are still expected to provide useful insights into this problem. 76 5.2 Behavior of Interior Columns in a Ten-Story Steel Moment Frame 5.2.1 Description of Frame To investigate the effect of the thermal restraint on the behavior of interior columns in steel buildings, an analysis was conducted on a moment frame with an interior column subjected to temperature increase. The moment frame is ten stories in height with four 30 ft bays. The first story is 15 ft in height, and the other nine stories are each 12 ft in height. All beam-to-column connections are moment resisting. The first story column bases are assumed to be fixed against rotation. The beam and column sizes were chosen somewhat arbitrarily as follows. All beams in the frame are W27x161 sections. Column sections for stories from 1 to 4 are W14x370, for stories from 5 to 8 are W14x257, for stories from 9 and 10 are W14x145. All beams and columns are assumed to be oriented for major axis bending in the plane of the frame. The material is modeled with a room temperature yield strength of 50 ksi. Stress-strain curves at elevated temperature follow Eurocode 3 (see Figure 2.3). The coefficient of thermal expansion is taken as 7.8*10-6 (1/ oF). An initial imperfection is introduced in these columns’ major axes following the sine function per Equation 4.2, with a maximum initial imperfection at midspan equal to 0.1% of column’s length. Only major axes bending of the columns are considered in the analysis. Figure 5.1 shows the frame elevation. In the figure, the letters A, B, C, D, E are used to indicate the grid lines of the frame. 77 9 x 12 ' = 1 08 ' 4 x 30ft = 120 ft A B C D E1 5' Figure 30.1 Ten Story Moment Frame with an Interior Column Subjected to Fire As described earlier, this frame was investigated using two different analytical approaches: load control and temperature control. In the load control method, room temperature was applied in the interior column and increased to a desired temperature and maintained at that value. Then the external load was applied and increased until a mechanism formed. In the temperature control method, an external load was applied to the frame and maintained at a specified value. Then a temperature was applied and increased to 2400 oF or until a mechanism formed. The load control method provides information the load resistance of the structure at different elevated temperatures. The temperature control method, on the other hand, is closer to the real situation of a fire in a building, and provides information on the temperature that causes the significant effect on a structure at a specified external load. With these definitions, thermally related analyses conducted in the previous chapter are based on the load control method. The 78 following sections provide two ABAQUS analyses for the moment frame, one using the load control method and one using the temperature control method. 5.2.2 Analysis of Ten Story Frame Using Load Control Three analytical cases on ABAQUS, similar to the first three cases in the truss problem in Chapter 4, were conducted to investigate the response of the interior column in the ten story moment frame. In these analyses, a temperature increase was applied to the column C, which is located at first story at grid line C of the moment frame, while other frame elements were maintained at room temperature. An external distributed load w was applied to all floor beams in the moment frame. This distributed load was intended to represent the self-weight of the frame as well as additional superimposed dead and live load. For comparison, the strength of the column C using the elevated temperature column strength equations in Appendix 4 of the 2010 AISC Specification was also computed. Because the base of column C is fixed the other end is connected to beams with moment connections, it end rotational restraints are significant. To account for this rotational restraint when computing the column axial capacity using AISC Appendix 4, an effective length factor of k = 0.6 was used in the calculations. Detailed descriptions of the three ABAQUS analysis cases are provided below. Case 1: An ABAQUS analysis was conducted of the moment frame to investigate the effect of both thermal restraint and thermal degradation on the axial force induced in column C. In this analysis, no external load was applied to the frame. The temperature in the column C was increased from room temperature up to 2400 oF, the temperature at which steel loses virtually all of its strength and stiffness. The analysis considers the effects of both the thermal expansion of the interior column as well as the temperature dependent degradation of material strength and stiffness. As the temperature of the interior column was increased, the axial force induced in the column was computed. 79 Case 2: ABAQUS analyses were conducted to investigate the effect of temperature dependent material degradation on the compressive strength of column C, when an external distributed load w was applied to all floor beams in the moment frame. For these analyses, thermal expansion of column C was not modeled, and so there was no thermally-induced force in the column. Thus, only the external load w produced axial force in the column. For each analysis, column C was assumed to be exposed to a specified elevated temperature and this temperature was kept constant during the analysis. Temperature dependent material properties for the specified temperature were used in the analysis, but as noted above, thermal expansion was not modeled. Thirteen ABAQUS analyses were carried out for temperatures of 68 oF, 200 oF, 400 oF, 600 oF, 750 oF, 800 oF, 1000 oF, 1200 oF, 1400 oF, 1600 oF, 1800 oF, 2000 oF, and 2200 oF. For each temperature, the distributed load w was applied to floor beams and increased until ABAQUS stopped running or a mechanism formed. For each analysis, the relationship between the axial force in column C and the out-of-plane deflection of the column at its midspan was determined, along with the maximum axial force sustained by column C at that temperature. As noted earlier, the axial force in column C for this case resulted only from applied load. Case 3: ABAQUS analyses were conducted to investigate the effect of both temperature dependent material degradation and thermal restraint on the behavior of column C. For these analyses, an external distributed load w was applied to all floor beams in the frame, and the thermal expansion of column C was modeled. Consequently, for these analyses, axial force in column C was generated both from the external load and from the restraint to thermal expansion. For each analysis, the temperature of column C was increased from room temperature (68° F) to a specified maximum temperature, and then maintained at that specified temperature. After that, the external load w was applied to the frame and increased until ABAQUS stopped running or a mechanism formed. Thirteen ABAQUS analyses were conducted for maximum temperatures of 68 oF, 200 oF, 400 oF, 600 oF, 750 oF, 800 oF, 1000 oF, 1200 oF, 1400 oF, 1600 oF, 1800 oF, 2000 oF, and 2200 oF. For each 80 analysis, the relationship between the axial force in column C and the out-of-plane deflection of the column at its midspan was determined, along with the maximum axial force sustained by column C at that maximum temperature. As noted above, the axial force in column C for this case resulted both from restraint to thermal expansion during the temperature increase and from the subsequently applied external load, w. Figure 5.2 shows the relationship between compressive strength and temperature for column C for ABAQUS analysis Case 2 and 3. These relationships are plotted in comparison with the curve representing column strength at elevated temperatures computed from the equations in AISC – Appendix 4 with an effective length factor k = 0.6. Also included in Figure 5.2 is the relationship between induced axial force in column C and temperature increase for Case 1 of the ABAQUS analysis. 0 1000 2000 3000 4000 5000 6000 7000 8000 0 500 1000 1500 2000 2500 3000 Temperature (oF) C ol um n C om pr es si ve S tr en gt hs o r In du ce d A xi al F or ce s (k ip s) ABAQUS - Case 1 - Temperature Increase with Thermal Restraint - No External Load ABAQUS - Case 2 - Elevated Temperature with External Load - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and External Load AISC - Appendix 4 - Individual Column with k = 0.6 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 31.2 Compressive Strength or Induced Axial Force for Column C in Ten Story Moment Frame 81 The plots in Figure 5.2 are similar to these in Figure 4.8 and indicate that ABAQUS can reasonably predict the response of the interior column in multi story building at elevated temperatures. For temperatures higher than 750 oF, column strength computed by ABAQUS matches well with that computed by Appendix 4 of AISC Specification with an approximate effective length factor. For temperatures lower than 750 oF, Appendix 4 of the AISC Specification provides column strength predictions which are lower than the ABAQUS solution. This phenomenon is consistent with ABAQUS analyses in Chapter 4 and with the statement of Takagi and Deierlein (2007) that equation for Appendix 4 of AISC Specification is suitable only for temperatures greater than 572 oF to 752 oF. Different from the analyses for the truss problem, in this case of analyses of the moment frame, the strength curve from ABAQUS Case 2 does not perfectly coincide with that from ABAQUS Case 3, although the two are still quite close. As with the truss problem, this result suggests that the maximum axial force that the column can resist at elevated temperature is about the same whether the axial force is generated by external load or thermally induced forces. The ABAQUS solutions in Figure 5.2 clearly show the dependence of column strength on temperature. The column retains much of its compressive strength when the temperature is lower than about 750 oF. When temperature is higher than 750 oF, column strength reduces significantly. Column strengths at 1000 oF, 1400 oF, 2000 oF are about 60%, 15%, 2% of those at room temperature, respectively. The plots in Figure 5.2, however, do not show the relationship between the applied load and the induced axial force. In Figures 5.3 to 5.6, ABAQUS Case 2 and Case 3 will be examined in greater detail to provide further insight to this relationship at temperatures of 200° F, 400° F, 800° F and 1200°F. In these figures, the horizontal axis, identified as “Load P” in the column, represents the axial force in the column due to the externally applied distributed load on the beams. This is computed by multiplying the total distributed load applied to all ten floor levels above column C with a tributary width. For column C, the tributary width is taken as 30 ft. 82 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 Load P (kips) A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 32.3 Interior Column Axial Force versus Equivalent Applied Load P at 200 oF 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 Load P (kips) A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 33.4 Interior Column Axial Force versus Equivalent Applied Load P at 400 oF 83 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 Load P (kips) A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 34.5 Interior Column Axial Force versus Equivalent Applied Load P at 800 oF 0 500 1000 1500 2000 0 500 1000 1500 2000 2500 3000 3500 4000 Load P (kips) A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Case 2 - Elevated Temperature with Applied Load P - No Thermal Restraint ABAQUS - Case 3 - Temperature Increase with Thermal Restraint and Applied Load P 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 35.6 Interior Column Axial Force versus Equivalent Applied Load P at 1200 oF 84 Similar to Figures 4.9 to 4.12 for the truss problem, Figures 5.3 to 5.6 illustrate the effect of thermal restraint on the induced axial force in the interior column. Due to thermally induced force, column C in Case 3 always sees a higher force than column C in Case 2. Because of that, column C in the Case 3 also fails at a lower magnitude of external load. Therefore, it can be concluded that forces in a column generated by restraint to thermal expansion will reduce the external load that the column can resist in a fire. The occurrence of load redistribution also can be seen in these figures by the change in the patterns of these curves. Another observation can be made is that, in the case of thermal degradation only (no thermally induced forces), at the lower temperatures such as at 200 oF, 400 oF, and 800 oF, where the stiffness of the column has not been significantly reduced, the values of the equivalent load P are close to the induced axial force in the interior column. Therefore, at these lower temperatures, the axial force induced in a column by external load can reasonably be estimated using the tributary width. Note, however, the total axial force in the column must still consider the addition of the thermally induced force. More information about the effect of fire on an interior column on the surrounding structural member and greater detail about load redistribution at high temperatures are presented in Figure 5.7. In this figure, the distributed external load w on the beams is plotted against the axial force in all five first story columns. These plots are all for ABAQUS analysis Case 3 that includes the effects of thermal expansion. In Figure 5.7, the axial forces in the first story columns are plotted for the case where the temperature of column C is 1000 oF. Note that because of the symmetry in the frame and in the loading,, the axial force in column A is equal to that in column E and the axial force in column B is equal that in to column D. 85 -1000 0 1000 2000 3000 4000 5000 0 2 4 6 8 10 12 14 Load w (kips/ft) A xi al F or ce o f C ol um n (k ip s) ABAQUS - Case 3 - T = 1000oF - Column C ABAQUS - Case 3 - T = 1000oF - Column A&E ABAQUS - Case 3 - T = 1000oF - Column B&D 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 36.7 Column Axial Force versus Distributed Load w at 1000 oF It can be seen from Figure 5.7 that the temperature increase in column C has a significant influence on only adjacent columns (columns B and D). For columns A and E, which are one bay further away from column C, the effect heating on column C is small. The effects of the temperature increase in column C on columns B and D are seen first by a tensile force developed in columns B and D prior to the application of external load. The tensile force in columns B and D is caused by the thermal expansion of column C. Columns A and E, on the other hand, see little impact from the thermal expansion of column C. The effect of heating column C is also illustrated by the pattern of the external load versus induced axial force curves, which show load redistribution. When the external load is lower than 12 kips/ft, the curve for column C already is nonlinear due to thermal degradation of the material and due to the thermally induced force. The curves for columns B and D, as well as for A and D, however, are nearly linear since these columns are not heated and therefore do not suffer thermal degradation of material properties. 86 When the external load approaches about 13 kips/ft, column C buckles and the axial force in this column subsequently decreases from that point. Columns B and D, at that time still possess axial strength and stiffness. The axial force in these columns increases at a higher rate, representing a redistribution of load from column C. There appears to be little redistribution of load to columns A and E. 5.2.3 Analysis of Ten Story Frame Using Temperature Control As discussed previously, it is convenient to use the load control method to investigate column strength under a specified elevated temperature. This method, however, does not realistically represent the conditions in a building during a fire. Another approach, closer to the real conditions that can be used to predict the response of a structure under fire is using the temperature control method. In this method, external load is applied in the structure and maintained at the desired value. Then a temperature increase is applied to selected members of the structure. The temperature is increased until a mechanism forms. This section describes an ABAQUS analysis of the same ten story moment frame using the temperature control method. Two steps were created and analyzed in the ABAQUS model. In the first step, a distributed load is applied to every floor beam and maintained at a specified value. The specified load used herein is about half of the load that causes the formation of a mechanism in the frame at room temperature. This is intended to represent the dead and live load that is present at the time of the fire. In the second step, the temperature in column C is increased from room temperature (68°F) up to 2400 oF, or until a mechanism is formed, while other surrounding members are maintained at room temperature. Figure 5.8 presents results of the analysis. In this plot, the step time on the horizontal axis indicates the loading proportions in each step in the ABAQUS analysis. The value of step time increasing from zero to 1 corresponds to the increase of external load from zero to the specified value, which is equal to half of the mechanism load. The values of step time 87 increasing from 1 to 2 corresponds to the increase of temperature from room temperature to 2400 oF, the temperature at which steel loses virtually all strength and stiffness. The vertical axis in this plot is the axial force in column C. 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Step Time A xi al F or ce o f I nt er io r C ol um n (k ip s) 9 x 12 ' = 1 08 ' 4 x 30' = 120' A B C D E15 ' Figure 37.8 Axial Force in the Interior Column of Ten Story Moment Frame for Temperature Control Analysis It can be seen from Figure 5.8 that increasing the temperature in column C dramatically increases the force in this column. Since the external load did not change during the temperature increase (i.e after Step Time 1), the large increase in axial force is the result of restraint to thermal expansion. When the external load was increased from zero to half of the mechanism load (Time Steps 0 to 1), the relationship between external load and the axial force in column C is linear, indicating that column C still was elastic under the applied load at room temperature. When the temperature was increased while maintaining a constant load, the axial force in the heated column increased rapidly due to thermal 88 restraint. At the step time of 1.2 or at the temperature of 534 oF, the magnitude of the force caused by restrained thermal expansion is about three times the force caused by the applied external load. At a temperature of about 1000 oF, column C begins buckling and its capacity starts decreasing. This analysis suggests that forces generated by restraint to thermal expansion can have a very large effect on the performance of columns in steel buildings subject to fire. These thermally induced forces can be large enough to cause failure of a column. 5.3 Behavior of Interior Columns in the SAC Buildings 5.3.1 General Information about SAC Buildings To further study the effects of thermally induced forces on columns in steel buildings subjected to fire, this section presents results of analyses of steel moment frames in the SAC Buildings. SAC is a Joint Venture which was established to investigate the damage to welded steel moment frame buildings in the 1994 Northridge earthquake (FEMA 2000). The building designs discussed herein were developed as part of the SAC steel moment frame project. In this project, 3-, 9- and 20- story model buildings were designed following the local code requirements of three cities: Los Angeles, Seattle, and Boston. Three different types of structural designs were carried out for these building. The first design used code provisions in-place before the 1994 Northridge earthquake and is called the Pre-Northridge Design. The second design used new design provisions developed after Northridge earthquake following the recommendations of FEMA 267 (1995) and is called the Post-Northridge Design. In this thesis, analyses are conducted for the external moment frames in the 3-, 9- and 20-story buildings in Boston using the Post-Northridge Design. Shown in Figures 5.9 are the elevations of these moment frames. In this figure, the letters A, B, C, D, E and F indicate the grid lines of the buildings. In the 3 story building, all beam-to-column connections are moment connections. In the 9 story building, moment connections are used at all locations between floor beams and the 89 column at grid line A. In the 20 story building, moment connections are used at all locations except at the first floor. The columns of the first floor of 3-story building are fixed at their base. For the 9 and 20 story buildings, the columns are pinned at their base. The first floor of the 9 story building and the first and second stories of the 20 story building are restrained against lateral translation. These lateral restraints are intended to model the presence of basement walls. A B C D E F 8 x 13 ' = 1 04 ' 5 x 30' = 150' 18 ' 12 ' 19 x 1 3' = 2 47 ' 18 ' 12 ' 12 ' A B C D E F 5 x 20' = 100' 3 - Story Frame 9 - Story Frame 20 - Story Frame 3 x 30' = 90' 3 x 13 ' = 3 9' A B C D Figure 38.9 Elevation of External Moment Frames of 3-Story, 9-Story and 20-Story SAC Buildings (FEMA 2000) Beam and column sections for the moment frames in 3-, 9- and 20-story SAC buildings are shown in Tables 5.1 to 5.3. 90 Table 4.1 Beam and Colum Sections of Moment Frame in 3-Story SAC Building (FEMA 2000) Story/Floor Columns Girders Exterior Interior ½ W14x82 W14x145 W21x62 2/3 W14x82 W14x145 W21x62 3/Roof W14x82 W14x145 W14x48 Table 5.2 Beam and Colum Sections in Moment Frame of 9-Story SAC Building (FEMA 2000) Story/Floor Columns Girders Exterior Interior -1/1 W14x283 W14x500 W12x53 ½ W14x283 W14x500 W33x141 2/3 W14x283, W14x257 W14x500, W14x455 W33x141 ¾ W14x257 W14x455 W21x101 4/5 W14x257, W14x211 W14x455, W14x398 W21x101 5/6 W14x211 W14x398 W21x101 6/7 W14x211, W14x159 W14x398, W14x311 W21x101 7/8 W14x159 W14x311 W18x97 8/9 W14x159, W14x109 W14x311, W14x193 W16x67 9/Roof W14x109 W14x193 W12x53 91 Table 6.3 Beam and Colum Sections in Moment Frame of 20-Story SAC Building (FEMA 2000) Story/Floor Columns Girders Exterior Next to Interior Interior -2/-2 W14x455 W36x393 W36x485 W12x14 -1/1 W14x455 W36x393 W36x485 W16x67 ½ W14x455, W14x455 W36x393, W36x393 W36x485, W36x485 W33x141 2/3 W14x455 W36x393 W36x485 W33x141 ¾ W14x455, W14x370 W36x393, W36x328 W36x485, W36x393 W33x141 4/5 W14x370 W36x328 W36x393 W33x141 5/6 W14x370, W14x342 W36x328, W36x300 W36x393, W36x359 W33x141 6/7 W14x342 W36x300 W36x359 W24x131 7/8 W14x342, W14x342 W36x300, W36x300 W36x359, W36x359 W24x131 8/9 W14x342 W36x300 W36x359 W24x131 9/10 W14x342, W14x311 W36x300, W36x260 W36x359, W36x300 W24x131 10/11 W14x311 W36x260 W36x300 W24x131 11/12 W14x311, W14x283 W36x260, W36x260 W36x300, W36x300 W24x131 12/13 W14x283 W36x260 W36x300 W24x117 13/14 W14x283, W14x283 W36x260, W36x260 W36x300, W36x280 W24x117 14/15 W14x283 W36x260 W36x280 W24x104 15/16 W14x283, W14x193 W36x260, W36x182 W36x280, W36x210 W24x104 16/17 W14x193 W36x182 W36x210 W24x104 17/18 W14x193, W14x159 W36x182, W36x150 W36x210, W36x150 W21x101 18/19 W14x159 W36x150 W36x150 W18x86 19/20 W14x159, W14x109 W36x150, W24x117 W36x150, W24x131 W18x76 20/Roof W14x109 W24x117 W24x131 W12x53 92 For the purposes of this thesis, the material was modeled with a room temperature yield strength of 50 ksi. Stress-strain curves at elevated temperature follow Eurocode 3 (see Figure 2.3). Only the major axis behavior was considered for analysis. 5.3.2 Load and Load Combinations for Structural Fire Design According to FEMA-355C (FEMA 2000), the design floor loads for the SAC buildings are as follows: Floor dead load for weight calculation: 96 psf Floor dead load for mass calculation: 86 psf Roof dead load: 83 psf Reduced live load per floor and for roof: 20 psf The load combination for the analysis of structures subjected to fire, according to AISC Appendix 4 is: U = 1.2D + 0.5L + T + 0.2S (5.1) Where: D is nominal dead load L is nominal live load S is nominal snow load T is nominal forces and deformations due to the fire. The snow load was assumed to be zero for this analysis. The tributary width for the 3- and 9-story frames is 15 ft and for the 20-story frame is 10ft. The loads applied to floor and roof beams were as shown in Table 5.3. 93 Table 7.4 Dead and Live Loads and Load Combinations Load Type Load (psf) 3 Story Building 9 Story Building 20 Story Building Trib. Area (ft) Distributed Load (k/in) Trib. Area (ft) Distributed Load (k/in) Trib. Area (ft) Distributed Load (k/in) Floor dead load 96 15 0.1200 15 0.1200 10 0.0800 Roof dead load 83 15 0.1038 15 0.1038 10 0.0692 Reduced live load 20 15 0.0250 15 0.0250 10 0.0167 Load Comb. for Floor per Eq. 5.1 0.1565 0.1565 0.1043 Load Comb. for Roof per Eq. 5.1 0.1370 0.1370 0.0913 5.3.3 ABAQUS Analyses for SAC Buildings Subjected to Fire ABAQUS analyses were conducted using the temperature control approach for the 3-, 9- and 20-story moment frames. An initial imperfection was introduced in the columns’ major axes following the sine function as in Equation (4.2) with maximum initial imperfection at midspan equal to 0.1% of columns’ lengths. Each ABAQUS analysis includes two steps. In the first step, the distributed loads provided in Table 5.4 were applied to the floor and roof beams and maintained at that value. In the second step, a temperature increase was imposed on an interior column. The temperature of the column was increased from 68 oF to 2400 oF, or until a mechanism forms, while other surrounding members were maintained at room temperature. The heated column in the 3- story frame was the column in the first story at grid line C. The heated column in the 9- story building was the column in the second story at grid line D. The heated column in the 20-story building was the column in the third story at grid line D. Figures 5.10 to 5.12 present results of the analyses. These figures plot step time versus axial force in selected columns. Results are included for the heated column in each frame, as well as for a column that is immediately adjacent to the heated column. Similar to 94 Figure 5.8, the step time increasing from zero to 1 corresponds to the increase of external load from zero to the full design values listed in Table 5.4. The step time values increasing from 1 to 2 correspond to the increase of temperature from room temperature to 2400 oF. 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Step Time A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Axial Force at Column C ABAQUS - Axial Force at Column B 3 x 30' = 90' 3 x 13 ' = 3 9' A B C D Figure 39.10 Column Axial Force in 3 – Story SAC Frame from ABAQUS Temperature Control Analysis 95 0 200 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Step Time A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Axial Force at Column D ABAQUS - Axial Force at Column E A B C D E F 8 x 13 ' = 1 04 ' 5 x 30' = 150' 18 ' 12 ' Figure 40.11 Column Axial Force in 9 – Story SAC Frame from ABAQUS Temperature Control Analysis 96 -500 0 500 1000 1500 2000 2500 0 0.5 1 1.5 2 2.5 Step Time A xi al F or ce o f I nt er io r C ol um n (k ip s) ABAQUS - Axial Force at Column D ABAQUS - Axial Force at Column C 19 x 1 3' = 2 47 ' 18 ' 12 ' 12 ' A B C D E F 5 x 20' = 100' Figure 41.12 Column Axial Force in 20 – Story SAC Frame from ABAQUS Temperature Control Analysis Figures 5.10 to 5.12 clearly illustrate the influence of forces developed due to restraint to thermal expansion. In all case, the force in the heated column increases substantially due to heating. The increase in force due to heating becomes more significant as the frame height increases. As shown in Figure 5.10 for the 3-story frame, the axial force induced in the heated column due to the design load at room temperature is 160 kips. With the applied load on the frame held constant the temperature of column C is increased. This column buckles at a temperature of about 1130 oF with the maximum induced axial force of 215 kips. Therefore, restraint to thermal expansion increased the force in the column by 55 kips, which is a 34% of the force in the column due to external load. For the 9- story and 20-story frames, the axial force generated in the heated columns from restraint to thermal expansion is 114% and 390%, respectively of the force in these columns due to external load. These numbers indicate that as the number of stories above the heated 97 column increases, the more significant is the thermally induced axial force in the heated column. Clearly, increasing the number of stories above the heated column increases the restraint to thermal expansion. Another observation from these figures is that the temperature at buckling of 20-story frame is about 900 oF, 1175 oF for the 9-story frame, and 1130 oF for the three story frame. Thus, an increase in the number of stories above the heated column decreases the temperature at which the column will buckle. This is likely related to the effects of thermal restraint, but may also be partly attributed to the proportion of column strength used by the externally applied load. More investigation is needed better understand this effect. Another observation that can be made from Figures 5.10 to 5.12 is the influence of the heated columns on the adjacent columns. When the heated columns are subjected to a temperature increase and expand, they initially reduce in the axial force in the adjacent columns. When the heated columns buckle and lose their loading resistance, the axial force in the adjacent columns increase, reflecting a redistribution of load. From the above analyses, it can be concluded that restraint to thermal expansion can significantly affect the performance of interior columns in steel buildings subjected to fire. Thermal restraint can cause a significant increase in the axial force in a heated column which may be large enough to cause yielding or buckling of the column. Failure of a heated column can lead to redistribution of load, causing an increase in the axial force in adjacent columns. Restraint to thermal expansion can cause a reduction in the critical temperature that causes buckling of a column. 5.4 Summary This chapter has provided the results structural fire analysis of multi story moment frames to investigate the effects of thermal restraint on heated columns. Four moment frames were studied, including a ten story moment frame with arbitrarily chosen beam 98 and column sizes, and the 3, 9 and 20 story SAC moment frames. In each of these frames, an individual column near the base of the frame was heated to simulate the effects of a localized fire. The results of the ABAQUS analyses on these frames clearly show that forces generated in the heated columns from restraint to thermal expansion was significant. These thermally induced axial forces were often larger than the axial force in the column from external load. Further, the thermally induced forces increases as the number of stories above the heated column increased. 99 CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Summary This thesis has presented the results of a research study on the behavior of columns in steel buildings subject to fire. When the temperature of a steel column increases in a fire, the strength and stiffness of the column are reduced. At the same time, thermally induced forces are generated in the column if thermal expansion is restrained. The overall objective of this research was to better understand the importance of these thermally induced forces to the performance of steel columns in fire. More specifically, the objective was to determine if it is safe to neglect these forces when assessing column performance in fire. The approach used in this research was to conduct a series of analyses of steel columns using the finite element computer program ABAQUS. Columns were modeled in ABAQUS using beam elements that included nonlinear geometry and nonlinear material properties. The model incorporated elevated temperature stress strain properties recommended in Eurocode 3 that accounts for the reduction in strength, stiffness and proportional limit of steel as a function of temperature. Columns were also modeled with initial geometric imperfections. The ABAQUS model used for the research was validated, to the extent possible, against available analytical and other published solutions for the columns at room temperature and at elevated temperature. These comparisons showed that the ABAQUS model was able to accurately predict thermally induced forces in columns, and was also able to accurately predict the buckling behavior of columns at elevated temperature. Using the validated ABAQUS model, a series of analyses were conducted on the behavior of columns under axial compression for temperatures varying from room 100 temperature up to 2400° F, the temperature at which steel has lost virtually all of its stiffness and strength. A series of individual columns were analyzed with and without restraint to thermal expansion. A column that was a part of a simple truss was also analyzed to study a simple case of a flexible restraint to thermal expansion. Finally, the behavior of columns that are part of a multi-story steel moment frame was investigated. Four different frame configurations were studied, including 3, 9, 10 and 20 story frames. In each case, the temperature of a column in a lower story of the frame was increased to simulate the effects of a fire. The axial force generated in the column due to externally applied gravity loads and due to restrained thermal expansion was computed in the analysis to assess the relative importance of the thermally induced forces. 6.2 Conclusions The analysis of an individual column with a flexible axial restraint showed that the axial force generated in the column from restrained thermal expansion significantly reduced the magnitude of external load that could be resisted by the column. The analyses of columns that were part of multi-story moment frames showed similar results. In each of these frames, an individual column near the base of the frame was heated to simulate the effects of a localized fire. The results of the ABAQUS analyses on these frames clearly showed that forces generated in the heated columns from restraint to thermal expansion was significant. These thermally induced axial forces were often larger than the axial force in the column from external load. Further, the thermally induced forces increased as the number of stories above the heated column increased. In summary then, all of the analyses conducted in this research indicate that forces generated by restraint to thermal expansion can have a very large impact on the performance of a steel column in fire. When evaluating the safety of a column in a fire, it is important to recognize that the total axial force in the column is the sum of the force 101 generated by external gravity load on the frame and the force generated by restraint to thermal expansion. The force generated by restrained thermal expansion can be very large, and neglecting this force can lead to unsafe designs. 6.3 Recommendations for Further Study The research in this thesis has demonstrated that forces generated by restrained thermal expansion can have a significant impact on the performance and safety of columns in steel buildings subjected to fire. This research analyzed columns in relatively simple two- dimensional frames. Additional studies are needed to study column performance in more realistic three-dimensional frames with analyses that include the effects of concrete floor systems on the restraint to thermal expansion of columns. Further, while the effects of restrained thermal expansion can be modeled using advanced finite element programs such as ABAQUS, there is a need for simpler design oriented analysis methods and tools that can permit a designer to reasonably estimate thermally induced forces in columns. 102 REFERENCES ABAQUS, 2008a. “Getting Started with ABAQUS: Interactive Edition”. Version 6.8. (2008). Dassault Systèmes Simulia Corp., Providence, RI, USA ABAQUS, 2008b. “ABAQUS Analysis User’s Manual: Volume IV- Elements”. Version 6.8. (2008). Dassault Systèmes Simulia Corp., Providence, RI, USA ABAQUS, 2008c. “ABAQUS Analysis User’s Manual: Volume II - Analysis”. Version 6.8. (2008). 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She received her Bachelor’s degree in Construction Project Management from National University of Civil Engineering in Vietnam in March 2005, and another Bachelor’s degree in Structural Engineering at the same university in March 2006. After graduation, she worked in construction industry as a structural designer. She started her Master’s program in Structural Engineering at the University of Texas at Austin in August 2008 under the sponsor of Vietnam Education Foundation (VEF). Permanent Address: 24 Nhan Hoa Street, Nhan Chinh Ward, Thanh Xuan District, Hanoi, Vietnam. This thesis was typed by the author.