SYSTEM-LEVEL TRANSIENT FIRE RESPONSE OF DOUBLE ANGLE
CONNECTIONS IN STEEL FRAMED STRUCTURES
By
Purushotham Pakala

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Civil Engineering – Doctor of Philosophy
2014

ABSTRACT
SYSTEM-LEVEL TRANSIENT FIRE RESPONSE OF DOUBLE ANGLE CONNECTIONS IN
STEEL FRAMED STRUCTURES
By
Purushotham Pakala
Steel framed structural systems are frequently used in high-rise buildings due to high structural
performance steel provides as compared to other construction materials. In these steel framed
structures, connections play a significant role in transferring forces from one member to another
member and influence the overall stability of the structural system at ambient and fire conditions.
Among different connection configurations available, double angle connections possess superior
tying resistance and rotational capacity as compared to other relatively brittle connection types.
Performance of connections under fire conditions is much more crucial to the stability of
structural system as they experience rapid degradation in load carrying capacity and stiffness,
due to deterioration of strength and elastic modulus of steel with temperature.
The current approaches for evaluating fire resistance of connections have many limitations and
do not take into consideration critical factors governing fire response. For example, connections
experience significant fire induced axial restraint forces when exposed to fire and these forces
are influenced by factors such as decay phase of fire, restraint to free thermal expansion imposed
by the unheated structural elements adjacent to the connection. However, current design
provisions do not consider the effect of fire induced restraint forces. Further, current provisions
are based on results from scaled/isolated connection tests. However, the response of double angle
connection at system-level is completely different from that of scaled/isolated connection
behavior.

iv

The system-level fire response of double angle connections is simulated by developing finite
element models using ANSYS. The finite element models account for material and geometrical
nonlinearities, degradation of constitutive material properties with temperature and complex nonlinear contact interactions, that have significant influence on the fire response of double angle
connections. The validated models are used to carry out parametric studies to quantify the
influence of critical parameters. Results from the parametric study show that fire performance of
double angle connections is affected by increased load level and fire characteristics, while fire
resistance gets enhanced when the presence of concrete slab in the structural frame is taken into
account.
To study the influence of system level interactions on the behavior of double angle connections
and to validate the finite element models, fire resistances tests on two double angle connection
assemblies were conducted. The test variables included load level, presence of slab, structural
continuity and fire scenario. Results from fire tests show that double angle connections are
highly ductile and have inherent rigidity to carry higher fire induced axial forces for which they
are not typically designed for.
Results from the fire tests, data from parametric studies were utilized to develop a rational
methodology for evaluating the fire resistance of connections. The proposed methodology,
developed based on equilibrium principles, uses temperature dependent moment-curvature-axial
force (M-κ-P) relationships to trace the response of double angle connection assemblies. The
proposed methodology accounts for critical factors such as fire induced axial forces, design fire
scenarios, material and geometric nonlinearities.

v

Dedicated to
my family and friends
who have always been with me
during various phases of my life.

iv

ACKNOWLEDGMENTS

I would like to thank a number of people who have contributed to the successful completion of
this thesis. Many thanks to Prof. Venkatesh Kodur for giving me the opportunity of doing
dissertation under his guidance. I would like to specially thank distinguished faculty members
Professors Alejandro Diaz, Rigoberto Burgueño and Dr. Nizar Lajnef for serving on my
committee. Thanks to all my committee members for their support, encouragement, useful
suggestions and guidance.
I am highly grateful to Dr. Mahmoud Dwaikat for his help and constant guidance with finite
element modeling without which this dissertation would not have been completed. Thanks to Dr.
Syed Haider for his advice and support during difficult times.
The success of this study was due to the support and help of faculty, staff, and personnel in the
Department of Civil and Environmental Engineering at Michigan State University. Special
thanks to the manager of Civil Infrastructure Laboratory, Mr. Siavosh Ravanbaksh for his help,
advice and fun attitude which made experiments an enjoyable experience.
Thanks to Laura Taylor, Margaret Conner and Mary Mroz for all the help they provided. Sincere
and warm thanks to my co-students who supported me during the course of my research and the
rest of my friends and colleagues in the department.
I would like to thank my wife Indumathy Jayamani who was supportive of me throughout the
course of the study.

iv

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................................... viii
LIST OF FIGURES ...................................................................................................................... x
KEY TO ABBREVIATIONS ................................................................................................... xvi
CHAPTER 1 .................................................................................................................................. 1
1. INTRODUCTION..................................................................................................................... 1
1.1 General .................................................................................................................................. 1
1.2 Effect of Temperature on Connection Behavior ................................................................... 4
1.3 Research Objectives .............................................................................................................. 7
1.4 Scope ..................................................................................................................................... 8
CHAPTER 2 ................................................................................................................................ 10
2. STATE OF THE ART REVIEW .......................................................................................... 10
2.1 General ................................................................................................................................ 10
2.2 Experimental Studies........................................................................................................... 11
2.3 Numerical Studies ............................................................................................................... 19
2.4 High Temperature Bolt Properties ...................................................................................... 26
2.5 Codes of Practice ................................................................................................................. 29
2.6 Summary ............................................................................................................................. 30
CHAPTER 3 ................................................................................................................................ 33
3. EXPERIMENTAL STUDIES ................................................................................................ 33
3.1 General ................................................................................................................................ 33
3.2 Test Specimens .................................................................................................................... 34
3.3 Instrumentation.................................................................................................................... 35
3.4 Test Apparatus..................................................................................................................... 36
3.5 Test Conditions and Procedure ........................................................................................... 37
3.6 Results and Discussion ........................................................................................................ 39
3.6.1 Thermal Response ........................................................................................................ 39
3.6.2 Structural Response ...................................................................................................... 41
Connection Rotations ......................................................................................................... 41
Forces in Connections ....................................................................................................... 42
Forces in Beams ................................................................................................................. 44
3.6.3 Failure Patterns............................................................................................................ 45
3.7 Summary ............................................................................................................................. 46
CHAPTER 4 ................................................................................................................................ 47
4. NUMERICAL MODELS ....................................................................................................... 47
4.1 General ................................................................................................................................ 47
4.2 Selection of Finite Element Program .................................................................................. 48
4.3 Finite Element Modeling ..................................................................................................... 50
4.3.1 General Approach ........................................................................................................ 50
4.3.2 Thermal Analysis .......................................................................................................... 53
v

4.3.2.1 General .................................................................................................................. 53
4.3.2.2 Governing Equations ............................................................................................. 53
4.3.2.3 Discretization of Beam........................................................................................... 55
4.3.3 Structural Analysis ....................................................................................................... 56
4.3.4 Modeling Contact Interactions ..................................................................................... 59
4.3.5 High-temperature Material Properties ........................................................................ 61
4.3.6 Failure Criteria and Numerical Convergence ............................................................. 62
4.4 Connection Models ............................................................................................................. 63
4.4.1 General ......................................................................................................................... 63
4.4.2 Assembly Level Connection Model (FEM1) ................................................................. 64
4.4.3 Validation of Assembly Level Connection Model (FEM1) ........................................... 66
4.4.3.1 Thermal Response .................................................................................................. 67
4.4.3.2 Structural Response ............................................................................................... 68
4.4.4 System Level Connection Model (FEM2) ..................................................................... 71
4.4.5 Validation of System Level Connection Model (FEM2) ............................................... 73
4.4.5.1 Thermal Response .................................................................................................. 75
4.4.5.2 Structural Response ............................................................................................... 75
4.5 Summary ............................................................................................................................. 77
CHAPTER 5 ................................................................................................................................ 79
5. PARAMETRIC STUDIES ..................................................................................................... 79
5.1 General ................................................................................................................................ 79
5.2 Characteristics of Double Angle Connections .................................................................... 80
5.3 Comparative Analysis ......................................................................................................... 81
5.3.1 Levels of analysis .......................................................................................................... 81
5.3.2 Analysis details ............................................................................................................. 83
5.3.3 Comparative Performance ........................................................................................... 84
5.3.4 Connection capacity ..................................................................................................... 86
5.3.5 Summary ....................................................................................................................... 87
5.4 Factors influencing fire resistance ....................................................................................... 87
5.5 Parametric Studies ............................................................................................................... 89
5.5.1 Selection of Connection Assemblies ............................................................................. 90
5.5.2 Analysis Details ............................................................................................................ 90
5.5.3 Range of Variables ....................................................................................................... 90
5.6 Results and Discussion ........................................................................................................ 92
5.6.1 Effect of Composite Action ........................................................................................... 92
5.6.2 Effect of Decay Rate ..................................................................................................... 93
5.6.3 Effect of Loading Type.................................................................................................. 94
5.6.4 Effect of Axial Restraint................................................................................................ 97
5.6.5 Effect of Steady-state and Transient Heating ............................................................... 99
5.6.6 Effect of Load Level .................................................................................................... 100
5.6.7 Effect of Fire Scenarios .............................................................................................. 102
5.6.8 Effect of High-temperature Bolt Properties ............................................................... 104
5.7 Summary ........................................................................................................................... 106
CHAPTER 6 .............................................................................................................................. 108
6. METHODOLOGY FOR EVALUATING FIRE RESISTANCE ..................................... 108
vi

6.1 General .............................................................................................................................. 108
6.2 Outline of Proposed Methodology .................................................................................... 109
6.3 Computing Temperatures .................................................................................................. 111
6.3.1 Standard Fire Exposure ............................................................................................. 111
6.3.2 Comparison to Other Methods ................................................................................... 113
6.3.3 Design Fire Exposure ................................................................................................. 117
6.4 Computing Fire Induced Axial Force................................................................................ 119
6.5 Strength Analysis .............................................................................................................. 122
6.5.1. General Analysis Procedure...................................................................................... 122
6.5.2 Generation of M-κ-P relationships............................................................................. 123
6.5.3 Iterative solution ......................................................................................................... 127
6.6 Computer Implementation................................................................................................. 128
6.6.1 Computer Program ..................................................................................................... 128
6.6.2 Material properties ..................................................................................................... 129
6.6.3 Beam, Connection Idealization .................................................................................. 129
6.6.4 Input data.................................................................................................................... 129
6.6.5 Output data ................................................................................................................. 130
6.6.6 Analysis of typical beams ........................................................................................... 130
6.7 Rotation Analysis .............................................................................................................. 131
6.8 Verification of the proposed methodology........................................................................ 132
6.8.1 Double Angle Connection........................................................................................... 132
6.8.2 Shear Tab Connection ................................................................................................ 133
6.9 Limitations ........................................................................................................................ 136
6.10 Design Applicability ....................................................................................................... 136
6.11 Summary ......................................................................................................................... 137
CHAPTER 7 .............................................................................................................................. 138
7. CONCLUSIONS AND RECOMMENDATIONS.............................................................. 138
7.1 General .............................................................................................................................. 138
7.2 Key Findings ..................................................................................................................... 139
7.3 Recommendations for Future Research ............................................................................ 143
7.4 Research Impact ................................................................................................................ 144
APPENDICES ........................................................................................................................... 146
APPENDIX A ......................................................................................................................... 147
APPENDIX B ......................................................................................................................... 149
APPENDIX C ......................................................................................................................... 156
APPENDIX D ......................................................................................................................... 174
APPENDIX E .......................................................................................................................... 190
APPENDIX F .......................................................................................................................... 192
APPENDIX G ......................................................................................................................... 207
REFERENCES .......................................................................................................................... 300

vii

LIST OF TABLES
Table B.1 – Constitutive relationships for high temperature properties of carbon steel
(Eurocode3, 2005b)..................................................................................................................... 149
Table B.2 – Values for the main parameters of the stress-strain relationships of carbon steel at
elevated temperatures (Eurocode3, 2005b)................................................................................. 151
Table B.3 – Constitutive relationships for high temperature properties of concrete(Eurocode2,
2004) ........................................................................................................................................... 152
Table B.4 – Values for the main parameters of the stress-strain relationships of normal strength
concrete at elevated temperatures (Eurocode2, 2004) ................................................................ 154
Table B.5 – Constitutive relationships for high temperature properties of bolt steel (Kodur et al.,
2012) ........................................................................................................................................... 155
Table C.1: Summary of double angle connection strength computed based on different limit
states ............................................................................................................................................ 161
Table C.2: Summary of single angle connection strength computed based on different limit states
..................................................................................................................................................... 167
Table C.3: Summary of shear tab connection strength computed based on different limit states
..................................................................................................................................................... 173
Table 2.1: Summary of the experimental program conducted by Lawson (Lawson, 1990) ....... 192
Table 2.2: Summary of test parameters used by Yu et al.(Yu et al., 2009a) .............................. 193
Table 2.3: Summary of experimental program conducted by Yu et al.(Yu et al., 2009b).......... 193
Table 2.4: Summary of specimen and loading details used by Daryan and Yahyai (Daryan and
Yahyai, 2009) .............................................................................................................................. 194
Table 2.5: Summary of specimen dimensions used by Wang et al. (Wang et al., 2011) ........... 194
Table 2.6: Summary of connection details tested by Al-Jabri (Al-Jabri et al., 2005) ................ 195
Table 2.7: Test parameters and specimen dimensions used by Yang et al.(Yang et al., 2009) .. 196
Table 3.1: Mechanical properties of steel members used in test assemblies .............................. 197
Table 3.2: Dimensions of beams and angles used in the test assemblies ................................... 197
Table 3.3: Mix-proportions of concrete used in the slab of test assembly S2 ............................ 197
Table 3.4: Input parameters used in the computation of design fire, DF1.................................. 198
viii

Table 3.5: Variables used in fire resistance tests on steel beam assemblies ............................... 199
Table 4.1: Summary of contact parameters used in the finite element models .......................... 200
Table 4.2: Summary of key thermal and mechanical properties of steel members used in the test
assemblies (S1, S2) used in the finite element model FEM1 ..................................................... 200
Table 4.3: Summary of key thermal and mechanical properties of concrete used in the assembly
S2 used in the finite element model FEM1 ................................................................................. 200
Table 4.4: Summary of thermal properties of insulation for test assemblies (S1, S2) used in the
finite element model FEM1 ........................................................................................................ 201
Table 4.5: Summary of thermal and structural response of assemblies S1 and S2 obtained from
finite element model FEM1and measured test data .................................................................... 201
Table 4.6: Ambient temperature mechanical properties of different steel members used in the
finite element model FEM2 ........................................................................................................ 201
Table 4.7: Summary of thermal and structural response of restrained steel frame assembly
obtained from finite element model FEM2 and measured test data ........................................... 202
Table 5.1: Summary of key mechanical properties of beam and connections used in comparative
analysis ........................................................................................................................................ 203
Table 5.2: Variation of connection capacity and fire induced forces in three connection types
with fire exposure time ............................................................................................................... 203
Table 5.3: Summary of parameters and their range used in the parametric studies ................... 204
Table 5.4: Compartment characteristics used for arriving at different design fire scenarios ..... 205
Table 5.5: Maximum value of fire induced forces and rotation in connection for different fire
scenarios ...................................................................................................................................... 205
Table 5.6: Ambient temperature mechanical properties for different bolt grades ...................... 205
Table 6.1: Geometric details of steel sections used in the analysis ............................................ 206
Table 6.2: Summary of thermal properties of insulation used in the analysis ............................ 206
Table 6.3: Ambient temperature mechanical properties of different components measured in
Cardington tests .......................................................................................................................... 206

ix

LIST OF FIGURES

Figure 1.1: Typical steel frame in an office building (a) before and (b) after failure of beam-tocolumn connection ...................................................................................................................... 207
Figure 1.2: Development of axial forces in connection .............................................................. 208
Figure 1.3: Illustration of standard and design fire scenarios in typical buildings ..................... 209
Figure 1.4: Connection details and moment-rotation curves for flush end-plate and double angle
connections ................................................................................................................................. 210
Figure 2.1: Schematic of the sub-frame assembly tested by Liu et al.(Liu et al., 2002) ............ 211
Figure 2.2: Schematic of connection configuration tested by Yu et al. (Yu et al., 2009b)......... 212
Figure 2.3: Connection details of the specimens tested by Daryan and Yahyai (Daryan and
Yahyai, 2009) .............................................................................................................................. 213
Figure 2.4: High-temperature stress-strain curves used in the FE model by Pakala et al. (Pakala et
al., 2012a).................................................................................................................................... 214
Figure 2.5: Comparison of predicted deformed shape of connection assembly with that observed
in fire tests (Part (b) adapted from Yu et al. (Yu et al., 2009b)) ................................................. 214
Figure 2.6: Geometrical details of the connection assembly used in the finite element model (all
dimensions in mm)...................................................................................................................... 215
Figure 2.7: Illustration of stresses in the (a) bolt shank region and (b) bolt head region at the time
of beam failure predicted by the model ...................................................................................... 217
Figure 2.8: Variation of connections axial force as a function of fire exposure time for different
scenarios ...................................................................................................................................... 217
Figure 2.9: Progression of connection axial force as a function of fire exposure time for (a) A325
and (b) A490 bolts types with different material models ........................................................... 218
Figure 2.10: Variation in shear strength of A325 and A490 bolts with temperature.................. 219
Figure 2.11: Comparison of high-temperature yield strength of A325 and A490 bolt steel with
conventional steel........................................................................................................................ 219
Figure 3.1: Layout of the steel beams used in test assemblies S1 and S2 (plan view) ............... 220
Figure 3.2: Connection details used in test assembly (all dimensions in mm) ........................... 221
Figure 3.3: Layout of the instrumentations and loading used in the steel beam assembly ......... 222
x

Figure 3.4: Structural fire furnace at Michigan State University................................................ 224
Figure 3.5: Predicted and measured time-temperature curves for design fires DF1 and DF2 .... 225
Figure 3.6: Recorded temperatures at center (mid-span) and quarter span of the Beam I, Beam II
of S1 as a function of fire exposure time (Refer to Fig. 3.1 for naming convention) ................. 226
Figure 3.7: Recorded temperatures at center (mid-span) and quarter span of the Beam I, Beam II
of S2 as a function of fire exposure time (Refer to Fig. 3.1 for naming convention) ................. 227
Figure 3.8: Recorded average temperatures at center (mid-span) and quarter span of the Beam I,
Beam II of assemblies (a) S1 and (b) S2 as a function of fire exposure time (Refer to Fig. 3.1 for
naming convention) .................................................................................................................... 228
Figure 3.9: Connection rotation recorded as a function of fire exposure time in test assemblies (a)
S1 and (b) S2 ............................................................................................................................... 229
Figure 3.10: Schematic illustration of evaluating the rotation at connection ............................. 229
Figure 3.11: (a) Local and (b) global buckling experienced by the secondary beam (Beam II –
North) in assembly S1 ................................................................................................................. 230
Figure 3.12: Warping and distortion of the metal deck in test assembly S1 .............................. 230
Figure 3.13: Variation of strains in secondary and primary beam as a function of fire exposure
time (Refer to Fig. 3.1 for naming convention) .......................................................................... 231
Figure 3.14: Variation of axial forces in secondary, primary beams and concrete deck as a
function of fire exposure time ..................................................................................................... 233
Figure 3.15: (a) Local and (b) global buckling experienced by the secondary beam (Beam IINorth) in assembly S1 ................................................................................................................. 234
Figure 3.16: Visual observations in the test assembly S2 during and after fire test ................... 235
Figure 4.1: Typical connection in a steel framed building along with different structural
components present in connection region ................................................................................... 236
Figure 4.2: Different types of finite elements used for thermal and structural analysis in the
model........................................................................................................................................... 240
Figure 4.3: Strength reduction factors for steel specified in Eurocode 3 .................................... 241
Figure 4.4: Discretized geometry (quarter) of the connection assembly (FEM1) modeled in
ANSYS ....................................................................................................................................... 242
Figure 4.5: Strength reduction factors for different aggregate concrete specified in Eurocode 2
..................................................................................................................................................... 243
xi

Figure 4.6: Comparison of measured and predicted secondary beam temperatures (top and
bottom flanges) in tested assemblies (a) S1 and (b) S2 .............................................................. 244
Figure 4.7: Comparison of measured and predicted secondary beam temperatures (web) in tested
assemblies (a) S1 and (b) S2 ....................................................................................................... 245
Figure 4.8: Comparison of predicted and measured connections rotation as a function of fire
exposure time in test assemblies (a) S1 and (b) S2 ..................................................................... 246
Figure 4.9: Illustration of (a) deformed shape (b) local buckling of secondary beam and (c)
lateral torsional buckling of secondary beam in assembly S1 predicted by the numerical model
..................................................................................................................................................... 247
Figure 4.10: Fire induced axial forces on connections predicted as a function of fire exposure
time (from model FEM1) in test assemblies (a) S1 and (b) S2 .................................................. 248
Figure 4.11: Vertical deflection of secondary beams (at loading actuator) predicted as a function
of fire exposure time (from model FEM1) in test assemblies (a) S1 and (b) S2 ........................ 249
Figure 4.12: Geometrical details of the connection assembly used in the second finite element
model FEM2 (all dimensions in mm) ......................................................................................... 250
Figure 4.13: Modeling of truss replacement and actuator pad.................................................... 252
Figure 4.14: Discretization of connection components (a) angles and (b) bolt .......................... 252
Figure 4.15: Standard and adjusted ISO834 standard fire time-temperature curves .................. 253
Figure 4.16: Predicted temperature distribution in the beam as a function of fire exposure time
..................................................................................................................................................... 253
Figure 4.17: Comparison of predicted and measured (a) beam mid-span deflection (b) axial force
in connection as a function of beam bottom flange temperature and (c) connection’s rotation as a
function of fire exposure time ..................................................................................................... 254
Figure 5.1: Connection details and moment-rotation curves for flush end-plate and double angle
connections ................................................................................................................................. 256
Figure 5.2: Illustration of the isolated connection assembly used in the finite element model (all
dimensions in mm)...................................................................................................................... 257
Figure 5.3: Geometrical details of different connection types (a) double angle (b) single angle
and (c) shear tab connection ....................................................................................................... 258
Figure 5.4: Parametric fire scenario (FS) used in the parametric studies ................................... 260
Figure 5.5: Variation of connections (a) axial force and (b) rotation as a function of fire exposure
time obtained from system-level and isolated connection analysis ............................................ 261
xii

Figure 5.6: Illustration of connection capacity and fire induced axial force with temperature .. 262
Figure 5.7: Comparison of rotation in connection as a function of fire exposure time in
assemblies S1 and S2 .................................................................................................................. 262
Figure 5.8: Time-temperature curves for design fire scenarios used to study the effect of decay
rate............................................................................................................................................... 263
Figure 5.9: Variation of connections rotation as a function of fire exposure time for different
decay rates in assembly (a) S1 and (b) S2 .................................................................................. 264
Figure 5.10: Illustration of different loading scenarios used in the analysis along with the
corresponding bending moments ................................................................................................ 265
Figure 5.11: Progression of connections rotation as a function of fire exposure time for
connection assemblies (a) S1 and (b) S2 with different loading types ....................................... 266
Figure 5.12: Illustration of (a) deformed shape and (c) lateral torsional buckling of secondary
beam in assembly S1 analyzed with uniformly distributed loading ........................................... 267
Figure 5.13: Variation of fire induced axial force in connections as a function of fire exposure
time for connection assemblies (a) S1 and (b) S2 with different loading types ......................... 268
Figure 5.14: Schematic of the spar elements used in the analysis .............................................. 269
Figure 5.15: Effect of restraining axial stiffness on the fire response of double angle connection
assemblies (a) S1 and (b) S2 ....................................................................................................... 270
Figure 5.16: Failure modes of connection assembly S1 for different values of axial restraint
stiffness ....................................................................................................................................... 271
Figure 5.17: Variation of (a) axial force and (b) rotation in connection as a function of fire
exposure time for transient and steady-state heating conditions ................................................ 272
Figure 5.18: Variation of (a) axial force and (b) rotation in connection as a function of fire
exposure time for different load ratios ........................................................................................ 273
Figure 5.19: Time-temperature curves for design fire scenarios used in the analysis ................ 274
Figure 5.20: Fire induced (a) axial force and (b) rotation in connection under different design fire
scenarios (see Fig. 5.14).............................................................................................................. 275
Figure 5.21: Progression of axial force in connection as a function of fire exposure time for (a)
A325 and (b) A490 bolt types ..................................................................................................... 276
Figure 5.22: Comparison of connections rotations as a function of fire exposure time for (a)
A325 and (b) A490 bolt types with different high-temperature material properties .................. 277
Figure 6.1: Flowchart summarizing the procedure to evaluate fire resistance of connection .... 278
xiii

Figure 6.2: Comparison of ISO834 and ASTM E119 standard fire curves ................................ 279
Figure 6.3: Schematic of (a) beam elevation and (b) cross-section used for computing
temperatures in fire protected steel sections ............................................................................... 280
Figure 6.4: Comparison of steel temperatures obtained using different methods for (a) W12x30
and (b) W24x76 beam sections with 15mm fire protection ........................................................ 281
Figure 6.5: Schematic of (a) beam elevation and (b) cross-section used for computing
temperatures in unprotected steel sections .................................................................................. 282
Figure 6.6: Comparison of steel temperatures obtained using different methods for (a) W12x30
and (b) W24x76 beam sections without fire protection .............................................................. 283
Figure 6.7: Comparison of steel temperatures obtained using step-by-step method for (a)
W12x30 and (b) W24x76 beam sections subjected to design fire exposure .............................. 284
Figure 6.8: Fire induced axial force development in connection................................................ 285
Figure 6.9: A schematic of beam idealization into segments and discretized cross-section ...... 285
Figure 6.10: Illustration of generalized stress and generalized strains oriented in the direction
corresponding to their positive values ........................................................................................ 286
Figure 6.11: Total strain along the beam cross-section along with different strain components 286
Figure 6.12: Flow chart illustrating the steps associated in the generation of moment-curvatureaxial force curve .......................................................................................................................... 287
Figure 6.13: Iterative solution technique for generating force-deformation curves ................... 288
Figure 6.14: Flowchart showing the steps associated with the analysis of double angle connection
..................................................................................................................................................... 289
Figure 6.15: Moment-curvature curves as a function of fire exposure time for (a) W12x30 and
(b) W24x76 steel section with 15mm of fire insulation (Note: The fire induced axial forces are
shown in the parenthesis) ............................................................................................................ 290
Figure 6.16: Progression of (a) temperature in beam (b) moment-curvature-axial force in the
beam, and (c) rotation at connection for double angle connection ............................................. 291
Figure 6.17: Illustration of the Cardington test (a) building plan and (b) shear tab connection
geometry (Note: all dimensions are in mm) ............................................................................... 293
Figure 6.18: Measured fire (gas) temperatures during Cardington test ...................................... 294
Figure 6.19: Comparison of measured and approximated fire temperature used for analyzing
shear tab connection .................................................................................................................... 294
xiv

Figure 6.20: Comparison of predicted and measured rotation for shear tab connection ............ 295
Figure E.1: Illustration of the steel framed building used in the analysis................................... 296
Figure E.2: Geometrical details of double angle connection ...................................................... 297
Figure E.3: Progression of (a) temperature in beam (b) moment-curvature-axial force in the
beam, and (c) fire-induced axial force in connection for the analyzed connection .................... 298

xv

KEY TO ABBREVIATIONS

[Q]

tangent stiffness matrix

{f}

generalized stress vector

{X}

generalized strain vector

a,n

regression coefficients for fitting the growth phase of fire scenarios

a1,a2,a3,a4

non-dimensional regression coefficients dependent on steel material model

As

cross sectional area of the steel section

c

specific heat

Cet

specific heat matrix

cs,cp

specific heat of steel and insulation, respectively

dA

area of element

E,Es

elastic modulus of steel at ambient temperature

Et

tangent modulus of steel at ambient temperature

Et

temperature dependent elastic modulus

Fen

applied nodal forces

Feth

element thermal load vector

Fp

heated perimeter of the cross-section

Fu,Fy

ultimate and yield tensile strength of steel at ambient temperature, respectively

h

thickness of insulation

hcon , hrad

convective and radiative heat flux, respectively

hf,ho

heat transfer coefficient on fire side and cold side, respectively

I

second moment of area of steel section

k

bending curvature

k

axial restraint stiffness

xvi

k,kp

thermal conductivity of steel and insulation, respectively

Ka,Kr

axial and rotational restraint stiffness, respectively

Ke

element stiffness matrix

Ket

thermal stiffness matrix

L

total length of the beam

LR

non-dimensional load ratio at room temperature

Mmax

maximum bending moment in the beam

Mo

maximum bending moment in the beam due to gravity load only

Mu

unfactored ultimate bending capacity of the section at ambient temperature

Mx

bending moment

My

unfactored yield bending capacity of the section at ambient temperature

ny,nz

components of vector normal to the boundary in the plane of cross-section

P

fire-induced axial force

Pc,max

maximum fire-induced compressive force

Pten,max

maximum catenary tensile force

qb

total heat flux on the boundary

Qe

applied nodal thermal load

s

coefficient used for computing steel temperature

s,t,r

isoparametric locus of a element

Sx

elastic section modulus

t

time

T,Ts

temperature of steel

Tc

temperature in steel at which fire-induced axial force P=0

Te

nodal temperatures

Tf

temperature of fire
xvii

Tlim

limiting temperature of steel

To

temperature of cold side

tp

insulation thickness

Tten,max

temperature in steel at maximum catenary axial force

Ty

yield temperature of steel section

u

vertical displacement of the beam at loading point

U

strain energy

u,v,w

nodal translations in x,y,z directions, respectively

ue

nodal displacements

V

external work

W/D

ratio of weight to heated perimeter of steel section

XA,XR

axial and rotational restraint factor

Zx

plastic section modulus

α

coefficient of thermal expansion

Δt

time increment

ΔT

thermal gradient (or) change in temperature in beam cross-section

ΔTs

temperature rise in steel

ε

strain in an element

εnom,εtrue

nominal and true strain, respectively

εo

axial strain

εr

residual strain in an element

εth,εme

thermal and mechanical strain, respectively

εy,εu

yield and ultimate strain of steel, respectively

θ

rotation at connection

μ

co-efficient of friction
xviii

ρ

density

ρs,ρp

density of steel and insulation, respectively

σ

stress in an element

σnom,σtrue

nominal and true stress, respectively

σy,σu

yield and ultimate strength of steel at ambient temperature

xix

CHAPTER 1
1. INTRODUCTION
1.1 General
Steel is one of the widely used construction material in high-rise buildings due to numerous
advantages such as high strength, stiffness, ductility characteristics and the reduced construction
time, it offers over other materials. However, steel has high thermal conductivity and loses its
strength and stiffness rapidly when subjected to high-temperatures such as those encountered in
building fires. Therefore steel structural members have to be designed for appropriate fire safety
measures when used in buildings. This is because fire represents one of the most severe hazards
to which buildings may be subjected to during their life span.
In a steel framed structure, various members are connected together by structural connections to
facilitate load transfer between them. A wide range of connection configurations such as all
bolted, all welded and welded-bolted are commonly used to provide connections. Different types
of bolted connections such as flexible end plate, flush end plate, shear tab (fin plate) and double
angle connections are available for facilitating connections in steel framed structures. Previous
observations show that end-plate and shear tab connections undergo brittle failure and can
1

sustain rotations upto 3.5° (60 milli-rad) only. On the other hand bolted double angle
connections possess higher ductility, tying capacity, superior rotational characteristics (Liu et al.,
2002; Pakala et al., 2012a; Wang et al., 2011; Yu et al., 2009b, c) and are economical, easy and
fast to erect. The higher ductility possessed by the double angle connections produces ductile
failure and they can undergo rotations up to 18° (≈ 315 milli-rad) before failure. For these
advantages, bolted double angle connections are often preferred over welded or other type of
bolted connections.
Failure of connections can lead to failure of connected structural members which in turn can
trigger progressive failure of the entire structure. Fig.1.1 illustrates the response of lower three
storeys of a multi-storey building. When fire occurs in a compartment, the temperatures increase
steadily in the structural members and the connection starts to lose its capacity and eventually the
connection fails. As a consequence of losing lateral members (beams), the unsupported length of
the column increases to twice the length of original column which in turn decreases the critical
buckling load by four times thus making the column more susceptible to early failure. Once the
column fails, the remaining connected beams and column might fail and might lead to the
progressive collapse of the entire structural system. Observations in previous fire induced
collapse of steel structures indicated that the failure most often gets initiated at the connections.
Since connections play a crucial role in maintaining integrity and stability of the entire structural
framing system, design of these connections to withstand loads under fire conditions is vital to
improving global performance of the structure.
It is well established from previous fire investigations and standard fire tests that unprotected
steel structural members can fail within 20-30 minutes of fire exposure. However, building codes
specify that structural members should be capable of maintaining structural integrity and load

2

bearing capability for duration of 30 minutes to 3 hours, during the fire event. This requirement,
known as fire resistance rating is a measure of quantifying the fire safety aspect of structural
members. Fire resistance is defined as the duration during which a structural member exhibits
adequate resistance with respect to structural integrity, stability and temperature transmission
under standard fire exposure.
In current practice fire resistance of connections is often evaluated through fire tests or numerical
models on connection assemblies. In this approach, the connection assemblies are assumed to be
isolated connections without considering them to be part of the structural framing system and are
subjected to uniform heating (steady-state temperature regime) or standard fire conditions.
Further, in most of the fire tests the connections are scaled down in size (due to size limitations
imposed by the testing furnace). However, in typical building fire scenarios, connections are
integral part of the structural framing system and experiences transient temperature conditions.
Therefore, the current approach of evaluating fire performance of connections do not accurately
capture the realistic behavior as they asses the behavior based on isolated connections subjected
to uniform temperature conditions. An alternative to this is to adopt a system-level approach. In
this approach the true transient connection behavior can be traced by considering the connections
to be an integral part of the entire structural framing system. Connections are then designed
based on the performance of the structural framing system as a whole rather than the results
generated based on isolated connections. Hence, there is a critical need to understand the realistic
fire behavior of connections and thus arrive at design guidelines and code provisions which can
account for all the significant factors affecting the connection fire performance and provide
realistic connection design strategies.

3

1.2 Effect of Temperature on Connection Behavior
At ambient temperature, bolted connections generally experience shear force and flexural tension
or compression force depending upon the configuration of the connection. These forces are
generated in the connection through force transfer that occurs from connected members (beams,
columns etc.). However, when these connections are subjected to fire conditions, connections not
only lose their strength and stiffness resulting from increasing temperature, but also experience
additional forces due to the expansion and contraction of connected steel members, as illustrated
in Fig. 1.2.
When fire occurs in a steel framed building (see Fig. 1.2(a)-(b)), all components of the frame
(beams, columns, connections) experience temperature rise (Fig. 1.2(c)). During initial stages of
fire, temperatures in beam increases (Fig. 1.2(c)) and this leads to expansion of the beam due to
significant thermal expansion of steel. The adjoining (cold) structural members, in the frame,
restrain the free thermal expansion of beam which leads to the development of axial compressive
forces as shown in Fig. 1.2(d). The axial force increases with time until the beam undergoes
initial yielding after which the restraint compressive force decreases in magnitude and finally
transforms to tensile force. This transformation in axial force can be attributed to deteriorating
strength and stiffness of beam with increasing temperatures which results in permanent
deformation of beam. At this stage, the beam is able to carry very little load through flexural
mechanism and is held in place by connections like a cable where load transfer happens through
catenary action. Finally, during the cooling phase the beam regains part of its initial strength and
undergoes thermal shrinkage leading to continuous increases in the tensile force until connection
fails.

4

In summary, the fire behavior of connections is different from that at ambient temperature on
two fronts namely, (a) the connection experience additional axial force from beams due to the
restraint imposed by the adjacent (cold) structural members on thermal expansion of heated
members, and (b) the nature and magnitude of the axial force can change with fire exposure time
(Bailey et al., 1996; Liu et al., 2002).
Further, the type of fire scenario significantly influences the fire induced axial forces in
connections. The fire exposure typically experienced in buildings comprises of a growth phase
that is followed by a decay phase. Fig. 1.3 illustrates different time-temperature curves
representing standard fire and realistic (design) fire scenarios. It can be seen from the figure that
in the case of design fire, temperature in building increases initially after which the compartment
enters cooling phase once the total available fuel is consumed and/or due to limited oxygen
supply. However the temperatures in a standard fire, typically used in standard fire resistance
tests, increases continuously with time without any decay phase. As mentioned earlier the
presence of decay phase and specifically the rate of decay significantly influence the rate at
which the beam recovers its strength and stiffness which in turn governs the magnitude and
transformation time of fire-induced axial forces from compression to tension. Hence, the
presence of decay phase plays a critical role in determining the failure time of connections as
well as the entire structural framing system. Therefore, the type of fire exposure (design vs.
standard) and the rate of decay have significant influence on the fire response of connections and
should be properly accounted for in modeling the connections.
The unique and superior performance characteristics of bolted double angle connections can be
better illustrated by comparing its rotation characteristics to that of other connection types, as
illustrated in Fig. 1.4. The details of two types of connections namely flush end-plate and double

5

angle connection along with their moment-rotation (M-ϕ) curves are shown in Fig. 1.4. A quick
glance of the ambient temperature M-ϕ response of flush end-plate connection indicates that the
response is characterized by three regions (Al-Jabri et al., 2005). Initially there is an
approximately linear response with increasing rotation, until the onset of yielding in one or more
of the connection components (such as bolts or angles). This is followed by curvilinear response
indicating the yielding of the connection. Finally, as the connection failure becomes impending,
the rate of rotation increases rapidly causing an almost flat plateau in the connection response.
However, the room temperature M-ϕ response of double angle connection consists of two stages,
before and after the beam bottom flange comes into contact with the column flange. The first
stage is again characterized by linear and bilinear response similar to that of flush end-plate
connections. During the second stage, when beam bottom flange makes contact with the column
flange, an increase in moment is accompanied by a smaller increase in rotation due to the
resistance offered by the column flange to free rotation of the beam flange. Finally, an increase
in moment is accompanied by increased rotation until the connection fails.
When connections are exposed to fire, they lose their strength and stiffness with increasing
temperature. Therefore, as the temperature increases, there is degradation of M-ϕ response (see
Fig. 1.4(c)-(d)) as the moment-carrying capacity of connection decreases and the resistance to
connection rotation decreases. Previous studies indicate that double angle connections can
undergo rotations up to 18° (≈ 315 milli-rad) before failure while other connection
configurations can sustain only 3.5° (60 milli-rad rotations) (Yu et al., 2009c). The higher
ductility possessed by the double angle connections produces ductile failure, rather than brittle
failure as experienced in other connection types, which enhances connection failure time.

6

1.3 Research Objectives
The above illustrations clearly indicate that temperature has a significant effect on the behavior
of double angle connections that are an integral part of structural frame. There is considerable
lack of understanding on the transient response of double angle connections under realistic fire,
loading and restraint conditions. Furthermore, the system-level response of double angle
connections is completely different from that of isolated connection behavior. To address some
of these knowledge gaps, it is proposed to undertake a comprehensive study on the transient
structural response of double angle connections under fire conditions with the ultimate objective
of developing a rational methodology for fire design of connections. The specific objectives of
this research are:


Carry out a detailed state-of-the-art review on the fire behavior of different connections
and identify knowledge gaps. The comprehensive review will cover both experimental
and numerical studies as well as current provisions in codes and standards.



Model the system level transient thermal and structural response of double angle
connections under realistic fire, loading and boundary conditions using the commercially
available finite element programs. The models for thermal and structural analysis will
account for high temperature properties of materials, geometric and material
nonlinearities as well as nonlinear contact interactions.



Carryout fire resistance experiments on assemblies with double angle connections to
develop needed data for validating numerical models.



Validate the computational models by comparing response predictions from the model
with test data obtained from fire resistance experiments on double angle connections.

7



Carry out parametric studies to quantify the effect of various factors on the performance
of double angle connections under realistic fire, loading and restraint conditions.



Utilize data from fire tests and parametric studies to develop a simplified design approach
for fire design of double angle connections.

1.4 Scope
The research presented as part of this dissertation is arranged in seven chapters. Chapter 1
outlines general background to fire response of double angle connections and presents key
objectives. Chapter 2 provides a detailed state-of-the-art review on the behavior of double angle
connections subjected to fire conditions. The review includes summary of experimental and
analytical studies, as well as fire design provisions in current codes and standards.
Chapter 3 deals with fire resistance experiments conducted on two full-scale double angle
connection assemblies under realistic fire, loading and restraint conditions. Results from fire tests
are used to discuss the response of double angle connections under realistic conditions. Chapter 4
presents details about the finite element model developed for tracing transient fire response of
steel double angle connections. The validation of the finite element models (thermal and
structural) is also presented in Chapter 4, where response parameters from the model are
compared with test data.
Results from parametric studies are presented in Chapter 5. Different parameters governing the
transient fire response of double angle connections are described along with a discussion of the
results from the parametric studies. Guidelines for the fire design of steel double angle
connections are presented in Chapter 6. Results from the parametric studies are applied to verify
the proposed approach for evaluating the fire resistance of double angle connections. Finally,

8

Chapter 7 summarizes the main findings arising from the current study and recommendations for
further research.

9

CHAPTER 2
2. STATE OF THE ART REVIEW
2.1 General
Bolted double angle connections are often used in steel-framed buildings due to their higher
tying resistance and rotational capacity. These connections play a crucial role in transferring
forces between members (beam to beam, girder or columns) in a structural framing system. The
behavior of these connections can be much different under fire conditions due to development of
large fire induced forces in structural members. Current provisions in codes and standards do not
specifically account for these fire induced forces, and therefore, double angle connections are
designed as simple shear connections. Further, the current approach for evaluating fire resistance
of connections is based on a prescriptive methodology where fire performance is evaluated based
on isolated connection response rather than considering overall response of the structural framing
system with embedded connections. While most of these studies aimed at developing the
moment-rotation characteristics of end plate connections, very limited studies focused on

10

performance of double angle connections under fire conditions. This section provides a state-ofthe-art review on experimental and numerical studies related to fire performance of connections.
Also, a review of fire resistance provisions in various codes and standards are provided.
2.2 Experimental Studies
The common approach for evaluating fire resistance of RC columns is through fire tests under
standard fire conditions. A number of fire resistance tests were conducted in the last two decades
to study the fire behavior of connections. The objectives, test methods, features and major
conclusions reached in some of the notable studies are summarized below.
The main objectives of most of the fire resistance experiments included:


Observing and monitoring the connection behavior under a standard fire exposure



Generating fire resistance ratings of connections with specific configuration and load
intensity



Generating data over a range of variables for verification of computer based models, and



Generating analytical, component based models and developing moment-rotation
relationships

Lawson (Lawson, 1990) conducted tests on different types of beam-to-column connection, by
subjecting them to ISO834 standard temperature-time curve, to quantify the moment capacity of
the connections at elevated temperature. A total of eight tests were conducted that included five
non-composite beams, two composite beams and one on a shelf angle floor beam. Three different
connection types namely extended endplate; flush endplate and double angle (web cleat) were
used. Beams and columns in four tests were fire protected for 1 hour by spraying with 20 mm
vermiculite-cement spray. A summary of the experimental program is presented in Table. 2.1.
Results from these tests indicated that the steel connections possessed significant strength at
elevated temperatures and were able to sustain large moments (two-thirds of ambient
temperature moment capacity) in fire. Lawson concluded that (a) the temperatures experienced
11

by bolts were always less than that in connecting beam indicating that bolt failure is unlikely to
happen before beam fails (b) bolt temperatures were significantly lower than the beam lower
flange, thus enhancing the performance of connections (c) mesh reinforcement present in
composite beams contributed to increasing moment capacity of the connections (d) moment
transferred through the connections helped in reducing the effective load ratio of simplysupported beams and thus increasing the limiting temperature of beams.
Liu et al. (Liu et al., 2002) experimentally studied the fire behavior of two types of beam-tocolumn connections namely, double angle (double web-cleats) and flush end-plate. The
connections were part of a sub-frame assembly and were fire-protected using 50 mm thick
ceramic fiber blanket. A schematic of the tested sub-frame assembly is shown in Fig. 2.1. The
authors conducted 20 fire tests, with two types of connections, by exposing the sub-frame
assembly to ISO834 standard fire temperature curves and varying load levels on the beam (a load
ratio of 0.3, 0.5 and 0.7). The behavior of the connection was assessed by measuring the moment
and force resisted and transmitted by the connection as well as the rotation of connections. The
authors observed that (a) the double angle connection resisted a minimum amount of moment
(15% of the beam moment capacity) despite being regarded as a pinned joint (b) the critical
temperatures of the beam with double angle connections were 20°C lower than those with endplate connections (c) the bolt holes in the beam web were elongated due to large bearing forces
resisted by double angle connections. The authors concluded that connections can improve the
fire resistance of beam by reducing the mid-span moment, during the heating phase of the fire. In
comparison to double angle connection, end-plate connections have larger moment transfer
capacity and higher catenary action which reduces the beam deflection.

12

Leston-Jones et al. (Leston-Jones et al., 1997) tested flush-end plate joints at elevated
temperatures in order to develop moment-rotation relationships. Results from the tests showed
that the strength and moment capacity of the connections decrease with increasing temperature
with a significant capacity reduction in 500-600°C temperature range. Al-Jabri et al.(Al-Jabri et
al., 1998) conducted experimental studies to investigate the degradation of steel and composite
connection characteristics at elevated temperature. As part of the experimental study, tests were
conducted on full end-plate, flexible end-plate bare steel connections and flexible end-plate
composite connections. The authors inferred that the failure modes of the connections at elevated
temperatures are similar to those at ambient temperature. Based on the tests, the authors
concluded that the presence of concrete slab in composite connections enhanced the connection
performance at elevated temperature by acting as a heat sink to the top of the beam. Using the
data generated in the elevated temperature tests conducted at constant load level, the authors
interpolated moment-rotation-temperature curves for different connection types (Al-Jabri et al.,
2005).
Wald et al. (Wald et al., 2006) undertook an experimental program to study the global structural
response of a fire exposed compartment in 8-storey steel-concrete composite frame building at
BRE’s Cardington test facility. The experimental program aimed at examining temperature
distribution within structural elements, distribution of internal forces, behavior of slab, beam,
columns and connections. . The structure was of 33 m height with five bays wide and three bays
deep. The steel frame consisted of two types of connections with M20 Grade 8.8 bolts namely,
flexible end plates for beam-to-column connection and fin plates for beam-to-beam connection.
Both the connections were fire protected with 15 mm of Cafco300 vermiculite-cement spray. A
total of seven tests were conducted and details about the natural fire used, thermal, structural

13

behavior of the steel frame is provided by the authors. The authors observed fracture in the endplate connections (along the welds) and attributed this to the large rotations, tensile forces
developed in the beam while it was cooling. No fracture was observed in the shear tab (fin plate)
connections though the bolt holes in the beam web underwent significant elongation and this was
due to the fact that the shear tab (10 mm) was thicker than the beam web (6 mm). Based on these
observations, the authors concluded that the elongation of the holes in the beam web leads to
increased connection flexibility thus allowing larger deformations without fracture.
Wang et al. (Wang et al., 2007) studied the capacity, failure characteristics and failure modes of
extended end-plate connections under fire exposure. In addition, the effect of rib stiffeners and
depth of end-plates on fire-resistance capacity was studied by comparing the connection capacity
with and without rib stiffeners. The authors developed a spring-component model and
established its validity by comparing the results obtained from the model with the experimental
results. Based on the test results, it was concluded that the extended end-plate joint has higher
rotation capacity at elevated temperatures. The rib stiffener and the thickness of end-plate have
significant and moderate influence, respectively, on the critical temperature of the extended endplate joints.
Yu et al.(Yu et al., 2009a) undertook experimental studies to characterize the behavior of fin
plate, flush endplates, flexible end plate, and web cleats (angle) connections in fire. The aim of
the study was to investigate the capacity and ductility of steel connections at elevated
temperatures, especially when the beams are in catenary phase. The shear tab connections were
made of different rows (1 row of 3 bolts, 2 rows of 3 bolts each), grade (8.8, 10.9) and diameter
(M20, M24) of bolts. The authors conducted 14 tests at four different temperatures (20°C,
450°C, 550°C, 650°C) and two loading angles (35°, 55°). A summary of the test program is

14

presented in Table 2.2. All the tests were steady-state tests in which the specimen was heated to a
target temperature and then the load applied until the specimens fractured. Experimental data
indicated that the resistance of fin plate connections are significantly affected by temperature rise
and bolt shear fracture tends to govern the failure of fin plate connections at elevated
temperatures. For the web cleat (angle) connections, the authors observed that these connections
have excellent rotational ductility, and resistance of these connections decrease rapidly with
increasing temperature.
Yu et al.(Yu et al., 2009b) also studied the high temperature rotational capacity, tying resistance
of double angle (web cleat) connections by subjecting them to a combination of shear and tying
forces. A schematic of the connection configuration is shown in Fig. 2.2. The connections were
heated to a uniform target temperature and then the beam was loaded (at different angles) away
from the connection region until the connection failed. A total of 14 steady-state temperature
tests were conducted with one and two rows of three bolts each, different loading angles and
temperatures. A summary of the test program along with the peak force and rotations
experienced by the connection assemblies are presented in Table 2.3. Based on these tests, the
authors concluded that for double angle connections the (a) tying capacities decreased rapidly
with temperature, with the connection having little resistance after reaching 650°C; (b) failure
mode is dependent mainly on temperature but not on the applied load combinations (shear and
tying forces); (c) fracture of double angle close to its heel and double shear bolts through beam
web are the two critical failure modes at elevated temperatures; and (d) rotational capacity that
develop in double angle connections is very high as compared to other connection types.
Daryan and Yahyai (Daryan and Yahyai, 2009) studied the response of bolted top-seat angle
beam-to-column connections by conducting 12 full-scale fire tests with two different connection

15

configurations (with and without web angle). The connection details of the specimens are
illustrated in Fig. 2.3. These experiments were aimed at studying the effects of temperature,
failure mode, bolt grade, angle thickness, web angle and load level on the fire behavior of
connections. Details of the specimens along with the applied load are shown in Table 2.4. In all
the tests, the specimens were subjected to a predetermined load level and then exposed to ISO
834 or ASTM E119 standard fire temperature. The applied loading (moment in this case) on the
connections was selected in such a way that the premature failure of connection would not occur
and the connections can sustain the effect of elevated temperatures. As the objective of the study
was to evaluate the behavior of connections, the column and beam of all the specimens were
wrapped with a 2.5 cm thick ceramic fiber blanket. The authors concluded that (a) increasing the
stiffness of connection does not improve temperature resistance (i.e., the temperature at which
failure happens) because the premature tension failure of bolts prevents the utilization of full
capacity of connection and web angle (b) the premature tension failure of bolts can be prevented
by using nuts of a higher strength (grade) or by using two or three nuts of less strength (c) the
onset of connection plastic behavior occurs between 500 and 650°C which is within the limiting
temperature for steel beams (d) high temperature strength of connections can be improved by
using temperature-resistant bolts, increasing the thickness of angles and decreasing the applied
moment on connections (e) bolts play a crucial role in determining the connection capacity at
elevated temperature and improper bolt behavior hinders the utilization of the full capacity of
other components. Based on the limited study, the authors concluded that the connections failed
at the same temperature at which beams are assumed to fail and hence the connections are not a
weak link in a braced frame.

16

Wang et al. (Wang et al., 2011) carried out an experimental study to evaluate the robustness of
different types of connection in restrained steel frames. The study was aimed at evaluating the
effect of varying axial restraint (provided by columns) on the connected beam and the connection
types on the overall response of the steel frame. The authors conducted fire tests with two
column sections and five connection types namely fin plate (shear tab), web cleat (double angle),
flush endplate, flexible endplate and extended endplate. Details of tests, beam, column and
connection dimensions are reproduced in Table 2.5. Due to the limitations of the furnace, the
sub-frame assembly did not have a concrete slab but the top flange of the beam was wrapped
with 15 mm thick ceramic fiber blanket to simulate the heat sink effect of the slab. During the
test, two point loads each of 40 kN (corresponding to a load ratio of 0.5) was applied on the
beam at a distance of 600 mm from the ends and then it was exposed to ISO 834 standard fire.
The columns were restrained at the ends and were free to move in longitudinal direction only
thereby enabling to study the effects of axial restraint on the beam and the connection. The
behavior of the connections was evaluated by beam mid-span deflection, failure modes and axial
forces in the connection. Based on the observations the authors inferred that the connections
experience different failure modes and the failure in connections happens only when the beam
enters the catenary action phase. The primary conclusion from this study is that the double angle
connection exhibited the highest performance, as compared to other types of connections, in
allowing the connected beam to develop catenary action without connection failure.
Al-Jabri et al. (Al-Jabri et al., 2005) carried out an experimental study to develop momentrotation-temperature curves for semi-rigid connections containing flexible and flush end-plate.
A total of twenty tests were conducted with five different connection details that included
composite and bare steel connections. Details about the beams, columns, connections used in the

17

tests are summarized in Table 2.6. All the connections were subjected to linearly increasing
temperature with a maximum temperature of 900°C achieved in 90 minutes. The applied loading
on the beam was varied between 20-80% of the moment capacity of the connection. The thermal
response was measured using thermocouples while the connection rotation was measured by
clinometers and displacement transducers. The authors observed uniform temperature
distribution across the depth in all the bare-steel connections. The presence of concrete slab in
composite connections created a non-uniform temperature profile with 19-30% reduction in the
average temperatures, compared to bare-steel connections. The author used the experimentally
observed connection rotation and derived empirical equations (by curve fitting method) for
predicting the moment-rotation of connections at different temperatures.
Yang et al. (Yang et al., 2009) conducted an experimental study to evaluate the fire performance
of welded flange-bolted web type moment connections. The program aimed at studying the
effect of fire-proofing, testing conditions and loading on the behavior of connections. A
summary of the test specimen dimensions and test parameters are presented in Table 2.7. The
authors concluded that, for welded flange-bolted web type moment connections under elevated
temperature (a) the connections exhibited ductile behavior, with necking and tearing at top flange
and local buckling at bottom flange (b) for temperatures beyond 500°C, deterioration of stiffness
is more significant than that of strength (c) by proper design of fire-proofing material, stability of
the connection can be maintained without loss of strength and stiffness.
Al-Jabri et al.(Al-Jabri et al., 1998) conducted experimental studies to investigate the
degradation of steel and composite connection characteristics at elevated temperature. A total of
five tests were conducted with two full end-plate, one flexible end-plate bare steel connections
and two flexible end-plate composite connections. Using the results obtained from the elevated

18

temperature tests conducted at constant load level, the authors interpolated moment-rotationtemperature curves for different connection types (Al-Jabri et al., 2005). The authors observed
that the failure modes of the connections at elevated temperatures are similar to those at ambient
temperature. Also, the authors concluded that the presence of concrete slab in composite
connections enhanced the connection performance at elevated temperature by acting as a heat
sink to the top of the beam.
The above review illustrates that there have been a number of experimental studies to
characterize the fire behavior of connections. In these fire tests, the effect of structural
continuity, fire protection, composite action, member characteristics and the connection
configuration, on the fire response of steel connections were studied. However, most of the tests
were limited to isolated and scaled connection specimens subjected to standard fire exposure and
uniform temperature (steady-state) field. Further these tests focused on end-plate connections
and developed moment-rotation relationships. Very few experimental studies focused on shear
tab (Yu et al., 2009a) and double angle connections (Yu et al., 2009b) or on the system-level
response (Wald et al., 2006) of the overall structural system. Thus, there is limited information
on the behavior of double angle connections under realistic fire, loading and restraint conditions.
2.3 Numerical Studies
A review of literature indicates that a large number of numerical studies have been carried out on
the fire behavior of connections. Initial modeling studies on the fire behavior of connections
were carried out by Liu (Liu, 1996) who developed a three dimensional model to simulate the
response of steel framed structures in fire. The model is based on tangential stiffness approach
and incorporated the material plasticity, non-uniform thermal expansion, large deformation and
degradation of the material properties at elevated temperatures. The model uses isoparametric

19

shell elements and carries out the analysis in different time steps using an iterative procedure.
Each iteration of the analysis starts by increasing the temperature which is followed by
computing the relevant stress-strain relationship and subsequently the stiffness matrix and the
thermal load vector. For a given load, using the updated stiffness and thermal load vector the
increment in the displacement at each time step is iteratively computed until the displacement
norm is within the allowable tolerance limit. Details about the derivation of the forcedisplacement equations, stress-strain curves and the nonlinear iterative procedure are described
by the author. Using the validated numerical model, Liu analyzed beams and beam-to-column
connection with different loading and structural continuity conditions. Based on the results, Liu
concluded that the fire resistance of the beams can be substantially enhanced by accounting for
continuity of the structural frame. In addition, the top tension bolts in connections were less
affected by loading and structural continuity and thus they enhance connection’s fire resistance.
Liu (Liu, 1998) conducted a study to evaluate the effect of connection flexibility on fire
resistance of steel beams. The study was aimed to quantify the beneficial effects of the endplate
type connections in terms of the reduction of effective load ratio in the beam. Liu analyzed
different types of connections, isolated beams with and without moment continuity, isolated
connection behavior and beams with endplate type connections. Based on the analyses, Liu
inferred that the connection details such as bolt size, endplate thickness does not substantially
affect the overall performance of the beam in fire.
Mao et al.(Mao et al., 2009) studied the fire response of steel semi-rigid beam-column moment
connections using ANSYS. The geometry of the specimen was discretized using 8 noded Solid70
and Solid185 elements for thermal and structural analysis, respectively. The authors performed a
sequentially coupled thermal-structural analysis of the connection. The FE model did not

20

consider the effects of residual stress in the material, welding and bolting material on structure
while the elevated temperature nonlinear material behavior is assumed to follow Eurocode 3
provisions. The numerical model was validated by comparing the results obtained from the
numerical model with those measured during full-scale fire tests reported by Ho et al.(Ho et al.,
2007). The validated model was applied to study the influence of different parameters. Results
obtained from these studies showed that the applied moment level have significant effect on the
stiffness of steel moment connection while the axial load of column, shear and axial force of
beam have less effect.
El-Rimawi et. al. (El-Rimawi et al., 1997) carried out an analytical study to examine the
influence of connection stiffness on the behavior of steel beams in fire. The authors proposed a
method for extending the ambient temperature connection characteristics to higher temperatures.
The analytical procedure is based on a secant stiffness approach in which each element of the
structure is divided into a number of line elements with three degrees of freedom at each node.
The authors validated the analytical model and conducted a parametric study to characterize the
influence of connection temperature, beam span and depth on the beam behavior. Based on the
results, the authors observed that the exact representation of the connection and its temperature
relative to the beam are not critical in determining failure and the proportions of beam span and
cross-section do not appear to have significant influence.
Liu (Liu, 1998) used a finite element model to study the effect of simple bolted connection
flexibility on the fire resistance of steel beams. The aim of the study was to quantify the
beneficial effect of the end plate type connections. The author presented a series of analyses on
isolated beams with or without moment continuity, isolated connection behavior and beams with
end plate type connections. The FE model, previously validated by the authors (Liu, 1996),

21

considered the effect of nonlinear geometric properties and stress-strain-temperature
relationships. Based on the analyses, the authors observed that the connection details such as bolt
size, endplate thickness does not substantially affect the overall performance of the beam in fire.
However, the authors expressed a word of caution when using the conclusions as the current
study ignored the axial restrains due to the column.
Pakala et al. (Pakala et al., 2012a) studied the fire performance of bolted double angle
connections using ANSYS. The model accounted for material and geometric nonlinearities, high
temperature properties of steel and nonlinear contact interactions. The double angle connections
were exposed to uniform temperature and then loaded until the failure. The steady-state hightemperature stress-strain relations used in the model are shown in Fig. 2.4. The FE model was
validated by comparing the predicting from the model with published test data. The failure mode
obtained from the numerical model and that observed in fire tests is shown in Fig. 2.5. The
validated model was used to study the influence of critical factors on the fire performance of
bolted double angle connections. Based on the results from parametric studies, the authors
concluded that bolt-hole size, edge-distance, thermal gradient and the slenderness of beam web
significantly influence the fire behavior of double angle connections.
Yang et. al. (Yang et al., 2009) conducted a finite element analysis to evaluate the performance
of welded flange-bolted web type moment connections under fire load. The authors modeled four
full-size steel beam-to-column specimens, with and without fire-proofing materials. The FE
model accounted for the material and geometric nonlinearities, degradation of material properties
with temperature and examined the load-displacement behavior. The FE model was validated by
comparing the results from the model with those obtained by the experiments conducted by Yang
et. al (Yang et al., 2009) and found a good correlation. The authors used the validated FE model

22

to conducted parametric studies to examine the strength reduction characteristics of unprotected
steel at elevated temperature. Based on FE analysis results, the authors observed that beam-tocolumn connections were able to retain their design strength up to 650 °C while the stiffness
dropped to 25% of the ambient temperature value. The authors also observed ductile behavior in
connections, with necking and tearing at top flange and local buckling at bottom flange. The
study concluded that the integrity and stability of the steel connections can be ensured by
providing proper fire-proofing materials.
Liu (Liu, 1999) used FE program FEAST to study the fire behavior of unprotected steel beams
and columns connected through bolted end-plate (extended and flush) connections. The author
examined the effect of bolt size, number of bolts and end-plate thickness on the behavior of endplate type connections. Based on the results obtained from FE simulations, the author concluded
that for a similar load ratio, beams with flush end-plate connections had similar fire resistance to
that of the simply-supported. On the other hand, beams with extended end-plate connections had
higher fire resistance than that of simply-supported beams. Also, the authors observed that only a
maximum of two-thirds of the ambient temperature moment capacity can help in increasing the
load capacity of the beam under fire conditions.
Garlock and Selamet (Garlock and Selamet, 2010) developed an ABAQUS FE model, of beamto-girder floor subassembly containing shear tab (single plate) connection, based on the full-scale
test performed at Cardington in 2003 (Wald et al., 2006). The effect of slab is represented with
linear springs using connector elements, which have temperature dependent stiffness. The
authors conducted an uncoupled thermo-mechanical analysis. The authors analyzed the floor
subassembly with zero, constant, temperature dependent spring stiffness’s and showed the effect
of varying composite action on the structural response. The validated FE model was used to

23

study the effects of fire characteristics such as the fast and slow fire growth curves, rate of
heating and cooling on the connections. Based on the FE results, the authors observed that the
beam web and flange buckled in the heating phase due to the development of large compressive
forces. Also, slow heating rate produces higher axial compressive forces in the connections due
to thermal elongation induced by more uniform temperatures in the section. The authors
concluded that the heating and cooling rates affect the beam stress distribution, maximum
temperatures and displacements but not the maximum beam axial force. Further, large tensile
forces develop in the connections during the cooling phase of fire.
Selamet and Garlock (Selamet and Garlock, 2010b) extended previously validated model
(Garlock and Selamet, 2010) of the floor subassembly to study the effect of modifying
connection details for improved fire performance of shear tab (single plate) connections. In the
study, the authors considered the effect of varying bolt grade, bolt hole type, doubler plate of the
beam, thickness of connection plate, bolt pretensioning, gap distance between beam and support,
bolt hole edge distance. Based on results from analysis, the authors concluded that the behavior
of single plate connections can be improved by (a) adding a doubler plate to the beam web, (b)
using a larger distance from the bolt-hole centerline to the beam end (c) increasing the gap
distance between the end of the beam to the connection member. In addition, the authors found
that larger bolt holes can improve the fire performance by imposing less axial restraint and
allowing the beam to move freely under fire-imposed thermal loads.
Pakala et al. (Pakala and Kodur, 2013) developed a three-dimensional finite element model using
ANSYS to assess the system-level transient fire response of double angle connections. The
geometry of the connection assembly along with the connection details are shown in Fig. 2.6
while the stresses in the bolts predicted by the model are illustrated in Fig. 2.7. The authors

24

analyzed the behavior of double angle connections under two different scenarios namely (a) the
connection assembly is assumed to be part of a structural framing system and is subjected to
uniform temperature (transient vs. steady-state heating) (b) the connection assembly is assumed
to be isolated and is subject to a design fire scenario (system-level vs. isolated behavior) as
illustrated in Fig 2.8. They observed that the behavior of double angle connections as part of a
structural system is completely different from that of an isolated connection. The authors used
the validated model to study the effect of system-level interactions, load level, non-standard fire
scenarios and high-temperature properties of bolts on the fire performance of double angle
connections. Based on the results from numerical simulation, the authors concluded that the
mechanical properties of high-strength bolts have a significant effect on the fire-induced axial
forces in connections as shown in Fig 2.9. Also, the magnitude of fire-induced compressive axial
force in connections depends on the fire scenario only and is independent of the loading level on
the connected beams.
Dai et al. (Dai et al., 2010) used general purpose finite element program ABAQUS to model the
fire behavior of ten restrained steel beam-column assemblies with five different connections
namely: fin plate (shear tab), flexible end plate, flush endplate, web cleat (double angle) and
extended endplate. Beam, column details along with connection dimensions are presented in
Table 2.5. The authors used true stress-strain relationships in conjunction with Eurocode 3
strength reduction factors to account for elevated temperature effects. The authors used artificial
viscous damping, which is defined using dissipated energy fraction, to counter the effects of
temporary structural instabilities caused by localized buckling and very large deformations. The
FE model of the connections was validated by comparing the predicted beam deflection, axial
forces, connection failure modes with those obtained from the tests and found a good correlation

25

between the two. The authors concluded that the FE model was able to predict the fire response
of all the five connection types with reasonable accuracy and can be used to conduct parametric
studies.
The above review illustrates that there have been a number of numerical studies on the fire
behavior of connections. These studies developed connection models, using finite element
computer programs such as ANSYS or ABAQUS, and studied the effect of connection
flexibility, moment rotation stiffness, moment continuity, catenary tension forces and fire
protection on the fire behavior of connections. These finite element models accounted for
material nonlinearity, friction and high temperature stress-strain relations. Most of these studies
focused on end-plate connections while only limited studies focused on bolted angle and simple
shear tab connections. Studies on double angle connections concluded that double angle
connections have higher tying resistance, rotational capacity and superior ductility in comparison
to other connection configurations.
2.4 High Temperature Bolt Properties
The fire performance of double angle connection depends on the type of material properties
used. The type of material properties is directly affected by the type of steel used to manufacture
connection components such as bolts. The types of steel used in construction can be broadly
classified as normal strength (mild steel) and high strength steel. These two types of steel have
different properties because of the differences in carbon content, presence of alloying elements
and type of alloying elements (Kodur et al., 2012). In addition, the type of heat treatment process
has significant influence on the strength properties. For example, normal strength steels are
produced by annealing and normalizing while high-strength steels are produced by quenching
and tempering process (Kodur et al., 2012).

26

Bolts are an integral part of a connection system and knowledge of high-temperature thermal and
mechanical properties of bolts is critical for evaluating fire resistance using numerical models. In
addition, high strength bolts are commonly used in construction to improve the overall
connection efficiency. However, due to lack of experimental data, high-temperature constitutive
material modes developed for mild steel are used by researchers in modeling the hightemperature behavior of bolts made with high-strength bolt steel. A review of carbon steel
properties can be found elsewhere (Kodur et al., 2010) . This section provides a review of the
studies involved in evaluating high temperature bolts properties.
Kirby (Kirby, 1995) studied the tension, double shear and residual strength behavior of highstrength Grade 8.8 M20 bolts in the temperature range of 20-800°C. Kirby used three sets of
bolts and two sets of nut with different lengths (overall, thread length) and manufactured through
different heat treatment processes. In all the tests, the bolts were heated to the desired
temperature at a specific rate and then allowed to stabilize before the loading was applied. Two
different heating rates of 5-10°C/min and 2-2.5°C/min and stabilization periods (15 min and 60
min) were used to assess the effect of heating rate on the strength reduction. All the tests were
displacement controlled with a nominal strain rate of 0.001-0.003/min in the elastic and plastic
regions. The residual strength of the bolts was evaluated by determining the hardness of a section
of bolt shank after the bolts were heated to elevated temperature. The author observed that for
bolts in tension and double shear (a) the ultimate capacity reduced drastically between 300700°C (b) the slow heating rate and increased stabilization period had little influence on ultimate
capacity (c) beyond 300°C, different processing conditions used in bolt manufacturing had little
influence on their ultimate capacity (d) premature failure of bolts in tension by thread stripping
depends on the interaction of threads between bolt and nut (e) the residual strength test data

27

indicates that further softening of bolts occur when the fire temperature exceeds the tempering
temperature used in its manufacturing process. Based on the test results and observations, the
author proposed strength reduction factors that can be used to predict the variation of ultimate
capacity of bolts (in shear and tension) with increasing temperature.
Yu (Yu, 2006; Yu and Frank, 2009) undertook an experimental program to evaluate the strength
and stiffness reduction of A325 and A490 bolts at elevated temperatures (20°C – 800°C). All the
tests were done by subjecting the bolts to a uniform constant temperature and then loading them
until failure occurred. Based on the test results, Yu observed that both A325 and A490 bolts
experience marginal strength and stiffness loss up to 300°C. However, the loss increases
significantly in the temperature range of 300-800°C. At 800°C, A325 bolts retained only 8% of
their initial capacity while A490 bolts retained 10% of their initial capacity. The authors also
conducted residual (post-fire) strength tests on both the bolt types and observed that both A325
and A490 bolts lose strength when heated above tempering temperature used in the heat
treatment of bolts. Residual strength tests conducted after exposing both bolt type to 800°C
indicated that A325 bolts experienced a maximum residual strength loss of 45% compared to
40% experienced by A490 bolts. In addition, test results indicated that the duration of exposure
has negligible effect while the cooling rate has no effect on the residual strength of A325 bolts.
Based on the test data, authors proposed high temperature shear strength reduction factors for
A325 and A490 bolts as illustrated in Fig. 2.10.
Lange et al. (Lange and Gonzalez, 2012) conducted experiments to study the elevated
temperature tension behavior of Grade 10.9 bolts in the temperature range of 20°C-700°C.
Tensile test specimens (coupons) were made from high strength grade 10.9 bolts with a diameter
of 16 mm and had a gauge length of 30 mm. The specimens were tested under a strain rate of

28

0.001/min till the strain in specimen was 2% (in order to compute proof stress) and then the
strain rate was increased to 0.025/min and maintained until failure. Based on the test results the
authors observed that the strength reduction of Grade 10.9 bolts was negligible till 300°C and
they undergo a drastic strength reduction between 300-700°C.
Kodur et al.(Kodur et al., 2012) carried out an experimental program aimed at evaluating the
high-temperature thermal and mechanical properties of Grade A325 and A490 bolts. Thermal
conductivity and specific heat of A325 and A490 bolts were measured in the temperature range
of 20-735°C while the thermal expansion was measured in the range of 20-1000°C in both
heating and cooling phases of fire. Single shear and tension tests were carried out under steadystate conditions in the temperature range of 20-800°C. The authors observed that (a) temperature
has significant influence on the thermal and mechanical properties of high-strength bolt steel (b)
the amount of carbon content influence the thermal properties of bolt steel (c) strength properties
of A325 and A490 bolt steel degrades faster than those of conventional steel (d) A490 bolt steel
exhibits slightly higher strength and stiffness properties than A325 steel in the temperature range
of 20-800°C. Based on the test results, the authors proposed high-temperature property
relationships for modeling thermo mechanical properties of bolt steel in numerical models. A
comparison of the proposed high-temperature bolt strength reduction properties along with
conventional steel strength reduction factors is shown in Fig. 2.11.
2.5 Codes of Practice
In most countries, specifications for fire resistant design of connections are included in steel
design standards. The current fire design provisions in most codes and standards are derived
from limited number of standard fire tests conducted on scaled and isolated connection
specimens.

29

In the US, specifications for the structural design of angle connections (simple shear
connections) under ambient conditions are given in the AISC steel construction manual (AISC,
2011). According to the manual, under ambient temperature conditions, simple connections
should be designed as flexible and are permitted to be proportioned for the reaction shear only.
The effect of temperature on the performance of connections is accounted indirectly through
strength reduction factors. The Appendix provides reduction factors for strength and modulus of
structural steel at high temperature. However, no specific procedures are specified for fire design
of connections.
Similarly, Eurocode 3: Part 1-8 (Eurocode3, 2005a) states that the bolted shear connections
loaded in shear should be designed as either bearing type or slip-critical connections. To account
for the effect of fire induced strength degradation, Eurocode 3: Part 1-2 (Eurocode3, 2005b),
specify the use of strength reduction factors to steel. However, Eurocode 3 does not provide
specific procedures for fire design of double angle connections. In addition, there are no special
design provisions to account for the effect of fire induced forces on the response of connections.
2.6 Summary
The state-of-the art review clearly indicates that there are limited studies on the system-level fire
performance of double angle connections. These studies observed that the bolted double angle
connections have good tying resistance and rotational capacity in comparison to other
connections. However, there is a lack of knowledge with respect to a number of key areas that
are critical for developing design strategies for fire safety design of double angle connections.
Based on the above review, the following conclusion can be drawn:


There have been limited studies on the fire performance of connections and most of them
focused on developing moment-rotation characteristics. Most of the experimental studies

30

tested isolated connections under standard fire or uniform temperature conditions. The
effect of realistic fire, loading and structural continuity on the behavior of connections as
part of structural framing system needs to be studied.


Numerical models for predicting the fire performance of connections are available in
literature. Almost all the models accounted for geometric, material nonlinearities as well
as the nonlinear contact interactions between connection components. However, most
models are validated for isolated connection configurations under standard fire or
uniform temperatures.



The ultimate capacity of bolts in tension and double shear reduces drastically between
300-700°C and softening of bolts occur when the fire temperature exceeds the tempering
temperature used in the bolt manufacturing process.



Connections can help in improving the fire resistance of beam by reducing the mid-span
moment during the heating phase of the fire. Different failure modes in connections were
observed when the beam entered catenary action phase.



Modeling of connections poses a significant challenge because of the presence of large
number of interacting contact surfaces and the need to select and define realistic contact
behavior, contact model parameters. An improper selection of contact surface behavior or
contact parameters can create solution convergence issues.



There is a need for the development of numerical models which are capable of
accounting realistic fire, loading and interactions between connected members (ex: beam,
column, slab) on the fire performance of connections. These models can be used to carry
out parametric studies and the results can be used to develop design guidelines for
enhancing the fire resistance of connections.

31



The chemical composition and the heat treatment processes that high strength bolts
undergo are different from those encountered by typical carbon steel. Therefore the hightemperature properties of bolt steel are completely different from that of carbon steel and
have significant influence on the connection response and these should be properly
account for in numerical models.



No fire resistance experiments have been conducted on double angle connections that are
considered to be part of structural framing system. Restraining effects arising from
structural continuity can have a significant influence on the fire performance of double
angle connections. Data from the fire tests are needed to validate finite element models.



The effect of fire scenario on the fire response of double angle connection was not
considered in previous studies. Fire scenario can vary from a standard fire curve with no
decay phase to a design fire curve with decay phase (Refer to Figure 1.3). The decay
phase can result in cooling of steel members and thus generate tensile forces due to the
thermal shrinkage of steel. The development of these tensile forces was not considered in
previous studies and needs further research.



There is a need to quantify and consider fire induced forces in modeling the fire
performance of double angle connections. The fire induced axial force can have a
significant influence on the performance of double angle connections. For example,
double angle connections (shear connections in general) are not designed to resist tensile
forces that develop due to fire induced catenary action or due to shrinkage of steel
members in cooling phase of a design fire. Generally, these fire induced forces are not
accounted for in fire safety design due to lack of tools for evaluating them.

32

CHAPTER 3
3. EXPERIMENTAL STUDIES
3.1 General
The state-of-the-art review indicated that there is lack of test data on the overall fire behavior of
double angle connections as part of a structural framing system. Though large number of
experimental studies were conducted on connections in recent years, almost all of them were on
isolated end-plate connections under uniform temperature conditions. To develop specific test
data on double angle connections forming part of structural framing system, two sub-frame
assemblies with double angle connections, were tested under fire conditions. The main objective
of these tests is to generate test data for validation of finite element models. Full details of the
fire resistance tests, specimen details, instrumentation, test procedure and measured parameters
are presented in this chapter.

33

3.2 Test Specimens
The experimental program consisted of fire resistance tests on two sub frame assemblies,
comprising of a network of beams connected through double angle shear connections. Two sets
of tests were carried out on sub frame assemblies S1 and S2. Assembly S1 did not have a slab
and had a lower fire decay rate and lower level of gravity loading than assembly S2, which had a
slab.
Both the test assemblies had W12x30 secondary beams (3505 mm long), W14x132 perimeter
beams (4216 mm long) and W14x74 primary beams (4293 mm long), as shown in Fig. 3.1. The
secondary beams were connected to perimeter beams which in turn were connected to primary
beams using two different double angle connection configurations. Connection details were
designed as per AISC specifications (AISC, 2005) and are shown in Fig. 3.2. All the beams and
angles used in the test assembly were made with ASTM A992 Grade 50 steel, while the bolts
were 22 mm (7/8 in.) in diameter and made of ASTM A490 Grade steel. The ambient
temperature mechanical properties of steel measured as per ASTM A370 (ASTM, 2011b)
specifications are presented in Table 3.1 while the cross-sectional dimensions of the beams and
angles are shown in Table 3.2.
Initially, each connection assembly was insulated with 12.7 mm (½ in.) spray-on fire resistive
material (SFRM) on all three sides of the secondary beam along the majority of its length.
During the application of fire insulation, the thickness was measured at several locations to
assure a uniform thickness along the length of the beam. The insulation was then allowed to
adequately dry for three weeks prior to fire testing. The insulation material that was used is
CAFCO 300 with a specified thermal conductivity of 0.078 W/m-K and a density of 240 kg/m3
at room temperature.

34

In assembly S1 the secondary beams were welded to the steel deck so as to provide rigidity
against buckling of the beam. After the steel beams were assembled, a corrugated steel deck, of
2286 mm long and 3048 mm wide, was spot welded to the top of the secondary beams at six
random locations. The deck was arranged such that the ribs are oriented perpendicular to the
secondary beams and primary to the perimeter beams. No concrete slab was casted on assembly
S1.
For assembly S2 a concrete deck was cast on steel frame to provide rigidity against lateral
torsional buckling. A steel deck of 3454 mm by 3658 mm was attached to the secondary beams
through shear studs spaced at 152 mm along the length of the secondary beam. The shear studs
were of 19 mm diameter and 99 mm long and were spaced at 152.4 mm along the length of the
secondary beam. The concrete slab of 114 mm was cast on 6x6-W1.4xW1.4 welded wire fabric
shrinkage reinforcement. The slab was cast with light weight concrete supplied from a local
concrete batch mix plant. Cross-sectional details of the beam and slab used in test assembly S2 is
shown in Fig. 3.3(d). The design compressive strength was 28 MPa with an actual strength 41
MPa on the day of testing of the slab. The batch proportions of concrete are shown in Table 3.3.
After placing the concrete, the concrete slab was covered with a vapor proof barrier and the slab
was allowed to cure at ambient conditions. The relative humidity of concrete was measured on
the day of testing at four different locations on top of the slab and the average relative humidity
was 88%.
3.3 Instrumentation
The sub frame assemblies were instrumented with thermocouples, strain gauges, and
displacement transducers to monitor thermal and structural response during the fire test.
Temperatures in steel beams were measured using Type-K Chromel-alumel thermocouples, 0.91

35

mm thick, installed on the lower and upper flanges as well as on the web of each secondary beam
at quarter and mid-span of the beam. Thermocouple layout in beam and concrete section, crosssectional details of the beam and slab in assembly S2 are shown in Figs. 3.3(a)-(d), respectively.
Vertically and horizontally oriented linear variable displacement transducers (LVDT’s) were
attached at six distinct locations on secondary beams and perimeters beams in order to calculate
axial and lateral beam displacements. The pressure in the vertical actuators was also recorded as
a measure of applied axial load. Ten high-temperature strain gages were attached closer to the
ends of secondary beams and at the mid-span of primary beams to directly measure the total
strains. Location of the LVDT’s and strain gauges in assemblies S1 and S2 are shown in Fig.
3.3(a).
Data from the above instrumentation network was recorded at five second intervals via a central
data acquisition system. Also, the furnace temperature was measured using six thermocouples
distributed spatially inside the furnace in accordance with the ASTM E119 test procedure. The
pressure in the vertical actuators of the furnace was also recorded as a measure of applied load.
Additionally, visual observations were made to record any important events (beam buckling,
insulation fall off) or at five minute intervals throughout the duration of the test.
3.4 Test Apparatus
The fire resistance tests on double angle connection assemblies were carried out using structural
fire testing facility at Michigan State University. The furnace has been specially designed to
produce varying conditions of temperature, loading and heat transfer, to which a structural
member might be exposed during a fire. The test furnace, shown in Fig. 3.4, has the capacity to
simultaneously apply both thermal and structural loading to the test specimen, to simulate
conditions experienced in a real structure. The furnace consists of a steel framework supported

36

by four steel columns, with the furnace chamber inside the framework. The furnace heating
chamber is 2.44 m (8 feet) wide, 3.05m (10 feet) long, and 1.78 m (5.8 feet) high. The maximum
heat power the furnace can produce is 2.5 MW. Six gas burners located within the furnace
provide the thermal energy, while six type-K Chromel-alumel thermocouples, distributed
throughout the test chamber, monitor the furnace temperature during a fire test as per ASTM
E119 standard. During the course of fire test, the gas supply is manually adjusted such that the
furnace temperatures follow a pre-determined standard or realistic fire curves. In this way, the
furnace temperature can be maintained along a designed curve.
Loading on the test specimen is applied through a furnace loading frame which consists of a
network of two large and four small vertical pressure actuators located above the furnace. The
large actuators (having an individual capacity of 2750 kN each) are primarily used for testing
columns while the small actuators (with an individual capacity of 240 kN each) are used for
testing beams and beam-slab assemblies. Each of the actuator is connected to a loading jack
which sits on the surface of test specimen (beam, column etc) and transfers load from actuator to
the test specimen. For testing current sub frame assemblies, four vertical pressure actuators were
used to apply load on the sub frame assembly. To facilitate the visual observation of the fire
exposed specimen during the fire test, two small view ports are provided on either side of the
furnace wall.
3.5 Test Conditions and Procedure
The test specimen was placed in the furnace such that only the secondary beams were inside the
furnace. The perimeter and primary beams were fully outside the heating chamber of the furnace
with perimeter beams resting on the cantilever supports, while the primary beams were
unsupported as shown in Fig. 3.2. The connections on the secondary beam are 152 mm to 178

37

mm outside of the furnace walls and are exposed to some heating indirectly through conduction
from secondary beams and also from some level of radiation from the furnace walls. The
connections were placed outside the heating chamber in order to measure the magnitude and
nature of forces experienced by the connections due to heating of the secondary beams. In order
to prevent the rotation of the assembly, pieces of wood were inserted between the perimeter
beam of S2 and the furnace external frame.
The secondary beams in the sub frame assemblies S1 and S2 were exposed to two design fires
DF1 and DF2 respectively, from three sides. Design fire, DF1, comprised of a growth phase for
the first 75 minutes as per ASTM E119 fire (ASTM, 2011a) and then a decay phase with a
cooling rate of 8°C/min (see Fig. 3.5). The decay rate was computed based on the Eurocode
parametric fire curve equation (Eurocode1, 2002), using the input parameters listed in Table 3.4.
Detailed calculations showing the computation of decay rate for design fire DF1 are presented in
Appendix A. This design fire scenario was arrived at based on the typical dimensions,
construction materials and fire loads encountered in a typical office building.
Similarly DF2 included a growth phase simulating the ASTM E119 standard fire for the first 90
minutes and then followed by a decay phase with a cooling rate of 15°C/min (see Fig. 3.5). The
higher cooling rate was chosen for assembly S2 to study the response of the assembly under
different cooling scenario and to quantify the effect of the cooling rate. The predicted and the
measured time-temperature relationship of DF1, DF2 are illustrated in Fig. 3.5. A summary of
test parameters considered in this experimental study are shown in Table 3.5.
Loading on both assemblies comprised of four point loads, applied at two points along each
secondary beam, as shown in Figs. 3.4(a)-(b). Assemblies S1 and S2 were subjected to an initial
load of 74.7 kN and 92.2 kN at each actuator loading point, respectively. These load values

38

correspond to a load ratio (level) of 40% and 50% of the secondary beam ultimate moment
capacity. Load ratio is defined as the ratio of applied loading under fire conditions to the ambient
temperature capacity of the beam. The loading was applied 30 minutes prior to the start of the
test to ensure that the deflections reached a steady state before the start of the fire. During the
test, the connection assemblies were exposed to heat controlled in such a way that the average
temperature in the furnace followed, as closely as possible, the targeted time temperature curve.
The load was kept constant throughout the test. The connection assemblies were considered to
have failed and the tests were terminated when the secondary beams suffered local and lateral
torsional buckling and could no longer carry the applied load.
3.6 Results and Discussion
Data generated from the fire tests were used to evaluate the thermal response, structural response
and failure patterns of double angle connections under fire conditions. The naming convention
used for various beams in each sub frame assembly (S1 and S2) is illustrated in Fig. 3.1.
3.6.1 Thermal Response
The progression of temperatures (at quarter and mid-span) in the secondary beams (Beam I and
Beam II) of assemblies S1 and S2 is plotted as a function of time in Fig. 3.6 and 3.7 respectively.
The temperatures at both mid-span and quarter span of both secondary beams follow closely with
fire temperature in the growth phase of fire. The temperatures at the bottom flange are higher
than that of the top flange and this is attributed to the closer vicinity of bottom flange to fire
exposure. In addition the bottom flange was exposed to fire from top and bottom surfaces as
opposed to bottom surface of top flange only.
The temperatures in the top flange of the secondary beams at mid-span are significantly lower
compared to that at the quarter span. This discrepancy in temperatures can be attributed to non-

39

uniform gas temperature distribution in the furnace. Specifically, the quarter span location is
closer to the furnace walls, and experiences heating (radiation effect) from two furnace walls as
opposed to mid-span receiving radiation from one furnace wall.
The maximum average temperatures measured in secondary beams, shown in Fig. 3.8, for
assemblies S1 and S2 are 700°C and 645°C respectively and this occurred at 90 minutes into fire
exposure. It can be seen from the figure that the general trend in temperature variation is similar
in both beams. However the maximum average temperatures reached in assembly S2 is lower
compared to S1 and this can be attributed to the shielding effect provided by the concrete slab to
the top flange of the beams.
Connections in the sub frame assemblies S1 and S2 experienced heating indirectly via
conduction from the secondary beam and radiation from the furnace wall. In addition, due to the
availability of limited number of thermocouple channels, all the thermocouple channels were
used to monitor the temperature progression in the secondary beams as well as the furnace
chamber. Therefore, no temperature measurements were made directly at the connections.
However, an indication of the maximum temperatures experienced by the connections could be
gauged (indirectly) through the high-temperature strain gauges placed on the connections. In
both tests, these strain gauges provided reliable readings throughout the duration of test without
any fluctuations (abrupt variations). Based on a series of fire tests (on different structural
assemblies) it has been established that these high-temperature strain gauges give reliable
readings when the temperatures are below 200°C (Dwaikat et al., 2011). This led us to conclude
that the temperatures in connections remained below 200°C. Thus, the connections retained most
of their ambient temperature strength in both the tested assemblies.

40

3.6.2 Structural Response
The structural response of double angle connections can be assessed by examining the
progression of rotation-time history, which in turn is related to the fire induced forces
development and strain levels attained in the connections.
Connection Rotations
The progression of rotation in the connections at the ends of secondary beams is shown in Fig.
3.9. The values of rotation (θ) are evaluated using vertical deflection (u) of the secondary beams
(measured by the LVDT’s at the loading points) and the distance from the connection to the
loading point (L), as illustrated in Fig. 3.10. Once, the vertical deflection and length to loading
point is known, the rotation at connection is evaluated using the following equation:
⁄

[3.1]

The north and south legends used in these figures indicate the direction with respect to the
secondary beams orientation where the measurements are made (Refer to Fig. 3.4(a) for the
location of LVDT).
It can be seen from Fig. 3.9(a) that the rotation at connections in assembly S1, gradually
increased with time during the early stages of fire. This increase in rotations is due to
deteriorating strength and modulus properties of steel, as well as increased thermal expansion of
the steel with temperature. However, the connection at one end of the secondary beams (Beam II
– North) underwent a sudden increase in rotations at 65 minutes into the fire test. This is because
the secondary beams (Beam II) at the location of the load actuator (North) experienced sudden
increase in vertical deflection due to local buckling of the top flange and web at the location of
the loading point, as shown in Fig. 3.11(a). In addition to local buckling, the secondary beam
(Beam II) at the location of the loading actuator (North) rotated and experienced global lateral

41

torsional buckling, as can be seen from Fig. 3.11(b). This led to sudden increase in rotations at
the connections. However due to safety concerns (to mitigate collapse of the test assembly in to
the furnace), the applied loading on the beam was removed and the test was continued to trace
the thermal response under cooling phase of fire. At 180 minutes, the metal deck, present on top
of the secondary beams, underwent significant warping (as shown in Fig. 3.12) which distorted
the entire assembly causing the connections to undergo rapid rotation.
The rotation at connections in assembly S2 progressed similar to that of S1, but the connections
underwent much higher (almost double) rotations without any local and global instabilities. This
can be attributed to the higher rigidity provided by the concrete slab and this enhanced the
overall strength of the beams and in turn the connections. After reaching a maximum rotation at
110 minutes (slightly after the start of cooling phase of fire) the rate of rotation at connections
decreased steadily. This is on expected lines since the beams regained some of their initial
strength and stiffness after the start of the decay phase of fire and thus the rotations started to
decrease.
Forces in Connections
The fire induced forces developed in the double angle connections were monitored by strain
measurements and these strains are plotted as a function of time in Fig. 3.13. Few of the strain
gauges were damaged (which is common in fire tests) so only the measurements obtained from
functional strain gauges are shown in Fig. 3.13. For assembly S1, strain gauges attached to top
flange of secondary beam (Beam II) and primary beams (Beam III) were damaged. Similarly in
assembly S2, strain gauges attached to top flange of secondary beam (Beam I) were damaged.
As can be seen from Figs. 3.13(a)-(b), strains in the connection gradually increase with fire
exposure time and are compressive in nature during the initial stages. This is on expected lines

42

because as the secondary beams are heated they undergo a significant thermal expansion which
results in the development of large compressive forces in the connection. However as the fire
enters the decay phase, the compressive strains decrease and finally transform to tensile strains.
This transformation in the direction of strains can be attributed to the deteriorating strength and
stiffness of beam with increasing temperature which results in permanent deformation of beam.
At this point the beam is no longer able to carry any load and it is held in place (like a cable) by
the connections alone through catenary action(Liu et al., 2002). Thus the connections experience
a shift in the nature of forces from compression to tension. The sudden increase in the
compressive force in S1 is due to global buckling (of Beam I). Nonetheless, the general nature of
forces experienced by the connection in two assemblies is similar except for the instabilities
(caused by local and global buckling) that occurred in the secondary beam of assembly S1.
The strain readings indicate that the top and bottom flanges of the beam did not carry significant
compressive (axial) forces in comparison to the beam web. This can be attributed to the fact that
the web of beam was connected directly to the perimeter beams through double angles. The
transfer of forces (during expansion or contraction of secondary beam) between secondary beam
and perimeter beam happened only through the beam web via double angle. Hence the web
experienced higher compressive (axial) force in comparison to the flanges.
Figs. 3.13(c)-(d) show the variation of strain in the primary beam (Beam III) as a function of fire
exposure time. It can be seen in the figure that the strains in the beam change from positive to
negative indicating that the connection between perimeter and primary beams experienced
tensile force in the initial phase and compressive force during the later stages of fire exposure.
This can be attributed to the fact that as the temperature of secondary beam increases (in initial
phase) it undergoes thermal expansion which tends to push the perimeter beams. Relatively

43

cooler perimeter beams resist the thermal expansion of secondary beam by pulling the primary
beams. This results in the development of tensile forces in connection at the location of perimeter
and primary beams. Similarly, the tensile forces transform to compressive forces when the
secondary beams starts to cool and subsequently begin to undergo thermal contraction. It is
interesting to note that the primary beams experienced a non-uniform strain indicating that the
primary beams are subjected to thermally induced moments despite the connections between the
beams (secondary-perimeter and perimeter-primary beams) being simple shear connections. This
shows that the double angle connections exhibit some rigidity and are capable of transferring
moments between the connected members under fire conditions. This is in contrast to simple
shear connections wherein the connection is designed to transfer shear forces only.
It can also be seen from Fig. 3.13(c) that the global buckling experienced by Beam II in
assembly S1 had an impact (caused sudden change in strain at 68 minutes) on the connection
between perimeter and primary beams. Strain readings for the primary beam (Beam IV) in S1
could not be recorded since these strain gauges were damaged early into fire test.
Forces in Beams
The progression of axial forces in secondary, primary beams and concrete deck are shown in Fig.
3.14 as a function of fire exposure time for sub frame assembly S2. A positive value indicates
tensile force in the beam while a negative value indicates compression in the beam. The axial
force in each beam is calculated using the degrading elastic modulus (E) and the measured strain
values. The axial resistance provided by the concrete deck is estimated by taking the difference
between the forces carried by perimeter and primary beams.
It can be seen from the figure that during early stages of heating (till 45 minutes), secondary
beam experienced compression while the concrete deck unloads (because of thermal bowing of

44

the deck, beam) through the development of composite action facilitated by the shear studs. As
the thermal gradient between the concrete deck and the bottom flange of secondary beam
increases (between 45-90 minutes into fire) the top flange of the beam goes into significant
tension, which brings the average secondary beam axial force into tension. This observation is
also confirmed by large tensile strains observed in the top flange of secondary beam (see Fig.
3.13(b)). The increase in tensile force continues till 90 minutes after which the secondary beam
stiffness gets significantly lower and the tensile force in the beam decreases. After 120 minutes,
the thermal contraction of secondary beam causes it to enter into tension phase again.
3.6.3 Failure Patterns
Visual observations were continuously made during the course of each fire test and also after fire
tests to trace the response of the double angle connections. Fig. 3.15 shows the local and lateral
torsional buckling experienced by the secondary beam in assembly S1. It can be seen from the
figure (Fig. 3.15(a)) that significant amount of insulation has fallen off from the web and top
flange of secondary beam. This abrupt loss of insulation led to rapid increase in the temperatures
in the web and top flange thereby decreasing the load carrying capacity of the beam. As a result,
the top flange and the web of the beam in assembly S1 buckled (Fig. 3.15(b)) leading to sudden
increase in deflections and subsequently increase in rotations at connections. This increase in
deflections together with local buckling, caused the beam to rotate at the ends through lateral
torsional buckling phenomenon, as illustrated in Figs. 3.15(c)-(d).
For test assembly S2, insulation fell-off (from the deck) and initial cracks began to form in the
concrete slab at around 30 minutes into the fire test. The number of cracks, as well as the
progress of these cracks in the slab gradually increased with steam and water vapor emitting out
of these cracks. At about 50 minutes in to fire exposure, a large crack formed in the middle of the

45

slab and small concrete fragments started to fly-off. By the end of the test, the deck had lost
almost all of the insulation and had undergone significant warping, as illustrated in Fig. 3.16.
Observations of the double angle connections after fire tests indicated that the connections
underwent permanent deformation in both assemblies and the extent of deformation in assembly
S2 was lower in comparison to that of assembly S1. It was also observed that the top flange of
the secondary beam in assembly S2 was intact with the concrete slab and it was free of local and
global instabilities as opposed to assembly S1. These observations indicate that the presence of
concrete slab stabilized secondary beam in assembly S2 throughout the fire test, which allowed
the double angle connections to develop significant (more than double) rotations, without failure,
in comparison to double angle connections in assembly S1.
3.7 Summary
Fire tests were conducted to evaluate the behavior of double angle connections as part of
structural framing system. The influence of concrete slab on the fire behavior of connections was
evaluated by comparing the thermal and structural response obtained from tested assemblies with
and without the slab. Double angle connections in both the assemblies did not experience failure
though they experienced permanent deformations demonstrating the inherent rigidity and the
robustness of these connections. The strain profile in primary beams was non-uniform which
indicates that double angle connections are capable of transferring fire induced moments
between connected members despite being designed as simple shear connections. Results of
these tests show that cooling rate of decay phase, load level, presence of slab and structural
continuity of the framing system has significant influence on the fire response of double angle
connection assemblies.

46

CHAPTER 4
4. NUMERICAL MODELS
4.1 General
Undertaking experiments to trace the fire behavior of connections is quite expensive, time
consuming and requires sophisticated test facilities. In fire tests, only limited number of
parameters can be monitored and interdependency of parameters cannot be traced. Further, for
undertaking such fire tests there are lack of reliable high temperature instrumentation (such as
strain gauges), that can monitor response parameters at elevated temperature. Therefore there is
limited experimental data in literature on connection behavior at elevated temperature.
At present there are limited specifications for fire design of connections in design standards such
as Eurocode 3. However, these design provisions are derived from limited number of standard
fire tests conducted on scaled and isolated connection specimens. These design provisions do not
specifically account for the effect of fire induced forces in evaluating rotation in connections. As
shown in Chapters 1, 2 and 5, connections, experience significant fire induced axial forces that

47

arise from the effect of structural continuity in a framing system. Hence, current design
provisions which are developed based on isolated connection response cannot be applied to trace
the system-level transient response of connections.
An alternative to overcome many of the shortcomings in fire tests is through applying numerical
models to trace the fire response of connections. In recent years, commercial finite element (FE)
programs became widely available thus making them an efficient alternative to model the
behavior of connections. The relatively inexpensive finite element method(s) provides effective
strategy to analyze fire response of connections.
This chapter presents the development of two finite element models (FEM1 and FEM2) to
simulate fire response of three steel framed assemblies with double angle connections. Both the
models employed system-level analysis (described later) to simulate the fire response of
connections. The first finite element model (FEM1) is developed to simulate fire response of two
sub-frame assemblies one with slab and the other without slab. The second finite element model
(FEM2) is developed to simulate the fire response of restrained steel frame with a different
double angle connection configuration compared to FEM1. Both the finite element models are
validated by comparing the thermal and structural responses obtained from the model with those
measured during fire tests. More details about the finite element models along with the validation
are presented in the following sections.
4.2 Selection of Finite Element Program
The general-purpose finite element program ANSYS was chosen to carry out the numerical
studies because of its diverse capabilities. ANSYS has the ability to efficiently model highly
sophisticated material and geometrical nonlinearities. ANSYS enables the user to specify

48

temperature-dependent thermal and mechanical properties of different material types such as
steel, concrete and fire insulation.
Since forces from different structural components are transferred through connections, there will
be contact interactions at the interface of each pair of these components (such as beam, column,
bolts and angles). These interactions at the interface play a major role on the way forces are
transferred and the development of failure modes in connection assemblies. Therefore, these
interactions are to be accounted for and properly simulated in the analysis. ANSYS provides a
wide range of contact interaction models and contact parameters to simulate contact interactions
in addition to allowing user-defined contact parameters. This enables the user to model any
specific type of contact interaction problem at hand.
In addition, ANSYS provides flexibility for creating geometry of the model using ANSYS script
language (APDL-ANSYS Parametric Design Language) in which different geometric
configurations and material models can be defined as parameters. Thus, APDL can be used as an
effective tool for creating new models and automating parametric studies. Further, ANSYS
contains a rich library of diverse categories of elements that are well suited for different purposes
of analysis.
The double angle connection assembly analysis was carried out by incorporating all significant
parameters that influence the response of connections, including material and geometric
nonlinearities, high temperature properties of steel, concrete and insulation, and nonlinear
contact interactions. Details of the finite element analysis are provided in the following sections.

49

4.3 Finite Element Modeling
4.3.1 General Approach
In finite element approach, fire resistance analysis of connection assemblies is generally carried
out through two stages of analysis, namely, thermal and structural analysis. The thermal analysis
provides temperature distribution in the connection assembly subjected to a given fire scenario.
The output of the thermal analysis, nodal temperatures, is then applied as an input thermal-bodyload in the structural analysis and a transient stress analysis is carried out. For practical cases of
analysis, the following assumptions are adopted in undertaking fire resistance analysis:


Fire temperature is independent of connection assembly response: As explained below,
fire temperature is dependent on geometry, fuel and ventilation characteristics of the
compartment only and is independent of the response of connection (and structural)
assembly.



Thermal analysis is independent of the structural analysis: The evolution of temperature
profile in the connection and structural members depends only on density, thermal
conductivity and specific heat of steel, concrete and insulation. However, the structural
response of the members depends on the mechanical properties, modulus of elasticity,
stress-strain relationships of steel and concrete. Hence, the progression of temperatures in
connection assembly occurs independently of the structural response.



Temperature distribution is uniform along the beam span length: The entire length of the
beam is subjected to the same fire exposure conditions and the cross-sectional geometry
remains the same throughout the member. Hence, cross-sectional temperature distribution
will be uniform along the beam span length.

50

The fire scenarios that are typically used in fire resistance analysis can be grouped under two
categories. Standard fire scenario; where fire temperature continues to increase without any
cooling phase (as in ASTM E119 and ISO 834 standard fire exposures), and realistic fire
scenario where fire temperature increase with time in growth phase and then gradually decrease
with time in the decay or cooling phase.
In case of a “realistic” fire, a decay phase follows after reaching a maximum fire temperature in
growth phase. Unlike in standard fire where temperature is expressed as a function of time only,
the growth and decay phases of realistic fires are dependent on the compartment geometry and
fuel and ventilation characteristics. The realistic fires can be approximated/modeled using timetemperature relationships provided in codes and standards (and referred to as design fire).
Examples of design fires are the Swedish fire curves and the parametric fire curves specified in
the Eurocode (Eurocode3, 2005b) and SFPE handbook.
In general, the rise of fire temperature is dependent on geometry, fuel and ventilation
characteristics of the compartment. Computational fluid dynamics (CFD) models are required to
compute the evolution of fire in a certain enclosure. Most of the CFD models are
computationally intense as they adopt iterative, finite-difference technique and are conditionally
stable (i.e., the time step depends upon the smallest element in the geometry). The goal of the
current study is to simulate the system-level fire response of connection assemblies accounting
for computationally expensive contact interactions. Adopting CFD analysis to predict the
evolution of fire temperature will add exponentially large computational time to already
computationally expensive finite element model (due to contact interactions). Therefore, CFD
analysis is beyond the scope of this study and the fire scenario is assumed to follow specific
time-temperature relationships specified in design standards.

51

The fire resistance analysis, with its thermal and structural sub models, is carried out via ANSYS
finite element program using the following general procedure:


Different structural components in the connection assembly are discretized into elements
and total fire exposure time is divided into a number of time steps and the fire resistance
analysis is carried out at each time step. A typical connection region in a steel framed
building along with different structural components present in the connection region is
illustrated in Fig. 4.1(f).



Ambient temperature structural response: Static structural analysis is performed on the
double angle connection assembly to obtain the structural response (deformations and
stresses) of all connected members at ambient conditions (room-temperature).



Thermal analysis: Temperature distribution in the beam cross-section is obtained at every
time step and for the entire fire exposure history. Since the beam and connections are
subjected to the same fire exposure conditions, the temperature progression at the
location of connection cross-section will be the same as that obtained for a beam crosssection.



Structural analysis: During the first time step, the temperature distribution of beam crosssection obtained from thermal analysis is applied as a thermal-body-load on the ambienttemperature deformed model of the connection assembly and a stress analysis is carried
out. For the subsequent time steps, the temperature distribution of beam cross-section
obtained from thermal analysis is applied as a thermal-body-load on the deformed model
from previous time step and a stress analysis is carried out

More details about the thermal and structural sub models are presented in the following sections.

52

4.3.2 Thermal Analysis
4.3.2.1 General
The cross-sections of beam, along its entire length, is subjected to the same fire exposure
conditions. Hence, the temperature distribution (profile) at any beam cross-section will be same
and the temperature distribution will be uniform along the beam span length. Therefore, the
beam cross-sectional temperatures can be obtained by carrying out a two dimensional heat
transfer analysis instead of conducting a three-dimensional heat transfer analysis. Connections
are not specifically accounted for in the thermal analysis. This is due to the fact that connections
in sub-assemblies S1 and S2 were placed outside the heating zone of the furnace (Refer to
Chapter 3). Therefore, the connections experienced heating indirectly via conduction from the
secondary beam. This conductive heating of connections through secondary beam is accounted
for in the thermal analysis.
4.3.2.2 Governing Equations
The two-dimensional governing partial differential heat transfer equation with in a structure can
be written as:
[4.1]
where k = conductivity matrix, c = heat capacity,  = density, c = specific heat, T =
temperature, t = time, and  = is the spatial gradient operator.
At the fire-beam interface, heat transfer occurs through radiation and convection. The total heat
flux on the boundary of beam due to convection and radiation is given by the following equation:
[4.2]
where hrad and hcon are the radiative and convective heat transfer coefficients, and are defined as:





hrad  4 T 2  T f2 T  T f
53



[4.3]

Tf = temperature of the atmosphere surrounding the boundary (i.e., fire temperature),
hcon = 25 W/m2.°K as recommended in literature (Eurocode3, 2005b)

 = Stefan-Boltzman constant = 5.6710-8 (W/m2.K4), and
 = emissivity factor for steel and it is related to the “visibility” of the exposed surface of
structural member to the fire.
Fire is assumed to be emanating from a point source (infinitesimal sphere) perfect black body
radiating heat equally in all spatial directions. To account for heat lost to surroundings as fire
travels from its source to beam; it is assumed that only 70% of the radiant heat from fire source
reaches the beam surface. Therefore, a steel emissivity factor of 0.7 specified in Eurocode
(Eurocode3, 2005b) is assumed in the analysis.
The heat flux and temperature gradient are related through Fourier’s law of heat conduction as:
q  kT

[4.4]

According to Fourier’s law, the governing heat transfer equation on any boundary of the beam
cross-section can be expressed as:
 T
T 
k 
ny 
n z    qb
z 
 y

[4.5]

where ny and nz = components of the vector normal to the boundary in the plane of the crosssection. The right hand side of Eq. [4.5] depends on the type of imposed boundary condition. As
the beam is exposed to fire from three sides, two types of boundary equations are to be
considered for thermal analysis, namely:


On fire exposed boundaries where the heat flux is governed by the following equation:
qb  h f T  T f

54



[4.6]



On unexposed boundary (top surface of the beam) where the heat flux equation is given
by:

qb  h0 T  T0 

[4.7]

where:
hf and h0 = heat transfer coefficient on fire side and the cold side, respectively, and
Tf and T0 = temperature at fire and cold side, respectively.
The nodal temperatures of any element are related by the appropriate shape functions matrix (N)
to arrive at the temperature of the element
T  N T  Te

[4.8]

Then the heat transfer equation (Eq.[4.1]) subjected to appropriate boundary conditions
(Eq.[4.5]) can be discretized as (Cook et al., 2002; Reddy, 2005):
̇
is the specific heat matrix.

[4.9]

is the thermal “stiffness” matrix and is the sum of conductivity

and convection matrices.

is the applied nodal thermal load and is composed of the convective

and radiative heat fluxes.

are the nodal temperatures.

4.3.2.3 Discretization of Beam
The thermal analysis of the beam cross-section was carried out using two types of elements,
namely SOLID70 and SURF152, as shown in Fig. 4.2. SOLID70 was used as a solid element
with three-dimensional thermal conduction capability. The element has eight nodes with a single
degree of freedom, temperature, at each node. The element is applicable to a three-dimensional,
steady-state or transient thermal analysis. The temperature within SOLID70 element is
interpolated from the nodal degrees of freedom (Te = Ti,j,k,l,m,n,o,p) using the following
isoparametric function:
55

Ti 1  s 1  t (1  r )  T j 1  s 1  t (1  r )  Tk 1  s 1  t (1  r )

1

T    Tl 1  s 1  t (1  r )  Tm 1  s 1  t (1  r )  Tn 1  s 1  t (1  r )
8


 To 1  s 1  t (1  r )  T p 1  s 1  t (1  r )



[4.10]

where s,t and r are the isoparametric locus of a point in the element domain. In order to carry out
the numerical integrations, 2x2x2 integration points are used in SOLID70 elements.
The SURF152 element is generally used for various load and surface effect applications. In the
thermal analysis, SURF152 element is overlaid onto the face of SOLID70 3D thermal solid
element to simulate the effect of both thermal radiation and heat convection from ambient air to
the exposed boundaries of the steel section (refer to Eq.[4.2]).
Heat convection load was applied around the section, and a convection coefficient of hcon = 25
W/(m2 .˚C) was assumed in the analysis. The ambient (bulk) temperature on the node M of
SURF152 element was assumed to be equal to either the fire temperature (Tf) in case the
boundary is exposed to fire, or to room-temperature, in case the boundary is not exposed to fire.
SURF152 uses linear interpolation functions and two integration points for numerical
integration.
4.3.3 Structural Analysis
The structural analysis is carried out by discretizing the geometry of connection assembly with
SOLID185 elements (see Fig. 4.2), suitable for 3-D modeling of solid structures. This element
has the capability of representing plasticity, stress stiffening, large deflections, large strains,
material and geometrical nonlinearities (ANSYS Inc, 2009c). Material nonlinearities can be
introduced through nonlinear temperature-stress-strain curves and temperature-dependent
thermal strain while geometric nonlinearities are introduced through accounting for large
deformations. SOLID185 (ANSYS Inc, 2009c) element has eight nodes with three degrees of
freedom at each node, namely translations in the nodal x,y and z directions. In the element
56

notation, the translations in x,y and z directions are referred to as u,v and w respectively. The
translation (in any direction) within SOLID185 is interpolated from the nodal degrees of freedom
(ue = ui,j,k,l,m,n,o,p ; ve = vi,j,k,l,m,n,o,p and we = wi,j,k,l,m,n,o,p) using the following isoparametric function:
u i 1  s 1  t (1  r )  u j 1  s 1  t (1  r )  u k 1  s 1  t (1  r )

1

u    u l 1  s 1  t (1  r )  u m 1  s 1  t (1  r )  u n 1  s 1  t (1  r )
8


 u o 1  s 1  t (1  r )  u p 1  s 1  t (1  r )



[4.11]

vi 1  s 1  t (1  r )  v j 1  s 1  t (1  r )  v k 1  s 1  t (1  r )

1

v    vl 1  s 1  t (1  r )  v m 1  s 1  t (1  r )  v n 1  s 1  t (1  r )
8


 vo 1  s 1  t (1  r )  v p 1  s 1  t (1  r )



[4.12]

 wi 1  s 1  t (1  r )  w j 1  s 1  t (1  r )  wk 1  s 1  t (1  r )

1

w    wl 1  s 1  t (1  r )  wm 1  s 1  t (1  r )  wn 1  s 1  t (1  r )
8


 wo 1  s 1  t (1  r )  w p 1  s 1  t (1  r )



[4.13]

where s,t and r are the isoparametric locus of a point in the element domain. In order to carry out
the numerical integrations, 2x2x2 integration points are used in SOLID185 elements.
The structural analysis is carried out based on the principle of virtual work. According to this
principle, any virtual change in the internal strain energy must be balanced by a change in the
external work due to the applied loads.
δU = δV

[4.14]

where U is the strain energy and V is the external work. Variation in strain energy (δU) can be
evaluated as:

U 

 δ εσdvol

[4.15]

vol

For structural members subjected to fire conditions, the strain vector (ε) is the sum of thermal
(εth) and mechanical strains (εm) in steel,i.e.:

   m   th
57

[4.16]

Creep strain is steel is not explicitly accounted for in the analysis. Creep strain is defined as the
time-dependent plastic strain under constant stress and temperature. The effect of creep in steel
becomes noticeable at temperatures above 400°C with the creep effects increasing with
increasing temperature. Including high-temperature creep in the analysis generally requires the
use of explicit solver which is computationally expensive. This is due to the fact that the time
increment in explicit solver is dictated by the size of the smallest element in the model. Hence,
accounting for creep in addition to already computationally intense contact interaction will
exponentially increase the overall computational time of the model. However, high-temperature
creep in steel is accounted for indirectly by using the high-temperature constitutive material
properties specified in Eurocode (Eurocode3, 2005b). This is because the steel properties
specified in Eurocode (Eurocode3, 2005b) , are based on transient-state tests, which accounts for
part of the high-temperature creep (Buchanan, 2002; Kodur et al., 2010). In fact, Eurocode
(Eurocode3, 2005b) explicitly states that the “effects of transient thermal creep need not be given
explicit consideration provided that the steel stress-strain relationships given in Eurocode are
used”.
n
Variation of external work (δV) due to the applied nodal forces ( Fe ) can be computed by

assuming a variation of nodal displacement u as:

 

V  uT Fen

[4.17]

The nodal displacements (ue) of the finite elements are related to the nodal displacement field
through shape functions matrix (N)as follows:

ue  N T  u
Then the virtual work equation (Eq. [4.14]) can be rewritten in matrix form as:

58

[4.18]

[4.19]
where Ke is the element stiffness matrix, and

Feth is the element thermal load vector.

4.3.4 Modeling Contact Interactions
In the connection assembly, contact interactions exist between different components (refer to
Figs. 4.1(d),(e) and (f)) such as bolt shank and bolt hole, angle and beam/column web, bolt
head/nut to the angle etc. Simulating these contact interactions is quite complex due to large
number of interacting surfaces (parts), highly nonlinear interaction response and numerical
convergence issues to reach a solution. Additionally, selecting a surface as contact or target
surface prior to analysis is often unclear in most cases and some trial runs have to be conducted.
Added to the above complexity, the behavior of interacting surfaces is highly dynamic because
the surfaces can move in and out of contact regions within or between iterations(ANSYS Inc,
2009b). This dynamic movement of surfaces leads to sudden change in stiffness matrix and more
equilibrium iterations (often with reduced time steps) have to be carried out to achieve
convergence. Thus, the computational time for structural analysis with contact elements is
generally much higher than those without contact elements. Further, the computational time
increases nonlinearly with each additional pair of interacting surfaces. This is due to the fact that
at any time step, if equilibrium conditions are not satisfied even for one contact pair the iterations
for all the contact pairs are to be repeated with a reduced time step.
The contact interactions between different parts are defined using “contact pairs” and are
modeled

as

“surface-to-surface”

contact

using

contact

elements

CONTA174

and

TARGE170(ANSYS Inc, 2009c). CONTA174 was used as a surface element and was overlaid
on the underlying SOLID185 elements. This element is capable of three-dimensional structural

59

and coupled field contact analyses. The element has eight nodes and is capable of modeling
three-dimensional surface-to-surface contact interactions.
The three-dimensional contact surface elements (CONTA174) are associated with corresponding
three-dimensional target segment elements (TARGE170) using a set of parameters called as real
constant set. In ANSYS, contact can happen only between CONTA174 and TARG170 elements
with the same set of real constants. Thus, the problem of spurious contact between noninteracting parts is taken care of.
Proper selection of contact pair parameters (contact algorithm, contact surface behavior, key
options, real constant set etc.) are crucial for avoiding most numerical convergence issues
(Pakala et al., 2012a) in the model. In addition, care should be taken to make sure that finite
element geometry is properly constrained (by appropriate boundary conditions) and there are no
rigid body modes. The presence of gap(s) between contact surfaces (ex: bolt holes are oversize
compared to bolt shank) is often the root cause for developing unconstrained/under-constrained
models. These gaps directly lead to rigid body modes and proper care must be taken to stabilize
and fully constrain the finite element model before starting the analysis. More details on the
guidelines for implementing nonlinear contact interactions in finite element programs can be
found elsewhere (Bursi and Jaspart, 1997a, b, 1998; Selamet and Garlock, 2010a; Van der Vegte
and Makino, 2004).
The contact between interacting surfaces is successfully incorporated using “surface-to-surface”
contact with “no separation but sliding permitted” option (ANSYS Inc, 2009b). The amount of
sliding depends on the frictional model, which is defined according to Coulumb’s frictional law
with a constant co-efficient of friction of µ=0.3 (assuming Class A faying surfaces) throughout
the analysis. To determine the optimum set of contact parameters, a sensitivity study was carried

60

out by varying the contact stiffness from 0.01 to 1 (as suggested in the ANSYS contact
technology guide) (ANSYS Inc, 2009b). The upper bound value of contact stiffness is more
appropriate for bulk deformation problems while the lower bound value is appropriate for
bending deformation dominated problems (similar to the current one) while ANSYS uses a
default value of 1 for the contact stiffness (ANSYS Inc, 2009b).

A summary of contact

parameters used are presented in Table 4.1.
4.3.5 High-temperature Material Properties
To accurately model the behavior of connection assemblies, appropriate thermal and mechanical
properties of the constitutive material should be specified as input in ANSYS. The thermal
properties include density, thermal conductivity and specific heat while the mechanical
properties include the stress-strain relationships of steel, all of which vary as a function of fire
temperature. For the thermal properties, empirical relations of steel provided in Eurocode 3
(Eurocode3, 2005b) are used, while the empirical relations of concrete specified in Eurocode 2
(Eurocode2, 2004) are used. The temperature-dependent thermal property relationships of steel
and concrete are presented in Appendix B. The thermal properties of insulation at elevated
temperatures are assumed to be same as that at room temperature due to lack of data on the
variation of insulation properties with temperature. Since the insulation material has significantly
low strength and stiffness, the strength contribution from the insulation is neglected.
For steel, the mechanical properties along with constitutive relationships and strength reduction
factors provided in Eurocode 3 (Eurocode3, 2005b) were used to generate nominal stress-straintemperature relationships. The strength reduction factors for carbon steel are shown in Fig. 4.3.
The constitutive relationships and the reduction factors for carbon steel are presented in
Appendix B. The strength reduction factors for bolt steel are based on the values proposed in a

61

recent study (Kodur et al., 2012) and are presented in Appendix B. The nominal stress-strain
relations were then converted into true stress-strain curves using the following relations:

 true   nom(1   nom) and

 true  ln(1   nom)

[4.20]

where σtrue,εtrue represent true stress and strain while σnom,εnom represent nominal stress and strain
respectively.
For concrete, the mechanical properties along with constitutive relationships and strength
reduction factors specified in Eurocode 2 (Eurocode2, 2004) were used to generate nominal
stress-strain-temperature relationships. The constitutive relationships and the reduction factors
for concrete are presented in Appendix B.
All the connection components (beam, column, double angle and bolts) are assumed to follow
elasto-plastic material behavior with Von-Mises plasticity yielding criterion and isotropic
hardening rule.
4.3.6 Failure Criteria and Numerical Convergence
The failure criteria adopted in the analysis is the one based on attaining ultimate plastic strain in
steel. This failure criterion was adopted in previous studies for modeling connection response at
elevated temperature (Selamet and Garlock, 2010b). Accordingly, the “failure” of the connection
(steel) is said to occur when the ultimate plastic strain of steel in any of connection component
exceeds 15% in 20-800°C temperatures range. The limiting strain of steel at failure is specified
as 15% in Eurocode3 (Eurocode3, 2005b) for all temperatures. When the strain in steel reaches
this limiting value, the governing finite element equations do not converge leading to nonconvergence of the solution.
Force (load) controlled solution technique is used in the finite element analysis as displacement
controlled technique cannot be used in conjunction with contact elements (ANSYS Inc, 2009b).
62

Numerically, convergence in the structural model is governed by the Newton-Raphson
equilibrium iterations (ANSYS Inc, 2009c). In the structural model simulations, force
convergence is said to be achieved if the error between successive iterations is less than 0.5%
(i.e., having an accuracy of 99.5%) (ANSYS Inc, 2009a). Similarly, in thermal analysis the
temperature convergence is assumed to be reached when the temperature difference at each node
between successive equilibrium iterations is less than 0.5 °C.
4.4 Connection Models
4.4.1 General
The response of connections under fire conditions can be studied at three levels, namely:
component-level, member-level (isolated member) and system-level. The component-level
analysis captures the response at the component (connection) level and does not capture the
effect of structural interaction on the response of connection. This is typically suitable for use in
design office environments and can be implemented using spreadsheet calculations. According to
this approach, the failure time of connection is determined by comparing the connection capacity
against the fire induced forces. Failure time of connection is defined as the time at which
connection capacity falls below that of fire induced forces. Though the component-level analysis
is relatively simple to use it has several drawbacks namely: (a) this analysis does not account for
the forces resulting from the interaction between structural members on the response of
connection; (b) the effect of local beam instabilities (such as local buckling and lateral torsional
buckling) on the connection response cannot be accounted for in this analysis, and (c) the
transformation of fire induced axial forces from compression phase to tension phase cannot be
captured. Therefore, component-level analysis cannot be used to predict the realistic fire
response of connection

63

The member-level analysis (or isolated member analysis) is commonly used to simulate the
response of connections which are not considered to be a part of the structural framing system.
As the structural members (ex: beams) are considered to be isolated implying that they are free to
expand thermally. Since the free thermal expansion is not restrained by surrounding structural
members no fire induced forces will be generated in the beam and subsequently in the
connections. However, in reality connections are always an integral part of a structural framing
system and the connected beams have restraint (from surrounding structural members) to their
free thermal expansion. This restraint to free thermal expansion, resulting in the development of
fire induced axial forces in beams and connections. Therefore, member-level analysis also does
not capture the realistic fire response of connections.
The system-level analysis refers to comprehensive analysis using finite element models. In this
approach connections are considered to be an integral part of the structural framing system.
Hence the connection region along with the surrounding structural members is modeled. In this
approach, the fire induced forces developed in the beam are computed through a second order
nonlinear analysis based on actual temperature profiles (in beam and connection), as well as
deformed geometry of beam and connection. Therefore, this type of analysis simulates realistic
response of the connection assembly and can capture the influence of surrounding structural
system on the fire response of connections.
Hence, system-level approach has been adopted in developing the numerical models presented in
this chapter and subsequent parametric studies presented in Chapter 5.
4.4.2 Assembly Level Connection Model (FEM1)
As mentioned earlier, two finite element models (FEM1 and FEM2) simulating the fire response
of three typical connection assemblies have been developed and validated. The first finite

64

element model (FEM1) is developed to simulate two tested sub-frame assemblies, namely S1 and
S2 (Refer to Chapter 3). This model, consisting of beam slab assembly with double angle
connections, was developed to simulate the response of sub-frame assemblies. Both the
connection assemblies, S1 and S2, had fire insulated secondary beam connected to perimeter
beam using double angle connections. However, assembly S1 did not have a slab and had a
smaller fire decay rate and gravity load ratio than S2, which had a slab.
As the geometry of sub-frame assemblies was symmetrical about two vertical planes, only one
quarter of the geometry was modeled, as shown in Fig. 4.4. The thermal analysis is carried out
using SOLID70 and SURF152 elements. The structural analysis is carried out by discretizing the
beams and connections in assemblies S1 and S2 using SOLID185 elements (see Fig. 4.2).
The boundary conditions of the sub-frame assemblies S1 and S2 were chosen to simulate
conditions present in the fire test (Pakala et al., 2012b). During the fire test, the stiffeners present
on the perimeter beam were placed on the furnace supports. In order to prevent the rotation of the
assembly, pieces of wood were inserted between the perimeter beam and the furnace external
frame. To simulate these conditions in the finite element model stiffener present on the perimeter
beam was fully fixed.
As mentioned in Chapter 3, concrete slab is added to test assembly S2 in order to avoid
local/global instabilities of the secondary beam (as observed in S1) and not to enhance its
composite action. Additionally, the concrete slab was not restrained in any direction, thus the
extent of composite action developed (if any) is minimal. Therefore, shear studs are not
explicitly modeled to avoid the need for defining and accounting for additional contact
interactions associated with studs. However, the beam-slab composite action is accounted for
indirectly by modeling surface-to-surface contact interactions between top flange and bottom

65

most layer of concrete slab. These contact interactions will ensure shear forces (which is the
primary mechanism of force transfer in composite action) are effectively transferred at the
interface of beam-slab.
The contact interactions between surfaces of different structural members are defined using
“contact pairs” and are modeled as “surface-to-surface” contact using contact elements
CONTA174 and TARGE170(ANSYS Inc, 2009c). In this model, there are one hundred and
seventeen contact pairs defined with approximately 42,113 contact elements for both assemblies
S1 and S2. In the current model, the contact between interacting surfaces is successfully
incorporated using “surface-to-surface” contact with “no separation but sliding permitted”
option. Based on the results of the sensitivity study, optimum normal contact stiffness (FKN)
value of 0.1 was assumed in the first iteration of the analysis. This value is updated in subsequent
iterations based on the current mean stress of the underlying element and the allowable
penetration of contact elements.
4.4.3 Validation of Assembly Level Connection Model (FEM1)
The finite element model FEM1 created in ANSYS was validated by comparing predictions from
the analysis with results obtained from fire experiments on connection assemblies S1 and S2
discussed in Chapter 3 (Fig. 3.6(a), Fig. 3.7(a) and Fig. 3.9). The validation process covered both
thermal and mechanical response. In order to accurately capture stress concentrations in the
connection region (angles, bolts, bolt holes), a relatively finer mesh was used within the vicinity
of these regions. During the fire test, static gravity loading was applied on secondary beam prior
to heating the connection assembly. Loading on the assembly comprised of four point loads,
applied at two points along each secondary beam. To circumvent the problems of high-stress

66

concentrations and numerical singularities due to external loading, the base of actuator pads was
modeled and the load is applied as nodal force in the vertical direction (y direction).
The properties for different steel members used in this model are based on ambient temperature
tensile coupon tests reported in Table 4.2. No coupon tests were performed to evaluate the
mechanical properties of bolts. Hence, the nominal yield strength, ultimate strength and modulus
of elasticity (as specified in the ASTM A490 standard (ASTM, 2012)) of bolts were assumed to
be 940 MPa, 1040 MPa and 210000 MPa respectively. The properties of concrete used for slab
are presented in Table 4.3. The reduction factors specified in Eurocode 2 (Eurocode2, 2004)
(Fig. 4.5) are used to arrive at the elevated temperature constitutive relations for concrete. Since
the insulation material has significantly low strength and stiffness, the strength contribution from
the insulation is neglected. However, thermal properties of insulation are accounted for in the
thermal analysis. For the current analysis thermal conductivity and dry density of fire insulation
are obtained from the values proposed for spray applied fire resistive material (SFRM) in a
recent study (Kodur and Shakya, 2013). It should be noted that there is limited data on the
variation of thermal properties with temperature of the insulation material; hence thermal
properties at elevated temperatures are assumed to be same as that at room temperature. A
summary of insulation properties used in the analysis is presented in Table 4.4. The following
sections provide details on the validation process.
4.4.3.1 Thermal Response
Figs. 4.6 and 4.7 shows the comparison between measured steel temperature and that predicted
by finite element model in fire exposed secondary beams in both connection assemblies. It can
be seen that there is good agreement between predicted and measured temperatures across the
depth of steel beam. In both the assemblies, temperatures at the bottom flange of the beam were

67

higher than that of the top flange and this can be attributed to the closer vicinity of bottom flange
to fire. However, the maximum temperatures reached in subassembly S2 were lower as
compared to that in subassembly S1, and this can be attributed to thermal shielding effect
provided by the concrete slab to the top flange of the beams.
Figs. 4.6 and 4.7 show that the predicted secondary beam temperatures in both assemblies were
initially conservative because the computational thermal model does not account for the
evaporation of the insulation’s residual water near 100°C, which slowed the initial increase of
temperature. The discrepancy in the top flange temperature during the cooling phase might be
due to the movement of thermocouple when the top flange experienced local buckling.
Temperature variation between the computational and experimental results throughout the time
series may also have been caused by variation in the actual insulation thickness as compared to
the idealized constant 12.7-mm thickness used for computational thermal analysis. In addition,
the discrepancy might also have been caused due to change in thermal properties of insulation
with temperature. However, due to lack of tests data, the elevated temperature properties of
insulation were assumed to be same as that at room temperature in the model. Therefore, it can
be concluded that the developed thermal sub model is capable of predicting steel temperatures
with reasonable accuracy.
4.4.3.2 Structural Response
The structural response of double angle connections can be assessed by examining the
progression of rotation at connection with fire exposure time. The values of rotation at
connections are evaluated using vertical deflection (u) of the secondary beams at the loading
points and the distance from the connection to the loading point (L), as illustrated in Fig. 3.10. A
comparison of predicted and measured rotations in connections is plotted as a function of fire

68

exposure time in Fig. 4.8 for both sub frame assemblies S1 and S2. It can be seen from the figure
that the predicted rotations from the model follows closely with those measured during fire tests.
In both assemblies, predicted rotation at connection gradually increases with time during the
early stages (till 60 minutes for S1 and 90 minutes for S2) of fire. This increase in rotation is due
to deteriorating strength and stiffness properties of steel, as well as increased thermal expansion
of the steel with temperature. However, during the fire test, assembly S1 experienced a sudden
increase in rotation at 65 minutes into the fire test. This is due to local and lateral torsional
buckling of the secondary beam which led to failure of assembly S1. However due to safety
concerns (to mitigate collapse of the test assembly in to the furnace), the applied loading on the
beam was removed and the test was continued to trace the thermal response (65 – 180 minutes)
under cooling phase of fire. The rapid change in rotation during the final stages (beyond 180
minutes), was caused by the significant warping (refer shown in Fig. 3.11) of the metal deck
present on top of secondary beams.
The current model captures the sudden increase in connection rotation and predicts a failure time
of around 60 minutes for S1, which is comparable to 65 minutes failure time measured during
the test. As seen in Fig. 4.9, a closer examination of the deformed shape indicates that the
secondary beam experienced local and lateral torsional buckling similar to that observed during
the test. As the assembly S1 failed at 60 minutes, the response of assembly beyond 60 minutes
cannot be simulated by the model.
The rotation response of assembly S2 progressed similar to that of S1, but the connection in this
assembly experienced higher rotation without any local or global instabilities. This can be
attributed to the higher rigidity provided by the concrete slab which enhanced the overall
strength of the beams and in turn the connections. The rate of increase in the rotation at

69

connection decreased steadily after reaching a maximum value. This is on expected lines because
as soon as the cooling phase of fire started, the beams regained some of their initial strength and
stiffness and thus the rotation at connection started to decrease.
Progression of connection axial forces, predicted from the model for both assemblies S1 and S2
is presented in Fig. 4.10. It can be seen from Fig. 4.10 that axial force in connection for both
assemblies S1 and S2 is compressive in nature initially and it increase gradually. This is due to
the fact that as the beam temperature increases, steel undergoes significant thermal expansion
leading to expansion of the beam. The free thermal expansion of the beam is restrained by the
surrounding (cold) structural members resulting in the development of compressive axial forces.
In assembly S1, the fire induced axial force starts to decrease (as can be seen from change in the
slope of the curve), till failure, after reaching a peak value at 52 minutes. This can be attributed
to degrading strength and stiffness properties of steel with increasing steel temperature. Similarly
in assembly S2, the axial force starts to decrease at about 75 minutes before continuing to
decrease further during the cooling phase of fire. During the cooling phase the beam regains part
of its initial strength and undergoes thermal shrinkage leading to continuous decrease in axial
force. The progression of axial force is in line with the expected trends presented in Chapter 1,
for the case of a typical connection assembly. A summary of predicted and measured thermal and
structural response of both connection assemblies S1 and S2 is presented in Table 4.5.
A comparison of predicted and measured vertical deflections (at loading point) in the secondary
beam for assemblies S1 and S2 are plotted in Fig. 4.11. It can be seen from the figure that the
vertical deflection of assembly S1 increases during early stages of fire (0-60 minutes). This is
due to gradual deterioration of strength and stiffness properties of steel with increasing
temperature. However, the model predicts a sudden increase in vertical deflection around 60

70

minutes. This sudden increase is due to the fact that secondary beam experienced failure by local
and lateral torsional buckling (see Fig. 4.9). The predicted failure time of around 60 minutes for
S1, is comparable to 65 minutes failure time measured during the test. As the assembly S1 failed
at 60 minutes, the response of assembly beyond 60 minutes cannot be simulated by the model.
Similar to S1, the vertical deflection of secondary beam in assembly S2 increases with time
during the initial stage (0-90 minutes) of fire. The increase in deflection can be attributed to
gradual deterioration of strength and stiffness properties of steel with increasing temperature.
The vertical deflection of secondary beam starts to decrease (beyond 90 minutes) after reaching a
maximum value. This is on expected lines because as soon as the cooling phase of fire starts, the
secondary beam regains some of its initial strength and stiffness and thus the vertical deflection
in secondary beam starts to decrease.
Based on the above comparisons, it can be concluded that the proposed finite element model is
capable of simulating the response of assemblies with double angle connections to a good degree
of accuracy.
4.4.4 System Level Connection Model (FEM2)
The second finite element model (FEM2), consisting of system level connection model, was
developed to simulate the fire response of restrained steel frame previously tested by Wang et al.
(Wang et al., 2011). The restrained steel frame has an unprotected beam (without any concrete
slab) connected to column with a different double angle connection configuration compared to
FEM1.
The tested steel frame assembly comprised of a column (UC 254x254x 73 section) connected to
beam (UB 178x102x19 section) through 130 mm deep angles (90x150x10 mm). The geometry
of the selected connection assembly along with the connection details are shown in Fig. 4.12.

71

Since the geometry of the connection assembly is symmetrical about the vertical plane, only one
column and half of the beam are modeled. SOLID70, SURF152 elements are used for the
thermal analysis while SOLID185 elements are used for structural analysis.
The boundary conditions of the connection assembly were simulated as present in the fire test
(Wang et al., 2011). During the test, the bottom of the column was attached to a base plate which
in turn was fixed to the ground using four bolts. The top of the column was allowed to freely
expand axially (longitudinally) in the gap between the column top and the reaction frame, while
it was restrained to move in the lateral direction. To simulate these conditions in the finite
element model, all the nodes at the bottom of the column were completely fixed in all three
directions (x,y and z) while the nodes at the top of the column were fixed in two directions (x
and z) only (free to expand axially in y direction).
In the fire test, the lateral restraining effect of the concrete slab was simulated by connecting
steel truss bars to the beam top flange. However, for simplicity in the current model FEM2 the
truss bars were replaced by 50x8 mm plates of 750 mm long on either side of the beam top
flange, as shown in Fig. 4.13. These plate dimensions were determined such that they contributed
the same amount of bending capacity of the beam (Dai et al., 2010) as that of lateral restraining
truss bars used in experiment. In addition, the top flange of the beam was prevented from any
lateral movement by restraining the nodes in lateral (X) direction. The bolt heads and nuts were
modeled as circular volumes and the root radius between the angle legs, beam/column web and
flange are ignored in the model.
The contact interactions between different parts are defined using “contact pairs” and are
modeled

as

“surface-to-surface”

contact

using

contact

elements

CONTA174

and

TARGE170(ANSYS Inc, 2009c). In this model, there are forty-five contact pairs with 28,171

72

contact elements. In the current model, the contact between interacting surfaces is successfully
incorporated using “surface-to-surface” contact with “no separation but sliding permitted”
option. Based on the results of the sensitivity study, optimum normal contact stiffness (FKN)
value of 0.1 was assumed in the first iteration of the analysis. This value is updated in subsequent
iterations based on the current mean stress of the underlying element and the allowable
penetration of contact elements. In order to accurately capture thermal and structural stress
concentration in the connection region (angles, bolts, bolt holes), a relatively finer mesh was
used within the vicinity of these regions, as depicted in Fig. 4.14.
4.4.5 Validation of System Level Connection Model (FEM2)
The finite element model FEM2 was validated by comparing predictions from the finite element
analysis with measured data on connection assembly tested by Wang et al. (Wang et al., 2011) .
In the fire test, double angles connected to an unprotected beam were subjected to ISO 834
standard fire conditions. The top flange of the beam was covered with a 15 mmm layer of
ceramic fiber blanket to simulate the heat-sink effect of the concrete slab. However, as the
connection assembly was relatively larger as compared to the volume of the furnace and due to
the unsymmetrical arrangement of the gas burners and exhaust inside the furnace, the connection
assembly was not subjected to symmetrical heating conditions(Wang et al., 2011). Further, the
average temperature difference inside the furnace, measured by six thermocouples, was about
200°C lower in comparison to ISO834 standard fire temperature conditions (Wang et al., 2011).
Thus, to account for the unsymmetrical heating and lower temperature distribution, the
connection assembly is analyzed by subjecting it to scaled time-temperature curve (shown in Fig.
4.15) from three sides. This modified fire scenario (referred to as scaled fire scenario hereafter)
is computed by scaling down the standard ISO834 temperatures by 25% i.e., using a scale factor

73

of 0.75. It can be seen from the figure that the maximum temperature difference remains around
200°C for the first 100 minutes of scaled ISO834 fire conditions.
During the fire test, static gravity load was applied on the beam before heating the connection
assembly. The load was applied using two loading actuators located at a distance of 600 mm
from the beam ends (refer to Fig. 4.12(a)). The magnitude of load in each actuator was 40 KN
which corresponds to a load ratio of 0.5 in the beam. Load ratio (LR) is defined as the ratio of
applied bending moment in the beam at the fire limit state to the ambient temperature plastic
bending moment capacity of the beam. To circumvent the problems of high-stress concentrations
and numerical singularities due to external loading, the base of actuator pads was modeled (refer
to Fig. 4.13) and the load is applied as nodal force in vertical direction (y direction).
The mechanical properties of steel used in the model are based on the temperature tensile coupon
tests reported by Wang et al. is shown in Table 4.6. These mechanical properties along with
constitutive relationships and strength reduction factors provided in Eurocode 3 (Eurocode3,
2005b) were used to generate nominal stress-strain-temperature relationships. The nominal
stress-strain relations were then converted into true stress-strain curves and used in the model. As
the connection assembly had an unprotected beam the properties of insulation were not
considered in the model.
Wang et al.(Wang et al., 2011) did not perform any tests to evaluate the mechanical properties of
bolts. Hence, the nominal yield strength, ultimate strength and modulus of elasticity (as specified
in the standards) of bolts were assumed to be 640 MPa, 800 MPa and 210000 MPa respectively.
The strength reduction factors specified in Eurocode 3 (Eurocode3, 2005b) for carbon steel at
elevated temperatures were used. The effect of high-temperature bolt properties on the system-

74

level transient fire performance of connection assembly is discussed in Chapter 5. The following
sections provide details on the validation process.
4.4.5.1 Thermal Response
The predicted beam temperatures from the model are shown in Fig. 4.16. The temperature
progression in the beam from fire tests was not reported by the authors. It can be seen from the
figure that the temperatures in the bottom flange and web of the beam were similar and increase
more rapidly with fire exposure time. However, the top flange experienced much slower rise in
temperature as compared to the web and the bottom flange. This is due to the fact that only top
flange of the beam is insulated and the beam is subjected to three-side fire exposure conditions.
The predicted temperatures follow expected trends within a typical unprotected steel section that
is heated. When a steel section is subjected to increasing temperature, the temperature in steel
section increases rapidly because of high thermal conductivity and specific heat of steel. The rate
of temperature increase depends upon the thickness of steel component (web, flange) as well as
the number of sides of fire exposure (3 side, 4 side exposure). For the same fire exposure
conditions, temperature increase in thicker steel components (ex: flanges) will be slower
compared to thin components (ex: web) because of the high thermal mass that needs to be heated
up.
4.4.5.2 Structural Response
The authors reported the variation of beam mid-span deflection and the development of axial
force in connection from the fire tests. A comparison of predicted and measured mid-span
deflections in the beam and the axial force in the connection is plotted in Fig. 4.17(a) and (b), as
a function of beam bottom flange temperature. It can be seen from Fig. 4.17(a) that the mid-span
deflection increases steadily with increasing bottom flange temperature up to 680°C. This is due
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to gradual deterioration of strength and stiffness properties of steel with increasing temperature.
However, the beam undergoes runaway deflection towards the later stages of fire exposure as
steel has lost most of its strength and can no longer contribute to capacity of the beam to sustain
external loading. For example, steel retains 27% of its ambient temperature capacity at 680°C
and retains only 11% of its initial capacity at 800°C (Eurocode3, 2005b).
The axial forces developed in the connection, predicted from the model, are compared to the
measured values from the fire test in Fig. 4.17(b). It can be seen from the figure that there is a
good agreement between the trends of the predicted and measured axial forces in the entire range
of fire exposure. However, some discrepancies in the values can be attributed to unsymmetrical
heating experienced by the connection assembly during the fire test as reported by Wang et.al.
(Wang et al., 2011). In contrast, in the finite element model, the connection assembly is analyzed
with symmetrical heating conditions.
A review of predicted axial force indicates that model predictions follow expected trends. The
variation of axial force in Fig. 4.17(b) can be grouped into three stages. The axial force in
connection is compressive in nature initially and it increases gradually with increasing beam
temperature during stage 1. After reaching a peak value, the fire induced axial force decreases
with fire exposure time (stage 2) due to temperature induced degradation in strength and stiffness
properties of steel. The axial force continues to decrease and finally transforms into tensile force
(stage 3) when the beam load carrying mechanism changes from “flexural action” to “catenary
action”. During this stage, the beam is held in place (like a cable) by connections.
In addition to the axial force, rotation characteristics of the connection better reflect the
connections behavior. Though no test data was reported by the authors on connection rotation,
the rotation characteristics predicted from the model are illustrated for the sake of completeness.

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The rotations are plotted in Fig. 4.17(c) and the response can be grouped into two stages, before
and after the beam bottom flange comes into contact with the column flange. In the first stage,
the rotation of the connection increases proportional to the external loading until the beam
bottom flange makes contact with the column flange (1.5° rotation and 8% plastic strain in steel
at 52 minutes). After making the contact, the connection exhibits stiff behavior because of the
additional resistance offered by the column flange to the free rotation of the beam flange. A
summary of predicted and measured thermal and structural response of restrained steel frame is
presented in Table 4.7.
Therefore, based on the assumptions considered in modeling the connection assembly and
response predicted from the ANSYS analysis, it can be concluded that the above developed finite
element model FEM2 is capable of simulating the fire response of connection assembly with
reasonable accuracy.
4.5 Summary
This chapter presents the development and validation of two numerical models for simulating the
system-level transient fire response of double angle connection assemblies. Both numerical
models, each comprising of thermal and structural finite element sub models, were developed
using commercially available finite element software ANSYS. All the stages associated with the
fire resistance analysis, namely: thermal and structural analyses, modeling contact interactions,
are explained. The developed models account for high temperature thermal and mechanical
properties, various fire scenarios, geometrical nonlinearities and non-linear contact interactions.
Both the finite element models used system-level approach to simulate the fire response of
connection assemblies. The first finite element model (FEM1) was developed to simulate the fire
response of connection assemblies with and without concrete slab. The second finite element

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model (FEM2) was developed to trace the fire response of restrained steel frame assembly with a
different double angle connection configuration compared to FEM1.
The validity of the thermal and structural models (for FEM1 and FEM2) is established by
comparing the prediction from the analysis with data generated from fire resistance tests. This
validation indicates that two models, FEM1 and FEM2, developed in ANSYS are capable of
simulating the fire response of double angle connection assemblies with good level of accuracy.
In the next chapter, these two finite element models will be applied to carry out a set of
parametric studies to quantify the effect of critical parameters influencing fire response of double
angle connection assemblies.

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CHAPTER 5
5. PARAMETRIC STUDIES
5.1 General
Fire response of connections in steel framed buildings is influenced by a number of factors
including fire induced restraint forces, composite action arising from slab, interactions between
different members in the structural frame. Although fire tests can provide better insight into the
behavior of connections, it is not feasible to undertake large number of fire experiments due to
high complexity, huge costs and time constraints. Further, there are limitations on the number of
variables that can be varied in fire tests, reliable instrumentation for monitoring response and test
facilities for undertaking tests. An alternative to fire tests is the use of numerical models for
tracing the performance of double angle connection. Such numerical models can be applied to
undertake large set of parametric studies and generate data to quantify the influence of various
factors on fire resistance.

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The validated numerical models presented in Chapter 4 are applied to quantify the effect of
various parameters on the fire response of double angle connection assemblies. The varied
parameters include: decay rate, loading type and load level, degree of restraint, composite action
arising from the presence of slab, system-level interactions, heating profile, fire scenarios and
high-temperature properties of bolts.
5.2 Characteristics of Double Angle Connections
Wide range of connection configurations such as all bolted, all welded and welded-bolted are
commonly used in practice. Most common types of bolted connections include endplate, shear
tab and double angle connections. However, bolted double angle connections are unique since
their higher tying capacity can sustain additional axial tensile forces and this is in contrast to
bolted shear connections which cannot withstand the axial forces. Also, double angle
connections experience ductile failure, due to higher ductility, as compared to brittle failure
experienced in other connection types. This ductile failure delays the failure time of connections.
A comparison of rotation characteristics of two types of connections, namely: flush end-plate and
double angle connection along with their moment-rotation (M-ϕ) curves is illustrated in Fig. 5.1.
As explained in Chapter 1 and shown in Fig. 5.1, the response of flush end-plate connections can
be grouped in one stage characterized by three regions. Initially there is an approximately linear
response with increasing rotation, until the onset of yielding in one or more of the connection
components (such as bolts or angles). This is followed by curvilinear response indicating the
yielding of the connection. Finally, as the connection failure is imminent, the rate of rotation
increases rapidly causing an almost flat plateau in the connection response.
However, response of double angle connections can be grouped into two stages as can be seen in
Fig. 5.1, before and after the bottom flange of beam comes into contact with the column flange.

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The first stage response is again characterized by linear and bilinear response similar to that of
flush end-plate connections. During the second stage, when beam bottom flange makes contact
with the column flange, an increase in moment is accompanied by a smaller increase in rotation
due to the resistance offered by the column flange to free rotation of the beam flange. Therefore,
double angle connections can carry additional load by undergoing higher rotations before
experiencing failure when compared to other connection types. For example, previous
experimental studies reported that double angle connections can undergo rotations up to 18° (≈
315 milli-rad) before failure, while other connection configurations can sustain only upto 3.5°
(60 milli-rad rotations) (Yu et al., 2009c). Despite, double angle connections have superior
performance characteristics compared to other connection types; very few studies (refer to
Chapter 2) focused on studying the behavior or isolated double angle connection. Therefore, this
chapter is focused on studying the effect of influencing parameters on the fire response of double
angle connections.
5.3 Comparative Analysis
Detailed information about different levels of analysis that are adopted to predict the fire
response of connection assemblies is discussed in this section. In addition, fire induced forces
obtained using different approaches are compared.
5.3.1 Levels of analysis
The response of connections under fire conditions can be studied at three levels, namely:
component-level, member-level (isolated member) and system-level. The component-level
analysis captures the response at the isolated connection and does not capture the effect of
member interactions (beams, columns) on the response of connection. This is typically used to
get a crude estimate of connection performance and this analysis can be implemented using

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spreadsheet calculations. In this type of analysis, the fire exposure time is divided into several
time steps. At a given time step, the capacity of the connection is calculated by knowing the
temperature of the beam, connection and the high-temperature strength-reduction factors for steel
(connection). Once the temperature in beam is known, the fire induced axial forces generated in
the beam (and in turn in connections) are calculated using the following equation:
[5.1]
where P(t) is the fire induced axial force at any time, ET is the temperature dependent elastic
modulus of beam, A is the cross-sectional area of the beam, α is the coefficient of linear thermal
expansion of steel and ΔT is the change in temperature.
The capacity of the connection is calculated by knowing the temperature of the beam, connection
and the high-temperature strength-reduction factors for steel (connection). Once the connection
capacity and fire induced forces are computed, the failure time of connection is taken to be the
time step at which the connection capacity against the fire induced forces. Failure time of
connection is defined as the time at which connection capacity falls below that of fire induced
forces.
Though the component-level analysis is relatively simple to use it has several drawbacks namely:
(a) this analysis does not account for the fire induced axial forces resulting from the interaction
between structural members on the response of connection; (b) the effect of local beam
instabilities (such as local buckling and lateral torsional buckling) on the connection response
cannot be accounted for in this analysis, and (c) the transformation of fire induced axial forces
from compression phase to tension phase cannot be captured. Therefore, component-level
analysis does not give realistic fire response of connection.

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The member-level analysis (or isolated member analysis) is commonly used to simulate the
response of connections which are not considered to be a part of the structural framing system. A
schematic of typical isolated connection is shown in Fig. 5.2. As can be seen from the figure, as
the beam end is free to expand thermally no fire induced forces will be generated in the beam
and transferred to the connections. However, in reality connections are always an integral part of
a structural framing system and connected beams have some level of restraint (from surrounding
structural members) to free thermal expansion. This restraint to free thermal expansion, leads to
development of fire induced axial forces in beams and connections. Therefore, member-level
analysis also does not capture the realistic fire response of connections.
The system-level analysis refers to comprehensive analysis where in the connections are
considered to be an integral part of the structural framing system. Hence the connection region,
along with the surrounding structural members, is modeled as a system. In the analysis, fire
induced forces developed in the beam are computed through a second order nonlinear analysis
based on actual temperature profiles (in beam and connection), as well as deformed geometry of
beam and connection. Therefore, this type of analysis simulates realistic response of the
connection assembly and can capture the influence of surrounding structural system on the fire
response of connections.
5.3.2 Analysis details
To demonstrate the varying response of connection assemblies, predicted using component-level,
member-level and system-level analysis, three different connection configuration were modeled
under fire exposure. The connection configurations analyzed are: a shear tab (fin plate), a single
angle, and a double angle connection. All the three different connection configurations are
assumed to be connected to same beam (W12x30) section, shown in Fig. 5.3. The beam is

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assumed to be unprotected for analysis purpose while the key mechanical properties of
connections are presented in Table 5.1. The coefficient of linear thermal expansion of steel is
assumed to be equal to 12x10-6 /°C. The bolts are of ASTM A490 grade with 19mm (3/4 inch) in
diameter. The ambient temperature capacity of shear tab, single angle and double angle
connection is equal to 211 kN, 211 kN and 306 kN, respectively. Detailed calculations for
evaluating the capacity of three connection types are presented in Appendix C. For comparative
purposes, the fire induced forces in all three connections are presented for the first 30 minutes of
exposure to parametric fire curve shown in Fig. 5.4.
5.3.3 Comparative Performance
The variation of connection capacity and progression of fire induced axial force for three
connection types is presented in Table 5.2. The fire induced axial forces obtained from the
system-level analysis (Refer to Fig. 4.10 assembly S1 data) is also presented in the table for
comparison purposes. As can be seen from Table 5.2, the fire induced axial force computed
using component-level analysis increases rapidly while the member-level analysis predicts no
axial force. This is on the expected lines because member-level analysis assumes that the
connections are not part of the structural framing system. Therefore, as beam temperature
increases the thermal expansion in the beam is not restrained by structural members resulting in
no axial force. Further, the fire induced forces predicted using component-level analysis
increases rapidly and this can be attributed to the fact that as beam temperature increases (ΔT
increases) the fire induced force increases according to Eq.[5.1].
It can be seen from Table 5.2, the fire induced forces computed using component-level and
system-level analysis increases with increasing fire temperature. However, the fire induced axial
force computed using component-level analysis increases rapidly and gives unrealistic values of

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fire induced forces compared to axial forces obtained from system-level analysis. This can be
attributed to the fact that component-level analysis does not consider the effect of stiffness of
connection and surrounding structural members on the development of fire induced forces in
beam. Hence, the predictions from component level analysis cannot be used for rational design
of connections.
To demonstrate the difference in the double angle connection fire response obtained by adopting
a system-level analysis and member-level (isolated) analysis, the connection assembly presented
in the second finite element model (FEM2) is used. However, the symmetrical boundary
conditions at the beam end, away from the connection region, is removed so that the beam is free
to move axially. Isolated connection assembly along with the modified boundary conditions is
shown in Fig. 5.2. The connection assembly is analyzed by subjecting it to the parametric fire
curve shown in Fig. 5.4.
The variation of axial force and rotation in connection obtained from the analysis is shown in
Fig. 5.5. It can be seen from Fig. 5.5(a) that the axial force in the connection (using system-level
analysis) exposed to parametric fire curve (see Fig. 5.4) is compressive in nature in initial stages
of fire exposure. After reaching a peak value, the fire induced axial force decreases with fire
exposure time and finally transforms to tensile force. On the contrary when the connection is
considered to be isolated (member-level) and exposed to fire it experiences no fire induced axial
force. This is on the expected lines because the connected beam is no longer restrained against
free thermal expansion which leads to zero fire induced axial force.
A comparison of rotation in connection (Fig. 5.5(b)) indicate that the rotation of connection
assembly subjected to parametric fire increases with increasing fire exposure time and then
decreases after reaching a maximum value. This is consistent with the fact that the assembly

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experiences increasing deflection as the temperature increases (during the growth phase of fire
curve). Once the fire enters the decay phase, the beam regains part of its initial strength and
stiffness which leads to reduced deflections.
However, when the connection assembly is analyzed as member-level it experienced much
higher rotation as compared to that analyzed using system-level analysis. This can be attributed
to the absence of the beneficial effect provided by the restraining effect. Beams analyzed at
system-level experience lower deflections compared to isolated ones because the restraining
force generally acts below the neutral axis of any cross-section along the span of the beam. This
develops an arch action which counters the effect of the applied external loading and reduces the
downward deflection of the beam. Therefore, the absence of restraining effect (due to the
structural continuity or system-level) produces higher beam deflections and hence higher
connection rotation.
5.3.4 Connection capacity
As can be seen from Table 5.2, a comparison of connection capacity of all three connection types
and the fire induced forces (obtained from system-level analysis) indicate that the capacity of
shear tab and single angle connection drops below the fire induced forces around 30 minutes. On
the other hand, double angle connection has capacity which is greater than the fire induced
forces. This implies that shear tab and single angle connections will experience failure by 30
minutes while double angle connections do not experience failure. Results from the system-level
analysis indicate that double angle connections experience failure around 60 minutes. This
indicates the robustness of double angle connections and therefore the current study focuses on
tracing the fire response of double angle connections.

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An illustration of the variation of connection capacity and fire induced axial force with
increasing temperature is shown in Fig. 5.6. It can be seen from the figure the connection
capacity decreases with increasing temperature due to degradation in strength and stiffness of
steel. On the other hand the fire induced axial force increases with temperature due to the
increase in thermal expansion of steel beam. The thermal expansion of beam is restrained by the
adjacent structural members resulting in an increase in axial forces. Failure of connection
happens when the connection capacity drops below that of the fire induced axial forces.
5.3.5 Summary
Based on the results, it can be inferred that the (a) response of connection analyzed using systemlevel approach is different from that obtained using member-level (or isolated, component level)
analysis, (b) development and progression of fire induced axial forces cannot be captured when
the connections are analyzed at member-level, (c) beneficial effect of arch action on the
connection is accurately captured through system-level approach (d) beneficial effect of arch
action developed due to structural interactions reduces rotation at connection (e) double angle
connections due to their inherent robustness can sustain additional fire induced forces compared
to shear tab and single angle connections.
Therefore, double-angle connection assemblies are considered as part of this study. Further,
numerical models presented in Chapter 4 and subsequent parametric studies presented in the
following sections are carried out using system-level approach.
5.4 Factors influencing fire resistance
A state-of-the-art review presented in Chapter 2 clearly indicates that several factors govern the
response of double angle connections under high temperature exposure. Previous sub-frame

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assemblies and isolated connection tests clearly show that the main factors influencing the fire
response of double angle connections are:


Decay rate of a design fire,



Loading type and intensity,



Axial restraint stiffness,



Presence of concrete slab,



Composite action of slab,



System-level interactions,



Heating profile (transient vs. steady-state heating),



Fire scenario (duration of growth phase, maximum temperature and total duration of fire),



High-temperature properties of bolts.

In previous studies, many of the above factors were not fully incorporated in modeling the
system-level behavior of connection assemblies. Two different numerical models (FEM1 and
FEM2), presented in Chapter 4, are used to study the influence of each of the above factors on
the fire resistance of sub-frame assemblies containing double angle connections. The first finite
element model (FEM1) simulates the fire response of two sub-frame assemblies one with slab
and the other without slab. The second finite element model (FEM2) simulates the fire response
of restrained steel frame with a different double angle connection configuration compared to
FEM1. The effect of decay rate, loading type, presence of concrete slab and axial restraint is
studied using first finite element model (FEM1) while the effect of heating profile, load level,
fire scenarios and high-temperature bolt properties is studied using second finite element model
(FEM2). The effect of composite action arising from the presence of concrete slab is studied
indirectly by comparing the response of connections in assembly S1 with that of assembly S2.

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Details about the parameters and the results obtained from numerical simulations are presented
in the following sections.
5.5 Parametric Studies
In this section, the parameters influencing transient fire response of double angle connections are
introduced and defined. Range of these parameters analyzed is also discussed. For the first finite
element model (FEM1), parametric studies were conducted by exposing the sub-frame
assemblies to the design fires shown in Fig. 3.5. In the case of second finite element model
(FEM2), parametric studies were conducted by exposing the restrained steel frame assembly to a
parametric fire curve (FS) as illustrated in Fig. 5.4. In this fire scenario there is a growth phase
(increasing temperatures) for the first 90 minutes which is followed by a decay phase with a
cooling rate of 3.5°C/min. This fire scenario is computed based on the parametric fire timetemperature curve proposed in Eurocode1 (Eurocode1, 2002). The fire scenario is assumed to
occur in a typical compartment having dimensions of 6mx5mx3m with an opening factor (Fv) of
0.011 m-1/2, fuel load energy density of 600 MJ/m2 and made up of materials having a thermal
inertia value of 1600 Ws0.5/m2K.
This fire scenario is selected such that the maximum temperature does not exceed 700°C due to
the fact that (a) the emphasis of this study is on the failure of the connections and any premature
failure of the beam should be avoided, and (b) the steel beam analyzed is an unprotected beam.
Beams with external fire protection can sustain fires with much higher temperatures, while
unprotected beams can fail when the fire temperatures exceed 700°C and steel temperatures
exceed 650°C.

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5.5.1 Selection of Connection Assemblies
To study the effect of influencing parameters on fire response of double angle connections, three
connection assemblies presented in Chapter 4 are selected. This includes connection assemblies
S1 and S2 sub-frame assemblies with and without slab, as shown in Fig. 4.5, and a restrained
steel-frame assembly modeled as shown in Fig. 4.11(a). Detailed description of all three
connection assemblies is presented in Chapter 4.
5.5.2 Analysis Details
The effect of influencing parameters (discussed in Section 5.4) on the fire behavior of double
angle connections is analyzed using the two validated finite element models (i.e., FEM1 and
FEM2) presented in Chapter 4. The influence of parameters on the fire performance of all three
(S1,S2 and restrained steel frame) connection assemblies is analyzed at system-level by
subjecting them to transient temperature conditions (except for the case of steady-state
conditions mentioned in Section 5.6.5). Details about the high-temperature constitutive material
properties and failure criteria used in the finite element models are discussed in Chapter 4. In all
the parametric studies, the connection assemblies are analyzed either for the total duration of fire
exposure time or till the connection assembly experienced failure, whichever occurs earlier. If
the assembly experiences failure before the end of fire duration, the failure time is mentioned in
the relevant sections.
5.5.3 Range of Variables
A list of parameters varied, together with the range of these parameters, used in the parametric
studies are summarized in Table 5.3. The range for these parameters is selected to reflect the
typical values encountered in steel framing systems. In practical situations, beams are subjected
to a load ratio in the range of 30% to 70%, under fire conditions, where load ratio (LR) is defined
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as the ratio of applied bending moment in the beam at the fire limit state to the ambient
temperature plastic bending moment capacity of the beam. Depending upon the size of the
connected members and end boundary conditions, beams in connection assemblies can
experience axial restraint ranging from nil to full (100%) restraint. The decay rate of the design
fire depends upon the fuel load, ventilation characteristics, firefighting activities and can vary
from rapid cooling to a slow cooling. Therefore, the decay rates are selected to represent a slow
cooling fire (decay rate of 2°C/min) to rapid cooling fire (decay rate of 22.5°C/min).
Typical fire exposure experienced in buildings consists of a growth phase during which fire
temperature increases and this is followed by a decay phase where the fire cools down. The
presence of cooling phase of fire influences the performance of connections due to the fact that
structural members regain part of their strength and stiffness after undergoing large
deformations. Hence, the effect of cooling phase is studies by subjecting the connection
assembly to design fire scenarios.
High strength bolts of Grade ASTM A325 and A490 are commonly used in construction to
improve the overall connection efficiency. The chemical composition and the heat treatment
processes that the high strength bolts undergo are different from those encountered by
conventional bolts made of carbon (mild) steel Hence, high-temperature properties and the
performance of bolts can be different from that of conventional steels (A36, A992) used for
structural members such as columns and beams. To assess the effect of bolt properties on the
transient fire performance of double angle connections, an analysis was carried out by
incorporating the high-temperature properties of steel specific to two types of bolt grades,
namely ASTM A325 and A490.

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5.6 Results and Discussion
Details about the results obtained from the parametric studies are presented in the following
section. The effect of each of the parameters on the fire response is discussed below.
5.6.1 Effect of Composite Action
The effect of composite action, arising from the presence of slab, on connection response can be
studies by comparing the rotation response of connection assemblies S1 and S2, as shown in Fig.
5.7. It can be seen from the figure that the rotation in both assemblies increases with fire
exposure time during the early stages. This increase in rotation is due to deteriorating strength
and stiffness properties of steel as well as increased thermal expansion of the steel with
temperature. However, assembly S1 experiences sudden increase in rotation and fails around 60
minutes. This sudden increase in rotation is due to local and lateral torsional buckling of the
secondary beam which led to failure of assembly S1 (refer to Fig. 4.9).
On the other hand, rotation at connection in assembly S2 continued to increase and assembly S2
experienced higher rotation without any local or global instabilities. The rate of increase in
rotation at connection decreased steadily after reaching a maximum value. The increase in
rotation can be attributed to the composite action of slab which enhanced the strength retention
of the secondary beam and this in turn the connections. In addition, higher rigidity of concrete
slab stabilized the secondary beam from onset of instabilities such as local and lateral torsional
buckling. The higher rigidity and lateral stability enhanced beam strength enabled connections in
assembly S2 to sustain 40% higher rotation compared to assembly S1. During the final stages of
fire, the decrease in rotation at connection is due to the fact that as soon as the cooling phase of
fire started, the beams regained some of their initial strength and stiffness and thus the rotation at
connection started to decrease.

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5.6.2 Effect of Decay Rate
It is well established that accounting for decay phase in a fire significantly influences the
connection behavior (Garlock and Selamet, 2010; Lennon and Moore, 2003; Wald et al., 2006).
This is due to the fact that the temperature level attained in the beam directly dictates the rate of
recovery of beam’s capacity, stiffness, thermal shrinkage and also the extent of tensile forces
developed in the connection region. To study the effect of decay rate, assembly S1 was subjected
to modified design fire DF1 (see Fig. 5.8(a)) which had a growth phase duration of 45 minutes
instead of 75 minutes used during the fire test (Refer to Fig. 3.5). Similarly, connection assembly
S2 was subjected to modified design fire DF2 (see Fig. 5.8(b)) which had a growth phase
duration of 120 minutes instead of 90 minutes used during the fire test (Refer to Fig. 3.5). Then,
the connection assemblies were analyzed by varying the decay rate of both design fires from
25% to 150% of the initial decay rates (8°C/min for DF1 and 15°C/min for DF2) i.e., the decay
rate of design fire DF1 was varied from 2°C/min to 12°C/min while that of DF2 from
3.75°C/min to 22.5°C/min.
Fig. 5.9 shows the predicted rotation in connections of both assemblies for varying decay rate
scenarios. For assembly S1, it can be seen from the figure that with changing decay rate the
connection response is similar in the initial stages of fire exposure. In all cases, except for
10°C/min, connections experienced failure due to instability in beam (local and lateral torsional
buckling) immediately after the start of decay phase of fire. Therefore, no specific conclusions
can be drawn for assembly S1.
For all decay rates, rotation in connection of assembly S2 progressed similar to that of S1, but
the connection experienced much higher rotation without any instabilities in the beam. This can
be attributed to higher rigidity provided by the composite action of slab which enhanced the
overall strength of the beam and in turn the connection. However, the failure time of the
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assembly decreased with increasing decay rate. This can be attributed to the fact that at higher
decay rates, the beam experiences large thermal gradients which in turn leads to the development
of higher thermal shrinkage forces in connections. Therefore, the connections are not able to
carry forces for a longer duration under higher decay rates as compared to the cases of lower
decay rates of fire.
Based on these observations, it can be inferred that a lower rate of decay (slow cooling rate of a
fire) improves the fire performance of the connection by decreasing the extent of thermal
shrinkage (restraint) forces in connections. The rigidity provided through the composite action of
slab increases the strength of beams and double angle connections which enables connections to
sustain higher rotation.
5.6.3 Effect of Loading Type
The type of loading on the beam can influence the performance of connection, especially under
fire conditions. This is due to effect of loading on resulting bending moment and deflection in
the connected beam and rotation in the connection. Further, the progression of contact forces
between bolt shanks and bolt holes, stress concentrations in the beam and in the connection
region are governed by loading on the beam.
During the fire test, the connection assemblies were tested by applying two point loads (see Fig.
3.4(b)) along each secondary beam. In the analysis, the assemblies are analyzed by applying two
loading types namely, mid-point load and uniformly distributed load along each secondary beam.
A schematic of two loading types, along with the corresponding bending moment diagram is
illustrated in Fig. 5.10. The magnitudes of concentrated loading at mid-span and distributed
loading are selected such that same level of mid-span bending moment is produced in the
secondary beam. The magnitude of concentrated load for the case of mid-point loading was

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111.2 kN and 138.9 kN, while that for uniformly distributed loading was 63.5 kN/m and 79.3
kN/m for connection assemblies S1 and S2, respectively.
The effect of loading type on the fire response of connection can be illustrated by reviewing the
results presented in Fig. 5.11. It can be seen from Fig. 5.11(a) that the connections in assembly
S1, with two point and mid point loading experienced gradual increase in rotation during the
early stages of fire till experiencing sudden failure. This increase in rotation is due to
deteriorating strength and stiffness properties of steel in connection and beam as well as
increased thermal expansion of steel with temperature. However, when the assembly was
analyzed with distributed loading it experienced failure with a sudden increase in rotation at the
start (less than a minute) of the analysis itself. As can be seen from Fig. 5.12(b) a review of the
deformed shape indicated that the secondary beam rotated and experienced lateral torsional
buckling which led to the failure of the assembly.
Compared to S1, rotation in connection for assembly S2 (Fig. 5.11(b)) increased with time
during the early stages (growth phase) of fire followed by a decrease during the later stages
(decay phase) of fire. The increase in rotation during early stages is due to the decreasing
strength and stiffness of the beam with increasing temperatures in steel. The decrease in
connection rotation during later stages can be attributed to the fact that beam regains part of its
initial strength due to decreasing steel temperatures during the decay phase of fire.

The

regaining strength in beam results in lower deflections in beam and which in turn leads to lower
rotation in connection.
It can also be seen from the Fig. 5.11(b) that the connections analyzed with distributed loading
were able to sustain higher rotations. The connections analyzed with distributed loading
experienced and sustained 50% more rotation compared to two point loading and 80% more

95

rotation compared to mid point loading. This can be attributed to the fact that the distributed
loading produced a quadratic increase in moment (Fig. 5.10(c)) as compared to linear variation in
the moment produced by two point and mid point loading. The quadratic moment leads to the
development of higher stresses and deflection in the beam which results in higher rotation in
connection.
The progression of axial force in connections for both assemblies S1 and S2 obtained from the
analysis are shown in Fig. 5.13. It can be seen from the Fig. 5.13(a) that the fire induced forces in
assembly S1 (with mid point and two point loading) is compressive in nature and increases
gradually till failure. This increase in axial force can be attributed to the fact that as the beam
temperature increases, steel undergoes thermal expansion resulting in expansion of the beam.
The thermal expansion of beam is restrained by surrounding structural members leading to the
development of compressive axial forces. However, when assembly S1 was analyzed with
distributed loading it experienced failure less than a minute into the analysis (by lateral torsional
buckling) at early stage of fire exposure.
On the contrary, the type of loading had marginal influence on the progression of fire induced
axial forces in assembly S2 (Fig. 5.13(b)). The fire induced axial forces in assembly S2 increased
with time during the early stages (growth phase) of fire followed by a reduction during later
stages (decay phase) of fire exposure. The increase in axial force is attributed to thermal
expansion of the steel beam (during growth phase of fire) while the decrease in axial force can be
attributed to recovery of strength and stiffness of beams (during decay phase of fire).
Additionally, fire induces forces are compressive in nature for the entire duration of fire
exposure time. The compressive nature of axial forces indicates that the primary load carrying
mechanism in beam is through “flexural action” for the entire duration and the load carrying

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mechanism did not change to “catenary action”. This can be attributed to the fact that the
presence of slab increased the strength of the beam (via composite action) and stabilized the
beam from local instabilities (such as lateral torsional buckling experienced by assembly S1).
The increased strength of beam and enhanced stability of beam enabled assembly S2 to sustain
the applied loading through flexural action.
Based on the progression of rotation in connection, it can be inferred that (a) distributed loading
represent the extreme loading case scenario for both connection assemblies, (b) for connection
assemblies without slab (or laterally unsupported), distributed loading creates instability in the
beam leading to early failure of the assembly, (c) composite action arising from the presence of
slab stabilizes and enhances the load carrying capacity of connection assembly, enabling the
connection assembly to sustaining higher rotations generated by distributed loading.
5.6.4 Effect of Axial Restraint
In a typical building, double angle connections form an integral part of structural framing system
and this fire performance of connections is highly influenced by the effect of adjacent structural
members. This is due to the fact that the adjoining structural members directly influence the
extent of thermal expansion of the heated beam, stresses generated in the beam and connection
region as well as the extent of restraint force applied on the connection. To study this effect, the
connection assemblies were analyzed by assuming the beam to be under four cases of axial
restraint stiffness, namely 25%, 50%, 75% and 100% of the secondary beam stiffness. The axial
restraint considered in this study is the one created by the flexural rigidity of the secondary beam
(Refer to Fig. 4.4) and is assumed to be arising from the presence of secondary beam in the
adjacent sub frame. The value of axial restraint (stiffness) of the beam can be computed as:
[5.2]
97

where k is the value of axial restraint, E,A and L are the room-temperature elastic modulus of
steel, cross-sectional area and length of the secondary beam, respectively.
In the finite element model, the presence of axial restraint is simulated through spar element
(LINK180), available in ANSYS. Nodes at one end of the spar elements were connected to the
perimeter beam while the nodes at the other end were completely fixed in all three directions (x,y
and z), as shown in Fig. 5.14.
Fig. 5.15 shows the progression of rotation in connection as a function of fire exposure time for
different values of axial restraint stiffness. It can be seen from Fig. 5.15(a) that the presence of
axial restraint significantly affects the fire response in assembly S1 and the failure time decreases
with increasing axial stiffness. This can be attributed to the fact that higher axial stiffness
(restraint) limits the extent of free thermal expansion that beam can undergo. The higher the
value of axial restraint, the lower will be free thermal movement in the beam. This restrained
movement in the beam results in the development of higher stresses and local instabilities (local
buckling and lateral torsional buckling) in the beam which ultimately lead to the failure in the
connection. Failure modes in assembly S1 for different values of axial restraint stiffness,
obtained from the analysis, are shown in Fig. 5.16. As can be seen from the figure, increasing
axial restraint from 25% to 100% increases the instability in beam from local buckling to flange
and web crippling (see Fig. 5.16(d)) of the entire beam.
A comparison of rotation in connection, presented in Fig. 5.15(b), indicates that the axial
restraint stiffness has marginal influence on the fire response of assembly S2. This is on expected
lines because assembly S2 had concrete slab which increased the rigidity and enhanced the
strength retention in the secondary beams and thus in turn enhanced the connection response.

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Further, in S2 the presence of slab stabilized secondary beam and prevented it from any local
instabilities compared to assembly S1.
5.6.5 Effect of Steady-state and Transient Heating
The type of heating conditions, steady-state or transient state heating, can influence the response
of connection. In most previous studies a steady-state heating that generates a uniform
temperature across the member and connection was adopted (refer to Chapter 2). To illustrate the
effect of heating conditions on the connection behavior the restrained steel frame assembly is
analyzed by subjecting it to a uniform temperature of 550°C (representing a steady-state heating)
and to a parametric fire (representing a transient heating). All other parameters remained
invariant in the analysis.
The variation of axial force and rotation, in connection, obtained from the analysis under steadystate and transient heating conditions are shown in Fig. 5.17. It can be seen from Fig. 5.17(a) that
the axial force developed in the connection exposed to transient heating conditions under
parametric fire (see Fig. 5.4) is compressive initially. After reaching a peak value, the fire
induced axial force decreases with fire exposure time and finally transforms to tensile force.
However, the connection assembly exposed to a uniform temperature (steady-state) experiences
a constant axial (compressive) force that is higher than that in transient heating conditions. This
change can be attributed to the fact that the connection assembly is subjected to constant
temperature rather than gradual temperature increase as encountered in the case of transient
heating under parametric fire. This higher initial temperature in steady-state heating produce
higher thermal expansion in the beam and this inturn generates higher axial forces in the
connection.

99

A comparison of rotation in connection shown in Fig. 5.17(b) indicate that the rotation of
connection assembly subjected to transient heating under parametric fire increases with
increasing fire exposure time and then decreases after reaching a maximum value. This is
consistent with the fact that the assembly experiences increase in deflection as the temperature
increases (during the growth phase of fire curve). Once the fire enters the decay phase, the beam
regains part of its initial strength and stiffness which leads to reduced deflections. In comparison,
the connection assembly subjected to uniform temperature (steady-state heating) experienced
lower rotation due to the fact that the beam it is subjected to a lower temperature (550°C) as
compared to the maximum temperature in parametric fire (≈700°C). The lower temperature
(heating) produces smaller deflection in the beam and this intun leads to lower rotation in the
connection. As can be seen from Fig. 5.17(b), connection subjected to steady-state heating had a
constant rotation of 0.57° at 60 minutes and 120 minutes into the fire exposure time. On the
contrary, connection subjected to transient heating experienced a rotation of 0.83° and 1.13° at
60 and 120 minutes, respectively.
5.6.6 Effect of Load Level
The level of loading on structural members can significantly influence the performance of
adjoining connection, especially under fire conditions, due to its influence on resulting deflection
in the beam, rotation in connection, contact forces between bolt shanks and bolt holes and the
stress concentration in the connection region. To quantify this effect a parametric study was
carried out by subjecting the beam to four load ratios. In the analysis, the beam was subjected to
an increasing loading corresponding to a load ratio (LR) of 30%, 40%, 60% and 70%, and was
exposed to parametric fire (see Fig. 5.4). Load ratio (LR) defined earlier is the ratio of applied
bending moment in the beam at the fire limit state to the ambient temperature plastic bending

100

moment capacity of the beam. In the fire tests the connection assembly, analyzed here, is tested
by applying a load of 40 kN on the beam and this corresponds to a load ratio of 50%.
The progression of rotation and fire induced axial force in connection is plotted, in Fig. 5.18, as a
function of fire exposure time for varying load levels. It can be seen from Fig. 5.18(a) that the
connections experienced same level of fire induced compressive force, generated due to the
thermal expansion of steel beam, regardless of the applied load ratio. This can be attributed to the
fact that the compressive force is primarily governed by the thermal expansion of the steel beam
which is solely dependent on the temperature and is independent of the applied load ratio.
However, when connection assembly is subjected to increasing load levels (above 50%), the
axial compressive force transformed into tensile force at an earlier times (i.e., the curve shifts
left). This can be attributed to the fact that increasing load level produces larger deflections and
higher stresses in beam. The higher deflections, stresses in beam along with the degraded
capacity of beam (steel) not only results in higher fire-induced (tensile) axial force but also
forces the beam to go into “catenary action” from “flexural action” at earlier stages resulting in
the early development of tensile axial force in connection.
A comparison of rotation in connection, shown in Fig. 5.18(b), indicates that the maximum
rotation experienced by the connection increases with increasing load intensity. Connections
analyzed with a load ratio of 40% experienced marginal increase in rotation as compared to the
connections analyzed with 30% load ratio. However, connections analyzed with load ratios of
50%, 60% and 70% experienced 27%, 61% and 67% higher rotations, respectively, compared to
connections analyzed with a load ratio of 30%. This is on expected lines because an increase in
load produces higher deflections, higher stresses and rapid run-away deflections in the beam,
which inturn generates higher restraint forces, especially in connections.

101

5.6.7 Effect of Fire Scenarios
The fire exposure typically experienced in buildings consists of a growth phase during which the
fire temperature increases with time followed by a decay phase when the compartment cools
down due to limited fuel and/or oxygen supply. Further, it is well established that accounting for
cooling phase of a fire can influence fire performance of connections due to the fact that
members and connection (beam, angle, bolts etc.) regain part of their initial strength and stiffness
after undergoing large deformations (during the growth phase). In order to study the effect of fire
exposure, specifically duration of growth phase, maximum temperature and total duration of fire,
the restrained steel framed connection assembly (in FEM2) was analyzed by subjecting it to three
different design fire scenarios (DF1-DF3), as illustrated in Fig. 5.19.
The parametric fire time-temperature curve proposed in Eurocode1 (Eurocode1, 2002) and the
recent modifications suggested by Feasey and Buchanan (Feasey and Buchanan, 2002) are
implemented to arrive at varying design fire scenarios. These design fires (DF1, DF2 and DF3)
are assumed to occur in a room of dimension 6x4x3 m3. Different compartment characteristics
utilized to establish the design fire scenarios are presented in Table 5.4.
Fig. 5.20 shows progression of fire induced axial force and rotation in connections exposed to
different design fire scenarios (see Fig. 5.19). It can be seen from Fig. 5.20 that the axial force in
is compressive in nature initially and it increases gradually due to increasing beam temperature
and restraint to free thermal expansion of beam. After reaching a peak value, the fire induced
axial force decreases with fire exposure time due to temperature induced degradation in strength
and stiffness properties of steel. The axial force continues to decrease and finally transforms into
tensile force when the beam load carrying mechanism changes from “flexural action” to
“catenary action”.

102

As can be seen in Fig. 5.20(a), the maximum axial compressive force experienced by the
connection changes marginally and is directly proportional to maximum temperature attained in
the growth phase of design fire (due to marginal difference in thermal expansion of steel).
However, the time to reach the maximum compressive force increases with increasing duration
of growth phase. This reinforces the earlier conclusion that the compressive forces (that develop
during the early stages of fire) resulting from the thermal expansion of steel beam solely depends
on the maximum temperature (fire scenario) attained in a steel beam. A summary of the
maximum fire induced forces and rotation experienced by connection for different design fire
scenarios is presented in Table 5.5.
During later stages of fire exposure the axial force in the connection transforms from
compression to tension and increase in magnitude throughout the decay phase. This is due to the
fact that large deflections in the beam (during the growth phase of fire) transform the load
carrying mechanism in the beam from “flexural action” to “catenary action”. In addition, steel
beam regains part of its initial strength and stiffness with decreasing temperatures and thus exerts
more axial force on the connections
Further, the time at which the axial force changes from compression to tension depends only on
the duration of growth phase. The shorter the growth phase duration, the earlier the
transformation time and this is due to the fact that beam loses its strength at a faster rate and
enters the “catenary action” phase earlier when subjected to higher temperatures within a short
period of time.
A comparison of rotations in Fig. 5.20(b) indicates that the maximum rotation (refer to Table
5.5) experienced by the connection is influenced by the duration of the growth phase i.e., the
longer the growth phase duration the higher is the maximum connection rotation. This can be

103

attributed to the fact that an increase in fire temperature decreases the capacity of the beam
leading to higher deflection in the beam and subsequently higher rotation in connection. These
results clearly indicate that the response of the connection and subsequently the entire structural
system is significantly influenced by fire characteristics.
5.6.8 Effect of High-temperature Bolt Properties
Bolts are an integral part of a connection system and thermal and mechanical properties of steel,
which the bolt is made of, plays a crucial role on the response of connections during fire
exposure. In recent years high strength bolts of Grade ASTM A325 and A490 are commonly
used in construction, in place of conventional ASTM A307 bolts, to improve the overall
connection efficiency. The chemical composition and the heat treatment processes that the high
strength bolts undergo are completely different from those encountered by conventional bolts
(A307) made of carbon (mild) steel (Kodur et al., 2011). Recent study has shown that hightemperature properties of A325 and A490 are different from that of conventional bolts made of
steel (A36). Due to lack of property data specific to high strength bolts, high-temperature
constitutive material models (strength degradation relations) developed for carbon steel (A36)
are used in modeling the high-temperature behavior of high strength bolts. To assess the effect of
bolt properties on the transient fire performance of double angle connections, an analysis was
carried out by incorporating specific high-temperature properties of steel for A325 and A490
bolts (Kodur et al., 2011). Two types of bolt grades, namely ASTM A325 and A490, (similar to
Grade 6.8 and Grade 10.9 in Europe practice) are considered to illustrate comparative connection
performance. In addition, analysis was carried out with properties of carbon steel provided in
Eurocode 3. Conclusions are drawn by comparing the axial force and connections rotation with

104

those obtained using carbon steel (Eurocode3) and high-strength bolt steel material properties
used for the connection.
The progression of axial force and rotation in connection for two types of bolts obtained from the
analysis are shown in Fig. 5.21 and Fig. 5.22, respectively. It can be seen from Fig. 5.21 that the
high-temperature properties of bolt steel significantly affects the axial force in connection
resulting in premature failure of the connection. Both bolt types analyzed with conventional
carbon steel bolt properties (those specified in Eurocode3) survived the entire duration of fire
exposure time. On the contrary, A325 bolts analyzed with high-temperature properties specific to
A325 bolts failed at 70 minutes while A490 bolts analyzed with high-temperature properties
specific to A490 bolts failed at 90 minutes into the fire exposure. This can be attributed to the
different heat treatment process that the high strength bolts undergo which significantly affects
thermal conductivity (Kodur et al., 2011) (refer to Appendix B). For example, at 100°C carbon
steel has approximately 8% and 14% higher thermal conductivity compared to high-strength
steel used in A325 and A490 bolt steel, respectively.
Furthermore, compared to conventional steel, high strength steel (used in A325, A490 bolts)
undergoes a rapid degradation in strength beyond 450°C (Kodur et al., 2011). For example, at
500°C, A325 and A490 bolt steel has approximately 55% and 42% lower residual strength,
respectively, compared to carbon steel. The rapid reduction in strength for high strength bolts is
attributed to its different chemical composition, manufacturing process and the microstructure
(Kodur et al., 2011). Therefore, the rapid reduction in strength and thermal conductivity led to
earlier failure of A325 and A490 bolts analyzed with high-temperature properties of bolt steel.
A comparison of rotation in connection (Fig. 5.22) indicates that the rotation tends to increase
with increase in temperature, during the growth phase, and then decreases steadily during the

105

decay phase of fire. However, analysis of the connection assembly using high-temperature bolt
properties lead to earlier connection failure. This can be attributed to the rapid reduction in bolt
strength which produces higher beam deflections and reduces the connection failure time.
Additional numerical simulations were carried out to study the effect of bolt strength (grade) on
the connection performance by using bolt grades commonly used in Europe, namely Grade 6.8
and Grade 10.9. These bolt grades are equivalent to ASTM A325 and ASTM A490 bolt grades
used in North America. The ambient temperature material properties of different bolt grades are
presented in Table 5.6. Results from the simulations indicated that the bolt strength has
negligible effect on the connections axial force and rotation. This indicated that bolts did not
experience forces in excess of their load carrying capacity and failure occurred in the beam prior
to that in bolts (either by tension, block-shear or bearing against beam web).
Based on the above results, it can be inferred that the response of connections made of high
strength bolts can be different from that of bolts with conventional steel due to variation in
properties of steel at elevated temperatures. Therefore, high-temperature steel properties, specific
to the bolt grade, should be accounted for in the analysis to evaluate the true connection
response.
5.7 Summary
This chapter presents results of parametric studies to illustrate the influence of various factors on
the system-level transient fire performance of double angle connection assemblies. The studied
parameters are: effect of decay rate, loading type, degree of axial restraint, effect of composite
action arising from the presence of slab, system-level interactions, heating profile, load level, fire
scenarios and high-temperature bolt properties. Data from parametric studies indicate that:

106



System-level analysis treats connections to be an integral part of structural framing
system and accounts for the effect of surrounding structural members in order to trace the
fire response of double angle connections.



Double angle connections are robust to a higher degree and can sustain additional fire
induced axial forces compared to other connection types (such as shear tab and single
angle).



Presence of concrete slab significantly enhances the rigidity of a steel frame assembly
under fire conditions through composite action and restrains the secondary beam against
instabilities such as lateral torsional buckling.



The failure time of connection assemblies without slab decreases with increasing axial
stiffness because a higher value of axial restraint stiffness restricts the free thermal
movement of the beam. This results in the development of higher stresses and local
instabilities in the beam which ultimately leads to the failure of the connection assembly.



The high-temperature properties of high-strength steel used in bolts are different from
that of carbon steel used in beams and columns and have significant influence on the
connection response. The mechanical properties of high-strength bolts (such as A325 and
A490 bolts) have a significant effect on the development of axial forces and connection’s
rotation while the thermal properties of A490 bolts influences the connection’s axial
force only.

Results from the parametric studies are utilized in Chapter 6 to develop guidelines for evaluating
fire resistance of double angle connections.

107

CHAPTER 6
6. METHODOLOGY FOR EVALUATING FIRE RESISTANCE
6.1 General
Beams under fire conditions can experience large fire induced forces. This is because of the
restraint provided by surrounding (cold) structural members to the free thermal expansion of the
beams. These fire induced forces gets transferred from the heated beam to the connected beam or
column through connections. Further, the fire induced axial forces change from compressive
forces during initial stages of fire to tensile forces in later stages of fire. These additional fire
induced forces on connections are to be considered in evaluating fire resistance of connection
assemblies. There are no simplified methods till date that account for fire induced forces in
evaluating fire response of connections. A methodology is developed in this chapter to account
for the effect of fire induced forces in evaluating fire resistance of double angle connection
assemblies.

108

6.2 Outline of Proposed Methodology
The proposed rational approach for evaluating the fire resistance of connection assemblies
comprises of two steps, namely: developing a methodology to predict the response of
connection. The validity of the proposed methodology is established by comparing the fire
resistance predictions from the methodology with values obtained from numerical studies for
double angle and shear tab connection assemblies. Predictions from the proposed empirical
equation are compared with fire resistance obtained from finite element models. Also the
applicability of the proposed approach to design situations is illustrated through numerical
example.
The proposed methodology uses moment-curvature-axial force (M-κ-P) relationships to trace the
response of double angle connection assemblies in fire over the entire rage of loading up to
collapse. The connected beam is divided into a number of segments along its length. The axial
and rotational stiffness provided by the connections is approximated by using axial and rotational
springs. The mid-section of each segment is assumed to represent the behavior of the whole
segment. The fire resistance analysis is carried out by incrementing time in steps. At each time
interval, the analysis is performed through two main stages, namely:


Pre strength analysis of connection assemblies, which includes:
o Establishing the fire temperature for a given fire exposure condition (ex:
ISO834 or ASTM E119),
o Computing temperatures in the cross-section of beam segment,
o Computing fire-induced axial force developed in the beam,



Perform a strength and rotation analysis on connected beam and connection,
through five sub-steps, namely:

109

o Generate moment-curvature-axial force relationships (using the crosssectional temperature and axial force computed above) for each beam
segment,
o Develop moment profile (bending moment diagram) of the beam using
external loading,
o Determine the curvature profile at each segment of the beam based on the
pre-established moment profile,
o Integrate the curvatures across the span of the beam to evaluate rotation in
connections, and
o Compute the failure time (fire resistance) of connection by checking
connection and beam limit states.
The above steps are illustrated through a flow chart as shown in Fig. 6.1. The first step of the
analysis is to establish fire temperatures, cross-sectional temperatures and fire-induced axial
forces that develop in the beam. Empirical equations developed by previous researchers are
utilized to evaluate temperatures in the beam.
Once the temperature distribution and fire induced axial force are computed, the next step is to
generate time dependent moment-curvature-axial force (M-κ-P) relationships in various beam
segments. These relationships are generated using an iterative approach, by increasing moment
so as to satisfy equilibrium and compatibility criteria for each segment along the span of the
beam.
The moment profile along the span of the beam can be computed knowing external loading.
Using the M-κ-P relationships and pre-established moment profile, curvature distribution is
established in each segment along the length of the beam. Once the curvature profile is

110

established, rotation at connection is calculated by integrating curvatures along the span of the
beam. Knowing all the response parameters, failure state of beam and connection at each time
step is checked by comparing the fire induced axial force in the beam and rotation in connection
against pre-determined failure criteria. The failure criteria include performance-based strength
criteria (for connection) and deflection (in the beam) considerations. Details of the analysis
procedure are presented in the following sub-sections.
6.3 Computing Temperatures
The fire resistance analysis is carried out by incrementing time in steps. At each time step, fire
temperatures are established based on standard or design fire relations given in standards such as
ASTM E119, ISO 834 or Eurocode. Next, for standard fire exposure conditions, the temperature
in beam (steel) cross-section and fire-induced axial forces are evaluated based on the empirical
equations previously developed by Dwaikat and Kodur (Dwaikat and Kodur, 2011, 2013). This
equation for computing steel temperatures is used because of its simplicity to use as well as the
fact that steel temperatures can be computed for any given time without employing an iterative
approach. For design fire exposure conditions, the temperature in beam (steel) cross-section is
computed using a lumped heat capacity approach. As explained below, the empirical equation
provided by Dwaikat and Kodur cannot be applied for design fire conditions.
6.3.1 Standard Fire Exposure
The fire temperatures are calculated by assuming three sides of a beam is exposed to a fire,
whose temperatures follow that of a standard fire exposure ISO834 (ISO, 1975) or ASTM E119
(ASTM, 2011a) fire exposure conditions. The ISO 834 standard fire is similar to ASTM E119
standard fire, as shown in Fig. 6.2. Time-temperature relationship of ISO 834 standard fire is
given by the following equation:

111

[6.1a]
where Tf is fire temperature (in °C) as a function of time t (in minutes).
Similarly, the time-temperature relationship of ASTM E119 standard fire is given by the
following equation:
√

(

)

√

[6.1b]

where Tf is fire temperature (in °C) as a function of time t (in hours).
The fire time-temperature curve can be approximated by a power function as following (Dwaikat
and Kodur, 2013):
[6.2]
where a and n are regression coefficients. For ISO 834 standard fire a = 469.9 and n = 0.1677.
Similarly, for ASTM E119 standard fire a = 496.5 and n = 0.1478.
Based on the standard fire temperatures (Eq. [6.2]), the temperatures in the steel (Ts) beam crosssection can be established by the following empirical relationship (Dwaikat and Kodur, 2013):
[6.3]
where Ts is the steel temperature (in °C) as a function of time t (min), s is a correlation
coefficient which is defined for protected steel sections as:

( ⁄ )
( ⁄

⁄

)(

(

⁄

)(

⁄

)(

where the terms are defined as below
Fp : heated perimeter of the cross section (or steel plate),
As : cross sectional area of the section (or the steel plate),
cs,cp : specific heat of steel and insulation, respectively,
112

⁄ ))

[6.4a]

ρs,ρp : density of steel and insulation, respectively,
h : convective heat flux coefficient,
tp : thickness of insulation,
kp : thermal conductivity of insulation, and
m : constant used for averaging temperature of insulation layer around steel section.
For unprotected steel sections, the value of s can be calculated by using a value of zero for the
insulation thickness (i.e., tp=0) in Eq. [6.4a]) resulting in the following equation:
( ⁄ )

[6.4b]

⁄

Therefore, the above equation can be used to evaluate the temperature of protected and
unprotected steel section. In addition, the above equation can evaluate the temperature of the
entire cross section or for individual parts of the section such as top flange, web and bottom
flange using appropriate parameters (Fp, As etc) in Eq. [6.3] and Eq. [6.4].
6.3.2 Comparison to Other Methods
To demonstrate the accuracy of Dwaikat and Kodur relationship (Eq. [6.3]) in predicting steel
temperatures, temperatures obtained using Dwaikat and Kodur relationship are compared to
temperatures calculated using two other widely used methods namely: best-fit method and stepby-step (lumped heat capacity) method.
The best-fit method for computing steel temperatures includes two equations that are derived
from statistical regression analysis of one-dimensional finite difference solutions of heat transfer
equation. According to this method, the temperature in a contour-insulated steel section can be
computed as follows (Buchanan, 2002):

113

⁄

((

) (

⁄

(

,or

))
⁄
⁄

)

[6.5a]

[6.5b]

Similarly, the temperature in unprotected steel section can be computed as follows:

(

⁄

,or

)

( ⁄

)

[6.5c]

[6.5d]

where t is time in minutes, Tlim is the required or given limiting temperature of steel in °C, Fp/As
is the section factor in m-1, tp is insulation thickness in m and kp is the thermal conductivity of
insulation material in W/m°K.
Similarly, the step-by-step method or lumped heat capacity method provides an equation for
predicting temperatures in a contour-insulated steel section. This method assumes that a steel
member is at a uniform temperature and adopts a quasi-steady-state lumped heat capacity
approach (Buchanan, 2002). According to this approach the increment in temperature of a
protected steel member can be computed as follows:
⁄

[6.6a]

where ΔTs is the temperature rise in steel in °C, kp is the thermal conductivity of insulation
material in W/m°K, cs is the specific heat of steel in J/kg °K, h is the thickness of the insulation
in m,W is the weight of steel per unit length in kg/m, D is the heated perimeter in m, Tf is the fire
temperature in °C, Ts is the steel temperature in °C and Δt is the time increment in seconds.
The increment in steel temperature for an unprotected steel member can be computed using the
following equation:

114

[6.6b]

⁄

, and

[6.6c]
[6.6d]

where ΔTs is the temperature rise in steel in °C,

is the heat transfer coefficient (W/m2 K), cs is

the specific heat of steel in J/kg °K, W is the weight of steel per unit length in kg/m, D is the
heated perimeter in m, Tf is the fire temperature in °C, Ts is the steel temperature in °C and Δt is
the time increment in seconds,

is the radiative heat transfer coefficient (W/m2 K),

is the

convective heat transfer coefficient (W/m2 K), σ is the Stefan-Boltzmann coefficient (5.67*10-8
W/m2 K4) and ε is the effective emissivity.
Temperatures in two representative steel sections, W12x30 and W24x76, are computed using the
best-fit method, step-by-step method and the Dwaikat and Kodur relationship (Eq. [6.3]).
Geometric details of both steel sections are presented in Table 6.1. The schematic of beam
section used in computing temperatures for protected and unprotected steel sections is illustrated
in Fig. 6.3 and Fig. 6.5, respectively. The temperatures of steel sections with fire protection are
presented in Fig. 6.4 while Fig. 6.6 shows the temperatures for unprotected steel sections. For the
case of insulated scenario, both the steel sections are assumed to be protected with 15mm thick
insulation having a thermal conductivity of 0.12 W/m°K.
As can be seen from Fig. 6.4 and Fig. 6.6, temperatures predicted by best-fit method are accurate
in a narrow range of temperatures (400-700°C). This can be attributed to the fact that the best-fit
method is based on statistical regression and is derived for calculating average steel temperatures
in the vicinity of the critical temperature of steel which is 538°C (Buchanan, 2002; Dwaikat and

115

Kodur, 2013). Critical temperature of steel is defined as the temperature at which steel loses
about 50% of its strength.
On the other hand, temperatures predicted by Dwaikat and Kodur relationship and step-by-step
method are similar and there are marginal differences between the two for the entire range of fire
exposure time. Step-by-step (lumped heat capacity) method is based on one-dimensional finite
difference technique and computes steel temperatures incrementally (Buchanan, 2002). Hence,
the accuracy of this approach depends on the time increment used. The smaller the time
increment the better the accuracy of the method, however as the time increment decreases the
computational time increases. Therefore, this method can be used with smaller time steps where
higher accuracy in predicting the steel temperatures is required and for comparison/calibration
purposes. Further, as this method uses incremental approach to compute steel temperatures, in
order to predict the steel temperature at any particular time (say 30 minutes) steel temperatures
should be computed incrementally up to this time. To achieve a high level of accuracy, an
incremental time increment of 10 seconds was used in computing the steel temperatures
presented in Fig. 6.4 and Fig. 6.6.
Temperatures predicted by Dwaikat and Kodur relationship are in good agreement with those
predicted using step-by-step method for the entire duration. The temperatures predicted by
Dwaikat and Kodur relationship are slightly lower compared to step-by-step method. This can be
attributed to the fact that Dwaikat and Kodur relationship approximates radiative heat transfer as
an equivalent convective heat transfer. However, it is important to note that the Dwaikat and
Kodur relationship can be used to predict steel temperature at any given time step, without the
need for incrementing time steps. Eq. [6.3] can be used to compute steel temperatures in one step
(no time-steps) and this is the advantage of using Eq. [6.3] compared to other incremental

116

approaches (such as step-by-step method). The simplicity of use and accuracy in predicting steel
temperatures makes the Dwaikat and Kodur relationship (Eq. [6.3]) a better approach to compute
steel temperatures. Hence, Dwaikat and Kodur relationship is adopted in the proposed approach
to compute steel temperatures.
6.3.3 Design Fire Exposure
Unlike standard fire, compartment temperatures in a design fire increases initially during growth
phase and then decrease once the total available fuel is consumed and/or oxygen supply gets
restricted. Dwaikat and Kodur relationship (Eq. [6.3]) cannot capture the temperatures in steel
when exposed to design fire scenario. This is due to the fact that the exponential term in Eq.[6.3]
increases asymptotically with time which will yield a continuous increase in steel temperature
during the growth phase of fire. During the decay phase of fire, according to Eq. [6.3] when the
time is large the temperature in steel tends to approximate fire temperature (i.e., as t →∞, e-st →
0 and Ts → Tf). However, in reality the temperature in steel increases and decreases slowly
compared to fire temperatures because of the lag effect of steel. Hence, the temperatures in steel
will be higher than fire temperatures during the decay phase of fire. But Eq. [6.3] will predict
steel temperatures equal to that of fire temperatures thereby underestimating the temperatures in
steel. In lieu of using Eq. [6.3] to predict temperatures in steel, the lumped heat capacity method
described earlier is used to predict temperatures in steel sections subjected to design fire
exposure.
To verify the applicability of lumped heat capacity approach temperatures in two representative
steel sections, W12x30 and W24x76 subjected to design fire scenarios are computed using the
step-by-step method. Geometric details of both steel sections are presented in Table 6.1. The
schematic of beam section used in computing temperatures for protected and unprotected steel

117

sections is illustrated in Fig. 6.3 and Fig. 6.5, respectively. The representative design fire selected
for the analysis is design fire DF1 described in Chapter 3. Design fire, DF1, comprised of a
growth phase for the first 75 minutes as per ASTM E119 fire (ASTM, 2011a) and then a decay
phase with a cooling rate of 8°C/min (see Fig. 3.5). The input parameters used in arriving at this
design fire scenario are listed in Table 3.4.
Both the sections are analyzed with and without fire insulation protection on steel. For the case
of insulated scenario, both the steel sections are assumed to be protected with 15mm thick
insulation having a thermal conductivity of 0.12 W/m°K. The progression of temperatures in
steel sections with and without fire protection is presented in Fig. 6.7. It can be seen from the
figure that the temperature in protected and unprotected steel sections increase during the growth
phase of fire and decrease during the decay phase of fire. However, the temperatures in protected
steel increases slowly compared to rapid rise in unprotected steel temperatures. In addition,
temperatures in unprotected steel sections are significantly higher compared to that of protected
steel sections. These observations are on the expected lines because the presence of fire
insulation decreases the amount of heat transmission to steel thereby limiting the temperature
rise in steel.
Next, temperatures in protected and unprotected steel increases slowly compared to fire
temperature during the growth phase of fire. Similarly, during the decay phase of fire steel
temperatures decrease slowly compared to fire temperature. Further, the steel temperatures are
always higher than that of the fire temperatures for the entire duration of decay phase of fire. As
described earlier, this can be attributed to the lag effect of steel which will delay the increase and
decrease of steel temperatures compared to fire temperature.

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Based on these observations, it can be concluded that the step-by-step method is capable of
predicting temperatures in steel with reasonable accuracy. Hence, step-by-step method is adopted
in the proposed approach to compute the temperatures in steel sections exposed to design fire
scenarios.
6.4 Computing Fire Induced Axial Force
Once temperatures in the beam cross-section are established, the fire-induced axial force (P)
generated in the beam can be computed using the relationships developed by Dwaikat and Kodur
(Dwaikat and Kodur, 2011). As shown in Fig. 6.8, this fire induced force acting on the
connection is compressive in nature during the initial stages of fire. The axial force increases
with time (stage 1) and the combined effect of restraining axial force and bending moment
results in initial yielding of the beam. After yielding of steel, the compressive force decreases in
magnitude (stage 2) and finally transforms into tensile force (stage 3). During the final stages of
fire exposure, the beam regains part of its initial strength and undergoes thermal shrinkage
leading to continuous increase in the tensile force until connection fails. During this stage, the
beam is held in place (like a cable) by the connections through catenary action.
The fire-induced axial force (P) can be calculated using the following equations developed by
Dwaikat and Kodur (Dwaikat and Kodur, 2011):
Stage 1: When steel temperature (Ts) is less than steel yield temperature (Ty) (i.e., Ts < Ty),
[6.7]
where P is the fire induced axial force (in kN), Fy is the yield strength (in MPa) and XA is the
axial restraint factor defined as:
(

⁄
⁄

where
119

)

[6.8]

α : coefficient of thermal expansion,
Es : elastic modulus of steel at ambient temperature,
Fy : yield strength of steel at ambient temperature,
As : cross-sectional area of steel section,
Ka : axial restraint stiffness of the connection,
L : total length of the beam, and
a1 : non-dimensional (constant) temperature-dependent reduction of steel elastic modulus.
-

For steel properties as specified in Eurocode 3: a1 = 0.6.

-

For steel properties as specified in ASCE manual: a1 = 0.6829.

The temperature (Ty) at which steel yeilds is given by:
⁄

[6.9]

where
M0 : maximum bending moment in the beam due to gravity load,
My : unfactored yield bending capcity of the section at ambient temperature,
ΔT : thermal gradient in the beam cross-section (difference in temperature of bottom
flange and top flange of beam),
a2 : factor (constant) that accounts for temperature-dependent reduction in steel yield
strength,
-

In case of Eurocode 3 steel properties: a2 = 0.0013.

-

In case of steel properties specified in ASCE manual: a2 = 0.0008.

XR : rotational restraint stiffness factor of the connection which is computed as
(

⁄
⁄

)

Kr : rotational restraint stiffness due to the connection, and
120

[6.10]

I : second moment of area of the section in the direction of the thermal gradient.
The maximum value of fire-induced compressive axial force (

in the connection can be

computed by substituting yield temperature (Ty) into Eq. [6.7]:
[6.11]
Stage 2: When steel temperature (Ts) is greater than steel yield temperature (Ty) but less than
steel catenary temperature (Tc) (i.e., Ty < Ts < Tc),
After reaching a maximum value (stage 2) (refer to Fig. 6.8), the axial compressive force
decreases in magnitude until it reaches zero (P=0) at catenary temperature (Tc), which is defined
as:
(

)

[6.12]

where Mu is the unfactored ultimate bending capacity of the section at ambient temperature.
At any temperature between the yield temperatures (Ty) and the catenary temperature (Tc) (i.e.,
stage 2), the fire induced force is computed by linear interpolation between the point of
maximum compressive force (Ty,

) and the point of catenary (Tc, P=0), as can be seen in

Fig. 6.8.
Stage 3: When steel temperature (Ts) is greater than steel catenary temperature (Tc) but less than
maximum tensile catenary force temperature (Tten,max) (i.e., Tc < Ts< Tten,max),
Beyond catenary temperature (stage 3), the axial force transforms to tensile force and continues
to increase till it reaches a maximum value, as can be seen in Fig. 6.8. The maximum tensile
catenary force (

is computed using the following equations:
(

)

[6.13]

and
[6.14]
121

where
Tten,max : temperature at the maximum tensile catenary force,
a3, a4: regression coefficients (constants) that are dependent on steel properties,
-

In case of Eurocode 3 steel properties: a3 = 1.139, a4 = 0.0013.

-

In case of steel properties specified in ASCE manual: a3 = 1.329, a4 = 0.0014.

The fire induced axial force in the temperature range between the catenary temperature (Tc) and
the maximum tensile catenary force temperature (Tten,max) (i.e., stage 3), is computed by linear
interpolation between the point of catenary (Tc, P=0) and the point of maximum tensile catenary
force (Tten,max, Pten,max).
Therefore, through this approach the fire induced forces generated in the beam and that gets
applied on the connection can be evaluated.
6.5 Strength Analysis
6.5.1. General Analysis Procedure
The next step in the methodology is the strength analysis at the mid-section of each segment of
the beam. The cross-sectional steel temperatures, as generated in the previous section, are used
as input to the strength analysis. For strength analysis, the following assumptions are made:


Plane sections before bending remain plane after bending,



The beam is subjected to uniform fire exposure throughout the span length,



At a given time step, fire induced axial force is constant along the span of the beam,



The connected beam is assumed to be free of instabilities such as local buckling and
lateral torsional buckling, till failure through loss of capacity in connection or excessive
beam deflections.

122

At each time step, the strength analysis is performed in five sub-steps, namely, (1) generate
moment-curvature-axial force relationships in each beam segment, (2) evaluate moment profile
along the length of the beam using external loading and beam length, (3) compute curvature
profile at each segment along the length of the beam using the generated moment-curvature-axial
force relationships and pre-established moment profile, (4) evaluate the rotation in connection by
integrating the curvatures along the span length of the beam, and (e) estimate the failure time (fie
resistance) of connection by applying failure limit states for connection and beam.
The M-κ-P relationships, at various time steps, are generated using the calculated axial force and
constitutive relationships for steel (refer to Appendix B). The interaction between fire induced
axial force and bending moment capacity is automatically accounted for in generating M-κ-P
relationships. In the nonlinear strength analysis, rotation of connection at each time step can be
calculated through an iterative approach that utilizes generated M-κ-P relationships to compute
curvature in each beam segment.
6.5.2 Generation of M-κ-P relationships
The M-κ-P relationships generation, at elevated temperatures, is carried out by dividing the beam
in different segments as described above. An illustration of the beam elevation and cross-section
along with the longitudinal segments and discretized cross-section is shown in Fig. 6.9.
Once the fire induced axial force is computed, the M-κ-P relationships are generated through a
tangential stiffness method proposed by Santathadaporn et al. (Santathadaporn and Chen, 1972).
The generalized stresses and strains in a beam cross-section are shown in Fig. 6.10. In addition,
Fig. 6.10 also shows the sign convention used in this approach for positive values of generalized
stresses and strains. As per this approach and illustrated in Fig. 6.10, if a steel section is
subjected simultaneously to generalized stresses (f) such as bending moment (Mx) and axial force

123

(P) then the corresponding set of generalized strains (X) are bending curvature (κ) and axial
strain (εo). The generalized stresses and strains can be written in vector form as follows:
{ }

{

{ }

{ }

}

[6.15a]
[6.15b]

This approach is based on sectional analysis and the steel cross-section is discretized into a
number of elements as shown in Fig. 6.9(d). Equilibrium state of the section is satisfied when the
applied external forces (generalized stresses) are equal to the generated internal forces i.e.,
- Mx = -∫σ.y.dA

[6.16a]

- P = -∫σ.dA

[6.16b]

where σ is the stress in an element of area dA. The sign for stress σ is assumed to be positive
when it produces tensile stress.
The stress in each element is computed using the constitutive stress-strain relationships of the
material (steel). For example, the ambient temperature stress-strain relationship for a strainhardening steel is given by

{

(

)

[6.17]

where E is the modulus of elasticity, Et is the tangent modulus of elasticity, ε is the strain in the
element, εy is the yield strain of the material, εu is the ultimate strain of the material, σy is the
yield strength of the material and σu is the ultimate strength of the material.
If it is assumed that the plane section remains plane before and after deformation, the strain at
any point in the cross-section can be evaluated as (refer to Fig. 6.11):
[6.18]

124

where y is the distance from centroid to the center of the elemental area dA, εr is the residual
strain in an element and εo is the axial strain in an element. Since the plastic behavior of material
(steel) is load path dependent, rate equations of equilibrium needs to be used to capture the effect
of history of loading on the material behavior. Therefore, once the existing state of stress and
strain for each element is known, the rate equations of equilibrium (derivate of equilibrium
equations i.e., Eq. [6.16]) for generalized stress and generalized strains can be evaluated as
follows:
̇

∫ ̇

̇

[6.19a]

∫ ̇

[6.19b]

Then, the rate of change of stress (derivative of Eq. [6.17]) is given by
̇
̇

̇

{

[6.20]

The strain rate equation (derivative of Eq. [6.18]) becomes
̇

̇

̇

[6.21]

Note that the residual strain (εr) is independent of time.
Equations [6.19], [6.20] and [6.21] can be combined so that they give a set of simultaneous linear
equations which can be written in the following matrix form:
{
where

̇
̇

}

] { ̇̇ }

[

[6.22]

is defined as
∫

[6.23a]

∫

[6.23b]
∫

Equation [6.22] can be rewritten as
125

[6.23c]

{ ̇}

[ ]{ ̇ }

[6.24]

Matrix [ ] is known as tangent stiffness matrix as it represents the tangent of force-deformation
curve as well as the stiffness of the cross-section. The above equation represents a general form
of linear force-deformation relationship which can be incorporated into a numerical program and
solved using an iterative technique(s). It should be noted that Equations [6.23] are applicable for
the elastic regime of the material only. For strain-hardening region, the value of E should be
replaced by the tangent modulus Et.
In this approach, the value of fire induced axial force (P) and the tangent stiffness matrix are
known at the beginning of each time step. Then the applied moment (Mx) is incremented and
Eq.[6.24] is solved to compute the value of curvature and axial strain. Once these values are
known, the internal forces are computed using Eqs. [6.19]-[6.21]. The curvature is iterated until
equilibrium of forces is satisfied i.e., internal force (Eq. [6.19]) must be equal to the external
applied forces. Once equilibrium is satisfied, the value of curvature corresponding to the applied
moment is calculated. Thus, the values of moment and curvature are stored to represent a point
on the moment-curvature curve for a given value of fire induced force. A flow chart illustrating
the steps involved in generating moment-curvature-axial force relationships is shown in Fig.
6.12.
The value of applied moment is incremented to generate subsequent points on the moment
curvature curve. This procedure is repeated at each preselected time step of fire exposure. The
generated M-κ-P relationships are used to trace the performance of connections through
nonlinear structural analysis. The generation of M-κ-P relationships is an important part of the
proposed methodology since these relationships form the basis for the fire resistance analysis of
connection.

126

6.5.3 Iterative solution
Equation [6.24] represents a general form of force-deformation relationship and can be solved
using an iterative technique. A graphical illustration of representative force-deformation curve is
shown in Fig. 6.13. The curve OABC represents the true force-deformation curve. Region OA
represents the linear portion of the force-deformation relationship. {fA} and {XA} be the force
and deformation vectors (Eq. [6.15]), at point A along the curve, which satisfy equilibrium
conditions. The next step is to compute the deformation at point B when the prescribed force is
{fB}. The incremental value of force from point A to point B is
{ ̇}

{ }

{ }

[6.25]

Till point A, the response is linear and thus the tangent stiffness matrix at point A [QA] (Eq.
[6.23]) is known and it represents the slope at point A on the curve. With the applied force
increment (Eq. [6.25]), the increment in deformation is obtained from
{ ̇ }

] { ̇}

[

[6.26]

where [QA]-1 is the inverse of matrix [QA].
The above equation is an approximation (predictor) to the solution as the tangent stiffness matrix
is computed before the force increment occurs. The accuracy of deformation, approximated by
the above equation, can be increased by applying a small increment in the external force.
The first estimated deformation value can be computed by adding {XA} and the incremental
deformation predicted by Eq. [6.26]
{
This deformation{

}

{

}

{ ̇ }

[6.27]

} develops an internal force {f1} that is not in equilibrium with the external

force {fB}. Therefore, the first unbalanced (residual) force can be computed as:
{ ̇}

{ }

{ }
127

[6.28]

Next, the unbalanced force is applied and the correction in displacement vector (corrector vector)
{ ̇ } is computed to eliminate unbalanced forces. Then, the corrector vector is added to the first
estimated deformation values {

} to compute the subsequent estimation of deformation vector.
{ ̇ }

[

] { ̇}

[6.29]

where [Q1]-1 is the inverse of new tangent stiffness matrix [Q1] which corresponds to the state
{f1}and {X1}.
This process of computing residual (unbalanced) force vector and the corresponding correction
in displacement vector is repeated until the unbalanced force becomes zero or is within the
predetermined tolerance limit, which is taken to be 0.1% (i.e., having an accuracy of 99.9%) in
the analysis.
The accuracy and convergence of this method can be improved by applying small increment of
force. If the force increment is small, the first estimate of the displacement vector will be
accurate and subsequent corrections will not be necessary.
6.6 Computer Implementation
6.6.1 Computer Program
The above outlined procedure requires a large amount of computational effort since it involves
use of an iterative approach. To facilitate numerical calculations the above procedure was
incorporated into a custom-built computer program written in MATLAB. Empirical relationships
for computing fire temperature, temperature in beam cross-section along with the fire-induced
axial forces are incorporated into the MATLAB program. The iterative scheme for generating
moment-curvature-axial force relationships, illustrated in Fig. 6.12 and Fig. 6.13, is implemented
in the program as well. Fig. 6.14 shows the flowchart of the numerical procedure associated with
the computer program. The MATLAB code is included in Appendix D.
128

6.6.2 Material properties
Steel properties suggested by Eurocode 3 (Eurocode3, 2005b), and presented in Appendix B,
were incorporated in the MATLAB program. Relevant thermal and constitutive material
properties (stress-strain relationships) of steel along with the high-temperature strength reduction
factors are built into the code. Since the insulation material has significantly low strength and
stiffness, the strength, stiffness contribution from the insulation is neglected. However, thermal
properties of fire insulation are accounted when computing the temperature of beam (steel). For
the current analysis thermal conductivity and dry density of fire insulation are obtained from the
values proposed by recent study (Kodur and Shakya, 2013). It should be noted that there is
limited data on the variation of thermal properties with temperature of the insulation material;
hence thermal properties at room temperature are used in the analysis.
6.6.3 Beam, Connection Idealization
The steel beam is idealized as a set of longitudinal beam segment and cross-section of each beam
segment is further discretized into a number of elements as shown in Fig. 6.9. The number of
longitudinal segments and the number of element divisions in each direction must be specified as
input to the program. Based on the number of elements, the program determines the element size.
Uniform and non-uniform element sizes can be input into the program. The connections (double
angle or any other connection configuration) at the beam ends are idealized as springs having
axial restraint stiffness (Ka) and rotational restraint stiffness (Kr) and these stiffness values need
to be specified as an input.
6.6.4 Input data
The input to the program consists of beam length, cross-sectional properties, number of divisions
in each direction of the cross-section, steel and insulation properties, type of fire exposure
129

(ISO834 or ASTME119) as well as the analysis time and tolerance limits for convergence. The
failure limit states are automatically computed based on the input data for beam and connection.
Thermal and mechanical properties of steel and insulation are built into the program. In addition,
the program also provides the flexibility to allow users to incorporate any general stress strain
relationships as a function of temperature.
6.6.5 Output data
At each time step, the output from the program includes temperatures distribution in the beam
cross-section, fire induced axial force (P), M-κ-P relationships. In addition, the program checks
for failure of connection against pre-defined failure criteria and outputs the failure time (fire
resistance) of the connection.
6.6.6 Analysis of typical beams
The above proposed approach, implemented in MATLAB, is used to generate momentcurvature-axial force relationships for steel beams. This is illustrated for two representative steel
sections, namely W12x30 and W24x76, with 15mm thick fire insulation having a thermal
conductivity of 0.12 W/m° K and specific heat of 900 J/kg K. Geometric details of both steel
sections are presented in Table 6.1. Both these steel sections are assumed to be exposed to
ISO834 standard fire and the generated moment-curvature relationships (which inherently
accounted for axial forces) are shown in Fig. 6.15, as a function of fire exposure time. The fire
induced axial forces corresponding to the moment-curvature relationships (in Fig. 6.15) are also
shown in the same figure. The computed temperatures in both steel sections are presented in Fig.
6.4 and Fig. 6.6.
It can be seen from Fig. 6.15, that there is only a marginal decrease in moment capacity of
W12x30 and W24x76 upto 45 minutes and 60 minutes of fire exposure time, respectively. This
130

is due to the fact that at these times the temperature in W12x30 and W24x76 beams does not
exceed 400°C (Refer to Fig. 6.4 and Fig. 6.6). As can be seen in Appendix B, steel retains almost
all of its strength when exposed to temperatures below 400°C which resulted in marginal change
of moment capacity in both the beams. After 45 and 60 minutes, the moment capacity of beam
decreases with increasing time of fire exposure (increasing time steps) in beam W12x30 and
W24x76, respectively. This can be attributed to the fact that the strength and stiffness of steel
deteriorates as a result of increasing temperatures, beyond 400°C.
6.7 Rotation Analysis
The M-κ-P relationships and the fire induced axial force generated for different segments are
used to trace the rotation of the connections exposed to fire. At each time step, moment profile
along the length of the beam is evaluated using externally applied load and beam length. Using
the moment profile and the M-κ-P relationships the curvature profile along the length of the
beam is established. The curvatures are then integrated about the beam end to compute the
rotation of the connection.
Thus, at any given time step, the temperatures, fire induced axial force, M-κ-P relationships and
connection rotation are known for a given fire exposure condition. These output parameters are
used to evaluate the failure of the connection by checking against connections bearing strength,
block shear strength, bolts shear strength, tension strength of beam, tension strength of
connection. Additionally, deflection limit of beam (beam span/20) is used to compute the
rotation at connection. The connection is assumed to be failed if the rotation at connection
(evaluated through integrating curvatures) exceeds the rotation limit obtained through beam
deflection limit state.

131

6.8 Verification of the proposed methodology
6.8.1 Double Angle Connection
The proposed methodology is verified by comparing is predicted response parameters from the
Matlab analysis with the results of the finite element model presented in Chapter 4. The
secondary beam and double angle connection in the connection assembly S2 is selected for the
validation of the methodology. The secondary beam is made up of W12x30 section and 3505
mm long. The beam is insulated with 12.7 mm (½ in.) spray-on fire resistive material (SFRM) on
all three sides. Geometric details of the beam are shown in Table 6.1 while a summary of
insulation properties used in the analysis is presented in Table 6.2. The beam is analyzed by
exposing it to ASTM E119 (ASTM, 2011a) standard fire time-temperature curve and subjecting
to an uniformly distributed load of 63.5 kN/m. This loading corresponds to a load ratio of 40% of
the secondary beam ultimate moment capacity. Load ratio (LR) is defined as the ratio of applied
bending moment in the beam under fire conditions to the ambient temperature plastic bending
moment capacity of the beam. In the MATLAB program, axial restraint stiffness (Ka) and
rotational restraint stiffness (Kr) were assumed to be equal to 0.1 and 2 times that of the
secondary beam axial and rotational stiffness, respectively. These values are commonly used in
the literature to approximate the axial, rotational stiffness of the connections (Dwaikat and
Kodur, 2011; Eurocode3, 2005a). In addition, the secondary beam is divided into 100
longitudinal segments for the analysis.
The progression of temperature, moment-curvature-axial force relationships in the beam as well
as the rotation at connection predicted from the proposed methodology is presented in Fig. 6.16.
Results from the proposed methodology predicted a failure time (fire resistance) of the
connection as 89 minutes while the finite element indicated that the connection failed in 78

132

minutes. The higher fire resistance predicted in the simplified methodology might have been
caused by the underlying assumptions made in predicting the steel temperatures as well as the
fire induced axial forces. In addition, in simplified Matlab analysis the beam is assumed to be
free of instabilities (such as local and lateral torsional buckling), while the finite element analysis
accounts for beam instabilities. Indeed, results from the finite element analysis show that the
beam web experienced local bucking which led to the failure of connection assembly. In order to
indirectly account for the beam instabilities, the failure time of connection is corrected by
assuming a 10% error margin and thereby reducing the failure time by 10%. Therefore, the
corrected failure time of connection assembly will be 80 minutes which is in close agreement to
that predicted by the finite element model. Hence, it can be concluded that the proposed
methodology is capable of predicting the fire resistance of double angle connection assemblies
with good accuracy.
6.8.2 Shear Tab Connection
To verify the robustness of proposed methodology in predicting fire response of other connection
types, a shear tab connection is also analyzed. The shear tab connection is a part of floor system
from a full-scale building tested as part of 2003 full-scale fire tests performed in Cardington
(Garlock and Selamet, 2010; Lennon and Moore, 2003; Selamet and Garlock, 2010b; Wald et al.,
2006). A plan of the tested compartment and the corresponding beam sizes and connection
details are shown in Fig. 6.17. The shear tab connection used as part of this verification is the
one connecting beam at gridline 1/2 to the girder at gridline D.
The tested assembly had UB365x171x51 girder (6000 mm long) connected to UB305x165x40
beam (9000 mm long), as shown in Fig. 6.17. The girder was connected to the beam using 10mm
thick shear tab (fin plate) connection containing four Grade 8.8 M20 bolts. Both the girder and

133

the beam do not have any external fire protection. The ambient temperature mechanical
properties of different components of the subassembly are presented in Table 6.3.
The sub frame assembly was tested by exposing it a design fire scenario, comprised of a growth
phase for the first 52 minutes and then a decay phase with an approximate cooling rate of
10°C/min. The measured gas temperatures during the test are shown in Fig. 6.18. The shear tab
connection assembly beam was tested by applying a uniformly distributed load of 13.5 kN/m.
This load value corresponds to a load ratio (level) of 56% of beam ultimate moment capacity.
As the proposed methodology needs an equation for fire temperature as an input, a nonlinear
regression analysis using least sum of square of errors analysis is carried out to obtain a best fit
for the fire temperatures measured during the fire test. Accordingly, the growth phase is
approximated using a power function (i.e., Tf = 47.3*t0.76) with a regression (R2) coefficient of
0.95 while the decay phase is approximated using an exponential function ((i.e., Tf = 2393.5*e0.018*t

) with a regression (R2) coefficient of 0.98. A comparison of measured and approximated

fire temperatures are shown in Fig. 6.19.
The Cardington experimental data by Wald et al.(Wald et al., 2006)

does not have any

information about the rotation at connections, and it only provides deflection data in the beam
However, a finite element model was developed by Selamet et al.(Selamet and Garlock, 2010b)
to simulate the response of the same tested shear tab connection assembly. The authors applied a
reduced fire load (with 10% reduced temperature) on the beam and analyzed the shear tab
connection assembly. The concrete slab was not explicitly modeled by the authors. However, the
effect of concrete slab was considered indirectly using vertical springs, whose stiffness is
calculated by conducting a calibration study on the results of Cardington fire test. The response
of the connection assembly predicted from the finite element model was slightly different from

134

that measured during the fire test. This can be attributed to the use of reduced fire load and the
approximation of concrete slab using linear springs. In addition, detailed finite element modeling
has limitations due to uncertainties in modeling highly non-linear contact interactions.
Nonetheless, the finite element model captured the trends in the behavior of shear tab connection
assembly. The authors provided data about the connection rotation at a distance of 100mm from
the connection location and this data is used for comparison purposed as described below. More
details about the finite element model can be found elsewhere (Garlock and Selamet, 2010)
(Selamet and Garlock, 2010b).
A comparison of the observed and predicted rotation of shear tab connection is shown in Fig.
6.20. The shear tab connection did not experience failure during the fire test, while the proposed
methodology predicts a failure time of 44 minutes. However, as can be seen from Fig. 6.20, a
review of the predicted connection rotation indicates that predictions from the proposed
methodology follow expected trends. The rotation in connection gradually increases with
increasing steel temperature and this is due to degrading strength and stiffness properties of steel
with increasing temperature. The connection experiences a sudden increase in rotation at 48
minutes indicating the failure of connection.
The discrepancy between experimental and predicted data can be attributed to (a) the fact that the
concrete slab is approximated as linear springs which might provide vertical restraint to the beam
but the highly non-linear interaction between concrete slab and beam cannot be captured using
linear springs, (b) the authors analyzed the connection assembly with a reduced fire load (with
10% reduced temperature), (c) the shear tab connection had a concrete slab during the fire test
while the shear tab connection analyzed did not have a concrete slab (d) the presence of concrete
slab provides a shielding effect to the beam top flange thereby reducing the beam top flange

135

temperatures. This shielding effect results in higher residual strength of the beam compared to
the beam without concrete slab even for the same fire exposure time, (e) the underlying
assumptions made in predicting the steel temperatures as well as the fire induced axial forces,
and (f) the higher temperatures predicted by the Dwaikat and Kodur relationship (Eq. [6.3])
results in the development of higher fire induced axial forces resulting in early failure time of
connections.
Therefore, based on the underlying assumptions considered in developing the methodology, it
can be concluded that the proposed methodology is capable of predicting fire resistance of shear
tab connections with a reasonable level of accuracy.
6.9 Limitations
The proposed equation expresses fire resistance as a function of structural parameters, and thus it
offer a convenient way to evaluate fire resistance. As the proposed equation is based on the
results from numerical studies, it is necessary to set limits of the applicability on the parameters
such that they are within the range of values used for developing the equation. The proposed
equation is valid for the following range of parameters:


Type of fire exposure: ASTM E119 standard fire, ISO 834 standard fire, or its equivalent.



Fire insulation thickness: 10 – 50 mm.



Load level on beam: 0.2 – 0.6.



Span of beam: 2.8 – 4.2 mm.

6.10 Design Applicability
To demonstrate the applicability of the proposed methodology in design situations, a numerical
example analyzing a typical double angle connection in presented in Appendix E.

136

6.11 Summary
The development of a methodology for evaluating the fire resistance of connection assemblies is
presented in this chapter. The approach is derived based on basic principles and is supported by
detailed parametric studies presented in this chapter. The proposed methodology uses
temperature dependent moment-curvature-axial force (M-κ-P) relationships to trace the response
of double angle connection assemblies in fire over the entire rage of loading up to collapse. To
proposed methodology is incorporated into a custom-built computer program written in
MATLAB. Empirical relationships for computing fire temperature, temperature in beam crosssection along with the fire-induced axial forces, constitutive material properties are incorporated
into the MATLAB program.
The methodology is validated by comparing the predicted response of double angle, shear tab
connections with those measured during fire test. The validation indicates that the methodology
is capable of tracing the fire response of connections with a reasonable level of accuracy.

137

CHAPTER 7
7. CONCLUSIONS AND RECOMMENDATIONS
7.1 General
The fire response of double angle connections was investigated in this study taking into
consideration of transient heating conditions and system-level interactions. In order to generate
experimental data on the behavior of double angle connections under realistic fire conditions, fire
resistance tests were carried out on two sub frame assemblies. The parameters that were varied in
the fire tests included load ratio, decay rate and the presence of concrete slab. Results from these
fire resistance tests, were used to validate finite element models. The finite element models,
developed in ANSYS, accounts for critical factors namely: material and geometrical
nonlinearities, fire induced material property degradation and non-linear contact interactions that
influence the fire response of double angle connections. The finite element models were
validated by comparing the prediction from the model with those measured during fire tests. It is
shown through the comparisons that the models can simulate the behavior, as well as the failure

138

modes, in double angle connection with good accuracy. The validated finite element models
were applied to conduct parametric studies to quantify the influence of various factors on the fire
response of double angle connections. Results from parametric studies and data from the fire
tests were utilized to develop a rational methodology for evaluating the fire resistance of
connections based on engineering principles.
The proposed methodology uses temperature dependent moment-curvature-axial force (M-κ-P)
relationships to trace the fire response of double angle connection at various time steps till
failure. At each time step, the fire induced forces generated from the beams on double angle
connection is taken into consideration in the analysis. The failure of connection is evaluated at
each time step by comparing fire induced axial force in the beam and rotation in connection
against connection capacity. This simplified approach for evaluating fire performance of
connections is implemented in a custom-built MATLAB code and the analysis can be carried out
without the need for complex finite element models. The validity of the proposed methodology is
established by comparing the fire resistance predictions from the methodology with values
obtained from numerical studies for double angle and shear tab connections.
7.2 Key Findings
Based on the information generated in this study, the following key conclusions can be drawn:
1. There have been only few studies on fire performance of double angle connections,
especially under realistic fire, restraint and loading scenarios. The current fire resistance
provisions, developed based on isolated connections under standard fire or uniform
temperature conditions, are prescriptive and do not yield reliable fire resistance. Also, to
date, no simplified rational methodologies for evaluating fire resistance of double angle
connections, based on system-level response, have been developed.

139

2. Double angle connections exhibit higher ductility, better tying capacity, superior
rotational characteristics, and are easy to erect. The higher ductility of the double angle
connections leads to attaining higher rotation prior to failure (about 18° (≈ 315 milli-rad))
rotation prior to failure as compared to conventional shear tab connections where failure
typically occurs at about 3.5° (60 milli-rad).
3. Fire resistance tests on double angle connection sub frame assemblies under realistic fire,
loading and restraint conditions indicate:
o Double angle connections, under fire conditions, can undergo significant rotations
prior to failure despite the connections undergoing permanent residual
deformations.
o Double angle connections are capable of transferring fire induced forces
(especially moments) between connected members despite being designed as
simple shear connections.
o Presence of concrete slab significantly enhances the rigidity of a steel frame
assembly under fire conditions and restrains the secondary beam against
instabilities such as local buckling and lateral torsional buckling.
4. The proposed finite element models takes into consideration transient heating conditions
and system-level response in modeling the response of double angle connections under
realistic fire and loading conditions. The models accounts for material and geometrical
nonlinearities, temperature induced degradation of material properties and highly
complex non-linear contact interactions. Results from finite element studies infer:

140

o The response of connections can be significantly different when modeled using
system-level approach as compared to that under isolated, component level
analysis.
o The development and progression of fire induced axial forces cannot be fully
captured when the connections are analyzed at member-level.
o The beneficial effect of arch action (developed due to structural interactions) on
the response of connection is accurately captured through system-level approach.
The effect of such arch action reduces rotations in a connection.
o Composite action arising from the presence of slab enhances the overall strength
of the connection in addition to eliminating instabilities (local and lateral torsional
buckling) in the connected beam.
5. Data from numerical studies infer that the main factors influencing the fire response of
double angle connections are decay rate of design fire, loading type, axial restraint
stiffness, system-level interactions, load level, fire scenarios and high-temperature
properties of bolts. Specifically:
o Connections analyzed at system-level experience significant fire-induced axial
forces while member-level analysis predicts no axial force in connection.
o Lower rate of decay in a design fire improves the fire performance of connections
by decreasing the amount of thermal shrinkage forces developed in the beam.
Lower shrinkage forces in beam leads to lower fire induced axial forces on
connections.
o Distributed loading on a beam represents the extreme loading case scenario by
producing higher stresses and deflection compared to two-point and four-point

141

loading scenarios. These higher stresses and deflections result in higher rotation in
connection.
o

Higher value of axial restraint stiffness decreases the failure time of double angle
connections without concrete slab by generating instabilities (local buckling,
lateral torsional buckling, web and flange crippling) in the connected beam.

o System-level interactions generate lower beam deflections due to the beneficial
effect of the arch action (developed by the restraining effect arising from
surrounding (cold) structural members). The arch action creates a counter moment
that reduces the beam deflections.
o Steady-state heating conditions produce higher axial forces compared to transient
heating conditions due to higher temperature change in steady-state heating
compared to gradual temperature increase in transient heating.
o The longer the duration of growth phase of fire, the higher is the maximum
rotation developed in the connection because of degradation in capacity of the
connection.
o Use of high-temperature properties specific to high-strength bolts lead to higher
axial forces and connection’s rotation as compared to general properties of
conventional (A36 or A992) steel.
6. The proposed simplified methodology for evaluating fire response of double angle
connections accounts for critical parameters influencing the fire response of connections.
Thus this methodology is capable of predicting the response of connections under
realistic fire, loading and support conditions.

142

7.3 Recommendations for Future Research
This study has advanced the state-of-the-art with respect to fire response of double angle
connections (a) by generating unique test data on the fire performance of full-scale double angle
connections, (b) implementing system-level approach to simulate the transient fire response of
double angle connections, and (c) developing a design methodology to evaluate the fire
resistance of connections. These experimental, numerical and analytical studies are a major step
forward in understanding the realistic (system-level) performance of double angle connections in
steel framed buildings. However, further research is required to fully characterize the complex
behavior of connections especially at system-level. The following are some of the key
recommendations for further research in this area:


The fire resistance test data generated as part of this study is the only test data available
on the system-level response of double angle connections exposed to fire. Restraining
effects arising from structural continuity have a significant influence on the fire
performance of connections. The fire induced restraining forces on connections arising
from multiple-bay or multiple-stories in steel framed building needs to be studied. Future
research can be directed towards generating such test data which is very valuable for
validating numerical models.



The proposed simplified methodology, developed herein, cannot account for the
composite action between concrete slab and the beam. However, the methodology can be
extended to include the effect of composite action arising from the presence of concrete
slab on top of the beam. This can be done by revising the equations to include the effect
of concrete slab and generating moment-curvature-axial force relationships.

143



The proposed methodology can be extended to include the effect of high-temperature bolt
properties in predicting the fire resistance of double angle connections. Improvements in
such constitutive models will help to enhance the accuracy of the model predictions.

7.4 Research Impact
Connections form critical part of a steel framed structure and the behavior of these connections
are governed by their interaction with surrounding structural members. The research undertaken
as a part of this study was aimed at evaluating fire behavior of double angle connections by
considering transient heating conditions and system-level interactions.
The current fire design provisions in most codes and standards are derived from limited number
of standard fire tests conducted on scaled and isolated connection specimens. These provisions
do not consider system-level interactions and thus have a number of drawbacks. Thus, the
current design approaches are not applicable for undertaking performance-based design which
provides rational and cost-effective fire safety solutions. Hence, numerical models are developed
to simulate the realistic fire performance of connections under realistic fire, loading and restraint
conditions. For validating the models, fire resistance tests on double angle connections are
undertaken as part of this study to generate unique test data. The validated models are applied to
study the influence of various factors on the fire response of double angle connections.
Results from the fire tests and parametric studies are utilized to develop a rational methodology
for evaluating the fire resistance of connections. The proposed methodology uses temperature
dependent moment-curvature-axial force (M-κ-P) relationships to trace the response of double
angle connections in fire over the entire rage of loading up to collapse.

144

In summary, the use of the proposed methodology can facilitate a rational fire resistance design
under a performance-based code environment. Such a rational design approach will contribute to
reduced loss of life and property damage in fire incidents.

145

APPENDICES

146

APPENDIX A

The rate of decay for design fire DF1 is computed using the following equation:

( )

√

⁄

√ ⁄

[A3.1]

where
: decay rate (°C/hr)

( )

: reference decay rate (°C/hr)
= 625°C/hr for fire with a growth period less than half an hour
= 250°C/hr for fire with a growth period greater than 2 hours
= for fires having growth phase duration between 0.5 to 2 hrs, the reference decay
rate can be evaluated by linearly interpolating the above two extreme case
reference values

Fv : ventilation factor (m-0.5)
b : thermal inertia (Ws0.5/m2K)
=√
k: thermal conductivity (W/mK)
ρ: density (kg/m3)
cp: specific heat (J/kg K)
The duration of growth phase for design fire DF1 is 75 min (1.25 hrs). Therefore, the reference
decay rate is equal to 437.5°C/hr. Using the compartment parameters presented in Table 3.4, the
rate of decay can be computed as:

147

⁄

√
√

⁄

⁄

148

APPENDIX B
Table B.1 – Constitutive relationships for high temperature properties of carbon steel (Eurocode3, 2005b)



 s E s ,T

2
2
 f sp,T  c  b / a  a  ( sy,T   s )

 s   f sy,T


   st ,T 

1  s

f
sy
,
T

 



su
,
T
st
,
T



0

Stress-strain relationships





0.5



 s   sp,T


 sp,T   s   sy,T 

 sy,T   s   st ,T 

 st ,T   s   su,T 


 s   su,T

Parameters

 sp,T 

f sp,T
E s ,T

 sy,T  0.02

 st ,T  0.15

 su,T  0.2

Functions

c
a 2  ( sy,T   sp,T )  sy,T   sp,T 

E s,T







b 2  c( sy,T   sp,T ) E s,T  c 2
( f sy,T  f sp,T ) 2
c 
( sy,T   sp,T ) E s,T  ( f sy,T  f sp,T )

Values of f sp,T , f sy,T and E s,T can be obtained from Table B.2

149

Table B.1 (cont’d)

(Strain-hardening region)

Thermal strain

Stress-strain relationships

For temperatures below 400°C
50 f su,T  f sy,T  s  2 f y ,T  f su,T

 f su,T
s  
 f su,T 1  20( s  0.15)
0


f u ,T

1.25 f sy,T

  f sy,T (2  0.0025T )

 f sy,T



300C  T  400C 

T  400C

T  300C

Specific heat

(J/kg K)
(W/m K)

1.2  10  5 T  0.4  10  8 T 2  2.416  10  4


 ths  1.1  10  2

2  10  5 T  6.2  10  3



Thermal conductivity

0.02   s  0.04 

0.04   s  0.15

0.15   s  0.2 

 s  0.2


20 o C  T  750 o C 


750 o C  T  860 o C 

860 o C  T  1200 o C 


425  7.73  10 1 T  1.69  10 3 T 2  2.22  10 6 T 3

666  13002

738  T
cs  
545  17820

T  731

650

2

54  3.33  10 T
ks  

27.3

20 o C  T  800 o C 


o
o
800 C  T  1200 C 


150



o
o
600 C  T  735 C 


o
o
735 C  T  900 C 


o
o
900 C  T  1200 C 

20 o C  T  600 o C

Table B.2 – Values for the main parameters of the stress-strain relationships of carbon steel at elevated
temperatures (Eurocode3, 2005b)

f yT

f sp *

Steel temperature T(°C)

fy

fy

E sT *
Es

20

1

1

1

100

1

1

1

200

1

0.807

0.9

300

1

0.613

0.8

400

1

0.42

0.7

500

0.78

0.36

0.6

600

0.47

0.18

0.31

700

0.23

0.075

0.13

800

0.11

0.05

0.09

900

0.06

0.0375

0.0675

1000

0.04

0.025

0.045

1100

0.02

0.0125

0.0225

1200

0

0

0

* f y and E s are yield strength and modulus of elasticity at room temperature

151

Stress-strain relationships

Table B.3 – Constitutive relationships for high temperature properties of concrete(Eurocode2, 2004)

c 

3  f c',T

 

 c1,T  2  
  c1,T









3

,    cu1,T






For ε c1(T) < ε ≤ ε cu1(T) , the Eurocode permits the use of linear as well as nonlinear
descending branch in the numerical analysis.
For the parameters in this equation refer to Table A.2
Specific heat (J/kg C)
for 20°C ≤ T ≤ 100°C

c= 900,
c = 900 + (T - 100),

for 100°C < T ≤ 200°C

c = 1000 + (T - 200)/2, for 200°C < T ≤ 400°C
for 400°C < T ≤ 1200°C

c = 1100,

Thermal capacity

Density change (kg/m3)
ρ = ρ(20°C) = Reference density
for 20°C ≤ T ≤ 115°C
ρ = ρ(20°C) (1 – 0.02(T - 115)/85)
for 115°C < T ≤ 200°C
ρ= ρ(20°C) (0.98 – 0.03(T - 200)/200)
for 200°C < T ≤ 400°C
ρ= ρ(20°C) (0.95 – 0.07(T - 400)/800)
for 400°C < T ≤ 1200°C
Thermal Capacity = ρ × c

152

Table B.3 (cont’d)

Upper limit:

(W/m K)

Thermal conductivity

All types :

kc = 2 – 0.2451 (T / 100) + 0.0107 (T / 100)2
for 20°C ≤ T ≤ 1200°C
Lower limit:
kc = 1.36 – 0.136 (T / 100) + 0.0057 (T / 100)2
for 20°C ≤ T ≤ 1200°C

Siliceous aggregates:
εth = -1.8×10-4+9×10-6T+2.3 × 10 -11T3
for 20°C ≤ T ≤ 700°C

Thermal strain

εth = 14 × 10-3
for 700°C < T ≤ 1200°C

Calcareous aggregates:
εth = -1.2×10-4+6×10-6T+1.4×10 -11T3
for 20°C ≤ T ≤ 805°C
εth = 12 × 10-3
for 805°C < T ≤ 1200°C

153

Table B.4 – Values for the main parameters of the stress-strain relationships of normal strength concrete
at elevated temperatures (Eurocode2, 2004)
Normal Strength Concrete
Temp.
(˚C)

Siliceous Aggregate

f c',T
f c' (20  C )

Calcareous Aggregate

εc1,T

εcu1,T

f c',T
f c' (20  C )

εc1,T

εcu1,T

20

1

0.0025

0.02

1

0.0025

0.02

100

1

0.004

0.0225

1

0.004

0.023

200

0.95

0.0055

0.025

0.97

0.0055

0.025

300

0.85

0.007

0.0275

0.91

0.007

0.028

400

0.75

0.01

0.03

0.85

0.01

0.03

500

0.6

0.015

0.0325

0.74

0.015

0.033

600

0.45

0.025

0.035

0.6

0.025

0.035

700

0.3

0.025

0.0375

0.43

0.025

0.038

800

0.15

0.025

0.04

0.27

0.025

0.04

900

0.08

0.025

0.0425

0.15

0.025

0.043

1000

0.04

0.025

0.045

0.06

0.025

0.045

1100

0.01

0.025

0.0475

0.02

0.025

0.048

1200

0

-

-

0

-

-

154

A325
A490
A325

(Yield and ultimate strength)

1

 y ,T   u ,T  

0.00001T  0.0137T  4.7

 y ,T   u ,T

2

20C  T  350C 

350C  T  800C 

20C  T  350C 
1


 0.000006T 2  0.0094T  3.65 350C  T  700C 
0.000006T 2  0.0094T  3.72 700C  T  800C 



 th  0.0017T  0.1

20 o C  T  1000 o C

 th  0.0015T  0.1

20 o C  T  1000 o C

20 o C  T  600 o C 


o
o
600 C  T  735 C 


2

490  0.39T  0.001T
cp  

15.274T  9438.8

kt  50  0.027T

20 o C  T  735o C

kt  47  0.023T

20 o C  T  735o C

A490

A490

2

490  0.19T  0.0014T
cp  
2

20.79T  0.0118T  7350

A325

A325

A490

(%)
(MJ/m3 °C)
(W/m °C)

Thermal conductivity

Specific heat

Thermal expansion

Strength reduction factors

Table B.5 – Constitutive relationships for high temperature properties of bolt steel (Kodur et al., 2012)

155

20 o C  T  600 o C 


o
o
600 C  T  735 C 


APPENDIX C
C1. Double angle connection strength calculations
1. Bearing strength of beam web
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 57.15 mm
Thickness of web (t) = 6.6 mm
Ultimate strength of beam steel (Fu) = 451 MPa
Clear distance (Lc)= Le –h/2 = 57.15 – 20.63/2 = 46.835 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*46.835*6.6*451
= 167.27 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*6.6*451
= 136.1 kN

Design bearing strength for 3 bolts =

∑

=0.75*3*136.1
= 306.23 kN

(Controls)

2. Bearing strength of double angle
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 31.75 mm
Thickness of angle (t) = 7.94 mm
Ultimate strength of angle steel (Fu) = 459 MPa
156

Clear distance (Lc)= Le –h/2 = 31.75 – 20.63/2 = 21.435 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*21.435*7.94*459
= 93.73 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*7.94*459
= 166.62 kN

Hence, bearing strength per bolt = 97.73 kN
Design bearing strength (per angle) for 3 bolts =

∑

= 0.75*3*93.73
= 210. 88 kN
Therefore, design bearing strength for double angle = 2*210.88 = 421.76 kN.
3. Block shear strength of beam web
Gross area in shear (Agv) = 754.38 mm2
Net area in shear (Anv) = 607.7 mm2
Net area in tension (Ant) = 712.47 mm2
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*451*607.7+451*712.47
= 485.77 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*375*754.38 + 451*712.47
= 491 kN
157

Hence, block shear strength of beam web = 485.77 kN
Design block shear strength of beam web =
= 0.75*485.77
= 364.33 kN

(Controls)

4. Block shear strength of double angle
Gross area in shear (Agv) = 504.19 mm2
Net area in shear (Anv) = 327.72 mm2
Net area in tension (Ant) = 857.12 mm2
Yield strength of angle steel (Fy) = 308 MPa
Ultimate strength of angle steel (Fu) = 459 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*459*327.72 + 459*857.12
= 483.67 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*308*504.19 + 459*857.12
= 486.59 kN

Hence, block shear strength of single angle = 483.67 kN
Block shear strength for double angle = 2*483.67 = 967.34 kN
Design block shear strength of double angle =
= 0.75*967.34
= 725.5 kN
5. Shear strength of bolts
Bolt diameter (d) = 19.05 mm

158

Area of bolt (Ab) =

= 285 mm2

Shear strength of bolt (single shear plane) = Fnv*Ab
= 0.5*1210*285
= 172.43 kN
Shear strength of bolt for double shear = 2*172.43
= 344.85 kN
Hence, the shear strength for 3 bolts (

) = 3*344.85

= 1034.55 kN
Design shear strength of bolts =
= 0.75*1034.55
= 775.92 kN
6. Tensile strength of beam web
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Gross area (Ag) = 5670.96 mm2
Area of holes (Aholes) = 440.06 mm2
Net area (An) = Ag – Aholes
= 5670.96 – 440.06
= 5230.9 mm2
Gross section tensile strength = Fy*Ag
= 375*5670.96
= 2126.61 kN

159

Net section tensile strength = Fu*An
= 451*5230.9
= 2359.14 kN
Design tensile strength of beam (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*2126.61, 0.75*2359.14]
= 1769.35 kN

7. Tensile strength of double angle
Yield strength of angle steel (Fy) = 308 MPa
Ultimate strength of angle steel (Fu) = 459 MPa
Gross area (Ag) per angle = 1714.25 mm2
Area of holes (Aholes) per angle = 529.25 mm2
Net area (An) per angle = Ag – Aholes
= 1714.25 – 529.25
= 1185 mm2
Gross section tensile strength per angle = Fy*Ag
= 308*1714.25
= 528 kN
Net section tensile strength per angle = Fu*An
= 459*1185
= 544 kN
Design tensile strength of single angle (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*528, 0.75*544]
= 408 kN

160

Design tensile strength of double angle (

) = 2*408
= 816 kN

(Controls)

Table C.1: Summary of double angle connection strength computed based on different limit states

Limit state

Component
Beam web
Bearing strength
Double angle
Beam web
Block shear strength
Double angle
Shear strength
Bolts
Beam web
Tensile strength
Double angle
Connection Capacity

161

Value (kN)
306.23
421.76
364.33
725.5
775.92
1769.35
816
306.23

C2. Single angle connection strength calculations
1. Bearing strength of beam web
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 57.15 mm
Thickness of web (t) = 6.6 mm
Ultimate strength of beam steel (Fu) = 451 MPa
Clear distance (Lc)= Le –h/2 = 57.15 – 20.63/2 = 46.835 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*46.835*6.6*451
= 167.27 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*6.6*451
= 136.1 kN

Design bearing strength for 3 bolts =

∑

=0.75*3*136.1
= 306.23 kN
2. Bearing strength of single angle
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 31.75 mm
Thickness of angle (t) = 7.94 mm
Ultimate strength of angle steel (Fu) = 459 MPa

162

Clear distance (Lc)= Le –h/2 = 31.75 – 20.63/2 = 21.435 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*21.435*7.94*459
= 93.73 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*7.94*459
= 166.62 kN

Hence, bearing strength per bolt = 97.73 kN
Design bearing strength (per angle) for 3 bolts =

∑

= 0.75*3*93.73
= 210. 88 kN
3. Block shear strength of beam web
Gross area in shear (Agv) = 754.38 mm2
Net area in shear (Anv) = 607.7 mm2
Net area in tension (Ant) = 712.47 mm2
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*451*607.7+451*712.47
= 485.77 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*375*754.38 + 451*712.47
= 491 kN

Hence, block shear strength of beam web = 485.77 kN
163

(Controls)

Design block shear strength of beam web =
= 0.75*485.77
= 364.33 kN
4. Block shear strength of single angle
Gross area in shear (Agv) = 504.19 mm2
Net area in shear (Anv) = 327.72 mm2
Net area in tension (Ant) = 857.12 mm2
Yield strength of angle steel (Fy) = 308 MPa
Ultimate strength of angle steel (Fu) = 459 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*459*327.72 + 459*857.12
= 483.67 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*308*504.19 + 459*857.12
= 486.59 kN

Hence, block shear strength of single angle = 483.67 kN
Design block shear strength of single angle =
= 0.75*483.67
= 362.75 kN (Controls)
5. Shear strength of bolts
Bolt diameter (d) = 19.05 mm
Area of bolt (Ab) =

= 285 mm2

Shear strength of bolt (single shear plane) = Fnv*Ab

164

= 0.5*1210*285
= 172.43 kN
Shear strength of bolt for double shear = 2*172.43
= 344.85 kN
Hence, the shear strength for 3 bolts (

) = 3*344.85

= 1034.55 kN
Design shear strength of bolts =
= 0.75*1034.55
= 775.92 kN
6. Tensile strength of beam web
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Gross area (Ag) = 5670.96 mm2
Area of holes (Aholes) = 440.06 mm2
Net area (An) = Ag – Aholes
= 5670.96 – 440.06
= 5230.9 mm2
Gross section tensile strength = Fy*Ag
= 375*5670.96
= 2126.61 kN

Net section tensile strength = Fu*An
= 451*5230.9

165

= 2359.14 kN
Design tensile strength of beam (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*2126.61, 0.75*2359.14]
= 1769.35 kN

7. Tensile strength of single angle
Yield strength of angle steel (Fy) = 308 MPa
Ultimate strength of angle steel (Fu) = 459 MPa
Gross area (Ag) per angle = 1714.25 mm2
Area of holes (Aholes) per angle = 529.25 mm2
Net area (An) per angle = Ag – Aholes
= 1714.25 – 529.25
= 1185 mm2
Gross section tensile strength per angle = Fy*Ag
= 308*1714.25
= 528 kN
Net section tensile strength per angle = Fu*An
= 459*1185
= 544 kN
Design tensile strength of single angle (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*528, 0.75*544]
= 408 kN

166

(Controls)

Table C.2: Summary of single angle connection strength computed based on different limit states

Limit state

Component
Beam web
Bearing strength
Single angle
Beam web
Block shear strength
Single angle
Shear strength
Bolts
Beam web
Tensile strength
Single angle
Connection Capacity

167

Value (kN)
306.23
210.88
364.33
362.75
775.92
1769.35
408
210.88

C3. Shear tab connection strength calculations
1. Bearing strength of beam web
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 57.15 mm
Thickness of web (t) = 6.6 mm
Ultimate strength of beam steel (Fu) = 451 MPa
Clear distance (Lc)= Le –h/2 = 57.15 – 20.63/2 = 46.835 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*46.835*6.6*451
= 167.27 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*6.6*451
= 136.1 kN

Design bearing strength for 3 bolts =

∑

=0.75*3*136.1
= 306.23 kN
2. Bearing strength of shear tab
Bolt diameter (d) = 19.05 mm
Bolt hole diameter (h) = 20.63 mm
Edge distance (Le) = 31.75 mm
Thickness of angle (t) = 7.94 mm
Ultimate strength of shear tab (Fu) = 459 MPa

168

Clear distance (Lc)= Le –h/2 = 31.75 – 20.63/2 = 21.435 mm
Bearing strength per bolt

= 1.2*Lc*t*Fu
= 1.2*21.435*7.94*459
= 93.73 kN

Upper limit

= 2.4*d*t*Fu
= 2.4*19.05*7.94*459
= 166.62 kN

Hence, bearing strength per bolt = 97.73 kN
Design bearing strength of shear tab for 3 bolts =

∑

= 0.75*3*93.73
= 210. 88 kN
3. Block shear strength of beam web
Gross area in shear (Agv) = 754.38 mm2
Net area in shear (Anv) = 607.7 mm2
Net area in tension (Ant) = 712.47 mm2
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*451*607.7+451*712.47
= 485.77 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*375*754.38 + 451*712.47
= 491 kN

Hence, block shear strength of beam web = 485.77 kN
169

(Controls)

Design block shear strength of beam web =
= 0.75*485.77
= 364.33 kN
4. Block shear strength of shear tab
Gross area in shear (Agv) = 504.19 mm2
Net area in shear (Anv) = 327.72 mm2
Net area in tension (Ant) = 857.12 mm2
Yield strength of shear tab (Fy) = 308 MPa
Ultimate strength of shear tab (Fu) = 459 MPa
Block shear strength (

) = 0.6*Fu*Anv + Fu*Ant

= 0.6*459*327.72 + 459*857.12
= 483.67 kN
Upper limit (

) = 0.6*Fy*Agv + Fu*Ant
= 0.6*308*504.19 + 459*857.12
= 486.59 kN

Hence, block shear strength of shear tab = 483.67 kN
Design block shear strength of shear tab =
= 0.75*483.67
= 362.75 kN (Controls)
5. Shear strength of bolts
Bolt diameter (d) = 19.05 mm
Area of bolt (Ab) =

= 285 mm2

Shear strength of bolt (single shear plane) = Fnv*Ab

170

= 0.5*1210*285
= 172.43 kN
Shear strength of bolt for double shear = 2*172.43
= 344.85 kN
Hence, the shear strength for 3 bolts (

) = 3*344.85

= 1034.55 kN
Design shear strength of bolts =
= 0.75*1034.55
= 775.92 kN
6. Tensile strength of beam web
Yield strength of beam steel (Fy) = 375 MPa
Ultimate strength of beam steel (Fu) = 451 MPa
Gross area (Ag) = 5670.96 mm2
Area of holes (Aholes) = 440.06 mm2
Net area (An) = Ag – Aholes
= 5670.96 – 440.06
= 5230.9 mm2
Gross section tensile strength = Fy*Ag
= 375*5670.96
= 2126.61 kN

Net section tensile strength = Fu*An
= 451*5230.9

171

= 2359.14 kN
Design tensile strength of beam (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*2126.61, 0.75*2359.14]
= 1769.35 kN

7. Tensile strength of shear tab
Yield strength of angle steel (Fy) = 308 MPa
Ultimate strength of angle steel (Fu) = 459 MPa
Gross area (Ag) per shear tab = 1714.25 mm2
Area of holes (Aholes) per shear tab = 529.25 mm2
Net area (An) per shear tab = Ag – Aholes
= 1714.25 – 529.25
= 1185 mm2
Gross section tensile strength shear tab = Fy*Ag
= 308*1714.25
= 528 kN
Net section tensile strength per shear tab = Fu*An
= 459*1185
= 544 kN
Design tensile strength of shear tab (

) = min [0.9* Fy*Ag, 0.75* Fu*An]
= min [0.9*528, 0.75*544]
= 408 kN

172

(Controls)

Table C.3: Summary of shear tab connection strength computed based on different limit states

Limit state

Component
Beam web
Bearing strength
Shear tab
Beam web
Block shear strength
Shear tab
Shear strength
Bolts
Beam web
Tensile strength
Shear tab
Connection Capacity

173

Value (kN)
306.23
210.88
364.33
362.75
775.92
1769.35
408
210.88

APPENDIX D
clc;
clear all;

Inputdata
%% Calling the function to compute the Node/Element numbering
[a,b,poin] =
Node_Num(mem,nodes,nxtf_tot,nytf_tot,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,xtf,yt
f,xw,yw,xbf,ybf);
%% Calculating the function to compute the x,y coordinate numbering
[x1,x2,x3,x4,y1,y2,y3,y4,x,y,xx,yyy] =
Coordinate_Num(mem,nxtf_tot,nytf_tot,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,xtf,yt
f,xw,yw,xbf,ybf,poin);
RLS = 0.1;
column = 0;
tt = 0:1:100;
data_arr
= cell(1,2,length(tt));
%% Computing the fire,steel temperature
for i=1:length(tt)
time = tt(i);
[p,tf,ts,fy,fu,es,sx,Ix,zx,xvec,Mvec] =
Axial_F(As,time,Props,w,h,T,t);
Axl_force(i) = p;
[Cap] = Connection_cap(t,As,ts,fy,fu);
Cnctn_cap(i) = Cap;
% Calling the function to compute the M-phi curves
[mom,phi,axl,eps] =
Moment1(p,tf,ts,fy,fu,es,a,b,x1,x2,x3,x4,y1,y2,y3,y4,x,y,mem,nxtf_tot,nytf_to
t,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,sx,Ix,As,zx);
end
column = column+1;
data_arr{1,1,i}
data_arr{1,2,i}

= mom/1e6;
= phi*1e6;

end
Moment = data_arr{1,1,1};
Curvature = data_arr{1,2,1};
grid off;
grid on;
plot(Curvature,Moment);

174

xlabel('Curvature (1/km)');ylabel('Moment (kN-m)');title('M-k
relationships');
for i=1:length(tt)
Moment = 0;
Curvature = 0;
Moment = data_arr{1,1,i};
Curvature = data_arr{1,2,i};
Curvvec(:,i) = interp1(Moment,Curvature,abs(Mvec),'spline');

Curvvec(isnan(Curvvec))=0;

% Checking to eliminate NaN

for i=1:length(tt)
Curvature = Curvvec(:,i);
Curvature = Curvature*(1e-6);
rotation(i) = trapz(xvec,Curvature);
rotation(i) = rotation(i)*(180/pi);
end
flag = 0;
for i=1:length(tt)
if((Cnctn_cap(i)-Axl_force(i))<=0)
failure_time(1) = tt(i);
flag = 1;
end
end

flag = 0;
for i=1:length(tt)
if(abs((rotation(i)*(pi/180)))>RLS)&&(flag==0)
failure_time(2) = tt(i);
flag = 1;
end
end
data = [tt' rotation' (rotation*(pi/180))'];

function [a,b,poin] =
Node_Num(mem,nodes,nxtf_tot,nytf_tot,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,xtf,yt
f,xw,yw,xbf,ybf)
%% Assigning values to temporary variables
nxtf
= nxtf_tot;
nytf
= nytf_tot;
nxw
= nxw_tot;
nyw
= nyw_tot;
nxbf
= nxbf_tot;
nybf
= nybf_tot;
%% Computing the node numbers and element dimensions
%$$$$$$$ TOP FLANGE

$$$$$$$$

175

% Loop to account for numbering in the top flange
for i = 1:nytf
for j = 1:nxtf
m
= (i-1)*nxtf+j;
a(m)
= xtf(j);
b(m)
= ytf(i);
poin(m,1)
poin(m,2)
poin(m,3)
poin(m,4)

=
=
=
=

i*(nxtf+1)+j;
i*(nxtf+1)+j+1;
(i-1)*(nxtf+1)+j+1;
(i-1)*(nxtf+1)+j;

end
end
% Tracking the numbe rof members, node numbers allocated in top flange
nm = nxtf*nytf;
nn = (nytf+1)*(nxtf+1);
% Temporary variables used to offset the node numbering
nxtf_left
= (nxtf-nxw)/2;
nxtf_right = nxtf_left;
nxbf_left
nxbf_right

= (nxbf-nxw)/2;
= nxbf_left;

%$$$$$$$$ WEB $$$$$$$$
% Loop to account for numbering in the web
m
= nm;
for i=1:nxw
m
= nm+i;
a(m)
= xw(i);
b(m)
= yw(i);
poin(m,1)
poin(m,2)
poin(m,3)
poin(m,4)

=
=
=
=

nn+i;
nn+i+1;
(nxtf+1)*(nytf)+(nxtf_left+1+1)+(i-1);
(nxtf+1)*(nytf)+(nxtf_left+1)+(i-1);

end
nn
nm

= nn+nxw+1;
= nm+nxw;

% Assigning node number from 2nd row to last but one (n-1) row of the web
for i=1:(nyw-2)
for j = 1:nxw
m
= nm+(i-1)*nxw+j;
a(m)
= xw(j);
b(m)
= yw(i+1);
poin(m,1)
poin(m,2)
poin(m,3)
poin(m,4)

=
=
=
=

((i-1)*(nxw+1))+j+nn;
(i-1)*(nxw+1)+j+1+nn;
(i-2)*(nxw+1)+j+1+nn;
(i-2)*(nxw+1)+j+nn;

176

end
end
nn
nm

= (nyw-2)*(nxw+1)+nn;
= m;

% Assigning node number in the last row of the web
for i=1:nxw
m
= nm+i;
a(m)
= xw(i);
b(m)
= yw(nyw);
poin(m,1)
poin(m,2)
poin(m,3)
poin(m,4)

=
=
=
=

nn+i+nxbf_right;
nn+i+1+nxbf_right;
nn+(i-nxw);
nn+(i-nxw)-1;

end
nm = nm+nxw;
temp_nn = nn+(nxbf+1);
%poin
%$$$$$$$ BOTTOM FLANGE
for i = 1:nybf
for j = 1:nxbf

$$$$$$$$

m = (nytf*nxtf)+(nyw*nxw)+(i-1)*(nxbf)+j;
a(m) = xbf(j);
b(m) = ybf(i);
poin(m,1)
poin(m,2)
poin(m,3)
poin(m,4)

=
=
=
=

temp_nn
temp_nn
temp_nn
temp_nn

+
+
+
+

(i-1)*(nxbf+1)+j;
(i-1)*(nxbf+1)+j+1;
(i-2)*(nxbf+1)+j+1;
(i-2)*(nxbf+1)+j;

end
end

function [x1,x2,x3,x4,y1,y2,y3,y4,x,y,xx,yyy] =
Coordinate_Num(mem,nxtf_tot,nytf_tot,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,xtf,yt
f,xw,yw,xbf,ybf,poin)
x1=0;
x2=0;
x3=0;
x4=0;
y1=0;
y2=0;
y3=0;
y4=0;

177

x=0;
y=0;
xx=0;
yyy=0;
width_tf
height_tf
width_web
height_web
width_bf
height_bf
nxtf
nytf
nxw
nyw
nxbf
nybf

=
=
=
=
=
=

=
=
=
=
=
=

0;
0;
0;
0;
0;
0;

nxtf_tot;
nytf_tot;
nxw_tot;
nyw_tot;
nxbf_tot;
nybf_tot;

%% Calculating the dimensions of the section
% Calculating the dimensions of top flange
for i=1:nxtf
width_tf = width_tf+xtf(i);
end
for i=1:nytf
height_tf = height_tf+ytf(i);
end
% Calculating the dimensions of web
for i=1:nxw
width_web = width_web+xw(i);
end
for i=1:nyw
height_web = height_web+yw(i);
end
% Calculating the dimension of bottom flange
for i=1:nxbf
width_bf = width_bf+xbf(i);
end
for i=1:nybf
height_bf = height_bf+ybf(i);
end
%% Calculating the coordinates of the nodes
%$$$$$$$ TOP FLANGE
temp_y1 = 0;
temp_y2 = 0;
temp_y3 = 0;
temp_y4 = 0;

$$$$$$$$

178

y_incr_tf = 0;
x_incr_tf = 0;
for i = 1:nytf
if(i==1)
temp_y3
temp_y4
else
temp_y3
temp_y4
end

= temp_y3;
= temp_y4;
= temp_y3 + ytf(i-1);
= temp_y4 + ytf(i-1);

temp_y1 = temp_y1 + ytf(i);
temp_y2 = temp_y2 + ytf(i);
temp_x1
temp_x2
temp_x3
temp_x4

=
=
=
=

0;
0;
0;
0;

for j = 1:nxtf
m = (i-1)*nxtf+j;
if(j==1)
temp_x1
temp_x4
else
temp_x1
temp_x4
end

= temp_x1;
= temp_x4;
= temp_x1 + xtf(j-1);
= temp_x4 + xtf(j-1);

temp_x2 = temp_x2 + xtf(j);
temp_x3 = temp_x3 + xtf(j);
x1(m,1)
x2(m,1)
x3(m,1)
x4(m,1)

=
=
=
=

x_incr_tf
x_incr_tf
x_incr_tf
x_incr_tf

+
+
+
+

temp_x1;
temp_x2;
temp_x3;
temp_x4;

y1(m,1)
y2(m,1)
y3(m,1)
y4(m,1)

=
=
=
=

y_incr_tf
y_incr_tf
y_incr_tf
y_incr_tf

+
+
+
+

temp_y1;
temp_y2;
temp_y3;
temp_y4;

end
end
%$$$$$$$ WEB

$$$$$$$$

x_incr_web
y_incr_web

= 0;
= 0;

x_incr_web
y_incr_web

= (width_bf-width_web)/2;
= height_tf;

179

nm

= nxtf*nytf;

% Accounting for the member number to start in the

web
temp_y1
temp_y2
temp_y3
temp_y4

=
=
=
=

0;
0;
0;
0;

for i = 1:nyw
if(i==1)
temp_y3
= temp_y3;
temp_y4
= temp_y4;
else
temp_y3 = temp_y3 + yw(i-1);
temp_y4 = temp_y4 + yw(i-1);
end
temp_y1 = temp_y1 + yw(i);
temp_y2 = temp_y2 + yw(i);
temp_x1
temp_x2
temp_x3
temp_x4

=
=
=
=

0;
0;
0;
0;

for j =1:nxw
m = nm+(i-1)*nxw+j;
if(j==1)
temp_x1
temp_x4
else
temp_x1
temp_x4
end

= temp_x1;
= temp_x4;
= temp_x1 + xw(j-1);
= temp_x4 + xw(j-1);

temp_x2 = temp_x2 + xw(j);
temp_x3 = temp_x3 + xw(j);
x1(m,1)
x2(m,1)
x3(m,1)
x4(m,1)

=
=
=
=

x_incr_web
x_incr_web
x_incr_web
x_incr_web

+
+
+
+

temp_x1;
temp_x2;
temp_x3;
temp_x4;

y1(m,1)
y2(m,1)
y3(m,1)
y4(m,1)

=
=
=
=

y_incr_web
y_incr_web
y_incr_web
y_incr_web

+
+
+
+

temp_y1;
temp_y2;
temp_y3;
temp_y4;

end
end
%$$$$$$$ BOTTOM FLANGE

$$$$$$$$

180

x_incr_bf = 0;
y_incr_bf = 0;
y_incr_bf = height_tf + height_web;
nm = nytf*nxtf + nyw*nxw;
start in bottom flange
temp_y1
temp_y2
temp_y3
temp_y4

=
=
=
=

%Accounting for the member number to

0;
0;
0;
0;

for i =1:nybf
if(i==1)
temp_y3
temp_y4
else
temp_y3
temp_y4
end

= temp_y3;
= temp_y4;
= temp_y3 + ybf(i-1);
= temp_y4 + ybf(i-1);

temp_y1 = temp_y1 + ybf(i);
temp_y2 = temp_y2 + ybf(i);
temp_x1
temp_x2
temp_x3
temp_x4

=
=
=
=

0;
0;
0;
0;

for j =1:nxbf
m = nm+(i-1)*nxbf+j;
if(j==1)
temp_x1
temp_x4
else
temp_x1
temp_x4
end

= temp_x1;
= temp_x4;
= temp_x1 + xbf(j-1);
= temp_x4 + xbf(j-1);

temp_x2 = temp_x2 + xbf(j);
temp_x3 = temp_x3 + xbf(j);

x1(m,1)
x2(m,1)
x3(m,1)
x4(m,1)

=
=
=
=

x_incr_bf
x_incr_bf
x_incr_bf
x_incr_bf

+
+
+
+

temp_x1;
temp_x2;
temp_x3;
temp_x4;

y1(m,1)
y2(m,1)
y3(m,1)
y4(m,1)

=
=
=
=

y_incr_bf
y_incr_bf
y_incr_bf
y_incr_bf

+
+
+
+

temp_y1;
temp_y2;
temp_y3;
temp_y4;

181

end
end

%% Calculating the location of the geometrical centroid
GC_x
= width_tf/2;
% x coordinate of
geometrical centroid
GC_y
= (height_tf + height_web + height_bf)/2;
% y coordinate of
geometrical centroid
%% Correcting the nodal coordinates to shift the origin to geomtrical
centroid
x1
x2
x3
x4

=
=
=
=

x1
x2
x3
x4

-

y1
y2
y3
y4

=
=
=
=

GC_y
GC_y
GC_y
GC_y

GC_x;
GC_x;
GC_x;
GC_x;
-

y1;
y2;
y3;
y4;

%% Calculate the centroid of each element
for i =1:mem
x(i) = (x1(i,1)+x2(i,1))/2;
y(i) = (y1(i,1)+y4(i,1))/2;
end

function [P,Tf,Ts,Fy,Fu,Es,Sx,Ix,Zx,x_vec,M_vec] =
Axial_F(As,time,Props,w,h,T,t)
%% Input data and temporary variables
alpha = 14e-6;
% Coefficient of thermal expansion of steel
% Steel
rhocs
hcon
Sx
Zx
Ix
Fy
Fu
Es
L
UDL

data
= (7850*450);
= 25;
= 0.6325e6;
= 0.7062e6;
= 99.0630e6;
= 375;
= 451;
= 187500;
= 3505;
= 63.5;

ka
= 0.1;
kr
= 2;
support = 0;
% Insulation data

182

tp
kp
rhocp

= 12.7;
= 0.1;
= (200*900);

% Fire parameters
m
= 2;
n
= 0.1677;
a
= 469.9;
%% Computing the section factors and
% Units: mm
Fp(1)
= 2*T+w-t;
% Fp
Fp(2)
= h-2*T;
% Fp
Fp(3)
= 2*T+w+w-t;
% Fp
Fp(4)
= w+2*h+2*(w-t);
% Fp
% Units: mm2
V(1)
= w*T;
V(2)
= (h-2*T)*t;
V(3)
= w*T;
V(4)
= As;
num
den_t1
den_t2
den

=
=
=
=

s

= num./den;

%
%
%
%

"s" for different parts of the section
for
for
for
for

Area
Area
Area
Area

of
of
of
of

the
the
the
the

the
the
the
the

top flange
web
bottom flange
total section

top flange
web
bottom flange
entire section

(Fp./V)*(10^3);
rhocs*((1/hcon)+((tp/1000)/kp));
1+(rhocp/rhocs)*(Fp./V)*(10^3)*((tp/1000)/m);
den_t1.*den_t2*(n+1);

Tf = a*(time^n);
Ts = Tf*(1-exp(-s*time*60));
Ts(Ts<20)=20;
%% Computing the moments at mid-span and the ends of the beam
if(support==0)
M_mid_span = (UDL*((L/1000)^2))/8;
%Units:kN.m
M_end
= 0;
else
M_mid_span = (UDL*((L/1000)^2))/24;
M_end
= (UDL*((L/1000)^2))/12;
end
Mo
My
Mu

= max(M_mid_span,M_end);
= (Fy*Sx)*(10^-6);
= (Fu*Zx)*(10^-6);

%% Computing the moment distribution along the length of the beam
x_vec = [linspace(0,0.211*L,20),linspace(0.211*L,L/2,31)];
x_vec(21) = [];
M_vec = (UDL/12)*((6*L.*x_vec)-(L*L)-(6.*x_vec.*x_vec));
M_vec = M_vec*(10^-6);

183

%% Axial force calculation
% Factor to compute the axial restraint factor
if (Props==1)
a1 = 0.6;
% Eurocode 3 factor
a2 = 0.0013;
a3 = 1.139;
a4 = 0.0013;
else
a1 = 0.6829;
% ASCE manual factor
a2 = 0.0008;
a3 = 1.329;
a4 = 0.0014;
end
Ka
= (ka*(Es*As/L));
num
= (a1*Ka*L)/(Es*As);
den
= 2*a1+ ((Ka*L)/(Es*As));
Xa = (alpha*Es/Fy)*(num/den);
Kr
= (kr*(Es*Ix/L));
num = 0;
den = 0;
num
= (a1*Kr*L)/(Es*Ix);
den
= 2*a1+ ((Kr*L)/(Es*Ix));
Xr = (alpha*Es/Fy)*(num/den);
deltaT = Ts(3)-Ts(1);
Ty
Tc

= (1-(Mo/My)-(0.5*Xr*deltaT))/(Xa+a2);
= (1/a2)*(1-(Mo/Mu)-(My/Mu)*((Xr*deltaT)/2));

Ttenmax = (Tc*Ty*Xa+a1*a3*(Tc-Ty))/(Ty*Xa+a1*a4*(Tc-Ty));
Pcmax = Fy*As*Xa*(Ty-20);
Ptmax = Fy*As*(a3-a4*Ttenmax);

if(Ts(4)<=Ty)
% Axial force will be in N
P
= (Ts(4)-20)*Xa*Fy*As;
elseif(Ts(4)>Ty) && (Ts(4)<=Tc)
P = Pcmax-Pcmax*((Ts(4)-Ty)/(Tc-Ty));
elseif(Ts(4)>Tc) && (Ts(4)<=Ttenmax)
P = Pcmax-Pcmax*((Ts(4)-Ty)/(Tc-Ty));
else
P = 0;
end

P

= P/1000;

% Converting axial force into kN

function [Cap] = Connection_cap(t_beam,As,ts,Fy_beam,Fu_beam)
n_bolts
n_angle

= 3;
= 2;

% Number of bolts
% Number of angles

184

Bolt_dia = (3/4)*25.4;
h_bearing = (1/16)*25.4;
h_blockshear = (1/8)*25.4;
Bolt_separation = 76.2;

%
%
%
%

Diameter of bolts in mm
Oversize clearance for bearing computations
Oversize clearance for block shear computations
Clear distance between bolt centers

Fy_angle
Fu_angle
t_angle
d_angle

%
%
%
%

Yield strength of angle in MPa
Ultimate strength of angle in MPa
Thickness of angle in mm
Depth of the angle in mm

=
=
=
=

308;
459;
7.94;
215.9;

Fy_bolt = 940;
Fu_bolt = 1210;

% Yield strength of bolt in MPa
% Ultimate strength of bolt in MPa

[Fyt_angle,Fut_angle] = Reduced_props(Fy_angle,Fu_angle,ts(4));
[Fyt_beam,Fut_beam]
= Reduced_props(Fy_beam,Fu_beam,ts(4));
[Fyt_bolt,Fut_bolt]
= Reduced_props(Fy_bolt,Fu_bolt,ts(4));
Le
Fu
Fy
t

=
=
=
=

[57.15, 31.75];
[Fut_beam,Fut_angle];
[Fyt_beam,Fyt_angle];
[t_beam,t_angle];

% Clear length of beam, angle
% Ultimate strength of beam, angle in MPa
% Yield strength of beam, angle in MPa
% Thickness of beam, angle in mm

%##########################################################
%
Calculating bearing strength in Beam web and angles
Lc = Le-(0.5)*(Bolt_dia+h_bearing);
Bearing_LL = 1.2.*Lc.*t.*Fu;;
Bearing_UL = 2.4.*Bolt_dia.*t.*Fu;
Bearing_str =
0.75*min(n_bolts*min(Bearing_LL(1),Bearing_UL(1)),n_angle*n_bolts*min(Bearing
_LL(2),Bearing_UL(2)));
%################################################################
%
Calculating block shear strength of beam web and angle
Agv = 2*t.*Le;
% Accounting for 2 shear planes
Lc_temp = Le-(0.5)*(Bolt_dia+h_blockshear);
Anv = 2*t.*Lc_temp; % Accounting for 2 shear planes

Lc_temp = 0;
Lc_temp = 2*(Bolt_separation-(Bolt_dia+h_blockshear));
Ant = t.*Lc_temp;
Blockshear_LL = 0.6.*Fu.*Anv+Fu.*Ant;
Blockshear_UL = 0.6.*Fy.*Agv+Fu.*Ant;
Blockshear_str =
0.75*min(min(Blockshear_LL(1),Blockshear_UL(1)),n_angle*(min(Blockshear_LL(2)
,Blockshear_UL(2))));

185

%################################################################
%
Calculating the shear strength of bolts
Ab = (pi/4)*(Bolt_dia.^2);
Rn = 2*(0.5*(Fut_bolt)*Ab);
shear
Shear_str = 0.75*n_bolts*Rn;

% Multiplying by 2 to account for double

%################################################################
%
Calculating the tensile strength of beam and angle
Ag = [As, (d_angle*t_angle)];
Aholes = n_bolts.*(Bolt_dia+h_blockshear).*t;
An = Ag-Aholes;
T_grossarea = 0.9*Fy.*Ag;
T_netarea
= 0.75*Fu.*An;
Tensile_str =
min(min(T_grossarea(1),T_netarea(1)),n_angle*min(T_grossarea(2),T_netarea(2))
);
%################################################################
%
Computing the connection capacity
Strength = [Bearing_str,Blockshear_str,Shear_str,Tensile_str]/1000;
Cap = min(Strength);

function [mom,phi,axl,eps] =
Moment1(p,tf,ts,fy,fu,es,a,b,x1,x2,x3,x4,y1,y2,y3,y4,x,y,mem,nxtf_tot,nytf_to
t,nxw_tot,nyw_tot,nxbf_tot,nybf_tot,sx,Ix,As,Zx)

%% Initializing temporary variables
Fyrt = fy;
Furt = fu;
Esrt = es;
Sx
= sx;
p
= p*1000;
% Converting p to N from kN
nxtf
nytf
nxw
nyw
nxbf
nybf

=
=
=
=
=
=

nxtf_tot;
nytf_tot;
nxw_tot;
nyw_tot;
nxbf_tot;
nybf_tot;

q11
q13
q31
q33

=
=
=
=

0;
0;
0;
0;

186

tolerance
error
iter
mx
loading

=
=
=
=
=

[1;1]*(1/100);
[1;1];
0;
0;
[mx;p];

incr

= [(10^6);0];

% Initial value of moment
% Initial laod vector
% Increment in moment (Mx) Units: N.mm

tolerance = [(10^2);0.1];
jj
mom
phi
axl
eps

=
=
=
=
=

%Tolerance

0;
0;
0;
0;
0;

f_app
= [0;0];
f_int
= [0;0];
f_res
= [1;1]*(10^50);
be > tolerance)

% Applied load vector
% Internal force vector
% Residual force vector (Initially it should

%%
%######################################
% Calculating initial K matrix
for i=1:mem
% Calculations for elements in the top flange
if (i<= nxtf*nytf)
[mod]

= Factors_Steel(ts(1),Esrt);

q11
= q11 + mod*(y(i)^2)*a(i)*b(i);
q13
= q13 + mod*y(i)*a(i)*b(i);
%q31
= q13;
q33
= q33 + mod*a(i)*b(i);
Ein(1) = mod;
% Calculations for elements in the web
elseif ((i > nxtf*nytf)& (i <=nytf*nxtf + nyw*nxw))
[mod]
q11
q13

= Factors_Steel(ts(2),Esrt);
= q11 + mod*(y(i)^2)*a(i)*b(i);
= q13 + mod*y(i)*a(i)*b(i);

q33
= q33 + mod*a(i)*b(i);
Ein(2) = mod;
% Calculations for elements in the bottom flange
elseif ((i >nytf*nxtf + nyw*nxw) & (i<=mem))
[mod]
= Factors_Steel(ts(3),Esrt);
q11
q13
q33
Ein(3)

=
=
=
=

q11 + mod*(y(i)^2)*a(i)*b(i);
q13 + mod*y(i)*a(i)*b(i);
q33 + mod*a(i)*b(i);
mod;

end

187

end
K(1,1)
K(1,2)
K(2,1)
K(2,2)

=
=
=
=

q11;
-q13;
-q13;
q33;

K_ref = K(1,1);
Mp = fu*Zx;
% Assigning axial force value
P = p;
e0(1:mem) = -p/(As*es);
e
= zeros(1,mem);
sigma = zeros(1,mem);
% Computing the reduced strength properties
[Est,Fyt,Fut,Telem] =
Factors_Stl(Esrt,Fyrt,Furt,ts,mem,nxtf,nytf,nxw,nyw,nxbf,nybf);
% Increase Mx
Mx = 0;
dMx = 10^6;
phix = 0;
ii = 1;
momentx(1) =0;
rotationx(1) = 0;
f = [0;0];
delta = [0;0];
aaaa = 1;

% Increment in moment (Mx) Units: N.mm

while(aaaa>0.01)
Mx = Mx+dMx;
aaaa=K(1,1)/(K_ref);
mmx=dMx;
pp=P;
Pint = 0;
Mxint = 0;
dif1 = 2;
dif2 = 2;
f(1) = mmx;
f(2) = pp;
while(dif1>1&&dif2>1)
delta=inv(K)*f;
phix = phix+delta(1);
e0 = e0+delta(2);
e = e0+(-1*phix*y);
epst = e;
q11 =0;
q33 = 0;
q13 = 0;
q31 = 0;

188

for i = 1:mem
[stress,modulus] =
Stress_stl(epst(i),Est(i),Fyt(i),Fut(i),Telem(i));
Et(i) = modulus;
sigma(i) = stress;
q11 = q11 + modulus*(y(i)^2)*a(i)*b(i);
q13 = q13 + modulus*y(i)*a(i)*b(i);
q33 = q33 + modulus*a(i)*b(i);
end
K(1,1) = q11;
K(1,2) = -q13;
K(2,1) = -q13;
K(2,2) = q33;
if q11==0
break
end
PPP = 0;
MMMx = 0;
for i =1:mem
MMMx = MMMx + (sigma(i)*y(i)*a(i)*b(i));
PPP = PPP + (sigma(i)*a(i)*b(i));
end
Pint = PPP;
Mxint = -MMMx;
dif1 =abs(-Mx-Mxint);
dif2 =abs(-P-Pint);
if(dif1<=1 && dif2<=1)
ii = ii+1;
momentx(ii)=Mx;
rotationx(ii)=phix;
mom(ii)=Mx;
phi(ii)=phix;
end
mmx =
pp =
f(1)=
f(2)=

-Mx-Mxint;
-P-Pint;
mmx;
pp;

end
end

189

APPENDIX E

The proposed methodology can be applied in fire design of connections. A numerical example is
presented here to demonstrate the applicability and rationality of the proposed methodology.
Step-by-step design procedure in analyzing a typical connection under fire is presented below:
Problem:


Evaluate the fire resistance of double angle connection in a steel framed building
subjected to ASTM E119 standard fire exposure.

Building details:


The steel framed building considered for this example is a typical 3-storey building
shown in Fig. E.1(a). The fire is assumed to occur in the first level of the building
subjecting the beam and the connections to elevated temperature, as shown in Fig. E.1(b).

Beam characteristics:


Beam length and section: 6000 mm, W24x76



Loading: Distributed dead and live service loads: WD = 35 kN/m and WL = 70 kN/m,
respectively



Steel properties; Grade 50 steel with Fy = 355 MPa and Fu = 445 MPa



Beam is assumed to be protected with 15mm thick insulation. The geometric details of
the beam are presented in Table 6.1 while the thermal properties of insulation are
presented in Table 6.2.

Connection details:


The double angle connection configuration connection the beam to column is illustrated
in Fig. E.2.

Response parameters:
190



Load combination under fire Wf = 1.0WD + 0.5WL = 70 kN/m (≈30% of beam ultimate
load carrying capacity at ambient temperature).



The computed beam temperatures along with the generated moment-curvature
relationships as well as evolution of fire-induced forces in connection are presented in
Fig. E.3.



The predicted failure time (fire resistance) of the connection is 66 minutes when
subjected to ASTM E119 standard fire exposure.

Summary:
To ensure that the connections will not experience premature failure when exposed to ASTM
E119 fire conditions the designer has to ensure that the connection capacity is always greater
than the fire-induced axial forces (from Fig. E.3(c)) experienced by the connection. Therefore,
this methodology will not only provide the designer with the tool to evaluate fire-induced axial
forces in connection but also enables the designer to design fail-proof connections for the
particular fire exposure conditions.

191

APPENDIX F
Chapter 2 tables
Table 2.1: Summary of the experimental program conducted by Lawson (Lawson, 1990)
Test
number

Connection type

Beam
section

Column
section

Bolt details

Fire
protection

Applied
moment

1

Extended end-plate
(bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

No

0.4 Mp

2

Flush end-plate
(bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

No

0.2 Mp

3

Extended end-plate
(bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

No

0.2 Mp

4

Flush end-plate
(bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

Yes

0.2 Mp

5

Double angle (bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

Yes

0.1 Mp

6

Flush end-plate
(composite)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

Yes

0.4 Mp

7

Double angle
(composite)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

Yes

0.2 Mp

8

Shelf-angle Flush
endplate (bare)

305x165x40
UB

203x203x52
UC

M20 Grade
8.8

No

0.2 Mp

Mp is the plastic moment capacity of steel beam

192

Table 2.2: Summary of test parameters used by Yu et al.(Yu et al., 2009a)
Test
Specimen Temperature
number geometry
(°C)
1
2
3
4
5
6
7
8
9
10
11
12
13
14

3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
6-8.8-20
6-8.8-20
3-10.9-20
3-10.9-20
3-8.8-24
3-8.8-24

20
450
550
650
20
450
550
650
550
550
20
550
20
550

Target
load angle
(°)
55
55
55
55
35
35
35
35
35
55
35
35
35
35

Observed
initial angle
(°)
53.85
51.47
53.44
53.09
33.8
39.04
40.94
40.5
41.56
55.99
36.53
40.85
37.38
42.1

Observed
final angle
(°)
32.41
41.37
42.68
44.02
34.06
33.52
31.51
30.6
32.21
46.6
29.8
23.9
29.67
29.06

Force
(kN)

Rotation
(°)

145.95
70.48
34.81
17.99
185.11
84.47
37.46
19.3
81.12
67.01
213
56.82
203.1
74.02

8.107
6.093
6.558
6.255
7.805
6.237
7.121
7.367
6.853
4.782
10.62
11.5
8.339
7.855

Table 2.3: Summary of experimental program conducted by Yu et al.(Yu et al., 2009b)
Test
Specimen Temperature
number geometry
(°C)
1
2
3
4
5
6
7
8
9
10
11
12
13
14

3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
3-8.8-20
6-8.8-20
6-8.8-20

20
450
550
650
20
450
550
650
20
450
550
650
550
550

Target
load angle
(°)
55
55
55
55
45
45
45
45
35
35
35
35
35
55

193

Observed
initial angle
(°)
55
55.8
56
56.5
45.7
46.7
47
48.1
37.4
41.1
41.4
40.9
40.2
55.7

Observed
final angle
(°)
34.4
43.5
42.2
34.4
32
37.3
36.8
34.5
21.2
29.1
26.6
21.6
27.2
41

Force
(kN)

Rotation
(°)

186.34
93.74
52.91
25.7
212.54
99.42
56.35
28.18
243.17
112.85
61.21
31.57
85.01
66.78

16.57
9.39
10.52
14.15
17.12
10.29
11.53
15.94
16.71
10.75
12.56
14.86
10.95
9.19

Table 2.4: Summary of specimen and loading details used by Daryan and Yahyai (Daryan and Yahyai,
2009)
Specimen Group
number number*

Angle
dimensions (mm)

Gap
(mm)†

Grade
of bolt

No of
nuts

Applied
moment‡

1

2

150x100x15

0

8.8

1

0.4 Mcc

2

1

150x100x15

0

8.8

1

0.6 Mcc

3

1

100x100x10

15

8.8

1

Mcc

4

1

100x100x15

15

8.8

1

0.6 Mcc

5

2

150x100x15

15

8.8

1

0.4 Mcc

6

1

100x100x15

15

10.9

1

0.6 Mcc

7

2

150x100x15

15

10.9

3

0.4 Mcc

8

1

100x100x10

15

10.9

1

0.5 Mcc

9

1

150x100x15

15

8.8

1

0.6 Mcc

10

1

100x100x15

15

10.9

3

0.6 Mcc

11

2

150x100x15

15

10.9

1

0.4 Mcc

12

2

150x100x15

15

10.9

1

0.2 Mcc

*1:Specimen without web angle; 2: Specimen with web angle
† Gap: Distance between beam and column flange
‡Mcc is the moment capacity of the connection

Table 2.5: Summary of specimen dimensions used by Wang et al. (Wang et al., 2011)
Test

Connection type

1
2
3
4
5
6
7
8
9
10

Shear tab (fin plate)
Flexible endplate
Flush endplate
Double angle (web cleat)
Extended endplate
Shear tab (fin plate)
Flexible endplate
Flush endplate
Double angle (web cleat)
Extended endplate

Connection
component
dimension (mm)
150x130x10
150x130x8
150x200x8
90x150x10*
150x250x8
150x130x10
150x130x8
150x200x8
90x150x10*
150x250x8

* depth = 130 mm

194

Column
section

Beam section

UC
254x254x73
UB 178x102x19
UC
152x152x23

Table 2.6: Summary of connection details tested by Al-Jabri (Al-Jabri et al., 2005)

Group

1

2

3

4

5

Connection
type

Flush end-plate
(bare)

Flush end-plate
(bare)

Flexible endplate (bare)

Flexible endplate
(composite)

Flexible endplate
(composite)

Beam
section

254x102x22
UB

356x171x51
UB

356x171x51
UB

356x171x51
UB

610x229x101
UB

Column
section

Bolt

152x152x23
UC

M16
Grade
8.8

254x254x89
UC

254x254x89
UC

254x254x89
UC

305x305x137
UC

M20
Grade
8.8
M20
Grade
8.8

M20
Grade
8.8

M20
Grade
8.8

Thickness
of endplate
(mm)

8

10

8

8

10

* indicates ambient temperature test; Mcc is the moment capacity of connection

195

Test

Moment
level

Test 1

0.2 Mcc

Test 2

0.4 Mcc

Test 3

0.6 Mcc

Test 4

0.8 Mcc

Test 1

0.2 Mcc

Test 2

0.4 Mcc

Test 3

0.6 Mcc

Test 4

0.8 Mcc

Test 1

0.1 Mcc

Test 2

0.2 Mcc

Test 3

0.5 Mcc

Test 0*

1.0 Mcc

Test 1

0.32 Mcc

Test 2

0.46 Mcc

Test 3

0.59 Mcc

Test 4

0.78 Mcc

Test 0*

1.0 Mcc

Test 1

0.27 Mcc

Test 2

0.46 Mcc

Test 3

0.77 Mcc

Table 2.7: Test parameters and specimen dimensions used by Yang et al.(Yang et al., 2009)
Test
no.

Beam Column
section section

3
4

H600x600x25x36

2

H600x300x12x25

1

Test
method

Fire
proofing

Heating
conditio
n

Steady
state

No

Steady
state
Transient
state

Applied loading
Column†

Beam‡

550°C

0.3 Pnc
(5390 kN)

Loaded to
failure

No

650°C

0.25 Pnc
(3920 kN)

Loaded to
failure

No

ISO 834

0.25 Pnc
(3920 kN)

0.6 Py
(333 kN)

ISO 834

0.25 Pnc
(3920 kN)

0.6 Py
(333 kN)

Transient
Yes (3 hrs)
state

†Pnc = nominal axial strength of column
‡ Py = yield strength of beam loaded at beam tip

196

Chapter 3 tables
Table 3.1: Mechanical properties of steel members used in test assemblies
Component
Elastic modulus (MPa)
Yield strength (MPa)
Maximum strength (MPa)
Ultimate strain (%)

Beam
187500
375
451
16

Angle
154000
308
459
23

Table 3.2: Dimensions of beams and angles used in the test assemblies

313.4
372.4
360.0

Width of the
flange (bf)
(mm)
165.6
374.1
255.8

Thickness of
flange (tf)
(mm)
11.2
26.2
20.0

101.6*

88.9†

8‡

Section

Depth (D)
(mm)

W12x30
W14x132
W14x74
L4x3.5x5/16

Thickness
of web (tw)
(mm)
6.6
16.4
11.4

Length (mm)
3505
4216
4293
215.9 and
292.1

*Length of long leg, † Length of short leg, ‡ Thickness of the angle

Table 3.3: Mix-proportions of concrete used in the slab of test assembly S2
Item
Type I Portland cement
Fly ash
Fine aggregate (sand)
Lightweight aggregate
Coarse aggregate (gravel)
Water
28 day compressive strength

197

Quantity
332 kg/m3
60 kg/m3
753 kg/m3
495 kg/m3
89 kg/m3
161 kg/m3
34 MPa

Table 3.4: Input parameters used in the computation of design fire, DF1
Energy density

qf,d

865 MJ/m2

Heat of combustion

Hc

17.3 MJ/kg

λ

1.75 W/m.K

Cp

960 J/kg.k

ρ

2150 kg/m3

Fire input parameters
Conductivity
Specific heat
Concrete properties

Density
Thermal inertia

λ

0.12 W/m.k

Cp

1200 J/kg.K

ρ

300 kg/m3

L

10 m

W

5m

h

3m

Floor area

Af

50 m2

Total surface area

At

190 m2

Average window height

Hv

2m

Average window width

Wd

1.25 m

Ventilation factor

Fv

0.0372 m-0.5

Conductivity
Fire protection
properties

Specific heat
Density
Length
Width
Height

Compartment
dimensions

1900 Ws0.5/m2.k

√

198

Table 3.5: Variables used in fire resistance tests on steel beam assemblies
Test
Specimen

Slab
presence

S1

No

S2

Yes

Fire Scenario
75 min of
ASTM E119
followed by a
decay rate of
8°C/min
90 min of
ASTM E119
followed by a
decay rate of
15°C/min

199

Composite
action (via
shear
studs)

Loading
(Load
ratio)

No

74.7 kN
(40%)

Yes

92.2 kN
(50%)

Chapter 4 tables
Table 4.1: Summary of contact parameters used in the finite element models

Contact parameter
FKN
FTOLN
ICONT
PINB
KEYOPT(2)
KEYOPT(5)
KEYOPT(9)
KEYOPT(10)
KEYOPT(12)

Value
0.1
0.1
0.1
0.2
0
1
1
2
2

Table 4.2: Summary of key thermal and mechanical properties of steel members used in the test
assemblies (S1, S2) used in the finite element model FEM1

Component
Elastic modulus (Mpa)
Yield strength (Mpa)
Maximum strength (Mpa)
Ultimate strain (%)
Thermal conductivity (W/m K)
Specific heat (J/kg K)

Beam
146258
367
451
16

Angle
122216
305
459
23

53
440

Density (kg/m3)

Hightemperature
properties

As per
Eurocode 3
provisions

7850

Table 4.3: Summary of key thermal and mechanical properties of concrete used in the assembly S2 used
in the finite element model FEM1

Property

Value

Hightemperature
properties

Thermal conductivity (W/m K)
Specific heat (J/kg K)
Density (kg/m3)
Compressive strength (Mpa)

1.3
900
2300
41

As per
Eurocode 2
provisions

200

Table 4.4: Summary of thermal properties of insulation for test assemblies (S1, S2) used in the finite
element model FEM1

Property
Thermal conductivity (W/m.K)
Dry density (kg/m3)
Specific heat (J/kg K)

Value
0.086
240
3000

Hightemperature
properties
Same as room
temperature

Table 4.5: Summary of thermal and structural response of assemblies S1 and S2 obtained from finite
element model FEM1and measured test data

S1
Maximum beam
temperature (°C)
S2
Maximum rotation
(rad)
Maximum axial force
(kN) (compressive)

Bottom flange
Web
Top flange
Bottom flange
Web
Top flange
S1
S2
S1
S2

FEM1

Test

766.73
834.23
660.55
726.90
769.55
490.66
0.1
0.03
233.4

764.39
722.00
617.78
719.22
664.39
554.39
0.12
0.03
NA

224

NA

Table 4.6: Ambient temperature mechanical properties of different steel members used in the finite
element model FEM2

Beam
Column
Angle
Bolt

Elastic modulus
(MPa)
226580
200000
228170
210000

Yield strength
(MPa)
344
390
342
640

201

Ultimate strength
(MPa)
514
553
493
800

Table 4.7: Summary of thermal and structural response of restrained steel frame assembly obtained from
finite element model FEM2 and measured test data

Bottom flange
Web
Top flange
Maximum axial force (kN) (compressive)
Maximum beam deflection (mm)
Maximum rotation (°)
Maximum beam
temperature at failure
(°C)

202

FEM2
738.24
738.83
217.04
78
229
12.8

Test
NA
NA
NA
82.9
251
NA

Chapter 5 tables
Table 5.1: Summary of key mechanical properties of beam and connections used in comparative analysis

Component

Beam

Elastic modulus (Mpa)
Yield strength (Mpa)
Ultimate strength (Mpa)

187500
375
451

Shear
tab
154000
308
459

Single
angle
154000
308
459

Double
angle
154000
308
459

Bolt (19mm
dia)
200000
940
1210

Table 5.2: Variation of connection capacity and fire induced forces in three connection types with fire
exposure time

Fire induced axial force (kN)
(compressive)
Time
(min)
0
5
10
15
20
25
30

Temperature
in
connections
(°C)
20
58
128
189
249
328
399

Connection capacity (kN)

Componentlevel analysis
(Eq. [5.1])

Memberlevel analysis

Systemlevel
analysis

Shear
tab

Single
angle

Double
angle

0
485
1336
1963
2490
3035
3391

0
0
0
0
0
0
0

0
18
50
70
100
124
167

211
196
196
196
196
185
157

211
196
196
196
196
185
157

306
292
292
292
292
275
234

203

Table 5.3: Summary of parameters and their range used in the parametric studies

No.

1

2

Value/Description Connection
of parameter
assembly

Parameter

Decay rate (°C/min)

Four point
loading (kN)

Loading
type

Distributed
loading (kN/m)

3

Axial restraint (% of secondary
beam stiffness)

4

Isolated
connection
System-level (transient heating)
interactions
System-level
connection
(uniform heating)

5

2
4
10
12
3.75
7
18.75
22.5

S1
S1
S1
S1
S2
S2
S2
S2

46.3

S1

57.8
63.5
79.3
25
50
75
100

S2
S1
S2
S1,S2
S1,S2
S1,S2
S1,S2

Parametric fire
curve (FS)
Uniform
temperature of
550°C
30%
40%
60%
70%

Load ratio

DF1
6

7

Design fire scenarios
Hightemperature
bolt
properties

DF2
DF3

A325 bolts

Eurocode 3

A490 bolts

Kodur et al.

204

Restrained
steel frame

Restrained
steel frame

Restrained
steel frame

Restrained
steel frame

Table 5.4: Compartment characteristics used for arriving at different design fire scenarios
Fire
scenario

Fuel load
(MJ/m2
floor
area)

DF1
DF2
DF3

510
750
725

Ventilation
Thermal
Maximum
factor
capacity
temperature
(m0.5)
(Ws0.5/m2k)
(°C)
0.014
0.014
0.010

1600
1900
1600

706
707
709

Duration
of growth
phase
(min)
60
92
120

Total
Decay
duration
rate
(min)
(°C/min)
192
291
432

5.4
3.5
2.3

Table 5.5: Maximum value of fire induced forces and rotation in connection for different fire scenarios

Fire induced forces (kN)
Fire scenario

Compressive

Tensile

DF
DF1
DF2
DF3

82
79
80
81

48
51
52
58

Rotation
(°)
1.27
1.35
1.53
1.65

Table 5.6: Ambient temperature mechanical properties for different bolt grades
Bolt grade
Grade 6.8
Grade 10.9
ASTM A325
ASTM A490

Yield strength
(MPa)
480
900
660
940

Ultimate strength
(MPa)
600
1000
830
1210

205

Ultimate strain
(%)
15
15
15
15

Elastic modulus
(MPa)
200000
200000
200000
200000

Chapter 6 tables
Table 6.1: Geometric details of steel sections used in the analysis

Depth (mm)
Flange width (mm)
Flange thickness (mm)
Web thickness (mm)
Yield strength (MPa)
Ultimate strength (MPa)

Section
W 12x30
W 24x76
313.4
607.1
165.6
228.6
11.2
17.3
6.6
11.2
375
345
451
448

Table 6.2: Summary of thermal properties of insulation used in the analysis
Property
Thermal conductivity (W/m.K)
Dry density (kg/m3)
Specific heat (J/kg K)

Value
0.086
240
3000

Table 6.3: Ambient temperature mechanical properties of different components measured in Cardington
tests

Component

Material
grade

Yield strength
(MPa)

Ultimate
strength (MPa)

Beam
(UB 305x165x40)

S375

303

469

Girder
(UB 356x171x51)

S355

396

544

Shear tab
(thickness = 10mm)

Grade 43

275

430

Bolt (M20)
(diameter = 20 mm)

Grade 8.8

695

869

206

APPENDIX G
Chapter 1 figures

Unsupported length

Unsupported length

Beam

Column

(a) Before connection failure

(b) After connection failure

Figure 1.1: Typical steel frame in an office building (a) before and (b) after failure of beam-tocolumn connection

207

Connection

Beam

Beam

Column

Fire
Column

(a) Steel framed building

(b) Fire exposed steel frame

Compression
Temperature

Fire induced axial force

Fire temperature

Beam temperature

Initial yielding
Catenary action

Tension

Steel temperature

Time
(c) Time-temperature curve

(d) Fire induced axial force development
in connection

Figure 1.2: Development of axial forces in connection

208

1200

Temperature(°C)

1000
ASTM E119 standard fire
Design fire1
Design fire2

800
600
400
200

0
0

50

100

150

200

250

300

Time (minutes)

Figure 1.3: Illustration of standard and design fire scenarios in typical buildings

209

B
End-plate

Column

W
eld

A

A
Beam
B

B-Bview

A-Aview

(a) Flush end-plate connection
B
Column

Double angle
A

A
B

B-B view

Beam

A-A view

(b) Double angle connection

20°C
400°C

Moment (M)

Moment (M)

20°C

400°C
600°C

600°C

Rotation (ϕ)

Rotation (ϕ)

(d) Double angle M-ϕ curves

(c) Flush end-plate M-ϕ curves

Figure 1.4: Connection details and moment-rotation curves for flush end-plate and double angle
connections

210

Chapter 2 figures

Furnace

152x152x30 UC

152x152x30 UC

Loading jack

178x102x19 UB

600 mm

800 mm

600 mm

Double angle/Flush end-plate connection

Figure 2.1: Schematic of the sub-frame assembly tested by Liu et al.(Liu et al., 2002)

211

20mm diameter bolts
UC 254x89
Loading angle
320

A

50 50
40

40
60
60
40

B

UB 305x165x40

P
300

400

10

103
B

L 90x8
A-A
B-B
All dimensions in mm

Figure 2.2: Schematic of connection configuration tested by Yu et al. (Yu et al., 2009b)

212

A
All dimensions in mm
10

9

22

1

B

B

27.5

A

A-A
B-B
(a) Specimen without web angle

A
10

9

22

10

1

B

B

27.5

A

A-A
(b) Specimen with web angle

B-B

Figure 2.3: Connection details of the specimens tested by Daryan and Yahyai (Daryan and
Yahyai, 2009)

213

600
20°C
500

400°C

Stress (N/mm2 )

500°C
400

600°C
700°C

300
200
100
0
0

0.1

0.2

0.3

0.4

0.5

0.6

Total strain

Figure 2.4: High-temperature stress-strain curves used in the FE model by Pakala et al. (Pakala et
al., 2012a)

(b) Fire test

(a) FE model

Figure 2.5: Comparison of predicted deformed shape of connection assembly with that observed
in fire tests (Part (b) adapted from Yu et al. (Yu et al., 2009b))
214

Figure 2.6: Geometrical details of the connection assembly used in the finite element model (all
dimensions in mm)

215

Figure 2.6 (cont’d)

216

(a) Stress in the bolt shank region

(b) Stresses in the bolt head region

Figure 2.7: Illustration of stresses in the (a) bolt shank region and (b) bolt head region at the time
of beam failure predicted by the model

Time (min)
80

Axial force (kN)

40
0
0

60

120

180

-40

240

300

Design fire

Isolated connection

-80

Steady-state heating

-120
-160

Figure 2.8: Variation of connections axial force as a function of fire exposure time for different
scenarios

217

Time (min)
60
Eurocode 3 (A325)

40

Kodur et al. (A325)

Axial force (kN)

20
0
0

60

120

180

240

300

240

300

-20
-40
-60
-80
-100

(a) A325 bolts
Time (min)
60
Eurocode 3 (A490)

40
Kodur et al. (A490)

Axial force (kN)

20
0
0

60

120

180

-20
-40
-60
-80
-100

(b) A490 bolts
Figure 2.9: Progression of connection axial force as a function of fire exposure time for (a) A325
and (b) A490 bolts types with different material models

218

1.2
A325

Reduction factor

1

A490

0.8
0.6
0.4
0.2
0
0

200

400
Temperature ( C)

600

800

Figure 2.10: Variation in shear strength of A325 and A490 bolts with temperature
1.2

Reduction factor

1
0.8
0.6
A325
A490
ASCE (conventional steel)
Eurocode (conventional steel)

0.4
0.2
0
0

200

400
Temperature ( C)

600

800

Figure 2.11: Comparison of high-temperature yield strength of A325 and A490 bolt steel with
conventional steel

219

Chapter 3 figures

S e c o n d a ry b e a m (B e a m I) (W 1 2 x 3 0 )

F u rn a c e

N

S e c o n d a ry b e a m (B e a m II) (W 1 2 x 3 0 )

P e r im e te r b e a m ( W 1 4 S o u th ) ( W 1 4 x 1 3 2 )

P e r im e te r b e a m ( W 1 4 N o r th ) ( W 1 4 x 1 3 2 )

P r im a r y b e a m ( B e a m I V ) ( W 1 4 x 7 4 )

P r im a r y b e a m ( B e a m I I I ) ( W 1 4 x 7 4 )

Figure 3.1: Layout of the steel beams used in test assemblies S1 and S2 (plan view)

220

Figure 3.2: Connection details used in test assembly (all dimensions in mm)
221

S

V LVDT

Strain gauges

W14x74

T

Thermocouple

S

Load points
V
V

T

V

W12x30

T

V

T

N

S W12x30
V

S

W14x132

W14x132

T

V

Load points

W14x74

S

(a) Location of instrumentation, loading in the test assemblies

Loading point

1321 mm

864 mm

1321 mm

= Thermocouple

(b) Schematic of load configuration on secondary
beam

(c) Location of thermocouples along
The secondary beam cross-section

Figure 3.3: Layout of the instrumentations and loading used in the steel beam assembly

222

Figure 3.3 (cont’d)

76 mm
38 mm

Shear stud

Secondary
Beam (W12x30)

(d) Cross-sectional details of the beam and slab in assembly S2

223

Figure 3.4: Structural fire furnace at Michigan State University

224

1100
1000
DF1 - Predicted

Temperature ( C)

900

DF1 - Measured

800

DF2 - Predicted

700

DF2 - Measured

600
500
400
300
200
100
0
0

30

60

90

120

150

180

210

240

Time (min)

Figure 3.5: Predicted and measured time-temperature curves for design fires DF1 and DF2

225

1000

Furnace average
Center-Top flange
Center-Web
Center-Bottom flange
Quarter-Top flange
Quarter-Web
Quarter-Bottom flange

900
800

Temperature ( C)

700
600
500

400
300
200
100
0
0

30

60

90

120

150

180

210

240

270

300

Time (min)

(a) S1- Beam I
1000

Furnace average
Center-Top flange
Center-Web
Center-Bottom flange
Quarter-Top flange
Quarter-Web
Quarter-Bottom flange

900
800

Temperature ( C)

700
600
500

400
300
200
100
0
0

30

60

90

120

150

180

210

240

270

300

Time (min)

(b) S1- Beam II
Figure 3.6: Recorded temperatures at center (mid-span) and quarter span of the Beam I, Beam II
of S1 as a function of fire exposure time (Refer to Fig. 3.1 for naming convention)

226

1000

Furnace average
Center-Top flange
Center-Web
Center-Bottom flange
Quarter-Top flange
Quarter-Web
Quarter-Bottom flange

900

Temperature ( C)

800
700
600
500
400
300
200
100

0
0

30

60

90
120
Time (min)

150

180

210

(a) S2- Beam I
1000

Furnace average
Center-Top flange
Center-Web
Center-Bottom flange
Quarter-Top flange
Quarter-Web
Quarter-Bottom flange

900

Temperature ( C)

800
700
600
500
400
300
200
100
0
0

30

60

90
120
Time (min)

150

180

210

(b) S2- Beam II

Figure 3.7: Recorded temperatures at center (mid-span) and quarter span of the Beam I, Beam II
of S2 as a function of fire exposure time (Refer to Fig. 3.1 for naming convention)

227

1000
S1-Furnace average
S1-Beam I-Center
S1-Beam I-Quarter span
S1-Beam II-Center
S1-Beam II-Quarter span

900

Temperature ( C)

800
700

600
500

400
300
200
100
0
0

30

60

90

120

150

180

210

240

270

300

Time (min)

(a) S1
1000
S2-Furnace average
S2-Beam I-Center
S2-Beam I-Quarter span
S2-Beam II-Center
S2-Beam II-Quarter span

900

Temperature ( C)

800
700
600
500
400

300
200

100
0

0

30

60

90

120

150

180

210

Time (min)

(b) S2
Figure 3.8: Recorded average temperatures at center (mid-span) and quarter span of the Beam I,
Beam II of assemblies (a) S1 and (b) S2 as a function of fire exposure time (Refer to Fig. 3.1 for
naming convention)

228

0.04

0.1

0.03
0.02

0.05
0
0

-0.05
-0.1
-0.15

60

120

180

Beam I-North
Beam I-South
Beam II-North
Beam II-South

240

300

Rotation (rad)

Rotation (rad)

0.15

0.01

0
0

60

-0.01

120

180

Beam I-North
Beam I-South
Beam II-North
Beam II-South

-0.02
-0.03

Time (min)

Time (min)

(a) S1

(b) S2

Figure 3.9: Connection rotation recorded as a function of fire exposure time in test assemblies (a)
S1 and (b) S2

Loading point
Undeformed shape
u

θ

Beam centerline
L

Perimeter beam

Deformed shape

Secondary beam

Figure 3.10: Schematic illustration of evaluating the rotation at connection

229

(a) Local buckling

(b) Global buckling

Figure 3.11: (a) Local and (b) global buckling experienced by the secondary beam (Beam II –
North) in assembly S1

(a) Top view

(b) Bottom view

Figure 3.12: Warping and distortion of the metal deck in test assembly S1

230

600
400
200

0

Strain (µ)

-200
-400
-600
-800

S1-Top flange
S1-Web
S2-Web
S1-Bottom flange
S2-Bottom flange

-1000
-1200

-1400
-1600
0

30

60

90

120 150 180
Time (min)

210

240

270

300

(a) Beam I (Secondary beam)
1200
1000
800

Strain (µ)

600
400
200
0
-200

S2-Top flange
S1-Web
S2-Web
S1-Bottom flange
S2-Bottom flange

-400
-600
-800
0

30

60

90

120 150 180
Time (min)

210

240

270

300

(b) Beam II (Secondary beam)
Figure 3.13: Variation of strains in secondary and primary beam as a function of fire exposure
time (Refer to Fig. 3.1 for naming convention)
231

Figure 3.13 (cont’d)
400
S2-Top flange

350

S1-Web

Strain (µ)

300

S2-Web

250

S1-Bottom flange

200

S2-Bottom flange

150
100
50
0
-50

-100
-150
0

30

60

90

120 150 180
Time (min)

210

240

270

300

(c) Beam III (Primary beam)
450
S2-Web

400

S2-Bottom flange

350

Strain (µ)

300
250
200
150
100

50
0
-50
0

30

60

90
120
Time (min)

(d) Beam IV (Primary beam)

232

150

180

210

500
400

Force (kN)

300
200
100
0
0

30

60

90

120

150

180

210

-100
-200

-300

Secondary beam
Primary beam
Concrete deck

Time (min)

Figure 3.14: Variation of axial forces in secondary, primary beams and concrete deck as a
function of fire exposure time

233

(a) Insulation fall off

(b) Local buckling of top flange and web

(c) Rotation of the beam end (top view)

(d) Rotation of the beam end (elevation)

Figure 3.15: (a) Local and (b) global buckling experienced by the secondary beam (Beam IINorth) in assembly S1

234

(a) Beginning of cracking in the slab

(b) Concrete fragment spalled-off from slab

(c) Warping of the deck

(d) Insulation fall-off from deck and beams

Figure 3.16: Visual observations in the test assembly S2 during and after fire test

235

Chapter 4 figures

Beam
Column

Fire exposed steel frame

(a) Three dimensional perspective view of steel framed building

Beam

Fire exposed
steel frame

Column

(b) Two dimensional view of steel framed building

Figure 4.1: Typical connection in a steel framed building along with different structural
components present in connection region

236

Figure 4.1 (cont’d)

Connection

Beam

Fire
Column
(c) Fire exposed steel frame

237

Figure 4.1 (cont’d)

Bolts
Beam

Double angle

Column

(e) Close up of connection region

(d) Geometry of fire exposed steel frame

238

Figure 4.1 (cont’d)

Beam
Column

Angles

Bolt

(f) Different structural components present in connection region

239

SOLID185

SOLID70

SURF152

TARGE170
CONTA174
Figure 4.2: Different types of finite elements used for thermal and structural analysis in the
model

240

1.2

Reduction factor

1
0.8

Yield strength

Modulus of elasticity

0.6
0.4

0.2
0
0

200

400

600

800

1000

1200

Temperature ( C)

Figure 4.3: Strength reduction factors for steel specified in Eurocode 3

241

Perimeter beam

Primary beam

Secondary beam

(a) S1

Perimeter beam
Primary beam

Concrete slab

Secondary beam

(b) S2

Figure 4.4: Discretized geometry (quarter) of the connection assembly (FEM1) modeled in
ANSYS

242

1.2

Reduction factor

1
0.8

Siliceous
Calcareous

0.6
0.4
0.2
0

0

200

400

600

800

1000

1200

Temperature ( C)

Figure 4.5: Strength reduction factors for different aggregate concrete specified in Eurocode 2

243

1000
Fire temperature

Temperature ( C)

900

Top flange (Test)

800

Bottom flange (Test)

700

Top flange (ANSYS)

600

Bottom flange (ANSYS)

500

400
300
200
100
0
0

30

60

90

120

150

180

210

240

270

300

Time (min)

(a) S1
1000

Fire temperature
Top flange (Test)
Bottom flange (Test)
Top flange (ANSYS)
Bottom flange (ANSYS)

900

Temperature ( C)

800
700
600
500
400

300
200
100
0

0

30

60

90

120 150 180
Time (min)

210

240

270

300

(b) S2
Figure 4.6: Comparison of measured and predicted secondary beam temperatures (top and
bottom flanges) in tested assemblies (a) S1 and (b) S2

244

Temperature ( C)

1000

900

Fire temperature

800

Web (Test)

700

Web (ANSYS)

600
500

400
300
200
100
0
0

30

60

90

120

150

180

210

240

270

300

Time (min)

(a) S1
1000
Fire temperature

900

Web (Test)

Temperature ( C)

800

Web (ANSYS)

700
600
500
400

300
200
100
0

0

30

60

90

120 150 180
Time (min)

210

240

270

300

(b) S2
Figure 4.7: Comparison of measured and predicted secondary beam temperatures (web) in tested
assemblies (a) S1 and (b) S2

245

0.15
Test

0.1

Rotation (rad)

ANSYS

0.05
0
0

50

100

150

200

250

300

-0.05
-0.1
-0.15

Time (min)

(a) S1
0.04
Test

0.035

ANSYS

0.03

Rotation (rad)

0.025
0.02
0.015

0.01
0.005
0
-0.005

0

30

60

90

120

150

180

210

240

270

300

-0.01
-0.015

Time (min)

(b) S2
Figure 4.8: Comparison of predicted and measured connections rotation as a function of fire
exposure time in test assemblies (a) S1 and (b) S2

246

Perimeter beam

Primary beam

Local buckling region

Secondary beam

(a) Deformed shape

(b) Local buckling

(c) Lateral torsional buckling

Figure 4.9: Illustration of (a) deformed shape (b) local buckling of secondary beam and (c)
lateral torsional buckling of secondary beam in assembly S1 predicted by the numerical model

247

Figure 4.10: Fire induced axial forces on connections predicted as a function of fire exposure
time (from model FEM1) in test assemblies (a) S1 and (b) S2

248

Vertical displacement (mm)

200
150
Test

100

ANSYS

50
0
0

50

100

150

200

250

300

-50
-100
-150
-200

Time (min)

(a) S1
50
Test
ANSYS

Vertical deflection (mm)

40
30
20
10
0
0

30

60

90

120

150

180

210

240

270

300

-10
-20

Time (min)

(b) S2
Figure 4.11: Vertical deflection of secondary beams (at loading actuator) predicted as a function
of fire exposure time (from model FEM1) in test assemblies (a) S1 and (b) S2

249

Figure 4.12: Geometrical details of the connection assembly used in the second finite element
model FEM2 (all dimensions in mm)

250

Figure 4.12 (cont’d)

251

Truss replacement
Actuator pad

Y
Z

Figure 4.13: Modeling of truss replacement and actuator pad

Y
Z

X

(a) Angles

(b) Bolt

Figure 4.14: Discretization of connection components (a) angles and (b) bolt

252

X

1200
1100
1000

Temperature ( C)

900

800
700
600

ISO834-Actual

500

ISO834-Scaled

400

Difference

300
200
100
0
0

20

40
60
Time (min)

80

100

Figure 4.15: Standard and adjusted ISO834 standard fire time-temperature curves

900
800

Temperature ( C)

700
600
Fire temperature

500

Bottom flange

400

Web

300

Top flange

200

100
0

0

30

60
Time (min)

90

120

Figure 4.16: Predicted temperature distribution in the beam as a function of fire exposure time
253

0

Beam bottom flange temperature ( C)
100
200
300
400
500
600
700

800

900

0

Deflection (mm)

-40
-80
-120

Test
ANSYS

-160
-200
-240
-280

(a) Beam mid-span deflection

Beam bottom flange temperature ( C)
60

40

Test

20

ANSYS

Axial force (kN)

0
-20 0

100

200

300

400

500

600

700

800

900

-40
-60
-80

-100
-120

Stage 2

Stage 1

-140

Stage 3

-160

(b) Connection axial force

Figure 4.17: Comparison of predicted and measured (a) beam mid-span deflection (b) axial force
in connection as a function of beam bottom flange temperature and (c) connection’s rotation as a
function of fire exposure time
254

Figure 4.17 (cont’d)

14

Rotation (degree)

After contact

Before contact

12
10
8
6
4
2
0
0

15

30

45
60
Time (min)

(c) Connection rotation

255

75

90

105

Chapter 5 figures
B
End-plate

Column

W
eld

A

A
Beam
B

B-Bview

A-Aview

(a) Flush end-plate connection
B
Column

Double angle
A

A
B

B-B view

Beam

A-A view

(b) Double angle connection

20°C
400°C

Moment (M)

Moment (M)

20°C

400°C
600°C

600°C

Rotation (ϕ)

Rotation (ϕ)

(d) Double angle M-ϕ curves

(c) Flush end-plate M-ϕ curves

Figure 5.1: Connection details and moment-rotation curves for flush end-plate and double angle
connections
256

Loading point

UC254x254x73

Furnace

UB178x102x19

Free end

Figure 5.2: Illustration of the isolated connection assembly used in the finite element model (all
dimensions in mm)

257

B
A

Column

Column

Beam

A

B
A-A view

B-B view

(a) Double angle connection
B

A
Column

Column

Beam

A
B
A-A view

B-B view

(b) Single angle connection

Figure 5.3: Geometrical details of different connection types (a) double angle (b) single angle
and (c) shear tab connection
258

Figure 5.3 (cont’d)
B

A
Column

Column

Beam

Weld

A
B
A-A view

B-B view

(c) Shear tab connection

259

800
700

Temperature ( C)

600
500

400
300

200
100
0
0

50

100

150
Time (min)

200

250

300

Figure 5.4: Parametric fire scenario (FS) used in the parametric studies

260

Time (min)
80

Axial force (kN)

40

0
0

60

120

180

-40

240

300

System-level analysis
Member-level analysis

-80

-120

(a) Axial force
8
7
Rotation (degree)

6
5
System-level analysis

4

Member-level analysis

3
2
1

0
0

60

120

180

240

300

Time (min)

(b) Rotation

Figure 5.5: Variation of connections (a) axial force and (b) rotation as a function of fire exposure
time obtained from system-level and isolated connection analysis

261

Connection capacity

Force (kN)

Fire induced
axial force

Temperature (°C)
Figure 5.6: Illustration of connection capacity and fire induced axial force with temperature
0.1
0.09
S1

0.08

S2

Rotation (rad)

0.07
0.06

0.05
0.04

S1 failure

0.03
0.02
0.01
0
0

30

60

90

120
150
Time (min)

180

210

240

Figure 5.7: Comparison of rotation in connection as a function of fire exposure time in
assemblies S1 and S2

262

1100

1000
S1 - 2°C/min

Temperature ( C)

900

S1 - 4°C/min

800

S1 - 10°C/min

700

S1 - 12°C/min

600
500
400
300

200
100
0
0

100

200
300
Time (min)

400

500

(a) S1
1100

Temperature ( C)

1000

S2 - 3.75°C/min

900

S2 - 7°C/min

800

S2 - 18.75°C/min

700

S2 - 22.5°C/min

600

500
400
300
200
100
0

0

100

200
300
Time (min)

400

500

(b) S2
Figure 5.8: Time-temperature curves for design fire scenarios used to study the effect of decay
rate

263

0.14
8°C/min (Test S1)

0.12

2°C/min
4°C/min

Rotation (rad)

0.1

10°C/min

12°C/min

0.08
0.06
0.04
0.02
0
0

30

60

90

Time (min)

(a) S1
0.045

0.04

Rotation (rad)

0.035
0.03

0.025
0.02
15°C/min (Test S2)
3.75°C/min
7.5°C/min
18.75°C/min
22.5°C/min

0.015
0.01

0.005
0
0

100

200
Time (min)

300

400

(b) S2
Figure 5.9: Variation of connections rotation as a function of fire exposure time for different
decay rates in assembly (a) S1 and (b) S2

264

Loading point

1752.5 mm

1752.5 mm

(a) Mid-point loading
Uniformly distributed load

3505 mm

(b) Distributed loading
120
Distributed loading

Moment (kN-m)

100

Mid-point loading

80
60
40
20
0

0

0.5

1

1.5
2
2.5
3
Distance from beam left end (m)

3.5

4

(c) Bending moment for different loading conditions

Figure 5.10: Illustration of different loading scenarios used in the analysis along with the
corresponding bending moments

265

0.1
Two point loading (Test S1)

0.09

Mid point loading

0.08

Distributed loading

Rotation (rad)

0.07
0.06
0.05
0.04
0.03
0.02
0.01
0

0

30

60

90

Time (min)

(a) S1
0.05
0.045
0.04

Rotation (rad)

0.035
0.03
0.025
0.02
0.015
Two point loading (Test S2)

0.01

Mid point loading

0.005

Distributed loading

0
0

30

60

90
120
Time (min)

150

180

210

(b) S2
Figure 5.11: Progression of connections rotation as a function of fire exposure time for
connection assemblies (a) S1 and (b) S2 with different loading types

266

Perimeter beam

Parallel beam

Secondary beam

(a) Deformed shape

(b) Lateral torsional buckling
Figure 5.12: Illustration of (a) deformed shape and (c) lateral torsional buckling of secondary
beam in assembly S1 analyzed with uniformly distributed loading

267

150
Two point loading (Test S1)

100

Mid point loading

Distributed loading

Axial force (kN)

50
0

0

30

60

90

-50
-100
-150

-200
-250

Time (min)

(a) S1
0

30

60

Time (min)
90
120

150

180

210

0

-50

Two point loading (Test S2)

Axial force (kN)

Mid point loading

-100

Distributed loading

-150

-200

-250

(b) S2
Figure 5.13: Variation of fire induced axial force in connections as a function of fire exposure
time for connection assemblies (a) S1 and (b) S2 with different loading types

268

Spar
elements
Perimeter beam
Parallel beam

Secondary beam

Figure 5.14: Schematic of the spar elements used in the analysis

269

0.1
AR = 0% (Test S1)

0.09

AR = 25%

0.08

AR = 50%

Rotation (rad)

0.07

AR = 75%

0.06

AR = 100%

0.05
0.04
0.03
0.02
0.01
0
0

30

60

90

Time (min)

(a) S1
0.035
0.03

Rotation (rad)

0.025
0.02
0.015
AR= 0%(Test S2)
AR = 25%
AR = 50%
AR = 75%
AR = 100%

0.01
0.005
0
0

50

100
150
Time (min)

200

250

(b) S2
Figure 5.15: Effect of restraining axial stiffness on the fire response of double angle connection
assemblies (a) S1 and (b) S2

270

AR=25%
AR=50%

(a) Local buckling

(b) Lateral torsional buckling

AR=100%
AR=75%

(d) Web and flanges crippling

(c) Lateral torsional buckling

Figure 5.16: Failure modes of connection assembly S1 for different values of axial restraint
stiffness

271

Time (min)
80

Axial force (kN)

40
0

0

60

120

180

240

300

-40
Transient heating

-80

Steady-state heating

-120
-160

(a) Axial force
1.4

Rotation (degree)

1.2
Transient heating

1

Steady-state heating

0.8
0.6
0.4
0.2

0
0

60

120

180

240

300

Time (min)

(b) Rotation
Figure 5.17: Variation of (a) axial force and (b) rotation in connection as a function of fire
exposure time for transient and steady-state heating conditions

272

Time (min)
120
80
Axial force (kN)

40
0
0

60

120

180

-40

240

300

LR = 30%
LR = 40%
LR = 50% (Test)
LR = 60%
LR = 70%

-80
-120

(a) Axial force
1.8

Rotation (degree)

LR = 30%

1.6

LR = 40%

1.4

LR = 50% (Test)
LR = 60%

1.2

LR = 70%

1
0.8
0.6
0.4
0.2
0
0

60

120

180

240

300

Time (min)

(b) Rotation
Figure 5.18: Variation of (a) axial force and (b) rotation in connection as a function of fire
exposure time for different load ratios

273

800
DF1

700

DF2

Temperature ( C)

600

DF3

500
400
300
200
100
0
0

100

200
Time (min)

300

400

Figure 5.19: Time-temperature curves for design fire scenarios used in the analysis

274

Time (min)
80
60

Axial force (kN)

40

20
0

0

60

120

180

240

300

360

420

-20
-40

DF

-60

DF1
DF2

-80

DF3

-100

(a) Axial force
1.8

1.6

Rotation (degree)

DF

1.4

DF1

1.2

DF2
DF3

1
0.8
0.6
0.4
0.2
0
0

60

120

180

240

300

360

420

Time (min)

(b) Rotation
Figure 5.20: Fire induced (a) axial force and (b) rotation in connection under different design fire
scenarios (see Fig. 5.14)

275

Time (min)
60
Eurocode 3 (A325)

40

Kodur et al. (A325)

Axial force (kN)

20
0
0

60

120

180

240

300

240

300

-20
-40
-60
-80
-100

(a) A325 bolts
Time (min)
60
Eurocode 3 (A490)

40
Kodur et al. (A490)

Axial force (kN)

20
0
0

60

120

180

-20
-40
-60
-80
-100

(b) A490 bolts
Figure 5.21: Progression of axial force in connection as a function of fire exposure time for (a)
A325 and (b) A490 bolt types

276

1.4
1.2
Rotation (degree)

Eurocode 3 (A325)
Kodur et al. (A325)

1

0.8
0.6
0.4
0.2
0

0

60

120
180
Time (min)

240

300

(a) A325 bolts
1.4
1.2

Eurocode 3 (A490)

Rotation (degree)

Kodur et al. (A490)

1

0.8
0.6
0.4
0.2
0

0

60

120
180
Time (min)

240

300

(b) A490 bolts
Figure 5.22: Comparison of connections rotations as a function of fire exposure time for (a)
A325 and (b) A490 bolt types with different high-temperature material properties

277

Chapter 6 figures

Input
data

Start
Calculate fire temperature

Calculate temperature in cross-section of each segment

Calculate fire-induced axial force in the beam

Generate M-κ-P relationships for each segment

Determine curvature profile at each segment

Calculate rotation in connection by
integrating curvature across beam span

Failure limit states for
beam/connection

No
Increment time step

Yes
Calculate fire resistance of connection
End
Figure 6.1: Flowchart summarizing the procedure to evaluate fire resistance of connection

278

1200

Temperature ( C)

1000
800
600
ISO834 fire curve

400

ASTM E119 fire curve

200
0

0

30

60

90

120

150

180

Time (min)

Figure 6.2: Comparison of ISO834 and ASTM E119 standard fire curves

279

w

Fire
L
(a) Elevation of fire exposed beam

Steel beam

Fire
insulation

(b) Beam cross-section

Figure 6.3: Schematic of (a) beam elevation and (b) cross-section used for computing
temperatures in fire protected steel sections

280

1200

Temperature ( C)

1000
800
600
ISO834 fire curve

400

Step-by-step method

Best-fit method

200

Dwaikat and Kodur

0
0

30

60

90

120

150

180

Time (min)

(a) W12x30
1200

Temperature ( C)

1000
800
600
ISO834 fire curve

400

Step-by-step method

Best-fit method

200

Dwaikat and Kodur

0
0

30

60

90

120

150

180

Time (min)

(b) W24x76
Figure 6.4: Comparison of steel temperatures obtained using different methods for (a) W12x30
and (b) W24x76 beam sections with 15mm fire protection

281

w

Fire
L
(a) Elevation of fire exposed beam

Steel beam

(b) Beam cross-section

Figure 6.5: Schematic of (a) beam elevation and (b) cross-section used for computing
temperatures in unprotected steel sections

282

1200

Temperature ( C)

1000
800
600
ISO834 fire curve

400

Step-by-step method
Best-fit method

200

Dwaikat and Kodur

0

0

20

40

60

80

100

120

Time (min)

(a) W12x30
1200

Temperature ( C)

1000
800
600
ISO834 fire curve

400

Step-by-step method

Best-fit method

200

Dwaikat and Kodur

0
0

20

40

60

80

100

120

Time (min)

(b) W24x76
Figure 6.6: Comparison of steel temperatures obtained using different methods for (a) W12x30
and (b) W24x76 beam sections without fire protection

283

1200
Fire temperature

Steel temeprature (protected)

Temperature ( C)

1000

Steel temperature (unprotected)

800
600

400
200
0

0

50

100

150

200

Time (min)

(a) W12x30
1200
Fire temperature

Steel temeprature (protected)

Temperature ( C)

1000

Steel temperature (unprotected)

800
600

400
200
0

0

50

100

150

200

Time (min)

(b) W24x76
Figure 6.7: Comparison of steel temperatures obtained using step-by-step method for (a)
W12x30 and (b) W24x76 beam sections subjected to design fire exposure

284

Compression

Fire induced axial force

Initial yielding

Catenary action
Tten,max

Tc

Ty

Steel temperature (Ts)

Stage 3

Stage 1

Stage 2

Tension

Figure 6.8: Fire induced axial force development in connection
w

L

(a) Elevation of beam
*Segment number

w

1* 2

3

4

5

6

(b) Cross-section

7

8

9

10 1112

13 14

L

(c) Idealization

(d) Discretized cross-section

Figure 6.9: A schematic of beam idealization into segments and discretized cross-section

285

Z

εo

Y

P
Mx

κ
X

Figure 6.10: Illustration of generalized stress and generalized strains oriented in the direction
corresponding to their positive values
w

Fire
L
(a) Elevation of fire exposed beam
Geometrical centroidal axis

Tension

Tension

Tension

Tension

εo

εr

κ
1

y

Compression

Compression

Tension

Tension

Neutral axis
Cross-section

Total strain

Bending strain

Axial strain

Residual strain

(b) Strain profile in beam cross-section

Figure 6.11: Total strain along the beam cross-section along with different strain components

286

Start

Tangent stiffness
matrix [Q] from
previous iteration

Fire induced
axial force (P)
Increment moment

Apply load vector {f}app

Calculate [Q] and {X} (Eq.[6.22])
Calculate strain in all elements (Eq.[6.21])
Calculate internal force vector {f}int (Eq. [6.19])
Update [Q]. Calculate unbalanced
force vector {f}unb = {f}app - {f}int

Unbalanced force
≤ Tolerance limit

No
{f}app = {f}unb

Yes
Write curvature and moment (point on
the moment curvature axial force curve)
End
Figure 6.12: Flow chart illustrating the steps associated in the generation of moment-curvatureaxial force curve

287

Force
First unbalanced
force
D

fB

Second unbalanced
force

E
2

Exact
solution

1
fA

A

O

XA

f1

X1

C

B

f2

X2

XB

Displacement

Figure 6.13: Iterative solution technique for generating force-deformation curves

288

Input
data

Start
Calculate fire temperature

Calculate axial force P

Increment moment

Apply load vector {f}app

Calculate [Q] and {X} (Eq.[6.22])
Calculate strain in all elements (Eq.[6.21])
Calculate internal force vector {f}int (Eq. [6.19])
Update [Q]. Calculate unbalanced
force vector {f}unb = {f}app - {f}int

Unbalanced force
≤ Tolerance

No
{f}app = {f}unb

Yes
Write curvature and moment (point on
the moment curvature axial force curve)
No

Each element in
[Q] ≠ 0
Yes
Calculate connections rotation
Failure limit
states
Yes

No
Increment time step

End

Figure 6.14: Flowchart showing the steps associated with the analysis of double angle connection

289

300

Moment (kN-m)

250
200
150
100

50

15 mins (-83 kN)

30mins (-110 kN)

45 mins (-23 kN)

60 mins (18 kN)

75 mins (35 kN)

0
0

20

40

60

80

Curvature (1/km)

(a) W12x30
1400
1200

Moment (kN-m)

1000
800
600
400
200
0
0

15 mins (-138 kN)

30 mins (-328 kN)

45 mins (-226 kN)

60 mins (-68 kN)

75 mins (23 kN)

90 mins (68 kN)

105 mins (86 kN)

120 mins (88 kN)

10

20

30

40

Curvature (1/km)

(b) W24x76
Figure 6.15: Moment-curvature curves as a function of fire exposure time for (a) W12x30 and
(b) W24x76 steel section with 15mm of fire insulation (Note: The fire induced axial forces are
shown in the parenthesis)

290

1200

Temperature ( C)

1000
800
600

400
ASTM E119 fire curve

200

Steel temperature

0

0

30

60

90

120

150

180

Time (min)

(a) Temperature
300

Moment (kN-m)

250
200
150
100

50

15 mins (-96 kN)

30 mins (-83 kN)

45 mins (-5 kN)

60 mins (28 kN)

75 mins (40 kN)

0
0

20

40

60

80

100

Curvature (1/km)

(b) Moment-curvature
Figure 6.16: Progression of (a) temperature in beam (b) moment-curvature-axial force in the
beam, and (c) rotation at connection for double angle connection

291

Figure 6.16 (cont’d)
50
ANSYS

40

Rotation (degree)

Proposed method

30

20

10

0
0

20

40
60
Time (min)

(c) Rotation

292

80

100

Figure 6.17: Illustration of the Cardington test (a) building plan and (b) shear tab connection
geometry (Note: all dimensions are in mm)

293

1000

Temeprature ( C)

800

600

400

200

0
0

30

60

90

120

150

180

210

240

Time (min)

Figure 6.18: Measured fire (gas) temperatures during Cardington test
1000

Temeprature ( C)

800

600

Fire temperature - measured
Fire temperature - approximated

400

200

0
0

30

60

90

120

150

180

210

240

Time (min)

Figure 6.19: Comparison of measured and approximated fire temperature used for analyzing
shear tab connection

294

800

Rotation (millirad)

700
600

Measured data

Predicted data

500
400
300
200
100
0
0

10

20

30

40

50

60

Time (min)

Figure 6.20: Comparison of predicted and measured rotation for shear tab connection

295

Appendix E figures

Beam
Column

Fire exposed steel frame

(a) Three dimensional perspective view of steel framed building

Beam

Fire exposed
steel frame

Column

(b) Two dimensional view of steel framed building

Figure E.1: Illustration of the steel framed building used in the analysis

296

Figure E.2: Geometrical details of double angle connection

297

1200

Temperature ( C)

1000
800
600
400
ASTM E119 fire curve
Beam temperature

200
0
0

30

60

90

120

150

180

Time (min)

(a) Temperature
1400
1200

Moment (kN-m)

1000
800
600
400
200
0
0

15 mins (-138 kN)

30 mins (-328 kN)

45 mins (-226 kN)

60 mins (-68 kN)

75 mins (23 kN)

90 mins (68 kN)

105 mins (86 kN)

120 mins (88 kN)

10

20

30

40

Curvature (1/km)

(b) Moment-curvature

Figure E.3: Progression of (a) temperature in beam (b) moment-curvature-axial force in the
beam, and (c) fire-induced axial force in connection for the analyzed connection

298

Figure E.3 (cont’d)
200

Axial force (kN)

100
0
0

20

40

60

80

-100
-200
-300
-400
-500

Time (min))

(c) Axial force

299

100

120

140

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