WHOLE RANGE BEHAVIOUR OF RESTRAINED ...

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WHOLE RANGE BEHAVIOUR OF RESTRAINED

REINFORCED CONCRETE BEAMS AND FRAMES

IN FIRE

A thesis submitted to The University of Manchester for the degree of

Doctor of Philosophy

in the Faculty of Science and Engineering

2017

Sherwan M.Z. Izzaddin Albrifkani

School of Mechanical, Aerospace and Civil Engineering

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CONTENTS

CONTENTS ................................................................................................................. 2

LIST OF TABLES ....................................................................................................... 5

LIST OF FIGURES ..................................................................................................... 7

NOTATION ............................................................................................................... 14

ABBREVIATION ...................................................................................................... 17

ABSTRACT ............................................................................................................... 18

DECLARATION ....................................................................................................... 20

COPYRIGHT ............................................................................................................. 21

DEDICATION ........................................................................................................... 22

ACKNOWLEDGEMENTS ....................................................................................... 23

LIST OF PUBLICATIONS ....................................................................................... 24

CHAPTER 1- INTRODUCTION ........................................................................... 25

1.1 Introduction ...................................................................................................... 25

1.2 Originality and Objectives of the Research ...................................................... 26

1.3 Thesis Structure ................................................................................................ 27

CHAPTER 2- LITERATURE REVIEW ............................................................... 29

2.1 Introduction ...................................................................................................... 29

2.2 Design for Fire Resistance ............................................................................... 29

2.3 Structural Response to Fire .............................................................................. 31

2.4 Behaviour of RC Beams in Fire ....................................................................... 34

2.4.1 Simply Supported RC beams (Axially and Rotationally Unrestrained at

Ends) ................................................................................................................... 34

2.4.2 Axially Restrained RC Beams ................................................................... 35

2.4.3 Rotationally Restrained RC Beams ........................................................... 36

2.4.4 Axially and Rotationally Restrained RC Beams ....................................... 38

2.5 RC Frames in Fire ............................................................................................ 41

2.6 Concrete Spalling ............................................................................................. 46

2.7 Catenary Action ................................................................................................ 48

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2.8 Simplified Calculation Methods of RC Beams in Fire ..................................... 56

2.9 Numerical Finite Element Models ................................................................... 57

2.10 Summary of Literature Review and Research Originality ............................. 59

CHAPTER 3- EXPLICIT MODELLING OF LARGE DEFLECTION

BEHAVIOUR OF RESTRAINED REINFORCED CONCRETE BEAMS IN

FIRE .......................................................................................................................... 61

3.1. Introduction ..................................................................................................... 61

3.2. Development of the Explicit Modelling Methodology ................................... 62

3.2.1 Brief Introduction to the Tests by Yu and Tan (2013) and Yu And Tan

(2014) .................................................................................................................. 62

3.2.2 Element Type, Boundary Conditions and Load Application ..................... 65

3.2.3 Material Constitutive Models .................................................................... 66

3.2.3.1. Concrete .............................................................................................. 66

3.2.3.2 Steel Reinforcement ............................................................................. 69

3.2.4 Mesh Sensitivity ........................................................................................ 70

3.2.5 Introduction to Dynamic Explicit Modelling............................................. 71

3.2.6 Reducing Computational Cost ................................................................... 72

3.2.6.1 Load Factoring ..................................................................................... 73

3.2.6.2 Material Damping ................................................................................ 76

3.2.6.3 Mass Scaling ........................................................................................ 78

3.2.7 Validation Against the Test Results of Yu and Tan (Yu and Tan, 2013, Yu

and Tan, 2014) .................................................................................................... 82

3.3 Comparison and Application of the Finite Element Model to RC Sructures in

Fire .......................................................................................................................... 92

3.3.1 Comparison Against the Fire Tests of Dwaikat and Kodur (Dwaikat and

Kodur, 2009a, Dwaikat, 2009)............................................................................ 92

3.4 Preliminary Investigation of the Large Deflection Behaviour of Axially

Restrained RC Beams in Fire ............................................................................... 100

3.5 Conclusions .................................................................................................... 105

CHAPTER 4- BEHAVIOUR OF AXIALLY AND ROTATIONALLY

RESTRAINED REINFORCED CONCRETE BEAMS IN FIRE ..................... 106

4.1 Introduction .................................................................................................... 106

4.2 Simulation Methodology ................................................................................ 106

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4.3 Case Studies ................................................................................................... 111

4.4 Beams with Symmetrical End Boundary Restraints ...................................... 111

4.4.1 Effect of End Rotational Restraint ........................................................... 111

4.4.1.1 Results and Discussions ..................................................................... 112

4.4.1.2 Effects of Rotational Restraint on Bending Resistance Limit Time .. 115

4.4.1.3 Effects of End Rotational Restraint on Beam Ultimate Failure Time 123

4.4.2 Effects of Changing Axial Restraint Stiffness ......................................... 126

4.4.3 Effects of Changing Beam Load Ratio .................................................... 128

4.4.4 Axially Unrestrained Beams .................................................................... 129

4.5 Beams with Asymmetrical End Boundary Restraints .................................... 133

4.5.1 Results and Discussions ........................................................................... 134

4.6 Conclusions .................................................................................................... 143

CHAPTER 5- DEVELOPMENT OF A SIMPLIFIED METHOD FOR

ANALYSIS OF AXIALLY AND ROTATIONALLY RESTRAINED

REINFORCED CONCRETE BEAMS IN FIRE ................................................ 147

5.1Introduction ..................................................................................................... 147

5.2 Key Features of Restrained Beam Behaviour ................................................ 147

5.3 Representative Beam Model .......................................................................... 151

5.4 Development of a Simplified Model .............................................................. 152

5.4.1 Beam Deflection Profile .......................................................................... 153

5.4.2 Simplified Structural Analysis Procedure................................................ 158

5.4.3 Compatibility and Equilibrium Conditions.............................................. 160

5.5 Maximum Concrete Compressive Strain (εcmax,T) .......................................... 163

5.6 Beam Bending Failure Modes ........................................................................ 164

5.7 Beam Bending Resistance Time When kr≤ kr,D .............................................. 164

5.8 Assumptions in Catenary Action Stage .......................................................... 169

5.9 Beam Ultimate Resistance Time (tUR) in Catenary Action ............................ 170

5.10 Limitation of applicability ............................................................................ 173

5.11 Verification of the Simplified Model ........................................................... 173

5.12 Effects of Changing Different Beam Parameters ......................................... 178

5.13 Conclusions .................................................................................................. 184

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CHAPTER 6- PERFORMANCE OF REINFORCED CONCRETE FRAMES

UNDER FIRE CONDITIONS .............................................................................. 187

6.1 Introduction .................................................................................................... 187

6.2 Simulation Parameters .................................................................................... 187

6.3 Parametric Study Cases .................................................................................. 193

6.4 Simulation Results and Discussions ............................................................... 195

6.5 Conclusions .................................................................................................... 213

CHAPTER 7- CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE

STUDIES ................................................................................................................. 214

7.1 Introduction .................................................................................................... 214

7.2 Explicit Finite Element Modelling Methodology........................................... 214

7.3 Behaviour of Restrained RC Beams in Fire ................................................... 215

7.4 Development of a Simplified Method for Analysis of Axially and Rotationally

Restrained RC Beams in fire ................................................................................ 217

7.5 Behaviour of RC Frames in Fire .................................................................... 218

7.6 Recommendations for Future Studies ............................................................ 219

APPENDIX ............................................................................................................. 221

A1 Beam Cross-Section Bending Moment Capacity ........................................... 221

A2 Calculation of Rotational and Lateral Stiffness of Supporting Columns ....... 221

A2.1 Rotational Stiffness KR,sup ........................................................................ 221

A2.2 Lateral Stiffness KA,sup.............................................................................. 222

A3 Values of the Stiffness and Gaps for the Horizontal Restraints during the Tests

by Yu and Tan (2013) and Yu and Tan (2014) .................................................... 223

A4 Material Properties According to Eurocode ................................................... 224

A4.1 Free Thermal Strain εth ............................................................................. 224

A4.2 Instantaneous Stress-Related Strain εσ ..................................................... 225

A4.3 Specific Heat ............................................................................................ 229

A4.4 Density ..................................................................................................... 230

A4.5 Thermal Conductivity .............................................................................. 230

REFERENCES ……………………………………………………………..…….231

Word count: 51,284

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LIST OF TABLES

Table 3.1: Reinforcement detailing ............................................................................ 64

Table 3.2: Mechanical properties of steel reinforcement ........................................... 64

Table 3.3: Parameters for definition of the concrete damaged plasticity model

(ABAQUS, 2013) ....................................................................................................... 67

Table 3.4: Comparison between stable time increment and CPU time using load

factoring and mass scaling for test S4 ........................................................................ 80

Table 3.5: Parameters used for modelling the tests of Yu and Tan (2013) and Yu

and Tan (2014) ........................................................................................................... 90


factoring and mass scaling ....................................................................................... 102

Table 4.1: Parametric study cases and summary of results of beams with

asymmetrical boundary restraints ............................................................................ 134

Table 4.2: Failure time results of beams with reinforcement details 1 and 2 .......... 139

Table 5.1: Selected parameter simulation cases for the validation study ................ 174

Table 6.2: Parametric study cases and summary of main results ............................. 193

Table 6.3: Ambient temperature capacities of beam and column members ............ 195

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LIST OF FIGURES

Figure 1.1: Collapse of an RC warehouse building in the port of Ghent-Belgium due

to fire .......................................................................................................................... 26

Figure 2.1: Non-uniform temperature distribution over the cross-section of a beam 32

Figure 2.2: Thermal expansion in a simply supported beam ..................................... 33

Figure 2.3: Thermal bowing in a simply supported beam ......................................... 34

Figure 2.4: Free body diagram of a beam with axial restraint (Buchanan, 2002)...... 35

Figure 2.5: Rotational restrained, axially unrestrained beam in fire .......................... 37

Figure 2.6: Test setup, dimensions and reinforcement details of test specimens by Lin

T.D et al. (1981) ......................................................................................................... 37

Figure 2.7: Test setup, dimensions and reinforcement details of test specimens by Shi

and Guo (1997) and Xu et al.(2015) .......................................................................... 38

Figure 2.8:Dimensions and reinforcement details of beams simulated by Dwaikat and

Kodur (2008), Riva and Franssen (2008) and Wu and Lu (2009) ............................. 39

Figure 2.9: Effect of axial restraint on beam bending failure time by Dwaikat and

Kodur (2008), Riva and Franssen (2008) and Wu and Lu (2009) ............................. 40

Figure 2.10: Lateral displacement of edge columns due to expansion of heated slabs

(Bailey, 2002) ............................................................................................................. 42

Figure 2.11: Test arrangement by Guo and Shi (2011) .............................................. 43

Figure 2.12: Position of plastic hinges of a frame specimen by Guo and Shi (2011) 44

Figure 2.13: Dimensions and reinforcement details of simulation structure of Lue

(2007) ......................................................................................................................... 45

Figure 2.14: Test setup (Raouffard and Nishiyama, 2015) ........................................ 46

Figure 2.15: Behaviour of beam-column sub-assemblage under column removal

scenario ...................................................................................................................... 49

Figure 2.16: Elevation view and reinforcing details of the three RC frames (Stinger

and Orton, 2013) ........................................................................................................ 51

Figure 2.17: Applied load versus displacement of the three test specimens by Stinger

and Orton (2013) ........................................................................................................ 51

Figure 2.18: Test arrangement and test specimen (Yu and Tan, 2013) ..................... 52

Figure 2.19: Crack patterns and failure modes of specimen S5-1.24/1.24/23 from Yu

and Tan (2013) tests ................................................................................................... 53

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Figure 2.20: Test arrangement (Yu and Tan, 2014) ................................................... 53

Figure 2.21: Detailing techniques of test specimens (Yu and Tan, 2014) ................. 55

Figure 3.1: Geometrical details of RC beam-column sub-assemblages and frames

(Yu and Tan, 2014, Yu and Tan, 2013, Yu, 2012)……………………………….....64

Figure 3.2: Boundary conditions applied in the FE models ....................................... 66

Figure 3.3: Concrete compressive stress-strain relationship ...................................... 68

Figure 3.4: Stress-strain relationship of concrete in tension ...................................... 69

Figure 3.5: Stress-strain relationship of reinforcing bars ........................................... 69

Figure 3.6: Sensitivity of FE simulation results to mesh sizes of concrete, for test S4

.................................................................................................................................... 71

Figure 3.7: Comparison between FE simulation and test results for different

simulation loading durations for test S4..................................................................... 74

Figure 3.8: Comparison between test and load-controlled simulation results for

different simulation loading durations (test specimen S4) ......................................... 75

Figure 3.9: Comparison between test results and FE simulation results using different

damping ratio 𝜉 for test S4 ........................................................................................ 78

Figure 3.10: Comparison between simulation results using load factoring and mass

scaling for test S4 ....................................................................................................... 80

Figure 3.11: The effect of applying mass scaling to a small region of fine mesh ..... 81

Figure 3.12: Comparison between modelling and test results (model S4) ................ 83

Figure 3.13: Variation of longitudinal steel reinforcement at critical regions (model

S4) .............................................................................................................................. 84



Figure 3.16: Comparison between modelling and test results (model F2) ................ 87

Figure 3.17: Comparison between modelling and test results (model F4) ................ 88

Figure 3.18: Variation of longitudinal steel reinforcement at critical regions (model

F4) .............................................................................................................................. 89

Figure 3.19: Deformed shape and failure mode of FE simulations and tests ............ 91

Figure 3.20: Details of test beams B1, B2 and B3 with the locations of

thermocouples (Dwaikat and Kodur, 2009a) ............................................................. 93

Figure 3.21: Comparison between predicted and measured temperature for B1 ....... 95

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Figure 3.22: Applying temperatures at nodes according to experimental

measurements of thermocouples for B2 and B3 ........................................................ 96

Figure 3.23: Tensile stress-strain relationship of concrete at elevated temperatures . 98

Figure 3.24: Mid-span deflection and kinetic energy versus fire exposure time for

different heating durations (Beam B2) ....................................................................... 99

Figure 3.25: Comparison between predicted and measured results ......................... 100

Figure 3.26: Details of the axially restrained beam ................................................. 101

Figure 3.27: General behaviour of axially restrained RC beam in fire .................... 102

Figure 3.28: Vertical reaction force, applied load and kinetic energy against time. 103

Figure 3.29: Strain profile in reinforcing bars against time ..................................... 104

Figure 3.30: Deformed shape and failure mode ....................................................... 104

Figure 4.1: Dimensions and boundary conditions of beam-column sub-frame model

.................................................................................................................................. 108

Figure 4.2: Temperature-time histories in the steel reinforcing bars, based on

numerical heat transfer modelling ............................................................................ 111

Figure 4.3: Effect of rotational restraint levels on beam behaviour (ka=0.166) ...... 113

Figure 4.4: Strain-fire exposure time relationship of longitudinal reinforcing bars at

beam mid-span (ka=0.166) ....................................................................................... 114

Figure 4.5: Effect of rotational stiffness level on beam fire resistance (ka=0.166) . 115

Figure 4.6: Deformed shapes and failure modes (kr=0.022, ka=0.166) ................... 116

Figure 4.7: Kinetic energy and beam axial force versus fire exposure time (kr=0.022

and 0.032, ka=0.166) ................................................................................................ 117


beam ends (ka=0.166) .............................................................................................. 119

Figure 4.9: Deformed shapes and failure modes (kr=0.064, ka=0.166) ................... 121

Figure 4.10: Strain-fire exposure time for concrete near the beam ends (kr=0.064,

ka=0.166) .................................................................................................................. 121

Figure 4.11: Behaviour of reinforcing bars at beam ends (kr=0.064, ka=0.166) ..... 122

Figure 4.12: Vertical reaction force, applied load and kinetic energy/internal energy

against fire exposure time (kr=0.064, ka=0.166) ..................................................... 123

Figure 4.13: Strain-fire exposure time longitudinal reinforcing bars at beam mid-span

(ka=0.166) ................................................................................................................ 124

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Figure 4.14: Comparison of beam axial force-fire exposure time relationships

(ka=0.0275) .............................................................................................................. 127


(ka=0.4) .................................................................................................................... 127

Figure 4.16: Effect of rotational stiffness levels on the beam fire resistance

(ka=0.0275) .............................................................................................................. 127

Figure 4.17: Effect of rotational stiffness levels on the beam fire resistance (ka=0.4)

.................................................................................................................................. 128

Figure 4.18: Comparison of beam mid-span deflections-fire exposure time

relationships (ka=0.166, LR=50%) .......................................................................... 128


(ka=0.166, LR=50%) ................................................................................................ 129

Figure 4.20: Effect of rotational stiffness level on beam bending resistance (ka=0)129


beam mid-aspan (ka=0) ............................................................................................ 130

Figure 4.22: Strains of longitudinal reinforcing bars at ends against fire exposure

time (ka=0) ............................................................................................................... 130

Figure 4.23: Sagging bending failure at mid-span (kr=0.013, ka=0) ....................... 131

Figure 4.24: Hogging bending failure at supports ................................................... 131

Figure 4.25: Definition of asymmetrical boundary restraints of beam-column sub-

frame model ............................................................................................................. 133

Figure 4.26: Effect of kr,L on beam bending and ultimate resistance times (kr,R=2,

ka,R=0.166, ka,L=0) .................................................................................................... 135

Figure 4.27: Effect of kr,R on beam bending and ultimate resistance time (ka,R=0.166,

kr,L=0.0075, ka,L=0) ................................................................................................... 135

Figure 4.28: Effect of ka,L on beam bending and ultimate resistance time (kr,R=2,

ka,R=0.166, kr,L=0.0075)............................................................................................ 135

Figure 4.29: Sagging moment failure mode (bending failure mode I) (kr,R=0.025,

ka,R=0.166, kr,L=0.0075, ka,L=0) ................................................................................ 136

Figure 4.30: Beam bending failure modes ............................................................... 136

Figure 4.31: Strain-fire exposure time relationship of top longitudinal reinforcing

bars at right support (ka,R=0.166, kr,R=2, ka,L=0) ..................................................... 137

Figure 4.32: Reinforcement details 1 and 2 ............................................................. 138

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bars at right support (ka,R=0.166, kr,R=2, ka,L=0) ..................................................... 139

Figure 4.34: Strain-fire exposure time relationship of longitudinal reinforcing bars

(ka,R=0.166, kr,R=2, ka,L=0)........................................................................................ 140

Figure 4.35: Strain-fire exposure time for concrete near the beam right end

(ka,R=0.166, kr,R=2, ka,L=0.08, kr,L=0.0075) .............................................................. 142

Figure 4.36: Strain of longitudinal reinforcing bars at right end and beam axial force

against fire exposure time with different ka,L values (ka,R=0.166, kr,R=2, kr,L=0.0075)

.................................................................................................................................. 142

Figure 5.1: Typical axial force-fire exposure time response of a restrained RC beam

in fire ........................................................................................................................ 148

Figure 5.2: Effects of rotational stiffness kr on RC beam behaviour in fire ............ 149

Figure 5.3: Representative restrained beam model .................................................. 152

Figure 5.4: Beam discretization for simplified analysis .......................................... 153

Figure 5.5: Comparison of beam deflection profiles in flexural action (kr=0.0064)155

Figure 5.6: Comparison of beam deflection profiles in flexural action (kr=0.032) . 156



Figure 5.9: Comparison of beam deflection profiles in flexural action (kr=2) ........ 157

Figure 5.10: Comparison of beam deflection profiles in catenary action (kr=0.0064)

.................................................................................................................................. 157

Figure 5.11: Comparison of beam deflection profiles in catenary action (kr=2) ..... 157

Figure 5.12: Distribution of total strain in a beam cross-section ............................. 159

Figure 5.13: Condition of beam at different stages .................................................. 161

Figure 5.14: Loading condition on deformed half beam ......................................... 162

Figure 5.15: Schematic M- 𝜑 at a given fire exposure time .................................... 166

Figure 5.16: Beam behaviour with kr=kr,B ................................................................ 167

Figure 5.17: Beam behaviour with kr,B<kr≤kr,D ........................................................ 168

Figure 5.18: Schematic diagram of a beam in catenary action with kr≥kr,E ............. 170

Figure 5.19: Strain profiles of top reinforcing bars at ultimate beam resistance time

.................................................................................................................................. 172

Figure 5.20: Compatibility condition for top reinforcing bars ................................. 172

Figure 5.21: Mesh density used in the simplified method for validation study ....... 175

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Figure 5.22: Comparison between simplified method results and ABAQUS results

for different end rotational stiffness ratios (kr) ........................................................ 176


for the effect of rotational stiffness on beam fire resistance .................................... 177


for different load ratios (LR) .................................................................................... 179


for different end axial stiffness ratios ka .................................................................. 180


for different span-to-depth ratios (L/D) ................................................................... 182


for different bottom reinforcement amount.............................................................. 183


for different top reinforcement amount .................................................................... 184

Figure 6.1: Dimensions, loading and boundary conditions of the simulation frame 188

Figure 6.2: Transvers reinforcement and beam cross-sectional details ................... 189

Figure 6.3: Different fire scenarios .......................................................................... 190

Figure 6.4: Mesh configuration used for fire scenario 1 .......................................... 191

Figure 6.5: Selective temperature-time histories of column reinforcement, based on

numerical heat transfer analysis (Column dimensions: 400×400mm) ..................... 192

Figure 6.6: Details of longitudinal steel reinforcement for different column sizes . 194

Figure 6.7: Determining maximum beam and column resistances in the frame at

ambient temperature ................................................................................................. 195

Figure 6.8: Typical deflected frame shapes (Deformation scale factor=3) .............. 196

Figure 6.9: Effects of column size and load ratio on frame failure times ................ 197

Figure 6.10: Beam B1 axial force-fire exposure time relationship with different

column sizes and load ratios (Fire scenario 1, beam reinforcement detail 1) .......... 198

Figure 6.11: Beam B1 axial force-fire exposure time relaitonship with beam

reinforcement details 1 and 2 (Fire scenario 1, LR=50%) ....................................... 198

Figure 6.12: Comparison of results between frame loadings 1 and 2 (Fire scenario 1,

beam reinforcement detail 2, column size=400x400mm, LR=50%) ....................... 199

Figure 6.13: Strain-fire exposure time relationships of longitudinal reinforcing bars at

ends of beam B1 with different column sizes (Fire scenario 1) ............................... 200

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Figure 6.14: Column-led failure mode in fire scenario 1 ......................................... 201

Figure 6.15: Beam-led failure mode in fire scenario 1 ............................................ 201


bars at right end of beam B1 (Fire scenario 1, LR=50%) ........................................ 202

Figure 6.17: Maximumn strain-fire exposure time relationship of longitudinal

reinforcing bars in beam B1 span with different column sizes (Fire scenario 1) .... 203

Figure 6.18: Horizontal displacement at the top of column C2 (Fire scenario 2) .... 203

Figure 6.19: Beam B2 deflection-fire exposure time relationship with different

column sizes and load ratios (Fire scenario 2) ......................................................... 204

Figure 6.20: Beam B2 axial force-fire exposure time relationship with different

column sizes and load ratios (Fire scenario 2) ......................................................... 204

Figure 6.21: Frame failure in fire scenario 2 (column-led failure) .......................... 205

Figure 6.22: Beam bending failure in fire scenario 2 and deformed shape of frame in

catenary action ......................................................................................................... 205

Figure 6.23: Strain-fire exposure time for concrete near beam B2 ends (Fire scenario

2, Frame F11) ........................................................................................................... 206

Figure 6.24: Maximumn strain-time relationship of longitudinal reinforcing bars in

beam B2 span with different column sizes and load ratios (Fire scenario 2) .......... 206

Figure 6.25: Strain-time relationship of longitudinal reinforcing bars at beam B2

ends with different column sizes and load ratios (Fire scenario 2) .......................... 206

Figure 6.26: Column-led failure of frame F11 in catenary action ........................... 208

Figure 6.27: Column-led failure of frame F12 in catenary action ........................... 208

Figure 6.28: Horizontal displacement of columns (Fire scenario 3, LR=30%) ....... 209


beam B1 and beam B2 ends with different column sizes and load ratios (Fire

scenario 3) ................................................................................................................ 210

Figure 6.30: Beam axial force-fire exposure time relationship with different column

sizes and load ratios (Fire scenario 3) ...................................................................... 211

Figure 6.31: Frame failure in fire scenario 3 (column-led failure) .......................... 212

Figure 6.32: Frame failure in fire scenario 3 (beam-led failure).............................. 212

Figure 6.33: Maximumn strain-time relationship of longitudinal reinforcing bars in

beam B1 span with different column sizes and load ratios (Fire scenario 3) .......... 212

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NOTATION

A Cross-sectional area of a member section

Am Area of a concrete element in the beam cross-section

As,m Area of a steel bar element in the beam cross-section

B Width of the beam cross-section

C Viscous damping

cp Specific heat of concrete

D Depth of the beam section

E Modulus of elasticity of a material

Ea Modulus of elasticity of steel at ambient temperature

Ea,T Modulus of elasticity of steel at temperature T

Ec1 Secant modulus of elasticity of concrete from the origin to the peak

compressive stress at ambient temperature

Ecm Modulus of elasticity of concrete at ambient temperature

Ecm,T Modulus of elasticity of concrete at temperature T

Ej Modulus of elasticity of the beam-column joint

Es Modulus of elasticity of steel at ambient temperature

F Axial force in the beam

f Mass scaling factor

fcm Compressive cylinder strength of concrete at ambient temperature

fcm,T Compressive cylinder strength of concrete at temperature T

fctm Tensile strength of concrete at ambient temperature

fctm,T Tensile strength of concrete at temperature T

Fint,i Internal axial force in the beam segment i

Fmid Axial compressive force in the beam at bending failure of the mid-span

fp,T Proportional limit stress of steel at temperature T

Fres Residual axial compressive force in the beam when stability of the

beam is regained again in bending after bending failure of the mid-span

fu Tensile strength of steel at ambient temperature

fy Yield strength of steel at ambient temperature

fy,T Yield strength of steel at temperature T

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H Height of the column

I Moment of inertial of a member section

K Stiffness matrix

KA Axial stiffness of the spring

ka,L Relative axil stiffness parameter at left beam end

ka,R Relative axil stiffness parameter at right beam end

KA,sup Lateral stiffness of the supporting columns

KR Rotational stiffness of the spring

kr Relative rotational stiffness parameter at beam ends

kr,L Relative rotational stiffness parameter at left beam end

kr,R Relative rotational stiffness parameter at right beam end

KR,sup Rotational stiffness of the supporting columns

kt,T Temperature-dependent reduction factor of concrete tensile strength

L Span length of the beam

Le Finite element characteristic length

Li Initial length of the beam segment i

Li′ Horizontal projected length of the deformed beam segment i

M Inertia mass matrix

MA Internal bending moment at the beam support

Mm Internal bending moment at the beam mid-span

MRd,fi,hog Reduced beam hogging moment resistance at elevated temperatures

t Fire time

T Temperature

T1 Temperature at the top fibre of a beam section

T2 Temperature at the bottom fibre of a beam section

tBR Beam bending resistance time at elevated temperatures

tmid Beam mid-span bending failure time at elevated temperatures

Tn Lowest natural period

tUR Beam ultimate resistance time at elevated temperatures

Ty Thermal gradient

u Concrete moisture content

w Uniformly distributed load

α Material thermal expansion coefficient

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β Rayleigh stiffness proportional damping factor

δ Deflection profile of the beam

Δa Axial dispacemnt at beam ends at the location of the axial spring

δBR,max Maximum mid-span deflection of the beam at bending resistance time

δCPL Deflection profile of the beam under a concentrated point load

δi,a Vertical deflection at the left end of the segment i

δi,b Vertical deflection at the right end of the segment i

δLP Linear deflection profile of the beam

δmax Maximum deflection at the beam mid-span

δmid,max Mid-span deflection of the beam at the mid-span failure

δst,max Maximum mid-span deflection of the beam when stability of the beam

is regained after bending failure of the mid-span

Δt Stable time increment

ΔT Mean temperature

δt Deflection profile of the beam with total end rotational restraint

δth Thermal bowing

δz Deflection profile of the beam with zero end rotational restraint

εc Strain of concrete at ambient temperature

εc,T Strain of concrete at temperature T

εc1 Strain of concrete at peak compressive stress at ambient temperature

εc1,T Strain value of concrete at peak compressive stress at temperature T

εcmax,T Concrete crushing strain at temperature T

εcr Cracking strain of concrete at ambient temperature

εcr,T Cracking strain of concrete at temperature T

εct Tensile strain of concrete at ambient temperature

εct,T Tensile strain of concrete at temperature T

εcu1,T Strain of concrete at zero stress in the softening stage at temperature T

εmec Mechanical strain

εs,T Strain of steel at temperature T

εsh Hardening strain of steel at ambient temperature

εt,T Maximum strain of steel while maintaining the yield stress at

temperature T

εth Free thermal strain

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εth,c Free thermal strain of concrete

εth,s Free thermal strain of steel

εtot Total strain in an element of the beam cross-section

εu Ultimate strain of steel at ambient temperature

εu,T Ultimate strain of steel at temperature T

εy,T Yield strain of steel at temperature T

λ̂ Lame’s constants

λc Thermal conductivity of concrete

μ Rayleigh mass proportional damping factor

μ̂ Lame’s constants

ν Poisson’s ratio

ξ Damping ratio

ξi Damping ratio in a mode of vibration i

ξmax Damping ratio associated with 𝜔𝑚𝑎𝑥

σc,m Stress in a concrete element in the beam cross-section

ρc,T Density of concrete at temperature T

σs,m Stress in a steel bar element in the beam cross-section

φi Total curvature in the beam segment i

φth Thermal curvature

ωi Natural frequency of the mode i

ωmax Higher natural frequency

ωmin Lowest natural frequency

ABBREVIATION

CPL Concentrated point load

DC Displacement control

HD Heating duration

IE Internal energy

KE Kinetic energy

L/D Span-to-depth ratio

LC Load control

LD Loading duration

LITS Load-induced thermal strain

LP Linear profile

LR Load ratio

MJD Middle joint displacement

MPC Multi-Point Constraint

UDL Uniformly distributed load

18

ABSTRACT

This thesis presents the results of a numerical investigation of the whole range, large

deflection behaviour of axially and rotationally restrained RC beams and interactions

between beams and columns in RC frame structures exposed to fire.

The dynamic explicit time integration algorithm implemented in the general finite

element package ABAQUS/Explicit solver was used so as to overcome various

modelling challenges including temporary instability, local failure of materials, non-

convergence and long simulation time. Either load factoring or mass scaling may be

used to speed up the simulation process. Validity of the proposed simulation model

was checked by comparison of simulation results against relevant test results of

restrained RC beams at ambient temperature and in fire.

The validated ABAQUS/Explicit model was then used to conduct a comprehensive

study of the effects of different levels of axial and rotational restraints on the whole

range behaviour of RC beams in fire, including combined bending and compression

due to restrained thermal expansion, bending failure, transition from compression to

tension when catenary action develops and complete fracture of reinforcement at

ultimate failure. The numerical results show that different bending failure modes

(middle span sagging failure, end hogging failure due to fracture of tensile

reinforcement, end hogging failure due to concrete crushing) can occur under

different levels of boundary restraints. Furthermore, release of a large amount of

energy during the rapid transition phase from compression to tension in a beam

prevents formation of a three hinge mechanism in the beam under bending. The

numerical results have also revealed that reliable catenary action develops at large

deflections following bending failure only if bending failure is governed by

compressive failure of concrete at the end supports whereby a continuous tension

path in the beam can develop in the top reinforcement.

To allow fire engineering practice to take into consideration the complex restrained

RC beam behaviour in fire, a simplified calculation method has been developed and

validated against the numerical simulation results. The proposed method is based on

sectional analysis and meets the requirements of strain compatibility and force

equilibrium. The validation study results have shown that the simplified method can

19

satisfactorily predict the various key quantities of restrained beam axial force and

beam deflection-fire exposure time relationships, with the simplified method

generally giving results on the safe side.

The validated explicit finite element model in ABAQUS was also used to investigate

structural interactions between beams and columns within an RC frame structure

with different fire exposure scenarios. When fire exposure involves beams and

columns located in edge bays of a frame, catenary action cannot develop. Also due to

thermal expansion of the connected beam, additional bending moments can generate

in the columns. Furthermore, very large hogging moments can be induced at the

beam end connected to the internal bay. It is necessary to include these bending

moments when designing beams and columns under such fire conditions. Catenary

action can develop in interior beams of the frame when fire exposure is in interior

bays where the beams have high degrees of axial restraint.

20

DECLARATION

No portion of the work referred to in the thesis has been submitted in support of an

application for another degree or qualification of this or any other university or other

institute of learning.

21

COPYRIGHT

The Author of this thesis (including any appendices and/or schedules to this thesis)

owns any copyright in it (the “Copyright”) and he has given The University of

Manchester the right to use such Copyright for any administrative, promotional,

educational and/or teaching purposes.

Copies of this thesis, either in full or in extracts, may be made only in accordance

with the regulations of the John Ryland’s University Library of Manchester. Details

of these regulations may be obtained from the Librarian. This page must form part of

any such copies made.

The ownership of any patents, designs, trade marks and any and all other intellectual

property rights except for the Copyright (the “Intellectual Property Rights”) and any

re-productions of copyright works, for example graphs and tables (“Reproductions”),

which may be described in this thesis, may not be owned by the author and may be

owned by third parties. Such Intellectual Property Rights and Reproductions cannot

and must not be made available for use without the prior written permission of the

owner(s) of the relevant Intellectual Property Rights and/or Reproductions.

Further information on the conditions under which disclosure, publication and

exploitation of this thesis, the Copyright and any Intellectual Property Rights and/or

Reproductions described in it may take place is available from the Head of School of

Mechanical, Aerospace and Civil Engineering.

22

DEDICATION

To My Parents

23

ACKNOWLEDGEMENTS

First and foremost, I wish to give all the praise to Almighty God for giving me the

strength and time to complete this research.

I would like to express my greatest gratitude to my supervisor, Professor Yong

Wang, for his constant encouragement, enthusiastic advice and professional guidance

through the undertaking of this research. He provided me with all kinds of support.

I wish to also thank Professor Martin Gillie for his advice on my yearly transfer

reports.

I would like to acknowledge the financial support given by my sponsor, Ministry of

Higher Education and Scientific Research-Kurdistan Region-Iraq. The efforts given

by the Kurdistan Regional Government UK Representation to assist with the

administration issues of my scholarship are really appreciated.

My Special thanks go to my colleagues in the research group for their help and

providing an inspiring environment.

Finally, I would like to express my deepest gratitude to my parents, my wife “Sara”,

my daughter “Leena”, my son “Mustafa”, my sisters and brothers for their endless

patience, support, encouragement and love during my PhD study. Without them, this

would not have been possible.

Sherwan Albrifkani

24

LIST OF PUBLICATIONS

1- ALBRIFKANI, S. & WANG, Y. C. (2016), “Explicit modelling of large

deflection behaviour of restrained reinforced concrete beams in fire”, Engineering

Structures Journal, Vol. 121, pp. 97-119.

2- ALBRIFKANI, S. & WANG, Y. C. (2016), “Investigation of progressive collapse

mechanisms of reinforced concrete frames in fire”, 18th Young Researchers

Conference, The Institution of Structural Engineers, April 2016, London. (Poster)

3- WANG, Y. C. & ALBRIFKANI, S. (2017), “Effects of rotational and axial

restraints on bending fire resistance of reinforced concrete beams”, ASCE EMI

conference (American Society of Civil Engineers, Engineering Mechanics Institute),

June 2017, University of California, San Diego.

CHAPTER 1 INTRODUCTION

25

CHAPTER 1

INTRODUCTION

1.1 Introduction

Fire is one of the most dangerous conditions to which a building may be subjected

during its life time. The most recent tragic fire in Grenfell building in London

graphically illustrates the destructive power of fire. Fortunately, the building

structure retained its integrity, but the fact that the fire was engulfing the entire

building puts in question one of the most basic assumptions of fire resistance:

compartmentation.

This fire, and along with many other historical fire events, makes it imperative for

the fire protection and structural engineering community to understand how building

structures perform under realistic fire conditions. Because building structures are

interconnected entities, it is necessary to understand how structural components

behave under different connecting (or restraint) conditions and how their interactions

affect fire resistance of entire structures. Whilst there have been such studies in

relation to steel and steel-concrete composite structures, reinforced concrete (RC)

structures have not undergone as much scrutiny. Consequently, fire resistance of RC

structures is based on isolated individual structural elements where preventing

degradations of material mechanical properties at elevated temperatures is of only

concern.

In a real RC frame structure in fire, restrained thermal expansion, interactions

between heated and unheated parts and large deflections lead to complicated

behaviour of RC elements that are different from that assumed in current design

methods, which may result in failure of RC structures in fire even though they were

deemed to have sufficient fire resistance according to member based design.

For example, Beitel and Iwankiw (2002) and Lue Taerwe (2007) reported a number

of RC structures that suffered collapse or partial collapse due to thermal expansion

effects in fire. In 1974, a fire developed in a three storey cast in situ RC warehouse

building in the port of Ghent in Belgium. The building was 50×50m on plan. It was

designed to satisfy tabulated fire resistance ratings regarding minimum concrete


26

cover over the reinforcement and minimum size of member cross-section. However,

collapse of part of the building began after about 1hr20min of fire exposure, as

shown in Figure 1.1. The main cause of the collapse was due to shear failure of some

columns caused by thermal expansion of the connected beams, which were exposed

to fire along three sides and restrained from elongation by the adjacent structure,

thereby generating additional forces.

Figure 1.1: Collapse of an RC warehouse building in the port of Ghent-Belgium

due to fire

On the other hand, for axially restrained beams, the alternative load carrying

mechanism of catenary action can develop after bending failure, which can

significantly enhance the beam survival time compared to the fire resistance

estimated based on bending resistance. Catenary action may be used to mitigate

against progressive collapse in fire situation.

1.2 Originality and Objectives of the Research

Proper consideration of the aforementioned two aspects of structural interactions in

RC structures in fire requires a thorough understanding of the whole range behaviour

of RC structures in fire. However, there is a lack of investigation on this subject. This

may be due to the perception that RC members are fire resistant due to their

relatively low thermal conductivity. However, the member based assessment method


27

cannot deal with structural interactions. Among research studies that have included

effects of structural interactions and restraints, they did not address the development

of catenary action at very large beam deflections. There is also an absence of

methods that may be used in design practice.

This project aims to develop thorough understanding of structural interactions in RC

frames in fire, focusing in particular on the whole range behaviour of axially and

rotationally restrained RC beams. Due to high cost of physical fire testing, this study

will be using numerical simulation. The principal objectives of this study are:

1. To develop and validate a 3D numerical simulation model using the general

finite element code ABAQUS/Explicit for analysing the highly complex,

whole range behaviour of RC structures in fire.

2. To perform extensive numerical simulations to investigate the effects of

different axial and rotational restraints on full history behaviour of RC beams

in fire until failure, including catenary action at large deflections.

3. To develop and validate a simplified calculation approach to predict the

whole history behaviour of restrained RC beams in fire.

4. To investigate interactions between RC beams and columns in RC framed

structures in fire.

1.3 Thesis Structure

This thesis is divided into seven chapters.

Chapter 2 reports the results of previous studies that are pertinent to this research

problem. After giving a brief introduction to existing design methods of RC

structures in fire, it reviews research into the behaviour of RC beams and frames

exposed to fire, followed by an introduction to the catenary action in RC beams. It

will also review existing simplified calculation and numerical simulation models to

trace fire performance of RC beams. The state of art review enables identification of

detailed gaps of knowledge in existing research.

Chapter 3 describes the development of a dynamic explicit finite element model for

simulating RC structural members subjected to large deformations at ambient and


28

elevated temperatures using the finite element package ABAQUS. It explains the

numerical simulation challenges, including temporary instabilities, local failure of

materials and long simulation time, and proposes methods to solve these challenges.

The developed simulation model is then validated against relevant test results.

Chapter 4 presents the numerical results, using the validated ABAQUS model

developed in Chapter 3, of a series of analyses of RC beams in fire with different

axial and rotational restraint levels, with either symmetrical or asymmetric restraint

conditions, for the entire range of behaviour . The effects of different boundary

restraints on beam bending resistance time and development of catenary action are

particular focuses of discussions.

Chapter 5 presents and validates a simplified calculation method that may be used in

practical design to include the effects of axial and rotational restraints on RC beams

in fire.

Chapter 6 presents the results of a series of simulations to investigate structural

interactions in RC framed structures in fire.

Chapter 7 concludes the present study and gives suggestions for future research work

on the topic.

CHAPTER 2 LITERATURE REVIEW

29

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

This chapter presents a background to the motivation for this research project. After

summarising the approaches for fire resistance design of RC structures, it reviews

previous research studies on RC beams and frames exposed to fire that provide

understanding leading to the performance based fire resistance design approach. The

concept of catenary action, developed in beams at large deflections, as an alternative

load-carrying mechanism to enhance structural robustness of RC framed structures at

ambient temperature and in fire is then introduced and relevant literature reviewed.

Finally, existing simplified calculation and numerical models to trace the high

temperature response of RC beams are reviewed.

2.2 Design for Fire Resistance

The design methods of specifying fire resistance of RC structures can be divided into

three principal levels based on their accuracy and complexity:

1- Standard fire resistance test;

2- Prescriptive method;

3- Performance-based method.

The standard fire resistance test is conducted by subjecting a structural element to

fire under the standard heating condition in a furnace. The furnace temperature is

controlled to follow the standard gas temperature-time curve as given in national and

international standards, such as ISO 834 (1975) and ASTM E119 (2008). Standard

fire resistance testing is costly. Also the test results are only used for grading purpose

because the standard fire test has many limitations, including idealised boundary

conditions and loading arrangements and limited dimensions (Purkiss, 2007). Testing

of a whole structure under a realistic fire is extremely expensive, impractical and

very rarely done, the fire tests on full-scale buildings at Cardington, UK (BRE, 2003)

being the only example so far.


30

The prescriptive approach for assessing the fire resistance of RC members, in codes

such as EN 1992-1-2 (CEN, 2004), relies on tabulated fire resistance ratings

established from standard fire test data. It simply specifies the minimum concrete

cover over the reinforcement and the minimum size of member cross-section

according to the required fire resistance period and load level of the member. The

minimum concrete cover is specified to ensure that the temperature of the main

reinforcement does not exceed a temperature of 500-550oC as the reinforcement

around this temperature loses about 50% of its strength. The prescriptive approach is

straightforward and simple to use, but in many cases, it is not likely to be accurate

and economic (Khoury, 2000). The effects of specific circumstances such as support

conditions, loading conditions and interaction of structural elements are not

considered.

For reasons that physical fire tests being very expensive and technically demanding

to conduct and prescriptive provisions having many drawbacks despite simplicity, a

more rational methodology, known as the performance-based design approach, has

been developed. This approach has the potential to offer more design flexibility,

reduced construction cost and more accurate representation of how a structure will

behave in fire (Wang et al., 2013, Kodur and Dwaikat, 2007). The performance-

based approach considers realistic fire scenarios, material properties at elevated

temperatures and structural behaviour at elevated temperatures with realistic support

and loading conditions. The performance-based fire resistance approach involves

determination of: (i) fire temperature; (ii) heat transfer to the structure and (iii) the

structural behaviour at elevated temperatures.

A simplistic performance-based fire resistance approach for RC structures is

presented in EN 1992-1-2 (CEN, 2004). At the required fire resistance time,

temperature distributions in RC sections of different sizes and for standard fire

resistance durations from 30 min to 240 min are given. The effects of high

temperatures are considered by removing layers of concrete in the heated cross-

section that have attained high temperatures. These calculations can be used to

predict the fire response of RC elements in isolation. They are not sufficient for

analysing RC members within realistic structures which exhibit complex interactions

between different structural members. To address this problem, sophisticated

numerical finite element analyses will be required. However, it would be unrealistic


31

to expect the majority of fire protection or structural engineers to use finite element

analysis in fire resistance design. In order to facilitate development of more

simplistic methods for performance-based fire resistance design of RC structures

with faithful prediction of structural interactions, it is necessary to develop a

thorough understanding of RC structures in fire.

2.3 Structural Response to Fire

Usmani et al. (2001) have laid out the most fundamental principles that govern the

behaviour of structures in fire where thermal strains play an important role. The total

stains (𝜀𝑡𝑜𝑡𝑎𝑙), which dictate the deformed shape of a structure, are the sum of the

thermal strains (𝜀𝑡ℎ) and the mechanical strains (𝜀𝑚𝑒𝑐), as shown in Equation 2-1.

Thermal strains develop due to the expansion behaviour of materials with increasing

temperatures and the stress state in the structure is only governed by the mechanical

strains.

𝜀𝑡𝑜𝑡𝑎𝑙 = 𝜀𝑡ℎ + 𝜀𝑚𝑒𝑐 (2-1)

With no externally applied load, when the element is unrestrained, thermal strains are

free to develop. No stresses induce in the element and the total strains are only a

function of thermal strains (Equation 2-2)

𝜀𝑡𝑜𝑡𝑎𝑙 = 𝜀𝑡ℎ , 𝜀𝑚𝑒𝑐 = 0 (2-2)

On the other hand, when the element is fully restrained against thermal expansion,

the total element deformation is zero. The thermal strains are therefore counter-acted

by equal and opposite thermal mechanicals, causing restraining stresses in the

element (Equation 2-3).

𝜀𝑡𝑜𝑡𝑎𝑙 = 0 = 𝜀𝑡ℎ + 𝜀𝑚𝑒𝑐 ⇒ 𝜀𝑡ℎ = −𝜀𝑚𝑒𝑐 (2-3)

Figure 2.1 depicts the temperature regime in a beam heated uniformly along its

length (L) from underneath. The temperature distribution over the beam depth (D)

can be split into two components: (1) a uniform mean temperature (∆T) and (2) a

thermal gradient Ty. Usmani et al. (2001) described the effects of thermal strains as

twofold: (1) pure thermal expansion, which leads to an increase in the length of the

heated structural member under the mean temperature rise and (2) thermal bowing,

which is induced due to the thermal gradient Ty.


32

Figure 2.1: Non-uniform temperature distribution over the cross-section of a

beam

Figure 2.2 shows a typical simply supported beam with no external load being

present. Assuming a linear temperature distribution over the beam depth, the mean

temperature (ΔT) and thermal gradient (Ty) can be described as:

∆𝑇 =(𝑇1+𝑇2)

2 (2-4)

𝑇𝑦 =(𝑇2−𝑇1)

𝐷 (2-5)

where, T1 and T2 are temperatures at the top and bottom fibres of the beam section. If

the temperature rise over the section is uniform (T1=T2=ΔT in Figure 2.1), the

thermal gradient is therefore zero (Ty=0), thus no thermal curvature is induced along

the beam length. When axial restraint is not present (Figure 2.2(a)), the beam will

expand freely by a magnitude of 𝛼𝐿∆𝑇 with no mechanical strains in the beam. If

this expansion is restrained, as in the pin ended beams (Figure 2.2(b)), the

mechanical strains are equal and opposite to the thermal expansion strains and an

axial restraining force (F) develops in the beam. The restraining force is

compressive, with a magnitude given by:

𝐹 = −𝐸𝐴𝛼∆𝑇 (2-6)

where, 𝛼 is the thermal expansion coefficient for the material, E is elastic modulus of

the material and A is the cross-sectional area of the beam.

T2

T1

D ΔT

∆𝑇 =(𝑇1 + 𝑇2)

2 𝑇𝑦 =

(𝑇2 − 𝑇1)

𝐷

= +

Temperature

distribution

over the depth

of a beam

cross-section

Uniform mean

temperature

Thermal

gradient


33

Figure 2.2: Thermal expansion in a simply supported beam

In structural elements made of materials with low conductivity, such as concrete,

thermal gradients inevitably induce over their depth. The bottom layers exposed to

direct fire are at significantly higher temperatures than the top layers. This causes the

hotter layers to expand more than the cooler layers, leading to curvature (bowing) in

the element. Figure 2.3 shows the same aforementioned simply supported beam but

this time subjected to non-uniform temperature over the depth (T2>T1 in Figure 2.1).

As a result of the thermal gradient, a uniform thermal curvature 𝜑𝑡ℎ = 𝛼𝑇𝑦 is

induced along the length of the beam. In axially unrestrained beam, the beam ends

contract to accommodate the deflection caused by thermal curvature (Figure 2.3(a)).

Restraint of this contract (inward displacement) when the beam is axially restrained

will generate a tensile axial force in the beam as shown in Figure 2.3(b). In addition,

in a rotationally fix-ended beam (Figure 2.3(c)), the uniform thermal curvature

𝜑𝑡ℎ = 𝛼𝑇𝑦, described above for the simply supported beam, is counteracted by the

end moments (M). In this case, the beam remains straight with a constant moment

𝑀 = 𝐸𝐼𝜑𝑡ℎ along its length (Usmani et al., 2001). In reality, the effects of thermal

expansion, thermal bowing and end boundary restraints act together. Thermal

expansion and inward displacement caused by thermal bowing may absorb each

other. This can affect the axial compression and axial tension generated in axially

restrained beams. Moreover, compressions generated due to restrained thermal

(a) Axially unrestrained

𝐹 = −𝐸𝐴𝛼∆𝑇

𝛼𝐿∆𝑇

L

F F

Uniform temperature rise ΔT

Uniform temperature rise ΔT

(b) Axially restrained


34

expansion and thermal bowing in fixed ended beams result in significant

compressions on the exposed layers of the beam and tensions or still small

compressions on the unexposed top layers.

Figure 2.3: Thermal bowing in a simply supported beam

2.4 Behaviour of RC Beams in Fire

2.4.1 Simply Supported RC beams (Axially and Rotationally Unrestrained at

Ends)

The fire behaviour of a simply supported RC beam is relatively straightforward.

When the beam is exposed to fire from underneath, it experiences downward

deflection due to loss of flexural stiffness and thermal bowing. During the initial

stages of fire exposure, the beam deflection is mainly due to thermal curvature

resulting from greater expansion of the bottom part of the beam than the top part. At

later stages of fire, the deflection caused by material property deterioration at high

M M

(c) Fully fixed

Uniform temperature gradient Ty

Δth

L

F F

𝛿𝑡ℎ




𝛿𝑡ℎ

(a) Axially unrestrained

(b) Axially restrained


35

temperatures dominates. The rate of increase in the beam deflection accelerates until

formation of a plastic hinge at the mid-span signified by run-away deflection of the

beam. At the fire limit state, the reduced moment capacity of the beam is equal to the

applied moment (Shi et al., 2004, Dotreppe and Franssen, 1985, Lin T.D et al., 1981,

Dwaikat and Kodur, 2009a, Choi and Shin, 2011, Zha, 2003, Kodur and Dwaikat,

2007). The fire resistance of the beam based on the beam strength can be accurately

quantified by calculating the positive (sagging) moment capacity of the beam using

the reduced strengths of concrete and streel reinforcement at elevated temperatures.

2.4.2 Axially Restrained RC Beams

When a beam is axially restrained, fire exposure generates additional axial forces in

the beam. Figure 2.4 shows the typical behaviour of a simply supported RC beam in

fire with axial restraint and free rotation at the ends. As the longitudinal movement of

the beam due to thermal expansion is restrained, an axial compressive force F

develops in the beam.

Figure 2.4: Free body diagram of a beam with axial restraint (Buchanan, 2002)

At small deflections, the position of the axial compressive force at the supports is

generally close to soffit of the beam because the concrete temperatures are higher in

the lower part of the beam. Also, the line of action of the compressive force is

expected to be below the neutral axis of the cross-section at the mid-span where the

highest sagging bending moment exists. Thus, the compressive force in the beam can

be beneficial to the beam through development of arch action.

However, compressive arch action can only develop at very small beam deflections

(less than half of the beam depth) and can be rapidly lost as beam deflections

F


36

increase (Buchanan, 2002, Wu and Lu, 2009). Afterwards, the compressive force in

the beam increases the sagging moment in the beam due to P-Δ effect. This

detrimental effect increases with increasing axial restraint due to higher compressive

force (Wu and Lu, 2009).

A review of the literature shows that among the fire resistance tests conducted on

individual RC beams with free rotation at ends, only the work by Dwaikat and Kodur

(2009a) has considered the effect of axial restraint. In their experimental work, they

examined the effects of the following variables: concrete compressive strength, fire

scenario, axial restraint condition and load level. Out of six tests, four beams were

simply supported with free expansion and rotation at ends. The other two beams

were axially restrained with a stiffness value of about 13 kN/mm but still free to

rotate at the ends. Results of these tests will be used in validating the finite element

model developed by the author in Chapter 3.

2.4.3 Rotationally Restrained RC Beams

End rotational restraint can improve the fire performance of RC beams in bending by

reducing the sagging bending moment in the span. Consider a beam with fixed-slide

end supports as shown in Figure 2.5, where the beam is free to expand but

rotationally restrained at the ends. When it is subject to fire exposure from

underneath, the beam tends to bow downward towards the fire due to non-uniform

temperature distribution within the depth of the beam. Owing to rotational restraint,

the hogging moment at the beam ends increase, causing a corresponding reduction in

sagging moment in the span. This lasts until formation of plastic hinges at the ends

when the hogging moment reaches the end moment capacities at elevated

temperatures. Afterwards, the hogging moment decreases and follows the bending

moment capacity-fire time relationship while the sagging moment increases until a

third plastic hinge forms in the span. At this stage, a complete plastic hinge

mechanism forms in the beam as shown in Figure 2.5.


37

Figure 2.5: Rotational restrained, axially unrestrained beam in fire

Lin T.D et al. (1981) conducted a series of experimental tests to investigate the effect

of beam continuity, type of concrete, load level and moment redistribution on the

behaviour of continuous RC beams in fire. Figure 2.6 shows details of the tested

beams with a fire exposed main span of 6.1m between the supports and two short

cantilever spans beyond the supports. Different degrees of continuity were simulated

by varying the applied loads P1 and P2 on the unheated cantilevers. The results

confirm full redistribution of moments between the negative (hogging) and positive

(sagging) moments under fire before flexural failure of the beams.

Figure 2.6: Test setup, dimensions and reinforcement details of test specimens

by Lin T.D et al. (1981)

Formation a full plastic hinge mechanism in continuous beams was also reported in a

number of fire tests on two-span continuous RC beams carried out by Shi and Guo

w

Plastic hinges

6#6

2#6

Section a-a

2#6

4#6

Section b-b

6#6

2#6

Section c-c

Beam width=305mm

Beam depth=356mm

P3

1.83m 1.83m

P3 P3 P3

6.1m

P1 P2

a

a

b

b

c

c


38

(1997) and Xu et al.(2015). Figure 2.7 provides details of the tested beams in each

study.

(a) Shi and Guo (1997) (b) Xu et al. (2015)

Figure 2.7: Test setup, dimensions and reinforcement details of test specimens

by Shi and Guo (1997) and Xu et al.(2015)

However, with axial restraint, the presence of an axial compressive force together

with bending moments in the beam will change the beam behaviour in fire. One of

the objectives of this present study is to examine whether it is possible to form a

complete hinge mechanism in axially restrained continuous beams (beams with both

axial and rotational restraints).

2.4.4 Axially and Rotationally Restrained RC Beams

In an RC frame, the adjacent structural members may impose both axial and

rotational restraints on the heated RC beam. Interactions between thermal expansion,

temperature gradient in the cross-section and material properties degradation at high

temperatures have important influences on the development of internal forces in the

beam and its failure modes. There are very few published studies on the influence of

axial and rotational restraints on the behaviour of RC beams in fire. The available

literature in this research field, which is of a numerical nature, is concerned with

l l

P P

a a

Φ8@200 Φ8@165

2Φ12

2Φ12

150

800

170

80

300

2Φ25

2Φ25

200

l=2400 mm

a=800 mm and 1600 mm l=1860 mm, a=930 mm

All dimensions in mm


39

rotationally fix-ended beams with different axial restraint levels. This section

highlights their key findings.

Dwaikat and Kodur (2008), Riva and Franssen (2008) and Wu and Lu (2009)

performed numerical analyses on rotationally restrained RC beams subjected to a

uniformly distributed load at top and fire on three sides. Figure 2.8 shows the beam

details in these three studies. The axial restraint had temperature-independent linear

elastic behaviour. These three studies have reached different conclusions on the

effect of axial restraint on bending resistance of axially restrained and rotationally

fixed beams, as shown in Figure 2.9: the results of Dwaikat and Kodur (2008)

showing monotonically increasing bending failure time at increasing axial restraint

stiffness while those of Wu and Lu (2009) showing exactly opposite trend, and those

of Riva and Franssen (2008) showing initially increasing and then decreasing

bending resistance time.

(a) Riva and Franssen (2008) (b) Dwaikat and Kodur (2008) (c) Wu and Lu, (2009)

Figure 2.8:Dimensions and reinforcement details of beams simulated by

Dwaikat and Kodur (2008), Riva and Franssen (2008) and Wu and Lu (2009)

a

a

b

b

KA

w

L

2Φ20

2Φ20

B=350 mm

D=500 mm

L=6000 mm

4Φ20

2Φ20

Section a-a Section b-b

3Φ20

3Φ20

Section a-a &

Section b-b

B=300 mm

D=500 mm

L=6000 mm

3Φ20

3Φ20

Section a-a &

Section b-b

B=250 mm

D=400 mm

L=6000 mm


40

(a) Wu and Lu (2009) (b) Riva and Franssen (2008)

(c) Dwaikat and Kodur (2008)

Figure 2.9: Effect of axial restraint on beam bending failure time by Dwaikat

and Kodur (2008), Riva and Franssen (2008) and Wu and Lu (2009)

Dwaikat and Kodur (2008) attributed their results to the beneficial effect of arch

action so the higher the axial restraint stiffness, the greater the arch action. However,

Wu and Lu (2009) explained their opposite trend results by focusing on the specific

beam bending failure mode at the ends. Due to restrained thermal expansion and

restrained thermal bowing, the lower parts of the beam at the ends have very large

stresses and hence initiate beam failure. The larger the axial restraint, the greater the

beam compression force, hence the lower the beam bending failure time. The results

of Riva and Franssen (2008) appear to indicate that beam bending failure can be

dominated by either of the two mechanisms identified by the above two

investigations.

0

100

200

300

400

500

0 40 80 120 160 200

Ben

din

g re

sist

ance

(m

in)

Axial restraint stiffness (kN/mm)

0

50

100

150

200

250

0 0.2 0.4 0.6 0.8 1

Ben

din

g re

sist

ance

(m

in)

× EA/L Axial restraint stiffness

0

50

100

150

200

250

300

0 0.5 1 1.5 2

Ben

din

g re

sist

ance

(m

in)

× EA/L Axial restraint stiffness


41

Clearly, the behaviour of axially and rotationally restrained RC beams is complex

and a thorough investigation is necessary. In addition, the rotational restraint to the

RC beams will not be able to achieve the idealised fully fixed end rotational restraint

condition. Furthermore, if the ends and longitudinal steel bars in RC beams are

adequately anchored to the adjacent structure, the load carrying mechanism of

catenary action may develop at very large deflection after bending failure. So far,

there have been no studies on catenary action behaviour of RC beams in fire. The

limited studies above on restrained RC beams were only concerned with the beam

behaviour at small deflections under the combined effects of moment and

compressive force. In performance-based fire engineering design, large deflections

are permitted, provided that fire integrity is maintained. Experimental and numerical

studies on robustness of RC structures at ambient temperature have revealed the

significant role of catenary action in RC beams in mitigating progressive collapse.

The present study will investigate large deflection behaviour of RC beams in fire and

their potential in developing catenary action.

Therefore, much research is necessary to improve understanding of how different

axial and rotational boundary conditions influence the fire response of RC beams.

This will be the main focus of this research work.

2.5 RC Frames in Fire

Interactions between different structural members of a structure greatly influence the

fire resistance of structural members and the structure. Some previous research

studies have shown that the fire resistance of restrained RC beams can be much

higher than that estimated based on tabular rules. However, this depends on the

capability of the adjacent columns to withstand the additional internal forces exerted

by the connected beams. Moreover, because of interactions between structural

members, support and loading conditions of a member can be altered during fire.

This can lead to different member behaviour from that predicted assuming fixed

support and loading conditions. It is important to understand how the overall

behaviour of concrete frames is affected by structural interactions between different

structural members in fire.


42

Full-scale fire testing was carried out on a seven-story cast in situ concrete building

at Cardington, UK (Bailey, 2002). Figure 2.10 clearly shows that the edge columns

were pushed out by the heated slabs. This needs to be considered by fire protection

designers.

Figure 2.10: Lateral displacement of edge columns due to expansion of heated

slabs (Bailey, 2002)

Guo and Shi (2011) carried out a number of fire tests on single-bay one-story RC

frames. They examined the influences of load level and relative beam to column

stiffness. Figure 2.11 shows the test specimen, loading pattern and reinforcement

details. In all cases, the widths of the beam and column were kept constant and equal

to 100mm, and the beams and columns had the same reinforcement details. For each

test specimen, the beam and two columns were exposed to high temperature from

three sides simultaneously using three independent electrical furnaces. Different

failure mechanisms were observed in frames with different depths of beams and

columns.


43

(a) Reinforcement details and dimensions of test specimens

(b) Test setup

Figure 2.11: Test arrangement by Guo and Shi (2011)

Frames with larger column depths failed by three plastic hinges in the beam. The first

plastic hinge was formed in the mid-span after significant reduction in the bending

moment capacity in that region. Bending moments at both ends of the beam then

increased as the load was redistributed. Two more plastic hinges formed at the ends

of the beam, resulting in failure of the frame. As the column rigidity was reduced,

due to a reduction in column depth, the bending moments at the beam ends were low

Section 1-1 Section 2-2

All dimensions

in mm

200

100

150 to

200


44

and therefore plastic hinges could not form at the beam ends. Instead, the failure

mechanism of the frame was observed to be formation of five plastic hinges, one in

the beam and four in the columns as sketched in Figure 2.12. Plastic hinges were

developed in the beam mid-span and at the bottom of the columns first. This was the

followed by formation of plastic hinges near mid-height of the columns.

Figure 2.12: Position of plastic hinges of a frame specimen by Guo and Shi

(2011)

Lue (2007) numerically studied evolution of internal forces in a single-bay “rugby

post” RC frame in fire. Figure 2.13 shows the structural details. The effects of beam

section types (rectangular-section, T-section and one-way slab section), beam span

and column fire exposure type were assessed. Two fire scenarios were considered for

the columns: (i) fire exposure on only the inner side and (ii) fire exposure on three

sides. Their results show that the heated beam caused the bending moment and shear

force in the columns to increase substantially. Due to thermal elongation and end

rotations of the beam, the bending moment and shear force in the lower columns

increased up to approximately seven and four times the initial values at ambient

temperature, respectively. Neglecting the detrimental effects of the beam thermal

expansion in the design of RC frames may lead to unsafe estimate of fire resistance

of structures.

Plastic hinge


45

Figure 2.13: Dimensions and reinforcement details of simulation structure of

Lue (2007)

Raouffard and Nishiyama (2015) conducted two one-third scale fire tests on two-

storey one-span moment resisting RC frames to study their fire behaviour and

resistance. One of the frames was heated until collapse, while the other was heated

for 60 minutes only to investigate post-fire behaviour of the frame. Figure 2.14

shows the details of the test frames. The test results of the frame heated to collapse

show appearance of flexural cracks in the mid-span and end sections. Many shear

cracks occurred in the lower columns, which was believed to be due to shear forces

caused by thermal expansion of the connected beam. The adopted seismic

reinforcement in the beam-column joints effectively contributed to prevent shear

failure in the joint and heated lower columns. At later stages of the fire test, a plastic

hinge formed in the mid-span of the heated beam. Hogging moments were noticed to

be small at the beam ends. Excessive deflections of the beam caused the beam to go

into catenary before failure of any member of the frame which was prevented from

happening by terminating the fire test to ensure safety of the test apparatuses.

400 mm

400 mm

8ϕ20

Section a-a

Beam width=350 mm

Beam depth=500 mm

3ϕ20

2ϕ20

Section b-b

2ϕ20

3ϕ20

Section c-c 6000 mm

DL=36 kN/m

LL=12 kN/m

3200 mm

1600 mm

b-b c-c

a-a

1000 kN ≈ 7 floors 1000 kN ≈ 7 floors


46

Figure 2.14: Test setup (Raouffard and Nishiyama, 2015)

The above studies on RC frames in fire confirm importance of considering the

interactions between different structural members in a fire. However, there is a lack

of detailed quantitative understanding and a lack of simplified method to predict such

structural interactions. In particular, understanding on catenary action, developed at

very large beam deflections, has not been systematically examined. These will form

parts of the investigations in this research project.

2.6 Concrete Spalling

Fire-induced spalling involves the breaking off of layers (pieces) of concrete from

the surface of a structural element at elevated temperatures. Spalling has detrimental

consequences through (i) reducing the cross-sectional area of the structural element,

(ii) reducing the concrete covering to the steel reinforcement and (iii) increasing heat

transfer to the inner parts of the concrete cross-section. Therefore, spalling may

result in substantial reduction in the structural element load bearing capacity to a

level where the element is no longer able to withstand the load in fire.


47

Spalling is broadly classified into three different types: explosive spalling, aggregate

spalling and corner spalling (Bailey, 2002). Explosive spalling occurs violently in the

early stages of fire due to pore water pressure and differential thermal stresses

induced within the concrete (Ali et al., 2010). The main factors influencing the

susceptibility to explosive spalling are moisture content, rate of heating, permeability

(or porosity of concrete) and mechanical stress levels (Bailey, 2002). During

exposure to fire, free and combined water in concrete close to the heated surface start

to evaporate. While some of the water vapour migrates, most of it is contained in the

gel pores, causing pore pressure to build-up. If this pore pressure exceeds the tensile

strength of the concrete, explosive spalling happens by taking chunks of concrete

from the structural element. Higher moisture content correspondingly gives rise to

increased pore pressure in the concrete, thus increasing the potential to explosive

spalling (Kodur and Dwaikat, 2012). Differential rates of thermal expansion within a

concrete cross-section are also thought to increase the risk of explosive spalling.

Furthermore, steep thermal gradient induces compressive stresses close to the heated

surface due to restrained thermal expansion and tensile stresses in the interior regions

where thermal expansion is lower (Khoury, 2000). The tensile (pore pressure)-

compressive (restrained thermal expansion) multiaxial stress state further reduces the

strength of concrete. However, pore pressure is believed to play a more significant

role in the occurrence of explosive spalling (Kodur and Dwaikat, 2012).

Aggregate spalling occurs due to aggregate failure near the fire exposed surface in

the early stages of fire. The damage of this type of spalling to a concrete member is

cosmetic and does not adversely influence its structural performance. Corner spalling

occurs in the late and often decay stages of fire, and is characterised by large pieces

of concrete at the corners of structural members falling off due to tensile cracks

forming at edges and corners. This type of spalling has minor effects on the

resistance of a concrete structure since the onset of this spalling is late, in which the

concrete and steel reinforcement have already been weakened substantially and the

fire temperature in the decay phase is lower (Bailey, 2002).

Explosive spalling is generally considered to be the most serious in terms of posing a

potential threat to structural integrity and instability. Therefore, the majority of

research and design codes are concerned with explosive spalling. Because of high

density and low permeability of high strength concrete that prevent the water vapour


48

from escaping during heating and thus increasing the likelihood of developing high

pore pressures within the concrete, high strength concrete has a high susceptibility to

explosive spalling compared to normal strength concrete (Dwaikat and Kodur,

2009b, Kodur and Dwaikat, 2012). According to EN 1992-1-2 (CEN, 2004),

explosive spalling is less likely to occur in traditional normal strength concrete when

moisture content of concrete is less than 3% by weight.

2.7 Catenary Action

Although there is little research on catenary action in RC beams under fire exposure,

there has been much research on catenary action of RC beams at ambient

temperature, as an alternative load carrying mechanism to reduce the risk of

progressive collapse of RC frames (Kim and Yu, 2012, Orton et al., 2009, Lew et al.,

2014, Yu and Tan, 2013, Yu and Tan, 2014). The results of these studies can be used

to provide an insight on its applicability under fire situation.

At ambient temperature, a typical scenario independent test of robustness of

structures is column removal. Such a situation is illustrated in Figure 2.15. After

removal of the centre column, a double-span beam is created as the two beams

framing into the removed column become a single structural member. As the beam

deflection increases, compressive arch action develops initially followed by catenary

action. Figure 2.15(c) plots the idealised load-displacement and axial force-

displacement curves of the beam.

Compressive arch action can only develop at small deflections. Therefore, due to

large span/depth ratio of the double span beam, the development of compressive arch

action is limited. On the other hand, under catenary action, the load that can be

resisted by the beam increases at increasing beam deflection. Since it is the

reinforcement that contributes to tension under catenary action, the failure mode of

an RC beam in catenary action is rupture of the steel longitudinal reinforcement.


49

(a) Schematic of compressive arch action (Qian et al., 2015)

(b) Schematic of tensile catenary action (Qian et al., 2015)

(c) Applied vertical load-displacement and axial force-displacement

Figure 2.15: Behaviour of beam-column sub-assemblage under column removal

scenario

Ax

ial

forc

e

Tension

Compression

Displacement

Compressive arch action Catenary action

Appli

ed v

erti

cal

load

Displacement

Compressive arch action Catenary action


50

Regan (1975) performed catenary tests at Imperial College, London, on precast floor

strips. Due to poor continuity, inadequate anchorage and concentrated rotations, full

development of catenary action was not observed in some specimens. As a

conclusion, Regan stated that “the successful development of catenary action

requires that the members in question possess not only tensile strength but also

ductility, which is largely determined by detailing of the longitudinal reinforcement.”

Choi and Kim (2011) tested four scaled down two-span RC beam-column sub-

assemblages to investigate structural robustness when the middle column was

removed. The specific purpose of that experimental work was to examine the

influence of reinforcement details and concrete strength on catenary action

activation. The test results show that a seismically designed specimen based on ACI

2005 with adequate concrete compressive strength (concrete compressive cylinder

strength=30 MPa) was able to develop effective catenary action through the top bars

even after the rupture of the bottom bars. However, for specimens designed for non-

seismic loading, because of inadequate reinforcement detailing in terms of shear

reinforcement and anchoring of longitudinal bars by standard hooks, they failed

before catenary action was mobilised. Their premature failure was mainly due to

pulling out of longitudinal bars and concrete crushing at the exterior beam-column

joints. Joint failure also occurred before developing catenary action in the specimen

constructed with low concrete compressive strength of 17 MPa even with seismic

reinforcement detailing.

Stinger and Orton (2013) carried out experiments on three one-quarter scaled, two-

bay and two-storey RC frames with removed centre column at the lower storey to

improve and expand understanding of alternate load path mechanisms for

progressive collapse resistance. The first frame had no continuous reinforcement,

representing old buildings designed in accordance with ACI 318-71, while the

second one was designed to satisfy ACI-318-08 continuity requirements. A frame

was constructed with partial-height infill wall to investigate whether it could make a

contribution to collapse resistance with the same reinforcing detail as in the first

frame. Figure 2.16 shows dimensions and detailing of the frames. As regards the first

frame, because the beams experienced moment direction reversal at sections over the

removed column, discontinuity of steel bars provided little resistance to bending

moments at these sections. Nevertheless, compressive arch action significantly


51

enhanced its flexural capacity by providing 40% of the ultimate flexural capacity of

the frame with continuous reinforcement. Another important finding was that no

noticeable increase from catenary action was achieved by the frame with continuous

bars over that without continuous bars, as demonstrated in Figure 2.17. This was due

to ability of the transverse reinforcement to transfer tension force from the positive to

negative longitudinal bars in the beam. Partial-height infill walls did not provide

apparent enhancement in progressive collapse resistance.

Figure 2.16: Elevation view and reinforcing details of the three RC frames

(Stinger and Orton, 2013)

Figure 2.17: Applied load versus displacement of the three test specimens by

Stinger and Orton (2013)


52

Yu and Tan (2013) tested six one-half scaled RC beam-column sub-assemblages

with different reinforcement and span-to-depth ratios to study alternate load path

mechanisms for progressive collapse resistance. Each sub-assemblage was comprised

of a column stub at ends, two single-bay beams and one middle joint tested quasi-

statically. Figure 2.18 shows the test arrangement. The results show that catenary

action resistance, which was reached at deflections more than twice the beam depth,

was 28-128% higher than the compressive arch action resistance. Figure 2.19

illustrates failure mode of one of the specimens (specimen S5-1.24/1.24/23) at

catenary action capacity. Based on their parametric study results, a significant

increase in structural resistance of the sub-assemblage from catenary action can be

attained with a larger span-to-depth ratio and a higher flexural reinforcement,

especially the top reinforcement. Catenary action of a specimen with small span-to-

depth ratio had almost no effect on structural resistance because the specimen

experienced severe damage under shear failure.

Figure 2.18: Test arrangement and test specimen (Yu and Tan, 2013)


53

Figure 2.19: Crack patterns and failure modes of specimen S5-1.24/1.24/23 from

Yu and Tan (2013) tests

Yu and Tan (2014) further tested four one-half scaled RC frames to investigate cost-

effective techniques for enhancing resistance of structures located in non-seismic

regions against progressive collapse due to loss of a column by accident. The frames

were designed with special detailing aimed at improving rotation capacity of the

beam-column connections. Figure 2.20 shows the test configuration. A monotonic

displacement-controlled pushdown load was applied at the middle of the two-span

beam over the removed column.

Figure 2.20: Test arrangement (Yu and Tan, 2014)


54

One frame was designed with conventional detailing according to ACI 318-05

(Figure 2.21(a)). However, the frame was unable to recover after reaching

compressive arch action resistance due to limited rotational ductility of the

connections that caused successive fracture of the steel reinforcing bars. Three

innovative detailing techniques were used for the other three frames, as shown in

Figure 2.21. In the first technique (Figure 2.21(b)), a layer of reinforcement was

added at the middle-height of the beam section throughout its length. This layer of

reinforcement contributed to enhancing flexural rotation at critical sections and

functioned with the top reinforcement as ties to increase catenary action resistance.

Debonding the bottom steel bars (Figure 2.21(c)) was employed in the second

technique to prevent premature rupture of the bars due to strain concentration in the

joint areas. This was achieved by using plastic sleeves to spread out strain

concentration. In the third technique (Figure 2.21(d), partial hinges in the beam ends

were designed. The location of the partial hinges was determined so that they should

not influence flexural capacity of the beam under conventional loading and column

removal scenarios. This technique resulted in large plastic rotations of the two-span

beam and shifted the crushing and cracking of concrete and rupture of reinforcing

bars form the beam-joint interfaces to the partial hinge region. This had the effect of

reducing the length of beams which rotates as rigid blocks, thus increasing the beam

rotation without increasing reinforcement strain, thereby allowing large beam

deflections and high catenary action resistance. All the three techniques displayed

well-developed catenary action and considerably improved progressive collapse

resistance. However, the authors did not recommend using the debonding technique

for progressive collapse resistance because it may affect the arch action resistance.

The detailing technique with partial hinges was found to provide the most cost-

effective design.


55

(a) The frame designed with conventional detailing

(b) First detailing technique (c) Second detailing technique

(d) Third detailing technique

Figure 2.21: Detailing techniques of test specimens (Yu and Tan, 2014)

The tests from the above two experimental studies by Yu and Tan (2013) and Yu and

Tan (2014) will be used for validation of the numerical models to be developed in the

current research in Chapter 3.


56

In summary, catenary action in RC beams can provide substantial extra load-carrying

capacity over flexural bending resistance. However, catenary action can only develop

at large beam deflections. The applied load on the beam is resisted by the vertical

components of the axial tensile catenary force sustained by the beam longitudinal

tensile reinforcement. Hence, whether catenary action will be completely developed

to resist the applied load depends on: (i) the amount of axial catenary force, (ii) the

beam deflection limit and (iii) the presence of a line of tension for the catenary force

to act throughout the beam. This study will further examine development of catenary

action in RC beams exposed to fire.

2.8 Simplified Calculation Methods of RC Beams in Fire

As an alternative to experimental tests and numerical simulations, simplified

approaches are essential and more practical for fire protection/structural engineers.

Recently, simplified methods have been developed to assess responses of RC beams

exposed to fire.

Kodur and Dwaikat (2011) derived simple analytical expressions for evaluating the

fire resistance of axially restrained RC beams. However, the approach is only aimed

at beams in flexural bending action with fully fixed rotational restraint at ends.

Kang and Hong (2004) developed an analytical model to trace the behaviour of RC

beams in fire. However, the model is limited for simply supported beams without end

axial or rotational restraint.

Kodur and Dwaikat (2008), Dwaikat and Kodur (2008) and (Kodur et al., 2009)

established a numerical computer model to simulate the response of RC beams at

elevated temperatures. In the structural analysis, sectional analysis is performed by

adopting a multi iteration approach and deriving moment-curvature relationships.

The total strain distribution of both concrete and reinforcing bars is expressed by a

reference strain and curvature. The model by Dwaikat and Kodur (2008) accounts for

the axial compressive force developed due to restrained thermal expansion. The

compressive force is predicated at each time step based on iterations through

satisfying the requirements of compatibility and equilibrium along the beam length.

The curvature at each time step is taken to be the same as in the preceding time step

for calculating the compressive force. Beam deflection is predicted through stiffness


57

approach. The validated model was used to undertake extensive parametric studies to

quantify the effect of various parameters on the behaviour of simply supported,

axially restrained and rotationally restrained RC beams under fire. The model does

not account for different rotational restraint levels and large deflection behaviour in

catenary action.

El-Fitiany and Youssef (2014b) proposed a simplified approach to trace the fire

response of continuous RC beams, following the moment-curvature approach.

However, their proposed approach is only suitable for continuous beams with no

axial restraint.

It is clear from the aforementioned studies that there is no simplified method

available in the literature to predict behaviour of RC beams at elevated temperatures

with different degrees of axial and rotational restraints and for the entire history of

beam behaviour. Therefore, one of the main goals of this research is to propose such

a simplified method.

2.9 Numerical Finite Element Models

Ellingwood and Lin (1991), Zhaohui and Andrew (1997), Zha (2003), Bratina et al.

(2003), Capua and Mari (2007), Huang et al. (2009), Huang (2010), Wu and Lu

(2009) and Ožbolt et al. (2014) proposed numerical models to trace the behaviour of

RC structural members under fire. However, the performance of RC beams at very

large deflection in catenary action after bending failure is not reported in these finite

element models.

Riva and Franssen (2008) simulated RC beams in fire using a fibre beam element

model in the code ABAQUS. The model was proposed to study the fire performance

of restrained RC beams in flexural bending only. It was capable of capturing detailed

structural behaviour of restrained RC beams in fire associated with different failure

modes of the beam under the combined action of bending moment and axial

compressive force due to restrained thermal expansion.

Ellobody and Bailey (2008) and Ellobody and Bailey (2009) developed a robust 3D

numerical model in ABAQUS to analyse bonded and unbonded post-tensioned slabs

exposed to fire. An uncoupled thermal–mechanical analysis was performed in the


58

model to obtain transient nodal temperatures for the structural analysis from a heat

transfer analysis conducted first. The numerical model was verified by comparing the

results against data of fire tests conducted by the authors and the numerical

predictions agreed well with the test results. The model efficiently predicts time-

temperature relationships in the slab, displacements, complex deformed shapes at

failure and fire resistance times of both unrestrained and restrained slabs.

Biondini and Nero (2011) developed a nonlinear finite element model for concrete

structures exposed to fire. The main difference between this proposed model and the

others above is the existence of a link between thermal and structural simulation

processes to allow automatic simulation. However, as with other models, their

analyses were conducted at small deflections when the beams are in flexural bending

action only.

Gao et al. (2013) used finite element program, ABAQUS, to model simply supported

RC beams under fire conditions. Their particular emphasis was effects of

temperature-dependent bond-slip between concrete and steel reinforcement. They

concluded that bond-slip behaviour in fire had a minor influence on the global

response of an RC beam and a perfect bond between the concrete and reinforcement

can be assumed.

In summary, the previous numerical models are only capable of investigating small

deflection behaviour of RC beams in fire up to flexural failure. They are not suitable

for investigating large deflection behaviour of RC beams in fire that is necessary to

understand catenary action following bending failure. At very large beam

deflections, severe cracking and crushing of concrete and fracture of reinforcement

steel occur. Their local material failures cause temporary loss of equilibrium of the

structure and dynamic behaviour. To overcome numerical convergence difficulties

and to enable the numerical model to capture the whole range of large structural

behaviour of RC beams in fire, it is proposed to employ an explicit dynamic solver in

this study.

In fact, some researchers have successfully modelled large deflection response of RC

beams at ambient temperature in catenary action under the column removal scenario

using an explicit dynamic solver (Bao et al., 2014, Hou and Song, 2016, Li et al.,


59

2016, Pham et al., 2017, Sadek et al., 2011). However, the structures in these studies

were loaded quasi-statically under a displacement-controlled pushdown force. This is

not suitable for RC beams under distributed loads. Furthermore, the explicit

simulation approach for elevated temperature modelling of restrained RC beams has

not been utilised to date.

Therefore, an important aim of this research is to develop a robust 3D explicit

dynamic finite element model to be implemented in the computer program ABAQUS

so that it is able to trace the highly nonlinear response of RC structures at ambient

temperature and in fire. Issues and key aspects of the simulation model, including

very small time increment and means to control local instability, will be discussed in

detail in Chapter 3.

2.10 Summary of Literature Review and Research Originality

Based on the findings of literature review, the following main conclusions can be

drawn:

Although many studies have been carried out to investigate the behaviour of

RC beams in fire, very few studies have considered the effects of axial and

rotational restraints. When axial restraint is present, there is a lack of clear

understanding of beam behaviour under different rotational restraints and

there are inconsistencies in the effects of axial restraint on beam’s flexural

bending resistance.

Catenary action in RC beams at very large deflection plays a crucial role in

enhancing robustness of RC structures. However, previous research studies

have only investigated catenary action behaviour in RC structures at ambient

temperature. All the existing studies on axially RC beams in fire are

concerned with beam behaviour at small deflections up to bending failure.

The existing fire resistance design approaches do not consider realistic

boundary conditions and structural interactions.

The existing numerical simulation models for concrete structures in fire are

based on static solvers which cannot capture structural behaviour involving

temporary loss of structural equilibrium caused by local material failure

(crushing and cracking of concrete and fracture of steel reinforcement).


60

Based on the above summary, the main objectives of this research are:

1. To develop and validate a 3D numerical simulation model using the finite

element code ABAQUS/Explicit so that it is suitable for simulating RC

structural members subject to large deformations in fire.

2. To conduct a numerical parametric study on restrained RC beams in fire to

gain a comprehensive understanding of the effects of axial and rotational

restraints on beam behaviour in flexural action and in catenary action.

3. To propose a simplified calculation approach to predict the behaviour of

axially and rotationally restrained RC beams exposed to fire throughout the

whole history of behaviour.

4. To investigate the effects of structural interaction on fire resistance of RC

framed structures using the finite element model developed in (1).

CHAPTER 3 EXPLICIT MODELLING OF LARGE DEFLECTION BEHAVIOUR OF

RESTRAINED REINFORCED CONCRETE BEAMS IN FIRE

61

CHAPTER 3

EXPLICIT MODELLING OF LARGE DEFLECTION

BEHAVIOUR OF RESTRAINED REINFORCED

CONCRETE BEAMS IN FIRE

3.1. Introduction

Faithful numerical simulation of the large deformation structural behaviour of RC

structural members presents serious challenges due to the material failures that can

occur, including concrete cracking, crushing and reinforcement fracture (which may

cause temporary loss of equilibrium of the structure and dynamic behaviour), and

very severe geometrical nonlinearities. Coupled with changing temperatures, it is not

feasible to numerically model the structure using a conventional static analysis.

The shortcoming of the static analysis in modelling structures that suffer temporary

loss of stability at large deformations can be overcome by performing a dynamic

analysis procedure with handling the problem quasi-statically. In the finite element

method, time integration algorithms for dynamic problems can generally be

classified as either implicit or explicit. In the implicit dynamic analysis, the actual

time of a quasi-static problem can be used and large time increments may be taken.

Inertia effects are introduced when loss of stability happens in the quasi-static

response. An iterative process is carried out at the end of each time increment to

satisfy the equilibrium equations. However, for problems with high geometric and

material nonlinearities, the unconditionally stable implicit procedure usually

experiences sever convergence difficulties, resulting in either taking many iterations

or premature numerical failure of the analysis. The convergence difficulties

encountered in the implicit method can be solved using explicit dynamic algorithm.

In the explicit method, central-difference time integration is employed. The solution

at the current time step is performed based on the kinematic state from the previous

time step. Local variables are directly predicted without the need of formation and

inversion of global matrices and no iterations are required.



62

This research explores the explicit modelling approach. Because the explicit time

integration algorithm is continually stable, the time increment has to be very small.

This makes it problematic when applying this modelling approach to fire conditions

because fire exposure has long durations. In addition, several controlling parameters

should be addressed carefully to obtain quasi-static response. Therefore, an important

task in implementing the explicit simulation approach is to resolve this challenge.

Two techniques may be considered, either separately or together, to achieve a

computationally economical explicit solution without compromising accuracy of the

simulation. These are (a) artificially increasing the loading speed (load factoring) and

(b) artificially increasing the mass of the structure to increase the stable time step

(mass scaling). This chapter will present details of the above mentioned two

approaches and provide guidance on their implementations for 3D restrained RC

beams at ambient and elevated temperatures using the ABAQUS/Explicit solver. To

validate the developed explicit modelling approach, the simulation results will be

compared against relevant test results, including the ambient temperature tests on

axially restrained RC beams by Yu and Tan (2013) and Yu and Tan (2014) and the

fire tests on axially restrained RC beams by Dwaikat and Kodur (2009a).

3.2. Development of the Explicit Modelling Methodology

The ambient temperature tests of Yu and Tan (2013) and Yu and Tan (2014) on

axially restrained RC beams will be used to explain the development of the explicit

modelling approach. These tests were selected owing to their comprehensive

reporting of the test arrangement and results.

3.2.1 Brief Introduction to the Tests by Yu and Tan (2013) and Yu And Tan

(2014)

Figures 2.18 and 2.20 show the test arrangements and Figure 3.1 shows the sub-

assemblage and the frame details. The three sub-assemblage test specimens, denoted

as S4, S5 and S7, comprised of two enlarged column stubs at the ends, two single-

bay beams and one middle joint. The experiments examined the effects of varying

reinforcement ratio and beam span-to-depth ratio. In the two frame specimens,

detonated as F2 and F4, the column stubs were replaced by side columns and beam



63

extensions. Each specimen was designed with a special technique of reinforcement

detailing, aimed at enhancing the ultimate load carrying capacity in catenary action.

All specimens were loaded by applying a displacement-controlled pushdown force at

a rate of 0.1mm/s at the top of the unsupported middle joint until failure. In the frame

tests, one of the side columns was subjected to an axial stress of 0.6 fcm and the other

to 0.4 fcm prior to load application on the middle joint and these column loads were

kept constant during the test. Table 3.1 lists the main specimen details and Table 3.2

gives the mechanical properties of the steel reinforcing bars. The compressive

cylinder strength of concrete (fcm) for specimens S4, S5 and S7 was 38.2 MPa, and

for specimens F2 and F4 was 29.69 MPa.

Ln =2750 for S4 and S5

Ln =2150 for S7

15

0 L1

ϕ 6 @ 100

250

L1=1000 for S4 and S5

L1=780 for S7

15

0

32

5

32

5

450

P

C.L.

B

B

A

A

A

A

(a) Specimens S4, S5 and S7

P

C.L.

15

0

11

75

1000 500

ϕ 6 @ 100

Ln=2750 250

1000

92

5

15

0

𝜙 6

@ 2

00

𝜙 6

@ 2

00

A

A

B

B

A

A

(b) Specimen F2

150

250



64

Figure 3.1: Geometrical details of RC beam-column sub-assemblages and

frames (Yu and Tan, 2014, Yu and Tan, 2013, Yu, 2012)

Table 3.1: Reinforcement detailing

Specimen A-A section B-B section

Top Bottom Middle Top Bottom Middle

S4 3ϕ13 2ϕ13 --- 2ϕ13 2ϕ13 ---

S5 3ϕ13 3ϕ13 --- 2ϕ13 3ϕ13 ---

S7 3ϕ13 2ϕ13 --- 2ϕ13 2ϕ13 ---

F2 3ϕ13 2ϕ13 2ϕ10 2ϕ13 2ϕ13 2ϕ10

F4 3ϕ13 2ϕ10 +

1ϕ13 --- 2ϕ13

2ϕ10 +

1ϕ13 ---

Table 3.2: Mechanical properties of steel reinforcement

Bar Type

Yield

strength,

fy (MPa)

Elastic

modulus,

Es (MPa)

Hardening

strain

𝜀𝑠ℎ(%)

Tensile

strength,

fu (MPa)

Ultimate

strain,

𝜀𝑢 (%)

ϕ 6 * 349 199177 ---- 459 ----

ϕ 6 **

442 209397 ---- 513 ----

ϕ 10 520 187090 4.12 595 13.7

ϕ 13 * 494 185873 2.66 593 10.92

ϕ 13 **

488 170125 2.86 586 11.00

* Specimens S4, S5 and S7

** Specimens F2 and F4


Ln=2750

(500)

ϕ 6 @ 50

500 125

(500)

ϕ 6 @ 50

(1750)

ϕ 6 @ 100

250

250

A

A

B

B

A

A

P

C.L.

C

C

A

A

(c) Specimen F4

150

250

Section C-C



65

3.2.2 Element Type, Boundary Conditions and Load Application

In this study, three-dimensional 8-node linear reduced-integration brick elements

(C3D8R) and two-node linear three-dimensional truss elements (T3D2) in ABAQUS

are used for modelling concrete and steel reinforcement, respectively. The optimal

mesh size will be identified in the following sections. Much attention was paid to the

simulation of the boundary conditions of the beam-column sub-assemblages and

frames. This is because there were several interactions at the supports, as illustrated

in Figure 3.2, and the accuracy of FE simulation results critically depends on

accurate specification of these interactions. In the laboratory tests, one end of the

specimen was restrained by an A-Frame and the other end by a reaction wall through

two horizontal pin-pin connections. In the simulation model, in order to avoid any

local stress concentration, the same assembly of steel plates and steel rods as in the

tests was created in the simulation model to anchor the beams to the connections by

using the ABAQUS “Tie constraint”. “Tie restraint” was also used to connect the top

and bottom column end plates to the concrete columns. In order to simulate the pin

boundary condition as in the actual tests and to make sure that each plate rotated

around the pin during the loading process, all the plates were modelled as rigid

bodies using “Rigid body constraint” in ABAQUS.

(a) Specimens S4, S5 and S7 (Yu and Tan, 2013)

ux=uy=uz=0

Pin

Connect

or

Pin

uy=uz=0

Rigid plate

Beam longitudinal

bar

Beam transverse

bar

Column transverse bar

Column longitudinal

bar

4 rods to avoid stress

concentration

Steel

roller



66

(b) Specimens F2 and F4 (Yu and Tan, 2014)

Figure 3.2: Boundary conditions applied in the FE models

Yu (2012) provided the linear elastic stiffness of the horizontal restraints, shown in

Figure 3.2, and the gaps between the restraints and the specimens. These stiffness

values and the gaps are presented in Appendix A3 and were used in the authors’

ABAQUS model when using the “Axial connector elements”. The axial loads in the

side columns of the frame models were firstly applied in one step up to the same

level as in the tests before application of the monotonically increasing load at the top

of the middle joint until failure. To save computation time, only half of the sub-

assemblage test specimen was modelled based on geometrical and loading symmetry.

For the frame tests, the whole frame specimen was modelled because the two side

columns had different applied loads.

3.2.3 Material Constitutive Models

3.2.3.1. Concrete

The concrete damaged plasticity CDP model in ABAQUS was used to define the

inelastic behaviour of concrete. Damage of concrete is associated with the two main

failure mechanisms, namely tensile cracking and compressive crushing, and

evaluation of the yield surface is controlled by the equivalent plastic strains in

tension and compression, respectively (ABAQUS, 2013). There are five parameters

which need to define the damaged plasticity model and they are: (1) 𝜎𝑏𝑜 𝜎𝑐𝑜⁄ : the

Load pin

ux=uy=uz=0

Pin

Connector

Pin

ux=uy=uz=0

Column axial load



67

ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive

yield stress, (2) Kc: the ratio of the second stress invariant on the tensile meridian to

that on the compressive meridian for the yield function, which defines the shape of

the yield surface in the deviatoric plane, (3) 𝜖: the eccentricity, which defines the rate

at which the flow potential function approaches its asymptote in the meridian plane,

(4) 𝜓: the dilatation angle at high confining pressure, measured in the meridian plane

and (5) 𝜇: the viscosity parameter that represents the relaxation time of the visco-

plastic system. In this mode, the default ABAQUS values of the above five

parameters were used as listed in Table 3.3. Figure 3.3 shows the uniaxial

compressive stress-strain relationship according to CEB-FIP mode code (CEB, 1991)

up to the peak compressive stress. The softening branch was approximately modelled

by a straight line to a stress of 0.2 fcm and the rate of strength decline was controlled

by varying the limit of maximum concrete strain 𝑛1𝜀𝑐1 corresponding to 0.2 fcm.

Values of n1 between 3 and 4 were chosen in this study because they gave consistent

numerical results.

Table 3.3: Parameters for definition of the concrete damaged plasticity model

(ABAQUS, 2013)

Parameter name Value

Dilatation angle, 𝜓 36

Eccentricity, 𝜖 0.1

Ratio of initial equibiaxial compressive yield

stress to initial uniaxial compressive yield stress

𝜎𝑏𝑜 𝜎𝑐𝑜⁄

1.16

Yield surface shape factor Kc 0.667

Viscosity parameter 𝜇 0



68

Figure 3.3: Concrete compressive stress-strain relationship

Many methods may be used to model cracked concrete. When the RC beam

deflections are small, satisfactory modelling results can be obtained by using any

shape as long as the area underneath the tension softening curve is kept constant

based on the tensile fracture energy Gf value and the crack band width. Furthermore,

the influences associated with slippage and bond between the reinforcement and the

concrete can be macroscopically represented by introducing tension stiffening effects

in the cracked concrete model. However, for problems involving large deflections

and severe tension cracks as in the current study, careful consideration should be

made when modelling tension softening behaviour in order to minimise numerical

simulation problems associated with strain localisation, stability loss and spurious

sensitivity of modelling results to mesh size. In addition, to allow the ABAQUS

option of embedding reinforcing bars in concrete, a more realistic tension stiffening

curve is necessary. This curve should implicitly consider the interactions between

reinforcement and concrete because explicit modelling of the reinforcement-concrete

bond is time consuming. In general, according to Maekawa et al. (2003), a power

form of tensile stress-strain curve, as shown in Figure 3.4, can be used. In Figure 3.4,

𝜀𝑐𝑟 is the cracking strain; c1 is a coefficient that controls the rate at which the tension

stress 𝜎𝑡 decreases with increasing strain 𝜀𝑡 after cracking. In general, the main

factors that influence the magnitude of coefficient c1 are tensile concrete fracture

energy, element mesh size and reinforcement ratio (Stevens et al., 1991, Salem and

Maekawa, 2004). The influence of fracture energy, which depends on element mesh

size, is important when dealing with propagation of cracks in plain concrete or

concrete with very little reinforcement, in which the tensile strength of concrete

𝐸𝑐𝑚 = Initial modulus of elasticity S

tres

s 𝝈𝒄

Strain 𝜺𝒄 𝜀𝑐1 𝜀𝑐𝑢 = 𝑛1𝜀𝑐1

0.2𝑓𝑐𝑚

𝑓𝑐𝑚

𝜎𝑐 =

𝐸𝑐𝑚𝐸𝑐1

𝜀𝑐𝜀𝑐1

− 𝜀𝑐𝜀𝑐1

2

1 + 𝐸𝑐𝑚𝐸𝑐1

− 2 𝜀𝑐𝜀𝑐1

𝑓𝑐𝑚

𝐸𝑐1 = 𝑓𝑐𝑚 0.0022⁄



69

quickly drops to zero. However, the influences of element size become small and are

usually ignored in reinforced concrete members because the concrete between cracks

can still carry tension stresses. Therefore, in this research, the mesh size effect was

not considered and a value of c1=0.4 was adopted as recommended for deformed

bars (Maekawa et al., 2003).

Figure 3.4: Stress-strain relationship of concrete in tension

3.2.3.2 Steel Reinforcement

The uniaxial stress-strain curve of the longitudinal steel reinforcement bars (ϕ10 and

ϕ13) is shown in Figure 3.5 according to the experimental data in Table 3.2. An

elastic perfectly plastic model was assumed for the transverse steel bars (ϕ6). The

classical metal plasticity model available in ABAQUS was used to model steel

materials.

Figure 3.5: Stress-strain relationship of reinforcing bars

𝜎𝑡 = 𝑓𝑐𝑡𝑚 𝜀𝑐𝑟𝜀𝑐𝑡

𝑐1

εcr Strain 𝜀𝑐𝑡

Str

ess 𝜎𝑡 𝑓𝑐𝑡𝑚 = 0.33 𝑓𝑐𝑚

εcr =𝑓𝑐𝑡𝑚𝐸𝑐𝑚

Str

ess

(MP

a)

Strain (mm/mm) 𝜀𝑢 𝑓𝑦 𝐸𝑠⁄

fy

fu

𝜀𝑠ℎ∗

* 𝜀𝑠ℎ: strain at the start of hardeining



70

3.2.4 Mesh Sensitivity

Before studying parameters in the explicit simulation approach, a mesh sensitivity

study was conducted to determine the type and size of finite elements that can be

used to achieve converged solutions. In this study, three-dimensional 8-node linear

reduced-integration brick elements (C3D8R) in ABAQUS were used for modelling

concrete and two-node linear three-dimensional truss elements (T3D2), as embedded

regions in the host concrete elements, were used to model steel reinforcement.

Perfect bond between steel and concrete was assumed. Displacement-controlled

(DC) method was used. The damping ratio (𝜉), loading duration (LD) and mass

scaling factor (f) were 0%, 5s and 1.0, respectively. Determination of 𝜉, LD and f

values will be described in the following sections.

Figure 3.6 compares the simulation results using different mesh sizes against the test

results of the test structure S4. It can be found that mesh sizes between 25 to 35 mm

give results in good agreement with the test results. In the following simulations, the

concrete mesh size is 30mm. For reinforcement, an element size for 50 to 60 mm

achieved accurate results.

(a) Load-middle joint displacement

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Load

(kN

)

Middle-joint displacement (mm)

25mm 30mm 35mm

40mm 50mm Test



71

(b) Beam axial force-middle joint displacement

Figure 3.6: Sensitivity of FE simulation results to mesh sizes of concrete, for test

S4

3.2.5 Introduction to Dynamic Explicit Modelling

The explicit solution procedure is simple to implement because no global tangent

stiffness and mass matrices need to be assembled and inverted and the internal forces

are determined on the element level. However, the explicit solver is only

conditionally stable and the time increment has to be very small so that the

acceleration throughout an increment can be assumed to be constant.

The maximum time increment that may be used is denoted as the stability limit. It is

initially defined as the time required by a dilatational wave across the smallest

elements in the mesh, and is estimated (ABAQUS, 2013) as:

∆𝑡≤ 𝑚𝑖𝑛 𝐿𝑒√𝜌

�̂�+2�̂� 3-1

where 𝐿𝑒 is the element characteristic length, 𝜌 is the mass density of the material.

�̂� and �̂� are Lame’s constants defined in terms of the material modulus of elasticity E

and Poisson’s ratio 𝜐 as:

�̂� = 𝐸𝜐

(1+𝜈)(1−2𝜈) 3-2

�̂� =𝐸

2(1+𝜈) 3-3

-300

-200

-100

0

100

200

300

0 100 200 300 400 500 600Axi

al f

orc

e (k

N

Middle-joint displacement (mm)

25mm 30mm 35mm 40mm 50mm Test



72

As the analysis proceeds, the stable time increment may be defined in terms of the

highest frequency of the entire model 𝜔𝑚𝑎𝑥, satisfying the following condition:

∆𝑡≤2

𝜔𝑚𝑎𝑥(√1 + 𝜉𝑚𝑎𝑥

2 − 𝜉𝑚𝑎𝑥) 3-4

where 𝜉𝑚𝑎𝑥 is the damping ratio associated with 𝜔𝑚𝑎𝑥.

Two techniques may be used to control the time increment in ABAQUS/Explicit:

fixed time increment and full automatic time increment. In the former technique, a

constant time increment size smaller than the stability limit may be used. In the latter

technique, the integration scheme uses the stable time increment as the time interval

to establish the numerical solution. In this research, the full automatic time increment

strategy is employed because it can efficiently control the solution procedure through

updating the stability limit. This is important for problems that experience very large

deformations and high material nonlinearity that cause continual changes in the

highest system frequency, thereby changing the stability limit.

3.2.6 Reducing Computational Cost

The explicit solver is intended for high speed transient events in which the inertial

effects play a significant role in the solution. For simulating the response of RC

structures under fire exposure, because the fire duration is long, the explicit

simulation becomes computationally very expensive. This is also problematic for

simulating static response of structures under monotonic loading at ambient

temperature until failure occurs and for applying the targeted constant vertical load

before thermal loading starts. Therefore, an efficient simulation strategy is necessary

to drastically reduce the cost of computation but still ensure the solution is quasi-

static. This may be done by either artificially increasing the loading/temperature

increase rate (load factoring technique) or increasing the material density (mass

scaling). In the load factoring approach, loads, boundary conditions and nodal

temperatures imported from a heat transfer analysis can be applied over a shorter

period of time compared to the actual event time. The mass scaling option allows for

the use of the same actual event time through increasing the density of the material

so that the dilatational wave speed within the elements is reduced, leading to

increased stable time step size and so reducing the number of increments to complete

the solution. In both approaches, it is most important to determine how much a



73

simulation can be accelerated before the inertial forces dominate the solution. The

inertia forces should be kept very small to the extent they can be ignored to ensure

that the solution is quasi-static.

To determine the appropriate quantities to be used, the simulation results will be

compared against the test results of Yu and Tan (2013) and Yu and Tan (2014). In

this study, the correlation between loading duration (LD) and the lowest natural

period (Tn) of the finite element model is taken as the basis for estimating the optimal

loading rate. It is assumed that the same ratio of the applied loading time to the

period of the lowest natural mode for the successful simulation of one structure can

be similarly used for other structures. This chapter will establish the minimum time

ratio that can be used. Determining this minimum ratio will also involve minimising

the kinetic energy of the structure.

3.2.6.1 Load Factoring

To illustrate this procedure, the beam-column sub-assemblage test S4 is used. Figure

3.7 compares the applied load-middle joint displacement (MJD) and beam axial

force-MJD relationships between the test data and simulations, and displays the ratio

of the kinetic energy to the internal energy using loading durations (LD) of 2.25, 3.5

and 5s. Displacement controlled loading method was used and the total applied

displacement during the whole LD was 700mm. The selected LD values correspond

to loading rates of 3110, 2000 and 1400 times the test rates respectively as the test

specimen was loaded at a speed of 0.1mm/s. The vertical load and the horizontal

axial force in the simulation results are summation of the reaction forces at supports.

It can be seen that numerical results converge to identical peak loads and are in a

close agreement with the test results. After fracture of the bottom reinforcement bars,

the results from LD = 2.25s exhibited vibration due to dynamic effects before they

disappeared due to increased stiffness. To obtain smooth results, a minimum loading

time of 3.5s is acceptable.



74



Figure 3.7: Comparison between FE simulation and test results for different

simulation loading durations for test S4

The simulations in the above example were displacement controlled as adopted in the

experimental tests. Displacement control is suitable if there is a single point loading.

In most cases, load control based simulation is necessary. In a load-controlled

system, if static equilibrium cannot be sustained, the structure may physically

undergo dynamic behaviour before returning to static stability. Therefore, it is

important that simulation of the subsequent static behaviour is not affected by the

temporary dynamic behaviour. Applying some artificial damping is often adopted to

minimise undesirable dynamic effects.

Figure 3.8 compares the test and load-controlled simulation results and displays the

ratio of the kinetic energy to the internal energy using LD values of 3, 4.5 and 6s for

0

0.01

0.02

0.03

0

25

50

75

100

125

0 100 200 300 400 500 600

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)

Middle joint displacement (mm)

Test LD=2.25s LD=3.5s

LD=5s LD=2.25s (KE/IE) LD=3.5s (KE/IE)

LD=5s (KE/IE)

KE: Kinetic Energy

IE: Internal Energy

-250

-150

-50

50

150

250

0 100 200 300 400 500 600

Axi

al f

orc

e (k

N)


Test

LD=2.25s

L=3.5s

LD=5s



75

the same sub-assemblage test (S4). The total linear applied load was 110kN and the

damping ratio was 35%. Although the same maximum load carrying capacity at the

same MJD was obtained as for longer loading durations when the loading duration

was 3s, numerical oscillation during the catenary action stage continued. Therefore,

a minimum loading duration of 4.5s would be more appropriate for load-controlled

simulation.



Figure 3.8: Comparison between test and load-controlled simulation results for

different simulation loading durations (test specimen S4)

It is assumed that successful explicit simulations of different structures have similar

minimum LD/Tn ratios, i.e.:

𝐿𝐷1

𝑇𝑛,1=

𝐿𝐷2

𝑇𝑛,2 3-5

-300

-200

-100

0

100

200

300

0 100 200 300 400 500 600 700

Axi

al f

orc

e (k

N)


Test LD=3s

LD=4.5s LD=6s

0

0.02

0.04

0.06

0.08

0.1

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)


Test LD=3s (Load) LD=4.5s (Load)

Ld=6s (Load) LD=3s (KE/IE) LD=4.5s (KE/IE)

LD=6s (KE/IE)

KE: Kinetic Energy IE: Internal Energy



76

The subscripts 1 and 2 denote two different structures. In the above examples, the

lowest natural frequency 𝜔𝑚𝑖𝑛1 of the model was 89.5 rad/s, giving the

corresponding 𝑇𝑛,1 of 0.070s. The value of 𝜔𝑚𝑖𝑛 can be determined from a frequency

analysis via “Static, linear perturbation” procedure in ABAQUS. Based on this

example, because the minimum LD for the displacement controlled simulation was

3.5s and that for the load controlled simulation was 4.5s, they give minimum LD/𝑇𝑛

of 50 for displacement controlled simulation and 64 for load-controlled simulation.

To demonstrate general applicability, these values will be used when comparing the

simulation results with the test results for the other structures tested by Yu and Tan

(2013) and Yu and Tan (2014) in the validation section of this chapter.

3.2.6.2 Material Damping

The temporary instability accompanying load-controlled simulation may lead to

significant increase in kinetic energy of the system. ABAQUS/Explicit introduces a

small amount of damping in the form of bulk viscosity. This damping helps to avoid

numerical issues such as element collapse in simulating extremely high-speed

dynamic problems (ABAQUS, 2013). Generally, predicting the exact value of

structural damping ratio is difficult. In the present research, Rayleigh damping in

ABAQUS/Explicit is used. It is described by a damping matrix in the following basic

form:

𝐶 = 𝜇𝑀 + 𝛽𝐾 3-6

where

C, M and K are the viscous damping, inertia mass and stiffness matrices of the

structure, respectively. 𝜇 is the mass proportional damping factor and 𝛽 is the

stiffness proportional damping factor.

For given values of 𝜇 and 𝛽, the damping ratio 𝜉𝑖 in a mode of vibration i can be

expressed as:

𝜉𝑖 =𝜇

2 𝜔𝑖+

𝛽𝜔𝑖

2 3-7

where 𝜔𝑖 is the natural frequency of mode i. The stiffness proportional damping

factor 𝛽 dramatically reduces the stable time increment and this would influence the

computational time (ABAQUS, 2013). Thus, it is more preferable to use 𝜇 to damp



77

out undesirable modes of the structures. Therefore, in this study, it is assumed that

𝛽=0. Equation 3-7 can now be written as:

𝜇 = 2 𝜔𝑖𝜉𝑖 3-8

To select an appropriate range of damping ratio, a series of models were run with

different values of 𝜉, introduced in the material model of both concrete and steel

reinforcement. Figure 3.9 compares the vertical reaction force versus MJD of sub-

assemblage test S4 for 𝜇 values of 0, 27, 45, 63 and 98, which introduce 0, 15%,

25%, 35% and 55% of the structural damping ratio 𝜉 in the lowest mode based on

𝜔𝑚𝑖𝑛 =89.5 rad/s, respectively. It can be seen that the simulation model with no

damping was not able to limit numerical oscillations before final failure of the sub-

assemblage is reached. A model with low damping ratio (𝜉=15%) regained stability

in catenary action at larger MJD. For models with high damping ratios (𝜉 =55%),

although the simulation models were stable, the high artificial damping dominated

the structural response and prevented the structure from further deformation (Figure

3.9(c)). This result is spurious and indicates that such high level of artificial damping

is not desirable.


(b) Kinetic energy-step time

-20

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Load

(kN

)

Middle joint displacement

Test 𝜉=0% 𝜉=15%

𝜉=25% 𝜉=35% 𝜉=55%

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6

Kin

etic

en

ergy

(J)

Step time (s)

𝜉=0%

𝜉=15%

𝜉=25%

𝜉=35%

𝜉=55%



78

(c) Middle joint displacement- step time

(d) Load-step time

Figure 3.9: Comparison between test results and FE simulation results using

different damping ratio 𝝃 for test S4

The above message is reinforced when the reaction force of the structure is compared

between the simulation and test results (Figure 3.9(d)). For high damping ratios (𝜉

=55%), the simulation reaction forces are lower than the applied load. Therefore, it

is recommended using damping ratio 𝜉 in the range between 25% ≤ 𝜉 ≤ 35%.

The acceptable damping factor (𝜇) is structure specific. However, on the assumption

that successful explicit dynamic simulations (numerical stability and minor influence

of damping on structural response) have similar damping ratios and from Equation

3-8, the acceptable range of Rayleigh mass proportional damping 𝜇 for different

structures may be taken as 0.5𝜔𝑚𝑖𝑛 ≤ 𝜇 ≤ 0.7𝜔𝑚𝑖𝑛.

3.2.6.3 Mass Scaling

An alternative method to improve the efficiency of explicit dynamic simulation is to

adopt the mass scaling technique. This technique is attractive if it is desirable to

-700

-600

-500

-400

-300

-200

-100

0

0 1 2 3 4 5 6

Mid

dle

join

t d

isp

lace

men

t (m

m)

Step time (s)

𝜉=0%

𝜉=15%

𝜉=25%

𝜉=35%

𝜉=55%

-20

0

20

40

60

80

100

120

0 1 2 3 4 5 6

Load

(kN

)

Step time (s)

𝜉=0% 𝜉=15% 𝜉=25%

𝜉=35% 𝜉=55% Applied Load



79

follow the real time structural behaviour. Furthermore, mass scaling can be applied

to some parts of structure where it is necessary to use non-uniform finite element

meshes for the structure. This is because the smallest elements will govern the stable

time for the whole system. In this case, selective mass scaling to the regions of small

elements can make the stable time increments more uniform.

From Equations 3-1 and 3-4, if the mass is scaled by a factor of say f, the stable time

increment increases by a factor of 𝑓. The number of time increments consequently

decreases by a factor of 𝑓. This would increase the dynamic side effects so the

value of f should be controlled to avoid spurious dynamic behaviour. It is necessary

to conduct a convergence study as for increasing the loading speed.

To investigate using adaptive mass scaling of structure, two analyses for the

benchmark model S4 were performed, one using displacement control and one using

load control. In the displacement control analysis, the loading speed was 0.1 mm/s

(test value) giving a loading duration of 7000s when the total imposed displacement

was 700mm. Replacing Tn in Equation 3-5 by 𝑓 gives:

𝐿𝐷1

𝑓1=

𝐿𝐷2

𝑓2 3-9

Therefore, if using mass scaling factor to achieve the same effect as shortening the

loading duration to LD1=5s, from Equation 3-9, a mass scaling factor of 1960000

(=(7000/5)2) would be necessary. In the load control simulation, assuming that the

actual loading duration was 3000s and mass scaling is applied to achieve the same

effect as shortening the loading duration to 6s but retaining the actual duration, then

the mass scaling factor would be 250000 (=(3000/6)2). Figure 3.10 compares the

simulation results and Table 3.4 shows the stable time increment and CPU

computation time for displacement-controlled and load-controlled simulations. The

closeness of the two sets of simulation results and CPU time in each case confirms

equivalence of the two simulation methods (loading factoring and mass scaling). As

stated before, adaptive mass scaling is preferred for performing an analysis with the

real process time. The results in Table 3.4 also show direct proportional reduction in

simulation time as the total loading duration is reduced (load factoring) or the stable

time step (mass factoring) is increased.



80

(a) Displacement-controlled method

(b) Load-controlled method displacement

Figure 3.10: Comparison between simulation results using load factoring and

mass scaling for test S4


factoring and mass scaling for test S4

Loading

method

Loading duration,

LD (s)

Mass scaling factor,

f

Stable time increment,

∆𝑡 (s)

CPU

time (s)

Displacement-

controlled

5 1 6.997E-6 4925

7000 1960000 8.88E-3 5294

Load-

controlled

6 1 6.997E-6 6887

3000 250000 3.174E-3 6966

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Load

(kN

)


Test

LD=5s, f=1, 𝜉=0%, μ=0

LD=7000s, f=1960000, 𝜉=0%, μ=0

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Load

(kN

)


Test

LD=6s, f=1, 𝜉=35%, μ=63

LD=3000s, f=250000, 𝜉=35%, μ=0.126



81

It should be pointed out that because the mass is scaled, the natural frequency of the

model is changed. Therefore, a new damping value μ had to be calculated according

to Equation 3-8 for the load-controlled simulation by keeping the damping ratio 𝜉

constant. This gives a value of μ=0.126 for the analysis using mass scaling factor of

250000.

The real benefit of using mass scaling is to increase the stable time increment for the

regions of structure with very fine meshes. Figure 3.11 shows a different mesh for

test structure S4, with a patch of elements whose smallest lengths is half that of the

regular elements (shown in Figure 3.2). The stable time increment for the regular

mesh was 6.997E-6s and that for the patch with small elements was 4.202E-6s. By

increasing the density of the small elements by a factor f=4 while keeping the density

of all other elements unaltered, a stable time increment ∆𝑡 of that of the regular

elements can be applied over the entire structure. Figure 3.11(b) compares the

simulation results between with and without locally applying mass scaling. Because

the region with mass scaling is small, the two sets of results are very close. However,

applying local mass scaling reduced the simulation time considerably.

(a) FE mesh for test S4 with one (b) Load-middle joint displacement

region of fine mesh

Figure 3.11: The effect of applying mass scaling to a small region of fine mesh

In summary, either load factoring or mass scaling may be used to reduce

computation time. A particularly useful benefit of mass scaling is the possibility to

0

20

40

60

80

100

120

0 200 400 600

Load

(kN

)


f=1 for uniform mesh, ∆t=6.997E-6 s

f=1 for non-uniform mesh, ∆t=4.202E-6 s

f=4 for non-uniform mesh, ∆t=6.997E-4 s



82

apply this technique to the regions of structural model with fine meshes.

Furthermore, when simulating the structure using load control, material damping is

necessary to overcome the temporary instability as a result of the restrained beam

transiting from flexural action to catenary action. How to select the appropriate load

factoring and mass scaling factors depends on the minimum nature period of

structure.

3.2.7 Validation Against the Test Results of Yu and Tan (Yu and Tan, 2013, Yu

and Tan, 2014)

Figures 3.12 to 3.18 compare the simulated beam responses with the test results,

using the load-controlled (LC) and displacement-controlled (DC) methods, giving

the vertical load, beam axial force and kinetic energy against MJD. Load factoring

was used. Table 3.5 presents the lowest natural periods, loading durations and

damping factors used for the models. Axial compression is developed in the beams

due to compressive arch action as the applied load (in LC simulations) or the applied

displacement (in DC simulations) increases. After reaching the maximum

compressive force, the structural resisting load decreases. Further increasing in the

displacement and load on the middle joint causes the beam axial compressive force

to decrease and the beam transits from compressive arch action to tensile catenary

action. Since the applied load continuously increases, the LC structures become

temporarily unstable when the resisting load drops below the applied load. The loss

of static stability is identified by the dramatic increase in the ratio of the kinetic

energy to the internal energy.



83

(a) Load and KE/IE-middle joint displacement


Figure 3.12: Comparison between modelling and test results (model S4)

-300

-200

-100

0

100

200

300

0 100 200 300 400 500 600 700

Axi

al f

orc

e (k

N)


Test ABAQUS (DC) ABAQUS (LC)

0

0.02

0.04

0.06

0.08

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)


Test

ABAQUS (Load, LC)

ABAQUS (Load, DC)

KE/IE (LC)

KE/IE (DC)

KE: Kinetic Energy

IE: Internal Energy



84

(a) Displacement-controlled simulation

(b) Load-controlled simulation

Figure 3.13: Variation of longitudinal steel reinforcement at critical regions

(model S4)

-0.04

0.00

0.04

0.08

0.12

0.16

-30

0

30

60

90

120

0 100 200 300 400 500 600

Stra

in (

mm

/mm

)

Load

(kN

)


Test

ABAQUS

Bottom bar strain (ABAQUS)

Top bar strain (ABAQUS)

Fracture of bottom bars near middle joint, 𝜀𝑢 = 0.109

Fracture top bars near side joint, 𝜀𝑢 = 0.109

-0.04

0.00

0.04

0.08

0.12

0.16

-30

0

30

60

90

120

0 100 200 300 400 500 600 700

Stra

in (

mm

/mm

)

Load

(kN

)


Test

ABAQUS



Fracture of bottom bars near middle joint, 𝜀𝑢 = 0.109

Fracture top bars near side joint, 𝜀𝑢 = 0.109



85




-300

-200

-100

0

100

200

300

0 100 200 300 400 500 600 700

Axi

al f

orc

e (k

N)


Test ABAQUS (LC) ABAQUS (DC)

0

0.01

0.02

0.03

0.04

0.05

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)


Test ABAQUS (Load, LC)

ABAQUS (Load, DC) KE/IE (LC)

KE/IE (DC)




86




-300

-200

-100

0

100

200

300

0 100 200 300 400 500 600 700

Axia

l fo

rce

(kN

)


Test ABAQUS (LC) ABAQUS (DC)

0

0.02

0.04

0.06

0.08

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Lo

ad (

kN

)


Test ABAQUS (Load, LC)

ABAQUS (Load, DC) KE/IE (LC)

KE/IE (DC)




87



Figure 3.16: Comparison between modelling and test results (model F2)

-100

-60

-20

20

60

100

140

180

220

260

0 100 200 300 400 500 600

Axi

al f

orc

e (k

N)


Test ABAQUS (DC)

ABAQUS (LC)

0

0.01

0.02

0.03

0.04

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)


Test

ABAQUS (Load, LC)

ABAQUS (Load, DC)

KE/IE (LC)

KE/IE (DC)

KE: Kinetic Energy

IE: Internal Energy



88



Figure 3.17: Comparison between modelling and test results (model F4)

-100

-50

0

50

100

150

200

250

300

0 100 200 300 400 500 600 700

Axi

al f

orc

e (k

N)


Test ABAQUS (DC) ABAQUS (LC)

0

0.01

0.02

0.03

0.04

0

25

50

75

100

125

0 100 200 300 400 500 600 700

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)


Test

ABAQUS (LC)

ABAQUS (DC)

KE/IE (LC)

KE/IE (DC)




89

(a) Displacement-controlled simulation

(b) Load-controlled simulation

Figure 3.18: Variation of longitudinal steel reinforcement at critical regions

(model F4)

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

-25

0

25

50

75

100

125

0 100 200 300 400 500 600 700

Stra

in (

mm

/mm

)

Load

(kN

)


Test

ABAQUS



Bent-up bar strain (ABAQUS)

Fracture of the bottom bars (near middle joint, 𝜀𝑢=0.137

Fracture of bent-up bar near middle joint, 𝜀𝑢 = 0.11

Fracture of the top bars near middle joint, 𝜀𝑢 = 0.11

-0.04

0.00

0.04

0.08

0.12

0.16

0.20

-25

0

25

50

75

100

125

0 100 200 300 400 500 600 700

Stra

in (

mm

/mm

)

Load

(kN

)


Test

ABAQUS



Bent-up bar strain (ABAQUS)

Fracture of the bottom bars (near middle joint, 𝜀𝑢=0.137

Fracture of bent-up bar near middle joint, 𝜀𝑢 = 0.11

Fracture of the top bars near middle joint, 𝜀𝑢 = 0.11



90

Table 3.5: Parameters used for modelling the tests of Yu and Tan (2013) and

Yu and Tan (2014)

Model

Lowest natural

frequency,

𝜔𝑚𝑖𝑛 (rad/s)

Lowest

natural

period, Tn (s)

Loading

duration,

LD (s)

Loading

speed LD/Tn

Damping

ratio, 𝜉

(%)

Mass

proportional

damping, 𝜇

Displacement-controlled (DC) method

S4,

S5, S7 89.5 0.070 5 140 mm/s 71 0 0

F2, F4 135.1 0.046 3.5 200 mm/s 76 0 0

Load-controlled (LC) method

S4,

S5, S7 89.5 0.070 6 18.35 kN/s 86 35 63

F2, F4 135.1 0.046 4 27.5 kN/s 87 35 95

During the catenary action stage, the resisting load rises again. The tensile catenary

force is withstood by the longitudinal steel bars. The sharp reductions in the applied

load are caused by the fracture of bars close to the middle joint interfaces.

Reinforcement bar fracture is accurately captured by the simulation model as

indicated by the reinforcement bar strains exceeding the bar fracture strains shown in

Figures 3.13 and 3.18 which plot the predicted strain versus MJD relationships for

sub-assemblage S4 and frame F4 for the sake of brevity. In the tests, complete

collapse of the sub-assemblage structures was due to fracture of the top bars near the

side joint interfaces and this was accurately simulated by comparison of the failure

modes between the simulation and experimental test of model S5 in Figure 3.19(a).

In order to prolong the catenary action phase, innovative reinforcement detailing

techniques were used by Yu and Tan (2014) in the frame tests, including adding a

reinforcement layer at the mid-height of the beam section in F2 and introducing a

partial hinge at the beam ends in F4 as shown in Figure 3.1. Complete collapse of the

specimens happened following rupture of all bars near the middle joint interfaces.

These failure modes were accurately captured by the simulation model, as shown in

the comparisons in Figure 3.19(b) for model F2.



91

Figure 3.19: Deformed shape and failure mode of FE simulations and tests

In all cases, the agreement between the numerical simulation results and the test

results is very good. In particular, the numerical simulation model reliably followed

the various temporary failure phenomena, including temporary loss of the applied

load, transition from compressive arch action to catenary action and fracture of

reinforcement until the final failure of structure. The proposed accelerated techniques

reduced the computational time by 3 orders of magnitude compared to simulations

using the actual loading speed of 0.1 mm/s.

(a) Model S5

(b) Model F2

At the side joint interface

At the middle joint interface

At the middle joint interface



92

3.3 Comparison and Application of the Finite Element Model to RC

Sructures in Fire

3.3.1 Comparison Against the Fire Tests of Dwaikat and Kodur (Dwaikat and

Kodur, 2009a, Dwaikat, 2009)

Three RC beams tested by Dwaikat and Kodur (2009a), named B1, B2 and B3, were

simulated. Figure 3.20 shows the details of the beams and the locations of

thermocouples installed to measure temperatures. The yield strength of reinforcing

bars for the three beams was 450 MPa, and the characteristic compressive cylinder

strength of concrete was 58.2 MPa for B1 and B2, and 106 MPa for B3. All the three

beams were loaded with two point loads of 50 kN each which produced a load ratio

of 55% of the beams’ bending moment capacities at ambient temperature determined

according to ACI 318. These loads were maintained constant during the subsequent

fire exposure. Beams B1 and B3 were exposed to the ASTM E119 standard fire

while beam B2 was exposed to a short fire scenario followed by a cooling phase. The

end support conditions for B1 and B3 were simply supported while B2 was axially

restrained with a stiffness value of about 13 kN/mm. “Surface-to-surface contact

(Explicit) interaction” and “axial connector element” in ABAQUS were used to

model the supports of beam B2 (Figure 3.20). The “Normal behaviour” and “hard

contact” in surface-to surface contact interaction options were used to apply physical

contact between the axial restraint system and the end section. The “Allow separation

after contact” option was activated in defining contact interaction since the end

section was not anchored to the adjacent frame during the test.



93

Figure 3.20: Details of test beams B1, B2 and B3 with the locations of

thermocouples (Dwaikat and Kodur, 2009a)

For beam B1, the average measured temperatures of thermocouples, reported in

detail in (Dwaikat, 2009), located on the exposed concrete surface, namely T3, T14,

and T19 were used as the initial thermal boundary condition in heat transfer analysis

to obtain the cross-section temperatures. A sequentially coupled thermal-structural

analysis method was adopted by firstly carrying out a 3D heat transfer analysis in

ABAQUS/Standard solver. Then, the cross-section temperatures were imported to

the subsequent structural analysis model in the ABAQUS/Explicit solver. Evrine et

al. (2012) experimentally investigated the effect of tensile cracking of RC beams on

the temperature field within the concrete cross-section. A minor difference was

noticed between the temperatures measured by thermocouples in the undamaged

concrete sections and those measured by thermocouples positioned in the damaged

sections close to cracks of width of the order of 101 mm at the exposed surface.

Besides, no spalling was observed in beam B1 during fire exposure. Hence, it is

reasonable to assume that the structural behaviour depends on the temperature

distribution, but there is no reverse dependency.

Accurate modelling of fire-induced spalling is very complicated and it is the subject

of research efforts by many researchers at present. In the proposed finite element


T13

254

406

44

2 𝜙 12

3 𝜙 19 T5

T11 T7,16

T18 T10

T9 T17 T6

T1,15 T8

102

38 26 37

65

37

254

406

7 37

57

T3,14,19

T12

T20,T4

150

2440 mm

1550

3660

1550 860

50 kN 50 kN

Span exposed to fire

Furnace

Flexible insulation

uy=uz=0

Axial connector

Steel plate

ux=uy=uz=0

Contact surface

Beam B2



94

model in this research, spalling is not explicitly modelled. Therefore, this research is

applicable only to RC structures made of traditional normal strength concrete with

moisture content ≤ 3% where spalling in this concrete is uncommon or very rare

(Hertz, 2003, CEN, 2004, Choi and Shin, 2011, Dwaikat and Kodur, 2009a).

Nevertheless, the present model can be used to simulate RC elements in tensile

catenary/membrane action that develops at very large deflections following bending

failure. This is because the applied load is sustained by the top cooler and more

protected reinforcing bars.

In heat transfer analysis, concrete and reinforcing steel were modelled using first-

order eight-node elements (DC3D8) and two-node link elements (DC1D2),

respectively. “Tie constraint” was used to transfer temperatures from the concrete

element to the embedded reinforcing steel element to indicate that the steel bars and

the surrounding concrete had the same temperature. A constant convective heat

transfer coefficient (hc) of 25 W/m2K and 9 W/m

2K was assumed for the exposed

and unexposed surfaces respectively according to EN 1992-1-2 (CEN, 2004). For the

radiative heat flux boundary condition, the resultant emissivity for concrete surface

was taken as 0.7. The required thermal properties of concrete, namely density,

thermal conductivity and specific heat as a function of temperature were defined

according to EN 1992-1-2 (CEN, 2004), as presented in appendix A4. The influence

of moisture evaporation in concrete was considered implicitly by modifying the

specific heat model suggested by EN 1992-1-2 (CEN, 2004). The measured moisture

content by weight was about 3% and this was used in the numerical model. Figure

3.21 compares the heat transfer analysis results with the test results for beam B1,

indicating good accuracy. The discrepancy during the initial period of fire exposure

up to 100 o

C for concrete can be attributed to the fact that the simulation model

ignored physical water evaporation.

Significant concrete spalling occurred in specimen B3, made of high strength

concrete, during testing because of low permeability of high strength concrete that

increases the build-up of pore pressures in concrete and thus possibility of explosive

spalling. Since the current numerical model cannot simulate spalling, the recorded

temperatures in the tests within the beam sections were input directly into the

numerical model and the thermal/structural behaviour were obtained without

conducting heat transfer analysis. This was also adopted for normal strength concrete



95

beam B2 although it suffered minor spalling. To allow direct use of the recorded

temperatures at the thermocouple locations, the cross-sections were divided into

elements according to Figure 3.22 so that the thermocouple locations coincide with

some nodes in the finite element model. The temperatures of other nodes on the

cross-section were scaled according to the temperature profiles recommended in EN

1992-1-2 (CEN, 2004).

Figure 3.21: Comparison between predicted and measured temperature for B1

0

100

200

300

400

500

600

700

0 50 100 150 200

Tem

per

atu

re (

oC

)

Time (min)

T6 (Test)

T6 (ABAQUS)

T5, T7, T16 (Test, Average)

T5, T7, T16 (ABAQUS)

0

100

200

300

400

500

600

0 50 100 150 200

Tem

per

atu

re (

oC

)

Time (min)

T10 (Test)

T10 (ABAQUS)

T11 (Test)

T11 (ABAQUS)

T12 (Test)

T12 (ABAQUS)

0

100

200

300

400

0 50 100 150 200

Tem

per

atu

re (

oC

)

Time (min)

T9 (Test)

T9 (ABAQUS)

T1, T4, T20 (ABAQUS)

T1, T4, T20 (Test, Average)

T18 (Test)

T18 (ABAQUS)



96

Figure 3.22: Applying temperatures at nodes according to experimental

measurements of thermocouples for B2 and B3

The subsequent elevated temperature structural analysis was performed in two steps.

In the first step, the mechanical loads on the beam were applied at ambient

temperature. In the second step, the beam was exposed to temperatures while

maintaining the applied mechanical loads constant. For B1, the FE mesh was the

same as those used in the corresponding heat transfer analysis, but the heat transfer

concrete elements DC3D8 and reinforcement elements DC1D2 were converted to

stress elements C3D8R and T3D2, respectively.

Eurocode 2 (EN 1992-1-2 (CEN, 2004)) was used to obtain the compressive stress-

strain relationships and the free thermal strains (𝜀𝑡ℎ) of concrete at elevated

temperatures. It was also used to obtain the elevated temperature properties (stress-

strain relationship, thermal strain) of reinforcement steel. The Eurocode 2 stress-

strain relationships of concrete and reinforcement steel are presented in Appendix

A4.

The thermal strain of concrete at elevated temperatures is complex. In addition to the

free thermal strain, there is also load-induced thermal strain (LITS). LITS develops

when concrete is heated under a state of compression. It is defined as the difference

between the thermal strain of stress-free concrete and the thermal strain when the

same concrete is heated while under a constant compressive stress. LITS is the sum

of several strain components. It consists of: changes in elastic strain due to

deterioration of elastic modulus as temperature increases, basic creep and drying

creep strains which develop by a rise in temperature and transient thermal creep

(Torelli et al., 2017, Torelli et al., 2016). Transient thermal creep is the largest

component of LITS. It develops during the first-time heating and is not recoverable.

It is thought to be the result of physical disintegration and chemical reactions within

T13, TB (Average)

T5, T7, T16 (Average) =TB

T13

T1, T4, T15, T20 (Average)

T14

T11

T11, TA (Average)

T2, T9, T10, T17, T18 (Average) =TA

T11, TB (Average)



97

the cement paste microstructure for temperatures up to 300-400oC and due to

mismatch of the thermal strains of cement past and aggregates (Torelli et al., 2017).

Hence, most of the LITS developed after a first heating cycle is irrecoverable on

cooling and does not develop further when the material is reheated if the first heating

temperature is not exceeded (Torelli et al., 2016).

In the Eurocode 2 model, the total strain of concrete at elevated temperatures is

considered as the sum of free thermal strain and mechanical strain. The mechanical

strain in the uniaxial compressive stress-strain relationship lumps together the

instantaneous stress-related strain and LITS, in which LITS is considered implicitly.

Previous studied have shown that using the Eurocode 2 model can properly simulate

concrete structures under constantly increasing temperature such as standard fire

exposures ISO 834 and ASTM E119 (Gernay and Franssen, 2012, Gernay, 2012).

This was revealed by comparing the structural behaviour results using the implicit

model of Eurocode 2 with those of structures with LITS considered explicitly as a

separate component of the total strain. However, the Eurocode 2 model has inherent

limitations for an accurate representation of the behaviour of structures subjected to

both heating and cooling. It has been shown that the Eurocode 2 model cannot

capture the irreversibility of the irrecoverable component of the LITS in the cooling

phase (Gernay and Franssen, 2012, Gernay, 2012).

In this study, the implicit Eurocode 2 model is used and LITS is not explicitly dealt

with. Hence, the current finite element model is limited to modelling RC structures

subjected to heating only. However, the model can be used to model RC beam and

slab members in tensile catenary/membrane action in both heating and cooling

phases because the member is fully in tension and LITS occurs only when concrete is

in compression.

The test beam B2 by Dwaikat and Kodur (2009a), used in the present validation

study, was subjected to heating with a short severe fire followed by a cooling phase.

However, this beam was simply supported with axial restraint. The behaviour of the

beam was mainly governed by the behaviour of the bottom (tensile) steel bars in the

beam mid-span and not by the behaviour of concrete. In addition, the compressive

zone at the critical mid-span section does not exhibit significant LITS because its

area is small and kept at relatively low temperatures. Therefore, using the implicit



98

Eurocode 2 model would only incur minor errors in structural behaviour predictions

during the cooling phase for beam B2. Using the same reasons above, and because of

the fact that the loss of concrete strength with increasing temperature is gradual and

that the maximum measured temperature of the tensile steel bar in the mid-span was

about 545oC, at which steel recovers almost all of its ambient stiffness and tensile

strength upon cooling (Neves et al., 1996), it was assumed that stress-strain curves of

concrete and steel in the cooling phase for beam B2 remain as they are in the heating

phase.

For the tensile stress-strain relationships of concrete at elevated temperatures, similar

relationships between 𝜎𝑡 and 𝜀𝑡 as at ambient temperature (Figure 3.4) were used.

However, the variation of concrete initial modulus of elasticity as a function of

temperature 𝐸𝑐𝑚,𝑇 was according to EN 1992-1-2 (CEN, 2004), and that of tensile

strength with temperature was according to Bazant and Chern (Youssef and Moftah,

2007, Bažant and Chern, 1987) as shown in Figure 3.23 and given below :

𝑘𝑡,𝑇 = −0.000526𝑇 + 1.01052 𝑓𝑜𝑟 20𝑜𝐶 ≤ 𝑇 ≤ 400𝑜𝐶 3-10

𝑘𝑡,𝑇 = −0.0025𝑇 + 1.8 𝑓𝑜𝑟 400𝑜𝐶 ≤ 𝑇 ≤ 600𝑜𝐶

𝑘𝑡,𝑇 = −0.0005𝑇 + 0.6 𝑓𝑜𝑟 600𝑜𝐶 ≤ 𝑇 ≤ 1000𝑜𝐶

Figure 3.23: Tensile stress-strain relationship of concrete at elevated

temperatures

𝑓𝑐𝑡𝑚,𝑇 𝑓𝑐𝑡𝑚,20𝑜⁄

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01

20

200

400

600

800

1000

𝜎𝑡 = 𝑓𝑐𝑡𝑚,𝑇 (𝜀𝑐𝑟,𝑇𝜀𝑐𝑡,𝑇

)

𝑐

𝜀𝑐𝑟,𝑇 =𝑓𝑐𝑡𝑚,𝑇

𝐸𝑐𝑚,𝑇 𝜀𝑐𝑡,𝑇

oC

oC

oC

oC

oC

oC



99

Evolution of nodal temperatures in the structural analysis model may be accelerated.

A series of analyses with various simulation heating durations (HD) for one hour of

the real fire exposure time were run for the axially restrained beam B2. The mass

scaling factor was f=1. Figure 3.24 compares the mid-span deflection versus fire

time relationship for each analysis against the test data and also displays kinetic

energy-time relationships of the models. The results from HD=0.25s display

oscillations during the initial period of fire loading application, which can also be

seen from the kinetic energy results. With HD ≥ 0.75s, the simulation results are

quasi-static, except at the start of the analysis which is associated with the transition

from mechanical load to temperature loading. The lowest natural period Tn of the

assembled beam model B2 was 0.039s. This means that a heating duration HD ≥

19Tn for one hour of the real fire exposure time could be considered appropriate for

restrained RC beams in fire if using the load factoring technique. The simulation

time may be changed for other heating durations pro-rata.

Figure 3.24: Mid-span deflection and kinetic energy versus fire exposure time

for different heating durations (Beam B2)

Figure 3.25 provides comparison for the axial force of beam B2 between the FE

simulation and the experimental results and between the mid-span deflections of

beams B1 and B3. The selected modelling parameters were: HD=19Tn, f=1 and 𝜉=0.

Damping was not required since the response of the tested beams by Dwaikat and

Kodur (2009a) were only investigated in flexural action and the numerical simulation

did not encounter any local instability. Overall, the comparisons are very good. The

simulation results display a higher deflection rate for beam B1 during the second half

0

50

100

150

200

250

0 50 100 150 200 250 300

-35

-30

-25

-20

-15

-10

-5

0

Kin

etic

en

ergy

(J)

Time (min)

Def

lect

ion

(m

m)

Deflection (Test)

Deflection (ABAQUS, HD=0.25s)

Deflection (ABAQUS, HD=0.75s)

Kinetic energy (HD=0.25s)

Kinetic energy (HD=0.75s)



100

of heating. The same discrepancy was also observed in the numerical analysis by

Dwaikat and Kodur (2009a) and Kodur and Agrawal (2016). Beams B1 and B3

failed in flexural mode and beam B2 did not fail. It can be concluded that the

developed FE model in ABAQUS/Explicit is able to capture the performance of RC

beams at elevated temperatures with satisfactory accuracy.

Figure 3.25: Comparison between predicted and measured results

3.4 Preliminary Investigation of the Large Deflection Behaviour of

Axially Restrained RC Beams in Fire

Although beam B2 of the tests by Dwaikat and Kodur (2009a) had axial restraint,

their fire test did not continue to the catenary action stage and did not reach structural

failure. Therefore, the validity of the modelling method could not be completely

demonstrated. A new restrained beam is used in this section as an example to

demonstrate the proposed modelling method. Figure 3.26 shows details of the beam.

Only half of the beam was analysed because of symmetry to save computation time.

The ambient temperature concrete compressive cylinder strength is 30 MPa, and the

steel reinforcement yield strength is 453 MPa with the ultimate strain as 0.05. An

extended ultimate strain was defined for the bottom bars for a distance of 1.5 times

the beam depth measured from the end sections to prevent false failure of the bars in

compression. The beam is exposed to the ISO 834 standard fire on three sides. The

density of the uniformly distributed load (w =23.1kN/m) gives a load ratio of 40% of

-500

-400

-300

-200

-100

0

0 25 50 75 100 125 150 175 200

Def

lect

ion

(m

m)

Time (min)

B1 (Test)

B1 (ABAQUS)

B3 (Test)

B3 (ABAQUS)

-35

-28

-21

-14

-7

0

7

14

21

28

35

-125

-100

-75

-50

-25

0

25

50

75

100

125

0 200

Def

lect

ion

(m

m)

Axi

al f

orc

(kN

)

Time (min)

Test (Axial force)ABAQUS (Axial force)Test (Deflection)ABAQUS (Deflection)

(a) Test beam B2 (2) Test beam B1 and B3



101

the rotationally fix-ended beam’s bending moment capacity at ambient temperature

(sagging capacity=hogging capacity=126 kN.m, calculated without consideration of

compression reinforcement (Appendix A1)).

Connectors were used to simulate the axial and rotation restraints at the supports.

The restraints were elastic with temperature-independent stiffness values of

0.125(𝐸𝐴 𝐿⁄ )𝑏𝑒𝑎𝑚, 20𝑜𝐶 and 0.5(4𝐸𝐼 𝐿⁄ )𝑏𝑒𝑎𝑚, 20𝑜𝐶 respectively, where

(𝐸𝐴 𝐿⁄ )𝑏𝑒𝑎𝑚, 20𝑜𝐶 and (4𝐸𝐼 𝐿⁄ )𝑏𝑒𝑎𝑚, 20𝑜𝐶 are the ambient temperature axial and

flexural stiffness of the beam, respectively. The displacement and rotation of nodes

at the end sections were constrained by a controlled reference point located at the

support point. This can be achieved in ABAQUS by using the Multi-Point Constraint

(MPC) type Beam function. The lateral translation of the beam at top was

constrained in order to prevent any twisting and torsional buckling at high

temperatures.

Figure 3.26: Details of the axially restrained beam

Based on the results of section 3.3.1 of this chapter, one hour of the real fire exposure

time was scaled down to 1s in the simulation, which is about 19 times the natural

period of the lowest mode (Tn=0.052s). The damping ratio was 25% (the mass

proportional damping factor 𝜇 = 60) and the mass scaling factor f=1.

Figure 3.27 shows the mid-span deflection and the beam axial force-fire exposure

time relationships. The general trend is as expected. For the deflection-time

relationship, the initial beam deflection is mainly due to thermal bowing. As the

𝜔𝑚𝑖𝑛=119 rad/s

L/2=3000

𝜙 10 @ 100

𝜙 10 @ 200

3#6

3#6

3#6

3#6

C.L.

KA KR

MPC



102

beam approaches its bending limit, it undergoes accelerated rate of deflection until

the activation of catenary action. This stage of behaviour corresponds to the

transition of the axial force from compression to tension. Afterwards, the beam

enters a stage of stable behaviour when the rate of deflection is steady and the

applied load on the beam is mainly resisted by tensile catenary action.

Figure 3.27: General behaviour of axially restrained RC beam in fire

If an analysis requires preservation of the same event time, adaptive mass scaling can

be used. As illustrated in Figure 3.27 and Table 3.6, identical response and CPU

computation time may be attained if the beam is simulated with HD=3600s (real

time) but with f= 12960000 determined using Equation 3-9. Figure 3.28 plots the

ratio of the kinetic energy to the internal energy using load factoring and mass

scaling techniques. The two speed-up techniques successfully demonstrate the quasi-

static behaviour as the kinetic energy remains bounded and is close to zero in the

stable periods. The applied damping, predicted based on Equation 3-8, was checked

by comparing the vertical reaction force with the applied load.


factoring and mass scaling

Heating duration, HD,

Simulation(s)/Real(s) Mass scaling factor, f

Stable time increment,

∆𝑡 (s)

CPU time

(s)

1/3600 1 8.807E-6 15771

3600/3600 12960000 3.170E-2 15567

-600

-500

-400

-300

-200

-100

0

100

200

-600

-500

-400

-300

-200

-100

0

100

200

0 100 200 300 400

Axi

al f

orc

e (k

N)

Def

lect

ion

(m

m) Time (min)

Deflection, HD=(1.0s/3600s) Axial force, HD (1.0s/3600s)

Deflection, HD=(3600s/3600s) Axial force, HD=(3600s/3600s)



103

(a) HD = 1.0s (simulation)/3600s (Real), f=1, 𝜉 =25%, μ=60

(b) HD = 3600s (simulation)/3600s (Real), f=12960000, 𝜉 =25%, μ=0.0167

Figure 3.28: Vertical reaction force, applied load and kinetic energy against

time

During transition from bending until final failure of the beam, the beam undergoes a

number of temporary failures due to severe concrete crushing and fracture of

reinforcement steel. Figure 29 presents the variations of strain in the reinforcement

against the fire exposure time and Figure 30 displays the deformed configuration of

the analysed beam. The proposed modelling method is able to follow temporary

failures and capture the beam behaviour at large deflections.

0.000

0.004

0.008

0.012

0.016

0.020

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)

Step time (s)

Vertical reaction

Applied load

KE/IE


0.000

0.004

0.008

0.012

0.016

0.020

0

20

40

60

80

100

120

140

160

0 6000 12000 18000 24000

Kin

etic

en

ergy

/In

tern

al e

ner

gy

Load

(kN

)

Step time (s)

Vertical reaction

Applied load

KE/IE




104

Figure 3.29: Strain profile in reinforcing bars against time

Figure 3.30: Deformed shape and failure mode

-1000

-800

-600

-400

-200

0

200

400

600

-0.15

-0.12

-0.09

-0.06

-0.03

0

0.03

0.06

0.09

0 100 200 300 400

Def

lect

ion

(m

m)

Stra

in (

mm

/mm

)

Time (min)

Top bars, ends Top bars, mid-span Bottom bars, ends

Bottom bars, mid-span Stirrups, ends Mid-span deflection

Limiting strain

Top bars, end

Stirrups, ends

Bottom bars, ends

Bottom bars, mid-span

Top bars, mid-span

At final failure

At beam mid-

span

At beam mid-

span

At beam ends

At beam ends

In catenary action



105

3.5 Conclusions

This chapter has presented a detailed explicit simulation methodology using

ABAQUS to model the whole range of large deflection behaviour of axially and

rotationally restrained RC beams at ambient and elevated temperatures. The main

challenges include material failure (concrete crushing and reinforcing bar rupture),

temporary instabilities and transition of the load-carrying mechanism from flexural

action to catenary action. A particular problem with explicit simulation is the very

small time step. To speed up the simulation process, load and mass scaling factors

have been examined. Damping was introduced in the load controlled loading method

to ensure numerical convergence. The proposed methodology has been validated by

checking the simulation results against relevant available test results. The following

key conclusions may be drawn:

(1) A concrete mesh size of between 25 to 35 mm may be adopted.

(2) When using explicit simulation to model static loading process, the dynamic

effects are negligible if the total loading duration does not fall below a

minimum value. For ambient temperature displacement-controlled and load-

controlled simulations, the minimum loading duration is about 50 and 65

times the structure’s lowest natural period respectively. For simulating

structural behaviour in fire, the minimum heating duration is 20 times the

lowest natural period for 60 minutes of real heating duration.

(3) Mass scaling may be used to achieve the results as above while keeping the

real loading/heating duration unaltered. To use mass scaling, the structural

mass should be scaled up (m)2 times, where “m” is the ratio of the real

loading/heating duration to the minimum simulation loading/heating duration.

A particular benefit of mass scaling is the possibility to apply this technique

in combination with the load-factoring technique to very fine meshes within

the structural model.

(4) To avoid premature final failure of beams due to significant dynamic effects

following bending failure in load-controlled simulation, a damping ratio of 25

to 30% should be applied to the simulation model.

CHAPTER 4 BEHAVIOUR OF AXIALLY AND ROTATIONALLY RESTRAINED

REINFORCED CONCRETE BEAMS IN FIRE

106

CHAPTER 4

BEHAVIOUR OF AXIALLY AND ROTATIONALLY

RESTRAINED REINFORCED CONCRETE BEAMS IN

FIRE

4.1 Introduction

An important objective of the present research is to gain a thorough understanding of

the influence of boundary conditions on the structural behaviour and failure modes of

RC beams in fire. This chapter presents the results of an extensive numerical

parametric study to examine the effects of different levels of axial and rotational

restraints on the full-history fire response of RC beams, including catenary action at

large deflections. The parametric studies were conducted on beam-column

assemblies. Linear temperature-independent axial and rotational restraints were

provided to the beam ends by using the connected columns and additional springs.

The proposed numerical model in Chapter 3 was employed for the parametric study

in this chapter and the results of this chapter will be the basis of assumptions for the

simplified calculation method of this thesis to be presented in Chapter 5.

4.2 Simulation Methodology

A schematic arrangement of a beam-column sub-frame shown in Figure 4.1 was

adopted to investigate the behaviour of restrained RC beams in fire under different

levels of end axial and rotational restraints. Details of only one-half of the sub-frame

are shown for clarity. In this parametric study, because the focus is on behaviour of

the restrained beam, linear behaviour of the axial and rotational restraints was

assumed. In addition, their levels were kept constant throughout the whole fire

exposure period. The end restraints, exerted by the supporting columns, axial and

rotational springs, were defined in a relative manner as follows:

𝑘𝑎 =𝐾𝐴,𝑠𝑢𝑝+𝐾𝐴

(𝐸𝑐𝑚𝐴 𝐿⁄ )𝑏𝑒𝑎𝑚 4-1



107

𝑘𝑟 =𝐾𝑅,𝑠𝑢𝑝+𝐾𝑅

(4𝐸𝑐𝑚𝐼 𝐿⁄ )𝑏𝑒𝑎𝑚 4-2

where ka, kr are non-dimensional relative axial and rotational stiffness parameters,

defined as the ratios of stiffness of the total axial and rotational restraints of the

adjacent structure, including the springs, to the ambient-temperature axial and

rotational stiffness of the beam under consideration, respectively. KA,sup and KR,sup are

the axial and rotational stiffness from the supporting columns, respectively. Ecm, I, A

and L are ambient-temperature elastic modulus of concrete, moment of inertia of the

beam section with respect to its geometric centre, beam cross-sectional area and

span, respectively. Calculations of KA,sup and KR,sup based on the sub-frame in Figure

4.1 are presented in Appendix A2.

In this study, the “axial connector element” in ABAQUS was used instead of the

spring element to model the axial and rotational springs as it can be easily operated

in ABAQUS CAE when the explicit solver is employed. To prevent stress

concentration and local failure of the column surface to which the springs are

attached, a rigid surface (Surface A in Figure 4.1) was introduced to the column and

this surface is at the same elevation position as the beam-column interface. The

location of the axial and rotational connector elements are at the geometric centre of

surface A. The nodes on this surface are coupled to a controlled reference point

(point O in Figure 4.1) using the Multi-Point Constraint (MPC) type function in

ABAQUS. This reference point is connected to a “Ground Point”. Meshes with an

element size of 30mm were used for the concrete beam-column sub-frame.



108

Figure 4.1: Dimensions and boundary conditions of beam-column sub-frame

model

The thermal properties and boundary conditions as suggested by EN 1992-1-2(CEN,

2004) were used. Siliceous concrete was selected, with a moisture content of 3% by

weight for calculating the specific heat of concrete and a density of 2300 kg/m3.

Normal strength concrete with low moisture content was assumed because of the

limitations of the numerical model employed in this study, which cannot account for

spalling of high strength concrete or concrete having high moisture content. The

ambient temperature concrete compressive cylinder strength is assumed to be 30

MPa and the modulus of elasticity of concrete at ambient temperature is 18000 MPa.

Surface A

Beam-column joint

panel zone

2#6

3#6

50mm

a-a

B=300 mm

3#6

3#6

D=400

mm

b-b

C

L/2=3000 mm

mm

H=3500 mm

H=3500 mm

KA

a

a

b

b L

c c

c-c

300 mm

300 mm

O

KR

KA : changed

Icolumn : constant

Ecolumn : changed to obtain different values of kr



109

The hot-rolled steel reinforcement bar yield strength at ambient temperature is

assumed to be 453 MPa (Grade 460), with the ultimate strain being 0.05, assuming

class A reinforcement as suggested by EN 1992-1-2 (CEN, 2004). In order to prevent

spurious fracture failure of reinforcement bars in compression, a higher value of

ultimate strain (0.25) was assumed for the bottom reinforcement bars within the zone

of a length of 1.5 times the beam depth measured from the column face.

The material behaviour of the column was assumed to be linear elastic and not

affected by temperature; however, the modulus of elasticity of the material was

varied to allow for a range of rotational stiffness ratios, defined in Equation 4.2, to be

investigated. In addition, the joint panel shown in Figure 4.1 was modelled with a

high elastic modulus value to prevent any local distortion of the concrete surrounding

the longitudinal steel bars in the beam-column joint; the elastic modulus of 18000

MPa (equal to the initial elastic modulus of the beam at ambient temperature) was

assumed irrespective of the value of the columns.

Transient modelling was carried out: the mechanical load on the beam was applied

first and kept constant and then the structural temperatures, imported from 3-D heat

transfer analysis, were increased The column was prevented from twisting and the

out-of-plane translation of the beam was not allowed so as to prevent lateral torsional

buckling. The beam was subjected to a uniformly distributed load of w=19.6 kN/m,

to give a load ratio (LR) of 35%, LR being defined as the ratio of total bending

moment in the beam (wL2/8) to the rotationally fix-ended beam’s bending moment

capacity at ambient temperature (sum of plastic bending resistance at the end and in

the span, sagging capacity=hogging capacity=126 kN.m, calculated without

consideration of compression reinforcement (Appendix A1)).

The beam length, cross-section dimensions and reinforcement details, shown in

Figure 4.1, were kept unchanged during the parametric study. All the top steel

reinforcement bars were continuous along the whole beam length without

curtailment.

In a performance-based approach for fire design, it is highly desirable to model a

structure taking into account realistic representation of the fire. This requires

assessing fire behaviour of the structure under different fire scenarios that depend on

the fuel load, ventilation conditions and properties of the wall lining in a



110

compartment. To date, a number of models for describing fire time-temperature

curves have been developed and are specified in design codes and standards, such as

Eurocode 1 provisions (CEN, 2002). There are standard and design (parametric) fire

time-temperature curves. The standard fire curves, such as those specified in ISO 834

(1975) and ASTM E119 (2008), do not include a decay (cooling) phase so fire

temperatures continue to increase, irrespective of compartment characteristics.

However, more realistic design fire curves incorporate a decay phase during which

the temperature of the fire decreases back to the ambient temperature. Previous

experimental and numerical studies have shown that the type of fire scenario has a

significant influence on the thermal and structural behaviour of RC members in fire

(Kodur et al., 2009, Dwaikat and Kodur, 2009a, Gao et al., 2017). Different fire

curves can give different thermal gradients that develop within the member cross-

section, which in turn result in different thermally-induced defamations and stresses

in the member and different rates of degradation of material properties at elevated

temperatures. Additionally, in the cooling phase, steel reinforcing bars recover parts

of their stiffness and strength. All these influence fire resistance.

The analysed beams in this chapter were exposed to the ISO 834 standard fire only,

where the time-temperature curve includes only the heating phase. The cooling phase

of the fire was not considered. The reasons for this are because: (1) the main purpose

of the study in this chapter is to understand the relative performance of RC beams

under different levels of axial and rotational restraints at elevated temperatures, (2)

understanding of catenary action developed at very large beam deflections after

bending failure is still immature and (3) as mentioned in Chapter 3, the finite element

model used in this study is not applicable to design fires that have a cooling phase as

it cannot capture irreversibility of the load-induced thermal strain (LITS) in analyses

involving concrete in compression. Figure 4.2 shows the temperature histories of

various reinforcing steel bars, obtained from the heat transfer analysis.



111

Figure 4.2: Temperature-time histories in the steel reinforcing bars, based on

numerical heat transfer modelling

4.3 Case Studies

Two sets of simulations were carried out to investigate the effects of end boundary

restraints on the behaviour of restrained RC beams in fire. The first set was

conducted on beams with symmetrical end boundary restraints to simulate interior

beams, while the second set was conducted on beams with asymmetrical end

boundary restraints to simulate edge beams. For the first set, to save computation

time, only half of the sub-frame was modelled to take advantage of symmetry.

However, for the second set, the whole sub-frame had to be modelled.

4.4 Beams with Symmetrical End Boundary Restraints

4.4.1 Effect of End Rotational Restraint

For presenting the general behaviour of restrained beam, the axial restraint level was

kept constant, ka equal to 0.166, in all analyses by changing the additional axial

stiffness of the spring (KA) to suit the change in column flexural stiffness. The

reference level of axial restraint stiffness of ka=0.166 was based on RC columns with

Young’s modulus E=20000MPa and cross-section size 500x500mm and height 3.5m

in the same sub-frame shown in Figure 4.1 without any additional spring stiffness

(KA=0 and KR=0). Additional simulations were then carried out for different levels of

axial restraint to confirm that the restrained beam behaviour discussed in this section

is applicable.

0

200

400

600

800

1000

1200

0 100 200 300 400 500

Tem

per

atu

re (

oC

)

Time (min)

1 2 3 4

5 6 7

1 2

4

6

1

3 3

5

6

7



112

The modulus of elasticity of the column material ranges from 25000 MPa to 63 MPa,

to reflect possible temperature effects of concrete in the range of ambient

temperature to about 1000oC.

In order to cover a range of possible end rotational restraints, different values of kr,

being 0.0064, 0.013, 0.022, 0.032, 0.045, 0.064, 0.125, 0.58, and 1.85, were

considered.

4.4.1.1 Results and Discussions

Figure 4.3 shows the overall global behaviour of the structural response of the

beams, presented for the beam mid-span deflection and beam axial restraining force

(tension positive)–fire exposure time relationships. All results show the same key

stages of global response as explained below. Before the start of fire, beams with low

rotational stiffness kr develop higher initial deflections. After fire exposure,

compressive forces are generated in the beams due to restrained thermal expansion.

The initial rate of increase of compression force depends on the axial restraint level.

This phase lasts until the combined bending moment and axial compression force

reach the capacity of the critical section, after which, the compression force in the

beam decreases. At the end of this process, the tension reinforcement bars in the

beam span fracture, as shown by the strain-time relationship in Figure 4.4. During

this stage, the beam deflection accelerates. As expected, beams with greater kr values

reach this stage later because the bending moment in the beam span is lower and the

rate of their deflection is lower.



113

(a) Comparison of beam mid-span deflections-fire exposure time relationships

(b) Comparison of beam axial force-fire exposure time relaitonships

Figure 4.3: Effect of rotational restraint levels on beam behaviour (ka=0.166)

-800

-700

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350 400 450

Def

lect

ion

(mm

)

Time (min)

kr=0.0064 kr=0.013 kr=0.022 kr=0.032 kr=0.045

kr=0.064 kr=0.125 kr=0.58 kr=1.85

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Time (min)

kr=0.0064 kr=0.013 kr=0.022 kr=0.032 kr=0.045

kr=0.064 kr=0.125 kr=0.58 kr=1.85



114

(a) Bottom corner bars (bars 1)

(b) Bottom middle bar (bar 2)

Figure 4.4: Strain-fire exposure time relationship of longitudinal reinforcing

bars at beam mid-span (ka=0.166)

One of the principal outputs of this research is the effects of restraints on fire

resistance times. In this research, two fire resistance times are assessed: beam

bending resistance time and beam ultimate resistance time. The bending resistance

-750

-600

-450

-300

-150

0

150

300

450

600

-0.08

-0.06

-0.05

-0.03

-0.02

0.00

0.02

0.03

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Def

lect

ion

(mm

)

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 (S) kr=0.013 (S) kr=0.022 (S)kr=0.032 (S) kr=0.045 (S) kr=0.064 (S)kr=0.125 (S) kr=0.58 (S) kr=1.85 (S)kr=0.0064 (D) kr=0.013 (D) kr=0.022 (D)kr=0.032 (D) kr=0.045 (D) kr=0.064 (D)kr=0.125 (D) kr=0.58 (D) kr=1.85 (D)

Mid-span

Limiting strain=5%

S: Strain D: Deflection

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Stra

in m

m/m

m)

Time (min)

kr=0.0064 kr=0.013 kr=0.022kr=0.032 kr=0.045 kr=0.064kr=0.125 kr=0.58 kr=1.85

Mid-span

Limiting strain=5%



115

time of the beam is the time when the compression force in the beam is decreased to

zero (i.e. the end of the flexural action), and the ultimate resistance time is when the

beam could not resist the applied load any longer. Figure 4.5 summarises how beam

bending resistance times and beam ultimate resistance times change with levels of

rotational restraint. It is expected that beams with higher rotational restraints, and

hence higher hogging moment resistance, should reach longer bending resistance

time, i.e. fire exposure time when the compression force in the beam decreases to

zero. However, this trend is not followed. This may be explained by the different

modes of bending behaviour of the beam, details of which are presented in section

4.4.1.2. Following bending failure, the beam tends to develop catenary action when

the axial compressive force begins to reverse to tension. The activation and

resistance of catenary action are highly affected by the rotational restraint stiffness,

but only for kr<0.064, indicated in Figure 4.5. The following two sections will

explain the effects of rotational restraint on beam bending resistance limit and beam

ultimate failure times.

Figure 4.5: Effect of rotational stiffness level on beam fire resistance (ka=0.166)

4.4.1.2 Effects of Rotational Restraint on Bending Resistance Limit Time

The results in Figure 4.5 show three phases of bending resistance time change with

increasing rotational stiffness: increasing initially, but then decreasing afterwards

before increasing slightly again. This trend reflects the three different possible modes

of bending failure of the beam: (I) beam bending failure governed by sagging

bending resistance at mid-span, (II) by combined fracture of top reinforcement at

ends and bottom reinforcement at mid-span, and (III) by concrete crushing at ends.

0

100

200

300

400

500

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3

Fire

res

ista

nce

(m

in)

Rotational stiffness ratio, (kr)

Bending resistance

Ultimate resistance

Resistance from catenary action



116

For beams with low rotational restraint levels, namely when kr ≤ 0.045, the sagging

bending moment reaches its limit value in the beam mid-span with no plastic hinge

formation at the beam ends, as shown by the typical example in Figure 4.6(a). When

the sagging bending capacity is attained, as indicated by first phase of rapid beam

deflection increase in Figure 4.3(a) and rapid reduction in beam compressive force in

Figure 4.3(b), the reinforcement strain increases rapidly to the fracture limit, as

shown in Figure 4.4. This causes the beam to temporarily lose static equilibrium, as

indicated by the jump in kinetic energy of the system shown in Figure 4.7. From here

onwards, the beam behaves as an axially and rotationally restrained one with a pin-

like joint at the mid-span.

Figure 4.6: Deformed shapes and failure modes (kr=0.022, ka=0.166)

Mid-span bending failure time=123 min

Beam bending resistance time=234 min

Total fire survival time=234 min

(b) At t = 234 min (final failure)

(a) At t = 123 min



117

Figure 4.7: Kinetic energy and beam axial force versus fire exposure time

(kr=0.022 and 0.032, ka=0.166)

With kr equal and lower than the first critical value of about 0.022, after the beam

mid-span has reached the sagging moment resistance, the hogging bending moment

resistances at the beam ends are small and cannot resist the released and redistributed

sagging moments at mid-span. This is confirmed by the small strains of the

longitudinal reinforcing bars at the beam ends, shown in Figure 4.8, when and after

bending failure of the mid-span section has occurred. Consequently, the beam has to

activate catenary action at large deflections to maintain equilibrium. This trend

continues until when the end rotational stiffness is increased to the critical value.

Because the bending resistance of the beam is controlled by the sagging moment

reaching the sagging moment resistance at mid-span, increasing the end rotational

stiffness increases the beam bending resistance failure time because it decreases the

sagging moment at mid-span. This bending failure mode is referred to mode (I)

mentioned in the previous paragraph. The bending failure limit is reached under

combined compression force (genrated due to restrained thermal expansion) and

sagging bending moment in the beam span.

-500

-400

-300

-200

-100

0

100

200

300

400

-500

-400

-300

-200

-100

0

100

200

300

400

0 30 60 90 120 150 180 210 240 270

Kin

etic

en

ergy

(J)

Axi

al f

orc

e (k

N)

Time (min)

kr=0.022, Axial force kr=0.032, Axial force

kr=0.022, Kinetic energy kr=0.032, Kinetic energy



118

(a) Top corner bars (bars 3)

(b) Top middle bars (bar 4)

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 kr=0.013 kr=0.022kr=0.032 kr=0.045 kr=0.064kr=0.125 kr=0.58 kr=1.85

Ends

Limiting strain=5%

-750

-600

-450

-300

-150

0

150

300

450

600

-0.08

-0.06

-0.05

-0.03

-0.02

0.00

0.02

0.03

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Def

lect

ion

(mm

)

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 (S) kr=0.013 (S) kr=0.022 (S)

kr=0.032 (S) kr=0.045 (S) kr=0.064 (S)

kr=0.125 (S) kr=0.58 (S) kr=1.85 (S)

kr=0.0064 (D) kr=0.013 (D) kr=0.022 (D)

kr=0.032 (D) kr=0.045 (D) kr=0.064 (D)

kr=0.125 (D) kr=0.58 (D) kr=1.85 (D)

Ends

Limiting strain=5%

S: Strain D: Deflection



119

(c) Bottom corner bars (bars 1)


bars at beam ends (ka=0.166)

When the end rotational stiffness is further increased from kr=0.022 (the first critical

kr) to kr=0.045 (the second critical kr), after the bottom bars at mid-span have

fractured on reaching the sagging moment resistance, the beam ends can still develop

sufficient hogging moment to continue to enable the beam to achieve equilibrium in

bending, as shown in Figure 4.3. This is because the applied bending moment

(LR=35%) is low, the top reinforcement at the ends are at low temperature and thus

retaining most of its ambient temperature resistance and the concrete has not crushed

at high temperatures.

When the mid-span has fractured, the beam deflection rapidly increases and this

causes the compressive force in the beam to decrease. The lower the end rotational

stiffness, the higher the beam deflection and therefore more of the compressive force

in the beam is released (Figure 4.3(b)). A compressive force in the beam (P-Δ effect)

is detrimental to the beam’s bending resistance. Therefore, the higher the end

rotational stiffness, the lower the period of time from mid-span sagging failure to

bending failure. The P-Δ effect is quite significant, so overall, increasing the end

rotational stiffness results in a reduction in the beam bending failure time.

Bending failure at the beam ends for 0.022 ≤ kr ≤ 0.045 is governed by fracture of the

top tensile reinforcement at ends. A typical example of this failure mode is shown in

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0 50 100 150 200 250 300 350 400 450

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 kr=0.013 kr=0.022

kr=0.032 kr=0.045 kr=0.064

kr=0.125 kr=0.58 kr=1.85

Ends



120

Figure 4.6(b) for the case kr=0.022. Strains of the beam end tensile reinforcement

bars, plotted in Figure 4.8 with time, confirm coincidence of beam bending failure

time with tensile strain reaching the fracture value. This failure mode is typical of

tension-controlled failure, which generates when stresses of the top bars reach their

yield strength before compressive failure of the concrete in the compression region.

This is referred to as bending failure mode (II). kr = 0.045 marks the end of this

failure mode in this case. At this level of stiffness, fracture of the top reinforcement

at the beam ends occurs at about the same time as bending failure of the beam mid-

span. The bending limit is governed by beam failure at ends due to combined total

bending moment in the beam and a small amount of residual compression force.

When the end rotational stiffness of the beam increases further (kr ≥ 0.064), the

hogging moments at the beam ends are higher than the mid-span bending moment

relative to their resistances and bending failure of the beam is governed by

compressive failure of concrete in the hogging moment region, as displayed in Figure

4.9(a). Concrete crushing is confirmed by observing that the strains of the bottom

layers of concrete exposed to fire, plotted in Figure 4.10, have exceeded the strains

corresponding to peak and zero compression stresses according to the concrete

stress-strain relationships at elevated temperature presented in Appendix A.4 in

Figure A4.7. However, once compressive failure of concrete occurs, the hogging

moments are released and the beam mid-span does not have sufficient sagging

moment resistance to achieve bending equilibrium. Even though increasing the beam

end rotational stiffness attracts more compressive stress in concrete at the beam ends,

this is offset by the reduced P-Δ effect because a higher beam end rotational stiffness

reduces the beam deflection. For such beams, before the end sections reach

compressive crushing of concrete on the fire exposed face, the sagging bending

moments has not reached its resistance, as indicated by the tensile strain being less

than the steel yield strain, shown in Figure 4.4.

After concrete crushing, the compressive reinforcement (bottom bars) at the same

location loses concrete restraint and buckles (Figure 4.11(a)), as shown and indicated

by rapid increase in compressive strain of the buckled bars in Figure 4.11(b).



121

Figure 4.9: Deformed shapes and failure modes (kr=0.064, ka=0.166)

Figure 4.10: Strain-fire exposure time for concrete near the beam ends

(kr=0.064, ka=0.166)

Beam bending resistance time=191 min

Total fire survival time=402 min

(a) At t = 190 min

(b) At t = 193 min

(c) At t = 402 min (final failure)

-600

-500

-400

-300

-200

-100

0

100

200

-0.24

-0.20

-0.16

-0.12

-0.08

-0.04

0.00

0.04

0.08

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Stra

in (

mm

/mm

) Time (min)

Strain, y=15 mm Strain, y=75 mm Strain, y=135 mm

Strain, y=195 mm Beam axial force

Section A-A




122

(a) Deformed reinforcement

(b) Strain-time relationships

Figure 4.11: Behaviour of reinforcing bars at beam ends (kr=0.064, ka=0.166)

Once concrete crushing near the beam ends happens, flexural action cannot be

sustained, as indicated by the sudden drop of axial compressive force in the beam to

zero in Figure 4.3(b). This is because the beam end moments and axial compressive

force are thereafter released so the crushed sections behave like pin joints and the

mid-span does not have sufficient bending resistance to maintain equilibrium of the

beam in bending at such high temperatures. This is demonstrated by the dynamic

behaviour shown in Figure 4.12 where the beam experiences a significant increase in

kinetic energy and a loss of the total vertical reaction force. As the mid-span is

unable to sustain any significant additional moments under the new bending load-

Bottom

bar, ends Stirrup, bottom, ends

Top bar, ends

Bottom bar, ends

Top bar, ends

-600

-500

-400

-300

-200

-100

0

100

200

300

-0.15

-0.125

-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Stra

in (

mm

/mm

)

Time (min)

Bottom bar, ends

Stirrup, bottom

Top bar, ends

Beam axial force

Limiting strain = 0.05



123

carrying configuration, bending failure at the mid-span region occurs immediately.

Catenary action has to be mobilised to reach static equilibrium of the beam. This

bending failure mode of the beam generated by compressive failure of the concrete at

the beam ends is referred to as “failure mode III” in the context of this research. For

this mode, the bending limit is governed by the end hogging bending moment, in

combination with a compression force due to restrained thermal expansion, reaching

the section capacity at the beam ends.

Figure 4.12: Vertical reaction force, applied load and kinetic energy/internal

energy against fire exposure time (kr=0.064, ka=0.166)

As a summary, the bending behaviour of axially restrained beams is complicated,

with the bending failure time being controlled by three failure modes. Due to the

compressive force induced by restrained thermal expansion, concrete crushing at the

beam ends prevents formation of a typical 3-plastic hinge mechanism in the beam.

This has important implications on calculating the bending limit time of axially and

rotationally restrained reinforced concrete beams. An analytical method will be

derived in Chapter 5.

4.4.1.3 Effects of End Rotational Restraint on Beam Ultimate Failure

Time

After completely releasing the compression force in the beam, the beam may

advance into catenary action stage when the axial force in the beam becomes tensile.

The transition into catenary action is accompanied by the strains of the top bars along

the beam length, whether they are initially in tension or compression in flexural

action, to become tension as shown in Figure 4.13. The duration of the catenary

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0

20

40

60

80

100

120

140

0 50 100 150 200 250 300 350 400

Kin

etic

en

ergy

/In

tern

al

ener

gy

Load

(kN

)

Time (min)

Vertical reaction

Applied load

KE/IE

KE: Kinetic energy IE: Internal energy



124

action stage may be considered to be the reserve in fire resistance of the beam that is

available to ensure robustness of the beam in fire. It is seen in Figure 4.5 that the

ultimate beam survival time deceases as kr increases and then slightly increases. The

rotational stiffness when the beam survival time reaches the minimum is kr=0.045,

which is the same as the second critical value of beam end rotational stiffness for the

bending resistance time. Different bending failure modes of the beam can be used to

explain the change in development of effective catenary action.

(a) Top corner bars (bars 3)

(b) Top middle bar (bar 4)

Figure 4.13: Strain-fire exposure time longitudinal reinforcing bars at beam

mid-span (ka=0.166)

For beams with kr < 0.022 (the first critical value of end rotational stiffness for beam

bending failure), when the beam mid-span fails in bending, the beam undergoes very

large deflections and enters into catenary action as the end bending moments are not

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 kr=0.013

kr=0.022 kr=0.032

kr=0.045 kr=0.064

kr=0.125 kr=0.58

kr=1.85Mid-span

Limiting strain=5%

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Stra

in (

mm

/mm

)

Time (min)

kr=0.0064 kr=0.013kr=0.022 kr=0.032kr=0.045 kr=0.064kr=0.125 kr=0.58kr=1.85Mid-span

Limiting strain=5%



125

adequate to achieve equilibrium in bending alone. Because the bottom tensile

reinforcement has fractured (at mid-span), the top reinforcement provides the

necessary tensile resistance for catenary action. The final survival time of the beam is

reached when the top reinforcement at the ends fractures, as shown by the strain plots

in Figure 4.8. The bending stresses are superimposed onto stresses due to catenary

force. Therefore, with increasing kr, the beam end hogging moments increase thus

reducing the catenary action duration. This is shown in Figure 4.5 by the decreasing

beam ultimate failure time with increasing end rotational stiffness.

For beams with 0.022 ≤ kr ≤ 0.045, after the mid-span bending moment has reached

its capacity, the induced end hogging moments balances the applied moment entirely

through bending action. This lasts until the top tensile reinforcement bars at the ends

of the beam fracture. Once this happens, the beam loses its load carrying capacity

completely and cannot enter catenary action because fracture occurs in both the top

(at ends) and bottom (at mid-span) reinforcement and there is no continuous tensile

catenary action path. This is further exhibited as the beam final failure time being the

same as the beam bending failure time in Figure 4.5.

For beams with kr ≥ 0.064, the beam ends experience concrete crushing in the

compression zone on the fire exposed side due to combined action of large hogging

bending moments and axial compressive force before the mid-span reaches its

bending resistance. Because concrete crushing is brittle, its compressive stress

decreases rapidly after reaching the peak. Therefore, the release of compressive force

at the beam ends is sharp and is accompanied by rapid increases in vertical

deflections of the beam, which activates catenary action in the beam. At this stage,

the top reinforcement has low tensile strains, as shown in Figure 4.8. Hence the top

reinforcement is almost un-stressed throughout the entire beam length at the onset of

catenary action. Because the top reinforcement provides catenary action and

rotational restraint does not change axial elongation of the reinforcement after the

beam ends start to behave like pin joints, the final failure time of the beam is

independent of the rotational stiffness, as shown in Figure 4.5.

To summarise, it is possible for axially restrained RC beams to develop catenary

action. Catenary action can only develop at either very low (kr ≤ 0.022) or quite high

(kr > 0.064) levels of end rotational stiffness when there is a continuous path for



126

developing tensile stress in the reinforcement. An end rotational stiffness level of kr ≤

0.022 to the beam is very small. For example, the elastic modulus E of the supporting

columns (with cross-section size 300x300mm and height 3.5m) that produces

kr=0.022 was as low as 218 MPa. According to EN 1992 1-2 (CEN, 2004), the

temperature at which the elastic modulus of concrete degrades from 18000 MPa at

ambient-temperature to 218 MPa is about 850oC. And this temperature has to be

reached everywhere in the column. This is unlikely to occur in practice. Furthermore,

at this low level of rotational stiffness, the catenary action resistance is very sensitive

to the rotational stiffness. Therefore, in analytical modelling of restrained beam

behaviour to be presented in Chapter 5, and it is to be recommended in practice, that

the contribution of catenary action to fire resistance of beams with low kr should be

disregarded. Therefore, if the benefit of catenary action is to be exploited, this should

only be considered for beams with high kr values. Fortunately, this level of rotational

restraint stiffness can be easily reached in most reinforced RC structures. Finally,

because the bottom reinforcement at mid-span will always fracture, only the top

reinforcement can be utilised to provide tension during the catenary action stage. In

Chapter 5, which presents development of an analytical model, the catenary action

stage will only be considered for beams with high values of rotational stiffness.

4.4.2 Effects of Changing Axial Restraint Stiffness

The above study was carried out for the axial restraint level of ka=0.166. Numerical

simulations were carried out for two other levels of axial restraint stiffness

(ka=0.0275 and 0.4). A selection of the results are presented in Figures 4.14 to 4.17,

showing axial force-time relationships (Figures 4.14 and 4.15) and variations of

bending and final failure times of the beams against rotational restraint stiffness

(Figures 4.16 and 4.17). Comparison with the results in Figure 4.3(b) and Figure 4.5

for ka=0.166, it is clear that the general trends for different levels of axial restraint are

identical.



127


(ka=0.0275)


(ka=0.4)


(ka=0.0275)

-300

-200

-100

0

100

200

0 50 100 150 200 250 300 350 400 450 500

Axi

al f

orc

e (k

N)

Time (min)

kr=0.0064 kr=0.013 kr=0.022 kr=0.032

kr=0.045 kr=0.064 kr=0.125 kr=1.85

-900

-750

-600

-450

-300

-150

0

150

300

0 50 100 150 200 250 300 350 400 450 500

Axi

al f

orc

e (k

N)

Time (min)

kr=0.0064 kr=0.013

kr=0.022 kr=0.032

kr=0.045 kr=0.064

kr=0.125 kr=1.85

0

100

200

300

400

500

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3

Fire

res

ista

nce

(m

in)


Bending resistance

Ultimate resistance




128


(ka=0.4)

4.4.3 Effects of Changing Beam Load Ratio

When the applied load (load ratio) is increased, large bending moments develop at

the beam ends; therefore, it becomes more difficult for the beam ends to resist the

applied load through bending action when the mid-span has reached its bending

capacity first. Therefore, mode II bending failure gradually disappears. The results in

Figure 4.18 show similar general trends of beam behaviour as for a lower load ratio

of 0.35, but the results in Figure 4.19 indicate missing of mode II bending failure.

Figure 4.18: Comparison of beam mid-span deflections-fire exposure time

relationships (ka=0.166, LR=50%)

-600

-500

-400

-300

-200

-100

0

100

200

300

0 40 80 120 160 200 240 280

Axi

al f

orc

e (k

N)

Time (min)

kr=0.013kr=0.022kr=0.035kr=0.045kr=0.064kr=0.125kr=0.365kr=1.85

050

100150200250300350400450500

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3

Fire

res

ista

nce

(m

in)


Bending resistance

Ultimate resistance




129


(ka=0.166, LR=50%)

4.4.4 Axially Unrestrained Beams

Although RC beams in realistic structures have axial restraint, the current fire

resistance design of RC beams assumes that there is no axial restraint and that at

failure, an RC beam can form a plastic hinge mechanism, as at ambient temperature.

Additional simulations have been carried out to examine whether the plastic hinge

mechanism is valid in fire. Figure 4.20 shows the variation of axially unrestrained

beam bending resistance time with increasing rotational stiffness level. From this

figure, it can be seen that the beam bending failure time initially increases with

increasing rotational stiffness kr from 0 to the first critical value of about 0.022, but

then remains almost constant with slight reduction at higher rotational stiffness.

Figure 4.20: Effect of rotational stiffness level on beam bending resistance

(ka=0)

0

50

100

150

200

250

300

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Ben

din

g re

sist

ance

tim

e (m

in)


0

50

100

150

200

250

300

0 0.05 0.1 0.15 0.2 0.25 0.3

Fire

res

ista

nce

(m

in)


Bending resistance

Ultimate resistance




130

The beam bending resistance failure time –rotational stiffness relationship in Figure

4.20 can be divided into three regions of rotational stiffness: kr<0.022,

0.022≤kr≤0.064 and kr>0.064. Figure 4.21 and Figure 4.22 plot representative mid-

span bottom reinforcement and support top reinforcement strains with increasing

time. Figures 4.23 and 4.24 show the failure modes for these three regions of

rotational stiffness.


bars at beam mid-aspan (ka=0)

Figure 4.22: Strains of longitudinal reinforcing bars at ends against fire

exposure time (ka=0)

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300

Stra

in (

mm

/mm

)

Time (min)

kr=0.013 kr=0.022 kr=0.045 kr=0.125 kr=0.58

Limiting strain=5%

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300

Stra

in (

mm

/mm

)

Time (min)

kr=0.013 kr=0.022 kr=0.045 kr=0.125 kr=0.58

Limiting strain=5%



131

Figure 4.23: Sagging bending failure at mid-span (kr=0.013, ka=0)

(a) kr=0.045, ka=0 (Hinge mechanism )

(b) kr=0.125, ka=0

Figure 4.24: Hogging bending failure at supports

At kr<0.022, beam bending failure is governed by sagging moment failure (failure

mode I) which causes fracture of the tensile reinforcement at mid-span. Increasing

the end rotational stiffness reduces the mid-span bending moment and hence

increases the beam bending failure time. However, it is not possible for the beam to

form a hinge mechanism. This is because forming a hinge mechanism would require

the supports to undergo large rotations in order to develop the necessary bending

moment. However, before this occurs, the bottom reinforcement at mid-span have

already fractured. This is shown in Figure 4.21 by the mid-span tensile reinforcement

reaching the fracture strain of 0.05 while the support tensile reinforcement strain

being low in Figure 4.22 for kr≤0.013. The failure mode for the beam of this

rotational stiffness in Figure 4.23 shows failure at the beam mid-span only.



132

At 0.022≤kr≤0.064, the mid-span reaches its sagging moment resistance first.

However, because the rotational stiffness is high, the end hogging moment reaches a

high level. Further deformation of the beam after mid-span bending failure allows the

supports to develop hogging moment. The beam failed by bending failure mode (II)

governed by fracture of the mid-span and support reinforcement, as shown in Figure

4.21 and 4.22 by the reinforcement reaching the fracture strain of 0.05. This is

further confirmed by the failure mode, formation of a hinge mechanism, shown in

Figure 4.24(a). Because a complete hinge mechanism is formed, the beam bending

resistance time is independent of the support rotational stiffness.

At high rotational stiffness (kr>0.064), the support region fails due to reaching the

beam the hogging moment resistance. Except of kr=0.125 where yielding of the mid-

span tensile reinforcement and hogging moment failure coincide, the beam cannot

develop a plastic hinge mechanism. This is because the sagging moment at the beam

mid-span is low. For the beam sagging moment to develop its sagging moment

resistance would require the beam to undergo further large deflections. However,

tensile reinforcement at the beam support fractures before this is reached. This is

confirmed by the mid-span tensile reinforcement strain being less than the yield

strain of 0.02 at the time of fracture of the tensile reinforcement at supports, for

kr>0.125 in Figure 4.21. Figure 4.24(b) confirms this showing support failure only.

The higher the support rotational stiffness, the higher the support hogging moment,

hence the lower the bending failure time as shown in Figure 4.20.

In summary, even without axial restraint, a reinforced concrete beam with rotational

restraints at ends may not behave as expected at ambient temperature by forming a

three hinge mechanism at failure. At high support rotational stiffness, a plastic hinge

mechanism does not happen because the support tensile reinforcement fractures

before the mid-span sagging moment can develop its sagging moment capacity.

Nevertheless, the reduction in fire resistance time from a hinge mechanism is small.



133

4.5 Beams with Asymmetrical End Boundary Restraints

The current parametric study was extended to investigate the behaviour of RC beams

with asymmetrical boundary restraints. The same beam-column sub-frame shown in

Figure 4.1 was employed. However, the whole structure, shown in Figure 4.25, was

modelled because of asymmetry of the boundary conditions. To simulate

asymmetrical restraints at the left and right ends of the structure, both the axial and

rotational restraints at the two ends (ka,R & kr,R and ka,L & kr,L) have different values. It

is assumed that the left end is more flexible (simulating connection to the edge of the

structure) than the right end (simulating connection to an internal bay of the

structure).

Dimensions of the sub-frame, beam reinforcement details, material properties and the

applied load were identical to those of the frame for symmetrical boundary

conditions.

Table 4.1 lists the parametric study cases, which were divided into three groups (G1,

G2 and G3). In all cases, the axial restraint stiffness level at the right hand is kept at

ka=0.166.

Figure 4.25: Definition of asymmetrical boundary restraints of beam-column

sub-frame model

KA,L

KR,L KR,R

KA,R

L R

ka,L

kr,L

ka,R = 0.166

(constant) kr,R



134

Table 4.1: Parametric study cases and summary of results of beams with

asymmetrical boundary restraints

Group Beam ka,R kr,R ka,L kr,L

Bending

resistance

time (min)

Ultimate

failure

time

(min)

G1

S1 0.166 2 0 0.0075 101 101

S2 0.166 2 0 0.025 133 133

S3 0.166 2 0 0.05 161 161

S4 0.166 2 0 0.1 203 203

S5 0.166 2 0 0.25 227 227

S6 0.166 2 0 0.5 248 248

S7 0.166 2 0 2 255 255

G2

S8 0.166 0.0075 0 0.0075 114 114

S9 0.166 0.025 0 0.0075 149 149

S10 0.166 0.05 0 0.0075 176 176

S11 0.166 0.1 0 0.0075 167 167

S12 0.166 0.25 0 0.0075 147 147

S13 0.166 0.5 0 0.0075 133 133

S14 0.166 1 0 0.0075 104 104

S1 0.166 2 0 0.0075 101 101

G3

S1 0.166 2 0 0.0075 101 101

S15 0.166 2 0.005 0.0075 121 121

S16 0.166 2 0.01 0.0075 142 142

S17 0.166 2 0.02 0.0075 157 157

S18 0.166 2 0.04 0.0075 159 424

S19 0.166 2 0.08 0.0075 150 413

S20 0.166 2 0.166 0.0075 141 427

4.5.1 Results and Discussions

Figures 4.26 to 4.28 present the trends of beam bending and ultimate failure times as

a function of the variable support axial or rotational stiffness for: (i) Group G1:

variable rotational stiffness ratio at the left support kr,L, (ii) Group G2: variable

rotational stiffness ratio at the right support kr,R and (iii) Group G3: variable axial

stiffness ratio at the left support ka,L. Figures 4.26 and 4.27 show that the bending and

ultimate resistance times coincide. This indicates that catenary action cannot develop

in these beams. This is because these beams are axially not restrained at the left

support.



135

Figure 4.26: Effect of kr,L on beam bending and ultimate resistance times (kr,R=2,

ka,R=0.166, ka,L=0)

Figure 4.27: Effect of kr,R on beam bending and ultimate resistance time

(ka,R=0.166, kr,L=0.0075, ka,L=0)

Figure 4.28: Effect of ka,L on beam bending and ultimate resistance time (kr,R=2,

ka,R=0.166, kr,L=0.0075)

0

50

100

150

200

250

300

0 0.25 0.5 0.75 1 1.25 1.5

Fire

res

ista

nce

(m

in)

Rotational stiffness ratio, (kr,L)

Bending resistance

Ultimate resistance

0

50

100

150

200

0 0.25 0.5 0.75 1 1.25 1.5

Fire

res

ista

nce

(m

in)

Rotational stiffness ratio, (kr,R)

Bending resistance

Ultimate resistance

050

100150200250300350400450

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Fire

res

ista

nce

(m

in)

Axial stiffness ratio, (ka,L)

Bending resistanceUltimate resistance



136

The trends of beam bending failure time varying with stiffness in Figures 4.26 to

4.28 reflect changes of the beam bending failure mode. When the rotational stiffness

at both end supports is low, the bending failure of the beam is controlled by span

sagging moment failure, as shown in Figure 4.29. Hence the beam bending failure

time increases with increasing support rotation stiffness.

For beams with higher support rotational stiffness, the beam bending failure is

controlled by hogging moment failure at the support with higher rotational stiffness

which attracts a bigger hogging bending moment. Hogging bending failure at support

is controlled by either fracture of the tensile reinforcement (Figure 4.30(a)) or by

concrete crushing in compression (Figure 4.30(b)).

Figure 4.29: Sagging moment failure mode (bending failure mode I) (kr,R=0.025,

ka,R=0.166, kr,L=0.0075, ka,L=0)

(a) Support failure mode due to tensile reinforcemenr fracture (kr,R=2, ka,R=0.166,

kr,L=0.0075, ka,L=0)

(b) Support failure due to concrete crushing (kr,R=2, ka,R=0.166, kr,L=0.0075,

ka,L=0.08)

Figure 4.30: Beam bending failure modes



137

It should be pointed out that because of nonlinear temperature distribution in the

beam cross-section, self-equilibrating stresses are induced in the cross-section. This

induces tension in the cooler middle bars at the top of the cross-section. To

demonstrate this, Figure 4.31 shows variations of strains of the top reinforcing bars at

the right support. It can be seen that the strain of the cooler middle bar (bar 4) is

always higher than those of the hotter corner bars (bars 3) for all investigated kr,L

values. This tensile thermal stress is additive to the tensile stress caused by hogging

moment at the beam ends. Premature facture of the top middle bar could occur before

the section attains its reduced bending resistance at elevated temperatures. This effect

is more pronounced in beams with no or small axial restraint and when the hogging

moment at one end of the beam is large and at the other end is small. However, the

effect of cooler middle bar decreases when the lower rotational stiffness ( kr,L)

increases. This is because the governing hogging moment at the support with higher

rotational stiffness (right support) decreases, thus prolonging the heating time

thereby the temperature difference is reduced. Therefore, in symmetrical rotationally

restrained beams, the effects of thermal stress due to non-uniform heating can be

ignored.

Figure 4.31: Strain-fire exposure time relationship of top longitudinal

reinforcing bars at right support (ka,R=0.166, kr,R=2, ka,L=0)

0

0.01

0.02

0.03

0.04

0.05

0.06

0 40 80 120 160 200 240 280

Stra

in (

mm

/mm

)

Time (min)

kr,L=0.0075 , bar3 kr,L=0.05 , bar 3 kr,L=0.5 , bar 3

kr,L=2 , bar 3 kr,L=0.0075 , bar 4 kr,L=0.05 , bar 4

kr,L=0.5 , bar 4 kr,L=2 , bar 4

Limiting strain=5%

Right end



138

The possible adverse effects of self-equilibrating strains in the top reinforcement bars

at beam ends on beam bending resistance can be minimised by locating the top

tensile reinforcing far from the mid-width of the beam to the corners. To illustrate

this, simulations with three typical kr,L values of 0.0075, 0.05 and 0.5 were re-run but

with a different top reinforcement arrangement. The three top bars (detail 1) were

replaced with two corner bars (detail 2) as shown in Figure 4.32, but keeping the

total reinforcement area the same. Table 4.2 summaries the results for the two cases

and Figure 4.33 plots the maximum strain in the top bars at the right end of the beam.

Although the hogging bending moment capacity of detail 2 is less than that of detail

1 at elevated temperatures because all detail 2 bars have higher temperatures,

because there was no thermal stress in the top bars in detail 2, the beam bending

resistance for detail 2 is higher. As the lower rotational stiffness (left end) increases,

the thermal stress effect is reduced. Therefore, the bending resistance time for the

case kr,L=0.5 is lower for detail 2 reinforcement arrangement than for detail 1 because

the cross-section resistance of detail 2 reinforcement arrangement is lower as

mentioned above.

(a) Detail 1 (a) Detail 2

Figure 4.32: Reinforcement details 1 and 2



139

Table 4.2: Failure time results of beams with reinforcement details 1 and 2

Detail * Beam ka,R kr,R ka,L kr,L

Bending

resistance

time (min)

Ultimate

failure

time

(min)

1 S1 0.166 2 0 0.0075 101 101

2 S21 0.166 2 0 0.0075 155 155

1 S3 0.166 2 0 0.05 161 161

2 S22 0.166 2 0 0.05 187 187

1 S6 0.166 2 0 0.5 248 248

2 S23 0.166 2 0 0.5 224 224

* Details 1 and 2 are as shown in Figure 4.32


reinforcing bars at right support (ka,R=0.166, kr,R=2, ka,L=0)

For group G1 beams, the rotational stiffness at one end (right end support) is

maintained at a higher value while the rotational stiffness at the other end (left end

support) varies. The end with higher stiffness rotational stiffness controls beam

failure. Therefore, when increasing the variable (left end) support rotational stiffness

kr,L, the hogging moment at the variable support is increased, leading to a reduction

in the hogging moment at the support with fixed rotational stiffness (right end). This

is shown by the tensile reinforcement strain at the right and left supports plotted in

Figure 4.34(a) for three typical levels of kr,L. Because the fixed end hogging moment

controls beam failure, the beam bending resistance time is prolonged with increasing

variable support rotational stiffness. Figure 4.26 shows that increasing kr,L results in a

0

0.01

0.02

0.03

0.04

0.05

0.06

0 40 80 120 160 200 240 280

Stra

in (

mm

/mm

)

Time (min)

kr,L=0.0075 (Bar 4, detail 1) kr,L=0.0075 (Bar 3, Detail 2)

kr,L=0.05 (Bar 4, detail 1) kr,L=0.05 (Bar 3, detail 2)

kr,L=0.5 (Bar 4, detail 1) kr,L=0.5 (Bar 3, detail 2)

Limiting strain=5%

3#6 As=855mm

2φ23.3mm As=855mm

Detail 1 Detail 2



140

considerable increase in the bending resistance time. Figure 4.34(b) demonstrates

that when tensile reinforcement strain at the right support reaches fracture (0.05), the

tensile reinforcement strains in the span have not reached yield (0.02).

(a) Top middle bar (bar 4)

(b) Bottom middle bar (bar 2) in the beam span


bars (ka,R=0.166, kr,R=2, ka,L=0)

In G2 group beams, the rotational stiffness at one support (left end, kr,L) is kept

constant at a very low level while the support stiffness at the other end (right end,

kr,R) is changed. For such boundary conditions, a hinge mechanism is considered to

develop by two plastic hinges, one at the right support and the other in the span.

When 0<kr,R<0.05, the beam bending failure is controlled by span bending moment

failure as shown in Figure 4.29. Because the sagging moment is reduced with

0

0.01

0.02

0.03

0.04

0.05

0.06

0 40 80 120 160 200 240 280

Stra

in (

mm

/mm

)

Time (min)

kr,L=0.0075 (right support) kr,L=0.0075 (left support)



Limiting strain=5%

0

0.01

0.02

0.03

0.04

0.05

0.06

0 40 80 120 160 200 240 280

Stra

in (

mm

/mm

)

Time (min)

kr,L=0.0075 kr,L=0.05 kr,L=0.5

Limiting strain=5%



141

increasing end rotational stiffness, an increase in kr,R results in higher bending failure

time. This trend is followed until the bending failure is attributed by formation of

plastic hinges at the right support and in the span when kr,R=0.05.

At kr,R>0.05, the variable end (right end) attracts high bending moment and therefore

governs beam bending failure. Therefore, increasing kr,R reduces the beam bending

failure time, as can be seen in Figure 4.27. The beam cannot develop a hinge

mechanism because with one support (left end) having a very low rotational stiffness,

the other support (right end) attracts a very high bending moment so the span sagging

moment is very low.

The bending failure time trend for group G3 beams is similar to that of group G1

beams: reaching the hogging moment resistance at the right support. However, the

presence of axial restraint in the beam introduces an axial compression force in the

beam. Therefore, the beam failure mode changes from reaching hogging moment

capacity due to tensile reinforcement fracture at the right support to concrete

compression failure at the right support (Figure 4.30(b)). This is shown by the

concrete strain plotted in Figure 4.35, which indicates that they have exceeded the

strains corresponding to peak and zero compression stresses according to the

concrete stress-strain relationships at elevated temperature presented in Appendix 4

in Figure A4.7. Increasing the axial restraint stiffness within its small amounts up to

ka,L=0.04 is beneficial to reduce the strain of the top tensile bars due to axial-flexural

interaction (Figure 4.36), thus decreasing the effect of self-straining strains on the

cooler middle bar and improving bending resistance of the beam. Increasing the axial

restraint stiffness further increases the compression force in the beam and thus

accelerates concrete crushing at ends, hence causing the beam to suffer a reduction in

bending failure time.

Only group G3 beams can develop catenary action because only these beams are

axially restrained.

In conclusion, for special cases when rotational stiffness at one end of the beam is

very high while at the other end is low and also the axial restraint is low, there is a

risk of premature hogging moment failure at the support having high rotational

stiffness. Tensile thermal stress in the cooler mid-bar at beam ends may cause



142

premature rupture of middle bars. This effect may be reduced by positioning the top

bars close to the corners of the beam in the hogging moment regions.

Figure 4.35: Strain-fire exposure time for concrete near the beam right end

(ka,R=0.166, kr,R=2, ka,L=0.08, kr,L=0.0075)

Figure 4.36: Strain of longitudinal reinforcing bars at right end and beam axial

force against fire exposure time with different ka,L values (ka,R=0.166, kr,R=2,

kr,L=0.0075)

-525

-450

-375

-300

-225

-150

-75

0

75

150

225

300

-0.350

-0.300

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0.200

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Stra

in (

mm

/mm

) Time (min)

Strain, y=15 mmStrain, y=75 mmStrain, y=135 mmBeam axial force

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Stra

in (

mm

/mm

)

Time (min)

ka,L=0 (S) ka,L=0.01 (S) ka,L=0.04 (S) ka,L=0.166 (S)

ka,L=0 (AF) ka,L=0.01 (AF) ka,L=0.04 (AF) ka,L=0.166 (AF)

Limiting strain=5%

Right end (Strain) S: Strain

AF: Axial force



143

4.6 Conclusions

This chapter has presented the numerical results, using ABAQUS, of a series of

analyses of RC beams in fire with different linear axial and rotational restraint levels,

with either symmetrical or asymmetrical restraint conditions, considering the large

deformation phase of structural response. Particular attention has been paid to the

bending failure modes generated under different boundary restraints and their effects

on beam bending resistance time and development of catenary action. The results of

this study indicate complicated beam bending behaviour and inability of the beam to

form the classical hinge mechanism. The following observations and conclusions can

be drawn from this numerical parametric study.

Beams with symmetrical restraint conditions:

As the rotational restraint stiffness at the beam ends increases, the beam may

experience three bending failure modes. At low level of rotational stiffness

(bending failure modes I and II), the beam bending failure occurs at mid-span

first, causing the bottom reinforcement at mid-span to fracture. After bending

failure of the mid-span, the applied load is resisted by the beam tensile

catenary force and/or end bending moments.

At very low level of rotational stiffness (bending failure modes I), because the

beam sagging moment at mid-span decreases at increasing rotational stiffness,

the bending failure time increases at increasing rotational stiffness. Since

catenary action is provided by tension in the top reinforcement and the top

reinforcement has not fractured, catenary action can develop in the beam.

Catenary action fully dominates the load resistance at zero end rotational

stiffness. With increasing rotational stiffness, the catenary action contribution

to the beam load resistance reduces, and meanwhile flexural action

contribution increases. However, the developed end bending moment due

flexural action contribution increases the tensile stress in the steel

reinforcement, causing earlier fracture of the top bars at the beam ends.

Hence, the beam final failure time decreases at increasing end rotational

stiffness. The above trend is followed until the applied load is entirely resisted

by the end bending moments.



144

Further increasing in beam end rotational stiffness changes the status of

internal force in the beam from pure bending to axial compressive force-

bending. After bending failure of the mid-span and release of the axial

compressive force, because of higher end bending moments in beams with

higher rotational stiffness, equilibrium in bending is regained at a smaller

increase in beam defection. Therefore, the axial compressive force is not

totally released when equilibrium is achieved. The higher the rotational

stiffness, hence the lower the beam deflection, the higher the residual

compressive force. The residual compressive force has an adverse influence

on the beam resisting time because of the second-order moment at the beam

ends due to P-Δ effects. Since bending failure of the beam is controlled by

reaching the hogging moment resistance at the beam ends, the beam bending

resistance time decreases at increasing end rotational stiffness. The release of

a significant amount of compressive force in the beam leads the end sections

to encounter tension-controlled failure and fracture of the tensile

reinforcement in bending (bending failure mode II). Because the bottom

reinforcement at the beam mid-span has already fractured, the beam does not

have a continuous path of tensile force to develop catenary action. Hence, the

beam bending failure time and final failure time coincide, both decreasing

with increasing end rotational stiffness.

As the rotation stiffness increases further, compressive failure of concrete

exposed to fire in the hogging moment region may occur first before the mid-

span reaches its sagging moment resistance. After concrete crushing, the beam

does not have sufficient mid-span bending resistance to maintain equilibrium

in bending and transits to catenary action immediately. Once this failure mode

governs the beam bending failure (bending failure mode III), increasing the

beam end rotational stiffness further reduces the beam deflection, and hence

the P-Δ effect, thus delaying concrete crushing at the beam ends and providing

the beam with a small increase in bending failure time. Because the top

reinforcement at end is at low strain when concrete crushing takes place

(compression-controlled failure) and the beam is relieved from end hogging

moments, the beam can develop the longest period of catenary action. The



145

beam behaves as one without rotational restraint, thus the beam final failure

time is unchanged with changing end rotational stiffness.

The aforementioned beam behaviour in bending and in catenary action holds

for different levels of axial restraint and different load ratios. The only change

is that mode (II) bending disappears when the load ratio is high. This is

because it is difficult for the beam ends to resist the high applied load after

failure of the beam mid-span.

The traditional approach for estimating fire resistance of beams in fire, which

ignores compressive force generated by restraint to thermal expansion and

hogging bending moment due to restrained thermal curvature, and assumes

that both the hogging and sagging bending moment resistances can be reached

together, may lead to overestimate of the bending resistance time of axially

restrained RC beams in fire. Crushing of concrete at the beam ends could

occur before the end hogging moments attains the bending resistance,

preventing the formation of a plastic hinge mechanism. A new design method

is needed to predict the fire resistance times of restrained RC beams under

bending. Such a method will be presented in Chapter 5.

For axially unrestrained beams, the beam bending failure is controlled by: (i)

sagging moment failure at low rotational stiffness, (ii) combined sagging and

hogging moment failures (three hinge mechanism) at medium rotational

stiffness and (iii) hogging moment failure before the span attains the sagging

moment resistance at higher rotational stiffness. Nevertheless, the reduction in

beam bending failure time under (iii) is small. Therefore, the three hinge

mechanism can still be applied.

If adequate axial restraints are provided, and typical reinforced concrete

structures are able to do so, tensile catenary action could reliably develop in

the beam at large deflections following bending failure. This may be used to

improve the beam survival time compared to the fire resistance under flexural

action. The applied load and tensile catenary forces are sustained by the top

reinforcing bars in catenary action. The beam in catenary action behaves as



146

one with zero restraint rotational stiffness. In Chapter 5, an analytical method

will be developed to calculate the beam final survival time in catenary action.

When the rotational restraint stiffness is low, there is no a reliable continuous

path to transmit the tensile force in tensile catenary action due to fracture of

the top reinforcement at beam ends and the bottom reinforcement at beam

mid-span. Catenary action would only develop when the rotational restraint

stiffness is very small (approaching zero). Also at low levels of end rotational

stiffness, the extent of catenary action development is very sensitive to the

beam end rotational stiffness. It would be better to disregard catenary action

for such low levels of beam end rotational restraint stiffness.

Beams with asymmetric restraints:

The flexural behaviour of beams with asymmetrical end restraints is more

complicated than that with symmetrical ones. When the supports have high

rotational stiffness, a hinge mechanism cannot happen in most cases because

the support with a higher rotational stiffness attracts a very high bending

moment and fails while the span bending moment is very low. There is no

time for the span moment to increase to its plastic moment capacity.

When rotational stiffness at one support is very high and at the other support is

low, the hogging moment resistance at the support with the higher rotational

stiffness can be adversely affected by tensile thermal stresses developed in the

cooler mid reinforcement bar at the top of beam in the hogging region. To

reduce this effect, top bars in the hogging moment region of a beam should be

located to the corners.

Similar to the beams with symmetrical boundary restraints, effective catenary

action develops following bending failure only if bending failure is initiated

by compressive failure of concrete in the hogging moment.

The trends observed in these beams will be used to explain the frame

behaviour to be presented in Chapter 6. However, due to constraint of time, no

simplified method has been developed for beams with asymmetrical restraint

conditions.

CHAPTER 5 DEVELOPMENT OF A SIMPLIFIED METHOD FOR ANALYSIS OF

AXIALLY AND ROTATIONALLY RESTRAINED REINFORCED CONCRETE BEAMS

IN FIRE

147

CHAPTER 5

DEVELOPMENT OF A SIMPLIFIED METHOD FOR

ANALYSIS OF AXIALLY AND ROTATIONALLY

RESTRAINED REINFORCED CONCRETE BEAMS IN

FIRE

5.1Introduction

In Chapter 3, a 3D dynamic explicit finite element model in ABAQUS was

developed to analyse the highly complex response of RC structures in fire. After

validation of this model, a comprehensive study of the behaviour of axially and

rotationally restrained RC beams in fire was carried out and the results have been

presented in Chapter 4. This numerical parametric study investigated influences of

different degrees of axial and rotational restraints on full history behaviour of RC

beams in fire until failure, including catenary action at large deflections. However, it

would be very difficult to use the sophisticated numerical model for design purpose.

For possible consideration of large deflection behaviour of axially and rotationally

restrained RC beams in fire in practice, a more simplified method would be

necessary. This is the aim of the current chapter. This method will make assumptions

based on observations of the numerical results in Chapter 4. The calculation

procedure is based on sectional analysis and uses simple concepts of compatibility

and equilibrium conditions. Validation of the simplified method comes from

comparison against the numerical modelling results of Chapter 4 and the additional

numerical results of the effect of beam parameters which are presented in this chapter

5.2 Key Features of Restrained Beam Behaviour

The numerical study results in Chapter 4 have shown that the response of axially and

rotationally restrained RC beams in fire is complex. Figure 5.1 illustrates

development of axial force versus fire exposure time of a typical beam. In the initial

stages of fire, an axial compressive force develops due to restrained thermal

expansion. The ultimate value of the compressive force marks formation of a plastic

hinge at a critical section (either in the span or at the ends), after which unloading of

the compressive force occurs until failure of the beam in bending is reached. If the



IN FIRE

148

beam is able to develop catenary action, featured by reversal of the axial force from

compression to tension, it could survive longer fire exposure beyond flexural

bending resistance.

Figure 5.1: Typical axial force-fire exposure time response of a restrained RC

beam in fire

The bending resistance time and ultimate resistance time are particularly important

quantities for design purpose. The numerical results in Chapter 4 have revealed the

bending resistance time/ultimate resistance time–relative rotational restraint stiffness

(kr) relationships as sketched in Figure 5.2(a). The typical beam axial force-fire

exposure time curves for different kr values are illustrated in Figure 5.2(b), in which

points tBR,A, tBR,B, tBR,C, tBR,D and tBR,E are bending resistance times and tUR,A, tUR,B, tUR,C,

tUR,D and tUR,E are ultimate resistance times at kr,A, kr,B, kr,C, kr,D and kr,E, respectively.

(a) Beam fire resistance–kr relationships

Ax

ial

Fo

rce

Tension

Compression

Time

Flexural action Catenary action

Ultimate

resistance time Bending

resistance time

Fir

e re

sist

ance

(ti

me)

Bending resistance time

Region 1 Region 3

Ultimate resistance time

Rotational stiffness ratio, kr

Region 2

kr,A

kr,B kr,C

kr,D

kr,E




IN FIRE

149

(b) Beam axial force–fire exposure time relationships for different kr values

Figure 5.2: Effects of rotational stiffness kr on RC beam behaviour in fire

The bending resistance time-rotational stiffness relationship in Figure 5.2(a) can be

divided into three regions according to rotational stiffness, namely 1, 2 and 3,

separated by two rotational stiffness kr,B and kr,D.

When kr is very low (kr=kr,A in region 1), the beam bending resistance is controlled

by the maximum sagging moment at mid-span reaching the sagging moment

resistance. Afterwards, the rotational restraint at ends cannot develop sufficient

hogging moment resistance to resist the applied load in bending. When kr = kr,B, the

rotational stiffness is still low enough to cause the mid-span to attain its sagging

moment resistance first. But after bending failure of the mid-span and release of the

axial compressive force, the beam ends can develop sufficient moments to continue

to enable the beam to achieve equilibrium in bending. Therefore, kr,B corresponds to

the rotational stiffness level when almost all of the axial force is released (F≈0). As

the rotational stiffness increases further, a smaller amount of the axial force is

released, as shown in Figure 5.2(b) for kr,C in region 2. The remaining unreleased

compressive force has adverse influence on the beam bending resistance time due to

P-Δ effects. The bending resistance of beams in region 2 with kr,B≤kr≤kr,D is reached

by fracture of tensile reinforcement at the beam ends.

tBR,E

tBR,C,tUR,C tBR,D,tUR,D

tUR,A tUR,E tBR,A tBR,B,tUR,B

Axia

l fo

rce,

F

Tension

Compression

Fire exposure time, t kr,A

kr,B

kr,C

kr,D

kr,E



IN FIRE

150

Further increase in kr (when kr>kr,D) causes compressive failure of the concrete

exposed to fire in the hogging moment region to occur first before the mid-span

reaches its sagging moment resistance. This requires determination of the lowest

rotational stiffness that leads the beam bending failure to be controlled by hogging

moment failure due to concrete crushing in compression. This critical rotational

stiffness in this analytical model is referred as kr,E (Figure 5.2(b)). After hogging

moment failure, the beam does not have sufficient mid-span bending resistance to

maintain equilibrium in bending. Hence, the bending resistance of beams that exhibit

this bending failure mode is governed by compressive failure of concrete at the beam

ends. This bending resistance could be lower than the resistance estimated based on

the traditional approach which assumes formation of a complete 3-plastic hinge

mechanism in the beam at failure.

Therefore, in order to develop a simplified method that may be adopted for design

purpose, it is necessary to quantify the critical rotational stiffness kr,B and kr,E, as well

as bending resistance time.

For catenary action development, when kr is very low (in region 1), tensile axial

force due to catenary action develops in the beam following bending failure of the

mid-span, as illustrates in Figure 5.2(b) for kr≤kr,A. The simulation results have

shown that the ultimate resistance time (catenary action resistance) is very sensitive

to kr for such low levels of beam end rotational restraint stiffness. Pure catenary

action would only develop when the rotational restraint stiffness is zero. When

kr,B≤kr≤kr,D, bending failure of the beam has a high potential to cause top bars at the

beam ends to fracture. Consequently, catenary action would not develop because

there will be no reliable continuous path to transmit the tensile catenary force in

catenary action. Therefore, within this region (regions 2), the ultimate resistance time

coincides with the bending resistance time. When kr≥kr,E in region 3, it was found

that the beam can develop effective catenary action. This is mainly because concrete

crushing occurs while the top bars at ends are at low strain and it releases the bars

from hogging bending stresses. Therefore, the contribution of catenary action to fire

resistance is only considered for kr≥kr,E.



IN FIRE

151

The aim of this chapter is to describe the methodology for developing a simplified

model to predict fire response of axially and rotationally restrained RC beams in fire.

The entire histories of beam mid-span deflection-fire exposure time and axial force-

fire exposure time in flexural action and catenary action will be evaluated under

different axial and rotational restraint levels and different key design parameters.

5.3 Representative Beam Model

The representative RC beam for simplified analysis is illustrated in Figure 5.3. The

isolated restrained beam is intended to represent a part of a reinforced concrete

frame. The beam has length L and is prevented from vertical movement at x=0 and

x=L. The restraints provided by the neighbouring members are accounted for by

equivalent axial and rotational springs respectively with stiffness of KA and KR at

both ends. Because the focus of this research is on developing a simplified method

for the RC beam, the stiffness values of the springs are assumed to be linear elastic

and constant throughout the fire exposure. Since an incremental approach is taken in

this development, in realistic situations, the changing stiffness of the restraining

structural members reflecting the effects of fire exposure on their mechanical

properties can be incorporated by using the reduced stiffness of the restraining

members at the start of each time increment. The simplified model can be used for

placing the axial restraints at any location at the beam ends with x=0 and x=L. The

end restraint stiffness is defined by non-dimensional relative axial and rotational

stiffness ratios ka and kr, respectively as:

𝑘𝑎 =𝐾𝐴

(𝐸𝑐𝑚𝐴 𝐿⁄ ) (5-1)

𝑘𝑟 =𝐾𝑅

(4𝐸𝑐𝑚𝐼 𝐿⁄ ) (5-2)

where Ecm, I, A, and L are ambient-temperature elastic modulus of concrete, moment

of inertia of the RC beam cross-section with respect to its geometric centre, cross-

sectional area and beam span, respectively.



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152

Figure 5.3: Representative restrained beam model

5.4 Development of a Simplified Model

The simplified model is one-dimensional (1D) based on sectional analysis. It has the

advantage of being very simple to use. Although the model requires iteration

procedures to predict beam deflection, axial force and fire resistance time, it does not

require solving a set of complex equations, not formation of matrices and element

nodal vectors at each time step as a 1D finite element model. In the analysis, the fire

exposure time is incremented in small time steps (t) and the fire response of the beam

is estimated at each time step. The beam is longitudinally divided into n segments, as

illustrated in Figure 5.4. For each segment i (i=1, 2,.., n), the behaviour at its mid-

length is taken as representative of the whole segment. Furthermore, the cross-

section of this mid-section is meshed into a number of rectangular elements (Figure

5.4) of uniform temperature, stress and strain and whose behaviour is represented at

the centre of the element.

Further assumptions are:

1. The beam is prevented from lateral torsional buckling.

2. There is perfect bond between steel reinforcing bars and concrete.

3. The effect of spalling is neglected.

4. The effect of shear deformation is not included.

5. The applied load is uniformly distributed and the boundary conditions are

symmetrical.

6. Flexural rigidity of the beam throughout its length is constant.

7. Force equilibrium of the beam is satisfied in a pointwise manner, not

throughout the beam length.

D

B

yo

w

Ka Kr Kr Ka

L

A B

x

z

y



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8. The effect of strain reversal of steel reinforcement from compression in

flexural bending action to tension in catenary action is ignored.

Figure 5.4: Beam discretization for simplified analysis

The analysis procedure is performed in two stages: thermal calculation and structural

analysis. However, if temperature information is available from alternative sources,

e.g. design charts of fire test results, the data can be used as input temperature field

for the segment cross-section.

This chapter will focus on structural analysis. The main assumption in the structural

analysis is the beam’s deflection profile. Afterwards, based on the assumption that

the beam flexural rigidity along the span is constant, the only unknown deflection of

the beam is the maximum deflection at mid-span. From the deflection profile, the

curvature 𝜑 in all segments is estimated. At the start of each time step, trial values of

the beam axial force and maximum deflection δmax are estimated. During each step,

an incremental-iterative process is carried out to determine the beam axial force and

maximum deflection to satisfy conditions of compatibility and force equilibrium.

5.4.1 Beam Deflection Profile

Under the combined action of bending moment and compressive axial force, the

beam deflection profile is assumed to be fourth order polynomial under a uniformly

A

A

Section A-A

Steel bar

area, As,m

Concrete element

area, Am

B

D

2 1 3

Li

L

w Beam segment

n



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154

distributed load w. The beam deflection profile is dependent on the end rotational

restraint stiffness. If the maximum deflection at the beam mid-span is δmax and

assuming that the flexural rigidity along the beam length is constant, for beams with

zero end rotational restraint (kr=0), the deflection profile is as assumed to be:

𝛿 =16𝛿𝑚𝑎𝑥

5𝐿(𝑥4

𝐿3−

2𝑥3

𝐿2+ 𝑥)

{

𝛿|𝑥=0

𝑥=𝐿= 0

𝛿|𝑥=𝐿/2 = 𝛿𝑚𝑎𝑥

𝜑|𝑥=0𝑥=𝐿

=𝑑2𝛿

𝑑𝑥2|𝑥=0𝑥=𝐿

= 0

(5-3)

For beams with complete end rotational restraint (kr=∞):

𝛿 =16𝛿𝑚𝑎𝑥

𝐿4𝑥2(𝐿 − 𝑥)2

{

𝛿|𝑥=0𝑥=𝐿

= 0


𝜃|𝑥=0𝑥=𝐿

=𝑑𝛿

𝑑𝑥|𝑥=0𝑥=𝐿

= 0

(5-4)

The deflection profile of beams with flexible end rotational restraint is approximated

by interpolation between that of the same beam with zero rotational restraint and that

of the beam with complete rotational restraint. Based on the ABAQUS simulation

results, an exponential function of the beam deflection curve in terms of the

rotational stiffness ratio kr will be used. The beam’s deflection profile equation is

therefore

δ =c × δz + (1-c) × δt (5-5)

where

δz is deflection profile of the beam with zero rotational restraint,

δt is deflection profile of the beam with total rotational restraint,

c is a factor dependent on the rotational stiffness ratio defined as:

𝑐 = 𝑒−14𝑘𝑟 for 0≤ kr ≤0.1

𝑐 = 0.36𝑒−4𝑘𝑟 for kr ≥ 0.1 (5-6)

In the catenary action stage, it has been shown in the previous chapter that it is

reasonable to assume that the beam behaves as one with zero rotational restraint at

the ends. Therefore, Equation 5-3 for beams with kr=0 needs to be modified. Based

on the numerical simulation results of Chapter 4, a better approximation of the

deflected shape was found to consist of equal proportions of the following three

deflection profiles: (i) the deflection profile of the beam under uniformly distributed



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155

load (UDL), (ii) the deflection profile of the beam under a concentrated point load

(CPL) at the mid-span, (iii) linear profile (LP). Deflection profiles (ii) and (iii) are:

𝛿𝐶𝑃𝐿 =𝛿𝑚𝑎𝑥

𝐿3(3𝐿2𝑥 − 4𝑥3)

{

𝛿|𝑥=0

𝑥=𝐿= 0



=𝑑2𝛿


= 0

(5-7)

𝛿𝐿𝑃 = {

2𝛿𝑚𝑎𝑥

𝐿𝑥 for 0 ≤ 𝑥 ≤ 𝐿/2

2𝛿𝑚𝑎𝑥

𝐿(𝐿 − 𝑥) for 𝐿/2 ≤ 𝑥 ≤ 𝐿

{

𝛿|𝑥=0

𝑥=𝐿= 0



=𝑑2𝛿


= 0

(5-8)

Thus, the assumed defection profile of the beam in catenary action is:

𝛿 =𝛿𝑈𝐷𝐿+𝛿𝐶𝑃𝐿+𝛿𝐿𝑃

3 (5-9)

To demonstrate suitability of the assumed deflection profiles, Figures 5.5 to 5.9

compare the ABAQUS simulation results of beam deflection profiles for combined

axial compression-bending behaviour, for different end rotational stiffness levels,

and Figures 5.10 and 5.11 for catenary action with the assumed deflection profiles.

The assumed deflection profiles were obtained by using the same maximum mid-

span deflections δmax as in the ABAQUS simulation results. These figures indicate

that the proposed deflection profiles are suitable at different fire exposure times.

Figure 5.5: Comparison of beam deflection profiles in flexural action

(kr=0.0064)

-250

-200

-150

-100

-50

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)

Deflection profile, t=0 ABAQUS, t=0Deflection profile, t=30min ABAQUS, t=30minDeflection profile, t=60min ABAQUS, t=60minDeflection profile, t=90min ABAQUS, t=90min



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Figure 5.6: Comparison of beam deflection profiles in flexural action (kr=0.032)



-300

-250

-200

-150

-100

-50

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)

Deflection profile, t=0 ABAQUS, t=0Deflection profile, t=30min ABAQUS, t=30minDeflection profile, t=60min ABAQUS, t=60minDeflection profile, t=90min ABAQUS, t=90minDeflection profile, t=120min ABAQUS, t=120min

-250

-200

-150

-100

-50

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)


-160

-140

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)




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Figure 5.9: Comparison of beam deflection profiles in flexural action (kr=2)

Figure 5.10: Comparison of beam deflection profiles in catenary action

(kr=0.0064)

Figure 5.11: Comparison of beam deflection profiles in catenary action (kr=2)

-120

-100

-80

-60

-40

-20

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)


-800

-700

-600

-500

-400

-300

-200

-100

0

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)

Deflection profile, t=200min ABAQUS, t=200minDeflection profile, t=300min ABAQUS, t=300minDeflection profile, t=400min ABAQUS, t=400min

-500-450-400-350-300-250-200-150-100

-500

0 1000 2000 3000 4000 5000 6000

Def

lect

ion

(mm

)

Length x (mm)

Deflection profile, t=300min ABAQUS, t=300min

Deflection profile, t=390min ABAQUS, t=390min



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158

5.4.2 Simplified Structural Analysis Procedure

The simplified structural analysis procedure at each time step is as follows:

1. The deflected shape of the beam is approximated by a deflection profile, as

described in section 5.4.1.

2. A mid-span deflection δmax is assumed.

3. From the assumed δmax, curvature in each segment 𝜑𝑖 and deflections (δ) at

the segment ends are calculated from the beam’s deflection profile.

4. The cross-section strains are calculated based on the plane section

assumption.

5. An axial force F is assumed.

6. At each segment I and based on the assumed fixed 𝜑𝑖, a reference strain εoi is

iterated until summation of the element forces in the cross-section (the

internal axial force (Fint,i)) is equal to the assumed F.

7. The reference strains εoi for all segments that fulfil Fint,i=F are then used to

check compatibility conditions as described in the following sections.

8. The assumed F is varied while the assumed δmax is kept uchanged and steps 6

an 7 are repeated for each new value of F until compatability conditions are

satisfied.

9. As soon as the compatability conditions are satisfied under the assumed δmax,

stresses in the concrete elements and steel bars at the beam mid-span and end

sections are used to check the equilibrium conditions described in the

following sections

10. Steps 2 to 9 are repeated until both compatability and equilibrium conditions

are satisfied.

11. The values of δmax and F that satisfy compatability and equilibrium

conditions are recorded as the beam mid-span deflection and axial force at

the specified time step

Based on plane section assumption, the total strain distribution in a beam section is

linear as shown in Figure 5.12. Based on the deflection profile, the total beam

curvature 𝜑𝑖 in each segment can be calculated using the following equation:

𝜑𝑖 =𝑑2𝛿

𝑑𝑥2|𝑥=𝑥𝑖 (5-10)



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159

The vertical coordinae of the reference strain εo is yo, and this should be the same as

that of the axial restraint at the beam end sections (see Figure 5.3). This is important

when checking the compatibility conditions explained later in section 5.4.3. Then,

the total strain εtot in each element of the beam cross-section can be calculated as:

휀𝑡𝑜𝑡 = 휀𝑜 + 𝜑(𝑦𝑜 − 𝑦) (5-11)

where coordinate y is measured from the top most fibre to the element centre, as

shown in Figure 5.12. It should be pointed out that the curvature is only dependent

on the maximum beam deflection δmax.

Figure 5.12: Distribution of total strain in a beam cross-section

The mechanical strain of the element can be calculated by subtracting the thermal

strain from the total strain:

휀𝑚𝑒𝑐 = 휀𝑡𝑜𝑡 − 휀𝑡ℎ (5-12)

where εth is the unrestrained thermal strain determined from the knowledge of

thermal expansion and temperature of concrete and steel reinforcement. Once the

mechanical strain of the element is obtained, its stress is calculated by using the

temperature-dependent stress-strain relationships for concrete and steel

reinforcement.

z

y 𝜑

-

휀𝑜

휀𝑡𝑜𝑡 휀𝑡𝑜𝑡

+

-

𝜑

휀𝑜

y yo yo

(a) Cross-section (b) Strain diagram

(𝜑 ≤ 0)

(c) Strain diagram

(𝜑 ≥ 0)

D

B

+ Tension

- Compression

+



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160

5.4.3 Compatibility and Equilibrium Conditions

With the aid of Equations 5-11 and 5-12, the stresses at the centre of all the concrete

elements and in all the reinforcing bars can be computed for any value of the

reference strain εo and maximum beam deflection δmax. The internal axial force acting

on a beam section can be calculated as stress resultant as:

𝐹𝑖𝑛𝑡 = ∑ 𝜎𝑐,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑚 +𝑘𝑖=1 ∑ 𝜎𝑠,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑠,𝑚

𝑞𝑖=1 (5-13)

where 𝜎𝑐,𝑚 and 𝜎𝑠,𝑚 are stresses in the concrete element and the steel bar,

respectively. 𝐴𝑚 is the area of the concrete element and 𝐴𝑠,𝑚 is the area of the steel

bar element.

To obtain the reference strains εoi, iteration is carried out until for each segement,

Fint,i = F is satisfied within a tolerance value of ± 1 kN.

After the total strain distribution within the mid-section of the segment has been

established (to produce Fint,i equal to the assumed value of F), the reference strains εoi

in the segments are utilised to check compatability conditions. The sum of the

differences between the horizontal projected length of the deformed line passing

through the reference points and the initial beam length should equal the total axial

displacemt between the entire beam ends (Dwaikat and Kodur, 2008), so that:

∑𝐿𝑖′ − 𝐿 = 2∆𝑎 (5-14)

where Δa is the axial dispacemnt at one end of the beam at the location of the axial

spring, 𝐿𝑖′ is the horizontal projected length of the deformed segment i, which can be

estimated from the the following compatability equation of segment i, according to

Figure 5.13 (Dwaikat and Kodur, 2008):

𝐿𝑖′ = √[𝐿𝑖(1 + 휀𝑜𝑖)]2−(𝛿𝑖,𝑎 − 𝛿𝑖,𝑏)2 (5-15)



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161

Figure 5.13: Condition of beam at different stages

where Li is the initial length of segment i. δi,a and δi,b are the vertical deflections at

the left and right ends of segment i, respectively, determined from the beam’s

deflecion profile. The total axial force F is related to the axial restraint stiffness and

the total beam axial deformation as follows:

F=KA. ∆ (5-16)

Substituding Equations 5-15 and 5-16 into Equation 5-14 yields the following

compatability equation:

1

2[∑√[𝐿𝑖(1 + 휀𝑜𝑖)]2−(𝛿𝑖,𝑎 − 𝛿𝑖,𝑏)2 − 𝐿]𝐾𝐴 − 𝐹 = 0 (5-17)

The compatibily condition in Equation 5-17 is checked by iteration by changing the

value of F until reaching a tolerance limit of ± 1 kN

The iterative process of adjusting the axial force is performed with a fixed mid-span

deflecion. To obtain the maximum mid-span deflection, the equilibrium condition for

the beam is invoked. Figure 5.14 shows the loading conditions on half of the

(a) Un-deformed beam

𝛿𝑚𝑎𝑥

𝛿𝑚𝑎𝑥

a

𝐿𝑖′

KA KR KA

L A B

∆

KA KR KR KA

A B

∆

∆

KA KR KR KA

A B

∆

KR

Li

𝐿𝑖′

𝛿𝑖,𝑏 𝛿𝑖,𝑎

b

𝛿𝑖,𝑏 𝛿𝑖,𝑎

F

F

F

F

(b) Deformed beam in flexural action

(c) Deformed beam in catenary action



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162

deformed beam. The equilibrium equation is checked only at the mid-span and end

ponits of the beam as:

Figure 5.14: Loading condition on deformed half beam

𝐹. 𝛿𝑚𝑎𝑥 +𝑀𝑚 +𝑀𝐴 −𝑀𝑒 = 0 (5-18)

where the bending moments are taken about the vertical coordinate of the axial

restraint (yo) at the beam end A as:

(𝐹. 𝛿𝑚𝑎𝑥 +𝑀𝑚)|𝑥=𝐿2

= ∑ 𝜎𝑐,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑚. (𝛿𝑚𝑎𝑥 + 𝑦 − 𝑦𝑜)𝑘𝑖=1 +

∑ 𝜎𝑠,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑠,𝑚. (𝛿𝑚𝑎𝑥 + 𝑦 − 𝑦𝑜)𝑞𝑖=1 (5-19)

𝑀𝐴|𝑥=0 = ∑ 𝜎𝑐,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑚. (𝑦 − 𝑦𝑜) +𝑘𝑖=1

∑ 𝜎𝑠,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑠,𝑚. (𝑦 − 𝑦𝑜)𝑞𝑖=1 (5-20)

The external bending moment is:

𝑀𝑒 =𝑤𝐿2

8 (5-21)

F is the axial force in the beam

MA is the internal bending moment at the support

Mm is the internal bending moment at mid-span

δmax is the maximum deflection at mid-span

Ideally, the end section bending moment MA should be in equilibrium with the

bending moment in the rotational restraint at the beam ends. This would require

precise calculation of the end rotations, which would not be possible using the

proposed deflection profiles. As simplification, the following process is

implemented:

With complete end rotational restraint, the whole MA calculated using Equation 5-20

should be used in the equilibrium Equation 5-18 because there is no end rotation;

L/2

Me

Mm

MA

𝛿𝑚𝑎𝑥

F

F

A

VA



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163

with no end rotational restraint, MA should be zero. For flexible rotational restraint,

the end bending moment is calculated as c.MA where the factor c is the same as

presented in Equation 5-6. Hence, equilibrium Equation 5-18 for flexible end

rotational restraint is rewritten as follows:

𝐹. 𝛿𝑚𝑎𝑥 +𝑀𝑚 + 𝑐.𝑀𝐴 −𝑀𝑒 = 0 (5-22)

The equilibrim condition in the simplified analysis is cocnsidered to be satisfied if

the difference between the internal and external bending moment is within a

tolerance of ± 0.02 Me. If the difference is grerater than this limit, the whole process

is repeated for a different δmax value.

The above solution procedure is programmed into Visual Basic integrated in Excel

spreadsheets.

5.5 Maximum Concrete Compressive Strain (εcmax,T)

Quantifying compressive failure of concrete under the combined action of bending

moment and axial compressive force is quite difficult due to non-uniform

temperature distribution in the beam cross-section. In literature, models predicting

the influence of temperature on the maximum concrete strain εcmax,T that causes the

concrete compressive zone to fail are very limited. El-Fitiany and Youssef (2014a)

conducted a parametric study on RC column members to evaluate εcmax,T

corresponding to the ultimate flexural capacity at high temperatures under different

axial compressive forces. It was suggested that reasonable results of εcmax,T, which

are higher than the strain value at peak compressive stress εc1,T , can be calculated

using the following expression:

휀𝑐𝑚𝑎𝑥,𝑇 = 𝑟(휀𝑐𝑢1,𝑇 − 휀𝑐1,𝑇) + 휀𝑐1,𝑇 (5-23)

where, r is a factor equal to 0.25, which locates εcmax,T in the softening branch of the

concrete stress-strain curve in compression and εcu1,T is the strain at zero stress

(complete concrete crushing). The same expression in Equation 5-23 is adopted in

this study to predict compressive failure of concrete at the beam ends. However, a

value of 0.2 is taken for the factor r instead of 0.25 because the values of εc1,T at

elevated temperatures used by El-Fitiany and Youssef (2014a) are lower than those



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164

recommended by EN 1992 1-2 (CEN, 2004) for temperatures above 500oC.

Therefore, Equation 5-23 is rewritten as:

휀𝑐𝑚𝑎𝑥,𝑇 = 0.2(휀𝑐𝑢1,𝑇 − 휀𝑐1,𝑇) + 휀𝑐1,𝑇 (5-24)

5.6 Beam Bending Failure Modes

In flexural action under combined compression and bending, if equilibrium equation

5-22 is not converged anymore while iterations are being carried out to find δmax and

F, it indicates bending failure. According to the different bending failure modes

described in section 5.2 which are based on the numerical simulation results in

Chapter 4, the following two bending failure scenarios should be checked:

1- Mid-span reaching its sagging moment capacity first before formation of

plastic hinges at the beam ends (kr≤kr,D in Figure 5.2). This failure mode

occurs when none of the extreme concrete elements at the beam ends has

reached the maximum concrete compressive failure strain (εcmax,T) and the top

bars at the beam ends have not reached the yielding strain. If this scenario is

observed, an extra check is needed to explore whether the beam ends can

sustain a prolonged bending period with reduced compressive force after

reaching failure of the mid-span. Details on this load redistribution

mechanism will be presented in section 5.7.

2- Compressive failure of concrete at the beam ends is reached before reaching

the mid-span its sagging moment resistance (kr≥kr,E in Figure 5.2). This

failure scenario is considered to have occurred when strain of any of the

extreme concrete elements at the beam ends has reached εcmax,T.

3- The maximum tensile strain of the top bars at the beam ends exceeds the

limiting strain of steel. In Chapter 4, this condition was found to happen for

beams failed by hogging moment failure caused by fracture of the tensile

reinforcement before sagging bending failure at the beam mid-span occurs, as

observed in axially unrestrained beams with high rotational stiffness.

5.7 Beam Bending Resistance Time When kr≤ kr,D

The numerical study results in Chapter 4 have shown that a beam with kr≤kr,D may

still balance the applied load through flexural action by the end hogging moment



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165

resistance when the mid-span has reached its sagging moment resistance. However,

this period of further bending resistance can only be achieved when the beam ends

can develop sufficient moments to balance the applied moment (kr≥kr,B) and the

reduced hogging moment resistance (MRd,fi,hog) is still higher than the applied moment

(MRd,fi,hog≥wL2/8). In this case, the compressive force in the beam is partially

released. To examine this condition, two critical parameters, kr,B and MRd,fi,hog need

to be quantified.

When the beam rotational stiffness is kr=kr,B, it has zero axial force (Figure 5.2).

Hence, the equilibrium equation for predicting kr,B is conservatively taken as follows:

𝑀𝐴 −𝑀𝑒 = 0 (5-25)

It should be pointed out that because the applied load is assumed to be entirely

resisted by the end sections, the MA value computed by Equation 5-20 is taken

without multiplying it by the factor c defined in Equation 5-6. However, the effect of

flexible end rotational restraint stiffness is taken into consideration when determining

the beam deflection and curvature. In realistic structures, MA may represent the

bending resistance of the surrounding structure.

The reduced section hogging moment resistance MRd,fi,hog with time cannot be

determined based on simple plastic analysis due to severe non-uniform distribution

of strength of concrete on the compression side which is directly exposed to fire. In

this analysis, it is determined by establishing moment-curvature (𝑀 − 𝜑)

relationships with zero axial force. Figure 5.15 shows a schematic 𝑀 − 𝜑 curve at a

given fire exposure time. The 𝑀 − 𝜑 curve is constructed by iterating the reference

strain εo at the mid-depth of the cross-section for any fixed value of curvature until

the resultant of internal axial force, calculated using Equation 5-13, is equal to zero.

The internal moment of the cross-section is calculated from stresses of the elements

using Equation 5-26. The maximum internal moment in the 𝑀 − 𝜑 curve defines the

hogging moment resistance at the given fire exposure time.



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166

Figure 5.15: Schematic M- 𝝋 at a given fire exposure time

𝑀𝑅𝑑,𝑓𝑖,ℎ𝑜𝑔 = ∑ 𝜎𝑐,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑚. (𝑦 − 𝐷/2) +𝑛𝑖=1

∑ 𝜎𝑠,𝑚(𝑦, 𝑧, 𝑡). 𝐴𝑠,𝑚. (𝑦 − 𝐷/2)𝑛𝑖=1 (5-26)

If MRd,fi,hog at the time when the sagging bending moment reaches its limit value is

lower than the applied moment, the rotational stiffness applied to the beam is not the

critical stiffness kr,B. For this case, the time to reach mid-span bending moment

capacity is recorded as the beam bending resistance time. Otherwise, kr,B should be

determined. The beam bending resistance time when kr=kr,B is tBR,B, as illustrated in

Figure 5.16, and equal to the time when MRd,fi,hog= Me based on the assumption made

in Equation 5-25. The value of kr,B is predicted by analysing the beam at t= tBR,B with

various values of kr using the conditions of compatibility (Equation 5-17) and

equilibrium (Equation 5-25). A kr value at which the axial force is zero is the

corresponding kr,B. The maximum mid-span deflection at kr=kr,B (denotes δBR,B,max as

shown in Figure 5.16) will be used to analyse beams with kr,B≤kr≤kr,D explained in

the following paragraphs.

Section moment

resistance

Mo

men

t, M

Curvature, 𝜑



IN FIRE

167

Figure 5.16: Beam behaviour with kr=kr,B

For beams with kr< kr,B (in region 1 shown in Figure 5.2(a)), they are unable to resist

the applied load through flexural action after bending failure of the mid-span. Hence,

tBR= tmid for kr< kr,B (5-27)

where, tBR is the beam bending resistance time and tmid is the time when the mid-span

bending failure reaches its sagging moment capacity.

For beams with kr,B<kr≤kr,D, the numerical results indicate that because the end

bending moment is higher compared to the case with kr,B, equilibrium in bending

after the mid-span has reached its sagging moment capacity can be regained after an

increase in beam defection. This releases the axial compressive force in the beam.

However, the amount of released axial compressive force depends on the increase in

beam deflection. The higher the value of kr, the lower the beam deflection, hence the

lower the released compressive force and the higher the residual compressive force.

Refer to Figure 5.17 which shows beam mid-span deflection – time and beam axial

force – time relationships for a typical kr in this range of rotational stiffness, based on

examining the numerical results and from the known values of tBR,B and δBR,B,max,

simple and reliable expressions have been proposed using linear interpolations to

predict the following quantities: (1) bending resistance time tBR, (2) maximum mid-

Axia

l fo

rce,

F

tmid,B

tmid,B tBR,B

Def

lect

ion, δ

δBR,B,max

δst,B,max

tBR,B

Fmid,B

δmid,B,max



IN FIRE

168

span deflection at bending resistance time δBR,max, (3) maximum mid-span beam

deflection when stability of the beam is regained after bending failure of the mid-

span δst,max and (4) residual axial compressive force Fres. Note tmid and δmid,max are just

before the beam experiences sudden deflection due to mid-span bending failure

(Figure 5.17)

Figure 5.17: Beam behaviour with kr,B<kr≤kr,D

tBR=tBR,B‒1.5(tmid-tmid,B) tBR ≮ tmid (5-28)

𝛿𝐵𝑅,𝑚𝑎𝑥 =𝑡𝐵𝑅.𝛿𝐵𝑅,𝐵,𝑚𝑎𝑥

𝑡𝐵𝑅,𝐵 (5-29)

𝛿𝑠𝑡,𝑚𝑎𝑥 = 𝛿𝐵𝑅,𝑚𝑎𝑥 − 0.1 𝑡𝐵𝑅.𝛿𝐵𝑅,𝑚𝑎𝑥

𝑡𝐵𝑅,𝐵 (5-30)

𝐹𝑟𝑒𝑠 = 𝐹𝑚𝑖𝑑,𝐵(𝛿𝑠𝑡,𝐵,𝑚𝑎𝑥−𝛿𝑠𝑡,𝑚𝑎𝑥)

(𝛿𝑠𝑡,𝐵,𝑚𝑎𝑥−𝛿𝑚𝑖𝑑,𝐵,𝑚𝑎𝑥) Fres ≯ 0 (5-31)

where Fmid,B and δmid,B,max are axial compressive force and maximum mid-span

deflection when bending failure of the beam mid-span section with kr= kr,B occurs.

Note Fmid,B is the compressive force just when bending failure of the mid-span

occurs, not the residual axial force in the beam, which is zero for the special case of

Compression

Axia

l fo

rce,

F

tmid

tmid tBR

Def

lect

ion, δ

Tension

δBR,max

δst,max

tBR

Fres Fres

Fmid

δmid,max



IN FIRE

169

kr=kr,B. The residual compressive force Fres is assumed to be constant with increasing

fire exposure time.

The lowest bending resistance time tBR for beams with kr,B<kr≤kr,D, which is for the

case kr=kr,D, is when the value tBR predicted using Equation 5-28 is close to tmid.

5.8 Assumptions in Catenary Action Stage

In catenary action, the same process of sectional analysis is performed but using the

assumed beam deflection profile for the catenary action phase (Equation 5-9). The

last time step in flexural action is taken as the first time step in catenary action. It is

assumed that the bottom bars at the mid-span have fractured during catenary action.

If catenary action is to develop in beams with kr≥kr,E, in which the beam bending

resistance is governed by compressive failure of concrete at the beam ends, the

following assumptions are made by referring to the schematic diagram shown in

Figure 5.18 for half an RC beam in actual and assumed catenary action:

1. The location of the vertical and horizontal supports at the ends is at the same

location of the top longitudinal reinforcement (point O in Figure 5.18(c)),

irrespective of their locations in flexural action. This assumption is

reasonable because after the beam end sections have lost their shear and

flexural resistance as well as buckling of the bottom reinforcing bars due to

concrete crushing, the applied load on the beam is entirely resisted by the

tensile catenary force developed in the top reinforcement bars.

2. The beam is rotationally free at the ends (MA=0) and thus not affected by the

end rotational stiffness. Hence, the equilibrium equation is expressed by

taking moments about the support position (point O in Figure 5.18(c)) as:

𝐹. 𝛿𝑚𝑎𝑥 +𝑀𝑚−𝑀𝑒 = 0 (5-32)

3. Because of fracture and buckling during the transition process from flexural

action to catenary action, the bottom reinforcing bars along the beam is not

considered in the sectional analysis.



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170

(a) Example of deformed shape of ABAQUS simulation in catenary action

(b) Actual beam in real catenary action

(c) Assumed beam in catenary action in the simplified model

Figure 5.18: Schematic diagram of a beam in catenary action with kr≥kr,E

5.9 Beam Ultimate Resistance Time (tUR) in Catenary Action

In the catenary action stage, the applied load is entirely sustained by the tensile force

developed in the top steel reinforcing bars. This stage ends once strain of the top

reinforcement exceeds its rupture limit, causing total collapse of the beam.

Numerical results in Chapter 4 have shown that large plastic deformation and bar

Top reinforcement

Bottom reinforcement

Buckled steel

bars

w

L/2

Crushed

concrete

Fractured

steel bars

F

Top reinforcement

Centre of

rotation

KA

w

L/2

F

O yo



IN FIRE

171

fracture are concentrated at the end zone. The assumed smooth beam deflection

profile is not able to accurately predict the maximum tensile strain in the top

reinforcement. Therefore, relying on using strains of the top reinforcing bars to judge

catenary action failure leads to overestimate of the ultimate resistance time of the

beam. Instead, this analysis proposes a maximum beam mid-span deflection for

terminating the catenary action stage. This deflection limit is calculated based on an

assumed strain profile for the top reinforcement.

Figure 5.19(a) shows the assumed strain profile. For each half of the beam, it consists

of two segments along the beam span. The first segment, from the support to a

position D away from the support, where D is the beam depth, the strain decreases

linearly from the limiting strain of steel (the maximum strain while maintaining the

yield stress, which is 5% for class A reinforcement in EN 1992-1-2 (CEN, 2004)) to

a value of 0.0075 which is kept constant in the reminder of the beam based on

observations of ABAQUS simulations. Figure 5.19(b) compares the assumed strain

profile with a number of ABAQUS simulation results, confirming good

approximation of the assumed strain profile.

(a) Assumed strain profile

b

L/2

εs' =0.0075

εt,T=5%

D

a

c

x

εx



IN FIRE

172

(b) From ABAQUS simulations

Figure 5.19: Strain profiles of top reinforcing bars at ultimate beam resistance

time

Assuming a linear deflection shape of the top bars according to Figure 5.20, the

maximum mid-span deflection of the beam can be estimated from the compatibility

requirements of the top bars as:

𝛿𝑚𝑎𝑥 = √(𝐿

2+ ∆𝐿

2

)2

− (𝐿

2−

𝐹

𝐾𝐴)2

(5-33)

∆𝐿2

= ∫ 휀𝑥𝑑𝑥𝐿

20

=𝐿

2휀𝑠′ +

1

2. 𝐷. (휀𝑡,𝑇 − 휀𝑠

′) (5-34)

where ∆𝐿2

represents the increase in half-length of the top reinforcing bars and F is the

axial tensile force in the beam taken from the sectional analysis.

Figure 5.20: Compatibility condition for top reinforcing bars

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5

Stra

in (

mm

/mm

)

× L (Length unit)

L=6m, kr=0.05, ka=0.05, LR=35%, ABAQUS

L=6m, kr=0.05, ka=0.166, LR=35%, ABAQUS

L=6m, kr=0.05, ka=0.25, LR=35%, ABAQUS

L=6m, kr=0.5, ka=0.166, LR=35%, ABAQUS

L=6m, kr=0.1, ka=0.166, LR=50%, ABAQUS

L=5m, kr=∞, ka=0.166, LR=40%, ABAQUS

L=6m, kr=∞, ka=0.166, LR=50%, ABAQUS

Assumed profile, L=6m

𝛿𝑚𝑎𝑥

w

Top reinforcement Centre of

rotation

Ka

L/2

O yo L/2+∆L/2

Fracture of

bottom bars



IN FIRE

173

The mid-span deflection δmax using Equation 5-33 is predicted at different increased

time steps. Final failure of the beam in catenary action is said to occur when δmax

from the iterative method (sectional analysis) exceeds the one predicted using

Equation 5-33.

5.10 Limitation of applicability

The applicability of the proposed simplified model has the following limitations:

1. The model is only applicable if the flexural rigidity along the beam length is

constant.

2. The load-induced thermal strain (LITS) is not considered explicitly as a separate

component in the total concrete strain at elevated temperatures. Thus, the

concrete constitutive law defined in the current simplified model is limited to

those which consider the LITS implicitly. Moreover, the simplified model is

applicable for predicting fire response of RC beams subjected to a heating phase

only. It is not applicable to fire scenarios that have a cooling phase after the

heating phase of a fire. The reason is because the model does not account for the

permanent and irreversibility effects of the LITS in the cooling phase.

3. The proposed model is limited to RC beams made of normal strength concrete

with low moisture content u (u ≤ 3% by weight) because the model does not

account for fire-induced spalling. The model may not yield accurate results for

high strength concrete and concrete with high moisture content which are more

susceptible to spalling.

4. The model may not achieve accurate and conservative results of the ultimate

beam resistance time in catenary action when the limiting strain of steel

reinforcement is larger than 0.05.

5.11 Verification of the Simplified Model

Validity of the proposed simplified model is checked by comparing its predictions

against the numerical simulation results of Chapter 4 and additional results of a

parametric study performed, using the sub-frame in Chapter 4, to investigate the

influence of some key parameters on the beam behaviour. Table 5.1 summarises the

chosen simulation cases for the validation study. All the beams have the same (i)

cross-section dimensions with width B=300mm and depth D=400mm, (ii) siliceous



IN FIRE

174

aggregate concrete with ambient temperature compressive cylinder strength of 30

MPa and steel reinforcement yield strength of 453 MPa and (iii) the axial spring

being located at the geometric centre of the beam cross-section at ends. The stress-

strain relationships and reduction factors for concrete under compression and steel

reinforcement at elevated temperatures were in accordance with EN 1992-1-2 (CEN,

2004), presented in Appendix 4. For concrete under tension, the same models as

described in section 3.3.1 in Chapter 3 were used.

Table 5.1: Selected parameter simulation cases for the validation study

Parameter LR

(%) kr * ka L/D

Top

bars

Bottom

bars L (m)

Rotational

stiffness ratio, kr

35 0.013 0.166 15 3#6 3#6 6

35 0.029 0.166 15 3#6 3#6 6

35 0.045 0.166 15 3#6 3#6 6

35 0.064 0.166 15 3#6 3#6 6

35 0.125 0.166 15 3#6 3#6 6

35 1.85 0.166 15 3#6 3#6 6

Load Ratio, LR

35 2 0.166 15 3#6 3#6 6

40 2 0.166 15 3#6 3#6 6

45 2 0.166 15 3#6 3#6 6

50 2 0.166 15 3#6 3#6 6

55 2 0.166 15 3#6 3#6 6

60 2 0.166 15 3#6 3#6 6

Axial stiffness

ratio, ka

35 2 0.015 15 3#6 3#6 6

35 2 0.05 15 3#6 3#6 6

35 2 0.1 15 3#6 3#6 6

35 2 0.166 15 3#6 3#6 6

35 2 0.4 15 3#6 3#6 6

Span-to-depth

ratio, L/D

50 2 0.166 10 3#6 3#6 4

50 2 0.166 12.5 3#6 3#6 5

50 2 0.166 15 3#6 3#6 6

50 2 0.166 17.5 3#6 3#6 7

Steel

reinforcement

50 2 0.166 15 3#5 3#6 6

50 2 0.166 15 3#6 3#6 6

50 2 0.166 15 3#7 3#6 6

50 2 0.166 15 3#8 3#6 6

50 2 0.166 15 3#6 3#5 6

50 2 0.166 15 3#6 3#6 6

50 2 0.166 15 3#6 3#7 6

50 2 0.166 15 3#6 3#8 6

* The relative stiffness of the rotational restraint is based on 6m beam length.



IN FIRE

175

Only half of a beam is analysed in the simplified model due to symmetry in geometry

and loading. The mesh size of the beam segment and the cross-section discretisation

of the beam section are shown in Figure 5.21 and are kept the same for all beams.

Figure 5.21: Mesh density used in the simplified method for validation study

Figure 5.22 compares the simplified method calculation results with numerical

simulation results for beam deflection-fire exposure time and beam axial force-fire

exposure time relationships under different end rotational stiffness. Figure 5.23

compares beam bending resistance time – rotational stiffness and beam ultimate

resistance time – rotational stiffness relationships between simulation results and

simplified method calculation results. It can be seen that in all cases, the full range

behaviour of restrained RC beams in fire can be predicted by the proposed simplified

method. Overall, in flexural action, the proposed model tends to underestimate the

beam deflection and the results diverge more from the numerical results in beams

with low to moderate rotational stiffness levels. This can be attributed to the

assumptions made to approximate the beam’s deflection profile and the end moment

MA by interpolation between the cases of zero and complete end rotation restraint. As

regards to beams with kr,B≤kr≤kr,D, the values of kr,B, tBR,B, δBR,B,max and kr,E were

found to be 0.029, 198min, 535mm and 0.075, respectively. As shown in Figure 5.22

and 5.23, the analytical model produces conservative predictions of the additional

bending resistance provided by the beam ends after bending failure of the mid-span.

L

Li=1/30 L Li=1/60 L

L/4

Li=1/60 L

L/2 L/4 A

A

(a) Beam segment discretisation

10×16 Elements

Section A-A

(b) Cross-section discretisation



IN FIRE

176

This is a direct result of the conservative way the value of tBR,B is estimated, in which

the applied load is assumed to be purely resisted by the beam end.

In general, prediction of the two key quantities needed for designing RC beams in

fire under the combined effect of the bending-compression, namely the fire-induced

axial force and the bending resistance time, agree well with the simulation results.

(a) Mid-span deflection-fire exposure time

(b) Axial force-fire exposure time

Figure 5.22: Comparison between simplified method results and ABAQUS

results for different end rotational stiffness ratios (kr)

-700

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350 400 450

Def

lect

ion

(mm

)

Time (min)

kr=0.013, ABAQUS kr=0.013, simplified kr=0.029, ABAQUS

kr=0.029, simplified kr=0.045, ABAQUS kr=0.045, simplified

kr=0.064, ABAQUS kr=0.064, simplified kr=0.125, ABAQUS

kr=0.125, simplified kr=1.85, ABAQUS kr=1.85, simplified

LR=35% k

a=0.166

L=6m

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Time (min)

kr=0.013, ABAQUSkr=0.013, simplifiedkr=0.029, ABAQUSkr=0.029, simplifiedkr=0.045, ABAQUSkr=0.045, simplifiedkr=0.064, ABAQUSkr=0.064, simplifiedkr=0.125, ABAQUSkr=0.125, simplifiedkr=2, ABAQUSkr=2, simplified

LR=35% k

a=0.166

L=6m



IN FIRE

177


results for the effect of rotational stiffness on beam fire resistance

In catenary action, the comparisons indicate that the proposed model is able to

provide reliable predictions of the maximum catenary force, beam deflection and

ultimate resistance time. In the simplified model, only beams with kr≥kr,E are

considered to be able to develop catenary action. The numerical results show that the

beam with kr=0.064 develops catenary action. However, because the value of kr,E

predicted by the simplified method is 0.075, catenary action is not considered for the

beam with kr=0.064 using the simplified method. Values of the catenary force and

deflection obtained from the simplified method are higher than the simulation results.

This is mainly because of the number of simplifications adopted to assess the rather

complex behaviour of the beam in catenary action after fracture of the bottom

reinforcing bars at the mid-span and severe crushing of concrete at ends. For

instance, in the proposed model, the centre of rotation of the beam ends is assumed to

be at the top reinforcement level at the end sections. In the numerical simulations,

however, the centre of the plastic hinge, where rotation of the beam ends is

concentrated, is at a distance about ½ to ¾ of the beam depth from the end sections

(Figure 5.18(a)). However, further refinement would make the simplified method

much more complicated. In any case, the simplified method produces results which

are on the safe side, in terms of both beam bending resistance time and beam

ultimate resistance time.

Figures 5.24 to 5.28 compare the simplified calculation results with numerical

simulation results for beams with different (i) load ratios, (ii) axial restraints, (iii)

050

100150200250300350400450500

0 0.05 0.1 0.15 0.2 0.25 0.3

Fire

res

ista

nce

(m

in)


Flexural action resistance (ABAQUS) Total resistance (ABAQUS)

Flexural action resistance (simplified) Total resistance (simplified)

LR=35% , ka=0.166 , L=6m



IN FIRE

178

span-to-depth ratios and (iv) bottom and top reinforcement amounts. It is verified

that the proposed model in this chapter is capable of capturing the essential features

of the beam behaviour with satisfactory accuracy in all cases.

5.12 Effects of Changing Different Beam Parameters

The following paragraphs briefly present the effects of additional parameters used for

the validation study on the behaviour of RC beams in fire. The relative rotational

stiffness of the beam was assumed to be kr=2 in all cases. As a result, bending failure

mode in all cases was observed to be controlled by hogging moment failure due to

concrete crushing in compression at beam ends. Table 5.1 presents all the

investigated parameters.

Figure 5.24 shows the effects of load ratio on the fire response of beams. As

expected, the decrease in the load ratio increases bending resistance time of the

beam. In catenary action, the higher the load ratio, the higher the beam deflection and

the higher the tensile catenary force. At a higher load ratio, because strains in the top

bars at beam ends are already higher immediately after development of catenary

action and because of higher catenary force, the fire resistance provided by catenary

action (i.e. the increase from the bending resistance time to the ultimate resistance

time) is lower. The results in Figure 5.24 hence suggest that there is a higher scope

for the development of catenary action when the applied load ratio is at low to

moderate levels.


-700

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350 400

Def

lect

ion

(mm

)

Time (min)

LR=35%, ABAQUS LR=35%, simplified LR=40%, ABAQUS

LR=40%, simplified LR=45%, ABAQUS LR=45%, simplified



kr=2

ka=0.166

L=6m



IN FIRE

179



results for different load ratios (LR)

Figure 5.25 shows the effects of the axial restraint levels. In flexural bending, the

higher the axial restraint (ka), the higher the maximum axial compressive force due to

restrained thermal expansion and the lower the bending resistance time. A reduction

in the bending resistance time with increasing ka is because of the higher

compressive force in the beam that causes concrete at beam ends to crush in

compression earlier. In catenary action, the beam with high axial restraint stiffness

has a higher tensile catenary force. Because the load ratio is the same, a smaller mid-

span deflection is required for a beam with higher ka to satisfy the equilibrium

between the external and internal moments. A beam with higher catenary force due

to higher ka has a lower ultimate resistance time. Further increase in ka above 0.166

does not cause a significant influence on the ultimate resistance time.

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

400

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N) Time (min)





kr=2

ka=0.166

L=6m



IN FIRE

180




results for different end axial stiffness ratios ka

Figure 5.26 compares the effects of the beam span-to-depth (L/D) ratio on fire

response of RC beams in fire. The relative stiffness values of axial and rotational

restraints for all beams were kept constant and based on 6m beam length. In flexural

-700

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350 400 450

Def

lect

ion

(mm

)

Time (min)

ka=0.015, ABAQUS

ka=0.015, simplified

ka=0.05, ABAQUS

ka=0.05, simplified

ka=0.1, ABAQUS

ka=0.1, simplified

ka=0.166, ABAQUS


ka=0.4, ABAQUS

ka=0.4, simplified

kr=2

LR=35%

L=6m

-1000

-800

-600

-400

-200

0

200

400

0 50 100 150 200 250 300 350 400 450

Axi

al f

orc

e (k

N)

Time (min)

ka=0.015, ABAQUS


ka=0.05, ABAQUS

ka=0.05, simplified

ka=0.1, ABAQUS

ka=0.1, simplified

ka=0.166, ABAQUS


ka=0.4, ABAQUS

ka=0.4, simplified

kr=2

LR=35%

L=6m



IN FIRE

181

action, because of larger rate of thermal expansion, longer beams have higher axial

compressive force. Although the applied load on the beam was adjusted to keep a

constant load ratio of 50%, longer beams have lower bending resistance times. The

reason for this is that identical boundary conditions were applied to the beam ends

irrespective of the span length. When the rotational stiffness is the same, longer

beams produce lager deflections and P-Δ effects, thus lowering bending resistance

time. In catenary action, since deflection of the beam with higher L/D ratio is larger

and the externally applied bending moment is the same for different L/D ratios,

smaller stresses develop in the top reinforcing bars to achieve force equilibrium.

Therefore, the catenary action phase is maintained to a longer fire exposure time for

beams with longer spans. Top bars at the ends of beams with L/D of 10 and 12.5

fractured immediately following bending failure of the beam, leaving no continuous

path of catenary force to develop. To conclude, there would be very limited scope for

the catenary action to develop in beams with low L/D ratios.


-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300

Def

lect

ion

(mm

)

Time (min)

L/D=10, ABAQUS

L/D=10, simplified

L/D=12.5, ABAQUS

L/D=12.5, simplified

L/D=15, ABAQUS

L/D=15, simplified

L/D=17.5, ABAQUS

L/D=17.5, simplified

kr=2

ka=0.166

LR=50%



IN FIRE

182



results for different span-to-depth ratios (L/D)

Figures 5.27 and 5.28 show the effects of the bottom and top longitudinal

reinforcement on the behaviour of RC beams in fire. Except for the value of the

applied uniform distributed load (w), all other parameters were maintained the same.

The value of w was adjusted to keep a constant load ratio of 50% at ambient

temperature since changing bottom and top reinforcement amounts would give

different sagging and hogging bending resistance capacities. The contribution of the

compressive reinforcement was neglected in predicting cross-section bending

resistance capacity.

As can be seen in Figures 5.27 and 5.28, increasing bottom or top reinforcement

amounts decreases bending resistance time of the beam. This is attributed to the fact

that bending failure of the beams under the applied end rotational restraints (kr=2)

were governed by concrete crushing in the lower layers exposed to direct fire at the

beam ends. It occurs without formation of a hinge mechanism in the beam as

assumed in predicting beam plastic bending resistance at ambient temperature.

Hence, with increasing reinforcement, the increased applied load to maintain the

same load ratio increases hogging moment and thus accelerates concrete crushing. In

catenary action, the increased applied load results in an increase in the tensile

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

400

0 50 100 150 200 250 300

Axi

al f

orc

e (k

N) Time (min)

L/D=10, ABAQUS L/D=10, simplified

L/D=12.5, ABAQUS L/D=12.5, simplified

L/D=15, ABAQUS L/D=15, simplified

L/D=17.5, ABAQUS L/D=17.5, simplified

kr=2

ka=0.166

LR=50%



IN FIRE

183

catenary force. This adversely affected beams with increased bottom reinforcement

(Figure 5.27) since the effect of additional bottom reinforcement to withstand

catenary force is very small due to its fracture at mid-span prior to activation of

catenary action. However, because fire survival time offered by catenary action is

mainly from the top reinforcement, it substantially increases by increasing the top

reinforcement (Figure 5.28).




results for different bottom reinforcement amount

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300

Def

lect

ion

(mm

)

Time (min)

T.r. 3#6, B.r. 3#5 (ABAQUS)

T.r. 3#6, B.r. 3#5 ( simplified)

T.r. 3#6, B.r. 3#6 (ABAQUS)

T.r. 3#6, B.r. 3#6 (simplified)

T.r. 3#6, B.r. 3#7 (ABAQUS)


T.r. 3#6, B.r. 3#8 (ABAQUS)


kr=2

ka=0.166

LR=50% L=6m

-600

-500

-400

-300

-200

-100

0

100

200

300

400

0 50 100 150 200 250 300

Axi

al f

orc

e (k

N)

Time (min)

T.r. 3#6, B.r. 3#5 (ABAQUS) T.r. 3#6, B.r. 3#5 (simplified)




kr=2

ka=0.166

LR=50% L=6m



IN FIRE

184




results for different top reinforcement amount

5.13 Conclusions

This chapter has presented a simplified calculation method aimed at capturing

structural behaviour of axially and rotationally restrained RC beams in fire,

considering the development of catenary action at large deflections. This method is

based on sectional analysis and uses the fundamental principles of equilibrium and

compatibility conditions.

The main assumption in the simplified analysis is the beam’s deflection profile. It

depends on the load-carrying mechanism action and end rotational restraint stiffness.

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350

Def

lect

ion

(mm

)

Time (min)

T.r. 3#5, B.r. 3#6 (ABAQUS)


T.r. 3#6, B.r. 3#6 (ABAQUS)


T.r. 3#7, B.r. 3#6 (ABAQUS)


T.r. 3#8, B.r. 3#6 (ABAQUS)


kr=2

ka=0.166

LR=50% L=6m

-700

-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

0 50 100 150 200 250 300 350

Axi

al f

orc

e (k

N)

Time (min) T.r. 3#5, B.r. 3#6 (ABAQUS)T.r. 3#5, B.r. 3#6 (simplified)T.r. 3#6, B.r. 3#6 (ABAQUS)T.r. 3#6, B.r. 3#6 (simplified)T.r. 3#7, B.r. 3#6 (ABAQUS)T.r. 3#7, B.r. 3#6 (simplified)T.r. 3#8, B.r. 3#6 (ABAQUS)T.r. 3#8, B.r. 3#6 (simplified)

kr=∞

ka=0.166

LR=50% L=6m



IN FIRE

185

In flexural action, the fourth order polynomial deflection profiles that satisfy the end

boundary conditions of beams under uniformly distributed load may be assumed for

zero and fixed end rotational restraints. The deflection profile of beams with flexible

end rotational restraint is approximated by exponential interpolation between that of

the beam with zero rotational restraint and that of the beam with complete rotational

restraint. In catenary action, the average of the following three deflection profiles

was used: (i) the deflection profile of the beam under uniformly distributed load

(UDL), (ii) the deflection profile of the beam under a concentrated point load (CPL)

at the mid-span and (iii) linear profile.

This chapter provides simplified and reliable formulations for analysing the period of

further bending resistance of the beam by the end hogging moment resistance after

the mid-span has reached its sagging moment resistance. Whether further resistance

is achieved or not depends on the applied load and end hogging moment resistance.

Identification of the critical rotational stiffness (kr,B in Figure 5.2) at which this

further bending resistance is the highest is important for analysing the beam

behaviour in bending under different rotational stiffness levels.

The proposed model predicts quite accurately the beam bending failure governed by

crushing of concrete in compression. This is also useful for considering the

development of catenary action as the numerical results in Chapter 4 have revealed

that effective catenary action can develop following only this beam bending failure.

A simple and reliable approach was suggested in this chapter to predict the lowest

rotational stiffness level (kr,E in Figure 5.2) that leads this bending failure mode to be

reached.

In catenary action, the following assumptions are made in the proposed model:

The beam is rotationally free at the ends.

The location of the vertical and horizontal supports at the ends is at the same

location of the top longitudinal reinforcement.

The bottom reinforcing bars along the beam is not considered in the sectional

analysis.

A strain profile for the top reinforcement based on the maximum steel strain

of 0.05 is assumed to help to predict the ultimate resistance time of the beam



IN FIRE

186

in catenary action, which is at the time when fracture of the top

reinforcement at beam ends is reached.

The proposed model in this chapter was validated against the numerical results of

Chapter 4. Additional results of numerical results were presented in this chapter to

provide more validation examples. The results of validation studies have confirmed

that the proposed model is generally applicable for predicting the full beam

deflection-fire exposure time and beam axial force-fire exposure time responses in

flexural bending action and in catenary action with good accuracy. Importantly, the

model is able to capture the bending resistance and ultimate resistance times with

satisfactory accuracy and reliability. This proposed calculation method can

potentially be applied, in design, to beams within RC frame systems provided

information on the end boundary restraints is available.

CHAPTER 6 PERFORMANCE OF REINFORCED CONCRETE FRAMES UNDER

FIRE CONDITIONS

187

CHAPTER 6

PERFORMANCE OF REINFORCED CONCRETE

FRAMES UNDER FIRE CONDITIONS

6.1 Introduction

Results of the numerical study on beam-column sub-frames presented in Chapter 4 of

this thesis have provided valuable insights into the fire response of restrained RC

beams. They have revealed considerable effects of boundary restraints on RC beam

behaviour and failure mechanisms. A beam undergoing large deflections in fire after

flexural bending failure may develop catenary action if axial restraints are present.

The focus of that study was only on the beam behaviour. In this chapter, the

numerical study, using the ABAQUS model developed in Chapter 3, is extended to

consider structural interactions between beams and columns within an RC frame

structure. Detailed investigations will be performed on two-story three-span planar

RC frames under different fire scenarios, loading levels and column cross-section

sizes.

6.2 Simulation Parameters

Figure 6.1 shows the RC frame structure used in the parametric study. The bases of

the columns on the ground floor are assumed to be totally fixed. Uniformly

distributed load (w) is applied to the beams. Additional loads from upper storeys are

represented by applying concentrated loads (P) at the top of the upper columns. The

clear height of all columns and the clear span of all beams are 3300 mm and 6000

mm respectively and were kept the same in all parametric study cases.


FIRE CONDITIONS

188

Figure 6.1: Dimensions, loading and boundary conditions of the simulation

frame

All the beams have identical cross-sectional sizes and reinforcement layouts, and so

have the columns. Figure 6.2 shows the beam cross-section dimensions and

reinforcement detail. The beam and column dimensions and their transverse

reinforcement arrangements were chosen to avoid shear failure. The column

longitudinal reinforcement arrangement is presented in the following paragraphs.

The beam cross-section detail 1, shown in Figure 6.2(b), is the same as for RC beams

in Chapter 4 to facilitate direct comparison between individual beams and beams in

frames. In Chapter 4, it was shown that for beams with low axial restraint, when

rotational stiffness at one end of the beam is much higher than that at the other end,

tensile thermal stresses developed in the beam cross-section could cause premature

fracture of the cooler top middle bars at the beam end with higher rotational stiffness.

It was found that this detrimental effect may be minimised by positioning the top

bars close to the corners of the beam in hogging moment regions. Such top

reinforcement detailing, illustrated in Figure 6.2(c), was also investigated in some

cases for the edge bay in fire. In an edge bay, the rotational stiffness at the beam end

connected to an internal bay is much higher than that connected to an edge column.

Furthermore, a beam located in an edge bay normally has low axial restraint. The

reinforcement detail 2 in Figure 6.2(c) replaces the three top bars in detail 1 with two

corner bars, but keeping the total top reinforcement area and all other conditions the

same.

w

w w w

w w

3300 mm

3300 mm

6000 mm 6000 mm 6000 mm

C1 C2 C3 C4

C5 C6 C7 C8

B1 B2 B3

B4 B5 B6

P P P P


FIRE CONDITIONS

189

(a) Transverse reinforcement in beams, columns and beam-column joints

Figure 6.2: Transvers reinforcement and beam cross-sectional details

Three fire scenarios were considered, namely fire scenarios 1, 2 and 3, as depicted in

Figure 6.3. Fire scenarios 1 and 2 represent situations where fire is confined to either

the edge or central bay on the ground floor, respectively. Fire scenario 3 represents a

fire taking place in the whole ground floor of the frame. Figure 6.3 also indicates the

beam and column faces exposed to fire in these different fire scenarios, based on

consideration of likely fire barriers provided by the adjacent compartment wall and

floor.


1500

#3@75

#3@125

#3@75 400

400

2400

a

a

b

b

a

a #3@100 #3@150 #3@100

1500 3000

400

300

2#6

Section a-a

3#6 50

Section b-b

400

300

400

300

2#6

Section a-a

2ϕ23.3mm

As=855mm

3#6

Section b-b

50

2ϕ23.3mm

As=855mm

400

300

(b) Beam cross-section detail 1 (c) Beam cross-section detail 2

3#6

As=855mm 3#6

As=855mm


FIRE CONDITIONS

190


(a) Fire scenario 1 (b) Fire scenario 2

(c) Fire scenario 3

Figure 6.3: Different fire scenarios

Fine meshes with a 3D element size of 30mm were used for the heated members and

for a length of 1.5 to 2 times the beam depth of the cool members adjacent to the

heated members. The remaining parts of the unheated columns directly above the

heated columns were constructed with a mesh size of 30mm in the transverse

direction and 50mm in the longitudinal direction. To save computational time by

reducing the number of nodes, larger mesh sizes were adopted for noncritical

unheated zones where no local stress concentration is expected. Figure 6.4 shows the

mesh division adopted for fire scenario 1. Furthermore, due to the symmetry and

increased number of heated members, only half of the frame was modelled for fire

scenario 3.

c c b b a

a

a-a b-b c-c

b b c c a

a

a-a b-b c-c

a

a

a

a

c c c c a

a

a-a b-b d-d

b b d d

c-c


FIRE CONDITIONS

191

Figure 6.4: Mesh configuration used for fire scenario 1

Cross-sectional temperature distributions of the heated structural members were first

predicted through numerical heat transfer analysis. The fire temperature-time curve

was assumed to follow the ISO 834 standard fire curve. The thermal properties and

boundary conditions as suggested by EN 1992-1-2 (CEN, 2004) were used. Siliceous

concrete was selected with a moisture content of 3% by weight and a density of

2300kg/m3. Fire-induced spalling was not considered. Figure 6.5 shows temperature

histories of the reinforcement of a typical column with dimensions of 400×400mm

exposed to fire on 3 and 4 sides. The same beam reinforcement temperatures, as used

for the sub-frames in Chapter 4 shown in Figure 4.2, were defined for the beams of

the frame.


FIRE CONDITIONS

192

(a) Fire on 3 sides (b) Fire on 4 sides

Figure 6.5: Selective temperature-time histories of column reinforcement, based

on numerical heat transfer analysis (Column dimensions: 400×400mm)

The ambient temperature compressive cylinder strength and modulus of elasticity of

concrete were 30 MPa and 18000 MPa, respectively. The ambient temperature yield

strength and modulus of elasticity of hot-rolled steel reinforcement were 453 MPa

and 200 GPa, respectively. The stress-strain relationships and reduction factors for

concrete under compression and steel reinforcement at elevated temperatures were in

accordance with EN 1992-1-2 (CEN, 2004) presented in Appendix 4. For concrete

under tension, the same models as described in section 3.3.1 in Chapter 3 were used.

According to EN 1992-1-2 for class A reinforcement, the limiting strain (εt,T) at

yield strength and the ultimate strain (εu,T) of steel were assumed to be 5% and 10%,

respectively.

Transient modelling was carried out: the mechanical loads on the RC frame were

applied first and kept constant and then the structural temperatures were increased

until structural failure, as indicated by inability of the frame to maintain the applied

mechanical loads. In the structural analysis, out-of-plane displacements of the frame

0

200

400

600

800

1000

1200

0 75 150 225 300 375 450

Tem

per

atu

re (

oC

)

Time (min)

1 2 3 45 6 7 8

0

200

400

600

800

1000

1200

1400

0 75 150 225 300 375 450

Tem

per

atu

re (

oC

)

Time (min)

1 2 3

2

1

1

2 1

2

1 2 3 3 3

7

2

1

1

3 4

5

4 3 7 6 8


FIRE CONDITIONS

193

at the beam-column joint panel zones were not allowed and lateral torsional buckling

of the beams was prevented. Based on the lowest natural period of the investigated

frames, a duration of 60min heating was scaled to 4s for explicit dynamic analysis.

6.3 Parametric Study Cases

For each fire scenario shown in Figure 6.3, the parametric study examined the

influence of changing column cross-section size and level of the applied load. Table

6.1 lists all the simulation cases, consisting of three column cross-section dimensions

of 300×300mm, 400×400mm and 500×500mm, two beam reinforcement details

(details 1 and 2 shown in Figure 6.2(b) and 6.2(c)) and two levels of loading (30%

load ratio (LR) and 50% load ratio on all beams and columns). Figure 6.6 shows the

column longitudinal reinforcement details for the three different cross-section sizes.

They were selected such that the same reinforcement ratio of 1.7% was maintained in

each of the three column cross-sections.

Table 6.1: Parametric study cases and summary of main results

Frame Fire

Case

Beam

cross-

section

detail

Beam

load

ratio

(%)

Column

load

ratio

(%)

Column

size

(mm)

Beam

bending

failure

time

(min)

Frame

failure

time

(min)

Frame

failure type

F1 1 1 30 30 300 -- 171 Column-led

F2 1 1 30 30 400 216 216 Beam-led

F3 1 1 30 30 500 254 254 Beam-led

F4 1 1 50 50 300 57 57 Beam-led

F5 1 2 50 50 300 -- 78 Column-led

F6 1 1 50 50 400 76 76 Beam-led

F7 1 2 50 50 400 -- 101 Column-led

F8 1 1 50 50 500 99 99 Beam-led

F9 1 2 50 50 500 -- 123 Column-led

F10 2 1 30 30 300 -- 204 Column-led

F11 2 1 30 30 400 259 362* Column-led

F12 2 1 30 30 500 231 384* Beam-led

F13 2 1 50 50 300 -- 94 Column-led

F14 2 1 50 50 400 -- 120 Column-led

F15 2 1 50 50 500 146 146 Beam-led

F16 3 1 30 30 300 -- 139 Column-led

F17 3 1 30 30 400 187 187 Beam-led

F18 3 1 30 30 500 229 229 Beam-led

F19 3 1 50 50 300 51 51 Beam-led

F20 3 1 50 50 400 73 73 Beam-led

F21 3 1 50 50 500 86 86 Beam-led

* Catenary action developed in the fire exposed beam


FIRE CONDITIONS

194

Figure 6.6: Details of longitudinal steel reinforcement for different column sizes

In this study, the load ratio of a member is defined as the ratio of the applied load to

its ambient temperature capacity. The ultimate capacities of the beam and column

members at ambient temperature were separately determined using ambient

temperature simulations in ABAQUS by monotonically increasing the load on the

member under consideration until its failure while assuming the loads on the other

members of the frame being zero, as illustrated in Figures 6.7(a) and 6.7(b) for a

frame with column dimensions of 300x300mm. Table 6.2 lists the ambient

temperature capacity values for all the investigated column sizes. As expected,

beams failed by forming a complete three plastic hinge mechanism (two plastic

hinges at the ends and a plastic hinge at the mid-span), as shown in Figure 6.7(a) and

6.7(c).

500

500

50

8#8

400

400

50

Corner bars:

4#7

Middle bars:

4#6

300

300

50

8#5

(a) For determining maximum beam

resistance at 20oC

(b) For determining maximum column

resistance at 20oC

w

Section a Section b

Section c

P


FIRE CONDITIONS

195

(c) Strains of beam longitudinal bars at mid-span and ends (column size:

300×300mm)

Figure 6.7: Determining maximum beam and column resistances in the frame

at ambient temperature

Table 6.2: Ambient temperature capacities of beam and column members

Column size Beam ultimate

load, wu (kN/m)

Column ultimate

load, Pu (kN)

300×300mm 61 3050

400×400mm 62.5 6066

500×500mm 64 9360

6.4 Simulation Results and Discussions

Heating causes a structural member to expand while material degradation at elevated

temperatures causes members to experience large deflections. Figure 6.8 shows

typical deflected shapes of frames under each fire scenario immediately prior to

failure of any member.

(a) Fire scenario 1 (column size: 400×400 mm, load ratio: 30%)

-75

-60

-45

-30

-15

0

15

30

45

60

-0.075

-0.060

-0.045

-0.030

-0.015

0.000

0.015

0.030

0.045

0.060

0 10 20 30 40 50 60 70

Def

lect

ion

(mm

)

Stra

in (

mm

/mm

)

Applied load, w (kN/m)

Strain, section a

Strain, section b

Strain, section c

Deflection

Limiting strain =5%

Yielding strain at

20oC=0.0022


FIRE CONDITIONS

196

(b) Fire scenario 2 (column size: 400×400mm, load ratio: 30%)

(c) Fire scenario 3 (column size: 400×400mm, load ratio: 30%)

Figure 6.8: Typical deflected frame shapes (Deformation scale factor=3)

As indicated in Table 6.1, failure of a frame can be either beam-led or column-led.

At the same load ratio in the beam and in the column, column-led failure tends to

happen when the column cross-section sizes are small. Figure 6.9 shows the trends of

frame failure time with changing column cross-section size and load ratio. In all

cases, increasing column sizes gives higher frame failure times.

Axis of

symmetry


FIRE CONDITIONS

197

(a) Fire scenario 1

(b) Fire scenario 2

(c) Fire scenario 3

Figure 6.9: Effects of column size and load ratio on frame failure times

For fire scenario 1, thermal expansion of beam B1 is highly constrained at the end

connected to the internal bay (right end) due to additional restraints by the cool

structural members in the adjacent bays. Hence, lateral displacements of the middle

columns C2 and C6 remain very small throughout the fire exposure. On the contrary,

0

50

100

150

200

250

300

200 300 400 500 600

Fram

e f

ailu

re t

ime

(min

)

Column cross-section size (mm)

LR=30%

LR=50% (beam detail 1)

LR=50% (beam detail 2)

0

100

200

300

400

500

200 300 400 500 600

Fram

e f

ailu

re t

ime

(min

)


LR=30%

LR=50%

0

50

100

150

200

250

200 300 400 500 600

Fram

e f

ailu

re t

ime

(min

)


LR=30%

LR=50%


FIRE CONDITIONS

198

significant outward movement of the edge columns C1 and C5 happen because of

very low level of lateral restraint. This can produce significant P-Δ effect in the edge

columns. At the same time, the compression force in the heated beam B1 due to

restrained thermal expansion imposes additional bending moments in the edge

columns. These can force the edge columns to deform laterally faster than thermal

expansion of the connected beam, thereby the compression force in the beam

changes to tension, giving a false sense that the beam is in catenary action. This can

be seen in Figures 6.10 and 6.11 which show beam B1 axial force-time curves for

different column cross-section sizes and load ratios.

Figure 6.10: Beam B1 axial force-fire exposure time relationship with

different column sizes and load ratios (Fire scenario 1, beam reinforcement

detail 1)

Figure 6.11: Beam B1 axial force-fire exposure time relaitonship with beam

reinforcement details 1 and 2 (Fire scenario 1, LR=50%)

To confirm this phenomenon, the frame with the highest tension force in beam B1

(frame F7, with 400x400mm column size, LR=50% and detail 2 beam

-150

-100

-50

0

50

100

0 50 100 150 200 250 300

Bea

m A

xial

fo

rce

(kN

)

Time (min)

300x300 mm, LR=30%300x300 mm, LR=50%400x400 mm, LR=30%400x400 mm, LR=50%500x500 mm, LR=30%500x500 mm, LR=50%

-150

-100

-50

0

50

100

0 30 60 90 120 150

Bea

m a

xial

fo

rce

(kN

)

Time (min)

300x300 mm (detail 1)300x300 mm (detail 2)400x400 mm (detail 1)400x400 mm (detail 2)500x500 mm (detail 1)500x500 mm (detail 2)


FIRE CONDITIONS

199

reinforcement) was re-run but with removed axial load on the edge columns C1 and

C5. Figure 6.12 compares axial forces of beam B1 between these two frame loading

cases. As can be seen, because the edge columns C1 and C5 in Frame loading 2 case

had no axial load, they did not suffer large lateral deformation so they acted as

restraints to beam B1 throughout the entire fire exposure, resulting in compression in

beam B1 in almost the entire fire exposure time.

(a) Frame loading 1 (b) Frame loading 2

(c) Beam B1 axial force-fire exposure time relaitonships

Figure 6.12: Comparison of results between frame loadings 1 and 2 (Fire

scenario 1, beam reinforcement detail 2, column size=400x400mm, LR=50%)

Due to high rotational restraint at the right end of beam B1, large hogging moments

were induced at this end, resulting in high hogging reinforcement strains, compared

to those at the left end of beam B1, as shown in Figure 6.13. In fact, beam B1

exhibited a similar behaviour to beams with asymmetrical boundary restraints

observed in Chapter 4.

-40

-20

0

20

40

60

80

100

0 30 60 90 120 150

Bea

m a

xial

fo

rce

(kN

)

Time (min)

Frame loading 1

Frame loading 2


FIRE CONDITIONS

200

(a) Load ratio =30%

(b) Load ratio =50%

Figure 6.13: Strain-fire exposure time relationships of longitudinal

reinforcing bars at ends of beam B1 with different column sizes (Fire scenario 1)

Except for the frame with column size of 300×300mm and LR of 30% where failure

of the frame is governed by failure of the edge column C1 (Figure 6.14), failure of

frames in fire scenario 1 with beam reinforcement detail 1 is beam-led. It is initiated

by hogging bending failure of beam B1 at its right end as shown in Figure 6.15 for a

typical frame. Premature fracture of the cooler middle bar at the top of the cross-

section occurred before the section attains its reduced hogging bending resistance at

elevated temperatures. Adopting beam reinforcement detail 2 to minimise the effects

of thermal stresses on the section bending moment resistance, as recommended in

Chapter 4 for beams with asymmetrical rotational restraints and low axial restraints,

enhanced flexural resistance of beam B1 (Table 6.1). Figure 6.16 further plots

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300

Stra

in (

mm

/mm

)

Time (min)

300x300 mm (1-1) 300x300 mm (2-2)400x400 mm (1-1) 400x400 mm (2-2)500x500 mm (1-1) 500x500 mm (2-2)

Limiting strain=5%

0

0.01

0.02

0.03

0.04

0.05

0.06

0 20 40 60 80 100

Stra

im (

mm

/mm

)

Time (min)

300x300 mm (1-1) 300x300 mm (2-2) 400x400 mm (1-1)

400x400 mm (2-2) 500x500 mm (1-1) 500x500 mm (2-2)

Limiting strain=5%


FIRE CONDITIONS

201

maximum strains in the top reinforcement at the right end of the beam for both

reinforcement details 1 and 2. As can be seen, the maximum strain for beam

reinforcement detail 1 increases rapidly and reaches fracture strain (0.05). However,

with reinforcement detail 2, the rate of increase in the reinforcement strain is low,

which changed the frame failure mode from beam-led to edge column C1 led failure.

Figure 6.14: Column-led failure mode in fire scenario 1

Figure 6.15: Beam-led failure mode in fire scenario 1

Column size: 300×300mm

Load ratio: 30%


Load ratio: 30%


FIRE CONDITIONS

202


reinforcing bars at right end of beam B1 (Fire scenario 1, LR=50%)

Increasing column size prolongs failure times of both beam-led and column-led

frame failure modes. For beam-led failure, increasing column size increases the

rotational restraint offered by the edge columns to the left end of beam B1.

Therefore, as indicated by the strain of the top bars at the beam ends in Figure 6.13,

the hogging bending moment is increased at the left end and decreased at the right

end which is less influenced by further increase in the rotational stiffness because of

being already at very stiff rotational restraint, thereby prolonging bending failure of

the beam. For column-led failure, while the axial load in the column is also increased

with increasing the column size to keep the same applied load ratio, the temperature

rise in the inner layers of a bigger concrete section is lower. Hence, a better

performance is observed for a bigger column size as the cooler inner layers

contribute more toward column capacity.

Because of high hogging moment developed at the right end of beam B1 that leads to

a reduction in the sagging moment, the beam cannot develop a hinge mechanism at

bending failure. This is confirmed by the maximum strains of the bottom

reinforcement in the beam span in Figure 6.17, which shows that the strains have not

reached the yielding strain (0.02). For enlarged column sections and lower load ratios

where the hogging moment at the critical right end of beam B1 is smaller, the results

in Figures 6.13 and 6.17 indicate a complete three plastic hinge mechanism has

formed in the beam at failure.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 30 60 90 120 150

Stra

in (

mm

/mm

)

Time (min)

300x300 mm (Bar 4, detail 1) 300x300 mm (Bar 3, detail 2)



3#6 As=855mm

2φ23.3mm As=855mm

Detail 1 Detail 2

Limiting strain=5%


FIRE CONDITIONS

203

Figure 6.17: Maximumn strain-fire exposure time relationship of longitudinal

reinforcing bars in beam B1 span with different column sizes (Fire scenario 1)

For fire scenario 2 when fire is confined to the internal bay, large axial compressive

force due to restrained thermal expansion is generated in beam B2 owing to high

axial restraints at both ends of the beam provided by the adjacent cool members.

Figures 6.18 to 6.20 show variations of the horizontal displacements at the top of

column C2, deflection of beam B2 and axial force in beam B2 against time with

different column sizes and load ratios. Higher beam axial compressive forces

develop for cases of larger column size. When bending failure of beam B2 happens

before failure of the adjacent columns C2 and C3, catenary action may develop in the

beam.

Figure 6.18: Horizontal displacement at the top of column C2 (Fire scenario 2)

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250 300Stra

in (

mm

/mm

)

Time (min)

300x300 mm, LR=30% 300x300 mm, LR=50%

400x400 mm, LR=30% 400x400 mm, LR=50%

500x500 mm, LR=30% 500x500 mm, LR=50%

0

5

10

15

20

25

30

0 100 200 300 400

Dis

pla

cem

ent,

U2 (

mm

)

Time (min)

300x300mm, LR=30% 300x300mm, LR=50%

400x400mm, LR=30% 400x400mm, LR=50%

500x500mm, LR=30% 500x500mm, LR=50%


FIRE CONDITIONS

204

Figure 6.19: Beam B2 deflection-fire exposure time relationship with different

column sizes and load ratios (Fire scenario 2)

Figure 6.20: Beam B2 axial force-fire exposure time relationship with

different column sizes and load ratios (Fire scenario 2)

Failure mechanism of the frame is strongly affected by the column size and loading

level owing to complex interactions between moments and axial forces in the beams

and columns. High rotational restraints are likely to be imposed to the ends of beam

B2 by the adjacent members. This can generate considerable hogging moment at the

beam ends due to restrained thermal curvature, which in turn increases the moment

in the adjoining heated columns C2 and C3 as well. Additional moments induce in

columns C2 and C3 under thermal expansion of the adjacent beam B2 (Figure 6.18).

Failure of frames with smaller column sizes is triggered by failure of columns C2

and C3 (Figure 6.21). It is reached while the heated beam B2 is at small deflections

in bending. For large column sizes, because of the improved fire resistance and

higher lateral restraint of the columns, bending failure of beam B2 happened first as

in frames F11, F12 and F15. It was initiated by hogging bending failure at beam

ends. The high axial compressive force in the beam led the hogging bending failure

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400

Bea

m d

efle

ctio

n (m

m)

Time (min)

300x300mm, LR=30% 300x300mm, LR=50%

400x400mm, LR=30% 400x400mm, LR=50%

500x500mm, LR=30% 500x500mm, LR=50%

-500

-400

-300

-200

-100

0

100

200

300

0 100 200 300 400

Bea

m a

xial

fo

rce

(kN

)

Time (min)

300x300mm, LR=30% 300x300mm, LR=50%400x400mm, LR=30% 400x400mm, LR=50%500x500mm, LR=30% 500x500mm, LR=50%


FIRE CONDITIONS

205

to be governed by concrete crushing of the lower layers directly exposed to fire as

illustrated in Figure 6.22. This is confirmed by excessive concrete compressive

strains at the beam ends plotted in Figure 6.23, which have exceeded the strains

corresponding to peak and zero compression stresses according to the concrete

stress-strain relationships at elevated temperature presented in Appendix 4 in Figure

A4.7. Increasing the column size increases the compression force in the beam, hence

causing the beam bending failure time to occur earlier before yielding of both the

tensile reinforcement at the ends and mid-span as shown in Figures 6.24 and 6.25.

Figure 6.21: Frame failure in fire scenario 2 (column-led failure)

Figure 6.22: Beam bending failure in fire scenario 2 and deformed shape

of frame in catenary action


Load ratio: 30%


Load ratio: 30%


FIRE CONDITIONS

206

Figure 6.23: Strain-fire exposure time for concrete near beam B2 ends (Fire

scenario 2, Frame F11)

Figure 6.24: Maximumn strain-time relationship of longitudinal reinforcing

bars in beam B2 span with different column sizes and load ratios (Fire scenario

2)

Figure 6.25: Strain-time relationship of longitudinal reinforcing bars at beam

B2 ends with different column sizes and load ratios (Fire scenario 2)

-250

-200

-150

-100

-50

0

50

100

150

200

-0.200

-0.160

-0.120

-0.080

-0.040

0.000

0.040

0.080

0.120

0.160

0 100 200 300 400

Bea

m a

xial

fo

rce

(kN

)

Stra

in (

mm

/mm

)

Time (min)

Strain, y=15 mm Strain, y=75 mm

Strain, y=135 mm Beam axial force

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 100 200 300 400

Stra

in (

mm

/mm

)

Time (min)

300x300 mm, LR=30%

300x300 mm, LR=50%

400x400 mm, LR=30%

400x400 mm, LR=50%

500x500 mm, LR=30%

500x500 mm, LR=50%

Limiting strain=5%

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 100 200 300 400

Stra

in (

mm

/mm

)

Time (min)

300x300 mm, LR=30%300x300 mm, LR=50%400x400 mm, LR=30%400x400 mm, LR=50%500x500 mm, LR=30%500x500 mm, LR=50%

Limiting strain=5%


FIRE CONDITIONS

207

When beam bending failure precedes column failure, following concrete crushing at

the ends of beam B2, the compressive force in the beam drops rapidly and changes to

tension due to the development of catenary action. Sagging bending failure caused by

fracture of the bottom tensile reinforcement (Figure 6.24) occurs immediately in the

beam span after concrete crushing because of the additional moments transferred

from the beam ends. As demonstrated in Chapters 4 and 5, the likelihood of a beam

to go into catenary action at large deflections following this bending failure mode of

the beam is high. However, the development of catenary action depends also on

whether the applied load on the beam can be sustained in catenary action by the

beam reinforcement provided for bending resistance. The higher the applied load, the

higher the tensile catenary force develops, as revealed in section 5.12 in Chapter 5.

For example, when the load ratio is LR=50%, the current amount of the

reinforcement in beam B2 was not adequate to sustain the applied load by catenary

action. Tensile strength of the reinforcement, which is largely from the top bars after

fracture of the bottom bars at mid-span and due to significant strength degradation of

the hot bottom steel bars at elevated temperature, was not sufficient to sustain the

tensile catenary force. The top bars fractured immediately after bending failure of the

beam. This can be confirmed in Figure 6.25 by a sudden jump in the bar strain to the

fracture limit (0.05) for the frame with column size of 500x500 mm and load ratio of

50% (frame F15).

In catenary action for frames with LR=30% and column sizes of 400×400mm and

500×500mm, tensile catenary force returned columns C2 and C3 to almost their

initial positions after they had been pushed away by thermal expansion of beam B2

in the flexural action (Figure 6.18). In addition, bending moments from the

connected beam B2 are released in columns C2 and C3 in catenary action after

hogging moment failure at the beam B2 ends. This is beneficial to survival of the

columns through reducing load eccentricity in the columns, which led the frame to

survive for prolonged fire periods in catenary action. Final failure of the frame with

column section 400x400mm (frame F11) was controlled by failure of columns C2

and C3 (column-led) as shown in Figure 6.26. For the frame having enlarged column

sections of 500×500mm (frame F12), the failure was beam-led controlled by fracture

of the top bars at ends of beam B2 under tensile catenary forces as shown in Figure

6.27 and indicated by the strain plots in Figure 6.25.


FIRE CONDITIONS

208

Figure 6.26: Column-led failure of frame F11 in catenary action

Figure 6.27: Column-led failure of frame F12 in catenary action

In fire scenario 3, the edge bays exhibit almost the same behaviour as in fire scenario

1. A significant reduction in rotational restraint from the edge columns to the left end

of beam B1 occurs at elevated temperatures. The reason is as explained previously

for fire scenario 1. A high level of rotational restraint is still imposed to the right end

of beam B1 although both the connected column C2 and beam B2 are heated. This is

because when beam B2 pushes columns C2 and C6 towards beam B1, as indicated in

Figure 6.28, additional moments generate in column C2 that follow the same

direction as the moment from the connected beam B2. Hence, rotations at the end of


Load ratio: 30%

Fire scenario 2


Load ratio: 30%

Fire scenario 2


FIRE CONDITIONS

209

beam B2 and top column C2 together act against the rotation of the connected beam

B1. In a consequence, rotational restraint at the right end of beam B1 stiffens.

Therefore, with deterioration in the rotational restraint at the left end, more hogging

moments are attracted to the right end of beam B1. Figure 6.29 compares evolutions

of beam steel reinforcement strains with time at critical sections. As can be seen,

strains of the top bars at the right end of beam B1 are substantially higher than strains

at the left end.

Figure 6.28: Horizontal displacement of columns (Fire scenario 3,

LR=30%)

(a) Load ratio=30%

0

20

40

60

80

100

0 50 100 150 200 250

Dis

pla

cem

ent

(mm

)

Time (min)

300x300mm, U1 300x300mm, U2 400x400mm, U1

400x400mm, U2 500x500mm, U1 500x500mm, U2

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250

Stra

in (

mm

/mm

)

Time (min)

300x300mm, (1-1)

300x300mm, (2-2)

300x300mm, (3-3)

400x400mm, (1-1)

400x400mm, (2-2)

400x400mm, (3-3)

500x500mm, (1-1)

500x500mm, (2-2)

500x500mm, (3-3)

Limiting strain=5%


FIRE CONDITIONS

210

(b) Load ratio=50%


bars at beam B1 and beam B2 ends with different column sizes and load ratios

(Fire scenario 3)

Axial forces in beams B1 and B2 against fire exposure time with different column

sizes are presented in Figure 6.30. Because of heating all the columns on the ground

floor, lower axial compressive forces due to restrained thermal expansion are

induced in the internal beams B2 compared to those for fire scenario 2. Moreover,

pushing apart the internal columns towards the edges columns by the expansion of

beam B2 reduces the relative displacement between the ends of beam B1, and

therefore stiffens the axial restraint in beam B1. Therefore, larger axial compressive

forces are induced in beams B1 compared with those in fire scenario 1. This delayed

reversal of the axial force from compression to tension caused by lateral movement

of the edge columns C1 and C5 explained above for fire scenario 1 (Figure 6.12).

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 20 40 60 80 100

Stra

in (

mm

/mm

)

Time (min)

300x300mm, (1-1)

300x300mm, (2-2)

300x300mm, (3-3)

400x400mm, (1-1)

400x400mm, (2-2)

400x400mm, (3-3)

500x500mm, (1-1)

500x500mm, (2-2)

500x500mm, (3-3)

Limiting strain=5%


FIRE CONDITIONS

211

(a) Load ratio=30%

(b) Load ratio=50%

Figure 6.30: Beam axial force-fire exposure time relationship with

different column sizes and load ratios (Fire scenario 3)

Failure of the frame with column size of 300×300mm and LR of 30% was column-

led. Although the edge column C1 carries higher moments, the frame failure was

triggered by failure of the internal column C2 (Figure 6.31). This may be due the

higher temperature rise in the internal column exposed to fire on four sides in fire

scenario 3 (Figure 6.3(c)). Failure of all other frames was beam-led and initiated by

hogging bending failure of beam B1 at its right end as shown in Figure 6.32 for a

typical frame. Similar to fire scenario 1, early fracture of the top middle bar at the

right end of beam B1 occurred due to the additional self-equilibrating stresses in the

beam cross-section. It was shown in fire scenario 1 and in the detailed investigations

of beam behaviour in Chapter 4 that this detrimental effect can be minimised by

locating the beam top longitudinal bars close to the cross-section corners (Figure

6.2(c)).

-300

-250

-200

-150

-100

-50

0

50

0 50 100 150 200 250B

eam

axi

al f

orc

e (k

N)

Time (min)

300x300 mm, Beam B1300x300 mm, Beam B2400x400 mm, Beam B1400x400 mm, Beam B2500x500 mm, Beam B1500x500 mm, Beam B2

-200

-150

-100

-50

0

50

0 20 40 60 80 100

Bea

m a

xial

fo

rce

(kN

)

Time (min)

300x300 mm, Beam B1 300x300 mm, Beam B2

400x400 mm, Beam B1 400x400 mm, Beam B2


FIRE CONDITIONS

212

Figure 6.31: Frame failure in fire scenario 3 (column-led failure)

Figure 6.32: Frame failure in fire scenario 3 (beam-led failure)

As illustrated in Figure 6.33, bending failure of the beam is reached while the

maximum tensile strain in the sagging moment region is significantly lower than the

yielding strain (0.02). It indicates that the beam failure is attained without formation

of a plastic hinge mechanism.

Figure 6.33: Maximumn strain-time relationship of longitudinal reinforcing

bars in beam B1 span with different column sizes and load ratios (Fire scenario

3)


Load ratio: 30%


Load ratio: 30%

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250Stra

in (

mm

/mm

)

Time (min)

300x300mm, LR=30% 300x300mm, LR=50%

400x400mm, LR=30% 400x400mm, LR=50%

500x500mm, LR=30% 500x500mm, LR=50%

Limiting strain=5%


FIRE CONDITIONS

213

6.5 Conclusions

This chapter has presented the results of a limited numerical parametric study on a

two story RC frame under different fire scenarios. From the results of this study, the

following conclusions may be drawn:

1. If fire exposure involves edge columns, there would be very limited scope for

the development of catenary action due to the small amount of axial restraint

that the edge columns could offer to the fire exposed beam.

2. Structural failure of a frame with small column sizes is mainly controlled by

column failure.

3. Due to thermal expansion of the connected beam, additional bending

moments can generate in a column. It is necessary to include these bending

moments when evaluating the column fire resistance.

4. Design of beams in edge bays for fire resistance on the assumption that

rotational restraints at both ends are identical and a full plastic hinge

mechanism forms may lead to non-conservative results. At elevated

temperatures, the end that is connected to edge columns could experience

deterioration in the rotational restraint stiffness. This results in more

redistribution of the load to the continuous end which may not have been

designed for. If this action is to be considered in design, top bars in the

hogging moment region of a beam should be located to the corners to reduce

the effects of tensile thermal stresses.

5. Catenary action can develop in a beam of an RC frame only when fire

exposure is in an interior bay so that the fire exposed beam has a high degree

of axial restraint. Furthermore, catenary action happens only when the

applied load ratio is at a moderate level.

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDIES

214

CHAPTER 7

CONCLUSIONS AND RECOMMENDATIONS FOR

FUTURE STUDIES

7.1 Introduction

The aim of this project is to gain thorough understanding of structural interactions in

RC frames in fire. Particular attention has been paid to investigating the whole range

behaviour of axially and rotationally restrained RC beams in fire. The study consists

of the following parts: (i) development and validation of a 3D numerical simulation

model using ABAQUS/Explicit, (ii) a numerical study of fire response of RC beams

under different axial and rotational restraint levels, (iii) development of a simplified

method for the full history behaviour of RC beams in fire and (iv) a numerical

investigation of the interactions between RC beams and columns in RC frames in

fire. This chapter summaries the main conclusions of the work and provides

recommendations for future studies.

7.2 Explicit Finite Element Modelling Methodology

Modelling the whole range behaviour of a structure until complete structural collapse

at large deformations presents serious challenges due to severe material and

geometric nonlinearities and temporary instabilities caused by local material failures,

including concrete crushing and reinforcement fracture. To overcome these

modelling challenges, this study has used a dynamic explicit analysis implemented in

ABAQUS/Explicit. A particular problem with explicit simulation is the very small

time increments. To speed up the simulation process, load and mass scaling factors

have been examined. To verify the effectiveness of the proposed simulation model,

results of explicit simulations were compared against relevant available test results at

ambient and elevated temperatures. Based on the validation results, the following

main conclusions may be drawn:

1- To obtain quasi-static response when using explicit simulation, the dynamic

effects are negligible if the total loading duration does not fall below a


215

minimum value. For ambient temperature displacement-controlled and load-

controlled simulations, the minimum loading duration is about 50 and 65

times the structure’s lowest natural period, respectively. For simulating

structural behaviour in fire, the minimum heating duration is 20 times the

lowest natural period for 60 minutes of real heating duration.

2- If an analysis requires preservation of the same real loading/heating duration

while achieving the results as above, adaptive mas scaling can be used. This

is achieved by scaling the structural mass up (m)2 times, where “m” is the

ratio of the real loading/heating duration to the minimum simulation

loading/heating duration.

3- A damping ratio of 25-30% should be introduced in the load controlled

loading method to avoid premature final structural failure of beams due to

significant dynamic effects following bending failure.

7.3 Behaviour of Restrained RC Beams in Fire

Using the validated numerical simulation model, this research then performed an

extensive study to investigate the effects of different axial and rotational restraints on

the behaviour of RC beams in fire, including catenary action at large deflections. The

results of this study indicate complicated beam bending behaviour and inability of

the beam to form the classical hinge mechanism. The following observations and

conclusions can be drawn from this study:

Beams with symmetrical restraint conditions:

As the rotational restraint stiffness at the beam ends increases, RC beams may

experience three bending failure modes: (I) beam bending failure governed by

sagging bending resistance at mid-span, (II) by combined fracture of top

reinforcement at ends and bottom reinforcement at mid-span and (III) by concrete

crushing at ends. These bending failure modes affect beam bending resistance time

and development of catenary action.

- At very low level of rotational stiffness, because the beam sagging moment at

mid-span decreases with increasing rotational stiffness, the bending failure

time of the beam increases with increasing rotational stiffness. Since catenary

action is provided by tension in the top reinforcing bars and these bars at


216

beam ends are at low tensile strains, catenary action can develop in the beam.

However, at such low levels of end rotational restraints, the extent of catenary

action development is very sensitive to the beam end rotational stiffness and

it would be better to disregard catenary action at low levels of rotational

restraints.

- With further increase in the beam end rotational stiffness, after the bottom

bars at mid-span have fractured on reaching the sagging moment resistance

and release of the axial compressive force, equilibrium of the beam in bending

can be achieved again by the end hogging moments resisting the externally

applied moment. Compression plays an increasing role when the rotational

stiffness is increased: a beam with higher rotational stiffness has higher end

bending moments so equilibrium can be achieved at smaller beam defections,

thus a smaller amount of the axial compressive force in the beam is released.

The remaining unreleased higher compressive force has adverse influence on

the beam bending resistance time due to P-Δ effects. Since bending failure of

the beam is controlled by hogging moment failure at the beam ends (bending

failure mode II), the beam bending resistance time decreases with increasing

end rotational stiffness. The hogging moment failure is controlled by fracture

of tensile reinforcement. Thus, the beam does not have a continuous path of

tensile force to develop catenary action. Hence, the beam bending failure time

and final failure time coincide, both decreasing with increasing end rotational

stiffness.

- As the rotation stiffness increases further, crushing of concrete layers exposed

to fire in the hogging moment region occurs first before the mid-span reaches

its sagging moment resistance. Increasing the beam end rotational stiffness

reduces the beam deflection and hence the P-Δ effects, and meanwhile

increases the compressive stresses at beam ends due to increased hogging

moments. Therefore, these two opposing mechanisms ensure that the beam

bending resistance time remains almost constant. Bending failure at the mid-

span occurs immediately after bending failure at the beam ends occurs

because the beam does not have sufficient mid-span bending resistance to

maintain equilibrium in bending. Because the top reinforcement at ends is at

low strain when concrete crushing takes place (compression-controlled

failure) and the beam is relieved from end hogging moments, the beam can


217

develop the longest period of catenary action. The beam behaves as one

without rotational restraint, thus the beam final failure time is unchanged with

changing end rotational stiffness.

- The aforementioned beam behaviour holds for different levels of axial

restraint and different load ratios. The only change is that the bending failure

mode II disappears when the load ratio is high. This is because it is difficult

for the beam ends to resist the high applied load after failure of the beam mid-

span.

- The traditional approach for estimating fire resistance of beams in fire, which

ignores compressive force generated by restraint to thermal expansion and

assumes that both the hogging and sagging bending moment resistances can

be reached together, may lead to overestimate of the bending resistance time

of axially restrained RC beams in fire.

Beams with asymmetrical restraint conditions:

The flexural behaviour of beams with asymmetrical end restraints is more

complicated than that with symmetrical ones. When the supports have high rotational

stiffness, a hinge mechanism may not happen in most cases because the support with

a higher rotational stiffness attracts a very high bending moment and fails while the

span bending moment is very low and has not reached its plastic moment capacity.

The hogging moment failure could be adversely affected by tensile thermal stresses

developed in the cooler mid reinforcing bars at the top of the beam in the hogging

region. To reduce this effect, top bars in the hogging region should be located close

to corner bars.

7.4 Development of a Simplified Method for Analysis of Axially and

Rotationally Restrained RC Beams in fire

Based on numerical simulation results in Chapter 4, a simplified calculation method

was proposed in this study to capture the whole history of structural behaviour of

axially and rotationally restrained RC beams in fire. The method is based on

sectional analysis and uses simple principles of equilibrium and compatibility

conditions.


218

One important assumption in the simplified analysis is the beam’s deflection profile.

In flexural action, the deflection profiles of simply supported and fixed end beams

under uniformly distributed load may be assumed for zero and fixed end rotational

restraints, respectively. An exponential interpolation between the deflection profiles

with zero and complete rotational restraints is assumed for the deflection profile of

beams with flexible end rotational restraints. In catenary action, the average of the

following three deflection profiles is used: (i) the deflection profile of the beam

under uniformly distributed load (UDL), (ii) the deflection profile of the beam under

a concentrated point load (CPL) at the mid-span and (iii) linear profile.

Simple and reliable approaches were suggested to predict the two critical rotational

restraints kr,B and kr,E (in Figure 5.2). kr,B corresponds to the rotational stiffness level

that gives the highest period of further bending resistance of the beam by the end

hogging moment resistance after the mid-span has reached its sagging moment

resistance. kr,E is the lowest rotational stiffness that leads the beam bending failure to

be controlled by hogging moment failure due to concrete crushing in compression.

Determination of kr,B and kr,E helps to predict the trends of changing beam bending

resistance time with increasing end rotational stiffness.

In the simplified model, only beams with kr≥kr,E are considered to be able to develop

catenary action. In catenary action, the following assumptions are made in the

proposed model:

The beam is rotationally free at the ends.

The location of the vertical and horizontal supports at the beam ends is at the

same location of the top longitudinal reinforcement.

The bottom reinforcing bars along the beam is not considered in the sectional

analysis.

A strain profile for the top reinforcement based on the maximum steel strain

of 0.05 is assumed to help to predict the ultimate resistance time of the beam

in catenary action, which is the time when fracture of the top reinforcement

at beam ends occurs.

7.5 Behaviour of RC Frames in Fire

The validated numerical simulation model was used to investigate structural

interactions between beams and columns within an RC frame structure with different


219

fire exposure scenarios. Structural failure of a frame with small column sizes is

mainly controlled by column failure. When fire exposure involves beams and

columns located in edge bays of a frame, due to thermal expansion of the connected

beam, additional bending moments can generate in the edge columns. Furthermore,

very large hogging moments may be induced at the beam end connected to the

internal bay. It is necessary to include these bending moments when designing beams

and columns under such fire conditions. Catenary action can develop in interior

beams of the frame when fire is in interior bays where the beams have high degrees

of axial restraint.

7.6 Recommendations for Future Studies

The following are some key recommendations for future research in this field:

1- The explicit numerical simulations conduced in this study have used solid

elements to model RC structural members in fire. Although solid elements

give more understanding of structural behaviour and accurate results, it

requires a substantial amount of computing time. For investigating global

behaviour of RC frames in fire, developing reduced finite element models

(macromodels) using fibre-beam elements and component-based method is

worthy of consideration.

2- The present finite element model is limited to RC structures subjected to

heating only. The model should be extended to enable both heating and

cooling phases to be modelled. This is achieved by modifying the concrete

constitutive law in order to predict irreversibility of the load-induced thermal

strain (LITS) occurring under compressive stress. Moreover, the model needs

modification to account for fire-induced spalling.

3- The present study has focused on behaviour of restrained RC beams in fire

under linear temperature-independent axial and rotational restraint stiffness.

Future studies should be performed using end boundary restraints offered by

adjacent restraining members in fire.

4- This study has ignored effects of large concrete cracks on temperature

distribution in RC member cross-sections. At large beam deflections in

catenary action, large cracks could alter insulation characteristics of concrete


220

to steel reinforcement. This should be considered since applied load in

catenary action is mainly sustained by steel reinforcement.

5- The effects of the cooling phase and spalling in the evaluation of the fire

resistance should be investigated as they are key requirements in the

performance-based approach.

6- Deriving simplified calculation methods to predict the effects of beam axial

compressive force in flexural action and beam axial tensile force in catenary

action on adjacent structures.

7- The study on the RC frames in fire should be extended to 3D frames.

8- Experimental investigation on fire response of RC beams in catenary action is

necessary to give more confidence to the numerical simulation models and

assumptions made in the simplified calculation method proposed in this

research.

APPENDIX

221

APPENDIX

A1 Beam Cross-Section Bending Moment Capacity

Figure A4.1: Determination of beam cross-section bending moment capacity

𝐴𝑠 = 3 ×𝜋

4× (

6

8× 25.4)

2

= 855𝑚𝑚2

𝑎 =𝐴𝑠 𝑓𝑦

0.85 𝑓𝑐𝑚 𝐵 =

855 × 453

0.85 × 30 × 300= 50.63𝑚𝑚

𝑀𝑢 = 𝐴𝑠 𝑓𝑦 (𝑑 − 𝑎 2⁄ ) = 855 × 453 (350 − 50.63 2⁄ ) = 126𝑘𝑁.𝑚

A2 Calculation of Rotational and Lateral Stiffness of Supporting

Columns

A2.1 Rotational Stiffness KR,sup

Figure A4.2: Determination of rotational stiffness of supporting columns KR,sup

Section a-a Beam ends

3#6 3#6

Section b-b Beam mid-span

B=300mm , D=400mm ,

d=350mm

fcm=30MPa , fy=453MPa

b

b

a

a

L/2=3000

C.L.

Ec

Ec 3

4

+

3

4𝐻

1

4

1

4

1

4

H

H

1 3

4𝐻(𝐻 − 𝐷/2)

3

4𝐻

D

_ θ

Ej

Compression reinforcement is not considered

Sagging moment=hogging moment

APPENDIX

222

𝜃 =

14(𝐻 − 𝐷/2)

𝐸𝑐 𝐼+

14𝐷2

𝐸𝑗 𝐼−

12 34𝐻

(𝐻 − 𝐷 2⁄ )(𝐻 − 𝐷 2)⁄

𝐸𝑐 𝐼−

34𝐻

(𝐻 − 𝐷 2⁄ )(𝐷 2)⁄

𝐸𝑗 𝐼

−

12 (

34−

34𝐻

(𝐻 − 𝐷 2⁄ ))𝐷2

𝐸𝑗 𝐼

𝐾𝑅,𝑠𝑢𝑝 = [1

𝜃]

𝐻 = 3500 𝑚𝑚 , 𝐷 = 400 𝑚𝑚, 𝐸𝑗 = 18000 𝑀𝑃𝑎 , 𝐼 =300×3003

12, 𝐸𝑐: 𝑐ℎ𝑎𝑛𝑔𝑒𝑑

Ej is the elastic modulus of the joint panel

A2.2 Lateral Stiffness KA,sup

Figure A4.3: Determination of lateral stiffness of supporting columns KA,sup

𝛿 =

12

12(𝐻 − 𝐷 2⁄ )(𝐻 − 𝐷 2⁄ ) (

𝐻 − 𝐷 2⁄3 + 𝐷 2⁄ )

𝐸𝑐 𝐼+

12(𝐻 − 𝐷 2⁄ )

𝐷2

𝐷4

𝐸𝑗 𝐼

+

12 (

𝐻2 −

12 (𝐻 −

𝐷2))

𝐷2

𝐷6

𝐸𝑗 𝐼−

𝐻4

(𝐻 − 𝐷 2⁄ ) ((𝐻 − 𝐷 2⁄

2 + 𝐷 2⁄ )

𝐸𝑐 𝐼−

𝐻4

𝐷2

𝐷4

𝐸𝑗 𝐼

𝐾𝐴,𝑠𝑢𝑝 = [1

𝛿]

𝐻 = 3500 𝑚𝑚 , 𝐷 = 400 𝑚𝑚, 𝐸𝑗 = 18000 𝑀𝑃𝑎 , 𝐼 =300 × 3003

12, 𝐸𝑐: 𝑐ℎ𝑎𝑛𝑔𝑒𝑑

𝐻

2

-

1

2

𝐻

4

𝐻

4

𝐻

4

H

H

1

1

2(𝐻 − 𝐷/2)

1

2

D

+

δ

Ec

Ec

Ej

APPENDIX

223

A3 Values of the Stiffness and Gaps for the Horizontal Restraints

during the Tests by Yu and Tan (2013) and Yu and Tan (2014)

Table A3.1 Sub-assemblages (Yu, 2012)

Horizontal

restraints

Tension

stiffness

(kN/m)

Compression

stiffness

(kN/m)

Tension Gap

(mm)

Compression

Gap (mm)

S4

Top 100571.92 – 1.8 –

Bottom 49255.37 175277.46 3.5 – 2.1

S5

Top 160834.3 – 3.0 –

Bottom 82649.59 195634.09 4.1 – 2.4

S7

Top 108723.92 – 0.3 –

Bottom 72955.43 157282.14 2.0 – 3.8

Table A3.2 Frames (Yu, 2012)

Horizontal

restraints

Tension

stiffness

(kN/m)

Compression

stiffness (kN/m)

Tension

Gap

(mm)

Compression

Gap (mm)

F2

At beam

extension

ends

At RW

side 153678 23941 4.2 0.8

At AF

side 60055 57985 0.8 2.3

At the

top of

side

columns

At RW

side 67634 ---- 3.5 1.9

At AF

side 54114 ---- 1.7 0.3

F4

At beam

extension

ends

At RW

side 121661 ---- 1.7 4.0

At AF

side 168161 ---- 4.4 2.3

At the

top of

side

columns

At RW

side 104312 6853.49 5.3 1.0

At AF

side 32479 6853.49 1.1 1.0

Top of side

columns Top of side

columns

RW side

Beam

extension ends Beam

extension ends

AF side AF: A Frame RW: Reaction wall

C. L. Top

Bottom

APPENDIX

224

A4 Material Properties According to Eurocode

A4.1 Free Thermal Strain εth

EN 1992 1-2 (CEN, 2004) gives the thermal strain for both siliceous and calcareous

normal-weight concrete (εth,c) through the following relationships as a function of

temperature:

For siliceous-aggregate normal-weight concrete:

휀𝑡ℎ,𝑐 = −1.8 × 10−4 + 9 × 10−6𝑇 + 2.3 × 10−11𝑇3 𝑓𝑜𝑟 20𝑜𝐶 ≤ 𝑇 ≤ 700𝑜𝐶

휀𝑡ℎ,𝑐 = 1.4 × 10−2 𝑓𝑜𝑟 700𝑜𝐶 ≤ 𝑇 ≤ 1200𝑜𝐶

For calcareous-aggregate normal-weight concrete:

휀𝑡ℎ,𝑐 = −1.2 × 10−4 + 6 × 10−6𝑇 + 1.4 × 10−11𝑇3 𝑓𝑜𝑟 20𝑜𝐶 ≤ 𝑇 ≤ 805𝑜𝐶

휀𝑡ℎ,𝑐 = 1.2 × 10−2 𝑓𝑜𝑟 805𝑜𝐶 ≤ 𝑇 ≤ 1200𝑜𝐶

The graphical representation of the respective relationship is presented in Figure

A4.4.

Figure A4.4: Thermal expansion of normal weight concrete with temperature

according to EN 1992 1-2 (CEN, 2004)

The free thermal strain of steel (εth,s) is directly related to the temperature rise and

gradually increases with temperature. EN 1992 1-2 (CEN, 2004) gives the following

relationships for computing thermal strain of carbon steels:

휀𝑡ℎ,𝑠 = −2.416 × 10−4 + 1.2 × 10−5𝑇 + 0.4 × 10−8𝑇2 for 20oC ≤ T ≤ 750oC

휀𝑡ℎ,𝑠 = 11 × 10−3 for 750oC ≤ T ≤ 860oC

휀𝑡ℎ,𝑠 = −6.2 × 10−3 + 2 × 10−5𝑇 for 860oC ≤ T ≤ 1200oC

The graphical representation of the respective relationship is presented in Figure

A4.5. Between 750-860oC, the gradual increase is halted, during which a phase

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 200 400 600 800 1000 1200

The

rmal

exp

ansi

on

(m

m/m

m)

Temperate (oC)

Siliceousaggregate

APPENDIX

225

transformation in the steel material takes place, before resuming again. The

linearised version of the model is defined in EN 1994-1-2 (CEN, 2005) as shown in

Figure A4.5.

Figure A4.5: Thermal expansion of carbon steel with temperature according to

EN 1992 1-2 (CEN, 2004)

A4.2 Instantaneous Stress-Related Strain εσ

Figure A4.6 shows the mathematical model for uniaxial compressive stress-strain

relationship of concrete at elevated temperatures according to EN 1992 1-2 (CEN,

2004). Tables A4.1 and A4.2 present the reduction factors of the parameters: ultimate

compressive stress (fcm,T), the corresponding strain (εc1,T) and the strain at zero stress

(εcu1,T) for normal-weight concrete with siliceous and calcareous aggregates and for

high strength concrete at high temperatures provided by EN 1992 1-2 (CEN, 2004).

For high strength concrete, fcm,T only varies with temperature, while εc1,T and εcu1,T

are the same as for normal strength concrete.

Figure A4.6: Mathematical model for uniaxial compressive stress-strain

relationship of concrete at elevated temperatures according to EN 1992 1-2

(CEN, 2004)

0

0.005

0.01

0.015

0.02

0 200 400 600 800 1000 1200

The

rmal

exp

ansi

on

(m

m/m

m)

Temperate (oC)

Carbon steelsCarbon steels (Linearised)

𝛆cu1,T 𝛆c1,T

𝒇𝒄𝒎,𝑻 𝜎𝑐,𝑇 = 3(

𝜀𝑐,𝑇𝜀𝑐1,𝑇

)/ 2 + (𝜀𝑐,𝑇𝜀𝑐1,𝑇

)

3

𝑓𝑐𝑚,𝑇

Compressive strain 𝛆c,T

Str

es

s 𝜎𝑐,𝑇

APPENDIX

226

Table A4.1: Parameters of stress-strain relationship of normal-weight concrete

at elevated temperatures according to EN 1992 1-2 (CEN, 2004)

Temperature

T (oC)

Siliceous aggregates Calcareous aggregates

𝑓𝑐𝑚,𝑇/𝑓𝑐𝑚 휀𝑐1,𝑇 εcu1,T 𝑓𝑐𝑚,𝑇/𝑓𝑐𝑚 εc1,T εcu1,T

20 1.00 0.0025 0.0200 1.00 0.0025 0.0200

100 1.00 0.0040 0.0225 1.00 0.0040 0.0225

200 0.95 0.0055 0.0250 0.97 0.0055 0.0250

300 0.85 0.0070 0.0275 0.91 0.0070 0.0275

400 0.75 0.0100 0.0300 0.85 0.0100 0.0300

500 0.60 0.0150 0.0325 0.74 0.0150 0.0325

600 0.45 0.0250 0.0350 0.60 0.0250 0.0350

700 0.30 0.0250 0.0375 0.43 0.0250 0.0375

800 0.15 0.0250 0.0400 0.27 0.0250 0.0400

900 0.08 0.0250 0.0425 0.15 0.0250 0.0425

1000 0.04 0.0250 0.0450 0.06 0.0250 0.0450

1100 0.01 0.0250 0.0475 0.02 0.0250 0.0475

1200 0.00 ----- ----- 0.00 ----- -----

Table A4.2: Strength reduction factors for high strength concrete at elevated

temperatures according to EN 1992 1-2 (CEN, 2004)

Figure A4.7 shows uniaxial compressive stress-strain relationship of concrete at

elevated temperatures for fcm=30MPa.

Temperature

T (oC)

𝑓𝑐𝑚,𝑇/𝑓𝑐𝑚

Class 1 Class 2 Class 3

20 1.00 1.00 1.00

50 1.00 1.00 1.00

100 0.90 0.75 0.75

200 0.90 0.75 0.70

250 0.90 0.75 0.675

300 0.85 0.75 0.65

400 0.75 0.75 0.45

500 0.60 0.60 0.30

600 0.45 0.45 0.25

700 0.30 0.30 0.20

800 0.15 0.15 0.15

900 0.08 0.113 0.08

1000 0.04 0.075 0.04

1100 0.01 0.038 0.01

1200 0.00 0.00 0.00

Class 1 for concrete C55/67 and C60/75

Class 2 for concrete C70/85 and C80/95

Class 3 for concrete C90/105

APPENDIX

227

Figure A4.7 Uniaxial compressive stress-strain relationship of concrete at

elevated temperatures (f′c=30 MPa) according to EN 1992-1-2 (CEN, 2004)

EN 1992 1-2 (CEN, 2004) provides a detailed model for specifying stress-strain

curves of steel at elevated temperatures which are defined by three temperature-

dependent parameters: Elastic modulus (Ea,T), proportional limit (fp,T) and effective

yield strength (fy,T). Figure A4.8 depicts the general model that can be constructed by

the following mathematical formulae:

where,

휀𝑝,𝑇 = 𝑓𝑝,𝑇/𝐸𝑎,𝑇 is the strain at the proportional limit

휀𝑦,𝑇 is the yield strain

휀𝑡,𝑇 is the limiting strain for yield strength

휀𝑢,𝑇 is the ultimate strain

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05

Tem

per

atu

re, T

(oC

)

Strain (mm/mm)

20

100

300

500

700

900

1100

f′c=30 MPa

APPENDIX

228

Values of parameters 𝐸𝑎,𝑇, 𝑓𝑝,𝑇 and 𝑓𝑦,𝑇 are obtained by applying the reduction

factors for hot-rolled carbon steels presented in Table A4.3.

Figure A4.8: General stress-strain relationship of steel reinforcement at

elevated temperatures according to EN 1992 1-2 (CEN, 2004)

Table A4.3: Values of the main parameters of the stress-strain relationships of

hot-rolled carbon steel according to EN 1992 1-2 (CEN, 2004)

Temperature

T (oC)

Yield strength

𝑘𝑦,𝑇 = 𝑓𝑦,𝑇/𝑓𝑦

Proportional limit

𝑘𝑝,𝑇 = 𝑓𝑝,𝑇/𝑓𝑦

Elastic modulus

𝑘𝐸,𝑇 = 𝐸𝑎,𝑇/𝐸𝑎

20 1.000 1.000 1.000

100 1.000 1.000 1.000

200 1.000 0.807 0.900

300 1.000 0.613 0.800

400 1.000 0.420 0.700

500 0.780 0.360 0.600

600 0.470 0.180 0.310

700 0.230 0.075 0.130

800 0.110 0.050 0.090

900 0.060 0.0375 0.0675

1000 0.040 0.0250 0.0450

1100 0.020 0.0125 0.0225

1200 0.000 0.0000 0.0000

where, Ea is modulus of elasticity of steel at ambient temperature

𝜀𝑢,𝑇 𝜀𝑝,𝑇 𝜀𝑦,𝑇

𝑓𝑦,𝑇

𝑓𝑝,𝑇

𝜀𝑡,𝑇

𝐸𝑎,𝑇

Strain 𝜀𝑠,𝑇

Str

es

s 𝜎

APPENDIX

229

A4.3 Specific Heat

The specific heat of concrete cp provided in EN 1992-1-2 (CEN, 2004), which is

valid for both types of normal-weight concrete siliceous and calcareous aggregate

concrete (in J/kg K) is as follows:

𝑐𝑝 = 900 𝑓𝑜𝑟 20𝑜𝐶 ≤ 𝑇 ≤ 100𝑜𝐶

𝑐𝑝 = 900 + (𝑇 − 100) 𝑓𝑜𝑟 100𝑜𝐶 ≤ 𝑇 ≤ 200𝑜𝐶

𝑐𝑝 = 1000 + (𝑇 − 200)/2 𝑓𝑜𝑟 200𝑜𝐶 ≤ 𝑇 ≤ 400𝑜𝐶

𝑐𝑝 = 1100 𝑓𝑜𝑟 400𝑜𝐶 ≤ 𝑇 ≤ 1200𝑜𝐶

EN 1992-1-2 (CEN, 2004) considers the effect of moisture in concrete explicitly

through adding a peak cp,peak to the above equations at 115𝑜𝐶 and then decaying 𝑐𝑝

linearly to 1000 kJ/kg K at 200𝑜. Table A4.4 gives cp,peak values for different

moisture contents (linear interpolation between them is permitted), and Figure A4.9

plots the evolution of specific heat with temperature.

Table A4.4: Values of cp,peak for different moisture contents given by EN 1992-1-

2 (CEN, 2004)

Moisture content u (%) cp,peak (J/kgK)

0 900

1.5 1470

3.0 2020

10.0 5600

Figure A4.9: Variation of the specific heat capacity of concrete with

temperature according to EN 1994-1-2 (CEN, 2004)

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 200 400 600 800 1000 1200

Spe

cifi

c h

eat

cap

acit

y (k

J/kg

.K)

Temperate (oC)

u=0%

u=3%

u=1.5%

APPENDIX

230

A4.4 Density

Th empirical model given in EN 1994-1-2 (CEN, 2005) that describes the density of

concrete 𝜌𝑐,𝑇 as a function of temperature is used as follows:

𝜌𝑐,𝑇 = 𝜌𝑐,20 𝑓𝑜𝑟 20𝑜𝐶 ≤ 𝑇 ≤ 115𝑜𝐶

𝜌𝑐,𝑇 = 𝜌𝑐,20[1 − 0.02(𝑇 − 115)/85] 𝑓𝑜𝑟 115𝑜𝐶 ≤ 𝑇 ≤ 200𝑜𝐶

𝜌𝑐,𝑇 = 𝜌𝑐,20[0.98 − 0.03(𝑇 − 200)/200] 𝑓𝑜𝑟 200𝑜𝐶 ≤ 𝑇 ≤ 400𝑜𝐶

𝜌𝑐,𝑇 = 𝜌𝑐,20[0.95 − 0.07(𝑇 − 400)/800] 𝑓𝑜𝑟 400𝑜𝐶 ≤ 𝑇 ≤ 1200𝑜𝐶

where, 𝜌𝑐,20 is concrete density at ambient temperature taken to be 2300 kg/m3

A4.5 Thermal Conductivity

The conductivity values (W/mK) lie between the following upper and lower limits,

which are graphically shown in Figure A4.10

upper limit: 𝜆𝑐 = 2 − 0.2451 (𝑇

100) + 0.0107(

𝑇

100)2

lower limit: 𝜆𝑐 = 1.36 − 0.136 (𝑇

100) + 0.0057(

𝑇

100)2

Figure A4.10 Thermal conductivity of concrete as a function of temperature

according to EN 1994-1-2 (CEN, 2004)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

0 200 400 600 800 1000 1200

The

rmal

co

nd

uct

ivit

y (W

/mK

)

Temperate (oC)

Upper limit

Lower limit

REFERENCES

231

REFERENCES

ABAQUS 2013. ABAQUS analysis user’s manual, version 6.13-1, ABAQUS Inc.

ALI, F., NADJAI, A. & CHOI, S. 2010. Numerical and experimental investigation of the

behavior of high strength concrete columns in fire. Engineering Structures, 32,

1236-1243.

ASTM E119 2008. Standard methods of fire test of building construction and materials.

American society of testing and materials, west Conshohocken, PA.

BAILEY, C. 2002. Holistic behaviour of concrete buildings in fire. Proceedings of the ICE -

Structures and Buildings, 152, 199-212.

BAO, Y., LEW, H. & KUNNATH, S. 2014. Modeling of Reinforced Concrete Assemblies

under Column-Removal Scenario. Journal of Structural Engineering, 140,

04013026.

BAŽANT, Z. & CHERN, J. 1987. Stress‐Induced Thermal and Shrinkage Strains in

Concrete. Journal of Engineering Mechanics, 113, 1493-1511.

BEITEL, J. & IWANKIW, N. 2002. Analysis of Needs and Existing Capabilities for Full-

Scale Fire Resistance Testing. NIST GCR 02-843.

BIONDINI, F. & NERO, A. 2011. Cellular Finite Beam Element for Nonlinear Analysis of

Concrete Structures under Fire. Journal of Structural Engineering, 137, 543-558.

BRATINA, S., PLANINC, I., SAJE, M. & TURK, G. 2003. Non-linear fire-resistance

analysis of reinforced concrete beams. Structural engineering and mechanics, 16,

695-712.

BRE 2003. Building Research Establishment. Client report: results an observations from

full-scale fire test at BRE Cardington.

BUCHANAN, A. H. 2002. Structural design for fire safety, John Wiley & Sons Ltd.,

Chichester, England.

CAPUA, D. D. & MARI, A. R. 2007. Nonlinear analysis of reinforced concrete cross-

sections exposed to fire. Fire Safety Journal, 42, 139-149.

REFERENCES

232

CEB 1991. Comite Euro-International du Beton (CEB). CEB-FIP model code 1990 design

code, Thomas Telford, London, UK.

CEN 2002. BS EN 1991-1-2: Eurocode 1: Actions on structures: Part 1-2: General actions –

Actions on structures exposed to fire. British Standards Institution.

CEN 2004. BS EN 1992-1-2: Eurocode 2: Design of concrete structures: Part 1-1: General

rules-structural fire design. British Standards Institution.

CEN 2005. BS EN 1994-1-2: Design of composite steel and concrete structures: Part 1-2:

General rules-Structural fire design . British Standards Institution.

CHOI, E. G. & SHIN, Y. S. 2011. The structural behavior and simplified thermal analysis of

normal-strength and high-strength concrete beams under fire. Engineering

Structures, 33, 1123-1132.

CHOI, H. & KIM, J. 2011. Progressive collapse-resisting capacity of RC beam-column sub-

assemblage. Magazine of Concrete Research, 63(4), 297-310.

DOTREPPE, J. C. & FRANSSEN, J. M. 1985. The use of numerical models for the fire

analysis of reinforced concrete and composite structures. Engineering Analysis, 2,

67-74.

DWAIKAT, M. 2009. Flexural response of reinforced concrete beams exposed to fire. PhD

thesis, Miscigan State University.

DWAIKAT, M. & KODUR, V. 2009a. Response of Restrained Concrete Beams under

Design Fire Exposure. Journal of Structural Engineering, 135, 1408-1417.

DWAIKAT, M. B. & KODUR, V. K. R. 2008. A numerical approach for modeling the fire

induced restraint effects in reinforced concrete beams. Fire Safety Journal, 43, 291-

307.

DWAIKAT, M. B. & KODUR, V. K. R. 2009b. Hydrothermal model for predicting fire-

induced spalling in concrete structural systems. Fire Safety Journal, 44, 425-434.

EL-FITIANY, S. F. & YOUSSEF, M. A. 2014a. Interaction diagrams for fire-exposed

reinforced concrete sections. Engineering Structures, 70, 246-259.

EL-FITIANY, S. F. & YOUSSEF, M. A. 2014b. Simplified Method to Analyze Continuous

Reinforced Concrete Beams during Fire Exposure. ACI Structural Journal, 111(1),

145-156.

REFERENCES

233

ELLINGWOOD, B. & LIN, T. 1991. Flexure and Shear Behavior of Concrete Beams during

Fires. Journal of Structural Engineering, 117, 440-458.

ELLOBODY, E. & BAILEY, C. G. 2009. Modelling of unbonded post-tensioned concrete

slabs under fire conditions. Fire Safety Journal, 44, 159-167.

ELLOBODY, E. A. M. & BAILEY, C. G. 2008. Modelling of bonded post-tensioned

concrete slabs in fire. Proceedings of the Institution of Civil Engineers - Structures

and Buildings, 161, 311-323.

ERVINE, A., GILLIE, M., STRATFORD, T. J. & PANKAJ, P. 2012. Thermal Propagation

through Tensile Cracks in Reinforced Concrete. Journal of Materials in Civil

Engineering, 24, 516-522.

GAO, W.-Y., DAI, J.-G. & TENG, J.-G. 2017. Fire resistance of RC beams under design fire

exposure. Magazine of Concrete Research, 69, 402-423.

GAO, W. Y., DAI, J.-G., TENG, J. G. & CHEN, G. M. 2013. Finite element modeling of

reinforced concrete beams exposed to fire. Engineering Structures, 52, 488-501.

GERNAY, T. 2012. Effect of Transient Creep Strain Model on the Behavior of Concrete

Columns Subjected to Heating and Cooling. Fire Technology, 48, 313-329.

GERNAY, T. & FRANSSEN, J. M. 2012. A formulation of the Eurocode 2 concrete model

at elevated temperature that includes an explicit term for transient creep. Fire Safety

Journal, 51, 1-9.

GUO, Z. & SHI, X. 2011. Chapter 10 - Behavior of Statically Indeterminate Structures at

Elevated Temperatures. Experiment and Calculation of Reinforced Concrete at

Elevated Temperatures. Boston: Butterworth-Heinemann.

HERTZ, K. D. 2003. Limits of spalling of fire-exposed concrete. Fire Safety Journal, 38,

103-116.

HOU, J. & SONG, L. 2016. Progressive Collapse Resistance of RC Frames under a Side

Column Removal Scenario: The Mechanism Explained. International Journal of

Concrete Structures and Materials, 10, 237-247.

HUANG, Z. 2010. Modelling of reinforced concrete structures in fire. Proceedings of the

ICE - Engineering and Computational Mechanics,163, 43-53.

REFERENCES

234

HUANG, Z., BURGESS, I. & PLANK, R. 2009. Three-Dimensional Analysis of Reinforced

Concrete Beam-Column Structures in Fire. Journal of Structural Engineering, 135,

1201-1212.

ISO 834-1975. 1975 Fire resistance tests - elements of building construction. International

Organization for Standardization.

KANG, S.-W. & HONG, S.-G. 2004. Analytical method for the behaviour of a reinforced

concrete flexural member at elevated temperatures. Fire and Materials, 28, 227-235.

KHOURY, G. A. 2000. Effect of fire on concrete and concrete structures. Progress in

Structural Engineering and Materials, 2, 429-447.

KIM, J. & YU, J. 2012. Analysis of reinforced concrete frames subjected to column loss.

Magazine of Concrete Research, 64, 21-33.

KODUR, V. & DWAIKAT, M. 2012. Fire-induced spalling in reinforced concrete beams.

Proceedings of the Institution of Civil Engineers - Structures and Buildings, 165,

347-359.

KODUR, V., DWAIKAT, M. & RAUT, N. 2009. Macroscopic FE model for tracing the fire

response of reinforced concrete structures. Engineering Structures, 31, 2368-2379.

KODUR, V. K. R. & AGRAWAL, A. 2016. An approach for evaluating residual

capacity of reinforced concrete beams exposed to fire. Engineering

Structures, 110, 293-306.

KODUR, V. K. R. & DWAIKAT, M. 2007. Performance-based Fire Safety Design of

Reinforced Concrete Beams. Journal of Fire Protection Engineering, 17, 293-320.

KODUR, V. K. R. & DWAIKAT, M. 2008. A numerical model for predicting the fire

resistance of reinforced concrete beams. Cement and Concrete Composites, 30, 431-

443.

KODUR, V. K. R. & DWAIKAT, M. B. 2011. Design equation for predicting fire resistance

of reinforced concrete beams. Engineering Structures, 33, 602-614.

LEW, YIHAI BAO, SANTIAGO PUJOL & METE, A. S. 2014. Experimental Study of

Reinforced Concrete Assemblies under Column Removal Scenario. ACI Structural

Journal, 111(4), 881-892.

REFERENCES

235

LI, S., SHAN, S., ZHAI, C. & XIE, L. 2016. Experimental and numerical study on

progressive collapse process of RC frames with full-height infill walls. Engineering

Failure Analysis, 59, 57-68.

LIN T.D, GUSTAFEROO A. H & ABRAMS M. S 1981. Fire Endurance of Continuous

Reinforced Concrete Beams. R&D Bulletin RD072.01B, Portland Cement

Association, IL, USA.

LUE TAERWE 2007. From Member Design to Global Structural Behaviour. Internaltion

workshop, Fire Design of Concrete Structures-From Materials Modelling to

Structural Performance, University of Coimbra-Portugal-November.

MAEKAWA, K., A. PIMANMAS & H. OKAMURA 2003. Nonlinear mechanics of

reinforced concrete. Spon Press Taylor & Francis Group.

NEVES, I. C., RODRIGUES, J. P. C. & LOUREIRO, A. D. P. 1996. Mechanical Properties

of Reinforcing and Prestressing Steels after Heating. Journal of Materials in Civil

Engineering, 8, 189-194.

ORTON, J. O. JIRSA & BAYRAK, O. 2009. Carbon Fiber-Reinforced Polymer for

Continuity in Existing Reinforced Concrete Buildings Vulnerable to Collapse. ACI

Structural Journal, 106(5), 608-616.

OŽBOLT, J., BOŠNJAK, J., PERIŠKIĆ, G. & SHARMA, A. 2014. 3D numerical analysis

of reinforced concrete beams exposed to elevated temperature. Engineering


PHAM, A. T., TAN, K. H. & YU, J. 2017. Numerical investigations on static and dynamic

responses of reinforced concrete sub-assemblages under progressive collapse.

Engineering Structures, 149, 2-20.

PURKISS, J. A. 2007. Fire safety engineering design of structures. Elsevier Press.

QIAN, K., LI, B. & MA, J.-X. 2015. Load-Carrying Mechanism to Resist Progressive

Collapse of RC Buildings. Journal of Structural Engineering, 141(2): 04014107-1.

RAOUFFARD, M. M. & NISHIYAMA, M. 2015. Fire Resistance of Reinforced Concrete

Frames Subjected to Service Load: Part 1. Experimental Study. Journal of Advanced

Concrete Technology, 13, 554-563.

REGAN, P. 1975. Catenary action in damage concrete structures. ACI Special Publication,

48, 191-224.

REFERENCES

236

RIVA, P. & FRANSSEN, J. M. 2008. Non-linear and plastic analysis of RC beams subjected

to fire. Structural Concrete, 9, 31-43.

SADEK, F., MAIN, J. A., LEW, H. S. & BAO, Y. 2011. Testing and Analysis of Steel and

Concrete Beam-Column Assemblies under a Column Removal Scenario. Journal of

Structural Engineering, 137, 881-892.

SALEM & MAEKAWA, K. 2004. Pre- and Postyield Finite Element Method Simulation of

Bond of Ribbed Reinforcing Bars. Journal of Structural Engineering, 130, 671-680.

SHI, X. & GUO, Z. 1997. Experimental investigation of behavior of reinforced concrete

continuous beams at elevated temperature. China Civil Engineering Journal, 30, 26-

34.

SHI, X., TAN, T.-H., TAN, K.-H. & GUO, Z. 2004. Influence of Concrete Cover on Fire

Resistance of Reinforced Concrete Flexural Members. Journal of Structural

Engineering, 130, 1225-1232.

STEVENS, S. M. UZUMERI, COLLINS, M. P. & WILL, G. T. 1991. Constitutive Model

for Reinforced Concrete Finite Element Analysis. ACI Structural Journal, 88(1), 49-

59.

STINGER, S. M. & ORTON, S. L. 2013. Experimental Evaluation of Disproportionate

Collapse Resistance in Reinforced Concrete Frames. ACI Structural Journal, 110(3),

521-530.

TORELLI, G., GILLIE, M., MANDAL, P. & TRAN, V.-X. 2017. A multiaxial load-induced

thermal strain constitutive model for concrete. International Journal of Solids and


TORELLI, G., MANDAL, P., GILLIE, M. & TRAN, V.-X. 2016. Concrete strains under

transient thermal conditions: A state-of-the-art review. Engineering Structures, 127,

172-188.

USMANI, A. S., ROTTER, J. M., LAMONT, S., SANAD, A. M. & GILLIE, M. 2001.

Fundamental principles of structural behaviour under thermal effects. Fire Safety

Journal, 36, 721-744.

WANG, Y. C., BURGESS, I. W., WALD, F. & GILLIE, M. 2013. Performance-based fire

engineering of structures. London, Taylor & Francis Group.

REFERENCES

237

WU, B. & LU, J. Z. 2009. A numerical study of the behaviour of restrained RC beams at

elevated temperatures. Fire Safety Journal, 44, 522-531.

XU, Q., HAN, C., WANG, Y. C., LI, X., CHEN, L. & LIU, Q. 2015. Experimental and

numerical investigations of fire resistance of continuous high strength steel

reinforced concrete T-beams. Fire Safety Journal, 78, 142-154.

YOUSSEF, M. A. & MOFTAH, M. 2007. General stress–strain relationship for concrete at

elevated temperatures. Engineering Structures, 29, 2618-2634.

YU, J. 2012. Structural behaviour of reinforced concrete frames subjected to progressive

collapse. PhD thesis, Nanyang Technological University.

YU, J. & TAN, K. 2013. Structural Behavior of RC Beam-Column Subassemblages under a

Middle Column Removal Scenario. Journal of Structural Engineering, 139, 233-

250.

YU, J. & TAN, K. 2014. Special Detailing Techniques to Improve Structural Resistance

against Progressive Collapse. Journal of Structural Engineering, 140, 04013077.

ZHA, X. X. 2003. Three-dimensional non-linear analysis of reinforced concrete members in

fire. Building and Environment, 38, 297-307.

ZHAOHUI, H. & ANDREW, P. 1997. Nonlinear Finite Element Analysis of Planar

Reinforced Concrete Members Subjected to Fires. ACI Structural Journal, 94(3),

272-281.

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