LIST OF CONTENTS - University of Manchester

CELLULAR BEAMS AT AMBIENT AND ELEVATED TEMPERATURES

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LIST OF CONTENTS

LIST OF CONTENTS 2 LIST OF TABLES 6 LIST OF FIGURES 7 NOMENCLATURE 14 ABSTRACT 16 COPYRIGHT 18 ACKNOWLEDGEMENTS 20

1 INTRODUCTION 21 1.1 BACKGROUND 21 1.2 OBJECTIVES 22 1.3 OUTLINE OF THE THESIS 23

2 LITERATURE REVIEW 26 2.1 INTRODUCTION 26 2.2 FIRE DESIGN OF STRUCTURES 26

2.2.1 Prescriptive method 27 2.2.2 Performance based method 29

2.3 BEAMS WITH WEB OPENINGS 36 2.3.1 Fabrication of cellular beams 37 2.3.2 Cellular beams versus castellated beams 37

2.4 INVESTIGATIONS ON BEAMS WITH WEB OPENINGS 39 2.4.1 Experimental studies on castellated and cellular beams 40 2.4.2 Experimental studies on cellular composite beams at ambient

temperature 42 2.4.3 Experimental studies on cellular composite floors at elevated

temperature 43 2.5 DESIGN OF CELLULAR BEAMS AT AMBIENT

TEMPERATURE 50 2.5.1 Web post buckling 51 2.5.2 Vierendeel mechanism 55

2.6 DESIGN OF CELLULAR COMPOSITE BEAMS AT AMBIENT TEMPERATURE 57

2.7 DESIGN OF CELLULAR COMPOSITE BEAMS AT ELEVATED TEMPERATURES 59 2.7.1 Temperature distribution in cellular composite beams 60

2.8 INTRODUCTION TO THE FINITE ELEMENT ANALYSIS 60

3 CELLULAR BEAMS AT AMBIENT TEMPERATURE 63 3.1 INTRODUCTION 63 3.2 MODELLING CELLULAR BEAMS 63 3.3 MODELLING NATAL BEAM 4B 64

3.3.1 Introduction 64


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3.3.2 Modelling using ANSYS and ABAQUS 65 3.3.3 Modelling in ABAQUS 66 3.3.4 Modelling in ANSYS 67 3.3.5 Sensitivity study on mesh size 68 3.3.6 Analysis types 69 3.3.7 Applying imperfections 71 3.3.8 Numerical modelling results for Natal Beam 4B 74

3.4 MODELLING LEEDS BEAM 2 76 3.4.1 Introduction 76 3.4.2 Modelling approach for Leeds Beam 2 77 3.4.3 Comparison between ABAQUS and ANSYS 79 3.4.4 Modified modelling approach for Leeds Beam 2 81 3.4.5 Importance of boundary conditions in modelling CBs 83 3.4.6 Importance of load increments in modelling CBs 98

3.5 MODELLING LEEDS BEAM 3 103 3.5.1 Introduction and modelling 103 3.5.2 Numerical versus test results 104 3.5.3 Internal stresses of web post while buckling 106

3.6 SUMMARY OF THE MODELLING OF CELLULAR BEAMS AT AMBIENT TEMPERATURE 108

4 CELLULAR COMPOSITE BEAMS AT AMBIENT TEMPERATURE 110 4.1 INTRODUCTION 110 4.2 MODELLING COMPOSITE SLABS USING ABAQUS 111

4.2.1 General 111 4.2.2 Modelling concrete material 112

4.3 MODELLING ULSTER BEAM A1 119 4.3.1 Introduction and objectives 119 4.3.2 Details of the modelling approach 119 4.3.3 Numerical results for the Beam A1 123 4.3.4 Investigating stress distribution at failure 128

4.4 MODELLING ULSTER BEAM B1 132 4.4.1 Modelling 132 4.4.2 Results of modelling for Beam B1 133

4.5 MODELLING RWTH BEAM 3 138 4.5.1 Details of the test 138 4.5.2 Numerical model 139 4.5.3 Numerical results against the test results 142

4.6 SUMMARY OF THE MODELLING OF CELLULAR COMPOSITE BEAMS AT AMBIENT TEMPERATURE 144

5 PARAMETRIC STUDIES ON CELLULAR COMPOSITE BEAMS AT AMBIENT TEMPERATURE 145 5.1 INTRODUCTION 145 5.2 EFFECT OF LOADING TYPE 145

5.2.1 Effect of loading type on symmetric beam 145 5.2.2 Effect of loading type on asymmetric beam 148


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5.3 EFFECT OF TENSILE BEHAVIOUR OF CONCRETE 150 5.4 EFFECT OF WEB IMPERFECTIONS 151

5.4.1 On symmetric Beam A1 151 5.4.2 On asymmetric Beam B1 152

5.5 SUMMARY OF THE CASE STUDIES ON THE CELLULAR COMPOSITE BEAMS AT AMBIENT TEMPERATURE 153

6 CELLULAR COMPOSITE BEAMS AT ELEVATED TEMPERATURE155 6.1 INTRODUCTION 155 6.2 ULSTER FIRE TESTS 156 6.3 NUMERICAL MODELLING AND RESULTS FOR ULSTER

BEAM A2 158 6.3.1 Modelling 158 6.3.2 Comparing modelling and test results for Beam A2 168 6.3.3 Investigating the internal forces 171

6.4 NUMERICAL MODELLING AND RESULTS FOR ULSTER BEAM B2 174 6.4.1 Introduction 174 6.4.2 Modelling and results 176

6.5 MODELLING AND RESULTS OF FIRE TEST FOR ULSTER BEAM A3 178 6.5.1 Introduction 178 6.5.2 Numerical modelling of Beam A3 179 6.5.3 Results of numerical modelling for Beam A3 180

6.6 NUMERICAL MODELLING AND RESULTS FOR ULSTER BEAM B3 184 6.6.1 Introduction 184 6.6.2 Numerical modelling and results for Ulster Beam B3 184

6.7 SUMMARY OF THE MODELLING AT ELEVATED TEMPERATURE 190

7 PARAMETRIC STUDIES ON CELLULAR COMPOSITE BEAMS AT ELEVATED TEMPERATURE 191 7.1 INTRODUCTION 191 7.2 WEB IMPERFECTIONS 191 7.3 TEMPERATURE DISTRIBUTION 194

7.3.1 Effect on symmetric beam 195 7.3.2 Effect on asymmetric beam 198

7.4 LOADING TYPE AND WEB STIFFENER 200 7.4.1 Effect on symmetric beam 200 7.4.2 Effect on asymmetric beam 206

7.5 LOAD RATIO 209 7.6 THICKNESS OF WEB STIFFENER 211 7.7 SUPPORT CONDITIONS 212

7.7.1 Effect on symmetric beams under point load 212 7.7.2 Effect on symmetric beams under a UDL 215 7.7.3 Effect on asymmetric beams under point load 216


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7.7.4 Effect on asymmetric beams under a UDL 218 7.7.5 Interaction between cellular beam and end supports (catenary

action) 224 7.8 SUMMARY OF THE CASE STUDIES ON CELLULAR

COMPOSITE BEAMS AT ELEVATED TEMPERATURE 229

8 CONCLUSIONS & RECOMMENDATIONS 231 8.1 CONCLUSIONS 231 8.2 RECOMMENDATIONS AND FUTURE WORK 236 REFERENCES 238

Main text word count: 42,672


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LIST OF TABLES

Table 3.1- Displacements obtained for Beam 3 93 Table 3.2- Displacements obtained for Beam 4 93 Table 4.1- Tensile test results for Beam A1 120 Table 4.2- Average values gained from the tensile test and used in modelling 140


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LIST OF FIGURES

Figure 2.1-Nominal fire curves based on BSEN1991-1-2 and PD7974 (Source: http://www.mace.manchester.ac.uk/project/research/structures/strucfire/default.htm)............................................................................................................... 27

Figure 2.2- Time-temperature curve of a natural fire and the 5 phases.................. 31 Figure 2.3- Angelina beam produced by Arcelomittal ........................................... 36 Figure 2.4- Cellular beams fabricated from steel plates used in Fabsec (Source:

www.fabsec.co.uk) ......................................................................................... 37 Figure 2.5- Cutting pattern of Castellated and Cellular beams............................... 39 Figure 2.6- Comparing use of infill plates in cellular and castellated beams......... 39 Figure 2.7- Geometry of the CTICM Beam P1 ...................................................... 44 Figure 2.8- Geometry of the CTICM Beam P2 ...................................................... 45 Figure 2.9- Applying various fire protections at various sections of the CTICM

Beam P2.......................................................................................................... 45 Figure 2.10- Failure of the protected CTICM Beam P2 due to web post buckling

after removing the fire protection (Source: test report (Joyeux, 2003)) ......... 46 Figure 2.11- Indicative furnace test on unloaded protected cellular and solid beams

with similar geometric details......................................................................... 47 Figure 2.12- Steel being exposed at the proximity of the holes as the protection fell

off.................................................................................................................... 48 Figure 2.13- Full scale compartment test by the University of Ulster with cellular

beams .............................................................................................................. 50 Figure 2.14- Forces applied to a web post and the critical section to check to web

post.................................................................................................................. 52 Figure 2.15- Design curves proposed in P100 to check the flexural and buckling

capacity of the web post.................................................................................. 53 Figure 2.16- The ‘Strut’ model for web post buckling proposed by Lawson......... 54 Figure 2.17- CTICM model for the web post buckling .......................................... 55 Figure 2.18- Different approaches to consider the critical section in the Tees ...... 56 Figure 2.19- The circular opening represented by an equivalent rectangle (for the

Vierendeel bending check) and applied forces ............................................... 58 Figure 2.20- The Riks method ................................................................................ 62 Figure 3.1- Details of Natal Beam 4B .................................................................... 65 Figure 3.2- Half of the Natal Beam 4B modelled in ABAQUS ............................. 66 Figure 3.3- Material specifications used in FE model ............................................ 67 Figure 3.4- Half of the Natal Beam 4B modelled in ANSYS................................. 68 Figure 3.5 - Sensitivity study on the mesh size, Natal Beam 4B............................ 69 Figure 3.6- Load steps applied to the Natal Beam 4B in which load has increased

with a high rate in elastic region and low rate while plasticity....................... 70 Figure 3.7-Comparison between General and Arc-length method ......................... 71 Figure 3.8- First buckling mode of Natal Beam 4B gained by ABAQUS buckling

analysis............................................................................................................ 73 Figure 3.9- Comparison of the load-deflection curves according to the test and the

results obtained from ANSYS and ABAQUS ................................................ 74


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Figure 3.10- Web post buckling of Natal Beam 4B using ABAQUS .................... 75 Figure 3.11- Development of S-shape web post buckling of the Natal Beam 4B

using ANSYS.................................................................................................. 75 Figure 3.12- Geometry of the Leeds Beam 2.......................................................... 76 Figure 3.13- (a) assumed stress-strain curve for Leeds Beam 2; (b) The (half) beam

geometry and B.C.s......................................................................................... 78 Figure 3.14- Test results versus ANSYS and ABAQUS results for Leeds Beam 2

assuming E= 205 kN/mm2 .............................................................................. 79 Figure 3.15- Test results versus ABAQUS for Leeds Beam 2 assuming E= 195

kN/mm2 ........................................................................................................... 81 Figure 3.16- Development of Von Mises stress with load increase ....................... 82 Figure 3.17- Boundary conditions applied to Beam 1 and Beam 2, for comparison

purposes .......................................................................................................... 83 Figure 3.18- Load-deflection curves for Beam 1, Beam 2 and the test .................. 84 Figure 3.19-Models developed to investigate the influence of the boundary

condition of the top flange on the behaviour of castellated beams................. 85 Figure 3.20- Failure due to excessive shear deformation of the web post in Model 1

and Model 2 .................................................................................................... 86 Figure 3.21- Failure due to S-shaped web post buckling in Model 3 ..................... 86 Figure 3.22- Out-of-web plane displacement contour ............................................ 87 Figure 3.23- Displacements of points A, B and C of Beam 1 and Beam 2 ............ 88 Figure 3.24- The beam and the applied material properties ................................... 89 Figure 3.25- Load-displacement comparison for different cases ........................... 90 Figure 3.26- Boundary conditions applied to beams 3 and 4 ................................. 90 Figure 3.27- Comparison of the load-deflection curves for the four beams........... 91 Figure 3.28- Sections investigated and naming used in details .............................. 92 Figure 3.29 Rotation of the top flange in Beam 3 (continuous restrain) compared to

Beam 4 (two point restraints).......................................................................... 94 Figure 3.30-Two samples of reaction compressive force applied by lateral supports

per node........................................................................................................... 95 Figure 3.31- Internal forces and interaction with lateral supports .......................... 96 Figure 3.32- Effect of lateral supports in the web buckling in numerical model ... 96 Figure 3.33- Expansion and shortening of the top and bottom flange respectively in

the solid webbed beam.................................................................................... 98 Figure 3.34- Out-of-web plane displacement contours indicating buckling and no

buckling in Beam 1 and Beam 2 respectively............................................... 100 Figure 3.35- Load-deflections curves of beams one 1 and 2 ................................ 100 Figure 3.36- Typical bilinear curve for steel material and importance of load

increment ...................................................................................................... 102 Figure 3.37- A conceptual model to look at web post buckling ........................... 103 Figure 3.38- Half of Leeds Beam 3 modelled in ABAQUS benefiting from

symmetry ...................................................................................................... 104 Figure 3.39- Experimental versus modelling results for Leeds Beam 3............... 105 Figure 3.40- Web Buckling of the web post near end support in the long span

Leeds Beam 3................................................................................................ 106


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Figure 3.41-Maximum principal stress which represent the tensile stress while buckling ........................................................................................................ 107

Figure 3.42-Minimum principal stress which represents the developed compression stress while buckling..................................................................................... 108

Figure 4.1 Typical Stress-strain relationship for concrete in compression to BSEN1994-1-2.............................................................................................. 113

Figure 4.2- Tension softening curves assumed for concrete................................. 113 Figure 4.3- Concrete in tension and fracture energy............................................. 114 Figure 4.4- Crack detection surface in concrete model ........................................ 116 Figure 4.5- Yield and failure surfaces defined for biaxial stress in concrete ....... 117 Figure 4.6- Details of the symmetric beam A1..................................................... 120 Figure 4.7- Providing high density of shear connectors to ensure 100% steel-

concrete interaction....................................................................................... 122 Figure 4.8- Test set up for Beam A1 .................................................................... 123 Figure 4.9- First buckling mode for Beam A1 from linear Eigenvalue analysis.. 124 Figure 4.10- Second buckling mode for Beam A1 from linear Eigenvalue analysis

...................................................................................................................... 125 Figure 4.11- Load-deflection comparison for Beam A1, FE model versus test ... 126 Figure 4.12- S-Shaped web post buckling of Beam A1 ....................................... 126 Figure 4.13- The web post bucking of the Beam A1............................................ 127 Figure 4.14- Von Mises stress contour plot to check Vierendeel mechanism...... 127 Figure 4.15- Max principal stress representing tensile stresses............................ 128 Figure 4.16- Maximum principal stress in tensile diagonals ................................ 129 Figure 4.17- Compressive force in the strut and tension in the opposite diagonal129 Figure 4.18- Min principal stress representing the compressive stress ................ 130 Figure 4.19- Min Principal stresses in compressive strut for Beam A1 ............... 131 Figure 4.20- Stress distribution after web buckling.............................................. 132 Figure 4.21- Details of the asymmetric beam B1 ................................................. 133 Figure 4.22- First buckling mode for Beam B1 based on linear Eigenvalue analysis

...................................................................................................................... 134 Figure 4.23- Second buckling mode for Beam B1 based on linear Eigenvalue

analysis.......................................................................................................... 134 Figure 4.24- S-shaped web buckling as the governing failure mode of Beam B1 135 Figure 4.25- Occurrence of the S-shaped web buckling of Beam B1 in the

ABQAUS model ........................................................................................... 135 Figure 4.26- Out-of-web plane displacement of the buckled web post ................ 136 Figure 4.27- Numerical and experimental results for Beam B1 ........................... 137 Figure 4.28-Plot of minimum principal stresses in Beam B1 while buckling...... 138 Figure 4.29- Geometric details of RWTH Beam 3............................................... 139 Figure 4.30-- Steel sheeting used in the RWTH Beam 3...................................... 141 Figure 4.31- The approach used for the slab thickness and steel decking............ 141 Figure 4.32- RWTH Beam 3 modelled in ABAQUS ........................................... 143 Figure 4.33-Web post buckling of the last web post gained in numerical and test

results ............................................................................................................ 143 Figure 4.34- Comparison of the model and test Load-Deflection curves for RWTH

Beam 3 .......................................................................................................... 144


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Figure 5.1- Web post buckling of Beam A1 without web stiffeners, under a UDL at ambient temperature ..................................................................................... 146

Figure 5.2- Web post buckling of Beam A1 with web stiffeners, under a UDL at ambient temperature ..................................................................................... 147

Figure 5.3- Load-deflection curve of Beam A1 under a UDL with and without web stiffener ......................................................................................................... 147

Figure 5.4- Web post buckling of Beam B1 without web stiffeners, under a UDL and at ambient temperature........................................................................... 148

Figure 5.5- Web post buckling of Beam B1 with web stiffeners, under a UDL and at ambient temperature.................................................................................. 149

Figure 5.6- Load-deflection curve of Beam B1 under a UDL with and without web stiffener ......................................................................................................... 149

Figure 5.7- Comparing the load-deflection curves for Beam A1 assuming ductile and brittle behaviour for concrete after cracking in tension ......................... 151

Figure 5.8-Effect of web post imperfection in the load-deflection curve for Beam A1.................................................................................................................. 152

Figure 5.9- Effect of web post imperfection in the load-deflection curve for Beam B1.................................................................................................................. 153

Figure 6.1- Standard and slow fire curve to BSEN1991 ...................................... 157 Figure 6.2- Geometry of Beam A2 (Also Beam A1 and Beam A3)..................... 158 Figure 6.3- Stress- strain transform of steel at elevated temperature ................... 159 Figure 6.4- Stress- strain transform of concrete in compression at elevated

temperature ................................................................................................... 160 Figure 6.5- Stress-Strain relationship of concrete in tension at elevated temperature

...................................................................................................................... 160 Figure 6.6- Drucker-Prager failure surfaces for different temperatures ............... 161 Figure 6.7- Temperatures at bottom, middle and top of the concrete slab ........... 162 Figure 6.8- Position of the thermocouples on steel Beam A2 (Nadjai, 2007) ...... 163 Figure 6.9- Temperatures recorded by thermocouples 4a to 4f in zone 4 of Beam

A2.................................................................................................................. 165 Figure 6.10-Temperatures averaged and applied equally all over the beam length

...................................................................................................................... 166 Figure 6.11- Temperature profile over the length of Beam A2 at 45 minutes ..... 167 Figure 6.12- Temperature profile over the length of Beam A2 at 55 minutes ..... 167 Figure 6.13-Time- temperature curves resulted from averaging in all over the beam

length ............................................................................................................ 168 Figure 6.14- Test results against the numerical results of Beam A2 considering the

recorded and extrapolated temperatures ....................................................... 169 Figure 6.15- Initiation of buckling at 50 minutes ................................................. 170 Figure 6.16- Prediction of web buckling of Beam A2 in numerical modelling ... 171 Figure 6.17- The struts in compression to investigate the internal stresses.......... 173 Figure 6.18- Minimum principal stresses (representing compression) in the six

nodes ............................................................................................................. 173 Figure 6.19- Geometric details of asymmetric Beam B2 ..................................... 174 Figure 6.20- Position of thermocouples for Beam B2 (Nadjai, 2007).................. 175


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Figure 6.21- Web post buckling in the Beam B2 (Source: The test report of the Ulster fire tests on cellular composite beams (Nadjai et al., 2007)) ............. 176

Figure 6.22- Web buckling of Beam B2 predicted by the developed numerical model ............................................................................................................ 177

Figure 6.23- Comparison of load-deflection curves for the test against numerical model ............................................................................................................ 177

Figure 6.24- Average of the temperatures recorded in the critical web post in Tests A2 and A3..................................................................................................... 179

Figure 6.25- Small areas used to calculate and apply the recorded temperatures more accurately............................................................................................. 180

Figure 6.26- Numerical versus experimental results for Beam A3 ...................... 181 Figure 6.27- Deformation at Time=24.16, Load increment number 262, web

buckling is developing .................................................................................. 182 Figure 6.28- Deformation at Time=24.16, Load increment number 263, Sudden

local buckling in the bottom Tee .................................................................. 182 Figure 6.29- Load-deflection curves by experiment and numerical model .......... 183 Figure 6.30- Final deformed shape of Beam A3 in ABAQUS............................. 184 Figure 6.31- Final deformed shape of Beam B3................................................... 185 Figure 6.32- Load-deflection curves of Beam B3 resulted from experiment and

model ............................................................................................................ 186 Figure 6.33- Time-deflection curves resulted from some of the approaches to

calibrate the numerical model for Beam B3 ................................................. 187 Figure 6.34- Furnace temperatures recorded by eight thermocouples for fire test on

Beam B3 ....................................................................................................... 188 Figure 7.1- Investigating the influence of web imperfection for Beam A2 at fire

conditions...................................................................................................... 192 Figure 7.2- Investigating the influence of web imperfection for Beam B2 at fire

conditions...................................................................................................... 193 Figure 7.3- Averaging the Temperatures along the beam length ......................... 196 Figure 7.4- Applying temperatures in details ....................................................... 197 Figure 7.5- The location of the critical web posts with regard to the moment and

shear: (a) Beam A2 and (b) Beam B2........................................................... 197 Figure 7.6- Comparing the time deflection curves resulted from employing the two

approaches to Beam A2 ................................................................................ 198 Figure 7.7- Temperature profile along the length of Beam B2 at 50 minutes ...... 199 Figure 7.8- Temperature profile along the length of Beam B2 at 65 minutes ...... 199 Figure 7.9- Comparing the time deflection curves resulted from employing the two

approaches to Beam B2 ................................................................................ 200 Figure 7.10- Out-of-web plane displacements for Beam A2 without stiffeners at 48

minutes.......................................................................................................... 201 Figure 7.11- Out-of-web plane displacements for Beam A2 without stiffeners at 63

minutes.......................................................................................................... 202 Figure 7.12- Final deformed shape showing the distorsional buckling at 75 minutes

...................................................................................................................... 202 Figure 7.13- A section of the mid-span of the beam at 61 minutes to schematically

present the P-∆ effect causing distorsional buckling.................................... 203


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Figure 7.14- Slight out-of-web plane displacements of the bottom Tee at the mid-span at 21 minutes......................................................................................... 204

Figure 7.15- Out-of-web plane displacements at 49 minutes highlighting the critical role of web stiffener to limit the lateral displacement in bottom Tee........... 204

Figure 7.16-Final deformed shape of the beam failed due to web buckling in which bottom and top Tees are still in a straight line .............................................. 205

Figure 7.17- Comparing the performance of Beam A2 under point load and a UDL at elevated temperature for load ratio of 50%............................................... 206

Figure 7.18- Distorsional Buckling of Beam B2 without stiffener at 62 minutes 207 Figure 7.19-Final deformed shape of Beam B2 with stiffener under a UDL ....... 208 Figure 7.20- Comparing the performance of Beam B2 under a point load and a

UDL at elevated temperature for load ratio of 50% ..................................... 208 Figure 7.21- Sensitivity of symmetric Beam A2 to load ratio at elevated

temperature ................................................................................................... 209 Figure 7.22- Sensitivity of asymmetric Beam B2 to load ratio at elevated

temperature ................................................................................................... 210 Figure 7.23- Effect of web stiffener thickness on performance of cellular beams in

fire ................................................................................................................. 211 Figure 7.24- Sensitivity studies carried out on the loading type and support

restraints at elevated temperature.................................................................. 212 Figure 7.25- Load-deflection curves for the Beam A2 under point load and various

axial end support stiffness............................................................................. 213 Figure 7.26- The maximum lateral deflection of the bottom flange of the Beam A2

versus time for three different K values........................................................ 215 Figure 7.27- Load-deflection curves for Beam A2 under distributed load and

various K values............................................................................................ 216 Figure 7.28- Load-deflection curve for Beam B2 under point load and experiencing

various axial restraints .................................................................................. 217 Figure 7.29- Load-deflection curve for Beam B2 under line load experiencing

various axial restraints .................................................................................. 218 Figure 7.30- Local flange buckling, distortional buckling and web post buckling of

CBs................................................................................................................ 219 Figure 7.31- Minimum in-plane principal stress representing the compressive

stresses .......................................................................................................... 220 Figure 7.32-Local buckling of the Top Tee of the last opening ........................... 221 Figure 7.33-Uz plot at 15 minutes which shows shaping of overall distorsional

buckling in Beam A2, due to high end restraints (K=100%)........................ 222 Figure 7.34- Uz plot at 24 minutes, which shows development of the overall

distorsional buckling in Beam A2 and how it perturbed the web post ......... 222 Figure 7.35- Shaping of the overall distorsional buckling in Beam A2, at 61

minutes, (K=100%)....................................................................................... 223 Figure 7.36- Failure of the internal cellular beams due to distorsional buckling at

the full-scale fire test carried out by University of Ulster in 2010 ............... 224 Figure 7.37-Position of the nodes through the beam height ................................. 225 Figure 7.38- Axial Deflection of the nodes through the height of Beam A2 for

K=0%............................................................................................................ 226


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Figure 7.39- Axial Deflection of the nodes through the height of Beam A2 for K=5%............................................................................................................ 226

Figure 7.40- Axial Deflection of the nodes through the height of Beam A2 for K=75%.......................................................................................................... 227

Figure 7.41- Axial Deflection of the nodes through the height of Beam B2 for K=25%.......................................................................................................... 228




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NOMENCLATURE

A The cross sectional area of the structural element

Atf Area of the top flange of the beam

Abf Area of the bottom flange of the beam

Do Opening diameter in cellular beams

Hp The perimeter of the section which is exposed to fire

Gf Fracture energy of concrete

GFo Base value of fracture energy

Mmax Maximum allowable web post moment

Mb,NV,Rd Bending resistance of the bottom Tee reduced for axial tension and

shear

Me Web post capacity at its critical section

Mt,NV,Rd Bending resistance of the top Tee, reduced for axial compression

and shear

Mmax Bending capacity the Tee section at the critical section

Mvc,Rd Local composite Vierendeel resistance moment

Ncr, θ Elastic critical buckling capacity of the strut at temperature θ

Npl, θ Plastic axial resistance of the effective strut at temperature θ

Pcr Buckling capacity of the web (Euler load)

Pmax Axial capacity of the Tee section at the critical section

Po Acting local axial force at Tee at the critical section

VEd Design value of the shear force

d Depth of the web of the steel beam

fb Buckling strength of the web post

fb,θ Buckling strength of the web post at temperature θ

fcm Mean compressive strength of concrete

fck Concrete cylinder strength at 28 days

ft Tensile strength of concrete

fu Ultimate strength of structural steel


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fy Yield strength of structural steel

hc Crack band width of concrete according to Bazant’s theory

ky,θ Strength reduction factor for steel at temperature θ

le Effective length of the equivalent rectangle web opening replacing

circular opening

t Thickness of the web of the beam

wi Associated scale factor for the ith buckling mode

θF,b Temperature at the bottom flange of the beam

χ Buckling coefficient

χfi Buckling coefficient at temperature θ

λ Non-dimensional slenderness

λfi Non-dimensional slenderness at temperature θ

ν Poissons ratio

φi ith mode shape of the buckling

δui Change of displacement


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ABSTRACT

Cellular beams (CBs) have become increasingly popular in the UK and other countries over the recent years. However, the research into the behaviour of these beams has not advanced at the same rate. There is still no robust codified guidance available to design cellular beams and cellular composite beams at ambient and elevated temperatures.

Meanwhile, numerical simulation approaches, such as Finite Element (FE) analysis, have enabled the researchers to advance their investigations into various behavioural aspects of these beams.

In this research, the developed numerical models using the ABAQUS package were able to predict, to a high accuracy, the failure mode and failure load (temperature) of CBs and cellular composite beams at ambient and elevated temperatures.

Within the investigations on cellular beams, it was found that predicting the correct failure mode by FE models can be extremely sensitive to the maximum load increment allowed in the software (for elastic-perfectly plastic stress-strain relationship for steel material) and also to the applied boundary conditions. In particular, slight changes in the boundary conditions applied to the top flange of the beam, can change the failure mode from web post buckling to Vierendeel mechanism.

The buckling resistance of the web post of cellular composite beams was found to be sensitive to the amplitude of web imperfections at ambient temperature. However, the ultimate resistance of these beams was not affected by the amplitude of web imperfections at elevated temperature. This suggests that the ‘Strut’ model used in current design method to estimate the buckling resistance of the web post is not reasonable at elevated temperature and needs to be modified.

The failure of cellular composite beams under a uniform distributed load (UDL) and at elevated temperatures, was governed by distorsional buckling before the development of web post buckling. Adding full-height web stiffeners to the beam in such cases improved their loading resistance at ambient temperature by up to 15% and prevented the occurrence of distorsional buckling at elevated temperature.

Increasing the end-restraints decreased the deflections of CBs which are governed by catenary action at elevated temperature. However, this also critically promoted the occurrence of web post buckling which could be due to the P-∆ effects and instabilities resulting from the restrained expansion of the beam.

Asymmetric beams showed a higher sensitivity and vulnerability to the magnitude of the load ratio than symmetric sections. This suggests a more prudent approach for the fire design of asymmetric beams as opposed to symmetric beams.


17

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.


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19

To My Parents


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ACKNOWLEDGEMENTS

I would like to express my appreciation and sincere gratitude to my supervisor,

Professor Colin Bailey, for his constant guidance, enthusiastic advice and critical

review of my work throughout this research project. Professor Bailey gave me the

opportunity to advance my knowledge up to a professional level by being an

excellent mentor and allowing me to attend international conferences. I am also

grateful to him for letting me work part-time at the period that I faced serious

financial problems and also fully supporting me towards getting an emergency

fund from the University in order to focus on my research with no delays and

hectic.

My Special thanks go to my colleagues and friends whom I have shared the same

office with, for their support and continued warm and honest friendships during my

studies at the University of Manchester. I would like to also thank Dr. Ehab

Ellobody for helping me with ABAQUS and the University staff, especially

Carmela Venosa-Ridyard and Christine Jinks, for their continued assistance and

support.

Finishing this course would have been very difficult without the support of the

Steel Construction Institute (SCI). In special, I am grateful to Dr Ian Simms and

Dr. Graham Couchman for being flexible and supportive and also giving me the

chance to focus on completing my thesis.

And last but no means least, I am eternally grateful to my parents, for their endless

patience, care, love and encouragement during this project. Without their

inspiration and support, I would not have made it. I gratefully dedicate the thesis to

them.


21

1 INTRODUCTION

1.1 BACKGROUND

It has always been one of the major concerns of construction engineers to optimise

the material usage in structural elements. One of the major advances in this regard

has been to split and expand steel beams. Perforated beams, and in particular

cellular beams, not only have notably higher bending capacity compared to the

parent section, but they are also more convenient in terms of passing service pipes,

wires and ventilating ducts through perforations (Chung, 2002). This crucially

reduces the height of floors, which is especially important in design of high-rise

buildings.

The performance and mode of failure of cellular beams varies depending on the

geometric details of the beam. Various simple design methods have been presented

for cellular beams to check the beam’s resistance against web post bucking,

Vierendeel mechanism and other failure modes. However, there is still no robust

codified design method available for the beams with web openings. The only

codified design guide was provided in the National Annex N of BSENV1993-1-1

(BSI, 1998) which was superseded later on due to reliability concerns.

The composite use of cellular beams with a concrete slab has become increasingly

popular within which the resulted section benefits from the concrete’s compressive

strength and steel’s tensile strength. This composite action has also added to the

complexities of implementing simple methods to design cellular composite beams

(Lawson and Hicks, 2009). Some tests were carried out to provide the researchers

with experimental data to calibrate the proposed analytical models and more

importantly to validate the numerical models to investigate the performance of


22

these beams with various combinations of loadings, geometries, material properties

and boundary conditions.

One of the controversial issues over the recent years has been to estimate the fire

resistance of cellular composite beams and fire protection requirements. Very

limited test data was available in this regard until 2006 when University of Ulster

conducted a series of 6 tests on unprotected cellular composite beams, 2 at ambient

temperature and 4 at elevated temperature (Nadjai, 2007). These tests provided the

first set of test data on the performance of unprotected cellular composite beams at

elevated temperature.

In order to obtain a better understanding on how cellular beams and cellular

composite beams behave, this research includes studies into multiple aspects of

their behaviour at ambient and elevated temperatures. The investigations are based

on numerical studies where after implementing a numerical approach to simulate

the performance of these beams, the calibrated models were utilised for case

studies to investigate the effect of some critical factors. Conclusions of this

research provide useful information to be considered in modifying the current

design approaches and include a design recommendation for cellular beams

subjected to a UDL.

1.2 OBJECTIVES

The basic objective of this research was to gain a better understanding on how

cellular composite beams behave at ambient and elevated temperatures. In order to

achieve this, finite element models were developed using the ABAQUS package

and validated against various test data. In brief, the main objectives of this research

were:

• Validate the numerical models developed for cellular beams and investigate

some of the modelling issues, which do not allow the developed numerical


23

models to be able to predict the correct failure mode and failure load for

cellular beams.

• Validate the FE models for cellular composite beams at ambient

temperature

• Investigate the role of web imperfection and loading type on the

performance of cellular beams with symmetric and asymmetric geometries

• Validate the FE models for cellular composite beams at elevated

temperature

• Investigate the effect of temperature distributions, loading type,

imperfections and also support conditions on the performance of these

beams at elevated temperature

• Improving the performance of cellular composite beams subjected to a

UDL at ambient and especially elevated temperatures by adding full height

web stiffeners to prevent the distortional buckling

• Proposing a modification to the current design approach to estimate the

buckling capacity of the web post at elevated temperature.

1.3 OUTLINE OF THE THESIS

This thesis is divided into 8 chapters and the following is a description of each

chapter:

This chapter (Chapter 1) reviews a general background to this thesis, the work

involved in the research and the outline of the project.

Chapter 2 provides a literature background of the fire engineering as a discipline

and the two common approaches to design structures for fire safety. This chapter

then presents the history of the development of beams with web openings and

cellular beams as the most popular product. Finally, this chapter reviews the failure

modes of these beams, the relevant experimental studies conducted on cellular

beams and cellular composite beams at ambient and elevated temperatures and the


24

analytical models to predict the failure load and mode of cellular beams at various

conditions.

Chapter 3 is dedicated to the numerical modelling of cellular beams at ambient

temperature where developed models have been validated against various test data.

In addition, this chapter investigates some of the issues in numerical modelling of

cellular beams which do not allow the correct prediction of failure mode and

failure load by numerical models and also suggests solutions to overcome these

issues.

Chapter 4 includes the numerical approach implemented within ABAQUS to

simulate some tests carried out on cellular composite beams at ambient

temperature. The approach to model the rather complicated concrete material

within these numerical studies is also discussed in this chapter.

Chapter 5 presents the results of the case studies conducted on calibrated models

for cellular composite beams at ambient temperature. These case studies included

the effect of web imperfections, loading types (point load(s) or a UDL) and tensile

characteristics of concrete.

Chapter 6 is dedicated to the numerical approach implemented to model the

cellular composite beams at elevated temperature. The numerical results are

compared against test data and discussed in detail.

Chapter 7, investigates the effects of temperature distribution, support conditions,

utilization factor and web imperfection on the performance of cellular composite

beams at elevated temperature and proposes a modification to the current ‘Strut’

model presented by SCI (SCI, 1985) to estimate the web buckling capacity. This

chapter also presents the investigations on the effect of the loading type and gives a

design recommendation for cellular beams under a UDL.


25

Chapter 8 summarises the main conclusions of the work carried out on this

research project along with recommendations for further research work.


26

2 LITERATURE REVIEW

2.1 INTRODUCTION

This chapter reviews the literature of the topics covered in this research. In

particular, Section 2.2 describes the principles of the fire safety engineering and the

two common approaches, the prescriptive and performance based approach, to

design structures for fire safety. Section 2.3 presents some details about cellular

beams and compares them against their rival castellated beams. Some experimental

tests have been carried out on these beams to investigate their performance, failure

modes and to validate the simple design methods. Section 2.4 presents these tests

which are carried out on castellated and cellular beams as well as cellular

composite beams at ambient and elevated temperatures. Finally, this chapter

describes the common failure modes in beams with web openings and development

of the simple design methods proposed to check the design of cellular beams and

cellular composite beams at ambient and elevated temperatures.

2.2 FIRE DESIGN OF STRUCTURES

The main objective of the fire safety regulations is to ensure the protection of

human lives (occupants and fire fighters) in the first place and partly environment

and the property (building and its contents). Through many measures, including a

combination of active and passive fire protection systems, the objectives are:

• To minimise the incidence of fire by controlling fire hazards in the building

• To provide safe escape routes for evacuation of occupants and prevent fire

spread from the fire compartment to other sections of the building

• To ensure that the building remains structurally stable for a time period

adequate to evacuate the occupants and for the fire fighters, to rescue the

trapped occupants.


27

The minimum fire resistance period that buildings should be designed for is

specified in the approved document B of the building regulations for England and

Wales (FTA, 2008) in 30 minutes intervals from 30 to 120 minutes. These values

are defined based on occupancy type, building height and size, and sprinklers.

Currently, design of the structures for fire safety is based on prescriptive or

performance based approaches. The following sections present these two

approaches and compare them briefly.

2.2.1 Prescriptive method

This method investigates the performance of individual structural elements when

they are subjected to a standard fire which is generally presented in ISO 834 (I.S.,

1985). For UK, standard fire curves are addressed in BSEN1991-1-2 (BSI, 2002b)

and PD7974 (BSI, 2003b) for external, standard, hydrocarbon and smouldering

fires (Figure 2.1). These curves are defined based on different fuel types and

ventilation conditions.

Figure 2.1-Nominal fire curves based on BSEN1991-1-2 and PD7974 (Source: http://www.mace.manchester.ac.uk/project/research/structures/strucfire/default.ht

m)


28

In the prescriptive method, the performance of the buildings against fire is

normally evaluated with three criteria: stability, integrity and insulation (BSI,

1987). Stability is the ability of the structural elements to resist the fire without

failing. In practice, the stability criterion is monitored by a limit for deflections

higher than span/30 with a maximum of span/20 for flexural members such as slabs

and beams. Integrity is assessed with ability of a separating element to prevent

passage of smoke and hot gas from the fireside to the other isolated sides. Finally,

insulation is the ability of an isolating element, to limit the temperature rise (to an

average of 140°C and a maximum of 180°C) on the unexposed face.

The prescriptive method does not require much expertise or advanced tools to be

exploited to design the structures for fire safety. However, there are increasing

concerns about this traditional approach:

• This method simply investigates the performance of one individual

structural element against the standard fire and excludes all the interactions

between different structural elements which can be very inaccurate and

conservative (non-economical), or sometimes even non-conservative(Bake

and Bailey, 2007a). A good example to show how non-economical this

approach can be is the development of membrane actions in the concrete

slab when the composite slab is subjected to fire. Utilising the effect of

enhancing membrane action in the fire design of composite floors has

significantly reduced the fire protection costs by excluding the costs

associated with fire protecting the secondary steel beams (Bailey and

Moore, 2000b; Bailey and Moore, 2000a) . However, if composite beams

are designed individually for fire then secondary beams need to be fire

protected or over designed to satisfy the design requirements for the fire

conditions.

• The standard fire curve, which is the basis of the prescriptive method, is an

ever-increasing curve and does not consider any cooling phases.

Nevertheless, in the fire tests and actual buildings engulfed in fire, it has


29

repeatedly been observed that a lot of structural damage and failures

actually happen due to high reversal strains that the structure experiences

during the cooling period (Peijun et al., 2008). Occurrence of ruptures in

the bolts and end plate connections (Wang, 2000) or occurrence of huge

tensile cracks in concrete slabs are some of the most common types of such

failures which are witnessed to take place during the cooling phase.

• The prescriptive method is generally restrictive and cannot provide reliable

measures of the robustness of the structures.

The above points have encouraged designers to incorporate a more realistic but

complicated approach, so called “Performance based method”, to design the

structures for fire safety.

2.2.2 Performance based method

The increasing concerns about the prescriptive method (Bailey, 2006) along with

the enhanced capabilities of numerical approaches to model integrated structures,

as opposed to individual elements, has raised interest in using the performance

based approach. This reliable and economical method is flexible and can be utilised

for structures of various types. However, extra expertise is needed to deal with the

complexities of this approach and designers are advised to consult experts in this

regard.

The performance based approach is constituted of three components: Fire

modelling, thermal modelling and structural modelling.

2.2.2.1 Fire behaviour

The aim of the fire modelling is to simulate the fire development to predict the

severity of the fire (gas temperatures and heat flux) on the structural elements.

Although the common approach is to represent the fire by a standard fire curve, the

fire safety design may also be based on more realistic design fires curves

(parametric fires) which are defined with regard to details of the compartment


30

specifically (Purkiss, 2007). Within these approaches, the fire curves and

corresponding thermal actions may be derived with regard to factors comprising

the fire load, heat release rate, size and geometry of the compartment, ventilation

factor and finally thermal characteristics of the boundaries of the compartment.

The identification of the relevant and realistic design fire scenarios is a vital aspect

of the fire safety design in the performance based approach. The design fire

scenarios used for the analysis of a building need to be deduced from the total

number of possible fire scenarios, which is infinite for most buildings. In other

words, only the ‘credible worst case’ fire scenarios need to be investigated.

When the design fire scenarios are chosen, a number of fire models are available to

assess the fire severity and calculate the corresponding thermal actions. Different

levels of fire models are relevant to the various stages of development of a natural

fire. Development of natural fires is generally divided into the 5 phases addressed

in Figure 2.2 and the severity of the fire in any of these phases depends on the

specifications of the compartment presented previously. When a fire is initiated, it

is localised within a compartment. Depending on the characteristics of the

compartment and the fire load, it can remain localised or become generalised to the

whole compartment. Localised fire refers to fires with no spread to the whole

compartment and happens when the fire spread is so slow. In localised fires, the

temperature increase is not adequate to cause flashover (BSI, 2003c). In

generalised fires however, the temperature increase is enough to cause flash over

and it is fair to assume a uniform temperature distribution throughout the fire

compartment. Generalised fires usually happen in small compartments or

compartments with low ventilation factors where the fire usually develops into a

fully engulfed fire.

There are various methods to define the fire severity for either localised or

generalised fires, which are briefly presented in the following sections.


31

Figure 2.2- Time-temperature curve of a natural fire and the 5 phases

2.2.2.1.1 Fire spread in localised fires

For localised fires, the fire severity can be defined by Fire Plume models, Zone

models or Computational Fluid Dynamics (CFD). The Fire Plume model (the

simplest of the three) uses the approach presented in BSEN1991-1-2 (BSI, 2002b)

which defines the thermal action depending on whether the flame reaches the

ceiling or not. However, the UK national annex does not accept this method and

refers to a method presented in PD7974-1 (BSI, 2003c) where the temperatures in a

fire compartment, in the fire growth phase, are predicted with a single formula.

The Zone models are computer models that divide a compartment into distinct

zones with uniform temperatures. This method is based on solving differential

equations for the conservation of energy and mass in a specific compartment.

Utilising CFD models is currently the most advanced and accurate approach to

define the fire severity and temperature distribution in a compartment. CFD models

are able to bring into account fluid flow and heat transfer effects to provide the

temperatures at every location within the compartment. This method is based on

solving thermodynamic and aerodynamic partial differential equations of the fluid

flow with regard to the boundary conditions, source and compartments details.


32

2.2.2.1.2 Fire spread in generalised fires

In generalised fires, all combustibles in the compartment are burning and the

amount of the heat being released is controlled by the ventilation or the type and

availability of the fuel in the compartment. There are also various approaches to

define the temperature growth in the compartment in generalised fires. Apart from

the simple way of considering standard fire curves in a fire design approach, the

designer can use the time equivalence method, natural fire curve method or

computer based zone or CFD methods to define the fire severity in a compartment

subjected to a generalised fire.

The time equivalence method is a simple approach to relate the actual temperature

of a structural member from predicted fire intensity to the time taken for the same

member to reach the same temperature when it is exposed to the standard fire

(Bailey, 2008). The time equivalence approach enables the designer to consider the

effects of the fire load as well as compartment size, ventilation and boundary

conditions.

The natural fire model defines the time-temperature curves by an energy equation

that deals with the balance of the heat in the compartment. This equation relates the

heat produced by the combustion to the heat loss by convection and radiation

through openings plus the heat loss by radiation and conduction through the

boundaries of the compartment (Drysdale, 1999; Karlsson and Quintiere, 2000).

Similar to the time equivalence method, this method also considers the effects of

the fire load as well as compartment size, ventilation factor and thermal properties

of the boundaries. In this approach, temperature is again assumed uniform inside

the compartment.

The basics of the zone models for generalised fires is similar to that described

earlier for the localised fires where the temperatures of each zone is defined based

on the mass and energy conservation. However, for generalised (post flashover)


33

fires, temperature is assumed to be uniform within the compartment and therefore,

one-zone models are used.

Similar to what was mentioned about the use of CFD models in localised fires,

CFD models can also be used for generalised fires as long as they are calibrated

against test data.

2.2.2.2 Thermal behaviour

Once the fire actions are calculated, the exchange of the heat between the gas and

the structural elements has to be calculated along with the temperature distribution

within the structural elements. The temperature distribution is generally dependent

to the radiation and convection characteristics of the surface of the members and

conduction of the heat within the members.

It may be reasonable to assume a uniform temperature distribution for materials

with a high thermal conductivity (such as steel) and ignore the thermal gradient

within members. However, this assumption is only valid when the member is not in

contact with a material with relatively low thermal conductivity (such as concrete).

In such cases, the material with low conductivity causes a heat-sink effect resulting

in a thermal gradient through the member with high thermal conductivity.

There are various ways to carry out heat transfer analysis for structural members.

There are simple design equations and tables provided in the codes (BSI, 2004b;

BSI, 2005b; BSI, 2005c) to define the temperature distribution within common

structural elements with regard to their thermal characteristics and fire exposure

type. However, estimating the heat transfer in materials such as concrete with a low

conductivity and high moisture content is difficult due to the high thermal

gradients. Therefore, use of simple design charts and tables (provided in the codes

for the heat transfer analysis) is generally not allowed with performance-based

approach, as these charts are resulted from standard fire tests. These charts can

only be used within the performance based approach if the fire behaviour is


34

represented by the standard fire curve. In any other cases where parametric curves,

zone models or CFD models are employed to assess the fire severity, the heat

transfer analysis should be used to investigate the thermal behaviour in structural

elements.

2.2.2.3 Structural behaviour

The final stage of the performance based approach is to assess the structural

performance of the structures subjected to the temperature distribution defined

within the heat transfer analysis. The structural behaviour can be assessed in three

levels: member analysis, sub-model analysis and global analysis.

In the member analysis approach, each member of the structure is assessed as

being separated from other members. The interaction with other members is

represented by appropriate boundary conditions. This simple method, which is

based on fundamental engineering principles, defines the loads at the fire limit state

using partial safety factors, which take into account realistic loads at the time of the

fire. BSEN1990 (BSI, 2002a) defines the load combination factors to be used in the

event of fire, which is categorised as an accidental condition, while the reduction

factors for the strength and stiffness at different temperatures for different materials

are also defined in the relevant design codes. The designer utilises the “Limiting

temperature” or “Moment capacity” method to define the fire resistance period of

individual elements and to decide on the fire protection requirements.

The limiting temperature method provides the designer with the maximum

temperature the section can reach before failure for a given load ratio (utilization

factors). This method can also be utilised to decide on the fire protection by

comparing the limiting temperature with the temperature of the hottest part of the

section at the required fire resistance time (the design temperature). BS5950 Part 8

(BSI, 2003a) includes a set of prepared design tables to check the fire design of

sections based on this approach.


35

The moment capacity method is based on assessing the fire resistance of a beam by

calculating its moment capacity given the temperature profile of the section at the

required fire resistance time. If the applied moment, which is calculated based on

the load combinations for the fire condition, is less than the moment capacity of the

beam at that time (temperature), the member provides the adequate fire resistance

without requiring fire protection. The method is limited only to beams with webs

which fulfil the section classification requirements for plastic or compact section.

However, observations based on full-scale fire tests (Bailey et al., 1999) have

proved that this method can be very conservative and unrealistic.

The idea of developing the structural model, from an individual element to a sub-

model, was followed after the full-scale Cardington fire tests. This approach takes

into account a limited part of the structure in the structural assessments and is

applicable with any fire models. The interactions with other parts of the structure

are again reflected by using appropriate boundary conditions. For Cardington test

(BSP, 1999), the developed simple sub-structure model (Newman et al., 2006;

Bailey, 2001) was based on considering the enhancing effects of the membrane

action of floor slabs along with the beneficial effect of the grillage of the secondary

beams acting compositely with the slab.

Finally, the most comprehensive approach is to model the whole structure using the

finite element method. This method is generally more accurate, reliable and

economical which provides a better evidence of the overall performance of the

structure, interaction between the structural elements and weak points of the

structures. However, this method is not necessarily detailed enough for localised

behaviour such as reinforcement fracture or failure of connections and extra

detailed models can be specifically developed to investigate local failures, which

are of special interest or have critical consequences.


36

2.3 BEAMS WITH WEB OPENINGS

In 1910, Horton, who was a member of the Chicago Bridge and Iron works, for the

first time proposed cutting the beam web and reassembling the two halves to

increase the section modulus, (Das and Seimaini, 1985). The idea of castellated

beam was proposed later in 1935 by Geoffrey Boyd who was a structural engineer

in the British Structural Steel Company (Knowles, 1991). Invention of castellated

beams which were previously known as ‘Boyd beams’, brought him the British

Patent award in 1939. Following the developments, cellular beams were first

introduced to the steel construction industry in 1987 by the steel manufacturer

Westok (Westok, 1985) who are the current world-wide patent holders of cellular

beams. These beams seem to be a significant development in steel construction in

the past 20 years. Since 1997, these beams have been used in over 4000 projects

and 20 countries.

Within this range Arcelomittal (Arcelormittal, 2001) has also introduced another

product called ‘Angelina Beam’ which is similar to castellated beams but with a

slightly different cut as shown in Figure 2.3. Beside these beams, which are

categorised as beams with multiple openings, it is also common to have beams

with single openings or multiple but isolated openings in which openings are too

far apart to worry about the failure of the web post or the interactions between the

openings.

Figure 2.3- Angelina beam produced by Arcelomittal


37

2.3.1 Fabrication of cellular beams

Cellular beams are currently manufactured using two methods. In the first method

which is used in Westok Company(Westok, 1985), they are manufactured by

cutting and welding a solid webbed beam by which the resulted cellular beams can

be up to 1.6 times deeper than its parent section. This procedure is similar to the

way the castellated beams are manufactured. Alternatively, the cellular beams are

fabricated as built-up sections from steel plates using automated flame-based

cutting and welding techniques. The web and flange plates are welded together

using a double-sided process with a thin wire submerged arc which creates a 7 mm

fillet weld in a single pass. This method is being used in the other major producer

company of cellular beam in the UK called Fabsec (Fabsec, 2004).

Figure 2.4- Cellular beams fabricated from steel plates used in Fabsec (Source:

www.fabsec.co.uk)

2.3.2 Cellular beams versus castellated beams

Both cellular and castellated beams provide passage for the utilities and have high

strength to weight ratio as opposed to solid beams, which results in lighter and

more economical structures. However, various differences between the cellular and

castellated beams have made the cellular beams the successor of the two over the

recent years. Cellular beams produce perpetually more options than castellated

beams due to their greater flexibility in geometry. The profile and cutting pattering

of the castellated section is fixed whereas the main dimensions of a cellular beam

(finished depth, cell diameter and cell spacing) are flexible. Figure 2.5 compares


38

the standard cutting pattern of the castellated and cellular beam according to

British standard. This flexibility results in cellular beams being generally lighter

than even the most efficient castellated beams.

In addition to using lighter sections, a cellular beam usually requires less infill

plates than castellated beams. Web openings of castellated beams need to be

infilled at the two following occasions (Figure 2.6):

1. At the position of high shear, such as beam-ends or point loads. However,

the geometry of cellular beams can be selected in a way to give higher

shear resistance, which reduces the use of infill palates.

2. At connections with secondary beams. In cellular beams, however,

adjusting the diameter and cell spacing can keep the added infill plates to

a minimum.

Cellular beams also have another advantage over the castellated beams. They can

be fabricated as highly asymmetric sections in which the bottom Tee can be much

stronger than the top Tee. Use of asymmetric beams is especially efficient when

these beams are acting compositely with concrete slab and therefore the top Tee

does not have much contribution in the beam resistance. In contrast, castellated

beams can only be constructed as symmetric sections.


39

Figure 2.5- Cutting pattern of Castellated and Cellular beams

Figure 2.6- Comparing use of infill plates in cellular and castellated beams

2.4 INVESTIGATIONS ON BEAMS WITH WEB OPENINGS

During extensive experimental and numerical investigations conducted on beams

with web openings, various failure modes are observed which are briefly discussed

here along with the design approaches.


40

2.4.1 Experimental studies on castellated and cellular beams

Research on the performance of beams with multiple web openings was seriously

followed in 1964 when Kolosowski (Kolosowski, 1964) tested one castellated

beam to study its deflection and failure mode. The overall height of this beam was

150% of its parent section, the span to depth ratio was 10 and the web post had the

angle of 56.3o (this angle is 60o in UK sections). Kolosowski was expecting a

behaviour similar to a Vierendeel truss, However, the beam eventually failed due

to overall lateral torsional buckling as there was no lateral restraints provided

within the supports. In 1973, Husain and Speirs (Hosain and Speirs, 1973)

conducted twelve tests to investigate the effect of the opening geometry on the

mode of failure of these beams. In their tests they observed three different failure

types, flexural failure, Vierendeel failure and web post buckling.

Nethercot and Kerdal (Nethercot and Kerdal, 1982) suspected the vulnerability of

castellated beams to lateral torsional buckling and set up two series of tests with

different boundary conditions and for each case investigated a wide range of web

slenderness ratios. They concluded that web openings do not have a critical effect

on the beam’s lateral stability. The following researches also confirmed

Nethercot’s conclusion (Radić and Markulak, 2007).

The web post buckling was only known as a major failure of such beams since

1996 when Zaarour and Redwood (Zaarour and Redwood, 1996) tested 12 short

span (3000 mm) castellated beams with thin webs with minimum web post width

to opening depth ratio ranging from 0.18 to 0.26. Most of these beams failed due to

web post buckling and the rest failed due to lateral-torsional buckling. Redwood

(Redwood and Demirdjian, 1998) also focused more on the web post buckling and

tested four short span castellated beams with the UK cutting details. In his tests, he

observed a double curvature buckling shape in the web post of the all but the

longest beam in which the web buckled with a single curvature. The test results

showed that web post buckling loads were not sensitive to the moment/shear ratio.


41

With the introduction of the cellular beams to the UK market by Westok (Westok,

1985), research was mostly focused on the performance of these beams. The first

series of full scale destructive tests was carried out at Bradford University under

the supervision of the Steel Construction Institute (SCI, 1985). The results of these

tests were not published but the relevant design guide published later by the SCI,

titled P100 (Ward, 1990), mentioned that web post flexural buckling was observed

in the tests. These tests were followed by another set of 7 tests carried out at Leeds

University (Surtees and Li, 1995) again under the supervision of the Steel

Construction Institute. The aim was to seek greater accuracy in modelling the

behaviour of cellular beams under normal service loading as well as at the point of

failure. Based on these tests it was shown that use of full height web stiffeners at

the location of point loads invariably increased the loading resistance of the beam.

However, most of the tests were carried out without web stiffeners to promote the

most critical conditions in the test. The test data of two of these beams are used in

this thesis to calibrate the numerical models developed by the author for cellular

beams at ambient temperature which will be presented in Sections 3.4 and 3.5 of

this thesis.

A total of 8 destructive tests was carried out on cellular beams spanning between

3.1 m to 8.2 m at the University of Natal in 2001 (Warren, 2001). The aim was to

assess the reliability of the existing method presented by SCI and improve it where

necessary. All of these beams failed due to Vierendeel mechanism apart from one

beam which failed due to web post buckling. Warren concluded that the SCI

method was accurate in predicting the failure mode but generally over

conservative. The test data of one of these beams is again used in this research to

validate the numerical approach used to model the cellular beams. This will be

presented in Section 3.3 of this thesis.


42

2.4.2 Experimental studies on cellular composite beams at ambient

temperature

Recently some tests were also carried out on composite floors with cellular beams

to provide the data for calibrating the simple design methods and the numerical

models. In 2002, RWTH (Institute of Steel Construction) coordinated the ECSC

project where 4 cellular composite beams were tested (ECSC, 2003) in

Kaiserslautern. The first two beams were tested twice. For the first beam, after the

beam failed due to Vierendeel mechanism, they tested the same beam again but

with a plate welded across the opening which had failed. Adding this plate changed

the failure mode to web post buckling and increased the failure load from 806 kN

to 844 kN. The second beam, which was the only beam with an elongated opening,

was also tested twice. In the first test, just one point load was applied to the beam

and beam failed due to Vierendeel mechanism. In the second test, the same beam

was subjected to four point loads which were applied through secondary beams.

This time the beam failed due to excessive plasticity in the bottom Tee at the

elongated opening.

The third cellular composite beam was a highly asymmetrical section which failed

due to web post buckling in the test. This composite beam was also modelled in

this research as part of validation of the numerical models which were developed

for composite cellular beams at ambient temperature and is presented in Section

4.5.

The scope of test on the 4th beam was to check the behaviour of ring-stiffened,

half-closed or closed circular web openings. More details about these tests can be

found in the test report (RWTH, 2002) which are also summarised in the SCI

report RT1025 (Simms, 2004b).

A total of five tests have also been carried out by Arcelormittal (Arcelormittal,

2001) where the main focus has been on highly asymmetric cellular beams acting

compositely with a concrete slab (Simms, 2004b). In four of the five beams, the


43

test was terminated due to development of a plastic hinge in the top Tee of the

section. In one of these four cases, development of the plastic hinge in the top Tee

was initiated by crushing of the concrete. The failure of the fifth case was due to

web post buckling.

University of Ulster carried out two tests on cellular composite beams (one

symmetric and one asymmetric sections) at ambient temperature in 2005 (Nadjai,

2005). These tests provided the base for the four fire tests carried out later on the

same geometries. Both of the beams failed due to web post buckling at ambient

temperature. Numerical models were developed for these two beams in this

research and Sections 4.3 and 4.4 presents the details of the numerical approach to

calibrate the models. Further case studies were then carried out based on the

calibrated beams, which will be presented in the Chapter 5.

2.4.3 Experimental studies on cellular composite floors at elevated

temperature

There are a very limited number of fire tests conducted on cellular composite

beams at elevated temperature which are summarised in this section.

2.4.3.1 CTICM Fire Tests

CTICM (CTICM, 1962) carried out the first set of fire tests on two cellular

composite beams in which two beams were tested at Mets in 2003 (Joyeux, 2003).

These beams were fire protected and the main aim of the tests was to assess the

performance of the fire protection material and how the protected cellular beam

performs at elevated temperature.

2.4.3.1.1 CTICM cellular composite beam P1

CTICM beam P1 was designed to investigate the performance of fire protected

asymmetric cellular beams for which a fire test was conducted on a 7200 mm long


44

cellular composite beam. Top and bottom Tees were based on IPE300 and HEB340

respectively. See Figure 2.7 for the geometric details.

Figure 2.7- Geometry of the CTICM Beam P1

2.4.3.1.2 CTICM cellular composite beam P2

The second fire test was carried out on a symmetric section based on IPE400 and

the mid-span openings were filled (Figure 2.8). The focus of this test was to

investigate the effect of fire protection on the edges of openings for which three

different details of fire protection (according to Figure 2.9) were applied in

different sections of the beam.

Both beams tested by CTICM failed due to web post buckling and Figure 2.10

shows the failure of the Beam P2 after removing the fire protection.


45

Figure 2.8- Geometry of the CTICM Beam P2

Figure 2.9- Applying various fire protections at various sections of the CTICM Beam P2


46

Figure 2.10- Failure of the protected CTICM Beam P2 due to web post buckling after removing the fire protection (Source: test report (Joyeux, 2003))

2.4.3.2 Indicative fire tests by the University of Manchester

The decision to carry out indicative fire tests on protected cellular beam was made

after the concerns raised about the temperature of the web post being higher than a

similar solid beam. This was contradicting the assumption of the relevant guidance

given by SCI in publication P160 (Yandzio et al., 1996) which assumed similar

temperatures for a web post and a solid web with similar geometric details.

Unfortunately, the test results which were the basis of the concerning comment

were commercially confidential. The only released information was published in a

technical note (Note, 2002) and an article in the Structural Engineer magazine

(Service, 2003). They stated that the fire tests on protected cellular beams

consistently showed higher temperatures in the web posts than flanges. The

technical note also provided a guidance on the required thickness of intumescent

which was up to 50% higher than the thickness recommenced by the SCI approach.

To investigate the rate of temperature increase in the web post and compare it with

solid beams, a series of tests were conducted on unloaded, protected and


47

unprotected cellular and solid beams. The furnace test (Figure 2.11) showed that in

the beams with no fire protection, the flange and web post temperatures of the

cellular beams were slightly lower than the corresponding temperatures of the solid

beams (Bailey, 2003a). It was also observed that the ratio of the web to flange

temperature did not increase at a faster rate in the cellular beam compared to the

solid beam (Bailey, 2004).

Figure 2.11- Indicative furnace test on unloaded protected cellular and solid beams with similar geometric details (Source: test report (Bailey, 2003a))

Based on these observations, it was concluded that the notable difference between

the web of the protected cellular and solid beams has to do with the performance of

the fire protection material. Therefore, three set of indicative fire tests were

conducted on unloaded protected cellular beams with 0.8 mm water-based, 0.8 mm

solvent-based and 2.1 mm solvent based intoumescent coatings (Bailey, 2003b).

These tests were repeated for similar solid webbed beams. The aim was to compare

the temperatures of the web post against the web and investigate the performance

of fire protection material.

It was witnessed that in all three cases the web post and flange temperatures were

higher in the cellular beam than the corresponding temperatures of the solid


48

webbed beams. Moreover, the relative temperature variation between the web post

and bottom flange of the cellular beam was similar to that of the solid beam.

In the two tests with solvent based coating the char was swept away from the edges

of the cellular beam exposing the steel web (Figure 2.12). These tests also

highlighted that the temperature difference between the web post and bottom

flange in protected cellular beams depends on type and thickness of the

intumescent material.

Figure 2.12- Steel being exposed at the proximity of the holes as the protection fell off (Source: test report beam (Bailey, 2004))

2.4.3.3 Fire tests by the University of Ulster

Four full-scale destructive fire tests were conducted on unprotected symmetric and

asymmetric cellular composite beams in 2006 by the University of Ulster (Nadjai,

2007). Before these tests there was no test data on how unprotected cellular beams

behave under fire conditions and these beams provided researchers with valuable

information. All four tests were also investigated in this research and numerically

modelled for further investigations. Chapter 6 of this thesis presents the details of

the 4 fire tests and the numerical approach used to model these fire tests.


49

2.4.3.4 Fire tests by CTICM

In 2009, CTICM carried out four full scale fire tests on cellular beams spanning 8.8

metres which covered asymmetric sections and elongated openings (CTICM,

2008). One of these beams was equipped with half-height web stiffeners for each

web post to investigate their impact on delaying the failure due to web post

buckling. Two of these beams clearly failed due to web post buckling while there

was no clear failure mode observed in the other two tests which suggested that

these beams have failed due to global bending.

2.4.3.5 Full scale compartment test with cellular beams

University of Ulster also carried out a full-scale fire test in 2010 on a 15 metres

long and 9 wide compartment (Figure 2.13) as part of a European project (FiCEB).

The aim was to investigate the development of enhancing membrane actions in the

concrete slab when it is acting compositely with unprotected secondary cellular

beams. It was also interesting to see how cellular beams perform in fire while

interacting with other structural elements as opposed to furnace tests where they

are simply supported and free to expand at the ends. The final report for this test is

not available at the time of writing.


50

Figure 2.13- Full scale compartment test by the University of Ulster with cellular beams

2.5 DESIGN OF CELLULAR BEAMS AT AMBIENT

TEMPERATURE

Because of the specific geometry of perforated sections (Kerdal and Nethercot,

1984) and in particular cellular beams, various failure modes are expected to

happen, which need to be checked and designed for. The following failure modes

are the most common ones for these beams:

1. Web post buckling (buckling failure, shear failure and flexural failure)

2. Vierendeel mechanism (Chung et al., 2001)

3. Global bending

4. Rupture of the web post weld

5. Lateral torsional buckling or distorsional buckling (in composite beams).

Among these failure modes, the web post buckling and the Vierendeel mechanism

are the two failure modes which has attracted most of the researcher’s attention


51

over the years. The reason is not only because these two failure modes are the most

dominant failure modes of cellular beams, but they have been proved to be more

complex to formulate and codify through a reliable simple design method

compared to the other failure modes. The only codified design approach was

presented in the National Annex N of the BSENV1993-1-1 (BSI, 1998) which has

been withdrawn due to the concerns raised about the reliability of this design

approach especially to check for the web post buckling and vierndeel failure.

The following section briefly describes these two failure modes and the design

methods developed to check the cellular beams against these failure types at

ambient temeratures.

2.5.1 Web post buckling

Buckling of the web post is the most common failure mode in cellular beams

especially at elevated temperature as steel loses its stiffness with a higher rate than

its strength. This section presents the design methods to check for web post

buckling in CBs at ambient temperatures and Section 2.7 discusses the design

approach for at elevated temperatures.

The first robust design method for the buckling of the web post was presented by

Ward in SCI publication P100 (Ward, 1990). His approach was that the ultimate

strength of a web post is governed by two modes, either a flexural failure caused

by the development of a plastic hinge in a web post, or buckling of the web post.

The type of failure in the web post was assumed to be dependant on the geometric

details of the web post. He carried out a series of nonlinear finite element analysis

to develop the design curves for the web post which resulted in proposing the

following formula for the web post capacity:

3

2

02

01 C

DSC

DSC

MM

e

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛= Equation 2-1


52

where:

S = Opening spacing

Do = Opening diameter

Mmax = Maximum allowable web post moment (Mmax= 0.9RVh)

Me = Web post capacity at section A-A of Figure 2.14

C1, C2 and C3 are constants defined as: 2

001 00174.01464.0097.5 ⎟

⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+=

tD

tDC Equation 2-2

200

2 000683.00625.0441.1 ⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+=

tD

tDC Equation 2-3

200

3 00108.00853.0645.3 ⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+=

tD

tDC Equation 2-4

where t is the thickness of the web post.

Figure 2.14- Forces applied to a web post and the critical section to check to web post

This method presented the simple design curves given in Figure 2.15 to check the

flexural and buckling capacity of the web post.


53

Figure 2.15- Design curves proposed in P100 to check the flexural and buckling capacity of the web post

One of the drawbacks of the method presented in P100 was that it was only

applicable to symmetric beams and did not allow treatment of web post moment in

asymmetric sections.

Lawson from SCI (Lawson et al., 2002) proposed a new approach based on a

compression zone or ‘Strut model’ schematically shown in Figure 2.16. This model

was reasonably calibrated against nonlinear finite element analyses (Lawson et al.,

2006). One of the complexities in this approach was to establish a reasonable

effective length for the strut in buckling because of the compound stress pattern

around the opening. The effective length, which considers the stress variations

around the critical opening, was represented by a simple geometric model with a

restraint point in the middle of the strut, and was calibrated against a range of finite

elements studies.


54

Figure 2.16- The ‘Strut’ model for web post buckling proposed by Lawson

For the effective web slenderness (λ) of the strut, this methods refers to the

approach given in BS5950-1 (BSI, 2000) or BSEN1993-1-1 (BSI, 2005a). This

approach checks the buckling of the strut against the applied compressive force,

which is equal to the horizontal shear at the web post (Vh in Figure 2.16).

In 2005, Bitar from CTICM presented another approach to estimate the buckling

capacity of cellular beams which was calibrated against the tests conducted by

CTICM and some numerical models (Bitar et al., 2005). In this approach the first

step is to identify the critical section for web post buckling (Figure 2.17) using

some factors which are drived from a curve fitting process based on the numerical

studies. The applied compressive force in this section is then compared to the

design resistance of the web post, which is defined as a function of imperfection,

slenderness of the web post and a factor dependent to the spacing of openings.


55

Figure 2.17- CTICM model for the web post buckling

2.5.2 Vierendeel mechanism

Vierendeel mechanism is caused by the failure due to the formation of four plastic

hinges in the top and bottom Tees. The need to transfer the shear force across the

opening causes some secondary moments in the Tee section. The interaction of

these moments with the global bending moments and the local axial force due to

global bending, dominate the formation of plastic hinges in the Tees (Sherbourne

and Oostrom, 1972). Therefore, top and bottom Tees should be checked for

flexural capacity.

Ward (Ward, 1990) proposed a linear interaction relationship between the local

axial force and moment of the Tees (Equation 2-5) to evaluate their moment

capacity. The forces were the ones acting at the critical section of the Tees (causing

the highest concentration of stresses) to form the plastic hinges.

0.1maxmax

≤+M

MPPo Equation 2-5


56

In this equation, Pmax and Mmax were the axial and bending capacities at the critical

section and Po and M were the acting local axial force and moment at the critical

section.

The moment capacity of the Tees depends on the location of the plastic hinge. On

the other hand, the moment capacity is also dependent on the axial force and shear

force, which is again a function of the location of the plastic hinge. This requires an

iterative approach to define the critical section, which makes it not very favourable

for simple hand calculations.

However, there are recommendations on how to consider the critical section

(straight or curved sections as shown in Figure 2.18) and also about the angle of

the critical section to the vertical line. This angle normally varies between 0o to 28o

for standard sections and Ward recommended 25o (Ward, 1990).

Figure 2.18- Different approaches to consider the critical section in the Tees

Investigations showed that Ward’s method is too conservative as it only considers

the most critical section while beams can carry higher loads until the formation of

4 plastic hinges to cause failure due to Vierendeel mechanism. Liu and Cheng (Liu

and Chung, 1999) conducted a wide range of numerical studies to investigate the

neglected capacity of the beam due to stress redistribution before the formation of

4 plastic hinges. They found that the Ward approach was about 15% conservative

and recommended a parabolic interaction formula (Equation 2-6).


57

0.12

max

2

max

≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛M

MPPo Equation 2-6

They also investigated the shear capacity of cellular beams in more detail using

numerical studies and suggested that a percentage of the flange (0.35tf) also

contributes in carrying shear stresses. They improved their proposed moment-shear

interaction to (Liu and Chung, 2003):

0.15.2

max

5.2

max

≤⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛M

MPPo Equation 2-7

2.6 DESIGN OF CELLULAR COMPOSITE BEAMS AT

AMBIENT TEMPERATURE

The basics of the design models proposed to check for web post buckling and

Vierendeel mechanism in cellular composite beams is identical to those proposed

for cellular beams. The difference is that for cellular composite beams, the

distribution of forces in the web post and Tees, as well as the flexural resistance of

the top Tees is modified to consider the contribution of the concrete slab.

The SCI publications P068 and P100 included the first guidance on the design of

composite beams with web openings (Ward, 1990). The SCI approach showed

good agreement with some tests on composite beams (Lawson and Chung, 1992)

and was also presented by a similar guidance published by AISC (Darwin, 1990).

This approach was rather generic and did not consider the composite action in the

checks for Vierendeel bending. Moreover, it did not provide any details about the

distribution of forces between the steel beam and concrete slab. The other

deficiency of this method was that there was no consideration of the slab type,

degree of shear connection and decking system.


58

Lawson proposed a semi-empirical simplified method for cellular composite beams

(Chung and Lawson, 2001; Chung et al., 2003) which was based on his approach

for cellular beams presented earlier in Section 2.5.1. In this approach, the load

applied to the strut was modified with regard to the contribution of the concrete

slab.

For Vierendeel bending checks, the effect of composite action of the top Tee was

included in the Vierendeel bending check and forces acting on the Tees were also

modified (Lawson et al., 2006). For Vierendeel bending check, the circular opening

is generally replaced by an equivalent rectangular opening (Lawson and Hicks,

2009) and the forces presented in Figure 2.19 should satisfy the Equation 2-8.

According to Lawson’s approach (Equation 2-8), sum of the Vierendeel bending

resistances at the four corners should not be less than the design value of applied

Vierendeel moment, which is expressed as:

Mc,RdMt,NV,Rd

Mb,NV,Rd Mb,NV,Rd

VEd VEd

Mc,RdMt,NV,Rd

Mb,NV,Rd Mb,NV,Rd

VEd VEd

Figure 2.19- The circular opening represented by an equivalent rectangle (for the Vierendeel bending check) and applied forces

VEdle ≤ 2Mb,NV,Rd + 2Mt,NV,Rd + Mc,Rd Equation 2-8

where:

Mb,NV,Rd is the bending resistance of the bottom Tee reduced for axial tension and shear, VEd


59

Mt,NV,Rd is the bending resistance of the top Tee, reduced for axial compression and shear

Mc,Rd is the local composite Vierendeel resistance moment

VEd is the design value of the shear force (taken as the larger of the values at either end of the length le)

le is the length of the equivalent rectangular opening.

2.7 DESIGN OF CELLULAR COMPOSITE BEAMS AT

ELEVATED TEMPERATURES

The strut model by SCI is extended to estimate the buckling resistance of the web

post at elevated temperature by applying reduction factors for the material at

elevated temperature. SCI report RT1085 (Simms, 2007) includes the details of this

approach for web buckling check. The CTICM method presented earlier in Section

2.5.3 is also updated to estimate the buckling capacity of the cellular beams at

elevated temperature (Vassart et al., 2008). This approach uses an iterative

procedure to define the critical section for the web post buckling, at temperature θ,

by comparing the applied force and the resistance.

For Vierendeel failure, the common approach at ambient temperature is again

extended to include the material reduction factors with the increase of temperature.

These approaches have been established in the software available to design cellular

beams (including fire) such as Cellbeam by Westok or FBEAM by Fabsec.

However, implementing a simple design method which can be followed by hand

calculations has proved to be difficult as the iterative procedure of considering the

interactions between moment, shear and axial force, distribution of the forces

within the composite section and finding the critical section can be tedious to

follow by hand calculations. Some research is now being carried out by SCI

focusing on the some pragmatic simplistic assumptions, which does not affect the

reliability of the approach but allows for a simplified method that can be followed

by hand calculations.


60

2.7.1 Temperature distribution in cellular composite beams

Along with the investigation on the structural side, it is also critical to assume a

reasonable temperature distribution within the sections in the fire design of cellular

beams. As steel has a high thermal conductivity and low thermal capacity, there is

not much time delay between the steel temperature and air temperature. However,

the thermal characteristics of concrete means that concrete is notably less

responsive to the surrounding gas temperature.

The relative steel to gas temperature also depends on the Section Factor (Hp/A)

which is the ratio of the area exposed to the fire to its volume that absorbs the heat

(Simms, 2004a). Equation 2-9 is an empirical formula that relates the temperature

of the bottom flange to the gas temperature based on the section factor of the

bottom Tee. Recommendations and simple methods are provided in the

BSEN1994-1-2 (BSI, 2005c) to estimate the temperatures in the steel section, shear

studs and concrete slab. However, comparisons between the output of this

approach and recent test data (Nadjai et al., 2008) suggested that improvements can

still be made in the temperature distribution assumed in the cellular composite

beams.

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛×+= 15065.0547bF,

BTM

p

AH

θ Equation 2-9

2.8 INTRODUCTION TO THE FINITE ELEMENT ANALYSIS

Use of the finite element analysis, which was implemented in 1943 (Pelosi, 2007),

has become increasingly popular over the recent years. Along with the advances in

computations, finite element packages are now playing a critical role in the design

of ordinary structures as well as research into new issues in structural engineering.

The basics of the finite element approach (Fagan, 1992) is to discretise the

structure into a number of nodes which are connected by elements. Displacement


61

interpolation functions (shape functions) (Zienkiewicz and Hinton, 1976) are

defined to simulate the variations within the elements and stress-strain

relationships to represent different materials. This method then derives the stiffness

matrix of elements and assembles the global stiffness matrix (K) which determines

a relationship between the nodal displacement vector (u) and the external forces

vector (F) presented in Equation 2-7. By introducing the boundary conditions, the

following equation can be solved for the unknown value, which is mostly

displacement in structural analyses. Equilibrium is attained when the virtual work

dissipated by the external forces is equal to the virtual strain energy for any

allowable virtual displacement (δu) of the deformed structure.

[ ] { } { }FuK =× Equation 2-7

The system of equations is solved in increments. Iterations in each increment

consider three types of nonlinearities associated with structural analysis.

Geometrical nonlinearities (Mackerle, 2002) to cover the P-∆ (second order)

effects, material nonlinearities which consider the materials with non-elastic

behaviours, and finally boundary nonlinearities which deal with contacts and

interactions between elements of separate bodies.

Geometrically nonlinear structures sometimes incur buckling or collapse

behaviour, where the load-displacement response shows a negative stiffness. This

post failure behaviour of the structure cannot be followed by the common Newton-

Raphson algorithm as the structure must release strain energy to remain in

equilibrium. The Riks method (Riks, 1984) allows for finding static equilibrium

states during the unstable phase of the response by decreasing the applied load to

find the maximum load that the failed structure can take at each stage (Figure

2.20). In this approach, the user just defines the load at the first iteration and has no

control over the load applied in the next iterations, which means this method

considers the load magnitude as an additional unknown. The Riks method solves

simultaneously for loads and displacements and another quantity is used to


62

measure the progress of the solution, which is the arc-length along the static

equilibrium path of the load-deflection curve. This approach provides solutions

regardless of whether the response is stable or unstable.

Figure 2.20- The Riks method


63

3 CELLULAR BEAMS AT AMBIENT TEMPERATURE

3.1 INTRODUCTION

This chapter presents the numerical approach used to model a Cellular Beam (CB)

at ambient temperature and compares the results of the numerical models

developed for various destructive tests against the test data. The calibration of the

numerical approach was conducted against test data of the tests carried out by

Leeds (Surtees and Li, 1995) and Natal (Warren, 2001) universities. In addition,

this chapter highlights some crucial points in the numerical modelling of CBs,

which might cause the numerical model to predict the wrong failure mode as

opposed to what was witnessed in the tests. In particular, this chapter investigates

the critical role of boundary conditions and time increments in the analysis of CBs,

especially when web post buckling occurs.

3.2 MODELLING CELLULAR BEAMS

The numerical models for single CBs were validated against three experiments

conducted on symmetric beams as follows:

1) Natal Beam 4B (Warren, 2001)

2) Leeds Beam 2 (Surtees and Li, 1995)

3) Leeds Beam 3 (Surtees and Li, 1995)

In this research, the following assumptions were taken within the nonlinear

analysis of CBs:

• Use of shell elements with a linear and quadratic deformation approach and

the ability to handle large strains, large deformations and plasticity.

• Geometrical as well as material non-linearity.


64

• Solver by iterations according to the Newton-Raphson method or using the

Riks (Arc-length) method, explained in Section 2.8, to investigate the post-

buckling behaviour.

• The stress-strain curve for steel was considered either according to the

results of the recorded one-dimensional tensile test, or the BSEN1992-1-1

(BSI, 2004a), BSEN1993-1-1 (BSI, 2005a) and BSEN1994-1-1 (BSI,

2004c) suggestions, were applied to the FE analysis when data were not

available.

• The section dimensions and thicknesses were assumed as nominal values

except the test samples in which actual section dimensions were measured

and available in the report.

• To reduce the computational time, only half of the beams were modelled

where possible and the symmetrical boundary conditions were applied.

Imperfections need to be considered in the modelling of CBs and Section 3.3.7

discusses the way they are defined and applied to the models.

3.3 MODELLING NATAL BEAM 4B

3.3.1 Introduction

A total of 8 cellular beams were tested in the University of Natal (Warren, 2001).

Two conveniently small sections, namely UB203X133X25 and UB305X102X25,

were used as parent sections to construct cellular beams with two ratios of opening

spacing to opening diameter. For each of the four resulted geometries, two loading

conditions were chosen: mid-point and third-point loading, in order to investigate

different web post slenderness ratios as well as loading patterns.

Out of these 8 beams, only one failed due to web post buckling which was Beam

4B. Since this research mainly focuses on the web post buckling failure mode, only

this beam was modelled in this study. The parent section of Natal Beam 4B was a

UB 305X102X25 and the cell diameters were 325 mm as shown in Figure 3.1.


65

Further details of the test are available in the test report (Warren, 2001) are not

repeated here.

Figure 3.1- Details of Natal Beam 4B

3.3.2 Modelling using ANSYS and ABAQUS

Two of the commercially available FE programs were used for this simulation,

ANSYS and ABAQUS. Using two well-known FE packages gave the opportunity to

compare the modelling approach used and their results. It was particularly useful to

compare the results knowing the fact that the approach used in the buckling of shell

elements is slightly different in the two software packages. Specifically, in ANSYS,

the occurrence of the web post buckling was probable even in the geometrically

perfect CBs. Whilst in ABAQUS, it was necessary to apply a minimum

imperfection in order to provoke the web post buckling failure mode. Various

approaches used to calculate and apply the web imperfections will be presented

and discussed later in this chapter.

Moreover, the mesh used in ABAQUS was generally finer than that of ANSYS. The

reason for this was that in ABAQUS, unlike ANSYS, there was no quadratic (8

noded) element with the ability to handle large strains and large deformations.

Hence, a finer mesh with 4 noded shells, especially in web posts, was applied to

compensate this deficiency and provide a better base for comparing the two

software packages. Sections 3.3.3 and 3.3.4 present the modelling approach within

ABAQUS and ANSYS respectively and Section 3.3.8 compares the results of the

numerical models developed within the two packages.


66

3.3.3 Modelling in ABAQUS

The finite membrane-strain, fully integrated, quadrilateral shell element S4 (Hibbitt

et al., 2004a) was used for the modelling of cellular beams within ABAQUS. In

order to have more accurate results, the ABAQUS manual (Hibbitt et al., 2004a)

recommends not to use elements with reduced integration (S4R) while shells are

loaded causing in-plane bending or distortion. Figure 3.2 shows the half-beam

modelled in ABAQUS. The boundary conditions applied to the beam complied with

those in the experiment. According to the test report (Warren, 2001), three lateral

supports were provided to the top and bottom flanges at the middle and both ends

of the beam and these were similarly considered in the simulation.

Figure 3.2- Half of the Natal Beam 4B modelled in ABAQUS

The stress-strain curve used for the material was taken from the results of the one-

dimensional tensile test conducted separately on the beam web and flange. Figure

3.3 shows the bilinear curves given in the test report and used in the finite element

modelling, presented in this thesis.


67

Figure 3.3- Material specifications used in FE model

3.3.4 Modelling in ANSYS

The 8-noded shell element, Shell93 (ANSYS, 2003) , with the ability to handle

large deformations, large strains and plasticity, was used in the simulation using

ANSYS. As mentioned, in ANSYS, unlike ABAQUS, the web post buckling of the

shell can occur with no imperfection. In other words, the solution algorithm in

ANSYS is more comprehensive and handles the buckling behaviour of perfect

structures at their critical load. Literally, a perfect plate, under a compressive force

in its plane, does not buckle in ABAQUS and either an imperfection or trigger load

needs to be used. Whereas in ANSYS, a plate buckles at its critical load. However,

this difference in the two software packages was not critical in modelling CBs, as

in most cases CBs were assumed imperfect due to their fabrication.


68

Figure 3.4- Half of the Natal Beam 4B modelled in ANSYS

3.3.5 Sensitivity study on mesh size

In general, to find the optimum element mesh size in numerical models, a number

of analyses are carried out considering various mesh densities and the load-

deflection graphs obtained from these analyses are compared. As an example

Figure 3.5 shows the load-deflection curves for the model developed for Natal

Beam 4B with different mesh densities. Since the same mesh was repeated in all

panels, the number of elements in each panel also represents the mesh size. The

lowest mesh density which led to similar results as the finer meshes, was the

chosen mesh size to achieve the computational efficiency. Therefore, Figure 3.5

suggests that the mesh density which results in 300 elements per panel was the

most efficient one in the numerical model developed for Natal Beam 4B.


69

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70Displacement (mm)

Load

(KN

)

420 Elements per panel

300 Elements p. p.

96 Elements p. p.

48 Elements p. p.

16 Elements p. p.

``

Figure 3.5 - Sensitivity study on the mesh size, Natal Beam 4B

The mesh density was also increased in some cases to help resolve convergence

problems. Increasing the number of elements (nodes and integration points) proved

to be useful in overcoming the divergence of the solution due to occurrence of

large displacements, local instability of web posts, and concrete cracking and

crushing, within very small time-increments before failure.

3.3.6 Analysis types

As mentioned in Section 3.2, two different types of analyses were used in the

simulations conducted at ambient temperature. The Newton-Raphson or general

method, which increases the load and controls the displacement, was used to

investigate the behaviour of the CB up to the failure (buckling) point. The Riks

(Arc-length) method was used in some cases in order to look into the pre as well as

the post failure behaviour. In the general method, models were analysed linearly

and then non-linearly. The initial elastic analysis gave a good indication of the load

at which the beam was likely to yield by comparing the von Mises stress

distribution against the yield stress. This information was needed in formulating


70

the load increments for the plastic analysis and resulted in saving time and space in

the finite element analysis (FEA). This is shown for the Natal Beam 4B in Figure

3.6.

0

20

40

60

80

100

120

140

160

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time step

Load

(KN

)

Beginning of the Plasticity for the Natal Beam 4B (P~88KN)

Figure 3.6- Load steps applied to the Natal Beam 4B in which load has increased

with a high rate in elastic region and low rate while plasticity

In general, when buckling happens in the structure, the load-displacement response

shows a negative stiffness and some strain energy should dissipate so that the

structure remains in equilibrium. Several approaches can be used for modelling the

buckling as well as post-buckling behaviour, one of which is the Riks method. The

Riks method uses both displacement and load magnitude as unknowns, and solves

the equilibrium equations for load and displacement simultaneously (Hibbitt et al.,

2004a). The only limit for using this method is that the loading should be

proportional; that is, the load magnitudes are increased by a single scalar

parameter. The Riks method uses the “arc length” along the equilibrium path in the

load-deflection curve, which provides solutions regardless of the response being

stable or unstable.


71

Figure 3.7 compares typical load-deflection curves obtained from the general and

Riks analyses which shows the general method has diverged with failure while the

Riks method, has handled the post-buckling behaviour of the failed beam.

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160 180Def. (mm)

Load

(KN

)

Arc-Length Method

General Method

Figure 3.7-Comparison between General and Arc-length method

3.3.7 Applying imperfections

Imperfections are usually introduced to the model as perturbations in the initial

geometry. They consist of multiple superimposed buckling modes applied to the

model, unless the exact shape of the imperfection is known. Considering

imperfections in cellular beams is vital because of several reasons. In practice, the

process of cutting and fabrication of cellular beams would typically cause

geometrical perturbations. Therefore, considering them as intact sections in

modelling is far from reality. Besides, as mentioned in most of the FE software

such as ABAQUS, in order to reach the post-buckling stage, the equilibrium path

should not branch (bifurcation point). In other words to analyse a buckling problem

it must be turned into a continuous response instead of bifurcation.


72

Two different methods were used in this research to apply the imperfection to the

model. One way was to apply a fictitious “trigger” load (or initial displacement) to

initiate the instability. This trigger load must be very small so that it does not affect

the overall buckling and post-buckling behaviour of the structure, but big enough

to avoid bifurcation. A sensitivity study on the magnitude of the trigger load was

carried out in this regard.

The trigger load must be located in a position that perturbs the structure in the

expected buckling mode, resulting from test observations or buckling analysis.

Therefore, in order to define the best location for the trigger load a buckling

analysis was carried out on the model, and global as well as local buckling modes

of the structure were defined. Figure 3.8 shows the first buckling mode from the

buckling analysis carried out on Natal Beam 4B. The imperfection was applied to

Natal Beam 4B by exerting a trigger load where the out-of-web plane displacement

was the maximum.

The second approach to introduce the imperfection is by applying a linear

superposition of local and global buckling modes to the model. In this method, the

rate of association of each mode is defined by imperfection amplitudes according

to Equation 3.1.

∑=

=∆M

iiii wx

1ϕ Equation 3-1

where:

iϕ is the ith mode shape

wi is the associated scale factor

There is no absolute regulation for the shape of the web post imperfection in

numerical modelling of perforated steel sections, or a unique formula to propose

the imperfection amplitude. The latest SCI report to design the cellular beams

(Simms, 2008) proposes to take the form of a half sine wave from the top to the


73

bottom of the section with a maximum amplitude equal to H/600 for numerical

modelling where H is the section height. The L/500 is also recommended by

BSEN1993-1-1 (BSI, 2004c) as the overall mid-span imperfection for lateral-

torsional buckling investigations. Schafer and Pekoz (Schafer and Pekoz, 1998)

have also proposed approaches in which the amplitude of the imperfections is

governed by the thickness of the steel plate for local bucking and the overall length

of the section for global buckling.

Figure 3.8- First buckling mode of Natal Beam 4B gained by ABAQUS buckling

analysis

In this method, the first step was to perform a buckling analysis on the beam to get

the first local and global buckling modes (the ones with the lowest Eigenvalues). In

the second analysis, the imperfection is applied to the perfect geometry by

introducing scale factors for buckling modes of the structure. This method was

used for most of the simulations of this research and also for the parametric studies

conducted later investigating the effect of web imperfections. Comparisons

between these two different ways of applying imperfections showed there was no

major difference between these methods at ambient temperature provided that they

were used properly i.e. the trigger load caused a deflection equal to what was

considered as the imperfection amplitude.


74

3.3.8 Numerical modelling results for Natal Beam 4B

It is mentioned in the test report (Warren, 2001) that the rate of vertical deflection

increased at 108 kN showing that the occurrence of Vierendeel mechanism was

imminent and the beam finally failed at 114 kN due to web buckling.

The failure mode predicted by both software packages was also web post buckling,

similar to the test and both models predicted exactly the same load-deflection

behaviour. Both models predicted the buckling load to be 125 kN, about 13%

overestimation compared to the test, which is a reasonable discrepancy. Figure 3.9

shows the load-deflection comparison between the test results and the ANSYS and

ABAQUS software predictions and Figure 3.10 shows the S-shape web post

buckling of Natal Beam 4B within ABAQUS.

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Load

(KN

)

Def. (mm)

ANSYSABAQUSExp.

Web Buckling (P=125 KN)

Web Buckling (P=114 KN)

Figure 3.9- Comparison of the load-deflection curves according to the test and the

results obtained from ANSYS and ABAQUS


75

Figure 3.10- Web post buckling of Natal Beam 4B using ABAQUS

The model developed in ANSYS also correctly predicted the occurrence of the S-

shape buckling for this beam as shown Figure 3.11.

Figure 3.11- Development of S-shape web post buckling of the Natal Beam 4B

using ANSYS


76

3.4 MODELLING LEEDS BEAM 2

3.4.1 Introduction

Five CBs were tested, on behalf of the Steel Construction Institute (SCI), at Leeds

University in 1995 (Surtees and Li, 1995), with the aim of seeking a better insight

into the behaviour of CBs. The beam geometries were designed to fail in various

modes including web post buckling, which was observed in Beam 2 and Beam 5.

The second finite element modelling presented in this research is based on Leeds

Beam 2. Figure 3.12 shows the overall specifications of this beam. More details

can be found in the test report (Surtees and Li, 1995). Notably, web stiffeners were

omitted in most of the beams tested in Leeds, including Beam 2, in order to

provoke the most critical circumstances of the tests.

Figure 3.12- Geometry of the Leeds Beam 2

One of the most important aspects of numerical modelling of CBs is to mimic the

load-deflection curve and the failure load of the test. However, identifying the

correct mode of failure is also critical. This is the area where the majority of the

existing software which are designed specifically for the analysis of CBs, such as

Cellbeam v5.0 (Simms, 2005), incur notable prediction errors. Cellbeam software

is developed by the Steel Construction Institute (SCI) to design cellular beams and

cellular composite beams and SCI report RT1025 (Simms, 2004b) includes the

design prediction obtained from this software for 22 tests carried out on composite

and non-composite cellular beams of various geometries. This report compares the

failure modes predicted by the software against the test results which reveals the

notable errors in the prediction of the correct failure modes observed in the tests. In


77

particular, Vierendeel mechanism was specified by Cellbeam as the failure mode in

some of the cases while the beam actually failed due to web post buckling in the

test. The utilization factors given by Cellbeam for Vierendeel mechanism and web

buckling were also conservative by 20% and 10% respectively.

3.4.2 Modelling approach for Leeds Beam 2

Following the methodology used for Natal Beam 4B, ABAQUS and ANSYS

software packages were used for the modelling of Leeds Beam 2. The Leeds Beam

2 failed due to web post buckling in the test (Surtees and Li, 1995). Unfortunately,

the test report for these beams suffers from the lack of data given for material

properties and the only available information was the yield strengths of the web

and the flange. Hence, some assumptions were made in modelling. The missing

values (ultimate stress, strain and Elastic modulus) assumed in the model were

determined with regard to the data available in the test report for other Leeds

beams or using BSEN1993-1-1 (BSI, 2005a) recommendations as a reference.

Figure 3.13 shows the bilinear stress-strain curve used for numeral modelling

which resulted from the following three initial assumptions:

Fu = 1.1 Fy

eu = 0.2

E = 205 (kN/mm2)

According to the test report, full lateral restraint was provided for the upper and

lower flanges of the beam at approximately one-metre intervals to avoid lateral-

torsional buckling of the beam. This was considered in the model as shown in

Figure 3.13.


78

(a) (b)

Figure 3.13- (a) assumed stress-strain curve for Leeds Beam 2; (b) The (half) beam geometry and B.C.s

Figure 3.14 compares the initial results of the nonlinear analysis conducted within

ABAQUS and ANSYS packages against the test result. The load-deflection curves

obtained from the two software packages were similar but the comparison with the

test result was poor. Moreover, this beam failed due to web buckling in the test but

the failure mode predicted by model was excessive plastic deformations in the top

and bottom Tee sections, known as Vierenedeel mechanism. According to Figure

3.14, this initial poor comparison between FE and test results was mainly because

the assumed elastic modulus (E) did not reasonably reproduce the initial (elastic)

part of the load-deflection curve, and needed to be modified (reduced).


79

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16 18Def. (mm)

Load

(KN

)

ExperimentABAQUS ANSYS

Figure 3.14- Test results versus ANSYS and ABAQUS results for Leeds Beam 2

assuming E= 205 kN/mm2

A decision should also have been taken on the software to be used for the

subsequent modelling to investigate cellular composite beams. Therefore, the

literature and capabilities of the ABAQUS and ANSYS software were investigated

and compared. The following section summarises these findings which proved

useful in choosing the software for the future numerical studies carried out in this

research.

3.4.3 Comparison between ABAQUS and ANSYS

In general, the studies conducted comparing ABAQUS and ANSYS in terms of the

ability to predict the correct failure mode of CBs, with no concrete slab involved,

showed no notable difference between the two software packages, as long as

similar approaches were used.

In order to make a sound decision regarding which software package to use in this

research project, their ability to model the rather complicated concrete material,

considering cracking and crushing behaviour, was investigated. This was done by

referring to the relevant literature and research (Barbosa and Riberio, 1998;

Fanning, 2001) which included modelling the concrete material, software manuals

and seeking advice from ANSYS and ABAQUS peers.


80

Based on these studies it was known that ANSYS is only capable of associating the

concrete material with a particular solid element (SOLID 65). In other words,

concrete material cannot be associated with any shell or beam elements. Despite,

ABAQUS enables the user to incorporate the concrete material with solid elements

as well as shell and beam elements. As the main focus of this research was the

performance of the steel beam, as opposed to the concrete slab, it was prefered to

model the concrete slab with shell elements to save computational time and avoid

the potential numerical issues associated with shell-solid interaction.

In terms of the available approaches to model the concrete material, ANSYS, only

offers the smeared cracking model whereas ABAQUS benefits from three concrete

models, namely smeared cracking model, damaged plasticity and crack model.

This equips the ABAQUS user with more options to tackle the foreseen and

unforeseen challenges associated with modelling concrete.

Moreover, ABAQUS offers notably more flexibility in the options available to

model and manipulate the concrete properties than ANSYS. An example of this

flexibility is related with defining the post cracking behaviour of concrete. In

ANSYS there is no option to control the post cracking behaviour of concrete, the

strength reduction rate and ultimate tensile strain and constant defaults have to be

used. However, in ABAQUS, post-cracking behaviour can be accurately defined

and manipulated as necessary. The definition of the post-cracking can be critical

considering that in many cases, minor changes in ultimate tensile strain (changing

from brittle towards ductile) could help overcome the convergence problems

associated with modelling concrete material. However, the user needs to ensure the

general response of the structure remains reasonably unaffected.

Consequently, the decision was made to use ABAQUS for future validations and

investigations presented in this research.


81

3.4.4 Modified modelling approach for Leeds Beam 2

Based on the comparison between the test and FE results, the elastic modulus

(which was an assumed value not a measured one, was then reduced to

195 kN/mm2 (from 205 kN/mm2) for a better fit with the elastic part of the load-

deflection curve. Figure 3.15 compares the modelling versus test result after this

modification.

It should be noted that the discrepancy between the numerical and test result in this

case could have been because of differences between the actual section sizes and

thicknesses, and the nominal ones that were used in the model. This discrepancy

can also be due to settlement of the support. Reducing the modulus of elasticity to

less than 200 kN/mm2 is not realistic and was done to get a better agreement

between test and numerical results, within the elastic range, in the absence of any

details about the actual section sizes and support settlements.

0

50

100

150

200

250

300

0 10 20 30 40 50 60Def. (mm)

Load

(KN

)

FE

Experiment

Buckling

Vierendeel Def.s

Figure 3.15- Test results versus ABAQUS for Leeds Beam 2 assuming E= 195

kN/mm2

This beam buckled at 195 kN in the test but in the FE model web post buckling

was notably delayed up to 270 kN. Meanwhile, development of plasticity in the

web post and Tee sections caused gradual plastic and Vierendeel deformations as


82

highlighted in Figure 3.15. Figure 3.16 shows the Von Mises stress distribution

with increasing load. The plasticity initiated in the web posts and then developed to

the Tee sections.

Figure 3.16- Development of Von Mises stress with load increase

The unconvincing results obtained from the numerical modelling approach led to

the need to carry out more comprehensive studies to investigate the effect of

material properties, boundary conditions, imperfections and time increments in the

behaviour of this beam.

Consequently, investigations focused in two significant areas of the numerical

modelling of CBs, which could influence the results notably. These two areas were


83

the boundary conditions and the time stepping and load increments while using the

automatic time stepping option to run the nonlinear analysis. These two issues are

discussed in the following two sections.

3.4.5 Importance of boundary conditions in modelling CBs

According to the test report (Surtees and Li, 1995), the top and bottom flanges of

the beam were laterally restrained at one-metre intervals. Here, two similar beams

are presented (Leeds Beam 2) and the only difference is the length in which the

lateral support is applied. Figure 3.17(a) shows approach 1 (Beam 1), where

supports were provided at single nodes (U3=0), the old approach also shown in

Figure 3.13. Figure 3.17(b) shows Beam 2, where lateral supports were applied

over 225 mm of the length of the flange, which relates to the test condition, rather

than just on a single node.

Both beams were expected to show a similar behaviour under the applied load, and

if different Beam 2, which benefits from more lateral support, was expected to be

slightly more stable against buckling and resist higher loads compared to Beam 1.

(a) (b)

Figure 3.17- Boundary conditions applied to Beam 1 and Beam 2, for comparison purposes


84

In order to give a comprehensive response of the beam behaviour in the pre and

post-buckling phases the Riks method was used for this analysis. Surprisingly the

results showed not only Beam 2, with more lateral supports, was more susceptible

to web post buckling compared to Beam 1, but also the difference was significantly

more than expected.

Figure 3.18 compares the load-deflection curves obtained for these two beams and

highlights that, in the numerical modelling of cellular beams, even minor changes

in the applied boundary conditions can considerably change the load-deflection

curve, failure load, and also the governing mechanism leading to failure. Beam 2

failed due to web post buckling, mimicking the correct failure mode observed in

the test, at P=180 kN, which was reasonably within 8% of the experimental value

(195 kN). In other words, just a slight modification in the way the lateral supports

were applied (from Beam 1 to Beam 2), changed the results significantly and

provided the necessary modelling validation for this beam.

0

50

100

150

200

250

300

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

Load

(KN

)

Beam 1ExperimentBeam 2

Figure 3.18- Load-deflection curves for Beam 1, Beam 2 and the test


85

This notable sensitivity of the models developed for beams with web openings, to

lateral supports was also observed in the previous investigations of the author into

the behaviour of castellated beams (Bake, 2004). Figure 3.19 shows three similar

beams with different support conditions applied to the top flange which were

investigated in these investigations by the author. Amongst these three models, the

one with full lateral support along the top flange (Model 3 in Figure 3.19) failed

due to web post buckling (Figure 3.20) whereas the other two models failed due to

excessive shear deformations in the weakest section of the web post which is

shown in Figure 3.21.

Figure 3.19-Models developed to investigate the influence of the boundary

condition of the top flange on the behaviour of castellated beams


86

Figure 3.20- Failure due to excessive shear deformation of the web post in Model 1

and Model 2

Figure 3.21- Failure due to S-shaped web post buckling in Model 3


87

The investigation subsequently focused on understanding the reason behind this

rather unexpected behaviour of this model. Looking at variations of out-of-web

plane displacements versus applied load was a useful indication. Three nodes with

the maximum out of plane displacements in each panel, (points A, B and C in

Figure 3.22), were investigated. The resulting curves are presented in Figure 3.23,

which shows a significant discrepancy (between Beam 1 and Beam 2) of the load

at which web post buckling was initiated.

Figure 3.22- Out-of-web plane displacement contour

The interesting point was that, according to Figure 3.23, the out-of-web plane

displacements curves in the two beams were also different in trend. Buckling

occurred gradually in Beam 2 but suddenly in Beam 1. In Beam 1, the webs had no

out of plane displacements prior to web buckling and failure happened abruptly at

265 kN. In contrast, in Beam 2, out-of-web plane displacements initiated much

sooner at 110 kN, and gradually developed until the web failed by buckling at

185 kN.


88

0

50

100

150

200

250

300

0 5 10 15 20 25Dis. (mm)

Load

(KN

)

Beam 1- Point ABeam 2- Point ABeam 1- Point CBeam 2- Point BBeam 2- Point CBeam 1- Point B

Figure 3.23- Displacements of points A, B and C of Beam 1 and Beam 2

Further investigations conducted on this beam were helpful in identifying the

reason behind this rather unexpected but interesting phenomenon in numerical

modelling of CBs.

In the first step to have a better understanding of how the CB behaves under the

applied load, four identical beams were modelled (based on Beam 2) with different

material properties. Elastic and elastic-plastic materials were considered for the

web and flange to cover all four possible cases and provide a better insight into the

behaviour of the web and flange.


89

Figure 3.24- The beam and the applied material properties

Figure 3.24 shows the elastic and elastic-plastic material properties used in the four

beams. Comparing the results of the four analyses, Figure 3.25, shows that

changing the material properties of the flange from elastic-plastic to elastic did not

affect the results confirming that the flanges were behaving completely elastically.


90

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30 35 40 45Def. (mm)

Load

(KN

)

Experiment

Web: E.P., Flange: E.P.

Elastic Beam

Web: E.P., Flange: E

Web: E., Flange: E.P.

Figure 3.25- Load-displacement comparison for different cases

In the second step to investigate the reason why negligible changes in boundary

conditions ended up in significant changes in results, the exaggeration approach

was used. In other words, to magnify the difference caused by the two types of

lateral supports, a third and forth boundary conditions (Beam 3 and Beam 4 as

shown in Figure 3.26) were introduced and compared to Beam 1 and Beam 2 to

facilitate understanding the mechanics and causes behind this notable sensitivity

Figure 3.26- Boundary conditions applied to beams 3 and 4

Beam 3 had full lateral support at the top and bottom flanges and in Beam 4 they

were kept to a minimum and were just provided at mid-span to prevent overall

lateral-torsional buckling of the beam.


91

Figure 3.27 compares the load-deflection curves obtained for these four beams and

highlights an interesting trend in the load-deflection curves. In fact, higher lateral

supports resulted in lower ultimate load resistance and promoted the web post

buckling. In particular, Beam 4, with the minimum lateral support carried the

highest load while Beam 3, which was the most laterally restrained one, buckled

first.

0

50

100

150

200

250

300

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

Load

(KN

)

Beam 1ExperimentBeam 2Beam 3Beam 4

Figure 3.27- Comparison of the load-deflection curves for the four beams

This suggests that although increasing the lateral support, in numerical modelling

of CBs, delays overall lateral-torsional buckling (global buckling), it can be

detrimental when considering web post buckling (local bucking) which is

governing the ultimate loading capacity.

Further studies were carried out on Beam 3 and Beam 4, the two beams with

extreme boundary conditions, and deflections were investigated in the two critical

zones where web buckling initiated. In particular, the vertical deflection of the

nodes at the top flange, and the horizontal deformation of the mid-web in critical

zones 1 and 2 (Figure 3.28) were investigated in both beams.


92

Figure 3.28- Sections investigated and naming used in details

Tables 3.1 and 3.2 provide the deflection values for the twelve nodes specified in

Figure 3.28 for Beam 3 and Beam 4 respectively. These values imply for both

beams that the top flange remains in a straight line (does not bend) during the

loading. Moreover, comparing the vertical deflections (U2) of the top flange nodes

(node A versus E or node G versus K) between the two beams highlighted that a

notable rotation takes place in the upper flange of Beam 3 whereas this rotation is

negligible in Beam 4, as reflected schematically in Figure 3.29.


93

Table 3.1- Displacements obtained for Beam 3

Table 3.2- Displacements obtained for Beam 4


94

Figure 3.29 Rotation of the top flange in Beam 3 (continuous restrain) compared to Beam 4 (two point restraints)

The reaction force of the lateral supports at the top flange of Beam 4 was

subsequently investigated. This compressive reaction force was directly

proportional to the applied load ending up in rather high values, depending on the

location on the selected node.

Figure 3.30 is the reaction forces induced at two of the lateral supports (denoted

node 1 and node 2) at different locations which shows this reaction force reached a

high value of 8 kN per restrained node.

The cause of such notable action and reaction forces between the top flange and

lateral supports was due to the lateral expansion of the top flange plate (depending

on the Possion’s ratio) while the applied vertical load causes compression in the

top flange. The resulting strain (also action and reaction force) depends on the

Poisson’s ratio and increases with the vertical load.


95

0

2

4

6

8

10

0 50 100 150 200 250

Total Vertical applied load (kN)

Late

ral s

uppo

rt R

eact

ion

(kN

)Reaction force of lateral support at Node 1

Reaction force of lateral support at Node 2

Figure 3.30-Two samples of reaction compressive force applied by lateral supports per node

This justification is well endorsed by looking at the lateral reaction forces in the

bottom flange which was under tension. Unlike the top flange, the reaction forces

applied from the supports to the bottom flange plate were tensile, supporting the

assumed behaviour.

The conclusion of all these investigations was that the internal forces induced in

the CBs with increasing loading, resulted in the unexpected significant effect of

lateral supports on the performance of CBs, in numerical modelling. Figure 3.31

shows the internal forces in the CB where the black arrows (in the left figure)

represent the force applied to lateral supports.


96

Figure 3.31- Internal forces and interaction with lateral supports

As mentioned, the CB sections were generally considered non-ideal and an

imperfection was applied to the web post. Therefore the eccentricities (e1) and (e2)

were not zero. According to Figure 3.32, the eccentricity (e1) times the load (P1)

causes the moment (M1) which makes the top flange rotate. As the web and flange

are welded together, the web also starts to rotate. The moment (M1) increases with

an increasing applied vertical load (load (P1) was up to 8 kN per restrained node as

stated) which promotes the instability of the web post. This explained why

increasing lateral supports, promoted failure by web buckling.

Figure 3.32- Effect of lateral supports in the web buckling in numerical model


97

In the bottom flange, the moment (M2) acts oppositely helping to stabilise the web

post. However, the effect is not enough to balance the web post perturbation caused

by (M1). Moreover, as the lateral supports were not welded to the bottom flange

and the fact that the bottom flange shortens in width, no interaction should be

considered between the lateral supports and the bottom flange. This was the case in

all simulations presented so far except for this particular model (for investigation

purposes).

This phenomenon also justifies the difference in the failure of Beam 1 and Beam 2,

as seen in Figure 3.23 and the differences in the shapes of out-of-web plane

displacements between Beam 1 and Beam 2. Due to the boundary conditions

applied, the moment (M1) was much higher in Beam 2 compared to Beam 1,

depending on the number of restrained nodes on the top flange. Therefore, the out-

of-web plane displacements in Beam 2 were the accumulation of the flexural and

buckling behaviour of the web. The out-of-web displacements due to bending

initiated at 110 kN and then increased gradually with the applied load, followed by

abrupt displacements due to buckling at 185 kN. However, the out-of-web plane

displacement in Beam 1 was solely caused by the buckling as no major moment

was applied to the top flange (and therefore to the web post) to cause a notable

bending prior to buckling.

A comparison was also conducted considering solid webbed beams to look into the

development of internal forces and the sensitivity of their behaviour to the

application of lateral supports. Figure 3.33 shows a beam with solid web having

similar specifications, comprising geometry and material properties, to that of

Leeds Beam 2. The lateral expansions of the top and bottom flange were similar to

that of the CB previously analysed and the same kind of action and reaction forces,

but relatively lower values were observed between the flange and the lateral

supports.


98

Figure 3.33- Expansion and shortening of the top and bottom flange respectively in the solid webbed beam

Adding or eliminating the lateral supports did not notably alter the performance of

the solid-webbed beam. This seemed reasonable because the failure of this beam

was not governed by any kind of web instabilities or local buckling and therefore,

solid-webbed beams were not very sensitive to lateral supports. Moreover, these

exerted lateral forces were also relatively smaller in the solid-webbed beam as

previously mentioned. This is because in CBs, the web openings cause a behaviour

similar to Vierendeel truss which intensifies the compression in the top flange,

compared to solid beams, and therefore, increases the strains in the lateral direction

(reaction force and consequently moment M1).

3.4.6 Importance of load increments in modelling CBs

As mentioned, two common analysis types were mostly used in this numerical

modelling, namely the general method and Riks method. In the general method,

which is based on the Newton-Raphson algorithm to solve the equilibrium

equations, the maximum and minimum allowed increment size in each step was

given by the user. The software then increases the load at each stage within the

defined range.

The limitations on the increment size help prevent numerical convergence

problems of the solution. However, within the investigations carried out on Leeds


99

Beam 2, it was highlighted that inappropriate application of the load increments

and allowing relatively large load increments, could result in not only erroneous

load-deflection curves, but also capturing the wrong failure mode.

Figure 3.34 shows a comparison between two beams, based on Leeds Beam 2, both

analysed with the general solver method using ABAQUS. The only difference

between Beam 1 and Beam 2, was the maximum allowable added load in each

increment which was limited to 16 kN and 18 kN respectively.

After running both models, the output showed that in Beam 1, out-of-web plane

displacement (U3 at node A) reached 13 mm at a load of 240 kN whereas in Beam

2 (Node B) this value was almost zero for a higher load (260 kN). This means that

no buckling occurred in the web post. Figure 3.34 shows the contours for the out-

of-web plane in these beams. Figure 3.35 compares the load-deflection curves

certifying the notable difference in the behaviour of the two beams.


100

Figure 3.34- Out-of-web plane displacement contours indicating buckling and no buckling in Beam 1 and Beam 2 respectively

0

50

100

150

200

250

300

0 5 10 15 20 25 30 35Dis. (mm)

Load

(KN

)

Beam 2

Beam 1

Experiment

Figure 3.35- Load-deflections curves of beams one 1 and 2


101

In Beam 1, the critical buckling load of about 240 kN was identified in the analysis

and the load increments applied in the solution at that stage were small enough to

handle the large displacements while buckling. On the contrary, in Beam 2, the

buckling point was not identified by the model and only some plastic and

Vierendeel deformations (Chung et al., 2001) occurred.

This behaviour implies how critical it is to limit the maximum allowable load

increment when conducting the numerical analysis of CBs using the general

method, particularly when web post buckling is expected.

This behaviour could be justified with regard to the specific geometry of the CB

and therefore, development of various mechanisms and failure modes with increase

of load. In particular, the web post buckling and Vierendeel mechanism are the

most probable ones to happen in this beam. Occurrence of web buckling is abrupt

whereas plastic deflections, causing the Vierendeel mechanism, are gradual and

begin to develop much before the plastic hinges in the Tee section cause failure.

A bilinear stress to strain conversion was assumed for the steel material. The web

buckling is more dependent on the elastic and plastic modulus while the Vierendeel

deformation is more related to the yield stress and strains. Figure 3.36 shows the

bilinear curve used for steel material. The step from point A leads to B or C,

depending on the load increment. Whilst there is not much difference in stress

between points B and C (web buckling), the development of plastic strains at point

B and C (Vierendeel deformations) are notably different.


102

Figure 3.36- Typical bilinear curve for steel material and importance of load increment

In other words, it seems that in Beam 2, between two increments that buckling was

expected, occurrence of plastic deformations and stresses dissipated by the plastic

deformations, while solving the simultaneous equilibrium equations, has reduced

the share of the web post stress in a way that not enough stress was applied to the

web to initiate buckling. Figure 3.37 shows a schematic model which helps to

present the justification behind the belated buckling in Beam 2. In this figure, the

stiffness (K) and buckling capacity (Pcr) of the web decrease as the load increases.

For example, assume a slender rod fixed with sponges at both ends and the entire

system being under compression. A compressive force is applied to the rod through

the two sponges. The possibility of buckling of the rod decreases if the sponge is so

soft so that the rod can plunge into it and cause deformation in the sponges by

dissipating strain energy, and releasing some of the compression load applied to the

rod.

This could be the reason why there was a remarkable difference between the results

of two similar beams. The equilibrium equations, in Beam 2 were satisfied by

excessive plastic deformations as reflected in Figure 3.37 and web buckling was

notably delayed.


103

Figure 3.37- A conceptual model to look at web post buckling

3.5 MODELLING LEEDS BEAM 3

3.5.1 Introduction and modelling

Figure 3.38 shows half of the long span Leeds Beam 3, which was subject to four

point loads and modelled in ABAQUS as the last validation against CBs tested at

ambient temperature. The beam height and span were 564 mm and 12250 mm

respectively and the parent section of this symmetric beam was UB564×140×39.

The opening diameter and spacing were 350 mm and 455 mm respectively. Further

details can be found in the test report (Surtees and Li, 1995). The material

properties were taken from the test report, which includes actual recorded

dimensions. These data were used in the modelling.


104

Figure 3.38- Half of Leeds Beam 3 modelled in ABAQUS benefiting from symmetry

3.5.2 Numerical versus test results

According to the test report, the test was preceded by an initial loading increment

of about 5% of the predicted maximum load. This was conducted to absorb

bedding down movements of the sample, and to ensure the efficiency of the lateral

supports. Nevertheless, the load-deflection curve resulted from this test presented

in Figure 3.39, shows the 5% pre-loading has not been efficient in this case and

some settlements were observed while loading in the elastic part of the deflection

curve. Applying a cyclic load could have been a better option to avoid bedding

down movements that was not the case in this test.

Comparing the load-deflection curves as obtained in the experiment and as

obtained from the analysis run in ABAQUS (Figure 3.39) suggests that if the test

curve is modified and shifted to the left, in a way that the extension of the elastic

behaviour (dotted line in Figure 3.39) intersects the origin of axes then the curves

conform reasonably.


105

0

50

100

150

200

250

0 20 40 60 80 100 120 140Def. (mm)

Load

(KN

)

Experiment

ABAQUS

Figure 3.39- Experimental versus modelling results for Leeds Beam 3

The beam failed due to web post buckling in the test and this was correctly

predicted by the developed model as shown in Figure 3.40. Web post buckling is

generally expected in high shear regions (near supports) as discussed in Chapter 2.

This hypothesis was well certified in this long span beam where moment and shear

effects are more distinguishable.


106

Figure 3.40- Web Buckling of the web post near end support in the long span Leeds Beam 3

3.5.3 Internal stresses of web post while buckling

Distribution of internal stresses while buckling was investigated by looking at

principal stresses. Figure 3.41 shows the maximum principal stress in which red

regions of the contour palette present the regions with the maximum tension.

Tensile stresses developed along a diagonal of the web post and were higher in the

two web posts where buckling had taken place, i.e. between openings 1 and 2, and

2 and3.


107

Figure 3.41-Maximum principal stress which represent the tensile stress while buckling

Similarly, looking at minimum principal stresses gave a good indication of the

distribution of compressive stresses (Figure 3.42) which developed in the other

diagonal, but not as clearly as tension. Again, the compression was higher in the

buckled web posts rather than in those further away from the beam support.


108

Figure 3.42-Minimum principal stress which represents the developed compression stress while buckling

3.6 SUMMARY OF THE MODELLING OF CELLULAR BEAMS

AT AMBIENT TEMPERATURE

This chapter presented the numerical approach implemented within ANSYS and

ABAQUS packages to model some of the tests on CBs at ambient temperature.

There was generally a good agreement between the numerical models and the test

results in terms of the failure load and the failure mode.

Within the numerical validations, it was found that the failure mode (and failure

load) of CBs can be very sensitive to boundary conditions. In particular, a slight

change in the way the lateral supports were applied to the top flange, changed the

failure mode from Vierendeel mechanism to web post buckling and reduced the

failure load by almost 40%.

Moreover, it was noted that the failure of CBs by web post buckling can easily be

missed, by mistake, in the numerical models. Vierendeel mechanism took over the


109

failure in the case presented in this chapter where the load increments were not

small enough to allow the model to identify the web buckling mode of failure.


110

4 CELLULAR COMPOSITE BEAMS AT AMBIENT

TEMPERATURE

4.1 INTRODUCTION

Nowadays, cellular beams are widely used in conjunction with concrete slabs

where the resulted composite section benefits from the concrete’s compressive and

steel’s tensile strength. The increasing popularity of cellular composite beams has

led to extensive experimental and numerical investigations into various aspects of

their behaviour. The experimental data of these tests allows researchers to validate

their numerical approaches and employ them to advance their investigation.

Meanwhile the reliability of the simple design methods proposed for these beams

can also be assessed and improved where necessary.

However, numerical modelling of cellular composite beams is rather complex due

to modelling of the rather complicated concrete material, steel-concrete interaction,

occurrence of various failure mechanisms, and convergence problems.

This chapter presents the details of the numerical modelling approach used to

validate the finite element models against the test data for three cellular composite

beams at ambient temperature. In addition, this chapter presents the numerical

results, comparative to the test results, and discusses the results where necessary.

In particular, Section 4.2 of this chapter presents the generics of the approach used

to model concrete material, steel decking, reinforcing mesh of slab, interaction

between the slab and top flange, the issues raised in modelling, and the way they

were tackled. Employing a systematic approach was vital in order to mimic the

experimental results, obtain the required software, validation and fully understand

the structural behaviour.


111

The other three sections of this chapter present the details and results of the three

models developed for cellular composite beams based on the two tests conducted

on cellular composite beams at University of Ulster, in 2005 (Nadjai, 2005), and

one at University of Kaiserslautern in 2002 (RWTH, 2002).

4.2 MODELLING COMPOSITE SLABS USING ABAQUS

4.2.1 General

The numerical modelling was carried out using ABAQUS and validated against the

following experiments.

1) Ulster Beam A1 (Nadjai, 2005)

2) Ulster Beam B1 (Nadjai, 2005)

3) University of Kaiserslautern, RWTH Beam 3 (RWTH, 2002)

The nonlinear analysis on cellular composite beams was carried out on the

following bases:

• The general solver option, which uses an iterative approach based on the

Newton-Raphson method, was used in most of the cases. However, the

Riks method was also used in some cases to avoid divergence problems and

to investigate the post-buckling behaviour.

• Stress-strain relationship for steel and concrete in compression were taken

according to results of one-dimensional tensile or crushing tests and

BSEN1992-1-1 (BSI, 2004a) recommendations.

• Geometrical and material nonlinearities were included in the models

developed for composite slabs. Software manuals (ANSYS, 2003)

generally recommend to exclude the large deformation (P-∆) effects in

models which include concrete material to minimise the convergence

problems associated with modelling concrete. However, this

recommendation was not satisfied in these modelling as geometrical


112

nonlinearities have a vital role in progressive instabilities of web posts

leading to buckling.

4.2.2 Modelling concrete material

This section presents the approach used to introduce the granular concrete material

and associated modelling issues comprising, behaviour in compression, tension,

cracking, crushing and post cracking.

4.2.2.1 Concrete in compression

The uniaxial stress-strain relationship for the concrete, with siliceous aggregates,

was taken from BSEN1994-1-2 (BSI, 2005c) which introduces the following

formula (Equation 4-1) for concrete in compression up to the crushing point.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=3

,1,1

,

2

3

θθ

θ

εεε

εσ

cc

cf

Equation 4-1

This equation is general and was also used at elevated temperature with reduction

factors presented in BSEN1994-1-2 (BSI, 2005c). Figure 4.1 presents the generic

stress-strain relationship for concrete material in compression for concrete grade

C30/35 with cylinder strength of 28 N/mm2. BSEN1992-1-2 (BSI, 2004a) suggests

the ultimate compressive strain of 0.02, and the strain at the peak of compressive

stress to be 0.0025 for room temperature. ABAQUS manual (Hibbitt et al., 2004a)

clarifies that concrete material loses all of its shear strength after crushing in

compression.

4.2.2.2 Concrete in tension

The uniaxial behaviour of concrete is assumed to be linear up to the yield point.

Various stress-strains curves are proposed for the post-cracking behaviour of

concrete (tension softening). The lower bound is to consider a brittle performance

for concrete after cracking where the material loses all its stiffness in tension after


113

cracking (Figure 4.2). The upper bound is the ductile behaviour where no reduction

occurs in tensile strength after cracking and the strength never reaches zero (Figure

4.2). Other tension softening models have also been suggested such as the Hoerdjik

nonlinear tension softening model (Figure 4.2) for post cracking behaviour

(Hordijk D.A., Cornelissen H.A.W. et al. 1986).

0

5

10

15

20

25

30

0.00 0.01 0.02 0.03Total strain [%]

Stre

ss [N

/mm

2 ]

Figure 4.1 Typical Stress-strain relationship for concrete in compression to BSEN1994-1-2

Figure 4.2- Tension softening curves assumed for concrete

In this research, the post cracking behaviour of concrete was generally assumed to

be a linear reduction to zero (Figure 4.3), unless specifically mentioned. Sensitivity


114

of the behaviour of composite beam slabs to the tension softening, is investigated

in Section 5.4. These sensitivity studies were used to also justify some

manipulations of the post cracking behaviour, which were done to help resolve the

numerical problems associated with concrete cracking.

Figure 4.3- Concrete in tension and fracture energy

The mean value of concrete tensile strength was calculated based on BSEN1992-1-

2, according to Equation 4-2, for concretes with grade less than C50/60.

)3/2(3.0 ckcm ff ×= Equation 4-2

In the absence of experimental data, the fracture energy (Gf) concept (Telford,

1993) was used together with the crack band width theory, presented by Bazant

(Bazant, 1983), to determine the ultimate tensile strain of concrete. Fracture

energy, which is the energy required to propagate a tensile crack on unit area, was

calculated from Equation 4-3, in which the maximum aggregate size defines the

base value of fracture energy (G Fo) (Telford, 1993).

7.0)/( cmocmFof ffGG = Equation 4-3


115

Based on the crack band width theory for fracture of concrete, the ultimate tensile

strain can be worked out according to Equation 4-4.

cut

fu

hG×

=σ

ε2

Equation 4-4

Where hc is crack band width and is dependent on the element size, in particular

the distance between integration points. As mentioned, S4 shell elements were used

to model the concrete slab, which has four integration points. The square root of

the distance between the integration points was taken as the crack band width,

according to the ABAQUS manual recommendation which was based on Bazant’s

theory (Bazant, 1983).

As mentioned the tension softening model was used in this research to consider the

post cracking behaviour of concrete. However, the post cracking stress-strain

relationship of the reinforced concrete can also be represented by another

approach, so-called “tension stiffening” model. It depends on the effects of the

reinforcement/concrete interaction such as dowel action and bond slip. The way the

tension stiffening is defined can be important knowing the fact that using a strain

softening approach can cause an unreasonable sensitivity (of the numerical results

and solution convergence) to the concrete mesh size in the areas with no or low

reinforcement. Crisfield (Crisfield, 1986) addressed this issue and recommended

the Hillerborg's approach (Hillerborg, 1976) as a solution. In Hillerborg's approach,

the energy required to open a unit area of crack is specified as a material parameter

using brittle fracture concepts. Using Hillerborg's approach enables us to address

the brittle behaviour of concrete by a stress-displacement transform, instead of a

stress-strain one.


116

4.2.2.3 Modelling cracking and crushing of concrete in ABAQUS

There are two common approaches available in ABAQUS to simulate the cracking,

post cracking and crushing of concrete which are the smeared crack and damaged

plasticity models (ACI, 1998).

4.2.2.3.1 Concrete smeared cracking model

The concrete smeared cracking method uses oriented damaged elasticity concepts

for the reversible part of the material response after cracking. Cracking is assumed

to occur when the stress reaches a failure surface (crack detection surface) which is

a linear relationship between the equivalent pressure stress (p) and the Mises

equivalent deviatoric stress (q) as shown in Figure 4.4.

Figure 4.4- Crack detection surface in concrete model

In the smeared cracking method, cracks are irrecoverable and limited to three at

any integration point. The orientation of the crack is stored for subsequent

calculations and next cracks are limited to be orthogonal to this direction. It is

because the stress components associated with the open crack are not incorporated

in defining the failure surface used for picking the additional cracks. In this

method, the crack is assumed smeared which means it does not track macro cracks

individually and the effect of the cracking in each integration point enters into the

calculations by modifying the associated stress and material stiffness.

"Crack detection" surface

"Compression" surface

σ c u q

σ c u p

1

1 4 32

2


117

Regarding the plasticity model incorporated to simulate the plasticity of concrete,

the options are generally limited and plasticity models which are based on plane

stress or strain could not be used where stresses and strains are considerable in all

three directions. Simulating the behaviour of the concrete in modelling is based on

the yield surface of the Drucker-Prager (Doran and Koksal, 1998) plasticity. It

represents a 3-dimensional conical failure surface in the principal stresses space.

This failure surface is schematically represented in Figure 4.5 in two dimensions in

which the tensile crack is detected based on the maximum principal tensile stress.

However, the failure surface in compression is dependent on the relative values of

the principal stress.

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.2

0.2

σf c

1

σf c

2

Uniaxial compression

Biaxial compression

Biaxial tension

Uniaxial tensionTension failure

Drucker - Pragerfailure surface

Figure 4.5- Yield and failure surfaces defined for biaxial stress in concrete

Some ratios had to be introduced into ABAQUS models to define the failure

surface. These values were generally defined based on the ones recommended by

ABAQUS user’s manual (unless mentioned) which are as follows:

• For the ratio of the ultimate biaxial compressive stress to the ultimate

uniaxial stress, the recommended value was 1.16.


118

• The absolute value of the ratio of the uniaxial tensile stress at failure to the

ultimate compressive stress was based on the formulation presented in

BSEN1992-1-1 (BSI, 2004a).

• The ratio of the magnitude of a principal component of plastic strain at

ultimate stress in biaxial compression to the plastic strain at ultimate stress

in uniaxial compression was assumed to be 1.28.

4.2.2.3.2 Concrete damaged plasticity model

The damaged plasticity model generally uses the concept of isotropic damaged

elasticity in combination with isotropic tensile and compressive plasticity to

express the inelastic behaviour of concrete. Unlike the smeared crack approach, it

assumes that the material properties after cracking degrade uniformly in all the

directions, regardless of the crack direction and pattern. More details about this

approach could be found in the ABAQUS materials manual (Hibbitt et al., 2004a).

In this research the smeared crack model (which is numerically more stable) was

the default approach for modelling the concrete material. The shear strength of

concrete after the crack is defined as the “shear retention” factor and relates the

shear stiffness of an open crack linearly to the magnitude of the crack opening. The

default value of 1, which means no reduction in shear resistance of concrete after

cracking, was used in modelling. This factor, and also the tension softening

behaviour of concrete, did not affect the results of modelling cellular composite

beams at ambient temperature (refer to Section 5.3) as most of the tension, if not

all, is resisted by the steel beam and concrete does not participate notably.

However, considering no decrease in shear stiffness after cracking was helpful in

avoiding convergence problems.

The following sections present the modelling details and results for the two cellular

composite beams tested by the University of Ulster (Nadjai, 2005) at ambient

temperature and one which was carried out by the Kaiserslautern University

(RWTH, 2002).


119

4.3 MODELLING ULSTER BEAM A1

4.3.1 Introduction and objectives

A total of six full-scale cellular composite beams were tested on behalf of Westok

(Westok, 1985) at University of Ulster in 2006, two at ambient and four at elevated

temperature. Two ambient tests were used to assess the analytical models for

cellular beams, Cellbeam V.5 (Simms, 2005) software and its accuracy in

predicting the failure mode and load, and finally calibrate numerical models for

further studies. The ambient tests also aimed to prepare the basis for the four fire

tests, on similar geometries, to investigate the critical temperature (the minimum

temperature which causes the failure) of web post and finally establish fire

protection recommendations. These destructive tests were carried out because of

the lack of data to validate models and design methods.

4.3.2 Details of the modelling approach

This section describes the details of the modelling of symmetric Beam A1 which

was subjected to two point loads. In addition, this section presents the numerical

results and compares them against the experimental data. Figure 4.6 shows the

geometry of Beam A1 spanning 4500 mm in which the opening diameter was

375 mm at 500 mm centres with a minimum web post width of 125 mm. The cell

diameter to pitch ratio was 1.33 having a slenderness ratio of d/t = 83.9 with a Tee

which satisfied the geometric limits recommended by SCI publication P355

(Lawson and Hicks, 2009) for cellular beams. Details about this test could be found

in the test report (Nadjai, 2005).


120

150 mm thick slab

4500mm

Detailing of Symmetric Composite Celluar Beam (A1, A2 and A3)

375 mm dia. cells at 500 mm centres

UB 406X140X39 Top & Btm Tee

3

Figure 4.6- Details of the symmetric beam A1

4.3.2.1 Steel beam

Top and bottom Tee sections of Beam A1 were produced from a section size

UB406×140×39, steel grade S355 with the finished depth of 575 mm. Table 4.1

shows the tensile test results carried out by Corus which were compared against the

results of the tensile tests conducted by the University of Ulster (Nadjai, 2005).

There was a good correlation between the two material tests and these values were

averaged and introduced to the ABAQUS model in accordance with the bilinear

stress-strain relationship proposed by BSEN1993-1-1 (BSI, 2005a) for structural

steel.

Table 4.1- Tensile test results for Beam A1


121

4.3.2.2 Concrete slab

The concrete slab was nominally 150 mm thick and 1200 mm wide and constructed

with normal weight concrete. The characteristic compressive strength of the

concrete was tested by three concrete cube samples, at the time of testing, which

gave an average strength of 35 N/mm2. The cylindrical strength is a better

representative of the actual strength as it excludes the stiffening effect due the

corners of the cube. The cylinder strength of 28.6 N/mm2 was considered in the

numerical model.

A composite shell element, incorporating the smeared crack approach for concrete,

was used to model the concrete slab. Using a composite shell to model the slab

enabled both the steel deck, as a bottom layer, and reinforcing mesh, as a layer

within the concrete layers, to be considered in this composite shell element.

Sections 4.3.2.3 and 4.3.2.4 discuss the modelling of the mesh and steel deck in

more detail. The fracture energy concept, as explained earlier, was used for post

cracking and ultimate tensile strain of concrete.

4.3.2.3 Steel reinforcing mesh

The slab reinforcement consisted of typical welded wire mesh reinforcement A142

(7 mm bars at 200 mm centres) with yield strength of 500 N/mm2. In order to

incorporate the reinforcing mesh into the shell element to model the slab, the

approach was to introduce the mesh as a mid-layer with an equivalent thickness

based on the diameter and spacing of the bars. Therefore, the composite shell for

the slab constituted of four layers, from the top concrete, reinforcing mesh,

concrete and finally the steel decking.

4.3.2.4 Steel deck

The steel decking used in the test was a Holorib (HR 51/150) with a thickness of

1.25 mm and yield stress of Fy=327 N/mm2 measured from a tensile test (Nadjai,

2005). Using a composite (layered) shell to model the slab gave the ability to

integrate the effect of the strength of the steel deck within the numerical model.


122

The deficiency of using shell elements (instead of solid elements) to model the slab

was that it did not allow the exact geometry of the deck to be modelled. However,

BSEN1994-1-2 (BSI, 2005c) states that the effective structural depth of a slab with

steel decking should be taken as the net concrete depth, when deck troughs are

oriented perpendicular to the beam direction.

4.3.2.5 Shear studs

Full interaction between the slab and beam was ensured by the use of a high

density of shear connectors (19 mm diameter studs at height 120 mm), as shown in

Figure 4.7. Shear studs were equally distributed in one row with a spacing of

150 mm over the beam length.

Figure 4.7- Providing high density of shear connectors to ensure 100% steel-concrete interaction (Source: the test report (Nadjai, 2005))

Therefore, full interaction was considered between the top flange and concrete slab

in the modelling. This assumption was well justified by the fact that no stud failure


123

occurred prior to overall failure of the beam due to web post buckling, according to

the test report (Nadjai, 2005).

4.3.2.6 Testing procedure

Load cycles at a load level of 20% and 60% of the pre-design load predicted by

Westok software were applied to the beam to avoid the slippage of the load cells

and bedding down of the supports. Each load step with a value of 10 (kN/step) was

kept for 3 minute intervals and a “Restraint frame” blocked the lateral deflection of

the test specimen as shown in Figure 4.8.

Figure 4.8- Test set up for Beam A1(Source: the test report (Nadjai, 2005))

4.3.3 Numerical results for the Beam A1

As mentioned, two numerical models were generally developed and run for each

beam. The first one was to do a linear buckling analysis to obtain the buckling

modes to allow an imperfection to be applied to the initial geometry by introducing

a reasonable buckling amplitude. This was then followed by the second model to

do the main nonlinear analysis.


124

Figures 4.9 and 4.10 present the first two buckling modes for Beam A1 from the

elastic buckling analysis. The two Eigenvalues were relatively close (1.10 and 1.19

in Figures 4.9 and 4.10 respectively). Since, the minimum Eigenvalue belonged to

the buckling mode witnessed in the test, the first mode was introduced to the main

model as an initial imperfection with the maximum imperfection amplitude of

1 mm. In some cases, it happened that the minimum Eigenvalue was for a buckling

mode which did not actually take place in the test. This is because the Eigenvalue

analysis simply ignores all the associated nonlinearities while they have notable

effects on the numerical results. In these cases, the buckling mode employed to

introduce the initial web post imperfection, was the mode similar to the buckling

shape observed in the test provided that its Eigenvalue was not notably higher than

the mode with the minimum Eigenvalue. If the difference in Eigenvalues was

notable, then two models were developed, based on the each imperfection mode, to

ensure they both end up with similar result as witnessed in the test and certify that

the approach has been reasonable.

Figure 4.9- First buckling mode for Beam A1 from linear Eigenvalue analysis


125

Figure 4.10- Second buckling mode for Beam A1 from linear Eigenvalue analysis

Figure 4.11 compares the load-deflection curves for numerical and test results for

Beam A1. The numerical model developed for this beam reached its ultimate

capacity at 415 kN, within 11% percent of the test result where failure occurred at

370 kN. Moreover, the numerical model developed for Beam A1 also predicted the

correct failure mode of web post buckling certifying the validity of the numerical

approach used to model this beam. Figure 4.12 shows the occurrence of the S-

shaped web post bucking in Beam A1 and Figure 4.13 shows the correct prediction

of the governing failure mechanism by the developed FE model. More pictures of

the tests carried out at the University of Ulster are available in Appendix A of this

thesis.


126

0

100

200

300

400

500

600

0 50 100 150 200Deflection (mm)

Load

(kN

)

Experiment

ABAQUS

WPB at 370 kN

WPB at 415 kN

Figure 4.11- Load-deflection comparison for Beam A1, FE model versus test

Figure 4.12- S-Shaped web post buckling of Beam A1 (Source: the test report (Nadjai, 2005))


127

Figure 4.13- The web post bucking of the Beam A1

The contour plot in Figure 4.14 shows the Von Mises stress distribution of this

beam while buckling. Investigating the Von Mises stress at critical bottom Tees

shows that the Vierendeel mechanism was not imminent.

Figure 4.14- Von Mises stress contour plot to check Vierendeel mechanism


128

4.3.4 Investigating stress distribution at failure

Figure 4.15 is the plot of maximum principal stresses (which represents tension)

while buckling, which shows tensile stresses have developed diagonally across the

openings. According to this figure, the maximum principal stress has been zero in

the opposite diagonal which means all principal stresses have been negative i.e.

compressive.

Figure 4.15- Max principal stress representing tensile stresses

The graph presented in Figure 4.16 is the maximum principal stress for the three

nodes of the two tensile diagonals addressed in Figure 4.15 (Nodes A to F).

Interestingly, the maximum tensile stress was higher in top web nodes A and D

than the bottom nodes C and F and this shows how dominant the secondary

moments (due to Vierencdeel action) can be compared to primary moments (due to

global bending). Expectedly, the ends of these diagonals experienced notably

higher tensile stresses than mid nodes B and E. In all six nodes, tension increased

smoothly until web buckling. At this stage, a sudden change was observed in these

curves associated with a new pattern of stresses distribution.


129

0

100

200

300

400

500

0 100 200 300 400 500Load (kN)

Max

Prin

cipa

l Stre

ssNode ANode BNode CNode DNode ENode F

Figure 4.16- Maximum principal stress in tensile diagonals

This developed tension helps stabilising the other diagonal (strut) which is under

compression (Figure 4.17) and acts to delay the web buckling. However, the

buckling resistance of the web post in the simple design methods is currently

formulated just based on the compression force (C) applied to the strut ends and

any effects of tension force (T) are conservatively ignored

Figure 4.17- Compressive force in the strut and tension in the opposite diagonal


130

Similarly, Figure 4.18 shows the distribution of minimum principal stresses, to

present the compressive stresses which are again developed in a diagonal pattern.

Six nodes are addressed in the strut in compression to look into their stresses

during and after buckling.

Figure 4.18- Min principal stress representing the compressive stress

According to Figure 4.19, which presents the minimum principal stresses, two ends

of the strut have experienced similar compressive stresses in the early stages of

loading, being notably higher than compression at the mid nodes B and E.

However, initiation of buckling was associated with a sudden reduction of

compressive stress at the bottom end of the strut (nodes H and J) and increase at

top end (nodes G and I) certifying a notable stress redistribution while buckling.


131

-500

-400

-300

-200

-100

0

100

0 100 200 300 400 500

Node H

Node B

Node G

Node J

Node E

Node I

Figure 4.19- Min Principal stresses in compressive strut for Beam A1

Figure 4.20 shows the stress distribution after the buckling which shows that the

diagonal stretch of compression and tension stresses has changed to high

compressive and tensile stresses at the top and bottom half of the web post

accordingly.


132

Figure 4.20- Stress distribution after web buckling

4.4 MODELLING ULSTER BEAM B1

4.4.1 Modelling

The same studies were conducted on the highly asymmetric Beam B1 spanning

4500 mm, in which the top Tee section was based on UB 406×140×39, the bottom

Tee was based on UB 457×152×52 and the beam had a finished depth of 630 mm

(Figure 4.21). This beam was subjected to one point load. The diameter of the

cellular openings was 450 mm, at 630 mm centres, with a minimum web post

width of 180 mm. Therefore, the cell diameter to pitch ratio was 1.4 (the minimum

permitted), and the ratio of the top to bottom Tee area was 1 to 1.43. Material

properties and other test details for this beam were identical to Beam A1 and more

details about this test could be found in the test report (Nadjai, 2005).


133

450 mm dia. cell at 630 mm centres

4500mm

150 mm thick slab

406x140UB39 Top & 457x152UB52 Btm Tee

Detailing of Asymmetric Composite Celluar Beam (B1, B2 and B3)

Figure 4.21- Details of the asymmetric beam B1

4.4.2 Results of modelling for Beam B1

The preliminary buckling analysis provided the two modes presented in Figures

4.22 and 4.23 as the ones with least Eigenvalues. In Beam B1, unlike Beam A1,

both modes induce the imperfection shape which was consistent with the buckling

mode observed in the test, and therefore could be used. The second mode was

employed to introduce an imperfection with a maximum amplitude of 1 mm.


134

Figure 4.22- First buckling mode for Beam B1 based on linear Eigenvalue analysis

Figure 4.23- Second buckling mode for Beam B1 based on linear Eigenvalue analysis

The nonlinear analysis was then followed after the Eigenvalue analysis. Figure

4.24 shows the occurrence of the web post buckling in the test which was correctly

predicted by the developed finite element model, as presented in Figure 4.25.


135

Figure 4.24- S-shaped web buckling as the governing failure mode of Beam B1(Source: the test report (Nadjai, 2005))

Figure 4.25- Occurrence of the S-shaped web buckling of Beam B1 in the ABQAUS model

Figure 4.26 reflects the out-of-web plane displacements for nodes A and B which

shows a sudden increase in the lateral displacements of the web post due to


136

buckling. This figure also shows that the S-shaped buckling mode has gradually

and constantly been forming from the first stages of loading.

-40

-30

-20

-10

0

10

20

30

40

50

60

0 100 200 300 400 500

Load (kN)

Out

of w

eb p

lane

Dis

. (m

m)

Node A

Node B

Figure 4.26- Out-of-web plane displacement of the buckled web post

Figure 4.27 compares the load-deflection curves for the experiment versus the

modelling results which shows the reasonable difference of 10% in predicting the

failure load.


137

0

100

200

300

400

500

600

0 50 100 150 200 250 300 350 400 450 500Def. (mm)

Load

(kN

)

Experiment

Numerical Model

Figure 4.27- Numerical and experimental results for Beam B1

Validation of the numerical models developed for Beam A1 and Beam B1 was

vital in the sense that they could be reasonably used for further parametric studies

not only to investigate, and compare, the behaviour of symmetric and asymmetric

beams, but also to compare their performance at ambient against elevated

temperature.

Figure 4.28 presents the contour plot for the minimum principal stress while

buckling which shows that development of compression at the two ends and along

the length of strut in asymmetric Beam B1 was not as obvious as that of Beam A1.

It also shows that bottom end of the strut experienced higher compressions than the

top end, while buckling.

Appendix B of this thesis presents the input file to model this beam using the

ABAQUS package.


138

Figure 4.28-Plot of minimum principal stresses in Beam B1 while buckling

4.5 MODELLING RWTH BEAM 3

4.5.1 Details of the test

A total of 4 composite beams were tested by the Institute of Steel Construction

(RWTH) in the University of Kaiserslautern (2002). These perforated composite

beams were particularly constructed to look into the composite action in the end

cell, behaviour of elongated openings, design of highly asymmetric sections and

the behaviour of stiffened web openings (RWTH, 2002).

As the main focus of this research was neither elongated openings nor stiffened

web openings, only Beam 3 of the four RWTH’s tested beams was modelled as the

last validation of composite beam slabs at ambient temperature. This beam was a

highly asymmetrical steel section and failed due to web post buckling. Figure 4.29

shows the general scope of this asymmetric beam in which the area ratio of the

bottom to top flange was 4, greater than the limit allowed in the BSEN1994-1-1,

which is 3. Moreover, the concrete slab was not taken to the support at one end, to

check the composite action in the end panel, and therefore, the full length of the


139

beam was modelled because of unsymmetrical geometry. More details about the

test is available in the RWTH test report (RWTH, 2002).

1800

484.

6 614.

6

130

Based on IPE300

Based on HEB340

Figure 4.29- Geometric details of RWTH Beam 3

4.5.2 Numerical model

This section covers the details of the approach used to model RWTH Beam 3 in

ABAQUS.

4.5.2.1 Steel beam

The actual section dimensions and thicknesses of the steel beam were measured in

the test and these values were considered in the numerical model, instead of

nominal values.


140

The upper and lower Tees were produced based on IPE300 and HEB340 steel

sections correspondingly, both with steel grade S355. Tensile tests were carried out

on the web and flange of the beam (two samples of the flange and one of the web)

and ultimate strains, ultimate strength and yield strength of the section were

derived from the test report to define the bilinear stress-strain curves. Table 4.2

shows the values used in modelling which were the averages of the data achieved

from the tensile test. The only value missing in the report was the yield strength of

the web of HEB340, highlighted with a “?” mark in the table. This value was

estimated based on the other available values measured at other sections which

suggested Fu/Fy~1.20 to estimate the missing value. The modulus of elasticity (E)

was not available in the report. It was initially assumed 210 kN/mm2 and then

reduced to 195 kN/mm2 for a better fit with the initial elastic part of load-deflection

curve.

Table 4.2- Average values gained from the tensile test and used in modelling

4.5.2.2 Steel decking

The overall thickness of the slab was 130 mm and Holorib sheets HR 51/150

(Figure 4.30) with a thickness of 1.25 mm were used as steel decking. The

measured yield stress was fy= 327 N/mm2 and deck troughs were oriented

perpendicular to the beam direction.


141

Figure 4.30-- Steel sheeting used in the RWTH Beam 3

The slab and Holorib heights were 130 mm and 51 mm respectively. Therefore, the

thickness of the concrete slab in modelling was reduced to an equivalent thickness

of 79 mm plus 1.25 mm, similar to the Ulster beams, to consider the contribution

of the steel decking. It should be noted that by default, the thickness of the shell

element in ABAQUS extends equally from both sides of the mid-shell plane.

Therefore, the position of the composite shell, relative to the top flange, was

adjusted to end up with an overall composite beam depth equal to that of the actual

beam (Figure 4.31).

Figure 4.31- The approach used for the slab thickness and steel decking


142

4.5.2.3 Reinforcement

The minimum isotropic mesh reinforcement of 0.4% in longitudinal and transverse

direction was provided in the slab which was located at 20 mm from top of the

concrete face. Reinforcing mesh bars were again considered as a layer in the

composite shell used to model the concrete slab.

4.5.2.4 Concrete

The concrete grade was C25/30 with the concrete cubic strength of 30.2 N/mm2.

The characteristic cylindrical strength of 25 N/mm2 was introduced into the

numerical model. The ultimate tensile strength of concrete was assumed 2.5

N/mm2 based on Equation 4-2. The post cracking behaviour of the concrete was

again based on the fracture energy with regard to maximum aggregate (16 mm) and

element size (distance between integration points) as explained in Section 4.2.2.2.

4.5.2.5 Shear studs

Full steel-concrete shear interaction was provided in the test by using sufficient

welded type of studs. Therefore, the degree of shear connection in modelling was

set to 100%, i.e. no slippage. There was no shear stud failure, prior to beam failure

by web post buckling, which justified assuming the full beam-slab interaction in

the model.

4.5.3 Numerical results against the test results

Figure 4.32 shows the full-length model developed for this beam. The web post

buckling between the openings 11 and 12 (where the slab was not taken) governed

the ultimate loading capacity in the developed model which conformed with the

test observations (see Figure 4.33).


143

Figure 4.32- RWTH Beam 3 modelled in ABAQUS

Figure 4.33-Web post buckling of the last web post gained in numerical and test results (Source: the test report (RWTH, 2002))

The load-deflection curves for the test and FE analysis are compared in Figure

4.34, which shows a relatively good conformity in all stages. This beam failed at


144

680 kN in the numerical model which was within 4% of the 658 kN observed in

the test.

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50Deflection (mm)

Load

(kN

)

Test

ABAQUS Model

Figure 4.34- Comparison of the model and test Load-Deflection curves for RWTH Beam 3

4.6 SUMMARY OF THE MODELLING OF CELLULAR

COMPOSITE BEAMS AT AMBIENT TEMPERATURE

The ABAQUS package was used to model the two composite beams tested at

University of Ulster and Beam 3 of RWTH tests at ambient temperature. There was

a good correlation between the numerical results and the test results in terms of the

failure mode and the failure load. This calibration allowed these models to be used

for some case studies, which are presented in Chapter 5.


145

5 PARAMETRIC STUDIES ON CELLULAR COMPOSITE

BEAMS AT AMBIENT TEMPERATURE

5.1 INTRODUCTION

This chapter presents the findings from the parametric studies conducted on

validated models for cellular composite beams at ambient temperature. In

particular, the symmetric Ulster Beam A1 and asymmetric Beam B1 were

employed to investigate the influence of web post imperfections, concrete tensile

behaviour, web stiffeners and finally loading type, in the performance of cellular

composite beams at ambient temperature.

5.2 EFFECT OF LOADING TYPE

Cellular beams are normally used as secondary beams in structures and mostly

experience distributed loads, rather than point loads. Therefore, it was important to

look into their performance under a distributed load (UDL). Examining Beam A1

and A2 under a UDL was not only useful in enlightening how different they

behave under a UDL as opposed to point loads, but also in providing the failure

load of these two beams which was necessary to apply the correct load ratio

(utilisation factor) for similar case studies in fire conditions.

5.2.1 Effect of loading type on symmetric beam

In this case study the loading type was changed from a point load to distributed

load and the two web stiffeners, located under the two point loads, were

eliminated. Figure 5.1 shows the occurrence of web buckling as the failure mode of

this beam. This was clearly more dominant in the end panel, which had the highest

combination of shear and moment.


146

Figure 5.1- Web post buckling of Beam A1 without web stiffeners, under a UDL at ambient temperature

However, similar studies conducted at elevated temperature, which are presented

later in Section 7.4, showed that eliminating the web stiffeners considerably

reduced the beam’s resistance against fire as the beam tends to fail notably sooner

than expected due to distorsional buckling (Bradford, 1992), instead of web

buckling. Re-adding the eliminated web stiffeners (returning to the original

geometry) proved very useful in avoiding the distorsional buckling at elevated

temperature.

Therefore, a UDL was also introduced to the original geometry (including web

stiffeners) to see the impact of web stiffeners at ambient temperature and to also

provide the failure load for future investigations in fire condition. Figure 5.2 shows

the failed shape of this beam, again due to web buckling and Figure 5.3 compares

the load-deflection curve between the beams with and without web stiffeners,

which underlines 14% difference between the failure loads. The beam without

stiffeners failed at a 126 kN/m while the beam with web stiffeners failed at


147

145 kN/m and this suggests that the existence of web stiffeners, in CBs under a

UDL, can notably improve their load resistance at ambient temperature.

Figure 5.2- Web post buckling of Beam A1 with web stiffeners, under a UDL at ambient temperature

0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50Displacement (mm)

Load

(kN

/m)

Without web stiffener

Witt web stiffener

Filure Load 145 kN/m

Filure Load 126 kN/m

Figure 5.3- Load-deflection curve of Beam A1 under a UDL with and without web

stiffener


148

5.2.2 Effect of loading type on asymmetric beam

Similar studies were conducted on applying a UDL this time on the asymmetric

Beam B1, with and without a web stiffener. This beam only had one stiffener,

located at the mid-span where the point load was applied.

Figure 5.4 shows the failed shape of the beam without a stiffener in which the web

post buckling is associated with some distorsional buckling at the bottom Tee.

Figure 5.5 presents the final deformed shape of the beam with a web stiffener in

which web post buckling dominated the failure. Finally, Figure 5.6 compares the

load-deflection curves for the cases without and with a web stiffener (original

geometry) in which the beam failed at 154 kN/m and 159 kN/m respectively.

Figure 5.4- Web post buckling of Beam B1 without web stiffeners, under a UDL and at ambient temperature


149

Figure 5.5- Web post buckling of Beam B1 with web stiffeners, under a UDL and at ambient temperature

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30

Displacement (mm)

Load

(kN

/m)

Without Web Stiffener

With Web Stiffener

159 kN/m154 kN/m

Figure 5.6- Load-deflection curve of Beam B1 under a UDL with and without web stiffener

In brief, a full-height web stiffener notably affected the load capacity of these

beams. The influence of existence or non-existence of web stiffeners, on the


150

loading capacity of Beam B1 was not as much as Beam A1 (affected by 3% and

14% respectively). This could be due to Beam B1 having 1 stiffener while Beam

A1 had 2 stiffeners. Excluding web stiffeners allows for the development of

relative displacements between the top and bottom Tees, in the out-of-web plane

direction. This case was more evident in the asymmetric beam in which stronger

bottom Tees resists higher vertical loads (P), transferred through web posts.

Therefore, the associated P-∆ effects between the top and bottom Tees become

more dominant. However, these displacements were not significant enough to

result in purely distorsional buckling at ambient temperature. Similar studies at

elevated temperature are presented later in Section 7.4 where distorsional buckling

completely dominated the failure.

5.3 EFFECT OF TENSILE BEHAVIOUR OF CONCRETE

Based on the fact that in cellular composite beams, most of tensile force, if not all,

is resisted by the steel beam, the tensile characteristics of concrete were not

expected to play a critical role in the overall behaviour of these beams at ambient

temperature.

The model was run twice to compare the influence of assuming brittle or ductile

behaviour for the post-cracking phase of concrete. Figure 5.7 compares the results,

which shows minimal difference between the two cases. Similar curves were

repeated for the case study on the shear retention factor of concrete (described in

Section 4.2.2.3.2) which shows the behaviour was not notably dependent on this

factor.


151

0

100

200

300

400

500

600

0 50 100 150 200Deflection (mm)

Load

(kN

)

Experiment

Brittle post cracking

Ductile post cracking

Figure 5.7- Comparing the load-deflection curves for Beam A1 assuming ductile and brittle behaviour for concrete after cracking in tension

5.4 EFFECT OF WEB IMPERFECTIONS

The impact of the web post imperfections on the performance of cellular beams at

ambient temperature was investigated by applying various imperfection amplitudes

to models developed for Beam A1 and Beam B1.

5.4.1 On symmetric Beam A1

Figure 5.8 compares the load-deflection curves resulting for various imperfection

amplitudes which shows that the load-deflection curve was slightly affected by the

magnitude of the imperfection value. This figure also highlights how important it is

to apply a reasonable web imperfection to the model. In particular, the model with

“Imp= 0.02 mm” has not still buckled in notably higher loads which shows how

this can cause notable differences between modelling and test results.


152

0

100

200

300

400

500

600

0 40 80 120 160 200Deflection (mm)

Load

(kN

)

Experiment

Imp=.02 mm

Imp=10 mm

Imp=1 mm

Figure 5.8-Effect of web post imperfection in the load-deflection curve for Beam A1

5.4.2 On asymmetric Beam B1

Similar studies were conducted on the asymmetric Beam B1 and the diagram in

Figure 5.9 presents these results for a wide range of imperfection amplitudes,

between 0.02 to 5 mm. This figure clearly shows that the load in which the beam

buckled has been notably affected by the imperfection amplitude. Comparing

Figure 5.9 with Figure 5.8 reveals that the ultimate loading capacity of the

asymmetric beam was more sensitive to web imperfections than the symmetric

one. This could be explained by the higher dependency of the ultimate capacity of

asymmetric (as opposed to symmetric) beams to the bottom Tee while the

contribution of the bottom Tee is through the web post. Therefore, an increase of

the web post imperfection affects the ultimate loading capacity of asymmetric

beams more than symmetric ones.


153

0

100

200

300

400

500

600

0 100 200 300 400 500Def. (mm)

Load

(kN

)

ExperimentImp=2 mmImp=0.2 mmImp=0.02 mmImp=5 mm

Figure 5.9- Effect of web post imperfection in the load-deflection curve for Beam B1

The importance of web imperfections at elevated temperature is investigated later

in Chapter 7, where comparisons are also made to see how important imperfections

are at elevated compared to ambient temperature.

5.5 SUMMARY OF THE CASE STUDIES ON THE CELLULAR

COMPOSITE BEAMS AT AMBIENT TEMPERATURE

This chapter presented the case studies on symmetric and asymmetric composite

beams at ambient temperature. The loading type was changed from point loads to a

UDL. This was associated with removing the web stiffeners, which resulted in the

asymmetric beam failing due to distorsional buckling. Re-adding the full-height

web stiffeners avoided the occurrence of distorsional buckling, by preventing the

relative displacements of the top to bottom Tee, and increased the loading capacity

by up to 14%.


154

Investigations also covered the role of web imperfections on the performance of

cellular composite beams at ambient temperature. Although the post failure

performance of the beam was almost unaffected by the imperfection amplitude, the

buckling (failure) load was found to be dependent to the imperfection amplitude.


155

6 CELLULAR COMPOSITE BEAMS AT ELEVATED

TEMPERATURE

6.1 INTRODUCTION

The great practicality and increasing use of cellular beams together with lack of

codified and robust design guides on the design of CBs at elevated temperature,

has underlined the necessity of further investigations in this regard. One of the

controversial issues has been the requirements for intumescent protection for these

beams. A rather unreasonable rule (ASFP et al., 1992) for beams with web

openings, asked for 20% extra coating thickness compared to solid-web beams,

which has been subjected to criticisms.

Researchers have also been trying to develop a simple but reliable design model for

these beams at elevated temperature. Meanwhile the experimental data have had a

critical and inevitable role in evaluation and validation of these approaches.

Only very limited experiments were carried out by 2006 on cellular composite

beams at elevated temperature. As explained in Section 2.4.3, the only

experimental data was from the two fire tests conducted by CTICM in 2002 and

one fire test by the University of Manchester. All three fire tests were carried out

on fire protected beams to investigate the efficiency and effect of fire protection

material rather than the performance of the beam itself.

In 2006, four unprotected cellular composite beams were tested at elevated

temperature at University of Ulster. In these tests, symmetric and asymmetric

cellular beams were tested under one and two point loads and two fire curves.

In brief, following is the list of fire tests which were carried out on cellular

composite beams by the time this research was carried out.


156

1. CTICM cellular composite beam P1 (Metz) (2003)

2. CTICM cellular composite beam P2 (Metz)

3. University of Manchester (2003)

4. Ulster Beam A2 (2006)

5. Ulster Beam B2

6. Ulster Beam A3

7. Ulster Beam B3

Among these fire tests, only the tests at University of Ulster were conducted on

unprotected steel sections. The results of these four fire tests were used to calibrate

the numerical models developed in this research for cellular composite beams at

elevated temperature. This chapter describes the details of these fire tests and

presents the details of the numerical modelling approach to model these four

beams. Finally, numerical results are presented and compared against the test

results to investigate the validity of the developed models for case studies at


6.2 ULSTER FIRE TESTS

Unlike the CTICM and Manchester fire tests, the purpose of the Ulster tests was

not to evaluate the performance of the intumescent coating. It was to provide data

on the web post failure temperature. Among the four fire tests on unprotected

symmetric and asymmetric composite beams, Beam A2 and Beam A3 were based

on the same geometry and loading as Beam A1. Similarly, Beam B2 and Beam B3

were based on Beam B1. Beam A2 and B2 were tested under a slow fire curve

shown in Figure 6.1 while Beam A3 and Beam B3 were tested under a standard

fire curve ISO-834 (Figure 6.1).


157

Figure 6.1- Standard and slow fire curve to BSEN1991

The slow fire curve was set up to produce lower peak temperatures but with longer

duration sufficient to permit considerable heat conduction. The applied loads for

the slow rate heating fire tests A2 and B2 were calculated based on 50% of failure

loads obtained from the ambient tests on Beam A1 and Beam B1. This ratio was

decreased to 30% for fire tests A3 and B3, which were tested using the standard

fire heating. In all tests, the beams were kept loaded to their respective applied load

for an hour before the furnace started functioning to eliminate any errors caused by

unwilling slippage or settlements.

These four Ulster tests provided valuable experimental data, which were used as

the bases of the validation for the numerical models developed in this research for

cellular composite beams at fire condition. Sections 6.3 to 6.6 of this chapter

present the details of modelling and results for each of these four fire tests.


158

6.3 NUMERICAL MODELLING AND RESULTS FOR ULSTER

BEAM A2

6.3.1 Modelling

The geometry of the Beam A2 was similar to that of Beam A1 (see Figure 6.2).

The fire tests were carried out in two steps. The mechanical load was applied in the

first step and the fire load in the second step and this procedure was also applied in

the numerical modelling approach.

The following sections describe the details of the modelling approach for each

section at elevated temperature.

150 mm thick slab

4500mm

Detailing of Symmetric Composite Celluar Beam (A1, A2 and A3)

375 mm dia. cells at 500 mm centres

UB 406X140X39 Top & Btm Tee

Figure 6.2- Geometry of Beam A2 (Also Beam A1 and Beam A3)

6.3.1.1 Modelling the steel beam and reinforcing mesh

The stress-strain relationship for structural steel and mesh reinforcement at

elevated temperature were based on the BSEN1994-1-2 (BSI, 2005c)

recommendations and Figure 6.3 shows the generic curves for steel grade S355

with the yield strength of 442 N/mm2.


159

0

50

100

150

200

250

300

350

400

450

500

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20Total strain [%]

Stre

ss [N

/mm

2 ]

20 C 100 C 200 C 300 C 400 C 500 C 600 C700 C 800 C 900 C 1000 C 1100 C 1200 C

Figure 6.3- Stress- strain transform of steel at elevated temperature

6.3.1.2 Modelling the concrete slab

The compressive part of stress-strain curve for concrete at elevated temperature

was again based on Equation 6-1 and applying the reduction factors defined in

BSEN1994-1-2 (BSI, 2005c). Figure 6.4 shows the generic stress-strains

conversion curves of concrete in compression (based on measured value on 28

N/mm2) for different temperatures.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=3

,1,1

,

2

3

θθ

θ

εεε

εσ

cc

cf

Equation 6-1


160

0

5

10

15

20

25

30

0.00 0.01 0.02 0.03 0.04 0.05Total strain [%]

Stre

ss [N

/mm2 ]

20 C 100 C 200 C 300 C 400 C 500 C600 C 700 C 800 C 900 C 1000 C 1100 C

Figure 6.4- Stress- strain transform of concrete in compression at elevated temperature

A similar approach to ambient temperature was generally used to consider the post

cracking of concrete at elevated temperature in which the tension softening was

assumed as a linear decrease of stress to zero after cracking (Figure 6.5).

Figure 6.5- Stress-Strain relationship of concrete in tension at elevated temperature


161

The Drucker-Prager yield surface was used to define the failure surface for

concrete material at elevated temperature, similar to ambient temperature, with

Figure 6.6 showing the yield surface for various temperatures.

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.2

0.2

20-200 C

400 C

600 C

σf c

1

σf c

2

800 C

Tension failure

Drucker - Pragerfailure surface

Figure 6.6- Drucker-Prager failure surfaces for different temperatures

6.3.1.3 Applying temperatures

Thermocouples positioned in the top, middle and bottom of the concrete slab

recorded the relevant temperatures (see Figure 6.7) and these were applied to the

model directly by defining temperature points through the thickness of the shell

elements representing the concrete slab.


162

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70 80 90

Time (mins)

Tem

pera

ture

in C

oncr

ete

( o C)

Mid-depth of the slab

Bottom face of slab

Top face of slab

Figure 6.7- Temperatures at bottom, middle and top of the concrete slab

In a fire test, the temperature distribution can be notably non-uniform across the

length and height of the steel beam. Changes of temperature through the height of

the steel beam are mainly a function of section thicknesses (section factors) and

fire exposure while the unavoidable non-uniform distribution of heat inside the

furnace, increases the temperature differences along the height of the section. This

implied having thermocouples all over the beam length and section with the most

focus in the critical regions where failure was expected.

Figure 6.8 shows the position of the thermocouples (1 mm sheathed Type K) on the

steel beam where a higher density of thermocouples was provided in the four

critical web posts expected to buckle. These recorded temperatures were averaged

and applied to the numerical model in limited areas.

Various approaches were used to calculate and introduce the temperatures to the

beam and the full length of the beam was modelled in some of these approaches

where the effect of temperature variations along the beam length was to be

considered in the model.


163

Figure 6.8- Position of the thermocouples on steel Beam A2 (Nadjai, 2007)

Zon

e 8

Zon

e 1

Zon

e 2

Zon

e 3

Zon

e 4

Zon

e 5

Zon

e 6

Zon

e 7

c d

e,f

g,h

ij

op

k r

s,t

lm

n

a,b 1

12

34

56

78

c

ab

gh

d

ef

stj m p

100

575 x

140 C

UB 3

9 k

g/m

Wes

tok

Fire

Tes

ting

The

rmoc

oupl

e Po

sition

sTes

t Bea

m 1

Sym

met

ic C

ompo

site

20

20

100

Zon

e 9

Zon

e 10

Zon

e 8

Zon

e 1

Zon

e 2

Zon

e 3

Zon

e 4

Zon

e 5

Zon

e 6

Zon

e 7

c d

e,f

g,h

ij

op

k r

s,t

lm

n

a,b 1

12

34

56

78

c

ab

gh

d

ef

stj m p

100

575 x

140 C

UB 3

9 k

g/m

Wes

tok

Fire

Tes

ting

The

rmoc

oupl

e Po

sition

sTes

t Bea

m 1

Sym

met

ic C

ompo

site

20

20

100

Zon

e 9

Zon

e 10


164

Some heat loss occurred through an observation window in the last minutes of this

test which affected the temperatures recorded by the thermocouples. Figure 6.9 is

an example of the temperatures recorded by thermocouples (zone 4 defined in

Figure 6.8) which clearly shows the thermocouples have been affected by this heat

loss. According to Figure 6.9, temperatures started to decrease at roughly 70

minutes of the test and then increased again. According to ABAQUS manual

(Hibbitt et al., 2004a), the smeared crack model which was used to model the

concrete, is only recommended for monotonic strains and is not very accurate in

unloading. To overcome this issue a second approach was also used in considering

the temperatures in the last minutes of the test. In this approach, temperatures were

linearly extrapolated for the last minutes as the gas temperature increased linearly

(10 Co/min, see Figure 6.1) and this was associated with a similar linear increase of

temperature in different sections of the beam before the heat loss (see Figure 6.9).

The numerical model was run for both cases with recorded and extrapolated

temperatures introduced into the model to give an upper and lower bound. The

numerical outputs are presented for these cases and compared against test results in

Figure 6.14.


165

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90

Time (mins)

Tem

pera

ture

(C)

4a

4b

4c

4d

4e

4f

Extrapolation

Figure 6.9- Temperatures recorded by thermocouples 4a to 4f in zone 4 of Beam A2

Applying a slow fire curve (compared to fire curve ISO 834) for this beam allowed

for more heat transfer and balance of temperature along the length and through the

section of the composite beam. This implied lower thermal gradient and a more

uniform temperature distribution and justified one of the two approaches used to

apply the recorded temperatures to numerical models. In particular, in this

approach (also used for Beam A2) a weighted average (based on the area) was

taken of the recorded temperatures for each minute of the test, at the bottom flange,

bottom web, top flange and top web, and applied over the whole length of the

beam. This approach is schematically shown in Figure 6.10 in which areas with

similar colour had identical temperatures at a time. Only half of the beam was

modelled in this approach and there was no need to model the full length.


166

Figure 6.10-Temperatures averaged and applied equally all over the beam length

The averaging was based on excluding the end panels (panel 1 and 9 in Figure 6.8)

because temperatures were notably lower in these two panels, due to the non-

uniform distribution of temperature in the furnace.

Figures 6.11 and 6.12 show the temperature profile along the beam length at 45

and 55 minutes into the test which illustrate that the temperature at the critical web

post was not more than 35oC different from the calculated average temperature

(which was uniformly applied to the model).


167

300

350

400

450

500

0 1 2 3 4 5 6 7 8 9Opening Number

Tem

p. (C

)

Average of Bottom Web

Average of Top Web

Point Load

Point Load

Critcal Web

Critical Web

Figure 6.11- Temperature profile over the length of Beam A2 at 45 minutes

350

400

450

500

550

0 1 2 3 4 5 6 7 8 9Opening number

Tem

p. (C

)


Average of Top Web

Point Load

Point Load

Critical Web

Critical Web

Figure 6.12- Temperature profile over the length of Beam A2 at 55 minutes

Figure 6.13 shows the consequent average temperature-time curves applied to

Beam A2.


168

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60 70 80 90Time (min)

Tem

p (c

)

Top Flange

Bot Flange

Web

Concrete

Extrapolation for last minutes

Figure 6.13-Time- temperature curves resulted from averaging in all over the beam length

6.3.2 Comparing modelling and test results for Beam A2

Figure 6.14 compares the test versus the numerical results for Beam A2 (Nadjai et

al., 2006) for the two cases when the recorded temperatures were applied to the

model in the first case, and in the other recorded temperatures were extrapolated in

the last minutes of the test. According to Figure 6.14, there was a very good

agreement between the test and numerical results in terms of the failure load and

overall behaviour of this composite beam in different stages of fire loading.


169

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90

Time (Min)

Def

. (m

m)

Experiment

Recorded Temps

Extrapolated Temps

Figure 6.14- Test results against the numerical results of Beam A2 considering the recorded and extrapolated temperatures

According to Figure 6.15, which shows the initiation of the web buckling at 50

minutes into the test, web post buckling initiated from the web posts located in the

high shear zone. The mid panel, under high bending moments, had no signs of any

instabilities.


170

Figure 6.15- Initiation of buckling at 50 minutes

Figure 6.16 shows the final deformed shape of the beam where the double

curvature S-shaped buckling of the web posts was evident. Interestingly, this figure

shows that the web post in the middle of the beam has finally buckled as well but

in a different mode. It has buckled with a single curvature as the beam is subjected

to two point loads and therefore there is no shear force in the mid web posts to

cause an S-shaped buckling mode. However, the buckling mode of the end web

posts is S-shaped as it is dominated by the shear.


171

Figure 6.16- Prediction of web buckling of Beam A2 in numerical modelling

Comparing the test with numerical results confirmed that this model was calibrated

against the test and can be used to look into some interesting aspects of the

behaviour of symmetric cellular composite beams in fire conditions, which is

presented in the next chapter.

6.3.3 Investigating the internal forces

The formation and development of the compressive force in the two diagonal

struts, crossing the two critical web posts, was investigated by looking at the six

nodes addressed in Figure 6.17. Comparing the contour plot in this figure against

the similar contour plot for ambient temperature (Figure 4.18) highlights that the

compressive stress block (the strut model presented in Chapter 2) is not as

developed and recognisable at elevated temperature as it was at ambient

temperature. Figure 6.18 presents the minimum principal stress values

(representing compression) in these six nodes (with a utilization factor of 50%)

which is the equivalent to Figure 4.19, for ambient temperature. Buckling has taken

place at a stress level of about 200 kN/mm2 at elevated temperature (compared to


172

300 kN/mm2 at ambient condition) which is mainly due to deterioration of the

material properties, in particular modulus of elasticity which define the resistance

against buckling, at high temperatures.

Similar to ambient conditions, the two nodes at the middle of the struts (nodes B

and E) had less compressive stresses than the other four nodes, before forming a

new stress distribution pattern due to web buckling.

In Chapter 4, it was observed that nodes G, H, I and J, experienced similar

compressive stresses prior to buckling at ambient temperature (see Figure 4.19).

However, investigating the compressive stresses at these nodes at elevated

temperature (Figure 6.18) showed that these stresses were not similar before and

during the buckling stages (unlike ambient conditions). These compressive stresses

were even notably different between nodes H and J (about 9%) or I and G (about

18%) which had rather similar conditions in terms of shear force (due to

mechanical loading), or applied temperatures (up to 20oC difference).

These differences between the behaviour at ambient and elevated temperatures,

which may possibly be due to redistribution of stresses based on temperature

distribution, adds to the complexity and uncertainty of developing a simple design

method for the cellular beams at elevated temperature. This also justifies

employing a more conservative approach in developing simple design models to

check the web post buckling (or possibly other failure modes) of cellular beams at



173

Figure 6.17- The struts in compression to investigate the internal stresses

-250

-200

-150

-100

-50

00 10 20 30 40 50 60 70 80

Time (min)

Stre

ss (N

/mm

^2)

Node I

Node E

Node G

Node J

Node B

Node H

Figure 6.18- Minimum principal stresses (representing compression) in the six nodes


174


BEAM B2

6.4.1 Introduction

The second fire test was conducted on asymmetric Beam B2 with the geometric

details presented in Figure 6.19.

This beam was also tested under the slow rate heating and with a utilization factor

of 50%, with an applied point load of 210 kN at the mid-span of the beam before

the fire loading.

Thermocouples were again positioned along the beam length, height and through

the depth of the concrete slab to record the temperatures. Figure 6.20 shows the

details of the position of these thermocouples, which were mostly focused in the

critical web posts, expected to buckle.

450 mm dia. cell at 630 mm centres

4500mm

150 mm thick slab

406x140UB39 Top & 457x152UB52 Btm Tee

Detailing of Asymmetric Composite Celluar Beam (B1, B2 and B3)

Figure 6.19- Geometric details of asymmetric Beam B2


175

Zon

e 8

Zon

e 1

Zon

e 2

Zon

e 3

Zon

e 4

Zon

e 5

Zon

e 6

Zon

e 7

Figure 6.20- Position of thermocouples for Beam B2 (Nadjai, 2007)


176

6.4.2 Modelling and results

Beam B2 was modelled based on the similar approach to Beam A2. Again

observations from the test showed that the beam and concrete slab acted fully

compositely with no sign of slippage. Therefore, the model was based on full shear

connection.

This asymmetric beam failed due to web post buckling in the test and Figure 6.21

shows the final deformed shape of this beam. The model developed in ABAQUS

also failed due to web buckling as shown in Figure 6.22. More importantly, the

numerical results showed very good conformity with the test data in all stages of

the fire test as well as the failure time (temperature) (Nadjai et al., 2007). Figure

6.23 shows this conformity by comparing the load-deflection curves from the

model against the test.

Figure 6.21- Web post buckling in the Beam B2 (Source: The test report of the Ulster fire tests on cellular composite beams (Nadjai et al., 2007))


177

Figure 6.22- Web buckling of Beam B2 predicted by the developed numerical model

0

50

100

150

200

250

300

350

60 70 80 90 100 110 120 130 140 150 160Time(mins)

Def

lect

ion

(mm

)

Experiment

Numerical Model

Figure 6.23- Comparison of load-deflection curves for the test against numerical model


178

Therefore, the numerical model developed for the fire test on Beam B2 could also

meet the necessary validation requirements to be used for parametric studies which

are presented in detail in the next chapter.

6.5 MODELLING AND RESULTS OF FIRE TEST FOR ULSTER

BEAM A3

6.5.1 Introduction

The two fire tests presented so far were based on a slow rate heating curve. For the

fire tests on Beam A3 and Beam B3 the standard heating rate ISO-834 (BSI,

2002b) to investigate how the rate in temperature increase, influences the

performance of cellular beams.

The symmetric Beam A3 was geometrically similar to Beam A2 (refer to Figure

6.2), but experienced the ISO-834 (BSI, 2002b) fire curve with the load factor of

30% and finally failed due to web post buckling in the test.

Heating according to ISO-834 increased temperatures in a considerably faster rate

so that Beam A3 and Beam B3 failed much sooner than Beam A2 and Beam B2 in

the fire test, even though the loading ratio was reduced from 50% to 30%. Figure

6.24 compares the average temperatures of the critical web post located at zone 3

in Beam A2 and A3 to give an idea of how different the rate of temperature

increase was in the two fire scenarios and how influential this has been in the fire

resistance period of this beam.


179

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60 70 80 90Time (Min)

Avr

age

Web

Tem

p (C

)

Beam A3

Beam A2

Web Buckling

Web Buckling

Figure 6.24- Average of the temperatures recorded in the critical web post in Tests A2 and A3

The higher heating rate meant that the role of thermal conduction was not as

dominant in balancing the temperatures. Therefore, the thermal gradients were

expected to be higher along the height of the section. Moreover, local effects due to

non-uniform temperature distribution in the furnace, and of course along the beam

length, were expected to be more influential.

6.5.2 Numerical modelling of Beam A3

The generics of the numerical approach used to model Beam A3 was similar to the

two beams already explained. However, comparing the temperatures certified that

in Beam A3, which experienced the fast growing fire, the variation of measured

temperatures in different sections of the beam were notably more than Beam A2.

This suggested employing a more detailed approach in applying the temperatures.

Therefore, the main difference in modelling this beam was to minimise the error

caused by averaging the temperatures for which notably smaller areas were


180

considered for calculating and applying the averaged temperatures. Figure 6.25

shows schematically the divisions used in this beam where the areas with similar

colours were applied similar temperatures based on an average of the relevant

recorded values.

Figure 6.25- Small areas used to calculate and apply the recorded temperatures more accurately

Further to what was explained in Section 6.6.2.1, item number 5, about

interpolating the temperatures to avoid numerical issues, this beam was a good

example. Figure 6.24 includes the average temperatures of Beam A3 in a critical

web post where the temperatures considerably increased in the first two minutes.

Therefore, temperatures at 1 and 2 minutes were ignored and a linear increase of

temperatures was considered from zero minutes to 3 minutes to avoid numerical

convergence problems in the first minutes.

6.5.3 Results of numerical modelling for Beam A3

The numerical results compared well with the test, as shown in Figure 6.26, and

web post buckling was also correctly recognised by the model (see Figure 6.27).

However, the solution stopped running before the last stages of the test. The actual

test lasted for 31 minutes but according to Figure 6.27, which compares the test

and FE results, the developed model stopped running at 24 minutes. This was due


181

to an abrupt local instability in one of the critical bottom Tees. The sequence of

sudden formation of this local buckling is shown in Figures 6.27 and 6.28, where

the web post clearly started to buckle at load increment number 262 (24 minutes).

However, a local buckling in one of the critical Tees, at load increment 263 (the

same 24 minutes) caused abrupt deformations which the ABAQUS solver was not

able to handle. This local buckling again took place at the high compression

diagonal but in the web post which was supported by the web stiffener.

0

40

80

120

160

200

240

0 4 8 12 16 20 24 28 32Time (Min)

Def

lect

ion

(mm

)

Experiment

Numerical Model

Solution terminated due to local instability

Figure 6.26- Numerical versus experimental results for Beam A3


182

Figure 6.27- Deformation at Time=24.16, Load increment number 262, web buckling is developing

Figure 6.28- Deformation at Time=24.16, Load increment number 263, Sudden local buckling in the bottom Tee

In order to overcome this issue, various options were explored. Among which,

slight changes in the amplitude and direction of the web imperfection, and refining


183

the areas used to apply the temperatures (around the buckled region) allowed the

model to pass this critical stage and run to the last minutes of the test.

This shows occurrence of various buckling modes could be competitive and close

together, especially at elevated temperature, and it is not always the web post

which is first to experience instabilities. However, it will be presented later in

Chapter 7 that such local instabilities in top and bottom Tees did not govern the

overall failure of these beams in most of the cases, as stresses redistribute. The

overall failure mode remained the web post buckling, in most of these cases.

However, web buckling was, of course, promoted by such preliminary instabilities.

Figure 6.29 shows the final numerical results against the test results, which

correlated relatively well. Moreover, the web post buckling again dominated the

failure mode of this beam (see Figure 6.30), which was also witnessed in the test.

0

40

80

120

160

200

240

0 4 8 12 16 20 24 28 32Time (Min)

Def

lect

ion

(mm

)

Experiment

FE Model

Figure 6.29- Load-deflection curves by experiment and numerical model


184

Figure 6.30- Final deformed shape of Beam A3 in ABAQUS


BEAM B3

6.6.1 Introduction

The last of the six Ulster beams was the asymmetric Beam B3 with a similar

geometry to Beam B2 (refer to Figure 6.19). This Beam was also tested under the

fast heating fire curve (ISO-834) and a load ratio of 30%.

6.6.2 Numerical modelling and results for Ulster Beam B3

This beam was modelled with an approach similar to Beam A3. Figure 6.31 shows

the failed shape of Beam B3 in the developed model which again matched well

with the web buckling observed in the test.


185

However, the load-deflection curve gained from the model did not match very well

with the test results. Figure 6.32 compares the two curves in which the developed

model has started to increase in deflection at about 12 minutes, whereas this

increase started notably later in the test (17 minutes).

Figure 6.31- Final deformed shape of Beam B3


186

0

50

100

150

200

250

0 4 8 12 16 20 24 28 32

Time (Min)

Def

lect

ion

(mm

) Experiment

Model

Figure 6.32- Load-deflection curves of Beam B3 resulted from experiment and model

As temperatures were already applied in detail, a wide range of studies were

conducted on some other factors aiming to improve the results and resolve the

validation issue for this beam by looking into some reasonable modifications.

Among these approaches it was tried to reduce the amplitude and change the shape

of the imperfection initially imposed to the beam geometry (approach one in Figure

6.33). In other approach (approach 2), the interaction between the top Tee and

concrete slab was changed in the model from two surfaces being in contact to two

lines. Another approach was to limit the lateral displacement and rotation of the

slab, where the point loads were applied, to see if any of these changes in the

model delayed the occurrence of web post buckling. Unfortunately, none of the

approaches, and their combinations, were notably helpful in reducing the gap

between the numerical and test curves (Figure 6.33).


187

0

50

100

150

200

250

0 4 8 12 16 20 24 28 32Time (Min)

Def

lect

ion

(mm

)

Test

Numerical Approach 1



Figure 6.33- Time-deflection curves resulted from some of the approaches to calibrate the numerical model for Beam B3

The following explains some of the main reasons behind this inaccuracy of the

numerical prediction, and pinpoint some of the differences that could negatively

affect the numerical results:

1. As mentioned the temperatures recorded by thermocouples at steel and

concrete sections were averaged and introduced to the numerical model

directly. On the other hand, the higher rate of fire growth is associated

with higher thermal gradients throughout the section, as transfer of the

heat due to conduction within the section becomes less dominant in

balancing the temperatures. Therefore, higher rate of temperature rise

increases the inaccuracies in the numerical model. This effect was more

influential in the concrete slab where the temperatures were recorded in

only one location of the slab (at three different depths) and there was no

record of the temperature variations over the length and width of the slab.

2. Eight thermocouples recorded the furnace temperature at different

locations to assess the performance of the furnace (in fire test on Beam


188

B3) in terms of the uniformity of temperature distribution (Figure 6.34).

The maximum values were up to about 100o C higher than the minimum

recorded temperatures. Although it has been tried to minimise the error by

refining the areas used to apply the temperatures, it still reduces the

accuracy.

3. The section sizes and thicknesses were based on the nominal ones (as no

measured values were available) which could have been different from

the actual thicknesses.

0

200

400

600

800

1000

0 5 10 15 20 25 30 35Time (Minute)

Furn

ace

Tem

pera

ture

(C)

Furnace Temp. 1

Furnace Temp. 2

Furnace Temp. 3

Furnace Temp. 4

Furnace Temp. 5

Furnace Temp. 6

Furnace Temp. 7

Furnace Temp. 8

Figure 6.34- Furnace temperatures recorded by eight thermocouples for fire test on Beam B3

6.6.2.1 Some empirical approaches used to tackle the convergence problems

It should be note that in some models developed for the furnace tests and for the

case studies (presented in Chapter 7), the numerical results were finally achieved

after various tries to overcome the solution convergence problems. Various reasons

caused these models to stop running due to numerical convergence problems


189

before the actual structural failure takes place. Some of these reasons are addressed

here:

• Accurate modelling of concrete, to include the cracking and crushing

behaviour, and the limited ability of the software to handle the occurrence

of cracking and crushing in one increment

• The interaction between the top flange and concrete slab as well as the local

stress concentrations caused by temperature gradients in different sections

of the composite beam

• Occurrence of high strains and deformations in the steel beam and concrete

slab due to fire and instabilities by web post or other sorts of buckling

The following empirical approaches were found useful to tackle the convergence

problems of the solution in the modelling of these beams:

1. Increasing the number of elements and integration points in the places

causing numerical failure (mostly in the concrete slab), was helpful in

overcoming the numerical issues. This is also recommended by the

software manual (Hibbitt et al., 2004b)

2. Sometimes manipulation of the maximum, minimum and starting time

increment in the automatic time stepping proved useful in letting the

model run further. It is commonly known that defining relatively high

maximum limits could cause problems because if the solution does not

converge, then the next increment will be half of the previous one.

Therefore, high maximum limits might not allow the load increment to

become small enough to satisfy the convergence criteria. This mostly

happened at increments associated with simultaneous cracking or

crushing of concrete material at many integration points, or at large

deformations during web buckling. Lowering the amplitude of the

minimum increment limit was also found helpful in cases where very

small increments were needed to let the solution run further.


190

3. Changing the starting load increment also affected how far the model runs

in some cases. This was to do with the iterative algorithm in the Newton-

Raphson to solve the simultaneous equilibrium equations, explained in

Chapter 3.

4. To ensure point loads were applied in an area as opposed to a node. This

was also helpful to avoid convergence problems knowing that these high

loads were applied to the concrete material, which is vulnerable to

crushing.

5. In some cases, there were notable fluctuations in temperature recorded by

thermocouples in a limited time, especially in the first couple of minutes

of the test. It was found useful to exclude these temperature impulses by

interpolating the temperatures in these specific cases, before inputting

them into the model.

6. Moreover, increasing the convergence criteria and the points already

mentioned in this regard, such as slight changes to the fracture energy of

the concrete also helped overcoming convergence problems.

6.7 SUMMARY OF THE MODELLING AT ELEVATED

TEMPERATURE

This chapter presented the details of the numerical modelling of the furnace tests

on four cellular composite beams carried out at University of Ulster. Two of the

fire tests were based on the ISO-834 fire and the other two were based on slow fire

rate of 10oC/min. The numerical results generally had a good agreement with the

test results in terms of failure mode, time (temperature) and overall performance.

The agreement of the numerical with test results was better in the two tests

subjected to slow fire. The calibration of the models against the fire tests allowed

these models to be used in some parametric studies, which are presented in Chapter

7.


191

7 PARAMETRIC STUDIES ON CELLULAR COMPOSITE

BEAMS AT ELEVATED TEMPERATURE

7.1 INTRODUCTION

There is still no simple design method available in the Eurocodes for cellular

beams. This highlights the necessity of carrying out numerical and experimental

investigations into the performance of these beams and implementing a robust

simple, but reliable, design method.

Therefore, in this research, a series of parametric studies were conducted on the

validated model to look at the sensitivity of the certain parameters and also to get a

better understanding of how these beams behave in fire. These parametric studies

investigated the influence of web imperfections, web stiffener, temperature

distributions, load ratios, loading types and finally support conditions, on the

performance of CBs at elevated temperature. This chapter focuses on these

investigations.

7.2 WEB IMPERFECTIONS

The sensitivity of the behaviour of cellular composite beams to web imperfections,

during fire conditions, was investigated by carrying out a series of analyses

introducing web imperfection amplitudes ranging from .001 mm to an

unreasonably high value of 10 mm. Figure 7.1 presents the results for the

symmetric Beam A2 and Figure 7.2 reflects the results for asymmetric Beam B2.

Comparing these results against the results of similar studies on these beams at

ambient temperature (refer to Chapter 5), highlights that web imperfections are

notably more influential at ambient temperature and their effect can reasonably be

ignored at elevated temperature. The reason why web imperfections are not as

dominant at elevated temperature as they are at ambient temperature can be


192

justified based on the fact that at elevated temperature, deformations (local and

global imperfections) due to different heat scenarios in different sections of the

beam are notably higher than the effect of the limited web imperfections.

0

20

40

60

80

100

120

140

160

180

200

0 10 20 30 40 50 60 70 80 90Time (Min)

Def

. (m

m)

ExperimentImp=0.001 mmImp= 0.1 mmImp=5 mmImp=10 mm

Figure 7.1- Investigating the influence of web imperfection for Beam A2 at fire conditions


193

0

50

100

150

200

250

300

350

60 80 100 120 140 160

Time (Min)

Def

lect

ion

(mm

)

ExperimentImp=2 mmImp=0.1 mmImp=-2 mmImp= 10 mm

Figure 7.2- Investigating the influence of web imperfection for Beam B2 at fire conditions

However, this contradicted the current method for designing cellular beams at

elevated temperature which is described in SCI report RT1187 (Simms, 2008).

This method bases the web post buckling strength on an equivalent strut, as

mentioned in Chapter 2, and the following equations are introduced to calculate the

buckling strength of the web post at ambient and elevated temperatures

accordingly:

yb ff χ= Equation 7-1

yθy,fi, fkfb ⋅⋅= χθ Equation 7-2

where: χ is the buckling coefficient

fiχ is the buckling coefficient at temperature θ

ky,θ is the strength reduction factor for steel at temperature θ


194

fb is the buckling strength of the web post

fb,θ is the buckling strength of the web post at temperature θ

Imperfection parameters for the strut are adopted depending on the web

slenderness.

The χ and χfi are calculated similarly as a function of non-dimensional slenderness

λfi (Equation 7-3) and the depth over thickness ratio (d/t).

cr,θ

pl,θfi N

N=λ Equation 7-3

To include the effect of an imperfection factor, this method introduces the buckling

curves “d” and “c” in BSEN1993-1-1 (BSI, 2005a), Section 6.3.1.2, for d/t ratios

higher and lower than 85 accordingly.

It seems reasonable to use two different buckling curves to include the effect of

web imperfections at ambient temperature as the parametric studies on symmetric

and asymmetric cellular beams, which failed due to web buckling, showed that the

web post capacity is dependent on imperfections at ambient temperature. However,

this was not the case at elevated temperature. The overall behaviour and failure

time (critical temperature) of the critical web post remained unaffected by web

imperfections. This suggests that it is not a very reasonable to introduce the two

buckling curves “c” and “d”, which give up to 9% difference in the reduction factor

(χ), to calculate the buckling capacity of the web post (strut) at elevated

temperature since, the failure due to web buckling was not affected by web

imperfections

7.3 TEMPERATURE DISTRIBUTION

In order to investigate the significance of the temperature distribution on the

behaviour of cellular beams two different temperature distributions, based on the


195

same fire curve, were applied to the validated models and the results were

compared (Bake and Bailey, 2007b). The total applied heating energy was constant

for both approaches (as explained later in Section 7.3.1), to have a reasonable

comparison, and the only difference was the way the temperatures were distributed

along the beam length.

7.3.1 Effect on symmetric beam

In the first approach, which is shown in Figure 7.3, the temperatures recorded by

the thermocouples were averaged and applied along the beam length. The

averaging was conducted within the areas with the same colour (Figure 7.3). As the

areas with the same colour also had the same areas quantity-wise, the averaging

was in fact the weighted average, which kept the total applied heat constant.

It should be noted that the end sections of the beam were defined with different

colours in Figure 7.3 as they were not taken into account in the averaging. This was

because firstly, the end sections were considerably colder and secondly, only one

thermocouple was positioned to record the temperatures in the two beam ends as

they were less critical.

In the second approach, the temperatures were averaged and applied to limited

regions as reflected schematically in Figure 7.4. Therefore, the full length of the

beam was modelled in this approach to consider the variations of the temperatures

along the length.

The difference of the response of Beam A2 to these approaches depended on the

temperature profile along the beam length. Figures 6.11 and 6.12 presented the

temperature variations of the top and bottom web posts, along the beam length, at

45 and 55 minutes of the test. Investigating the temperature profiles showed that

temperatures were generally slightly higher in the critical web post (up to about

30oC) than the average value. This meant that the second approach applied slightly

higher temperatures to the critical web post than the first approach.


196

It should also be noted that the critical web posts for Beam A2 and Beam B2

(shown in Figure 7.5) were specified with regard to the dominancy of shear in web

post buckling and also the influence of moment-shear interaction. Between the two

critical web posts defined for each beam in Figure 7.5, only the one that

experienced higher temperatures was considered as the dominant critical web post

to compare its temperatures against the average value. This approach was justified

based on the fact that in the fire tests, one side of beam failed while the other side

had not failed yet. This is clearly shown in Figure 6.21 where half of Beam B2

completely failed due to web buckling in the test but the other half did not show

any sign of failure.

Figure 7.3- Averaging the Temperatures along the beam length


197

Figure 7.4- Applying temperatures in details

(a)

(b)

Figure 7.5- The location of the critical web posts with regard to the moment and shear: (a) Beam A2 and (b) Beam B2

Figure 7.6 compares the curves resulting from these two approaches in which the

load-deflection curve of the approach which applied slightly higher temperatures to

the mid panels, slightly shifted to the left (failed sooner). Even though this shift


198

was nominal in this case, it still could be a trend suggesting high temperatures are

more decisive in the critical panels (particularly web posts).

Comparing the two approaches of applying the temperatures was also useful in

justifying the reasonability of the first approach of applying the average

temperatures for this beam.

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90Time (Min)

Def

. (m

m)

Experiment

Temperatures Averaged

Temps Applied in Detail

Figure 7.6- Comparing the time deflection curves resulted from employing the two

approaches to Beam A2

7.3.2 Effect on asymmetric beam

The two approaches of introducing the temperatures were also applied to the

asymmetric Beam B2. Figures 7.7 and 7.8 present the temperature profile along the

beam length at 50 and 65 minutes respectively. According to these graphs,

considering the accurate temperatures applied slightly higher temperatures (up to

about 35oC difference) to the critical web posts, as opposed to applying the average

temperature. Figure 7.9 compares the time-deflection curves resulting from the two

approaches for this asymmetric beam where again (similar to the symmetric beam),


199

higher temperatures in the critical web posts shifted the time-deflection curve to

the left (beam failed sooner), this time more clearly.

250

300

350

400

450

500

0 1 2 3 4 5 6 7Opening Number

Tem

p. (C

)


Average of Top Web

Critical Web

Critical Web

Point Load

Figure 7.7- Temperature profile along the length of Beam B2 at 50 minutes

450

500

550

600

650

0 1 2 3 4 5 6 7Opening Number

Tem

p. (C

)


Average of Top Web

Critical Web

Point Load

Critical Web

Figure 7.8- Temperature profile along the length of Beam B2 at 65 minutes


200

0

50

100

150

200

250

300

350

0 10 20 30 40 50 60 70 80 90 100

Time (Min)

Def

lect

ion

(mm

)

ExperimentAveraged temps

Accurate temps

Figure 7.9- Comparing the time deflection curves resulted from employing the two

approaches to Beam B2

This certifies that the performance of cellular composite beams is sensitive to the

temperatures at critical web posts and applying even slightly higher temperatures to

critical web posts visibly promoted the failure by web buckling.

7.4 LOADING TYPE AND WEB STIFFENER

Parametric studies were conducted on both symmetric and asymmetric geometries

of Beams A1 and B1 to look into their performance in fire conditions under a

UDL. The load ratio was again set to 50%, similar to the point load case, to allow a

valid comparison between the two cases.

7.4.1 Effect on symmetric beam

The results at elevated temperature are presented in two sections, with and without

web stiffeners, where the beam experienced two different failure modes depending

on the existence of the web stiffeners.


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7.4.1.1 Without web stiffeners

This beam without web stiffeners failed at 126 kN/m at ambient temperature and

the mechanical load for the fire analysis was based on this load and the load ratio

of 50%. Figure 7.10 shows the contour palette presenting out-of-web plane

displacements at 48 minutes which shows high out-of-web plane displacements

were not just limited to web posts. This was also the case in the bottom Tee, with

higher values toward the mid-span of the beam.

Figure 7.11 is the similar output for 63 minutes which shows how the development

of web post buckling turned to distorsional buckling (Santiago et al., 2008) of the

beam. Finally, Figure 7.12 presents the final deformed shape of the beam where the

beam has failed due to distorsional buckling.

Figure 7.10- Out-of-web plane displacements for Beam A2 without stiffeners at 48

minutes


202

Figure 7.11- Out-of-web plane displacements for Beam A2 without stiffeners at 63

minutes

Figure 7.12- Final deformed shape showing the distorsional buckling at 75

minutes

The occurrence of the distorsional buckling seems to be initiated by some lateral

displacement of the bottom Tee and intensified due to P-∆ effects (see Figure 7.13)

leading to failure. Lateral displacements of the bottom Tee could be due to the

warping effect caused by higher temperatures of the top compared to the bottom


203

Tee, non-uniform temperature distribution in different sections along the beam

length and height, and finally local compression forces in the bottom Tee (due to

Vierendeel moments).

The load-deflection curve for this case is presented in Figure 7.17 where it is also

compared against the case with web stiffeners, and also the case where point loads

were applied, to allow an overall comparison.

Figure 7.13- A section of the mid-span of the beam at 61 minutes to schematically

present the P-∆ effect causing distorsional buckling

7.4.1.2 With web stiffeners

Similar studies were conducted on Beam A2 under a UDL but this time with

stiffeners (original geometry). The mechanical load was calculated by applying the

utilization factor of 50% to a failure load of 145 kN/m at ambient temperature

(refer to Section 5.2).

Figure 7.14 is the out-of-web plane displacements at the early stages of the test (21

minutes) which shows slight lateral deformation of the bottom Tee at mid-span.

Figure 7.15 is the same contour palette at a later stage of the test (49 minutes)

which highlights the critical role of web stiffeners in limiting the lateral

displacement of the bottom Tee and avoiding distorsional buckling. Figure 7.16 is

the final deformed shape of this beam, failed due to web buckling, in which the

axes of the bottom and top Tees are still in a straight line. Figure 7.10 can be


204

compared against Figure 7.15, as both are at 49 minutes. The same applies to

Figures 7.12 and 7.16 as both present the last minutes of the test.

Figure 7.14- Slight out-of-web plane displacements of the bottom Tee at the mid-

span at 21 minutes

Figure 7.15- Out-of-web plane displacements at 49 minutes highlighting the critical role of web stiffener to limit the lateral displacement in bottom Tee


205

Figure 7.16-Final deformed shape of the beam failed due to web buckling in which

bottom and top Tees are still in a straight line

Figure 7.17 presents the time-deflection curve for symmetric Beam A2 under a

UDL, with and without web stiffeners, and also compares these curves against the

similar curve obtained for the point load, presented earlier.


206

0

50

100

150

200

250

300

350

0 10 20 30 40 50 60 70 80 90Time (Min)

Dis

plac

emen

t (m

m)

UDL - Load ratio=50% - With stiffener

Point Load - Load ratio=50%

UDL - Load Ratio=50% - Without stiffener

Figure 7.17- Comparing the performance of Beam A2 under point load and a UDL at elevated temperature for load ratio of 50%

This diagram suggests that Beam A2 showed a better performance in fire

conditions, based on a similar load ratio, while point loads were applied to the

beam rather than a UDL. It also shows that adding web stiffeners where a UDL is

applied to the beam, not only increased the loading capacity at ambient temperature

(as presented earlier in chapter 5), but it can also increase the beam’s resistance

against fire. This is supported because the beam without stiffeners had applied

lower loads compared to the one with stiffeners (126 kN/m versus 145 kN/m) to

have a similar load ratio of 50% and still failed notably sooner.

7.4.2 Effect on asymmetric beam

A similar study was conducted on asymmetric Beam B2 at elevated temperature,

under a 50% load ratio. This section presents the results for the two cases, with and

without stiffeners.


207

7.4.2.1 Without web stiffeners

Figure 7.18 presents the deformed shape of this beam without stiffeners where

again distorsional buckling dominated the failure of this beam. The resulting load-

deflection curve is presented later in Figure 7.20 and is compared against the case

with stiffeners and the case where a point load was applied to the beam.

Figure 7.18- Distorsional Buckling of Beam B2 without stiffener at 62 minutes

7.4.2.2 With web stiffeners

Similar studies were conducted on asymmetric Beam B2 with stiffeners (original

geometry), under a UDL and a load ratio of 50%. Figure 7.19 is the final deformed

shape of this beam where web post buckling governed the failure. Figure 7.20

presents the time-deflection curve for this case and compares it against the point

load case. This asymmetric beam also buckled sooner in the case where a UDL was

applied.


208

Figure 7.19-Final deformed shape of Beam B2 with stiffener under a UDL

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90Time (Min)

Def

lect

ion.

(mm

)

UDL with Stiffener ( Load Ratio = 50%)

Point Load ( Load Ratio = 50%)

UDL without Stiffener ( Load Ratio = 50%)

Figure 7.20- Comparing the performance of Beam B2 under a point load and a

UDL at elevated temperature for load ratio of 50%

Repeating similar results and trend for the two beams supports this hypothesis that

in the fire conditions and for a constant utilization ratio (50%), the web post

buckling occurred first in the beams subjected to UDLs (mostly the case in


209

practice) than point loads (mostly the case in furnace tests). Therefore, it is non-

conservative to simply rely on the performance of these beams on fire tests. This is

especially the case where no stiffeners are used in beams under a UDL and needs

to be considered while developing design guides for these beams at elevated

temperature.

7.5 LOAD RATIO

The load ratio was 50% for the point loads applied in the fire tests on Beam A2 and

Beam B2. In this case study, load ratios varying between 10% to 85% were applied

to the models developed for Beam A2 and Beam B2 to investigate the sensitivity of

varying to load ratio at elevated temperature. Figures 7.21 and 7.22 reflect the

resulted time-deflection curves for the symmetric Beam A2 and Asymmetric Beam

B2 where a reasonable trend was observed in both cases.

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90Time (Min)

Def

lect

ion

(mm

)

Experiment (Load Ratio=50%)Load Ratio=30%Load Ratio=70%Load Ratio=10%Load Ratio=50%

Figure 7.21- Sensitivity of symmetric Beam A2 to load ratio at elevated

temperature


210

0

50

100

150

200

250

300

350

0 10 20 30 40 50 60 70 80 90Time (min)

Def

. (m

m)

Experiment (Load Ratio=50%)

Load Ratio=30%

Load Ratio=70%

Load Ratio=10%

Load Ratio=85%

Load Ratio=50%

Figure 7.22- Sensitivity of asymmetric Beam B2 to load ratio at elevated

temperature

However, comparing the sensitivity of the time-deflection curves to the load ratio

in the two diagrams highlights that asymmetric Beam B2 was more sensitive to

load ratio, compared to symmetric Beam A2. This could be justified based on the

fact that the bottom Tee plays a more significant role in the ultimate load capacity

of asymmetric cellular composite beams compared to symmetric beams, at room

temperatures. Deterioration of the steel material is higher than concrete at elevated

temperature, which means that asymmetric composite beams loses greater overall

strength compared to symmetric beams. Moreover, the bottom Tee of CBs

experience higher temperatures than the top Tee and this also causes composite

asymmetric beams to lose strength and stiffness at a higher rate compared to

symmetric beams. Therefore, asymmetric beams are more sensitive to the

magnitude of the load ratios at elevated temperature. In other words, generally in

steel-concrete composite members, the higher the role of steel in the ultimate load

capacity of the composite member, the higher the rate of stiffness and strength loss

in the event of fire which results in more sensitivity to the load ratio.


211

7.6 THICKNESS OF WEB STIFFENER

The model developed for Beam A2 was employed to investigate the impact of web

stiffeners on web buckling of cellular beams at fire conditions. Various thicknesses

were considered for the web stiffeners within the study. The reason to select this

geometry was because in Beam A2 (refer to Figure 7.5), the critical web posts,

with the highest shear and moment combination, were located next to the web

stiffener, unlike Beam B2 (see Figure 7.5), Therefore web buckling was expected

to be influenced more notably by web stiffeners. Figure 7.23 shows the results of

this case study for the stiffener thickness varying between 5 mm to 25 mm.

According to this figure, even unreasonably high web stiffener thicknesses did not

delay the web post buckling and beam failure.

0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50 60 70 80 90Time (min)

Def

lect

ion

(mm

)

Web Stiffener Th.= 25 mm




Figure 7.23- Effect of web stiffener thickness on performance of cellular beams in

fire


212

7.7 SUPPORT CONDITIONS

Another case study investigated the influence of support conditions in the

performance of cellular composite beams at elevated temperature by introducing

some linear springs to the beam-ends to apply various axial stiffnesses. Figure 7.24

shows schematically the sensitivity studies presented in this section. The stiffness

of the springs (represented by ‘K’ in Figure 7.24) was based on ratios (5% to

100%) of the total axial stiffness (EA/L) of a cellular composite beam at room

temperatures. Springs were introduced to symmetric and asymmetric beams, each

under point load and a UDL (four cases). This section presents and compares the

results obtained from these four cases.

It should be noted that both point loads and UDL were applied to the original

geometry to avoid the possible distorsional buckling and thus focus on web post

buckling.

Figure 7.24- Sensitivity studies carried out on the loading type and support

restraints at elevated temperature

7.7.1 Effect on symmetric beams under point load

Figure 7.25 shows the load-deflection curves for symmetric Beam A2 while

springs with various stiffnesses were applied to the beam-ends. According to the

results of these parametric studies, higher ‘K’ values resulted in slightly less


213

deflections in the pre-buckling as well as the post buckling phases of the load-

defection curve. However, the buckling phase of the load-deflection curves and the

fire resistance period of this beam were not affected by changes in the K value.

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90

Time (Min)

Def

lect

ion

(mm

)

K=100%

K=15%

K=50%

EXPERIMENT

Pre Buckling

Buckling

Post Buckling

Figure 7.25- Load-deflection curves for the Beam A2 under point load and various

axial end support stiffness

Generally, these changes to the load-deflection curve, due to changes to the support

conditions from free horizontal restraint to full restraint, could be justified by

referring to the three factors, which generally affect the load-deflection curve in

this case. In other words, the changes in pre-buckling, buckling and post-buckling

stages of this curve (Figure 7.25) and similar curves for the other similar case

studies presented later, were influenced by the resultant of these three factors and

their dominancy at each stage of the test.

The first factor acts similar to ambient temperature while higher values of spring

stiffness, provides more restraint at the beam ends resulting in less deflections.

This effect is more dominant in the pre-buckling phase and could also be

influential in the post buckling face.


214

The second factor is caternary action, which plays a major role in the post-buckling

phase of the curve. Section 7.7.5 of this chapter looks into this in more detail and

presents the associated results. The catenary action (Yu and Liew, 2005) refers to

conditions where the flexural behaviour of CBs which is usually governed by the

web post resistance and Vierendeel actions, turns into tensile behaviour to sustain

the applied load, after web buckling. Investigations have shown that this

phenomena plays a critical role in the performance of steel beams in fire and

efforts have been made trying to utilise the catenary action to eliminate fire

protection of steel beams (Wang et al., 2005). Although this phenomenon is long

known, utilising it within fire design calculations of CBs is not as easy as it is for

common solid webbed beams. This is because after web posts has buckled, the

bottom Tee will not fully contribute to the beam resistance. Especially after the

full-scale fire test in Belfast (presented in Section 2.4.3.5) where bottom Tee

moved considerably out of web plane due to distorsional buckling (see Figure

7.36), researchers are also considering to conservatively include only the effect of

the top Tee in the calculations of the catenary action.

On the other hand, applying higher K values (end restraints), increases the axial

compressive force applied to the beam (reaction against the longitudinal thermal

expansion), especially in the bottom Tee which is hotter and also not constrained to

the concrete slab. This causes P-∆ effects, decreases the beam stiffness, especially

in the bottom Tee, and in some cases causes instabilities, comprising local

instabilities in the Tees or distortional buckling, which results in higher deflections

and quicker failure. This effect is counted as the third factor. Figure 7.26 presents

the maximum lateral deflection of the bottom flange of Beam A2 against time for

three different K values. It shows that the development of distortional buckling,

which has been followed by the web buckling, was accelerated by the increase of

the K value.


215

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90Time (Min)

Late

ral D

efle

ctio

n of

Bot

tom

Fla

nge

(mm

)

K=15%

K=50%

K=0 %

Figure 7.26- The maximum lateral deflection of the bottom flange of the Beam A2

versus time for three different K values

These three factors can justify the behaviour observed in different phases of the

load-deflection curve for Beam A2.

7.7.2 Effect on symmetric beams under a UDL

The same case studies on the role of end restraint conditions were carried out on

Beam A2 but under a UDL and Figure 7.27 compares the resulting load-deflection

curves. This chart again highlighted that increasing the stiffness of the springs

resulted in slightly less deflection in the pre and post-buckling phases, but not in

the buckling phase.


216

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90Time (Min)

Dis

plac

emen

t (m

m)

K=0%

K=75%

K=5%

Figure 7.27- Load-deflection curves for Beam A2 under distributed load and

various K values

Consequently, the two case studies on Beam A2, Figures 7.25 and 7.27, suggest

that the overall behaviour of symmetric cellular beams in fire was not greatly

affected by the K value, compared to asymmetric Beam B2, which is presented in

the next section.

7.7.3 Effect on asymmetric beams under point load

The same studies on end restraints were conducted on asymmetric Beam B2 and

major differences were noticed between the behaviour of asymmetric against the

symmetric beams in this respect. Imposing a higher K value to springs resulted in

lower deflections in pre and post-buckling phases, similar to the symmetric beam.

However, the web post buckling was notably and critically accelerated. This is

shown in Figure 7.28, which compares the load-deflection curves for different

support conditions and indicates that the buckling phase has initiated notably

earlier with the increase of K values.


217

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90Time (min)

Def

lect

ion

(mm

)

K=12.5%K=50%K=100% K=75%ExperimentK=0%K=5%

Pre Buckling

Post Buckling

Buckling

Figure 7.28- Load-deflection curve for Beam B2 under point load and

experiencing various axial restraints

The interesting point is that providing higher support rigidities promoted the web

post buckling by up to about 10% in terms of time. According to Figure 7.28, the

mid-span deflection of 100 mm occurred at 67 and 59 minutes for 0% and 50% of

axial stiffness (K) respectively, which shows a relatively notable difference in the

buckling phase. This chart also shows that the buckling phase was relatively

sensitive to the existence, or non-existence of some axial restraint (shown by

curves for K=0% and K=5%). However, this sensitivity faded for higher K values

whilst the curves for K=50% and K=100% hardly diverged from each other.

One of the major reasons why the performance of asymmetric beams is more

sensitive to end restraint is again hidden behind the fact that the bottom Tee plays a

more critical role in the ultimate loading capacity of these beams as opposed to

symmetric beams. Experiencing the high temperatures in the bottom Tee and

excessive compressive forces applied by end supports is more influential in

reducing the loading capacity of asymmetric beams, which is more dependent on

the bottom Tee. This leaves the asymmetric beam more sensitive to support

conditions (K value).


218

7.7.4 Effect on asymmetric beams under a UDL

A similar case study was conducted on the asymmetric Beam B2 under a UDL and

Figure 7.29 presents the results, which show a similar trend to that obtained for the

asymmetric beam under a point load.

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90Time (Min)

Def

lect

ion.

(mm

)

K=0%

K=50%

K=75%

K=25%

Figure 7.29- Load-deflection curve for Beam B2 under line load experiencing

various axial restraints

The presented results at elevated temperature shows that asymmetric beams are

more vulnerable to support conditions, in terms of failure by web post buckling,

compared to symmetric beams. This needs to be considered in the design methods

proposed for asymmetric cellular beams. Moreover, this underlines the fact that it

is not conservative for asymmetric beams to develop calculation methods relying

on their performance in fire tests as in practice CBs are ideally connected to

primary beams which means some axial restraint is applied to the ends.

Finally, it is critical to note that the investigations presented in this section were all

based on one particular form of variation of support stiffness (K) though the beam

height, in which the stiffness was assumed to be relatively uniform through the


219

height of the composite beam. In practice, the stiffness of the support could vary

through the height, depending on the torsional stiffness of the primary beams,

whether the main beam is on the edge of the floor slab or internal, and the

distribution of the fire load. This means more critical conditions could also be

expected in accelerating the buckling of the beam and more research could be done

in this regard.

7.7.4.1 Some observations within carrying out case studies on support conditions

The case studies to increase the restraint stiffness (K) were associated with some

local buckling in the Tees in some of the models (see Figure 7.30a). Moreover,

some models experienced distortional buckling of the bottom Tee (see Figure

7.30b) prior to web post buckling (Figure 7.30c). As explained before, these

instabilities prior to web buckling were intensified by high compression forces

applied to the beam by end supports due to restrained thermal expansion.

(a) Local buckling (b) Distortional

buckling

(c) Web post buckling

Figure 7.30- Local flange buckling, distortional buckling and web post buckling of CBs


220

Top Tees generally experience lower temperatures than the bottom Tees, and

therefore lower compression due to thermal expansion which was restrained.

Nevertheless, local instabilities were not always limited to the bottom Tee. They

also occurred in the top Tees as they already experience lower tension due to

overall bending compared to the bottom Tee. Moreover, an additional compression

is also applied to the top Tee due to considering the full shear interaction between

the slab and the top flange knowing that the concrete slab has lower temperatures

and thermal expansion than the top Tee.

The local instabilities of the top flange occurred in the relatively early stages of fire

loading when flexural behaviour was still the dominant load bearing mechanism.

Figures 7.31 and 7.32 present a good example in which the first figure shows the

last panel of Beam A2 with excessive local compressive force (at 35 minutes), and

the next figure shows the local buckling of the top Tee at the next loading stage

(again 35 minutes).

Figure 7.31- Minimum in-plane principal stress representing the compressive

stresses


221

Figure 7.32-Local buckling of the Top Tee of the last opening

Nevertheless, in none of the case studies on axial restraint, presented in this section

(where the beams were modelled with web stiffeners), local buckling or

distorsional buckling, governed the ultimate loading capacity, as stresses

redistributed, and web post buckling dominated the overall load capacity in all

cases. However, these events were not totally independent and interactions were

evident in some cases. In other words, the web post buckling was promoted by the

occurrence of these instabilities. Figures 7.33, 7.34 and 7.35 present the gradual

formation of distorsional buckling of the bottom Tee in Beam A2 when K=100%

(as explained in section 7.7.1) and how the instability of the web post is promoted

resulting in earlier web buckling.

The fact that in none of these case studies distorsional buckling governed the

overall beam’s failure, emphasises the critical role of web stiffeners on avoiding

distorsional buckling. This is achieved by limiting the lateral drifts of the bottom

Tee, even when high K values increases instability of bottom Tee (as explained)

and its tendency to move out of the web plane.


222

Figure 7.33-Uz plot at 15 minutes which shows shaping of overall distorsional buckling in Beam A2, due to high end restraints (K=100%)

Figure 7.34- Uz plot at 24 minutes, which shows development of the overall distorsional buckling in Beam A2 and how it perturbed the web post


223

Figure 7.35- Shaping of the overall distorsional buckling in Beam A2, at 61

minutes, (K=100%)

It should be noted that the effect of end-restraints in changing the failure mode

from web post to distorsional buckling (in cellular beams with no web stiffeners)

was observed in the recent full-scale fire test conducted by the University of Ulster

in 2010 (report to be published). In this fire test both of the internal unprotected

cellular beams, which were restrained at the beam-ends by protected primary

beams, failed clearly due to distorsional buckling. Figure 7.36 shows the final

deformed shape of the two internal beams showing distorsional buckling.


224

Figure 7.36- Failure of the internal cellular beams due to distorsional buckling at the full-scale fire test carried out by University of Ulster in 2010

7.7.5 Interaction between cellular beam and end supports (catenary

action)

The axial displacement of Beam A2 and Beam B2, was also investigated at the

beam-ends to see how any of these beams interacted with the end supports and how

this relates to the development of catenary action. This has been carried out by

looking at five nodes through the beam height which are highlighted in Figure

7.37.

7.7.5.1 Interaction between the symmetric Beam A2 and end-supports

Figures 7.38, 7.39 and 7.40 show the axial displacements (also the spring forces as

springs were linear) of the five nodes for Beam A2, under a point load, for K=0%,

K=5% and K=75% respectively. According to these figures, the axial deformation


225

of the top Tee (Point E) was governed by the interaction of the concrete slab with

the top flange. As concrete material is less sensitive to temperatures and loses its

axial stiffness at a lower rate than steel, the top Tee experienced similar axial

deflections in all cases, regardless of the K value.

However, the axial deflection of the other nodes through the beam height and

development of catenary action at elevated temperature, particularly in the bottom

Tee, was greatly affected by the K values. Figures 7.38, 7.39 and 7.40 show that

catenary action becomes more dominant, and also started to develop earlier, with

higher axial restraints. In particular, catenary action started at 68, 58 and 25

minutes for K=0%, K=5% and K=75%, respectively, which shows higher end

restraints were associated with a earlier web buckling (as explained) and earlier

development of catenary action (in the post-buckling phase). These figures also

show that catenary action was initiated first in the bottom Tee and then developed

in the top Tee.

These studies were conducted for both loading types (UDL and point loads) and no

major difference was observed.

Figure 7.37-Position of the nodes through the beam height


226

-30

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80 90 100

Time (Min)A

xial

Dis

plac

emen

t (m

m) Point A

Point BPoint CPoint DPoint E

Domination of catenary action

Figure 7.38- Axial Deflection of the nodes through the height of Beam A2 for

K=0%

-30

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80 90

Time (Min)

Axi

al D

efle

ctio

n (m

m)

Point APoint BPoint CPoint DPoint E



K=5%


227

-30

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80 90

Time (Min)A

xial

Def

lect

ion

(mm

) Point APoint BPoint CPoint DPoint E



K=75%

7.7.5.2 Interaction between the asymmetric Beam B2 and end-supports

The same studies were conducted on the asymmetric Beam B2 and Figures 7.41,

7.42 and 7.43 presents the results for this beam for K=25%, K=50% and K=75%

respectively. These figures show a trend in asymmetric beams, which is very

similar to what was observed for the symmetric beam.


228

-14

-12

-10

-8

-6

-4

-2

0

2

0 10 20 30 40 50 60 70 80 90

Time (Min.)A

xial

Dis

plac

emen

t (m

m)

Point APoint BPoint DPoint E


Figure 7.41- Axial Deflection of the nodes through the height of Beam B2 for

K=25%

-9

-8

-7

-6

-5

-4

-3

-2

-1

00 10 20 30 40 50 60 70

Time (Min)

Axia

l Dis

plac

emen

t (m

m)

Point APoint BPoint CPoint DPoint E



K=50%


229

-7

-6

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60 70 80 90

Time (Min)

Axi

al D

ispl

acem

ent (

mm

)

Point APoint BPoint CPoint DPoint E Domination of

catenary action


K=75%

7.8 SUMMARY OF THE CASE STUDIES ON CELLULAR

COMPOSITE BEAMS AT ELEVATED TEMPERATURE

This chapter presented the parametric studies conducted on models validated at

elevated temperature. It was shown that the magnitude of web imperfection does

not affect the performance of these beams at elevated temperature (unlike ambient

temperature). Therefore, the current approach by the “Strut” model, which

calculates the web buckling resistance at ambient and elevated temperatures

similarly, is not very reasonable at elevated temperature.

Case studies also presented the effect of loading type (a UDL as opposed to point

loads) and excluding the full height web stiffeners on the performance of cellular

composite beams at fire conditions. For a constant utilization factor, beams

performed better against fire when they were subjected to point loads, as opposed

to a UDL. In some cases excluding the web stiffeners (for beams subjected to a

UDL) resulted in distorsional buckling which promoted the failure of the beam.


230

Moreover, the sensitivity of cellular beams to the utilization factor and support

conditions was investigated at elevated temperature for both symmetric and

asymmetric sections. Asymmetric beams were found to be more sensitive and

vulnerable to the utilization factor than symmetric beams as explained in Section

7.5. Increasing the end-restraints caused local and global instabilities and promoted

the failure by either web post or distorsional buckling. However, increasing the

end-restraints was also associated with a better post failure performance, which

was dominated by the catenary action.


231

8 CONCLUSIONS & RECOMMENDATIONS

After an extended review of the literature and an in depth investigation into the

behaviour of cellular beams and cellular composite beams at ambient and elevated

temperatures, conclusions could be made and future investigations recommended.

8.1 CONCLUSIONS

1. Two of the well-known commercial finite element packages, ANSYS and

ABAQUS were initially used and their capabilities were compared in

modelling cellular composite beams. ABAQUS was proved notably more

equipped and capable in modelling the complex concrete material and the

associated numerical problems. ANSYS, only allows to introduce concrete

material to a particular solid element whereas ABAQUS enables the user

to incorporate the concrete material to solids as well as shell and beam

elements. Besides, ABAQUS benefits from three material models to

represent the concrete performance while these options are limited to only

one in ANSYS. Finally, ABAQUS enables the user to manipulate the post

cracking behaviour of concrete, whereas ANSYS leaves no options. This

was critical to overcome the associated convergence problems.

2. The numerical models developed for cellular beams at ambient

temperature showed a good conformity with the test results in terms of

failure mode, failure load and load-deflection behaviour. However,

validation of the numerical models for cellular beams against test results

can be difficult when the two common failure modes, web post buckling

and Vierendeel bending were both imminent. In such cases, recognizing

the correct failure mode by numerical models can be very sensitive to

boundary conditions and load increment. In particular, supports applied to

restrain the lateral displacement of top flange also apply an additional


232

moment to web post. This moment, which depends on the number of

supports and increases with vertical load, promotes the misbalancing and

therefore buckling of the web post. This can be very decisive in

modelling the correct behaviour by numerical models. In the case

presented in this research, slight modifications of the lateral supports

applied to top flange, in particular changing the lateral restraint from

being applied to a node to being applied over a limited length of flange,

changed the failure mechanism from Vierendeel bending to web buckling.

Subsequent to this change, the failure load was also reduced by 45% and

the numerical model was validated against the test results.

Moreover, if load increments are not small enough at the loads which

buckling is expected to happen, web post buckling could be easily missed

by the software and Vierendeel mechanism takes over the failure. This

mistake in the numerical modelling can happen due to the dependency of

Vierendeel action to yield stress, and web buckling to elastic and plastic

modulus while stress-strain transform was almost an elastic-fully plastic

bilinear curve for structural steel (based on the BSEN1993-1-1).

3. The numerical models developed for cellular composite beams generally

had a very good conformity with the results of the tests conducted at

ambient temperature. This agreement between the numerical and test

results was also good at elevated temperature in terms of the load-

deflection curve, failure mode and failure load (temperature). This

calibration of the numerical models allowed these models to be used for

case studies on cellular composite beams at ambient and elevated

temperatures.

4. Various amplitudes of web imperfections were introduced to symmetric

and asymmetric cellular composite beams at ambient and elevated

temperatures that had failed due to web buckling. Comparing the failure


233

loads underlined that imperfections could have an impact at ambient

temperature but they were not influential at elevated temperature. This

suggests that the current method to calculate the buckling strength of web

posts at elevated temperature (SCI publication RT1187), does not seem

reasonable. In particular, this method makes the web buckling capacity at

elevated temperature dependent on web imperfections (similar to ambient

temperature), by introducing the two different buckling curves “c” and

“d" (BSEN1993-1-1) which gives up to 9% difference in the reduction

factor (χ) of the web post (strut). However, this approach is incorrect as

the case studies showed that failure due to web buckling at elevated

temperature remained unaffected by web imperfections.

5. Case studies on the performance of cellular composite beams under a

UDL showed that their resistance against web buckling at ambient

temperature can significantly increase by adding web stiffeners. In the

absence of web stiffeners, some relative displacements were witnessed

between the top and bottom Tees, in the out-of-web plane direction.

However, they were not major enough to cause distorsional buckling and

web post buckling was the dominant failure mode in all case studies at

ambient temperature.

The performance of cellular composite beams at elevated temperature

was investigated under point loads and UDLs, based on a slow fire curve

of 10oC/min, and fixed load ratio of 50%, which showed that they resisted

notably higher temperatures when point loads were applied, rather than a

UDL. In all cases where a UDL was applied, excessive relative

displacements took place between the top and bottom Tees, which caused

failure due to distorsional buckling (and not web buckling). Adding web

stiffeners, which although increased the applied load (to keep the load

ratio constant), avoided distorsional buckling and enhanced the overall

resistance of beam against high temperature.


234

This hypothesis that these beams are more vulnerable against fire

conditions if they are designed based on UDLs (which is mostly the case

in practice) than point loads (mostly the case in experiments) emphasises

that it is non-conservative to rely on performance of these beams on fire

tests as is the case in current design methods. This seeks particular

attention in fire design of these beams remembering that web stiffeners

are not normally used when beams are subjected to a UDL. These results

also recommend the use of web stiffeners in cellular beams, not only to

notably increase their resistance at ambient temperature but also to avoid

the distorsional buckling and a better overall performance at elevated

temperature.

6. Parametric studies were conducted on the role of different temperature

distributions along the beam length, based on the same fire curve and a

constant heating energy. It was found essential to apply the temperatures

accurately in critical web posts to achieve the required validation against

experimental data, especially in fast growing fires. Moreover, the ultimate

loading capacity was found to be sensitive to even slight changes in

distribution of temperature along the beam length. In particular, higher

temperatures in critical panels, where the shear and its interaction with the

moment were highest, promoted web buckling.

7. The asymmetric cellular beam showed a higher sensitivity and

vulnerability to the magnitude of the load ratio compared to the

symmetric beam. For a similar load ratio (utilization factor), a symmetric

beam resisted higher temperatures before failure and this seeks further

research and prudency in fire design of asymmetric cellular beams. This

was justified based on the fact that generally in steel-concrete composite

members, the higher the role of steel in the ultimate load capacity of the

composite member, the higher the rate of stiffness and strength loss in the


235

event of fire. This results in more sensitivity to the load ratio.

Experiencing the highest temperatures in bottom Tee, compared to other

sections also contributes to this result.

8. Applying semi-restraints at beam-ends (by introducing linear springs to

apply a uniform stiffness based on percentages of overall axial stiffness of

the beam) in the fire situation decreased the initial and also final

deflections of CBs which was governed by catenary action. On the other

hand, applying higher restraints constantly promoted the failure due to

web buckling and reduced the resistance duration compared to simply

supported beams. In some cases with increasing the end-restraints, the

web buckling was followed after the occurrence of distortional buckling

(which did not govern the failure unless in beams with no web stiffener)

or local buckling in the bottom or top Tees, caused by high axial forces

due to thermal expansion. These instabilities interacted with web posts

towards destabilising it and promoted web buckling. This effect was

again more evident in asymmetric cellular beams. This is especially

critical since relying on the findings from standard fire tests (in which

beams are mostly simply supported) to propose general fire design guides

for these beams is non-conservative, and these beams can be more

vulnerable to web post buckling in fire conditions in practice compared to

test conditions.

9. Although applying restraints at the beam-ends promoted failure due to

web post buckling (especially in asymmetric beams) it also allowed for

the development of catenary action in the post failure phase. Catenary

action was more dominant when higher restraints were provided at the

beam-ends. The axial displacement (catenary action) in the bottom Tee,

within the thermal loading, was quite dependent on the K value (axial

stiffness), while the top Tee did not show much sensitivity to K values

and was mostly dominated by the full interaction with the concrete slab.


236

8.2 RECOMMENDATIONS AND FUTURE WORK

The author would like to make to following recommendations on the basis of this

research.

• It was observed that adding full-height web stiffeners to cellular composite

beams subjected to a UDL, improved the web buckling resistance at

ambient temperature by up to 14%. However, this was just examined in two

geometries. The author recommends investigating this on beams with

various spans and geometric details and looking at how the stiffener affects

the stress distribution and delays the web buckling. At elevated

temperature, cellular composite beams subjected to a UDL (with no web

stiffener) tend to fail due to distorsional buckling as opposed to web

buckling. This tendency became more dominant when some restraints were

applied at beam supports. It was only when the web stiffeners were added

that beams became equipped to resist the distorsional buckling. Based on

this, the author recommends adding full-height web stiffeners to cellular

beams under a UDL to improve their performance at both ambient and

elevated temperatures. However, these investigations are only conducted

for two geometries and investigating a wide range of geometers and spans

will assess the justifiability of this design recommendation.

• This research was based on fire curves used with the Ulster tests. It would

be interesting to investigate the performance of the same beams when they

are subjected to other fire scenarios. In addition, future research can focus

on considering a part of the beam length being subjected to fire, as opposed

to the whole beam length, and investigating whether changing the fire

exposure conditions notably delays the failure of the beam.

• It was presented that the current “Strut” model to estimate the buckling

resistance of the web post (using buckling curves “c” and “d”) is not very

reasonable at elevated temperature as this model makes the buckling

resistance dependent on web imperfections which is not reasonable at


237

elevated temperature. However, more research needs to be done in this

regard to specify the best buckling curve to be used in the “Strut” model to

estimate the web buckling capacity at elevated temperature.

• Finally, future research needs to also focus on implementing a simple but

reliable design model, which can predict the failure mode and load

(temperature) of cellular beams and cellular composite floors at elevated

temperature. The way to validate this analytical model would be to

numerically model a single web post (2 half-panels) of one beam, apply the

critical loads and moments proposed by the analytical model for each

failure mode, and calibrate it against the numerical results of the full-length

beam.


238

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University of Leeds. TELFORD, T. (1993) CEB-FIP Model Code 1990- Design Code. VASSART, O., BOUCHER, A., MUZEAU, J.-P. & NADJAI, A. (2008)

ANALYTICAL MODEL FOR THE WEB POST BUCKLING IN CELLULAR BEAMS UNDER FIRE. Proceedings of the Fifth International Conference on Structures in Fire (SiF’08).

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WANG, Y. C., YIN, Y. Z., SHEN, Z. Y., LI, G. Q. & CHAN, S. L. (2005) Feasibility of utilising catenary action to eliminate fire protection to steel beams. Fourth International Conference on Advances in Steel Structures. Oxford, Elsevier Science Ltd.

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WARREN, J. (2001) Ultimate load and deflection behaviour of cellular beams. School of Civil Engineering. Durban, University of Natal.

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YANDZIO, E., DOWLING, J. J. & NEWMAN, G. M. (1996) P160 - Structural Fire Design: Off-Site Applied Thin Film Intumescent Coatings – Part 1: Design Guidance and Part 2: Model Specification, Steel Construction Institute.

YU, H. X. & LIEW, J. Y. R. (2005) Considering catenary action is designing End-restrained steel beams in fire. Advances in Structural Engineering.

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244

Appendix A Some pictures of the tests carried out by the University of Ulster on cellular composite beams

This appendix presents some pictures of the failure of the Ulster beams at ambient

and elevated temperatures.

Figure A.1- Failure of the Ulster Beam A1 at ambient temperature

Figure A.2- Failure of the Ulster Beam B1 at ambient temperature


245

Figure A.3- Failure of the Ulster Beam B1 at ambient temperature

Figure A.4- Failure of the Ulster Beam A2 at elevated temperature


246

Figure A.5- Failure of the Ulster Beam B2 at elevated temperature

Figure A.6- Failure of the Ulster Beam B2 at elevated temperature


247

Figure A.7- Rupture taking place in the test on Beam B3 at elevated temperature within the cooling phase

Figure A.8- The other half of the Beam B3 which did not fail in the fire test


248

Figure A.9- Thermocouples positioned to record temperatures and occurrence of web buckling and rupture


249

Appendix B ABAQUS input file for Beam A1

This appendix includes the input file to produce the numerical model for Beam A1

within ABAQUS at ambient temperature.

*Heading ** Job name: Beam1AAmb Model name: Model-1 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Final *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Final-1, part=Final *Node 1, -2250., 143.100006, 0. ! The rest of the nodes until 2064, -2137.5, 641.200012, -494.179993 *Element, type=S4 1, 1, 190, 1107, 197 ! The rest of the elements until 1880, 2064, 1083, 188, 1084 *Nset, nset=_PickedSet2, internal, generate 1, 2064, 1 *Elset, elset=_PickedSet2, internal, generate 1, 1880, 1 *Nset, nset=_PickedSet3, internal 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 992, 993, 994, 995, 996, 997 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1044, 1045 1046, 1047, 1048, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074, 1075, 1076, 1077 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1090, 1091, 1092, 1093 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1875, 1876, 1877 1878, 1879, 1880, 1881, 1882, 1883, 1884, 1885, 1886, 1887, 1888, 1889, 1890, 1891, 1892, 1893


250

1894, 1895, 1896, 1897, 1898, 1899, 1900, 1901, 1902, 1903, 1904, 1905, 1906, 1907, 1908, 1909 1910, 1911, 1912, 1913, 1914, 1915, 1916, 1917, 1918, 1919, 1920, 1921, 1922, 1923, 1924, 1925 1926, 1927, 1928, 1929, 1930, 1931, 1932, 1933, 1934, 1935, 1936, 1937, 1938, 1939, 1940, 1941 1942, 1943, 1944, 1945, 1946, 1947, 1948, 1949, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957 1958, 1959, 1960, 1961, 1962, 1963, 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973 1974, 1975, 1976, 1977, 1978, 1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030, 2031, 2032, 2033, 2034, 2035, 2036, 2037 2038, 2039, 2040, 2041, 2042, 2043, 2044, 2045, 2046, 2047, 2048, 2049, 2050, 2051, 2052, 2053 2054, 2055, 2056, 2057, 2058, 2059, 2060, 2061, 2062, 2063, 2064 *Elset, elset=_PickedSet3, internal, generate 1601, 1880, 1 *Nset, nset=_PickedSet4, internal 5, 6, 9, 10, 15, 16, 19, 20, 23, 25, 28, 30, 31, 32, 33, 34 35, 36, 37, 38, 39, 40, 41, 42, 49, 50, 51, 52, 54, 56, 57, 59 60, 62, 63, 64, 65, 66, 67, 68, 71, 73, 74, 75, 76, 81, 82, 83 84, 85, 86, 87, 89, 91, 92, 94, 95, 97, 98, 99, 100, 101, 102, 103 106, 108, 109, 110, 111, 116, 117, 118, 119, 120, 121, 122, 124, 125, 127, 128 130, 131, 132, 133, 134, 137, 139, 140, 141, 142, 145, 146, 147, 148, 150, 151 153, 154, 156, 157, 158, 159, 160, 161, 162, 165, 166, 168, 169, 171, 172, 173 174, 179, 201, 215, 230, 244, 256, 261, 273, 278, 282, 283, 284, 285, 286, 287 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319 320, 321, 322, 323, 324, 325, 368, 369, 370, 371, 372, 373, 374, 375, 379, 380 381, 388, 389, 390, 391, 392, 396, 397, 398, 405, 406, 407, 408, 409, 410, 411 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 428, 429, 430 437, 438, 439, 440, 444, 445, 446, 450, 451, 452, 453, 454, 455, 456, 457, 458 459, 460, 461, 492, 493, 494, 495, 496, 497, 498, 499, 503, 504, 505, 506, 507 508, 509, 510, 511, 512, 513, 514, 518, 519, 520, 521, 522, 523, 524, 528, 529


251

530, 531, 532, 533, 534, 535, 539, 540, 541, 548, 549, 550, 551, 552, 556, 557 558, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579 580, 581, 582, 583, 584, 588, 589, 590, 597, 598, 599, 600, 604, 605, 606, 610 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 652, 653, 654, 655, 656 657, 658, 659, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 678 679, 680, 681, 682, 683, 684, 688, 689, 690, 691, 692, 693, 694, 695, 699, 700 701, 705, 706, 707, 708, 709, 713, 714, 715, 722, 723, 724, 725, 726, 727, 728 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 744, 745, 746, 753 754, 755, 756, 760, 761, 762, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775 776, 777, 802, 803, 804, 805, 806, 807, 811, 812, 813, 814, 815, 816, 817, 818 819, 820, 821, 822, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837 838, 839, 843, 844, 845, 849, 850, 851, 852, 853, 857, 858, 859, 866, 867, 868 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884 885, 889, 890, 891, 898, 899, 900, 901, 902, 906, 907, 908, 915, 916, 917, 918 919, 920, 921, 922, 923, 924, 925, 926, 927, 958, 959, 960, 961, 962, 963, 967 968, 969, 970, 971, 972, 976, 977, 978, 979, 980, 981, 982, 986, 987, 988, 989 990, 991, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168 1169, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1251, 1252, 1253, 1263, 1264, 1265 1275, 1276, 1277, 1278, 1279, 1280, 1281, 1282, 1283, 1293, 1294, 1295, 1305, 1306, 1307, 1308 1309, 1310, 1347, 1348, 1349, 1350, 1351, 1352, 1362, 1363, 1364, 1365, 1366, 1367, 1377, 1378 1379, 1380, 1381, 1382, 1392, 1393, 1394, 1395, 1396, 1397, 1407, 1408, 1409, 1419, 1420, 1421 1431, 1432, 1433, 1434, 1435, 1436, 1437, 1438, 1439, 1449, 1450, 1451, 1461, 1462, 1463, 1464 1465, 1466, 1503, 1504, 1505, 1506, 1507, 1508, 1518, 1519, 1520, 1521, 1522, 1523, 1533, 1534 1535, 1536, 1537, 1538, 1548, 1549, 1550, 1551, 1552, 1553, 1563, 1564, 1565, 1575, 1576, 1577 1587, 1588, 1589, 1590, 1591, 1592, 1593, 1594, 1595, 1605, 1606, 1607, 1617, 1618, 1619, 1620 1621, 1622, 1659, 1660, 1661, 1662, 1663, 1664, 1674, 1675, 1676, 1677, 1678, 1679, 1689, 1690 1691, 1692, 1693, 1694, 1704, 1705, 1706, 1707, 1708, 1709, 1719, 1720, 1721, 1731, 1732, 1733 1743, 1744, 1745, 1746, 1747, 1748, 1749, 1750, 1751, 1761, 1762, 1763, 1773, 1774, 1775, 1776


252

1777, 1778, 1815, 1816, 1817, 1818, 1819, 1820, 1830, 1831, 1832, 1833, 1834, 1835, 1845, 1846 1847, 1848, 1849, 1850, 1860, 1861, 1862, 1863, 1864, 1865 *Elset, elset=_PickedSet4, internal 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192 321, 322, 323, 324, 325, 326, 327, 328, 345, 346, 347, 348, 349, 350, 351, 352 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384 385, 386, 387, 388, 389, 390, 391, 392, 409, 410, 411, 412, 413, 414, 415, 416 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624 641, 642, 643, 644, 645, 646, 647, 648, 665, 666, 667, 668, 669, 670, 671, 672 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704 705, 706, 707, 708, 709, 710, 711, 712, 729, 730, 731, 732, 733, 734, 735, 736 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944 961, 962, 963, 964, 965, 966, 967, 968, 985, 986, 987, 988, 989, 990, 991, 992 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200


253

1217, 1218, 1219, 1220, 1221, 1222, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1230, 1231, 1232 1249, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1258, 1259, 1260, 1261, 1262, 1263, 1264 1281, 1282, 1283, 1284, 1285, 1286, 1287, 1288, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312 1329, 1330, 1331, 1332, 1333, 1334, 1335, 1336, 1337, 1338, 1339, 1340, 1341, 1342, 1343, 1344 1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1369, 1370, 1371, 1372, 1373, 1374, 1375, 1376 1393, 1394, 1395, 1396, 1397, 1398, 1399, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408 1473, 1474, 1475, 1476, 1477, 1478, 1479, 1480, 1481, 1482, 1483, 1484, 1485, 1486, 1487, 1488 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520 1537, 1538, 1539, 1540, 1541, 1542, 1543, 1544, 1545, 1546, 1547, 1548, 1549, 1550, 1551, 1552 1569, 1570, 1571, 1572, 1573, 1574, 1575, 1576, 1577, 1578, 1579, 1580, 1581, 1582, 1583, 1584 *Nset, nset=_PickedSet5, internal 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 190, 191 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 1107, 1108, 1109, 1110, 1111, 1112 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154 *Elset, elset=_PickedSet5, internal, generate 1, 128, 1 ** Region: (Stiffener:Picked) *Elset, elset=_PickedSet5, internal, generate 1, 128, 1 ** Section: Stiffener *Shell Section, elset=_PickedSet5, material=Web 10., 5 ** Region: (FLANGE:Picked) *Elset, elset=_PickedSet4, internal 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160


254

161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192 321, 322, 323, 324, 325, 326, 327, 328, 345, 346, 347, 348, 349, 350, 351, 352 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384 385, 386, 387, 388, 389, 390, 391, 392, 409, 410, 411, 412, 413, 414, 415, 416 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624 641, 642, 643, 644, 645, 646, 647, 648, 665, 666, 667, 668, 669, 670, 671, 672 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704 705, 706, 707, 708, 709, 710, 711, 712, 729, 730, 731, 732, 733, 734, 735, 736 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944 961, 962, 963, 964, 965, 966, 967, 968, 985, 986, 987, 988, 989, 990, 991, 992 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200 1217, 1218, 1219, 1220, 1221, 1222, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1230, 1231, 1232 1249, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1258, 1259, 1260, 1261, 1262, 1263, 1264 1281, 1282, 1283, 1284, 1285, 1286, 1287, 1288, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312 1329, 1330, 1331, 1332, 1333, 1334, 1335, 1336, 1337, 1338, 1339, 1340, 1341, 1342, 1343, 1344


255

1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1369, 1370, 1371, 1372, 1373, 1374, 1375, 1376 1393, 1394, 1395, 1396, 1397, 1398, 1399, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408 1473, 1474, 1475, 1476, 1477, 1478, 1479, 1480, 1481, 1482, 1483, 1484, 1485, 1486, 1487, 1488 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520 1537, 1538, 1539, 1540, 1541, 1542, 1543, 1544, 1545, 1546, 1547, 1548, 1549, 1550, 1551, 1552 1569, 1570, 1571, 1572, 1573, 1574, 1575, 1576, 1577, 1578, 1579, 1580, 1581, 1582, 1583, 1584 ** Section: FLANGE *Shell Section, elset=_PickedSet4, material=Flange 8.6, 5 ** Region: (WEB:Picked) *Elset, elset=_I3, internal 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640


256

649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072 1089, 1090, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152 1169, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 1182, 1183, 1184 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1210, 1211, 1212, 1213, 1214, 1215, 1216 1233, 1234, 1235, 1236, 1237, 1238, 1239, 1240, 1241, 1242, 1243, 1244, 1245, 1246, 1247, 1248 1265, 1266, 1267, 1268, 1269, 1270, 1271, 1272, 1273, 1274, 1275, 1276, 1277, 1278, 1279, 1280 1289, 1290, 1291, 1292, 1293, 1294, 1295, 1296, 1297, 1298, 1299, 1300, 1301, 1302, 1303, 1304 1313, 1314, 1315, 1316, 1317, 1318, 1319, 1320, 1321, 1322, 1323, 1324, 1325, 1326, 1327, 1328 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1362, 1363, 1364, 1365, 1366, 1367, 1368 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384, 1385, 1386, 1387, 1388, 1389, 1390, 1391, 1392


257

1409, 1410, 1411, 1412, 1413, 1414, 1415, 1416, 1417, 1418, 1419, 1420, 1421, 1422, 1423, 1424 1425, 1426, 1427, 1428, 1429, 1430, 1431, 1432, 1433, 1434, 1435, 1436, 1437, 1438, 1439, 1440 1441, 1442, 1443, 1444, 1445, 1446, 1447, 1448, 1449, 1450, 1451, 1452, 1453, 1454, 1455, 1456 1457, 1458, 1459, 1460, 1461, 1462, 1463, 1464, 1465, 1466, 1467, 1468, 1469, 1470, 1471, 1472 1489, 1490, 1491, 1492, 1493, 1494, 1495, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504 1521, 1522, 1523, 1524, 1525, 1526, 1527, 1528, 1529, 1530, 1531, 1532, 1533, 1534, 1535, 1536 1553, 1554, 1555, 1556, 1557, 1558, 1559, 1560, 1561, 1562, 1563, 1564, 1565, 1566, 1567, 1568 1585, 1586, 1587, 1588, 1589, 1590, 1591, 1592, 1593, 1594, 1595, 1596, 1597, 1598, 1599, 1600 ** Section: WEB *Shell Section, elset=_I3, material=Web 6.4, 5 ** Region: (CONCRETE:Picked) *Elset, elset=_PickedSet3, internal, generate 1601, 1880, 1 ** Section: CONCRETE *Shell Section, elset=_PickedSet3, material=Concrete 150., 5 *Rebar Layer BAR1, 28.27, 200., -40., Reineforcement, 90., 1 BAR2, 28.27, 200., -40., Reineforcement, 0., 1 *End Instance ** *Nset, nset=_PickedSet119, internal, instance=Final-1 161, 169, 170, 175, 178, 181, 182, 185, 186, 189, 909, 910, 911, 928, 929, 930 955, 956, 957, 973, 974, 975, 1011, 1051, 1052, 1053, 1054, 1079, 1103, 1104, 1105, 1106 *Elset, elset=_PickedSet119, internal, instance=Final-1 1389, 1390, 1391, 1392, 1412, 1416, 1420, 1424, 1457, 1461, 1465, 1469, 1533, 1534, 1535, 1536 1620, 1640, 1660, 1680, 1700, 1720, 1740, 1741, 1761, 1781, 1801, 1821, 1841, 1861 *Nset, nset=_PickedSet120, internal, instance=Final-1 5, 6, 28, 201, 273 *Elset, elset=_PickedSet120, internal, instance=Final-1 15, 16, 111, 112 *Nset, nset=_PickedSet123, internal, instance=Final-1, generate 186, 189, 1 *Nset, nset=_PickedSet127, internal, instance=Final-1 20, *Elset, elset=__PickedSurf115_E3, internal, instance=Final-1 63, 64, 95, 96 *Surface, type=ELEMENT, name=_PickedSurf115, internal __PickedSurf115_E3, E3 *Elset, elset=__PickedSurf116_E3, internal, instance=Final-1 63, 64 *Surface, type=ELEMENT, name=_PickedSurf116, internal __PickedSurf116_E3, E3 *Elset, elset=__PickedSurf124_SPOS, internal, instance=Final-1


258

1601, 1602, 1603, 1604, 1605, 1606, 1607, 1608, 1609, 1610, 1611, 1612, 1613, 1614, 1615, 1616 1617, 1618, 1619, 1620, 1621, 1622, 1623, 1624, 1625, 1626, 1627, 1628, 1629, 1630, 1631, 1632 1633, 1634, 1635, 1636, 1637, 1638, 1639, 1640, 1741, 1742, 1743, 1744, 1745, 1746, 1747, 1748 1749, 1750, 1751, 1752, 1753, 1754, 1755, 1756, 1757, 1758, 1759, 1760, 1761, 1762, 1763, 1764 1765, 1766, 1767, 1768, 1769, 1770, 1771, 1772, 1773, 1774, 1775, 1776, 1777, 1778, 1779, 1780 *Surface, type=ELEMENT, name=_PickedSurf124, internal __PickedSurf124_SPOS, SPOS *Elset, elset=__PickedSurf126_E3, internal, instance=Final-1 95, 96 *Surface, type=ELEMENT, name=_PickedSurf126, internal __PickedSurf126_E3, E3 *Elset, elset=__PickedSurf130_E3, internal, instance=Final-1 63, 64 *Surface, type=ELEMENT, name=_PickedSurf130, internal __PickedSurf130_E3, E3 *Elset, elset=__PickedSurf135_SPOS, internal, instance=Final-1 161, 162, 163, 164, 165, 166, 167, 168, 185, 186, 187, 188, 189, 190, 191, 192 321, 322, 323, 324, 325, 326, 327, 328, 345, 346, 347, 348, 349, 350, 351, 352 385, 386, 387, 388, 389, 390, 391, 392, 409, 410, 411, 412, 413, 414, 415, 416 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560 641, 642, 643, 644, 645, 646, 647, 648, 665, 666, 667, 668, 669, 670, 671, 672 705, 706, 707, 708, 709, 710, 711, 712, 729, 730, 731, 732, 733, 734, 735, 736 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880 961, 962, 963, 964, 965, 966, 967, 968, 985, 986, 987, 988, 989, 990, 991, 992 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200 1281, 1282, 1283, 1284, 1285, 1286, 1287, 1288, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312 1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1369, 1370, 1371, 1372, 1373, 1374, 1375, 1376 1473, 1474, 1475, 1476, 1477, 1478, 1479, 1480, 1481, 1482, 1483, 1484, 1485, 1486, 1487, 1488 1505, 1506, 1507, 1508, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520 *Surface, type=ELEMENT, name=_PickedSurf135, internal __PickedSurf135_SPOS, SPOS


259

** Constraint: Constraint-1 *Tie, name=Constraint-1, adjust=yes _PickedSurf135, _PickedSurf124 *End Assembly *Imperfection, file=beam1abuckle , step=1 1, 1 ** ** MATERIALS ** *Material, name=Concrete *Concrete 0.1, 0. 4.199, 0.0002 7.34, 0.00035 10.458, 0.0005 13.531, 0.00065 16.529, 0.0008 19.417, 0.00095 22.156, 0.0011 24.706, 0.00125 27.027, 0.0014 29.084, 0.00155 30.85, 0.0017 32.305, 0.00185 33.439, 0.002 34.256, 0.00215 34.765, 0.0023 35., 0.0025 0., 0.02 *Failure Ratios 1.16, 0.2, 1.28, 0.3333 *Tension Stiffening 1., 0. 0., 0.004 *Density 2.3544e-05, *Elastic 20000., 0.2 *Material, name=Flange *Density 7.7e-05, *Elastic 210000., 0.3 *Plastic 265.2, 0. 305.177, 0.000637143 323.268, 0.00123714 334.889, 0.00173714 379.95, 0.00473714 406.751, 0.00773714 424.281, 0.0107371 435.288, 0.0137371 440.942, 0.0167371 442., 0.0187371 442., 0.0287371 442., 0.0487371 442., 0.0687371


260

442., 0.0887371 442., 0.108737 442., 0.128737 442., 0.148737 353.6, 0.158737 265.2, 0.168737 176.8, 0.178737 88.4, 0.188737 0.001, 0.198737 *Material, name=Reinforcement *Elastic 210000., 0.3 *Plastic 265.2, 0. 305.177, 0.000637143 323.268, 0.00123714 334.889, 0.00173714 379.95, 0.00473714 406.751, 0.00773714 424.281, 0.0107371 435.288, 0.0137371 440.942, 0.0167371 442., 0.0187371 442., 0.0287371 442., 0.0487371 442., 0.0687371 442., 0.0887371 442., 0.108737 442., 0.128737 442., 0.148737 353.6, 0.158737 265.2, 0.168737 176.8, 0.178737 88.4, 0.188737 0.001, 0.198737 *Material, name=Web *Density 7.70085e-05, *Elastic 210000., 0.3 *Plastic 265.2, 0. 305.177, 0.000637143 323.268, 0.00123714 334.889, 0.00173714 379.95, 0.00473714 406.751, 0.00773714 424.281, 0.0107371 435.288, 0.0137371 440.942, 0.0167371 442., 0.0187371 442., 0.0287371 442., 0.0487371 442., 0.0687371 442., 0.0887371 442., 0.108737 442., 0.128737


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442., 0.148737 353.6, 0.158737 265.2, 0.168737 176.8, 0.178737 88.4, 0.188737 0.001, 0.198737 ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES, inc=1000 *Static, riks 5., 20., 0.0002, , , ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Symmetry/Antisymmetry/Encastre *Boundary _PickedSet119, XSYMM ** Name: BC-2 Type: Displacement/Rotation *Boundary _PickedSet120, 2, 2 _PickedSet120, 3, 3 ** Name: BC-3 Type: Displacement/Rotation *Boundary _PickedSet123, 3, 3 ** ** LOADS ** ** Name: Load-1 Type: Concentrated force *Cload _PickedSet127, 2, -500. ** ** OUTPUT REQUESTS ** *Restart, write, frequency=0 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field, variable=PRESELECT ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history, variable=PRESELECT *End Step

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