DEVELOPMENT OF A COMPONENT-BASED FINITE...

UNIVERSITY OF SHEFFIELD

Department of Civil and Structural Engineering

& School of Architectural Studies

DEVELOPMENT OF A COMPONENT-BASED FINITE

ELEMENT FOR STEEL BEAM-TO-COLUMN

CONNECTIONS AT ELEVATED TEMPERATURES

by Florian Mauricius Block

A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of

Philosophy

December 2006

Summary

A component based connection element has been developed which is able to predict

the behaviour of bolted flush and extended endplate beam-to-column connections.

The new element represents the behaviour of such connections under the influence of

bending moment, normal force in the beam and column, and increasing and

decreasing temperatures. It has been implemented into the specialised finite element

software Vulcan which has been developed to predict the behaviour of steel,

composite and concrete structures in fire.

The research presented in the thesis has been inspired by the observation that beam-

to-column connections in a fire are exposed to a combination of forces and moments,

significantly different to the shear loading of the connection assumed in ambient

design. The additional moments and axial forces in the beam originate from restraint

thermal expansion, large vertical deflections and rotations and the effects of cooling

on a plastically deformed structure. As connections are normally not designed to

withstand those additional forces they could fail to transfer the beam forces to the

column which could lead to a progressive collapse of the frame structure.

Due to the combination of axial load and moment, together with the large variety of

possible connections, the Component Method has been used as a basis of the new

element. This method separates a joint into its zones of fundamental behaviour i.e.

tension, compression and shear which are then represented by force-displacement

curves calculated from simplified mathematical models. Finally, the joint is

reassembled as a spring model in which each zone is represented as a spring. The

spring model responds very similar to the real connection.

A large part of this thesis concentrates on the development of a simplified component

model for the compression zone in the column web including the effects of elevated

temperatures and axial load in the column. Experimental, numerical, statistical and

mechanical studies have been used to develop and validate this model.

The new connection element has been compared with a number of ambient and

elevated temperature experiments on connections with good success.

Finally, a plane frame connected with different endplate connections has been

analysed during the heating and cooling phase of a fire, showing the vulnerability of

connections and bolts in particular during the cooling phase of a fire.

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Table of Contents

Table of Contents ....................................................................................................... i

List of Figures ......................................................................................................... vii

List of Tables ......................................................................................................... xiv

Acknowledgment .....................................................................................................xv

1 Introduction ....................................................................................................... 1

1.1 Fire resistance of buildings......................................................................... 1

1.2 Full frame analysis..................................................................................... 3

1.3 Scope of Research...................................................................................... 4

1.4 Thesis layout.............................................................................................. 5

2 Modelling of semi-rigid joints in fire ................................................................. 6

2.1 Joint definition........................................................................................... 6

2.2 Joint classification...................................................................................... 6

2.2.1 Stiffness classification........................................................................ 7

2.2.2 Strength classification ........................................................................ 7

2.2.3 Rotation capacity classification .......................................................... 8

2.3 Idealisation of semi-rigid joints.................................................................. 8

2.3.1 Curve-fit models ................................................................................ 9

2.3.2 Mechanical models........................................................................... 10

2.3.3 Finite element models ...................................................................... 12

2.4 The ‘Component Method’........................................................................ 13

2.4.1 Identification of the active components ............................................ 14

2.4.2 Specification of the component characteristics ................................. 15

2.4.3 Assembly of the active joint components.......................................... 16

2.4.4 Moment – Normal force interaction.................................................. 20

2.5 Joint behaviour in fire .............................................................................. 23

2.5.1 Application of the Component Method in fire................................... 26

2.6 Conclusion............................................................................................... 28

3 Experimental work on the compression zone ................................................... 29

3.1 Introduction ............................................................................................. 29

3.2 Scope of the experiments ......................................................................... 29

3.3 Earlier experimental work on the compression zone................................. 30

3.4 Methodology of testing ............................................................................ 31

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3.5 Test rig .................................................................................................... 31

3.5.1 Reaction Frame and Loading devices ............................................... 32

3.5.2 Furnace ............................................................................................ 34

3.5.3 Measurement of the temperatures ..................................................... 35

3.5.4 Measurement of the displacements ................................................... 36

3.5.5 Measurement of the forces ............................................................... 37

3.5.6 Material and geometrical properties of the specimens....................... 37

3.6 Initial calibration tests.............................................................................. 38

3.6.1 Horizontal stiffness of reaction frame............................................... 38

3.6.2 Restraint of thermal expansion ......................................................... 39

3.6.3 Vertical stiffness of reaction frame ................................................... 39

3.6.4 Temperature Distribution Tests ........................................................ 41

3.7 Observations and Results ......................................................................... 42

3.7.1 Observations .................................................................................... 42

3.7.2 Temperature distribution in the specimens........................................ 43

3.7.3 Correction of test results – Thermal expansion of the loading plates . 44

3.7.4 Correction of test results – Elastic deformation of the loading plates 45

3.7.5 Testing speed ................................................................................... 47

3.7.6 Summary of the tests conducted ....................................................... 48

3.7.7 Force-displacement behaviour .......................................................... 50

3.8 Strain-rate effects on steel at elevated temperatures.................................. 54

3.9 Discussion and Conclusion ...................................................................... 58

4 Finite element modelling of the compression zone........................................... 60

4.1 Introduction ............................................................................................. 60

4.2 Previous FEM modelling of the compression zone ................................... 60

4.3 2-3D shell element model ........................................................................ 63

4.3.1 Modelling of the material for ambient temperatures.......................... 63

4.3.2 Modelling of the material for elevated temperatures ......................... 64

4.3.3 The ANSYS material model ............................................................. 65

4.3.4 Solution options ............................................................................... 65

4.3.5 Consideration of imperfections......................................................... 67

4.3.6 Geometry of the model..................................................................... 68

4.3.7 The finite element type used............................................................. 69

4.3.8 The boundary conditions and the load introduction used................... 69

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4.3.9 Mesh study....................................................................................... 70

4.4 Stress distributions in the 2-3D model...................................................... 72

4.5 Comparison of the numerical model with experimental data..................... 73

4.5.1 Spyrou’s experiments at elevated temperatures................................. 73

4.5.2 Comparison with the author’s test results ......................................... 75

4.6 Parametric study on the effects of axial load ............................................ 77

4.7 Further FEM study on the compression zone............................................ 79

4.7.1 Geometry of the 3D model ............................................................... 79

4.7.2 Load and Boundary Conditions ........................................................ 80

4.7.3 Pre-deformation in accordance with the eigenvalue buckling shape.. 80

4.7.4 Mesh study....................................................................................... 81

4.7.5 Deformed shape and stress and strain patterns in the 3D model ........ 85

4.8 Comparison of the axial load sensitivity of the two models ...................... 88

4.9 Discussion and Conclusion ...................................................................... 90

5 Simplified modelling of the compression zone................................................. 92

5.1 Introduction ............................................................................................. 92

5.2 Force-displacement curves at ambient temperature................................... 93

5.2.1 Force-displacement model after Tschemmernegg et al. .................... 93

5.2.2 Force-displacement model after Eurocode 3–1.8 .............................. 94

5.2.3 Force-displacement model after Kühnemund.................................... 94

5.2.4 Conclusion ....................................................................................... 95

5.3 The resistance of the compression zone at ambient temperature ............... 95

5.3.1 Statistical comparison of the resistance approaches at ambient

temperature...................................................................................................... 96

5.3.2 Resistance approach after Eurocode 3............................................... 98

5.3.3 Resistance approach using an empirical equation.............................. 99

5.3.4 Resistance approach after Block......................................................100

5.3.5 Resistance approach after Lagerqvist and Johansson .......................101

5.3.6 Resistance approach after Kühnemund ............................................104

5.3.7 Conclusion ......................................................................................105

5.4 Initial stiffness of the compression zone at ambient temperature .............105

5.4.1 Statistical comparison of existing approaches for the initial stiffness

106

5.4.2 Initial stiffness approach after Eurocode 3-1.8.................................107

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5.4.3 Initial stiffness approach after Aribert et al. .....................................108

5.4.4 Conclusion ......................................................................................110

5.5 The deformation capacity of the compression zone .................................110

5.5.1 Statistical comparison of existing approaches for the deformation

capacity 111

5.5.2 Deformation capacity approach after Huber and Tschemmernegg ...112

5.5.3 Deformation capacity approach after Vayas et al. ............................113

5.5.4 Deformation capacity approach after Block.....................................114

5.5.5 Conclusion ......................................................................................116

5.6 Design approaches at elevated temperatures............................................116

5.6.1 Spyrou’s approach to the compression zone at elevated temperature116

5.6.2 Statistical comparison of existing approaches at elevated temperatures

117

5.6.3 Resistance of the compression zone at elevated temperatures ..........117

5.6.4 Initial stiffness of the compression zone at elevated temperature .....119

5.6.5 Ductility of the compression zone at elevated temperature...............122

5.7 Force-displacement curve model for elevated temperatures.....................124

5.8 Validation of the simplified model at elevated temperatures....................126

5.8.1 Comparison between the simplified model and tests by Spyrou .......126

5.8.2 Comparison between the simplified model and tests by the author...129

5.9 Conclusion..............................................................................................132

6 The influence of axial load on the compression zone ......................................134

6.1 Introduction ............................................................................................134

6.2 Previous research ....................................................................................134

6.3 Proposed analytical approach for the ultimate load..................................138

6.3.1 Plastic hinge mechanism in the compression zone with axial load ...138

6.3.2 Consideration of the stability of the column web with axial load .....142

6.3.3 Validation of the new approach .......................................................144

6.4 Parametric study on the effects of axial load at high temperature.............146

6.4.1 Reduction of the ultimate load due to compression in the column....146

6.5 Reduction factor for the displacement under ultimate load ......................150

6.6 Comparison between the simplified model and the experiments ..............155

6.7 Conclusion and discussion ......................................................................157

7 The component based connection element ......................................................159

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7.1 Introduction ............................................................................................159

7.2 Selected previous connection elements at ambient temperature ...............159

7.3 The existing spring models in Vulcan......................................................161

7.4 The proposed connection element ...........................................................161

7.5 Derivation of the stiffness matrix of the connection element ...................163

7.5.1 The stiffness matrix.........................................................................163

7.5.2 Incorporation of the stiffness matrix into Vulcan .............................168

7.6 Relocation of the reference axis ..............................................................171

7.7 Spring Component model used................................................................173

7.7.1 Tension zone ...................................................................................174

7.7.2 Yield line approach for the effective length .....................................178

7.7.3 Compression zone ...........................................................................180

7.8 Ambient temperature behaviour of the connection element .....................181

7.8.1 Comparison of the connection element with Eurocode 3-1-Annex J 181

7.8.2 Comparison of the connection element with test results...................184

7.9 Elevated Temperature behaviour of the connection element ....................189

7.9.1 Temperature distribution .................................................................189

7.9.2 Comparison of the connection element with high temperature tests .190

7.9.3 Anisothermal connection responses of the example connections......192

7.10 Discussion and Conclusion .....................................................................195

8 Unloading and cooling of the connection element...........................................196

8.1 Unloading of the connection element at constant temperatures................196

8.2 Unloading of the connection element at changing temperatures...............199

8.2.1 The Reference Point concept ...........................................................199

8.2.2 Unloading and heating in tension.....................................................202

8.2.3 Unloading and heating in compression ............................................204

8.3 Cooling behaviour of the connection element..........................................206

8.3.1 Assumed material behaviour for bolts..............................................206

8.3.2 Example of a connection under cooling ...........................................207

8.4 Discussion and conclusion ......................................................................209

9 Preliminary application of the connection element ..........................................210

9.1 Connection element together with an isolated beam ................................210

9.1.1 Flush endplate with two bolt rows ...................................................212

9.1.2 Flush endplate connection with three bolt rows ...............................213

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9.1.3 Extended endplate ...........................................................................215

9.2 2D Sub-frame .........................................................................................217

9.2.1 Geometry and Loading....................................................................217

9.2.2 Results ............................................................................................218

9.2.3 Cooling response of the sub-frame ..................................................223

9.3 Conclusion..............................................................................................227

10 Discussion, conclusion and further recommendations .....................................228

10.1 Summary and discussion.........................................................................228

10.1.1 Step 1 of the development process of the connection element..........229




10.2 Recommendations for the usage of the connection element .....................234

10.3 Recommendations for further work.........................................................235

10.3.1 Extension to compression zone model .............................................235

10.3.2 Further development to the connection element...............................235

10.4 Concluding remark .................................................................................236

List of references....................................................................................................237

A. Temperature distribution during the test..........................................................250

B. Force-displacement curves..............................................................................252

C. Results from the parametric study...................................................................260

D. Experimental data used in the statistical analysis of the compression zone ......261

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List of Figures

Figure 2.1 Definition of joint and connection – double-sided joint configuration ....... 6

Figure 2.2: Stiffness classification of joints after EC3-1.8.......................................... 7

Figure 2.3: Active components of a joint with one extended endplate connection .... 14

Figure 2.4: Different complexities of component approximations ............................ 15

Figure 2.5: Spring model of an extended endplate joint after EC3-1.8 ..................... 16

Figure 2.6: Equivalent (a) and simplified (b) spring model after EC3-1.8 ................ 18

Figure 2.7: Moment-rotation curves after EC3-1.8................................................... 20

Figure 2.8: Spring model of an extended endplate joint for moment and axial force

after Cerfontaine and Jaspart (2002, 2005)............................................................... 22

Figure 2.9: Spring model of a flush endplate connection (a) and equivalent model (b)

after Leston-Jones ................................................................................................... 26

Figure 3.1: Overview of the test setup ..................................................................... 32

Figure 3.2: Lateral bracing system........................................................................... 34

Figure 3.3: Cross section of the furnace................................................................... 35

Figure 3.4: Thermocouple location in the cross section............................................ 36

Figure 3.5: Transducer arrangement ........................................................................ 37

Figure 3.6: Frame stiffness test ................................................................................ 40

Figure 3.7: Stiffness of the reaction frame ............................................................... 40

Figure 3.8: Initial temperature test ........................................................................... 42

Figure 3.9: Failure mode 1: Symmetric.................................................................... 43

Figure 3.10: Failure mode 2: Asymmetric................................................................ 43

Figure 3.11: Typical time-temperature curve ........................................................... 44

Figure 3.12: Assumed temperature distribution in load-introduction plate................ 45

Figure 3.13: Elastic deformation of the loading plates ............................................. 46

Figure 3.14: Assumed points of rotation in Test 14.................................................. 49

Figure 3.15: Normalised resistance again temperature ............................................. 50

Figure 3.16: Force-displacement curves at 20°C...................................................... 51

Figure 3.17: Force-displacement curves at 450°C .................................................... 52



Figure 3.20: Various force-displacement curves ...................................................... 54

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Figure 3.21: Overview of the tensile test rig (from Renner (2005)) .......................... 55

Figure 3.22: Experimental isothermal stress-strain curves after Renner (2005) ........ 56

Figure 3.23: Influence of the strain-rate on the material strength at elevated

temperatures after Renner (2005)............................................................................. 57

Figure 4.1: Ambient temperature material model ..................................................... 64

Figure 4.2: Elevated temperature model for S275 mild steel after EC3-1.2 .............. 65

Figure 4.3: Newton-Raphson and the Arc-Length for a single degree of freedom (from

ANSYS 8.0 User Manual) ....................................................................................... 67

Figure 4.4: Typical imperfection of the finite element model with sine waves ......... 68

Figure 4.5: Thickness estimation of the 2-3 D model ............................................... 69

Figure 4.6: Mesh of the 2-3D model (with (a) and without (b) element thickness) ... 69

Figure 4.7: Element meshes used for convergence study.......................................... 70

Figure 4.8: Mesh study on the 2-3D model .............................................................. 71

Figure 4.9: Mesh study on the 2-3D model - detail .................................................. 71

Figure 4.10: Stresses in transverse direction (x) under the peak load........................ 72

Figure 4.11: Stresses in longitudinal direction (z) under the peak load..................... 73

Figure 4.12: Comparison of the FEM model and Spyrou’s tests on UC203x203x46

sections ................................................................................................................... 74

Figure 4.13: Comparison of the FEM model with the author’s tests at 20°C ............ 75

Figure 4.14: Comparison of the FEM model with Test 8 at 558°C ........................... 76

Figure 4.15: Comparison of the FEM model with Test 9 at 591°C ........................... 76

Figure 4.16: Comparison of the FEM model with Tests 13 at 546°C........................ 77

Figure 4.17: Force-displacement curves for UC 203x203x46 at different temperatures

in combination with different axial loads ................................................................. 78

Figure 4.18: Typical element mesh of the 3D models (only one quarter is shown) ... 80

Figure 4.19: Typical finite element model with eigenvalue buckling imperfections . 81

Figure 4.20: 3D model with different numbers of elements through the flange

thickness ................................................................................................................. 82

Figure 4.21: Influence of the number of elements through the flange thickness........ 82

Figure 4.22: 3D model with different numbers of elements through the web thickness

................................................................................................................................ 83

Figure 4.23: Influence of the number of elements through the web thickness........... 83

Figure 4.24: Influence of the number of elements through the flange and web

thickness on the peak load ....................................................................................... 84

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thickness on the peak displacement ......................................................................... 84

Figure 4.26: Influence of the magnitude of imperfection.......................................... 85

Figure 4.27: Deformed shape of test specimen of Test 9 .......................................... 85

Figure 4.28: Deformed shape of the 3D model......................................................... 86

Figure 4.29: Stresses in transverse direction (x) under the peak load........................ 86

Figure 4.30: Stresses in longitudinal direction (z) under the peak load..................... 87

Figure 4.31: Von Mises strains at peak load............................................................. 87

Figure 4.32: Comparison between the force-displacement curves from the 2-3D and

the 3D model........................................................................................................... 88

Figure 4.33: Effects of axial load on the force-displacement curves (3D model) ...... 89

Figure 4.34: The axial load reduction factors found in the 2-3D and the 3D model .. 89

Figure 5.1: Force-displacement curve after Tschemmernegg et al............................ 93

Figure 5.2: Force-displacement curve after EC3-1.8 ................................................ 94

Figure 5.3: Force-displacement curve after Kühnemund .......................................... 95

Figure 5.4: Comparison of the resistance after EN 1993-1-8:2005 with tests............ 99

Figure 5.5: Comparison of the resistance after the empirical equation with tests .....100

Figure 5.6: Comparison of the resistance after Block with tests ..............................101

Figure 5.7: Assumed plastic mechanism in the column flange after Lagerqvist .......102

Figure 5.8: Comparison of the resistance after Lagerqvist et al. with tests ..............103

Figure 5.9: Comparison of the resistance after Kühnemund with tests ....................105

Figure 5.10: Comparison of the initial stiffness after EN 1993-1-8 with tests..........108

Figure 5.11: Comparison of the initial stiffness after Aribert et al. with tests ..........109

Figure 5.12: Comparison of the initial stiffness after Aribert and Younes with tests109

Figure 5.13: Definition of the deformation limit of the compression zone...............110

Figure 5.14: Comparison of the displacement capacity after Huber and

Tschemmernegg with tests .....................................................................................112

Figure 5.15: Comparison of the displacement capacity after Vayas et al. with tests 114

Figure 5.16: Comparison of the displacement capacity after Block with tests .........115

Figure 5.17: Sequential yielding of the compression zone after Spyrou...................117

Figure 5.18: Comparison of the resistance after Lagerqvist and Johansson with tests at

elevated temperatures .............................................................................................118

Figure 5.19: Reduction of the initial stiffness with temperature ..............................120

Figure 5.20: Elevated-temperature initial stiffness after Aribert and Younes...........121

x

Figure 5.21: Corrected high-temperature initial stiffness after Aribert and Younes .121

Figure 5.22: Reduction of the ductility with temperature ........................................122

Figure 5.23: Comparison of the displacement capacity after Block with high

temperature tests.....................................................................................................123

Figure 5.24: Comparison of the corrected displacement capacity after Block with high

temperature tests.....................................................................................................124

Figure 5.25: Force-displacement curve for the compression zone at high temperature

...............................................................................................................................125

Figure 5.26: Comparison between the simplified model and tests on UC152x152x30

sections ..................................................................................................................127


sections ..................................................................................................................127


sections ..................................................................................................................128


sections ..................................................................................................................128

Figure 5.30: Comparison between the simplified model and Test 4.........................129

Figure 5.31: Comparison between the simplified model and Test 8.........................130

Figure 5.32: Comparison between the simplified model and Test 12.......................130



Figure 6.1: Comparison between tests and different reduction factors kwc ...............137

Figure 6.2: Reduced plastic moment of the inner plastic hinge due to axial load .....138

Figure 6.3: Notations of the T-section in the outer plastic hinge..............................139

Figure 6.4: Reduced plastic moment of the outer plastic hinge due to axial load .....139

Figure 6.5: Plastic mechanism in the column flange ...............................................141

Figure 6.6: Comparison of tests on HEA 240 sections with the new approach ........144

Figure 6.7: Comparison of tests on HEB 240 sections with the new approach.........145

Figure 6.8: Comparison between ambient temperature test results with axial load and

the new simplified model .......................................................................................146

Figure 6.9: Reduction of the peak load due to axial loads in an UC203x203x46 .....147

Figure 6.10: Reduction of the peak load due to axial loads in an UC152x152x37 ...148

Figure 6.11: Reduction of the peak load due to axial loads in an UC254x254x167 .148

xi

Figure 6.12: Comparison of the proposed approach with tests at elevated temperatures

without axial load...................................................................................................149

Figure 6.13: Comparison of the proposed approach with tests at elevated temperatures

including axial load ................................................................................................150

Figure 6.14: Comparison of test series 1 and 2 on HEA 240 with the approaches....151

Figure 6.15: Comparison of test series 1 and 2 on HEB 240 with the approaches....151

Figure 6.16: Reduction of the ductility due to axial loads in an UC203x203x46 .....153

Figure 6.17: Reduction of the ductility due to axial loads in an UC152x152x37 .....153

Figure 6.18: Reduction of the ductility due to axial loads in an UC254x254x167 ...154

Figure 6.19: Comparison of the new approach with tests at 20°C (a) and over 450°C

(b) with axial load ..................................................................................................155

Figure 6.20: Comparison of the simplified model and Test 7 at a LR of 0.36 ..........156

Figure 6.21: Comparison of the simplified model and Test 9 at a LR of 0.41 ..........156

Figure 6.22: Comparison of the simplified model and Test 13 at a LR of 0.30 ........156



Figure 7.1: Assumed position and components of the new connection element .......162

Figure 7.2: The forces and displacement on the 2D connection element..................164

Figure 7.3: Deformation modes of node i of the connection element.......................164

Figure 7.4: Stiffness matrix test – normal force ......................................................169

Figure 7.5: Comparison between the new connection element and theory...............170


Figure 7.7: Offset arrangement of the connection element ......................................172


Figure 7.9: The three failure modes of a T-stub ......................................................174

Figure 7.10: Assembly of the individual springs to the final tension zone spring.....175

Figure 7.11: Tension zone model and tests CA1 and CA4 at 660°C and 530°C.......176

Figure 7.12: Tension zone model and tests CB1 and CB5 at 650°C and 505°C.......177

Figure 7.13: Tension zone model and tests CE1 and CE4 at 610°C and 410°C .......178

Figure 7.14: Considered yield line patterns in a column flange or an endplate ........179

Figure 7.15: Yield line patterns in the plate extension of an extend endplate...........179

Figure 7.16: Example endplate connections............................................................181

Figure 7.17: Comparison of the connection element and CoP for Type A ...............183

Figure 7.18: Comparison of the connection element and CoP for Type A, B, C ......183

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Figure 7.19: Comparison of the new element with test FS1a by Girão Coelho ........185



Figure 7.22: Comparison of the new element with the 20°C test by Leston-Jones ...187

Figure 7.23: Comparison of the new element with Test 1 by Bailey and Moore ......188

Figure 7.24: Comparison of the new element with Test 2 by Bailey and Moore ......188

Figure 7.25: Comparison of the new element with test BFEP 5 and BFEP 15 .........191

Figure 7.26: Comparison of the new element with test BFEP 10 and BFEP 20 .......191

Figure 7.27: High temperature behaviour of connection Type A – tp = 12 mm........192

Figure 7.28: High temperature behaviour of connection Type B .............................193

Figure 7.29: High-temperature behaviour of connection type C ..............................194

Figure 7.30: Summary of the failure temperature – load ratio relationships.............194

Figure 8.1: Hysteresis behaviour of the tension (a) and the compression zone (b) ...197

Figure 8.2: Force transfer between the different components ..................................198

Figure 8.3: Definition of the Reference Point and the Intersection Point .................200

Figure 8.4: Behaviour of connection Type A under tension-heating-unloading .......202

Figure 8.5: Force-displacement curves for tension zone in heating and unloading...203

Figure 8.6: F-δ curves for the tension zone heating-unloading example - detail ......203

Figure 8.7: Behaviour of connection Type A in compression-heating-unloading.....204

Figure 8.8: F-δ curves for connection Type A in compression-heating-unloading...205

Figure 8.9: F-δ curves for compression zone heating-unloading example - detail....205

Figure 8.10: Temperature-rotation plot of a connection in heating and cooling .......207

Figure 8.11: Temperature-spring displacement plot of a connection in cooling .......208

Figure 8.12: Spring force-temperature plot of a connection in cooling ....................208

Figure 9.1: Isolated beam with connection elements ...............................................211

Figure 9.2: Analysed endplate connections .............................................................211

Figure 9.3: Flush endplate (two bolt rows) at a LR = 0.6 (simple beam) .................212

Figure 9.4: Flush endplate (two bolt rows) at the ‘real’ LR = 0.6 ............................213

Figure 9.5: Flush endplate (three bolt rows) at a LR = 0.6 (simple beam) ...............214

Figure 9.6: Flush endplate (three bolt rows) at the ‘real’ LR = 0.6 ..........................214

Figure 9.7: Extended endplate (three bolt rows) at LR = 0.6 (simple beam) ............215

Figure 9.8: Extended endplate (three bolt rows) at the ‘real’ LR = 0.6 ....................216

Figure 9.9: Restraint sub-frame ..............................................................................218

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Figure 9.10: Vertical displacement at mid-span of the heated beam ........................219

Figure 9.11: Temperature (Tb) - rotation curve of the connection element ..............220

Figure 9.12: Axial force in the connections and the heated beam ............................221

Figure 9.13: Moment at the connections and at mid-span of the heated beam .........222

Figure 9.14: Connection component forces.............................................................223

Figure 9.15: Vertical deflection at mid span of the beam ........................................224

Figure 9.16: Temperature-rotation curve including the cooling phase .....................224

Figure 9.17: Axial force in the connections and the heated beam ............................225

Figure 9.18: Mid-span bending moment in the heated beam ...................................226

Figure 9.19: Connection and beam-end moment .....................................................226

Figure B.1: Force-Displacement curve of Test 1 at 446°C and 265 kN axial load ...252


Figure B.3: Force-Displacement curve of Test 3 at 20°C and 394 kN axial load .....253

Figure B.4: Force-Displacement curve of Test 4 at 447°C and 3 kN axial load .......253




Figure B.8: Force-Displacement curve of Test 8 at 553°C and 2 kN axial load .......255


Figure B.10: Force-Displacement curve of Test 10 at 20°C and 265 kN axial load .256



Figure B.13: Force-Displacement curve of Test 13 at 546°C and 266kN axial load 258


Figure B.15: Force-Displacement curve of Test 16 at 448°C and 274kN axial load 259


xiv

List of Tables

Table 2.1: Active components in an extend endplate joint........................................ 14

Table 3.1: Material properties of the specimens....................................................... 38

Table 3.2: Temperature distribution across the section............................................. 44

Table 3.3: Loading speed of the tests ....................................................................... 47

Table 3.4: Summary of the tests conducted.............................................................. 48

Table 3.5: Stress reduction factors for 1% and 2% strain ......................................... 57

Table 3.6: Stress reduction factors for 5% and UTS strain ....................................... 58

Table 5.1: Statistical comparison of ambient-temperature design approaches........... 97

Table 5.2: Statistical comparison of ambient-temperature stiffness approaches.......106

Table 5.3: Statistical comparison of the displacement approaches...........................111

Table 5.4: Deformation capacity after Tschemmernegg and Huber .........................112

Table 5.5: Statistical comparison of elevated-temperature resistance approaches ....118

Table 5.6: Statistical comparison of elevated-temperature stiffness approaches ......119

Table 5.7: Statistical comparison of elevated-temperature resistance approaches ....122

Table 6.1: Statistical comparison of the new approach and tests at 20°C.................145

Table 6.2: Comparison of the new approach and tests at high temperatures ............149

Table 6.3: Statistical comparison of the new approach and tests with axial load......154

Table 7.1: Summary of the connections behaviour at ambient temperature .............182

Table A.1: Temperature distribution of the experiments 1-8 ...................................250

Table A.2: Temperature distribution of the experiments 9-18 .................................251

Table C.1: Results from the parametric study on the compression zone ..................260

Table D.1: Ambient temperature tests data without axial load.................................261

Table D.2: Ambient temperature tests data with axial load .....................................265

Table D.3: Elevated temperature tests data without axial load.................................266

xv

Acknowledgment

The author would like to thank Professor Ian Burgess, Dr. Buick Davison and

Professor Roger Plank for their supervision and support throughout this research

project. Without their guidance and excellent human approach, this project would

have not been completed in time. The financial support of Buro Happold Ltd and the

free steel from Corus is gratefully acknowledged.

Special thanks go to the technical staff at the University of Sheffield for their advice,

comments and commitment of time and resources in the not always easy development

of the experimental setup and the conduction of the tests. Furthermore, my research

colleagues in Room D120 shall be thanked for their company and support during my

time in Sheffield.

Very special thanks go to my parents for their support and trust without I would have

never reached this level and finally to Sonja and my daughter Thalia, who was born

during the course of this work, for their continuing support and joy.

Declaration

Except where specific reference has been made to the work of others, this thesis is the

result of my own work. No part of it has been submitted to any University for a

degree, diploma, or other qualification.

Florian Mauricius Block

Chapter 1: Introduction

1

1 Introduction

From the beginnings of humanity, fire was used to serve the needs of humans, but it

also formed one of the greatest dangers to life and property. This fear of fire remains

throughout the development and civilization of humans up to the present day,

although today the statistical likelihood of dying through a fire in a building is a lot

smaller than that of death in a road accident or as a result of smoking related illnesses.

This is partially the achievement of the high standard of fire safety implemented in

current building regulations.

1.1 Fire resistance of buildings

In recognition of the danger of fire to life, the authorities throughout the world have

developed requirements and guidelines to minimise the risk of death due to building

fires. As an example, the British Building Regulations (ODPM, 2000) state in section

B3.-(1) that “The building shall be designed and constructed so that, in the event of

fire, its stability will be maintained for a reasonable period.”

Buildings require different fire resistance periods in accordance with the risk to life

and difficulty for the fire brigade to carry out seek and rescue operations. This risk is

converted into fire resistance periods which are dependent on the type of usage and

the height of the building. In the UK, these fire resistance periods are specified in

Approved Document B (ODPM, 2000) (ADB), which is an interpretation of the

British Building Regulations concerned with fire safety in buildings.

The traditional way of achieving the required fire resistance periods is based on the

fire performance of single structural elements and the application of fire protection to

the members. This procedure is outlined in ADB, which states in Section B3.ii that

“The fire resistance of an element of construction is a measure of its ability to

withstand the effects of fire in one or more ways, as follows:

a. resistance to collapse, ie the ability to maintain loadbearing capacity […];

b. resistance to fire penetration, ie an ability to maintain the integrity of the

element;

c. resistance to transfer of excessive heat, ie an ability to provide insulation from

high temperatures.”


2

Each of these three criteria is tested experimentally in the so-called ‘Standard Fire

Test’, which is specified in BS476 Part 21 (BSI, 1999). In this standard the testing

procedure, including a time-gas temperature curve, the so-called ‘Standard Fire

Curve’, the geometry of the specimens, the loading conditions and the failure criteria,

in accordance with the three points outlined above, are specified. During the test,

these failure criteria are closely monitored and if any of them is violated the time with

respect to the ‘Standard Fire Curve’ is measured, and this time is the fire resistance

for the criterion. If the failure time in the Standard Fire Test is less than the required

fire resistance period for the structural member some sort of fire protection has to be

applied to the member, thus delaying the time at which the member reaches

temperatures high enough to violate the failure criterion.

However, in the UK the Building Regulations do not prescribe the use of this

traditional method, which opens up the possibility of achieving the required structural

performance with alternative means; i.e. through calculations and engineering

judgement. These alternative means are commonly called Structural Fire

Engineering. Structural Fire Engineering follows a performance-based route

assessing the real behaviour of the structure during the cause of a fire. It is used more

and more throughout the world. The advantage of a performance-based design is that

it can be tailored to the building, accounting for the exact geometry, the gravity

loading, the fire load, natural fire behaviour and the actual behaviour of the structure.

This helps to increase the safety of the building and can also reduce the required

amount of fire protection to the structure, which can reduce the cost of a building

considerably, not only by direct costs for material and labour but also in terms of

construction time and therefore less interest costs for borrowed capital. Furthermore,

the maintenance costs of the building reduce if the amount of fire protection measures

is reduced and the fire resistance is built into the structure.

However, performance-based design also brings a lot more work and responsibility to

the engineer. Instead of just following the prescribed rules, it is necessary to

understand and calculate the real behaviour of a structure in detail, which demands

great skill and knowledge by the engineer and also requires valid design methods and

tools.


3

The four basic steps of performance-based structural fire engineering are:

1. specification of the failure criteria;

2. specification of the fire load and the fire behaviour;

3. a heat transfer analysis from the fire to the structure;

4. calculation of the structural performance during the full duration of the fire.

In this work, only the fourth point shall be considered. In general, this extends from

the calculation of single elements up to the prediction of the behaviour of multi-storey

3D frames including beams, columns and floors. The latter can only be analysed with

the help of non-linear finite element programs.

1.2 Full frame analysis

The development of finite element programs capable of predicting the response of

structures in fire started over 20 years ago at different research institutions, mainly in

Europe. A series of full-scale fire tests at the BRE large test facility at Cardington in

the mid 1990s gave researchers validation cases for their numerical models, and from

there on the development speed increased significantly, helped by the rapid

development in affordable computers enabling the analysis of larger and more

complex models.

A good overview of the different programs used to model structures at elevated

temperatures is given by Wang (2002). He divides the available codes into two

different groups; the first are special programs developed at university and research

organisations and the other group consists of commercial finite element programs.

Into the first group falls the non-linear finite element software ADAPTIC developed

at Imperial College in London, UK by Izzuddin. Initially, the program was used to

predicted the ambient-temperature dynamic behaviour of frame structures, and was

later extended to elevated temperatures by Song, Izzuddin and Elghazouli. The

program was used to model the Cardington fire tests. Another specialist program is

FEAST developed at the University of Manchester, UK by Liu. This program

includes shell, solid and contact elements, which makes detailed modelling of joints in

fire possible. The program SAFIR developed by Franssen at the University of Liége,

Belgium is also a finite element software specialising on steel and composite

structures in fire. It includes 2D and 3D solid elements as well as shell and beam

elements. Whereas in other programs the temperature distribution in the structure has

to be calculated external to the software, SAFIR combines a thermal and structural


4

analysis. The finite element code Vulcan has been developed at the University of

Sheffield, UK, and is capable of predicting the behaviour of steel, composite and

concrete structures at elevated temperatures. The program includes 3D beam-column-

, spring-, shear connector- and slab-elements. It has been extensively validated

against test results including all seven fire tests at Cardington.

In the second group of finite element software, the commercial programs ABAQUS

and DIANA are found. ABAQUS has been extensively used at the University of

Edinburgh, UK to model the Cardington fire tests and also large multi-storey

buildings similar to the World Trade Center. DIANA has been developed at TNO,

Netherlands and has the ability to model concrete at ambient and elevated

temperatures including discrete cracking analysis.

What is common to all the programs described above is that realistic connection and

joint behaviour is ignored. Only very recently has some work been published (Ramli

Sulong, 2005, 2006) on the incorporation of connection behaviour into ADAPTIC.

Although fire tests on isolated beam-to-column and beam-to-beam connections, which

will be described later, have shown acceptable performance of the connections and an

increase of rotational ductility, the tests at Cardington and frame analyses showed

force combinations during and after the fire which were far from the pure moments

investigated in the isolated connection tests. These additional forces were caused by

the restraint of thermal expansion of the beams in the initial stage of a fire generating

large axial compression forces, and at later stages by catenary forces when the beams

undergo large deflections. After the fire ended and the structure cooled down, large

tensile forces occurred in the beams, which damaged the connections. Therefore,

realistic full-frame analysis has to include a sophisticated representation of the

connection behaviour, otherwise the safety of the performance-based solution cannot

be guaranteed. The research work presented in this thesis is concerned with this

problem.

1.3 Scope of Research

The scope of this research project is the understanding and modelling of connections

in fire conditions. It is concentrated on the detailed behaviour of flush and extended

endplate connections, as well as the incorporation of these into global frame analysis

programs. In particular, the behaviour of unstiffened column webs under the


5

influence of transverse and longitudinal compression as well as high temperatures will

be studied experimentally, numerically and analytically. Further, these findings,

together with research results by Spyrou (2002, 2004a) will be used as a basis for a

two-noded finite element based on the principles of the Component Method. This

element is implemented into Vulcan, and is able to predict the behaviour of flush and

extended endplate connections with up to five bolt rows. It is further able to deal with

the moment-axial force combinations, cooling and unloading needed for the

successful prediction of beam-to-column connection behaviour in steel frames.

1.4 Thesis layout

The thesis is divided into ten chapters. The first chapter has presented an overview of

the requirements a structure has to fulfil to be safe with respect to fire, and

summarised performance-based design as well as the programs, which are used for

high-temperature structural modelling in the context of performance-based design.

Further, the importance of the joint and connection behaviour in fire has been

highlighted. The second chapter gives an introduction to joint modelling at ambient

and elevated temperatures. The calculation of moment-rotation curves using the

Component Method is shown in detail. Chapter Three, describes the experimental

programme performed to investigate the effects of axial column load on the

compression zone in the column web at high temperatures. The fourth chapter

describes numerical study of the compression zone, as well as a parametric study on

the influence of axial load in the column. Chapter Five, reviews the available

calculation approaches for the force-displacement behaviour of the compression zone

and develops from the most accurate and logical approaches one for elevated

temperatures. In chapter Six, the newly developed force-displacement model for the

compression zone is extended to account for axial column loading. Chapter Seven,

uses this model and an approach for the tension parts of a connection to develop a

finite connection element. In the eighth chapter, this element is extended to be able to

predict the unloading and cooling behaviour of a connection. Chapter Nine, uses this

new finite element in combination with beam-column elements to predict the

connection behaviour in a steel frame.

The final chapter discusses and concludes the presented work. It also gives further

recommendations on how to use and extend the new connection element.

Chapter 2: Modelling of semi-rigid joints in fire

6

2 Modelling of semi-rigid joints in fire

After the importance of joints in fire in general, and the incorporation of joint

behaviour into frame analysis programs in particular, was highlighted in the

introductory chapter, this chapter will focus on previous attempts to model joints at

ambient and elevated temperatures.

2.1 Joint definition

To avoid confusion the definition of a joint and a connection will be repeated here, as

it is stated in BS EN 1993-1-8:2005 (EC3-1.8) (CEN, 2005a). A ‘connection’ is

defined as the location where two or more members meet, and a ‘joint’ is defined as

the zone where two or more members meet. This means that a ‘connection’ is

considered as the parts which mechanically fasten the connected members; in the case

of an endplate connection these are the endplate, the bolts, the welds and the column

flange. A ‘joint’ further includes the column web and the beam-end. For example, a

beam-to-column joint can include up to four connections, two major-axis connections

attached to the column flanges and two minor-axis connections attached to the column

web. One can see a joint with two major-axis endplate connections in Figure 2.1

below

Figure 2.1 Definition of joint and connection – double-sided joint configuration

2.2 Joint classification

In general, joints are defined by their rotational stiffness, their strength and their

rotation capacity.

Joint

Connections


7

2.2.1 Stiffness classification

The rotational stiffness of a joint is defined as the initial slope of the moment-rotation

curve. In traditional design, joints have been assumed either ‘pinned’ or ‘rigid’ in

frame analyses. In the ‘pinned’ case the rotational stiffness is zero and no rotational

continuity exists between the beam and the column, and therefore no moment can be

transferred. The ‘rigid’ case refers to a joint with infinite rotational stiffness, which

allows no relative rotation between the beam and the column, and therefore the full

beam-end moment is transferred. Over eighty years of research on joints has shown

that the real behaviour is neither rigid, nor pinned. However, there are some cases

which are close to the two extremes: a fully welded joint with web stiffeners in the

column is almost rigid and a web cleat connection with slotted holes is almost pinned.

Nevertheless, the majority of joints, which are used economically in practice, are

semi-rigid. This means some relative rotation occurs between the beam and the

column, and a moment dependent on the stiffness of the connection and the connected

members is transferred. It is the task of the design engineer to find the balance

between sufficient rotational stiffness to reduce the mid-span moment and deflection

of the beam, which enables the use of larger spans or smaller beams, giving the client

more flexibility or reducing the storey height, respectively. To simplify design, EC3-

1.8 specifies the boundaries between joints which are assumed as pinned or rigid.

Figure 2.2: Stiffness classification of joints after EC3-1.8

2.2.2 Strength classification

The strength classification as it is used in EC3-1.8 refers to the moment resistance of

the connection in relation to the plastic bending moment of the connected members.

If the bending resistance of the joint is larger than or equal to the plastic moments of

M

Φ

Semi-rigid

Pinned

Rigid Rigid: Sj,ini ≥ kb EIb / Lb

where kb = 8 for braced frames and

kb = 25 for other frames

Pinned: Sj,ini ≤ 0,5 EIb / Lb

where Ib is the second moment of area

of a beam and

Lb is the span of a beam


8

the members then the joint is called ‘full-strength’. If the bending resistance of the

joint is less then 25% of one of the members and has sufficient rotational capacity it

can be nominally called ‘pinned’. A joint that falls between these two boundaries is

called a ‘partial-strength’ joint. Most endplate connections used in practice fall into

this range, which means that, if plastic design is used, the plastic hinges will form in

the joints and not in the adjacent members. Therefore, sufficient rotational capacity in

the joint is required to form a plastic mechanism and thereby to achieve the assumed

plastic moment at the mid-span of the beam.

2.2.3 Rotation capacity classification

The ductility of a joint represents the ability to maintain its plastic moment over a

sufficient rotation. If this rotation capacity is large enough, to enable the development

of a plastic mechanism in the adjacent members the joint is categorised as Class 1 –

‘ductile’, in accordance with Jaspart (2000) with reference to member classes. The

lower bound of the ductility classification is Class 3, indicating ‘brittle’ joint

behaviour, and should only be used in elastic frame design. Class 2 lies between the

ductile and the brittle behaviour and is called ‘semi-ductile’. However, the

boundaries between the classes are not defined generally. In EC3-1.8, the rotational

capacity or ductility of joints is treated in a very approximate manner, reflecting the

sparse research conducted in the field before the publication of the document. It has

only been in recent years that researchers have focused on the available ductility of a

joint. Girão Coelho and Simões da Silva (2001), Kühnemund (2003), Girão Coelho

(2004), Beg et al. (2004) and Girão Coelho et al. (2005) all have used the Component

Method, which will be described later, to predict the ductility of semi-rigid joints.

The first two authors can be particularly recommended to interested readers for

further studies.

Now that the general terms used in joint design have been explained, an overview of

the different ways in which joints and connections can be represented in a frame

analysis will be given.

2.3 Idealisation of semi-rigid joints

In general, five different ways of representing the moment-rotation response of semi-

rigid joints can be categorised:


9

1. mathematical expressions – curve-fit models,

2. simplified analytical models,

3. mechanical models – spring models,

4. finite element models and

5. macro-element models.

This is one more than in the classic publication by Nethercot and Zandonini (1989),

which summarised the early work in this area very well. At the time, they published

their paper, macro-element models had not yet been developed. These are a

combination between mechanical models and finite element models, as they use finite

element formulations to incorporate mechanical models into frame analysis. It is on

this type of modelling that this work is concentrated, and the relevant publications at

ambient temperature will be mentioned in Chapter 7. In the following, only the

modelling techniques that have been used at elevated temperatures will be discussed.

These are curve-fit models, mechanical models and finite element models; for the

remaining ones the reader should refer to the literature.

2.3.1 Curve-fit models

Curve-fit models are mathematical expressions fitted to moment-rotation curves

found in experiments. The expressions range from linear, bi-linear to tri-linear, and

polynomial, power expressions and B-spline techniques have also been used. For

more detailed information on these different techniques, the interested reader is

referred to Jones et al. (1983) and Nethercot and Zandonini (1989). Another curve-fit

model, which has been used at ambient as well as at elevated temperatures, is based

on the work by Ramberg and Osgood (1943). It uses the so-called Ramberg-Osgood

curve to describe the stress-strain behaviour of metallic materials. This expression

was modified by Ang and Morris (1984) to represent the moment-rotation curves of

joints, and was extended by El-Rimawi (1989) to elevated temperatures. The

approach is shown in equation 2.1:

.n

c c

c

M M0 01

A B

Φ = +

...2.1

where Φc is the joint rotation, Mc is the corresponding level of moment, and A, B and

n are temperature-dependent factors.


10

This expression has subsequently been used by Leston-Jones (1997) and Al-Jabri

(1999) to model their elevated-temperature test data for bare steel and composite

joints. It can be applied at elevated temperatures by making the terms A and B

temperature-dependant. These factors control the stiffness and capacity of the joint

respectively, whereas the index n defines the shape of the moment-rotation curve.

Although curve-fit models are very easily implemented into frame analysis, normally

via a rotational spring at the end of the beam, they can only be used to represent joints

which have been previously investigated experimentally. For the fire case, important

axial forces acting on the connection cannot be represented with a curve-fit approach

unless all different combinations of moment, rotation, axial force and temperature

have been tested, which is impracticable due to the high expense of experiments and

the vast number of connection configurations used in practice. A more practical

approach is the use of mechanical models, as discussed in the next section.

2.3.2 Mechanical models

Mechanical models divide a joint into zones of fundamental behaviour such as

tension, shear and compression. Each of these zones is then represented by a number

of translational springs, called ‘components’, which are linked by rigid bars to

produce a simplified model of the joint, which is able to predict the full moment-

rotation curve. The accuracy of these models depends highly on the force-

displacement curves adopted for the translational springs and the number of

components included.

Mechanical models rely on the definition of force-displacement curves for the

components, which can be found in a three-step process in which a component is first

studied in isolation, experimentally and then numerically, using the finite element

method for parametric studies. Finally, the knowledge gained is used to develop

models, based on mechanics, describing the required curve.

The principles of the Component Method are based on the experimental and analytical

work by Zoetemeijer (1983) which was conducted from 1974 to 1983. After that the

method was developed further by a number of researchers, but the work of

Tschemmernegg and his co-workers at the University of Innsbruck, Austria was


11

particularly important. The work was carried-out in the 1980s and 90s as a series of

PhD projects and is summarised in a series of papers by Tschemmernegg et al. (1987,

1988, 1989) and Huber and Tschemmernegg (1998) for welded and bolted bare-steel

endplate joints. For composite joints the same techniques were applied, and the work

was published by Tschemmernegg et al. (1994, 1995). Further, the work of Jaspart

(1991, 1997) from the University of Liége, Belgium should be mentioned. He

combined the available component data to a practical design concept for joints at

ambient temperature.

From 1990 onwards, the COST C1 workgroup (a Europe-wide project) focused on the

investigation of semi-rigid joints. As a result of this workgroup, the so-called

Component Method was standardised and incorporated into Eurocode 3, first as

Annex J, and in the final version of the code as EC3-1.8. After the Cost C1 action

finished a number of open questions remained and some of them were investigated by

Simões da Silva et al. (2000, 2001a, 2002) at the University of Coimbra, Portugal.

They focused on the post-limit stiffness and the ductility of several components.

Further Faella et al. (2000) published a book on structural steel semi-rigid

connections. It summarises comprehensively the research work in the field of semi-

rigid joints.

In order to describe the behaviour of an isolated connection at elevated temperatures,

the component method has been used successfully by a number of researchers.

Leston-Jones (1997) was the first to apply the method to his cruciform tests; Al-Jabri

(1999) used the method to model the flexible endplate behaviour of his high

temperature experiments. Simões da Silva et al. (2001b) used the component models

given in EC3-1.8 in combination with the temperature reduction factors given in

EC3-1.2 (CEN, 2005b) to model the cruciform tests conducted by Leston-Jones and

Al-Jabri. Spyrou (2002, 2004a, 2004b) conducted a large number of high-temperature

component tests and combined the investigated components using a simple two-spring

model. However, none of these studies combined directly the Component Method

with whole-frame action. Furthermore, apart from a limited study by Spyrou, the

effects of axial load on the connection in fire were not considered.


12

2.3.3 Finite element models

In finite element models, a joint is divided into a large number of either shell or solid

elements of finite size, which are able to represent the exact geometry and the

materials of the joint. The model is then solved numerically with respect to the

applied forces, temperatures and boundary conditions. As output from a joint

analysis, the deformations, strains, stresses and contact forces between surfaces can be

gained. Based on this information the full non-linear response of the joint can be

predicted. With the use of finite element models, it is possible to have an insight into

the stress and strain fields of a joint in much greater detail than is possible with

experiments, especially at elevated temperature. However, the application of the

method is complex and requires careful construction of the finite element joint

representations. Nevertheless, the method has the potential to describe the joint

behaviour in a very detailed manner, and it is also cheaper to perform large numbers

of different joint geometries and loading conditions than it would be the case with a

series of experiments. However, due to the level of details required and the long

computing times, the finite element method is not yet suitable for practical frame

design.

The finite element method has been extensively used to describe the joint response at

ambient temperatures. At elevated temperatures however, the models are still rather

rare. As was mentioned in the Chapter 1, Liu (1996, 1998, 1999) developed the finite

element code FEAST. The program was capable of incorporating non-linear high-

temperature material properties, non-uniform thermal expansion across a section, and

large deformation at high temperatures. The response of bolts and the contact ‘link’

between the column flange and end plate was simulated using a beam element with

special characteristics to take into account the behaviour of bolts during their thermal

expansion at elevated temperatures. Liu validated his program against the endplate

joints previously tested at elevated temperatures by Lawson (1989, 1990), Leston-

Jones (1997) and Al-Jabri (1999). Good correlations between the tests and

predictions were found.

A semi-rigid extended endplate joint at ambient and elevated temperatures has been

analysed by El-Houssieny et al. (1998). After validation against test data, they


13

performed an extensive parametric study to predict simple equations for the moment-

rotation stiffness, the bolt forces and stresses, in order to contribute to the

understanding of the behaviour of different joint components at elevated

temperatures. These equations were used, with considerable accuracy, for the design

of common joint types at different temperatures.

Al-Jabri et al. (2005) published a paper on the modelling of flush endplate

connections in fire using the commercial finite element package ABAQUS. A

detailed 3D model using solid and contact elements was developed and validated

against test results (Al-Jabri, 1999). The conclusion of the study was that the finite

element analysis is capable of predicting the behaviour of joints in fire accurately.

Recently, Lou and Li (2006) performed an ANSYS analysis of an extended endplate

cruciform joint with two major-axis connections. The analysis was divided into two

steps; the first was a thermal analysis of the joint in order to find the temperature

distribution, which was then transferred to the structural model using 3D solid

elements. The model was validated against high-temperature experiments conducted

in the same project and good correlation was found. It was concluded that the

ANSYS model could be used in future to predict moment-rotation-temperature

curves.

Comparing the three approaches discussed for modelling joints, it is evident that only

the mechanical models are suitable for the consideration of realistic joints in frames at

ambient and elevated temperatures. Therefore, the standardised version of the

mechanical models for endplate and cleat connections, the Component Method, will

be discussed in more detail in the next section.

2.4 The ‘Component Method’

As was stated above, the Component Method has become the standard tool for the

calculation of semi-rigid joint behaviour with its inclusion into the Eurocodes in

Chapter 6 of EC3-1.8. It represents a relatively easy way suitable for hand

calculations to predict the initial stiffness and the moment resistance for endplate and

cleat joints, as well as column bases. As this work is focused on flush and extended

endplate connections, the three basic steps of the Component Method will be


14

explained using the example of a major-axis single-sided joint with an extended

endplate connection.

2.4.1 Identification of the active components

The first step of the Component Method is the identification of the active components

in a joint. Active components are those which either contribute to the deformation of

a joint or limit its strength. For a joint with an extended endplate connection under

pure bending moment, EC3-1.8 considers the components summarised in Table 2.1 to

be active.

Table 2.1: Active components in an extend endplate joint

Component Index Resistance Stiffness

Column web in shear (cws) � �

Column web in compression (cwc) � �

Beam web and flange in compression (bfc) � -

Column web in tension (cwt) � �

Column flange in bending (cfb) � �

Bolts in tension (bt) � �

Endplate in bending (epb) � �

Beam web in tension (bwt) � -

In Figure 2.3, the location of these active components is shown. For more clarity, the

column and the beam have been drawn separated.

Figure 2.3: Active components of a joint with one extended endplate connection

The next step in the Component Method is to specify the force–displacement

behaviour of each active component.

(bfc)

M (bwt)

(cwc)

(cws)

(cwt) (cfb)

(epb)

(bt)


15

2.4.2 Specification of the component characteristics

This step is the most important part of the design process of a joint, because the

accuracy of the final moment-rotation curve largely depends on the quality of the

force–displacement curves of the individual components. In general, there are a

number of different options (elastic-plastic, bi-linear, multi-linear or non-linear) to

approximate the real component behaviour. Some of these different options are

shown in Figure 2.4.

Figure 2.4: Different complexities of component approximations

In EC3-1.8, each component is characterised by an initial stiffness k and a design

resistance FRd, which are linked in an elastic-perfectly plastic fashion. This simple

approximation allows a direct calculation of the moment-rotation curve of the joint.

However, for higher accuracy in the joint approximation more complex force-

displacement models can be used, derived from test results, finite element models or

preferably from simplified mechanical models. Unfortunately, the increase in the

complexity of the component representations makes it necessary to solve the final

spring model iteratively, which is not a problem if the spring model is incorporated

into a non-linear finite element program.

Although, the Component Method associates each component with a certain internal

force in the joint, in reality some components are exposed to stresses in more than one

direction. EC3-1.8 specifies reduction factors for the presence of shear, ω, and

longitudinal stresses, kwc, in the column web in compression, and for the presence of

shear stresses ω in the column web in tension.

It is in this second step, in which the Component Method can be extended to elevated

temperatures by using high-temperature material properties with the ambient-

temperature component models, as conducted by Leston-Jones (1997), Al-Jabri

δ

F

δ

F

δ

F

Experimental behaviour Approximation

Elastic-plastic Multi-linear Non-linear


16

(1999) and Simões da Silva et al. (2001) or by developing new multi-linear elevated-

temperature components models as was done by Spyrou (2002, 2004a, 2004b).

2.4.3 Assembly of the active joint components

The final step of the Component Method is the assembly of the components and the

calculation of the resulting moment-rotation curve. Each component is represented as

a translational spring interconnected by rigid links. The spring model for the example

joint is shown in Figure 2.5 below.

Figure 2.5: Spring model of an extended endplate joint after EC3-1.8

Bending moment resistance of the joint

The first parameter needed to predict the moment-rotation curve of a joint is the

bending moment resistance Mj,Rd, which can be calculated from Equation 2.2:

, ,1

n

j Rd ti Rd i

i

M F z=

=∑ ...2.2

where Fti,Rd is the design tension resistance of bolt row i, zi is the distance from bolt

row i to the centre of compression and i is the bolt row number, starting from the bolt

row furthest away from the centre of compression.

The resistance of each bolt row Fti,Rd is equal to the weakest component in this row or

the compression resistance of the two components of the bottom row (cwc, bfc) or the

resistance of the shear panel:

,, , , , , , , , , , , , ,min , , , , , , ,cws Rd

ti Rd cwt i Rd cfb i Rd bt i Rd epb i Rd bwt i Rd cwc Rd bfc Rd

FF F F F F F F F

β

=

...2.3

(bfc)

(bwt,1)

(cwc) (cws)

(cwt,1) (cfb,1) (epb,1) (bt,1)

(bwt,2) (cwt,2) (cfb,2) (bt,2) (epb,2)

M z2 z1


17

where β is a transformation parameter for the consideration of the beam-end moments

on the shear panel, ranging from zero for double-sided joints with equal beam-end

moments to one for a single-sided joint, to two for double-sided joints with equal but

opposite beam-end moments. For more detail the interested reader should refer to

EC3-1.8.

However, equation 2.3 only gives the design value for bolt row i if the distance to the

next bolt row is sufficiently large, and the column flange and the endplate develop

individual failure mechanisms. Otherwise, two or more bolt rows fail as a group and

the resistance is lower than the sum of the individual rows. The procedure to account

for such ‘group effects’, suggested by the Eurocode, considers each bolt row

individually at first (Equation 2.3) and then in combination with the successive rows

above. The procedure can be summarised as follows:

(1) Calculate the resistance of bolt row 1 ignoring the bolt rows below:

,, , , , , , , , , , , , ,min ( , , , , , , , )cws Rd

t1 Rd cwt 1 Rd cfb 1 Rd bt 1 Rd epb 1 Rd bwt 1 Rd cwc Rd bfc Rd

FF F F F F F F F

β= ...2.4

(2) Calculate the resistance of bolt row 2 ignoring the bolt rows below but including

the possibility of rows 1 and 2 combining:

, , , , , , , , , , , ,( ), , ,

,( ), , , ,( ), , , ,( ), , ,

,,( ), , , , ,

min ( , , , , , ,

, , ,

, ,

t 2 Rd cwt 2 Rd cfb 2 Rd bt 2 Rd epb 2 Rd bwt 2 Rd cwt 1 2 Rd cwt 1 Rd

cfb 1 2 Rd cfb 1 Rd bt 1 2 Rd bt 1 Rd epb 1 2 Rd epb 1 Rd

cws Rd

bwt 1 2 Rd bwt 1 Rd t1 Rd cwc R

F F F F F F F F

F F F F F F

FF F F F

β

+

+ + +

+

= −

− − −

− − , , ,, )d t1 Rd bfc Rd t1 RdF F F− −

...2.5

(3) Calculate the resistance of bolt row 3 ignoring the bolt rows below but including

possible interaction with the rows above:

, , , , , , , , , , , ,( ), , ,

,( ), , , ,( ), , , ,( ), , ,

,( ), , , ,( ), ,

min ( , , , , , ,

, , ,

,

t 3 Rd cwt 3 Rd cfb 3 Rd bt 3 Rd epb 3 Rd bwt 3 Rd cwt 2 3 Rd cwt 2 Rd

cfb 2 3 Rd cfb 2 Rd bt 2 3 Rd bt 2 Rd epb 2 3 Rd epb 2 Rd

bwt 2 3 Rd bwt 2 Rd cwt 1 2 3 Rd cwt 1

F F F F F F F F

F F F F F F

F F F F

+

+ + +

+ + +

= −

− − −

− − , , ,

,( ), , , , , ,( ), , , , ,

,( ), , , , , ,( ), , , , ,

,, , , , ,

,

, ,

, ,

, ,

Rd cwt 2 Rd

cfb 1 2 3 Rd cfb 1 Rd cfb 2 Rd bt 1 2 3 Rd bt 1 Rd bt 2 Rd

epb 1 2 3 Rd epb 1 Rd epb 2 Rd bwt 1 2 3 Rd bwt 1 Rd bwt 2 Rd

cws Rd

t1 Rd t 2 Rd cwc Rd t1 Rd t 2 Rd

F

F F F F F F

F F F F F F

FF F F F F

β

+ + + +

+ + + +

−

− − − −

− − − −

− − − − , , , )bfc Rd t1 Rd t 2 RdF F F− −

...2.6

and so forth.


18

Rotational stiffness of the joint

For the calculation of the second parameter of the moment-rotation curve, which is

the rotational stiffness of the joint, the fairly complex spring model shown in Figure

2.5, can be simplified by replacing each bolt row with an equivalent spring with the

stiffness ket,i, calculated in equation 2.7,

,

, , , , ,

11 1 1 1 1et i

cwt i cfb i bt i epb i bwt i

k =

k k k k k+ + + +

...2.7

where i is the number of the bolt row and the stiffness of the beam web in tension kbwt

is assumed to be infinite. Furthermore, the compression and shear components can

also be represented by an equivalent spring. Again, the equivalent stiffness of the

compression components kec can be calculated by Equation 2.8:

1

1 1 1ec

cws cwc bfc

k =

k k k+ +

...2.8

where the stiffness of the beam flange and web in compression kbfc is assumed to be

infinite. This simplifies the spring model in Figure 2.5 to the one shown in Figure

2.6(a).

Figure 2.6: Equivalent (a) and simplified (b) spring model after EC3-1.8

The equivalent spring model can be simplified even further and the springs for each

bolt row can be replaced with a single equivalent tension spring, as shown in Figure

2.6(b). The corresponding equation for the stiffness of this equivalent tension spring

is shown in Equation 2.9,

M z2

(ket,1)

z1 M

zeq

(a) (b)

(ket,2)

(kec) (kec)

(keqt)


19

,

1

n

et i i

i

eqt

eq

k z

k =z

=

∑ ...2.9

where zi is the distance between bolt row i and the centre of compression, which is

assumed to be at the centre of the compressed beam flange, and zeq is the equivalent

lever arm, which is calculated from Equation 2.10.

2,

1

,1

n

et i i

i

eq n

et i i

i

k z

z

k z

=

=

=∑

∑ ...2.10

After simplifying the spring model this far, it is easy to calculate the initial rotational

stiffness of the joint with Equation 2.11.

2,

ec eqt

j ini eq

ec eqt

k kS E z

k k=

+ ...2.11

or more generally the secant stiffness as:

2

eq

j n

i 1 i

E zS

1

kµ

=

=

∑ ...2.12

where E is the Young’s modulus, µ is the stiffness ratio Sj,ini / Sj (defined in the next

section) and ki the stiffness of each active component, regardless if the component is

in tension or compression. It is also possible to use the stiffness of the equivalent

springs instead of the once of each individual component.

Moment-rotation curve

After having calculated the bending moment resistance and the initial rotational

stiffness of the joint, EC3-1.8 offers two different options to approximate the

moment-rotation curve. The first option is an elastic-perfectly plastic curve, for

which the stiffness ratio is assumed to be µ = 2 for bolted endplate connections. The

second option is a non-linear curve with an initial elastic part up to 2/3 of the design

moment resistance, and continuing with a curve of continually changing secant

stiffness Sj entering into a horizontal. The stiffness ratio µ for the non-linear option is

defined by Equations 2.13 and 2.14,

1µ = for , ,

2

3j Ed j RdM M≤ ...2.13


20

,

,

1.5 j Ed

j Rd

M

M

ψ

µ

=

for , , ,

2

3 j Rd j Ed j RdM M M< ≤ ...2.14

in which the coefficient ψ is taken as 2.7 for bolted endplate connections. The two

moment-rotation options are shown in Figure 2.7 below.

Figure 2.7: Moment-rotation curves after EC3-1.8

These moment-rotation curves can then be introduced as rotational springs into frame

analysis program.

As one could see, the calculation process of the Component Method is quite lengthy

and therefore a number of programs have been developed to simplify the application

in engineering practice. The software CoP developed by the University of Liège,

Belgium and RWTH Aachen, Germany has been used in this project for all ambient-

temperature joint calculations.

2.4.4 Moment – Normal force interaction

As was mentioned earlier, in the fire case axial force will occur in the beam due to the

restraint conditions in a structure. The Component Method however, as it is described

in EC3-1.8, is only valid for combinations of bending moment and axial load in the

beam up to a maximum of 5% of the design resistance of the beam. This is because

the force distribution within the connection is strongly influenced by axial force in the

beam. If one imagines a compression force in the beam, some part of the resistance of

the compression components is used to resist this axial force and Equations 2.3 – 2.6,

Φ

M

Φ

M

Experimental behaviour Approximation

Elastic-plastic Non-linear

Mj,Rd

2/3 Mj,Rd

Sj,ini

Sj

j,iniS2

Mj,Rd


21

which compare the design values of the bolt rows with the resistance of the

compression components, might become unconservative. The same can be imagined

for the tension components, if the axial force in the beam is tensile. Furthermore, the

centre of rotation changes due to the influence of axial load, which influences the

active components. In a case where the bending moment is relatively small and the

axial force is relatively large, situations can occur where the connection is fully

compressed or fully tensioned. In such cases, either a second compression zone

located at the upper flange of the beam is developed, or the lower bolt row, normally

dedicated to resisting the shear force in the connection, will be in tension and has to

be included in the analysis as a tension component.

If the axial load in the beam is larger then 5% of Npl, EC3-1.8 considers the influence

of axial load in a simple interaction equation,

, ,

, ,

1.0j Ed j Ed

j Rd j Rd

M N

M N+ ≤ ...2.15

where Mj,Rd is the design moment resistance of the joint assuming no axial load, and

Nj,Rd is the axial design resistance of the joint assuming no applied moment.

Since the inclusion of this equation into the Eurocode, more research has been

conducted across Europe. At the University of Liège, Jaspart et al. (1997) and

Cerfontaine and Jaspart (2002, 2005) have investigated the influences of axial load in

the beam on the joint behaviour theoretically using the principles of the Component

Method. They developed a software called ASCON, which is able to take account of

the group effects between bolt rows and stress interaction in the appropriate

components. With this program they were able to generate M-N interaction diagrams

for different joints. This work showed clearly that the moment resistance of a joint is

significantly reduced by the presence of axial forces. A typical spring model

suggested by Cerfontaine and Jaspart is shown in Figure 2.8 below.


22

Figure 2.8: Spring model of an extended endplate joint for moment and axial

force after Cerfontaine and Jaspart (2002, 2005)

The differences from the original spring model suggested in EC3-1.8 are obvious, the

shear spring has been arranged diagonally in a linkage which can only deform in the

shape of a parallelogram. This modification had to be made to prevent the axial force

from deforming the horizontally placed shear spring. Further, the lower bolt row and

the upper compression zone have been included. However, the additional springs are

only active under certain combinations of axial force and moment.

At the University of Prague, Wald and Švarc (2001) conducted a series of five tests on

eccentrically loaded beam-to-column joints (2 tests) and beam-to-beam splices (3

tests). Sokol et al. (2002) used these tests to validate a calculation approach based on

the column base approach developed by Wald and Jaspart (2005). At the University

of Coimbra, an experimental study on single-sided major-axis joints with flush and

extended endplate connections was conducted by Simões da Silva et al. (2004) and

De Lima et al. (2004), respectively. They conducted in total 16 tests, 9 on flush

endplate joints and 7 on extended endplate joints. The axial loads varied from

compression of - 27% to tension of + 20% of the plastic axial capacity of the beam.

An application of a nonlinear calculation procedure developed by Simões da Silva and

Coelho, (2001c) based on the same principles as the work conducted in Liège, has

shown good agreements with the tests.

Although none of the tests or analytical models considers the effects of elevated

temperatures, it will be possible to use the general ideas developed in these studies in

the development of a finite connection element. As a first step towards the

(bfc,2)

(bwt,1)

(cwc,2)

(cws)

(cwt,1) (cfb,1) (epb,1) (bt,1)

(bwt,2) (cwt,2) (cfb,2) (bt,2) (epb,2) M

(bfc,1) (cwc,1)

(bwt,3) (cwt,3) (cfb,3) (bt,3) (epb,3)

N


23

development of such an element, the existing experimental studies on joints in fire

will be reviewed.

2.5 Joint behaviour in fire

Although the investigation of semi-rigid joints started over 80 years ago, it was not

until the mid 1970s that the research community became interested in the high

temperature behaviour of joints. However, the focus was more on the high-

temperature performance of high-strength bolts than to establish the full non-linear

behaviour of the joint. These first experimental fire tests on six different types of

joints, ranging from fin-plate connections, to cleat angle connections to flush and

extended endplate connections, were conducted by Kruppa (1976) at CTICM in

France. The results showed that, due to the deformation of other elements the bolt

failure was inevitable.

In the early 1980s, British Steel (1982) performed two elevated-temperature tests on

‘rigid’ moment resisting joints with cleat connections in order to observe their

behaviour. Despite the limited number of tests and tested connection types, the

conclusion was that bolts and their connected elements could undergo considerable

deformation in fire.

Lawson (1989, 1990) was the first to measure time-rotation curves of eight cruciform

joints with different major-axis connections exposed to the Standard Fire, at the Steel

Construction Institute in the UK. The joints were loaded to different load ratios

before heating, which created a transient test configuration. Failure of the joints was

assumed at a beam-end rotation of 100mrad or 6°, which is equal to a mid-span

deflection of the beam of span/20, the failure criterion in the Standard Fire test for

beams. Five of the eight tests were conducted on non-composite connections, and

two used small composite slabs connected with shear studs to the beams and one test

used a shelf angle beam. Three different types of connection were studied of different

rotational stiffness: a) extended end plate b) flush end plate and c) double-sided web

cleat. It was the aim of the test programme to investigate the beneficial influence of

rotational restraint provided by the joint to the fire performance of the beams. Once

more, these tests demonstrated that failure of the connecting bolts or welds did not

occur, even under considerably larger rotations than at ambient temperature.


24

However, to investigate the full moment-rotation-temperature relationship of a joint,

the same configurations have to be tested under a larger number of different moments.

This was the main concern of the work by Leston-Jones et al. (1997) at the University

of Sheffield, UK. In total eleven tests were carried out on cruciform joints with

major-axis flush endplate connections, including two ambient-temperature tests, one

for bare-steel and one for composite joints. The experiments confirmed that the joint

stiffness and moment capacity decrease with increasing temperature, especially at

temperatures above 500°C. Although the tests were conducted on small beam and

column sections, they gave good insights into the moment rotation behaviour of joints

in fire. This was the first time that a number of moment-rotation curves at different

temperatures were derived, describing the full non-linear joint response when exposed

to fire.

Continuing Leston-Jones’s work, Al-Jabri (1999) extended the scope of the

experiments to study the influence of parameters such as member size, connection

type and different failure mechanisms. In total twenty tests were conducted on flush

endplates with different section sizes and on flexible endplate joints, for both cases:

bare-steel joints and composite joints of all types used in the composite building at

Cardington. However, it was during this project that it was realised that experiments

on isolated joints are not sufficient to describe the behaviour of joints and connections

in frame structures, due to the lack of axial forces in the beams caused by the restraint

from the surrounding structure.

Nevertheless, two axially unrestrained cruciform joints with extended endplate

connections have been tested in China by Lou and Li (2006) recently. Relatively

large sections for the beams (H300x160x8x14) and for the column

(H240x240x10x16) were connected by a 16mm endplate and four M20-8.8 bolts. The

moment resistance at ambient temperature was 203 kNm and load ratios of 0.55 and

0.40 were used. Whereas the connections failed by buckling of the column web at

ambient temperature the failure mode changed at elevated temperatures to fracturing

of the bolts and yielding of the column web in tension, even though the endplate

temperatures close to the bolts were lower then the column web temperature.


25

Despite the obvious importance of predicting the behaviour of unprotected joints at

elevated temperature in a restrained condition, no experimental studies concerned

with this matter have been published at the time of writing of this thesis. However, a

so far unpublished experimental series of six internal extended endplate joints have

been tested at temperatures between 400°C and 700°C by Z.H. Qian and K.H. Tan in

Singapore. The tests were designed to fail in shear at the end of the beam. The first

three of the tests were conducted at 700°C with different amounts of axial restraint to

the beams, which caused axial forces in the joints. These tests failed in a combination

of endplate bending and shear deformation of the beam-end. In the remaining three

tests, the endplate thickness was increased to 40mm in order to isolate the shear

component in the beam-end. The tests were conducted at 400°C, 550°C and 700°C

and it could be observed that the capacity of the beam-end reduced in the expected

way with increasing temperature. Once this test data is published, it will provide a

good opportunity to validate the different approaches of modelling steel connections

at elevated temperature including the effects of axial force in the beams.

At the University of Manchester (Liu et al. 2002), in conjunction with the University

of Sheffield (Allam, 2003), some experiments on restrained beams were conducted.

A 2 m long small beam was placed between two columns creating a ‘rugby goal post’

inside a furnace. The columns and the connections were fire-protected and remained

at sufficiently low temperatures. The aim of this study was to investigate the effects

of translational and rotational restraint to the beam. Large axial compressive forces

were recorded in the early stages of the fire, but after the vertical beam deflections

had increased these compressive forces changed to tensile (or catenary force) and

increased the failure temperature of the beam considerably compared with an

unrestrained beam. Although, the connections were protected the tests gave good

insights into the axial forces acting on a connection at elevated temperatures.

However, the tests followed the standard fire curve and no information on the cooling

forces in the beam of the connections was recorded.

At the BRE Cardington Laboratories in Bedfordshire, UK, a series of seven full-scale

fire tests on an eight-storey composite building were conducted in 1995-6 and 2003.

The seventh test (Wald, 2004) focused, amongst other things, on the beam-to-column

and the beam-to-beam connection behaviour. The connections were instrumented


26

with a large number of thermocouples, giving temperature profiles of the connections.

Furthermore, strain gauges were used to monitor the forces acting on the joints and

the bolts. The damage mechanisms observed near the joints were the following: local

bottom-flange buckling of the beams, shear buckling of the beam webs, plastic

deformation of the column flange around the tension bolts of the connections, fracture

of the partial depth endplates along the welds and finally elongation of the bolt holes

in beam webs as part of the beam-to-beam fin-plate connections. All these damages

suggest that considerable axial forces are acting in the beams, and support the case for

the inclusion of realistic joint models into full-frame analysis used in practice.

As a second step towards the development of a component-based connection element,

the different attempts to use the Component Method for the prediction of high

temperature joint behaviour will be reviewed.

2.5.1 Application of the Component Method in fire

As mentioned above, the Component Method has been applied to elevated

temperature conditions. In general, this has been done by using ambient-temperature

component models in combination with elevated-temperature material properties.

Leston-Jones (1997) developed a spring model based on four basic components:

column flange in bending (cfb), bolts in tension (bt), endplate in bending (epb) and

column web in compression (cwc). The spring model for a flush endplate connection

with two bolt rows, as used by Leston-Jones, is shown in Figure 2.9.

Figure 2.9: Spring model of a flush endplate connection (a) and equivalent model

(b) after Leston-Jones

(cwc)

M

(epb,1)

(cwc)

M

(teq) (a) (b)

(cfb,1) (bt,1)

(epb,2) (cfb,2) (bt,2)


27

For situations with more than one bolt row, Leston-Jones replaced the bolt rows with

an equivalent bolt row in a very similar way to that described above in the Eurocode

section. The force-displacement curves of the components were represented in a tri-

linear way. The spring model was validated against the elevated-temperature tests by

Leston-Jones.

Al-Jabri (1999, 2005) used the same principles as Leston-Jones for his component

models for the modelling of his high-temperature experiments on flush-endplate

connections. He extended the use of the Component Method to partial-depth or

flexible endplate connections. He achieved a good agreement between his

experiments and the spring model.

Simões da Silva et al. (2001b) included all components as defined in EC3-1.8

together with the extension of the post limit component stiffness as derived by Simões

da Silva et al. (2000). Their approach was to assess a joint at ambient temperatures

and then to multiply the moment resistance and the stiffness of the joint by the high-

temperature material reduction factors given in EC3-1.2 for the yield stress and the

Young’s modulus to predict the high-temperature behaviour of the joint. Al-Jabri’s

tests were used to validate the approach, and with the help of a global temperature

correction factor, equal to 0.925 Texperiment, good agreement between the tests and the

model was reached.

Spyrou (2002) and Spyrou et al. (2004a, 2004b) conducted a large number of tests on

single components at ambient and elevated temperatures. He tested T-stubs, which

are used to represent the behaviour of bolt rows in tension, and developed simplified

analytical models for the force-displacement curves of this component. Furthermore,

Spyrou tested column webs in transverse compression at temperatures up to 765°C,

and developed a semi-empirical force-displacement model for this component at high

temperatures. However, in the column web tests it was assumed that no shear and no

axial loads were present in the column. Spyrou further used his new component

models in a spring model similar to the one used by Leston-Jones to predict the

elevated-temperature connection tests by Leston-Jones and Al-Jabri with good

success. After validating his model, he conducted a short study on the effects of axial


28

force in the beam on the moment-rotation behaviour of the modelled connection.

However, there were no experimental studies to validate this study.

2.6 Conclusion

In this chapter, different ways of modelling the response of joints at ambient and

elevated temperatures have been summarised. From the initial comparison between

curve-fit approaches, mechanical models and finite element models it became clear

that the mechanical models, standardised in EC3-1.8 as the Component Method,

appear most suitable for the inclusion of detailed modelling of joints into frames.

The principles and the calculation procedure of the Component Method have been

described, including recent studies on moment-normal force interaction. Attempts to

apply the component method to elevated temperatures have been summarised.

A number of important points in this chapter can be highlighted:

• the Component Method is suitable to model high-temperature joint behaviour;

• the spring models and the component models of the Component Method can

be used to form a component-based macro element to include joint behaviour

into frame analysis;

• in order to model the effects of axial load in the beam, which is very important

in the fire case on the joint, all bolt rows and both compression zones (i.e. the

column web in line with the top flange of the beam, as well as the compression

zone adjacent to the beam bottom flange) have to be included;

• the high-temperature model for the compression zone in the column web

developed by Spyrou does not include the effects of axial load in the column.

In the following chapters of this thesis, these points will be investigated further.

Chapter 3: Experimental work on the compression zone

29

3 Experimental work on the compression zone

3.1 Introduction

In the previous chapter, the principles of the Component Method have been explained

together with the application of the method at elevated temperatures. It was found

that there are certain shortcomings in the force-displacement models for the individual

components at elevated temperatures. In particular, the effect of axial column load on

the behaviour of the compression zone in a column web has been ignored. Therefore,

an experimental programme has been devised to investigate this effect at elevated

temperatures. This experimental work on the compression zone will be described in

this chapter. Firstly, a short review is given on the experimental work on column

webs under concentrated loads conducted by previous researchers. Secondly, the test

setup and initial calibration tests are described. Finally, the results of the experiments

are presented.

The high-temperature experiments have been conducted in the Heavy Structures

Laboratory at the University of Sheffield. Small British Universal Column sections

(UC 152x152x37) were chosen because of their relatively low axial capacity, which

reduces the size of the loading and reaction gear. The specimens were tested

horizontally in a purpose-built electric furnace, loaded both axially and transversely.

This loading arrangement simulated the beam flanges in an internal joint.

The work described in this chapter has been presented by the author at Eurosteel 2005

in Maastricht (Block et al., 2005a) and at ICASS 2005 in Shanghai (Block et al.,

2005b).

3.2 Scope of the experiments

The purpose of the experiments was to produce validation cases for FEM studies and

simplified models, describing the high-temperature behaviour of the column web in

bi-axial compression. In the fire case, the transverse compressive forces are increased

by the restraint of the surrounding structure against thermal expansion as well as

beam-end rotations. The columns must also resist axial load from superstructure

loading. In order to simulate this condition a furnace was designed to be capable of


30

generating steel temperatures up to 600°C and loading a short column section in both

the longitudinal and the transverse direction.

3.3 Earlier experimental work on the compression zone

At ambient temperature, experimental testing of the effects of concentrated loads on

steel beams started in the early 1930s, which has been summarised by Vellasco and

Hobbs (2001). These early studies investigated the effects of concentrated loads on

beams i.e. bearing of beams onto other beams and points of support. The studies

focused on the bearing capacity of slender webs rather then on the full force-

displacement behaviour of column webs, which are rather stocky. Because the

researchers concentrated on beams rather than columns, the effect of axial load in the

sections was mainly ignored.

With the inception of semi-rigid connection design, the response of the column webs

in compression moved into focus. Zoetemeijer (1980) investigated this zone

experimentally with and without axial load in the section. Aribert et al. (1990)

summarised 24 tests they conducted since 1977 on different European sections (in

steel grade S235 and S460) without axial load. Unfortunately, only the ultimate

resistances from the tests can be found in the literature. As was mentioned in the

previous chapter Tschemmernegg and his co-workers improved the component

method in the late eighties and nineties in a series of PhD projects. As part of these

projects, further tests on the load introduction in column webs were conducted. Some

of these testes included axial loading. Kuhlmann and Kühnemund (2000) conducted

in total 16 tests on European column sections of the size HE 240 A and HE 240 B,

with axial loads up to 67% of the squash load. All of these tests were performed

using isolated, relatively short column sections.

The compression zone in the column web was also investigated as part of a complete

beam and column assembly. As part of Tschemmernegg’s work in Innsbruck, a

number of tests on welded beam and column subassemblages were conducted,

simulating external joints. Some of these experiments included axial load in the

column. Bailey and Moore (1999) conducted a series of cruciform tests at BRE,

varying the size of the column section, the amount of axial load in the column and the

ratio of beam loads generating different amounts of shear in the column web. The


31

purpose of these tests was to investigate the conservatism in the British design rules

for the influence of axial loading in the column on the moment-rotation behaviour of

semi-rigid joints. Similar tests have been reported by Jaspart (1997).

The first elevated-temperature tests on the compression zone were conducted by

Spyrou (2002, 2004b) at the University of Sheffield. These experiments concentrated

on the effect of temperature on the force-displacement behaviour of a wide range

British Universal Column sections. The effects of axial load in the column were

ignored in the study.

3.4 Methodology of testing

The experimental procedure comprised three steps: firstly, the specimen was loaded

axially; it was then heated to the test temperature, maintaining its axial load; finally, it

was loaded transversely until failure occurred in the column web. This testing

procedure resulted in a steady-state experiment with a finite loading speed, which was

designed to be similar to the increase of the compressive force in the beams in the

seventh natural fire test at Cardington, as found by Wald et al. (2004).

Tests were planned at 20°C, 450°C, 550°C and 600°C with axial loads equal to 20%

and 30% of the squash load of the sections. For reference purposes tests with no axial

load were also conducted. A maximum axial load ratio of 0.3 may appear low, but

considering the tested column section as part of a multi-storey building, it would have

a buckling length of about 3.0m which would mean that the column could only utilize

about 60% of its squash load at ULS due to buckling. Considering further the partial

safety factors at the Fire Limit State (FLS), the design load would be reduced by

about 50%, resulting in an axial load of about 30% of the plastic capacity of the

column.

3.5 Test rig

The test rig was situated within a reaction frame in the Heavy Structures Laboratory at

the University of Sheffield. It should be noted that the funding for these experiments

was very limited, and therefore it was necessary to reuse as much equipment from

previous projects to manufacture as many parts of the test-rig in the university’s

workshop as possible. The specimens were tested in a horizontal alignment in a


32

purpose-built electric furnace, loaded in axial and transverse directions. The test

arrangement can be seen in Figure 3.1 below.

Figure 3.1: Overview of the test setup

3.5.1 Reaction Frame and Loading devices

In the ambient temperature tests by Kuhlmann and Kühnemund (2000), a closed

system with four pre-tensioned steel rods running parallel to the specimen, was used

to generate axial load in the specimen. The advantage of such a system is that no

axial loads have to be reacted against outside the rig. In order to use such a technique

at elevated temperature the furnace would need to sit within the rods, which would

make the transfer structure at the end of the specimen very substantial and therefore

heavy. Furthermore, changing specimens would become more difficult. Therefore, it

was decided to use the existing reaction frame to support the axial loads in the

specimen, and to attach as many heavy parts of the rig to it as possible.

The axial load, simulating the superstructure load in the column, was introduced by a

hydraulic jack attached to the reaction frame. The jack was powered by a pressure-

controlled pump, which kept the axial load constant as the specimen expanded due to

increasing temperature.

Furnace

Hydraulic jack

Specimen

Actuator

Roller block

Bottom support Roller block


33

A displacement-controlled actuator with a capacity of 500kN applied the transverse

compression to the section. The load introduction plate had to be removed completely

from the inside of the furnace to prevent the load-cell from over-heating, and

therefore only 25mm of the 150mm of available travel in the actuator could be used to

load the specimens. The actuator was controlled by an advanced control device made

by Kelsey, allowing programming of the displacement rate to a constant rate of

1.5mm/min for the first 15mm and 2.0mm/min for the remaining 10mm.

The transverse load was introduced to the section by opposed 20mm thick steel plates.

The edges of the plates were rounded, with a radius of 3mm in order to reduce stress

concentrations at the edges, resulting in a load-introduction width of about 14mm

initially, but at larger displacements of the compression zone the load introduction

width increased to the full thickness of the plates. The upper plate was attached to the

actuator and could be moved out of the furnace during the heating phase of the

experiment, as mentioned above. The plates had to stay cold enough for the edge

stresses to remain lower than the proportional limit of the material (i. e. elastic). To

achieve this, a glove of commercial fire blanket was slid over both load-introduction

plates. The bottom plate could also be moved out of the furnace during the heating

phase of the test. Just before the test, the bottom plate was moved into the furnace

and located in place with a steel plate and shims.

In order to enable the test rig to perform in the assumed way the ends of the specimen

needed to be able to move vertically. That prevents additional vertical support at the

ends, which would reduce the force in the compression zone at the side opposite the

vertical load actuator. To achieve this target, roller blocks were situated between the

ends of the specimen and the reaction frame allowing the required vertical movement.

Each roller block consisted of two hardened steel plates in a guiding frame which

allowed only translation in vertical direction. One plate was attached to the reaction

frame and the other was connected to the specimen via a spherical seat. In order to

minimise the friction, 120 stainless steel rollers were placed between the plates.

These rollers were kept in place by a steel plate between the hardened plates with 12

square holes. Each of the holes grouped 10 rollers together. All parts of the roller

block were well lubricated using oil. This construction allowed for vertical

movements of the specimen almost frictionless, even under the high axial loads.


34

At the beginning of the test series, lateral movements and rotations of the jack and the

upper flange of the section occurred, causing the column web to fail in an asymmetric

shape, as described later in this chapter. After a number of unsuccessful attempts to

prevent this movement and a global failure of the test rig due to minor-axis buckling

of a specimen, a bracing system was developed, supporting the upper load

introduction plate and the flanges of the specimen laterally. An independent frame

next to the furnace was built, to which the upper load introduction plate was fixed

with a strut. Around the furnace two square ring-frames were constructed and again

braced to the independent frame. Within the ring frames large bolts were situated,

penetrating the furnace walls and bracing the flanges of the specimen laterally from

both sides. A picture of this bracing system can be seen below in Figure 3.2.

Figure 3.2: Lateral bracing system

3.5.2 Furnace

An electrically heated furnace was constructed around the specimens, which

protruded through shaped holes in removable panels at both ends. The furnace was

insulated with 50mm fibreboard. Commercial electric heating elements, closely

arranged around the specimen, were used to heat up the specimen. At the sides, long

2 kW heating elements could be used, but at the top and bottom of the furnace two 1

kW elements per side had to be used in order to allow the load introduction plates to

enter the furnace. The elements had a total power output of 8 kW. Each of the four

sides was controlled independently by variable high-current resistors to achieve

uniform heating of the section. Cross sections of the furnace are shown in Figure 3.3

below.

Bolts

Ring frame Independent frame

Bracings


35

Figure 3.3: Cross section of the furnace

Special care had to be taken over the last 150mm of the heating elements, as they

needed to stay at a temperature below 200°C. Therefore, the holes in the insulation

were cut in such a way that there was about 10mm air-gap around the heating

elements. The ends of the elements continued outside the furnace walls into mesh

boxes where the power cables leading to the heating control devices were connected

to the heating elements. During the tests these boxes were cooled by two fans, which

guaranteed that the ends of the heating elements remained below the critical

temperature.

3.5.3 Measurement of the temperatures

Initially nine K-type thermocouples were used to measure the steel temperature at

different points across the section, near the transverse load introduction area and at

two points to the left and right of the loaded area to measure the longitudinal

temperature distribution in the specimen. After the initial temperature tests, which are

described later, the number of thermocouples was reduced to seven; five

thermocouples close to the transverse load-introduction zone, as shown in Figure 3.4,

and one thermocouple each in the middle of the web at the outer locations.

B

B

A

A

Section A-A Section B-B

1 kW heating element

2 kW heating element


36

Figure 3.4: Thermocouple location in the cross section

3.5.4 Measurement of the displacements

Measurements inside furnaces are always a problem because of the temperature

sensitivity of standard instrumentation. Displacement transducers capable of resisting

elevated temperatures are available but very expensive. Therefore, it is usual to try to

position the instrumentation outside the furnace. Initially, the use of a measurement

system based on digital cameras and the automated picture processing software

developed by Spyrou (2002) was considered. Unfortunately, the automated part of

the picture processing software was faulty, and therefore the pictures would have

needed to be processed by hand. Therefore, the displacement in transverse directions

was measured by two LVDTs outside the furnace, which made it possible to use

standard transducers. These LVDTs were mounted onto a frame which was bolted

onto the upper end of the load-introduction plate. With this arrangement the relative

displacement and rotation between the ends of the load-introduction plates could be

measured. Additional to these external LVDTs the displacement of the actuator was

measured within the actuator itself. To use this displacement data, the stiffness of the

reaction frame had to be measured, which enabled the author to calculate the

additional displacement of the frame and to correct the readings.

The out-of-plane movement of the column web was measured with a ceramic rod

protruding through the hole in one of the furnace doors. The rod was attached to a

transducer located outside the furnace. Ceramic rods are ideal for measuring

displacements in furnaces because of their relatively low thermal expansion

coefficient. The transducer had to be spring-loaded to be able to track web

displacements in both out-of-plane directions. The arrangement of the LVDTs can be

seen from Figure 3.5 below.

Initial thermocouple locations Final thermocouple locations


37

Figure 3.5: Transducer arrangement

Additional to the transducers which measured the compression zone response two

LVDTs were situated at the ends of the specimen in order to record the vertical rigid

body movement of the column section. Monitoring the results from these transducers

during the test it was possible to see if the roller blocks were working. If not, slight

taps with a hammer helped to overcome the friction.

3.5.5 Measurement of the forces

The axial force in the specimen was recorded by a calibrated pressure transducer

fitted between the pump and the hydraulic jack. The load in the transverse direction

was measured by a calibrated load cell attached to the actuator.

3.5.6 Material and geometrical properties of the specimens

In order to have comparable test results and to reduce the number of material tests

required, all specimens were cut by the manufacturer from the same batch of column

section. Four standard tensile coupon tests on the first set of specimens were

conducted by Sheffield Testing Laboratories Ltd. For the material tests, two

specimens from the web and two from the flanges were used. From the second set of

specimens, only a coupon each from the web and the flange has been tested. The

results of these tests are shown in Table 3.1 below.

Actuator Load

introduction

plates Furnace

Specimen

LVDT

Ceramic rod


38

Table 3.1: Material properties of the specimens

Yield Stress [N/mm2]

Ultimate Stress [N/mm2]

Ultimate strain [%]

Young’s modulus [N/mm2]

Specimens 1 Flange 1

291 493 31.5 160560

Specimens 1 Flange 2

285 497 35.0 173010

Specimens 1 Flange average

288 495 33.25 166785

Specimens 2 Flange

295 510 35.0 165680

Specimens 1 Web 1

292 489 28.5 161640

Specimens 1 Web 2

287 483 32.5 178380

Specimens 1 Web average

289.5 486 30.5 170010

Specimens 2 Web

306 504 31.5 181550

It is worth noting that the measured Young’s modulus of this steel was significantly

lower than the normally assumed Young’s modulus of 205000 N/mm2. Furthermore,

none of the test diagrams showed an upper yield stress. These results appear unusual

for certified S275 steel, but no explanation could be found. A material study by

Renner (2005), which will be described later in this chapter in more detail, has

confirmed these unusual values.

The geometric properties were measured using a digital calliper, in a number of

different locations on six different specimens. Averaging resulted in the following

geometrical dimensions (the values in parentheses are the standard values according

to BS 4-1 (BSI, 2005)): depth of the section d = 161.4 mm (161.8 mm), flange width

bf = 154.2 mm (154.4 mm), web thickness tw = 7.6 mm (8.0 mm) and flange thickness

tf = 11.0 mm (11.5 mm). Based on these values the cross-sectional area of the

specimens was calculated as A = 4510 mm2 (4711 mm2).

3.6 Initial calibration tests

3.6.1 Horizontal stiffness of reaction frame

First, the lateral movement of the reaction frame was investigated to evaluate its

stiffness. The specimen was carefully loaded axially up to 400kN, which was the

maximum test load. During this loading, the lateral movement of the frame in the


39

middle was monitored. The test showed a linear elastic lateral displacement of about

2mm, which recovered after unloading almost back to its origin, demonstrating that

the frame possessed sufficient stiffness over the testing range.

3.6.2 Restraint of thermal expansion

The axial load introduction system had to allow for thermal expansion while keeping

the axial load in the specimen constant. In order to account for this phenomenon a

pressure-controlled pump was used in combination with a hydraulic jack. To

guarantee that the pump kept the pressure in the jack, and therefore the axial force in

the specimen constant, the jack was situated in a testing machine and loaded until a

load of 50kN was reached by increasing the pressure in the jack via the pump. Then it

was attempted to increase the load further using the displacement controlled testing

machine but the pump kept the pressure in the jack constant and the cylinder of the

jack was pushed back. This test demonstrated that the axial load introduction system

was appropriate for the experiment.

3.6.3 Vertical stiffness of reaction frame

To guarantee redundancy in the transverse displacement measurement the two LVDTs

and the internal displacement gauge of the actuator were used in parallel. Therefore,

it was necessary to determine the vertical stiffness of the test-rig to be able to use the

displacement readings from the actuator. In order to evaluate the frame stiffness,

custom-made stiffeners were fitted between the flanges of a specimen at the centre-

line of the load. This eliminated any non-elastic displacement of the specimen. The

elastic part of the displacement of the specimen was measured using the two external

LVDTs. The arrangement can be seen in Figure 3.6 below.


40

Figure 3.6: Frame stiffness test

The load was applied in cycles, firstly increasing to 50kN, then back to 0kN, up to

100kN, then back to 0kN and so on up to 350kN. In total three tests were conducted

to be able to average the stiffness of the frame. The force-displacement plot of the

first test is shown in Figure 3.7 below.

Figure 3.7: Stiffness of the reaction frame

0

50

100

150

200

250

300

350

400

-1 0 1 2 3 4 5

Displacement [mm]

For

ce [

kN

]

Specimen

Actuator

Calculated frame

response


41

It can be seen that the behaviour of the system is elastic up to the maximum tested

load. To evaluate the response of the frame the displacement of the specimen has to

be subtracted from the displacement measured by the actuator. From the results of the

three tests, an average reaction frame stiffness of SFrame = 85626 N/mm could be

calculated.

3.6.4 Temperature Distribution Tests

In order to give good comparability within the test series and with numerical and

analytical predictions, a uniform temperature distribution in the cross-section is vital.

Therefore, two initial temperature tests were conducted; the first to investigate the

longitudinal temperature distribution in the specimen and the second to explore the

temperature distribution across the middle section of the specimen. In the first test,

nine thermocouples in total were attached at three locations along the section, at about

the third-points of the specimen and in the middle. This test revealed a temperature

difference of about +5% from the middle to the outer section.

In the second test, nine thermocouples were distributed across the middle of the

specimen; three each in the middle and the tips of the upper and lower flanges, and

three in the web. It could be seen that the flange tips heated up quicker initially but

after a short while a uniform temperature across the specimen could be reached.

Another purpose of the second test was to test the temperature control device, which

should enable the user to generate a constant temperature in the specimen for the

duration of the transverse loading phase. The initial increase in flange temperature

and the uniform temperature distribution in the specimen can be seen in Figure 3.8

below.


42

050

100150200250300350400450500550600

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Time [min]

Ste

el t

empe

ratu

re [

C]

Figure 3.8: Initial temperature test

From the high-temperature tests it can be concluded that the purpose-built furnace

generates a very uniform temperature distribution in the specimen. Furthermore, it

was shown that with the control device it is possible to keep the specimen temperature

constant for the proposed duration of a test.

3.7 Observations and Results

3.7.1 Observations

Two different failure modes occurred in the experiments. In the first failure mode, the

column web failed in a single buckle; in the second failure mode the web deformed

into a S-shape with simultaneous lateral displacement of the upper flange. The

second failure mode can be explained by the relatively low lateral stiffness of the

vertical actuator, which was not sufficient to restrain the column flanges laterally, so

the specimens failed in a S-shape. After the introduction of the lateral bracing system

the asymmetric failure mode did not occur anymore. Both failure modes can be seen

in Figure 3.9 and Figure 3.10 below.

Flange tips


43

Figure 3.9: Failure mode 1: Symmetric

Figure 3.10: Failure mode 2: Asymmetric

The problem of these two different failure modes had occurred before in the

experiments by Spyrou (2002), in which two tests at the same temperature failed

under similar loads in the two different shapes, which indicates that there is no large

difference in the resistance of the two modes. That indicates that both failure modes

can be used as comparisons for the numerical and analytical results.

3.7.2 Temperature distribution in the specimens

As shown before, it is possible to heat the specimens uniformly with the purpose-built

furnace within a tolerance of ± 2.5% in the web and ± 5% over the whole cross-

section. The temperatures for all elevated-temperature tests conducted are given in

Table 3.2. The table shows the temperature readings of a particular thermocouple at

the time of maximum load. These values can be recommended for numerical

modelling of the tests. More detailed temperature data is given in Appendix A.

k > kcrit

k < kcrit


44

Table 3.2: Temperature distribution across the section

Web Upper flange Upper Middle Lower

Lower flange Test

# [°C] [°C] [°C] [°C] [°C]

1 455.2 454.1 460.8 443.8 425.8

2 480.6 507.0 531.5 508.8 483.9

4 430.7 449.5 462.3 439.0 418.2

5 516.3 529.2 537.4 528.4 516.1

7 439.4 459.2 475.5 457.5 418.0

8 517.0 553.3 562.2 533.4 508.8

9 576.4 595.5 598.4 580.7 548.5

12 601.7 606.3 607.6 580.0 563.2

13 539.7 550.2 558.8 530.9 500.8

15 436.2 443.2 453.0 435.2 420.3

16 448.0 453.9 458.5 440.2 429.3

17 547.2 551.3 554.9 528.4 531.1

18 590.3 598.1 604.3 583.0 571.4

In all tests, it was noted that the temperatures of the flanges decrease during the

application of the transverse load. This is caused by the heat-sink effect of the cold

load-introduction plates touching the flanges. This effect can be seen in Figure 3.11

below, showing a typical time-temperature curve.

Figure 3.11: Typical time-temperature curve

3.7.3 Correction of test results – Thermal expansion of the loading plates

The thermal expansion of the load introduction plates caused additional displacements

in the compression zone as they heated up. This expansion had to be calculated and

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100 110Time [min]

Ste

el t

emp

erat

ure

[o C

]

4

1 2, 3

1

3

4 5

2

5


45

the displacement readings of the LVDTs and the actuator had to be corrected.

Therefore, the temperature of the top load-introduction plate was measured with two

thermocouples, one 20mm from the bottom and the other 140mm from the bottom of

the plate.

Figure 3.12: Assumed temperature distribution in load-introduction plate

Assuming a temperature distribution as shown in Figure 3.12 above, the additional

displacement in the compression zone can be calculated by using equation 3.1 below

in combination with Figure 3.12.

( )( ) ( ), , , ,

, ,

1 i 1 0 2 i 2 0

i T 1 1 i 1 0 2

T T T Tx T T x

2δ α

− + − = − +

…3.1

Where αΤ is the thermal expansion coefficient of steel, i indicates the temperature at

which the additional displacement is calculated and 0 indicates the temperature at the

beginning of the test.

3.7.4 Correction of test results – Elastic deformation of the loading plates

The displacement of the compression zone is measured at the ends of the loading

plates outside the furnace, as described above. Therefore, the elastic deformation of

the loading plates has to be considered when the initial stiffness and force-

displacement curves are measured. As the plates remain elastic during the test, elastic

theory can be used to calculate the stiffness of the plates. The stiffness distribution of

the specimens is not uniform, because the web in compression is a lot stiffer than the

flanges in bending. Therefore, the stress in the plates is not uniform, and a 45° stress

distribution, as shown in Figure 3.13 below can be assumed.

T2

T1 x1

x2


46

Figure 3.13: Elastic deformation of the loading plates

The stiff length s is calculated by assuming a load spread of 45° from the end of the

web by equation 3.2 below.

( )2w f

s t r t= + + …3.2

where tw and tf are the thicknesses of the web and the flange of the specimen,

respectively and r is the root radius.

The load spreads further into the plate, and an effective width of the plates accounting

for the non-uniform stresses can be calculated using equation 3.3 below.

1 2

1 2

2eff

w sx w x

wx x

+ +

=+

…3.3

This can be done for the upper and lower plates and weff,u = 144 mm and weff,l = 146

mm can be found. The elastic stiffnesses of both plates can now be calculated using

the well-known equation 3.4.

EA

kl

= …3.4

where E is the Young’s Modulus of the plates, A the cross sectional area accounting

for the effective width and l the length of the plates. With this equation a stiffness of

k = 1378 kN/mm for the stiffness of the plates was calculated. This approach ignores

the reduction of the Young’s Modulus when the temperature of the plates increases.

x1

x2

s

we

45°

we

45° x2

x1

w


47

3.7.5 Testing speed

During this research project it became evident that the material properties of steel at

elevated temperatures depend significantly on the speed of loading. In a later section

of this work this matter will be discussed in more detail. However, the loading speeds

of the conducted tests are given in Table 3.3 below. The column “Average speed”

contains the overall average speed of the tests, the columns “Average speed – pre

peak” and “Average speed – post peak” give the average speed of the tests between

the beginning of a test and the maximum load, and the maximum load and the end of

a test, respectively.

Table 3.3: Loading speed of the tests

Average speed Average speed –

pre-peak Average speed –

post-peak Test # [mm / min] [mm / min] [mm / min]

1 1.84 1.55 4.03 2 2.88 1.58 5.93 3 1.38 0.81 2.58 4 1.13 0.93 2.02 5 2.56 1.75 3.33 6 0.71 0.59 1.00 7 0.94 0.58 1.29 8 5.10 3.53 6.91 9 1.58 1.09 1.91

10 1.21 0.70 2.05 11 1.27 0.70 2.20 12 1.41 1.04 1.71 13 1.36 0.90 1.70 15 1.33 0.90 2.15 16 1.31 0.84 1.83 17 1.49 1.00 2.08 18 1.44 1.01 1.82

For most of the tests, the loading speed was between 0.6 and 1.0 mm/minute.

However in Tests 1, 2, 5, 8 the speed was considerably higher, which was caused by a

manually operated displacement control and resulted in quite uneven force-

displacement curves, as can be seen later. In Test 8, an attempt was made to program

a constant displacement rate into the control device of the actuator. Unfortunately,

the displacement rate in the Kelsey has to be based on relative displacements and not

on absolute values as was being attempted in Test 8. This mistake resulted in a very

high for a quasi-static experiment average displacement speed of 5.10mm/minute.


48

3.7.6 Summary of the tests conducted

A summary of the tests conducted is shown in Table 3.4 below, ordered by increasing

temperature and axial load. The axial load ratio is calculated using the reduced yield

stress at 2% strain, based on EC3-1.2 temperature reduction factors. Tests 15 to 18

were conducted using specimens from a different batch of steel, and therefore the

values cannot be compared directly with the other tests.

Table 3.4: Summary of the tests conducted

Test # Web temp. [°C]

Flange temp. [°C]

Axial load [kN]

Axial Load ratio

Failure mode

[-]

Fu,exp [kN]

δu,exp [mm]

Initial stiffness [kN/mm]

11 20 20 3 0.00 2 418.3 6.2 336

10 20 20 265 0.20 2 421.7 6.6 356

3 20 20 394 0.30 2 413.7 6.1 336

6 20 20 398 0.31 2 423.4 7.2 351

4 450.3 424.5 3 0.00 2 331.7 5.5 177

1 452.9 440.5 266 0.23 2 331.0 5.8 256

7 464.1 428.7 403 0.36 1 331.4 5.7 141

2 515.8 482.3 390 0.41 1 280 5.0 134

8 549.6 512.9 2 0.00 2 257.4 6.0 131

5 531.7 516.2 266 0.30 1 246.4 6.1 82

13 546.6 520.3 266 0.30 1 240.2 5.8 148

12 598.0 582.5 5 0.00 1 183.3 7.1 114

9 591.5 562.5 266 0.41 1 175.1 6.7 70

15 443.8 428.3 2 0.00 1 377.6 8.1 148

16 450.9 438.7 274 0.23 1 369.3 7.2 135

17 544.9 539.2 5 0.00 1 271.6 8.4 90

18 595.2 580.8 267 0.42 1 176.9 6.1 76

From Table 3.4 a significant reduction of the ultimate transverse load due to

increasing temperatures can be seen, but the displacement to reach this load seems to

be similar in all tests. Tests 5, 9, 17 and 18 have an unusually low initial stiffness. A

closer inspection after these tests showed some permanent curvature in the specimens.

From the readings of the vertical LVDTs at each end of the specimens, it could be

seen that the actuator moved more than twice the displacement of the roller blocks.


49

Therefore, it can be said that in the tests with the low stiffness the roller blocks carried

enough load to cause the section to act partially as a beam until the bottom flange

reached the lower load introduction plate. In general, only a very small influence of

the axial load could be seen in some tests at higher temperatures.

Test 14 failed before the transverse loading phase of the test could be reached, due to

minor-axis buckling with a buckling length larger than the length of the specimen.

The test damaged the test rig significantly, and additional repair work had to be done.

After this test the lateral bracing system was installed. In order to take some

advantage from this situation the column buckling approach for elevated temperatures

given in the Eurocode 3 Part 1.2 will be used to examine Test 14.

Figure 3.14: Assumed points of rotation in Test 14

From measurements on the deformed test rig the buckling length of the specimens

could be found, which can be seen in Figure 3.14. The distance between the assumed

points of rotation was measured as about 2075mm. Some end restraint can be

assumed, and as the assumed strut was only heated over about a metre, it seems

justifiable to reduce this distance to about 85%, which gives a buckling length of

about 1760mm. The average steel temperature at which buckling occurred was

between 590°C and 600°C. The calculation will assume the temperature to be 590°C.

Having performed the calculation an axial failure load could be calculated to be about

396kN, which is very close to the recorded axial force of about 395kN. Therefore,

from Test 14 it can be concluded that the column approach in the Eurocode predicts

the real behaviour quite realistically.

Assumed point of rotation Assumed point of rotation


50

The results of Tests 1 to 13 have been normalised with respect to Test 11, resulting in

Figure 3.15 below. The chart shows also the temperature reduction factors for yield

stress and Young’s Modulus as they are found in the EC3-1.2.

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500 600 700 800

Temperature [°C]

Nor

mal

ised

for

ce [

-]

Tests

Yield stress reduction factor

Young's modulus reduction factor

Average

Figure 3.15: Normalised resistance again temperature

It can be seen that the ultimate load of the tests reduced with increasing temperature

in a very similar way to the average between the reduction factors for the yield stress

and the Young’s Modulus of mild steel. This confirms the findings by Spyrou (2002).

However, the axial load seems to have little influence on the resistance at

temperatures of 20°C and 450°C, whereas for the higher temperatures a small

reduction can be seen. Unfortunately, uncertainties exist about the amount of friction

which was generated in the roller blocks at the ends of the specimens, the influence of

the two different failure modes on the ultimate load, and the effect of the

displacement speed. Furthermore, at temperatures between 400°C and 600°C the

material properties of the steel reduce significantly, and therefore a small difference in

the steel temperature will have a lager effect on the resistance of the specimen.

3.7.7 Force-displacement behaviour

As was highlighted in the previous chapter, the force-displacement behaviour of the

column web is essential for the development of the component method at elevated

temperatures. It represents the stiffness and the resistance of the compression spring


51

and provides the necessary data for the validation of the numerical and analytical

models.

In the following figures the force-displacement curves of all conducted tests are

shown, sorted by temperature. The displacements shown represent the average of the

two vertical transducers outside the furnace. In cases where, under large

displacements, one of the transducers slipped off its bottom reaction point, the curves

had to be reconstructed based on the displacement reading of the actuator. The

displacements have been corrected to account for thermal expansion of the load-

introduction plates.

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Displacement [mm]

For

ce [

kN

]

Test 3 - 20°C - N = 394kN

Test 6 - 20°C - N = 398 kN

Test 10 - 20°C - N = 265 kN

Test 11 - 20°C - N = 3 kN

Figure 3.16: Force-displacement curves at 20°C

From Figure 3.16 almost no influence of axial load on the resistance of the

compression zone can be seen at ambient temperature.


52

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Displacement [mm]

For

ce [

kN

]

Test 4 - 447°C - N = 3 kN

Test 7 - 454°C - N = 403 kN

Test 15 - 448°C - N = 2 kN

Test 16 - 448°C - N = 274 kN


In Figure 3.17, the test results around 450°C are shown. However, no reduction due

to axial load can be seen between Tests 4 and 7, but by looking at Tests 15 and 16,

performed under very similar conditions, a reduction of 2% for the ultimate load and

about 11% for the displacement under ultimate load can be seen.

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Displacement [mm]

For

ce [

kN]

Test 8 - 553°C - N = 2 kN

Test 13 - 546°C - N = 266 kN

Test 17 - 549°C - N = 5 kN


In Figure 3.18 the test results around 550°C are shown. Comparing Test 8 with Test

13 a difference of 6.7% in the ultimate load can be seen. Unfortunately, it is not

certain wether this reduction is fully caused by the axial load, because Test 8 was

conducted at an average loading speed of 3.53mm/minute and Test 13 at


53

0.90mm/minute, which makes a considerable difference at elevated temperatures.

Furthermore, Test 8 failed in the first failure mode and Test 13 in the second, which

will also make some difference to the capacity of the column web. Test 17 was

conducted using a different batch of steel sections, and can therefore not be compared

with the other tests at this temperature, but will be used for validation purposes.

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Displacement [mm]

For

ce [

kN]

Test 9 - 595°C - N = 266 kN

Test 12 - 601°C - N = 5 kN

Test 18 - 600°C - N = 267 kN


In Figure 3.19, the test results at 600°C are shown. A reduction of 4.4% in the

ultimate load and 5.6 % in the displacement under the ultimate load can be seen. The

loading speed of the two tests was almost identical at 1.09 mm/minute and 1.04

mm/minute, and the failure mode was the same. The only difference is that the

average steel temperature in the web in Test 9 is about 1% lower than in Test 12. If

the rule that the ultimate load reduces due to temperature in a way similar to the

average of the yield stress and the Young’s modulus, as shown in Figure 3.15, this

would result in a 4.3% reduction of the ultimate load of Test 9. That would mean a

reduction of 8.7% due to the axial load in the specimen. It is interesting to note that

Test 18, which has been conducted using a specimen from the same steel batch as

Test 17, had almost the same capacity as Test 9. At about 550°C Test 17 was about

5% stronger than the equivalent test done on a specimen from the first steel batch.


54

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Displacement [mm]

For

ce [

kN

]

Test 1 - 446°C - N = 266 kN

Test 2 - 524°C - N = 390 kN

Test 5 - 546°C - N = 266 kN

Figure 3.20: Various force-displacement curves

Some of the tests gave erratic results, or the temperatures dropped considerably

during the tests. These tests are shown in Figure 3.20.

3.8 Strain-rate effects on steel at elevated temperatures

It should be noted that, due to constraints of the test rig, the tests by the author have

been performed with a considerably higher displacement rate than the tests by Spyrou.

As a consequence of this, the reduction of material strength is lower at elevated

temperatures and strain-hardening is present up to higher temperatures. This contrasts

with observations in material tests with a lower displacement rate (Renner 2005) or in

creep tests (Kirby and Preston, 1988). After this was realised, a research dissertation

was initiated by the author to investigate the influence of strain-rates on the elevated

temperature behaviour of structural steel and also the strain hardening range of the

stress-strain curve. This research dissertation was conducted by Renner, who

performed a series of isothermal tensile tests with varying displacement speeds and

temperatures. She conducted in total 21 tests on coupons taken partly from one of the

column sections of the compression zone test series at ambient temperature presented

in this thesis, and partly from a smaller column section also in S275 steel. Three

different displacement speeds were investigated, 0.7 mm/min, 3.1 mm/min and 6.0

mm/min, which can be converted into strain-rates, if the gauge length of 60 mm is

used, of 0.0122 min-1, 0.0516 min-1 and 0.096 min-1. The two higher strain-rates are

in the range between 0.02 min-1 and 0.2 min-1 given by the appropriate elevated


55

testing standard EN10002-5:1991 (CEN, 1991) for the evaluation of the ultimate

tensile stress of steel at elevated temperatures. However, for the evaluation of the

stresses at small strains a much lower strain-rate of 0.001 min-1 and 0.005 min-1 is

recommended. Therefore, it is questionable if the stresses at low strains found in

these tests are reliable. An overview of the test setup can be seen in Figure 3.21

below.

Figure 3.21: Overview of the tensile test rig (from Renner (2005))

The specimens were heated in a small furnace powered by a 1kW halogen lamp,

developed by Spyrou (2002) and Theodorou (2001), at temperatures between 400°C

and 700°C. The temperature of the specimens was controlled by a dummy coupon

next to the specimen, instrumented with thermocouples, so that the strength of the

specimens was not affected by the thermocouple holes. A considerable effort was

made to archive a uniform temperature distribution within the gauge length, which

finally was achieved by shielding the specimens from the direct radiation of the lamp

with a small U-shaped assembly of firebricks standing over the specimens. With this

technique, a temperature difference within the gauge length of below 10°C was

achieved. The elongation in the gauge length was measured with a digital camera

system developed by Spyrou (2002). The tests were conducted in three phases:


56

firstly, the specimens were heated up to the test temperature. Secondly, the

temperature was kept constant for about 10 minutes before the displacement was

applied, to guarantee a uniform temperature through the cross-section of the coupon,

and finally the specimen was pulled with a constant displacement speed until failure

occurred. The resulting stress-strain curves are shown in Figure 3.22 below for the

material taken from a used specimen of the compression zone test series, tested at the

medium strain-rate. For comparison, the stress-strain curves of EC3-1.2 are shown as

the dashed lines.

0

50

100

150

200

250

300

350

400

450

500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Strain [-]

Str

ess

[N/m

m²]

20°C

400°C

450°C

500°C

550°C

600°C

700°C

Figure 3.22: Experimental isothermal stress-strain curves after Renner (2005)

From the figure above one can see that the stress-strain curves gained from an

isothermal test are quite different from those assumed in the Eurocode. Especially at

temperatures between 400°C and 550°C, the assumption that no strain hardening

exists seems very conservative. Further, it was highlighted by Renner that the

displacement speed has a great influence on the material behaviour. In order to

illustrated that, three tests with different strain-rates at 500°C are compared in Figure

3.23.


57

Figure 3.23: Influence of the strain-rate on the material strength at elevated

temperatures after Renner (2005)

From the figure above, one can see the large influence of the strain-rate on the

material strength. For the slow speed, the experimental stress-strain curve even falls

below the Eurocode curve. For a full description of the testing method and the results

of all tests, the interested reader should refer to the report by Renner (2005). Based

on these material tests at the medium speed level and 240 isothermal tests published

by Kirby and Preston (1988), temperature reduction factors for the stress at different

strain levels could be derived, and are shown in Table 3.5 and Table 3.6.

Table 3.5: Stress reduction factors for 1% and 2% strain

Temperature 1% Strain 2% Strain

°C Renner Kirby and

Preston Renner Kirby and Preston

20 1.000 1.000 1.000 1.000

400 0.875 0.969 0.954 1.000

500 0.726 0.782 0.727 0.773

550 0.567 0.619 0.558 0.590

600 0.435 0.432 0.426 0.397

700 0.202 0.188 0.205 0.178

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Strain [-]

Stre

ss [

N/m

m²]

0.096 min-1

0.0122 min-1

0.0516 min-1

BS EN1993-1-2:2005


58

Table 3.6: Stress reduction factors for 5% and UTS strain

Temperature 5% Strain Ultimate Tensile Stress (UTS) Strain

°C Renner Kirby and Preston Renner Kirby and Preston

20 1.000 1.000 1.000 1.000

400 0.947 1.000 0.922 0.939

500 0.616 0.682 0.588 0.639

550 0.463 0.500 0.439 0.462

600 0.347 0.337 0.318 0.309

700 0.164 0.152 0.146 0.138

From these tables, in general a good agreement between the factors derived from the

results by Kirby and Preston and the values based on the experiments by Renner can

be seen. However, at lower temperatures the results by Renner drop more rapidly

than the values by Kirby and Preston. This may be due to the fact that the reduction

factors by Kirby and Preston are based on the average of all 240 tests and also

because the strain-rate used was not published but may be higher than the medium

rate in Renner’s tests, which is at the bottom end of the recommended strain-rate

range (EN10002-5:1991, CEN, 1991).

3.9 Discussion and Conclusion

This chapter has presented the results of a test series on the force-displacement

behaviour of the compression zone component in the column web, at elevated

temperatures and under the influence of axial column load. Firstly, a brief summary

of other experimental studies on this component was given, the test rig and initial test

were described and finally the test results were presented.

The experiments showed the expected reduction in ultimate load with increasing

temperature due to the loss of strength and stiffness of the steel. No significant

reduction due to axial load was observed at lower temperatures, and only a slight

reduction in resistance and displacement was found at high temperatures, which can

be explained by the increased relative axial load ratio.

Considering the amount of available funding and the fact that, apart from the

hydraulic jacks and the instrumentation, every part of the test rig was developed and


59

designed by the author, good results were generated from the experiments. It should

be kept in mind that the purpose of the experiments was not to give a full and

comprehensive study of the effects of axial load at high temperatures on the

transverse behaviour, but to generate validation cases for numerical and analytical

models. Considering the number of parameters, which had to be controlled during the

series of experiments, namely the specimen temperature and its distribution over the

cross section, the axial load, the friction in the roller blocks and the testing speed, it

proved quite difficult to vary only the axial load and therefore isolate the reduction

due to this parameter.

An investigation associated with the project presented in this thesis by Renner showed

a strong influence of the testing speed, and therefore the strain-rate, on the strength of

steel at elevated temperatures. From this study and from earlier work by Kirby and

Preston a set of strength reduction factors for different strain values up to the ultimate

load could be derived. However, the available experimental data on the material

behaviour of steel under different strain-rates at high temperature is limited, and more

research is necessary in this area.

In order to extend further the experimental study on the influence of axial column

loading on the compression zone behaviour, it would be necessary to investigate

different cross-sections and to increase the axial load ratio of the specimens. To

address these issues one would need a large number of additional experiments and a

strengthened test rig, as the axial capacity of the setup was almost reached. An easier

way of investigating a larger number of parameters is given by the possibilities of the

Finite Element Method, which will be used in the next chapter to fill in the gaps the

experiments left open, to describe the force-displacement behaviour of the column

web under axial loading, transverse loading and elevated temperatures.

Chapter 4: Finite element modelling of the compression zone

60

4 Finite element modelling of the compression zone

4.1 Introduction

In this chapter, a finite element model will be developed and compared with test

results by Spyrou and the tests described in the previous chapter. The purpose of the

model was to perform a parametric study to extrapolate from the tests conducted to a

larger variety of column sections and axial load-temperature combination than was

possible in the experimental study. Towards the end of the project, an alternative way

of modelling the compression zone using 3D solid elements was pursued. The

response of this alternative model was then compared with the initial finite element

model used in the parametric study. However, before the development of the finite

element model is described a brief literature review will be conducted.

4.2 Previous FEM modelling of the compression zone

The numerical modelling of the compression zone in a column web, together with

patch loading on beams and plate girders, has attracted researchers around the world

since the early days of the finite element method. One of the first was Bose et al.

(1972), who used a self-written 3D finite element code to model beam-to-column

joints with special emphasis given to the column web behaviour. Although, the

number of elements used to model the problem was low, a fairly good approximation

of experimental data was achieved. He further conducted an extensive parametric

study on different parameters, including the axial load in the column as well as

stiffeners. Further, Bose proposed a number of design equations based on the

parametric study.

Hendrick and Murray (1984) used an inelastic two dimensional finite element

program to predict the stress distribution and the yield pattern in the column web of

an endplate connection. However, the comparison between the numerical model and

the experiments they conducted was not very good, over-predicting the experimental

stiffness by a factor of about two and failing to develop much plastic deformation.

In 1998 Ahmed and Nethercot (1998) published a paper on the effects of column axial

load on composite joint behaviour in which they used ABAQUS to create shell

models of bare-steel and composite joints with different amounts of axial load in the

column. Unfortunately, how they modelled these beam-to-column joints is not


61

explained in the paper. However, they conclude that the axial load in the column can

be neglected for the moment capacity of a joint.

The first to model the compression zone component at elevated temperatures was the

author (Block 2002) in a previous study. The author used ANSYS 5.7 to model the

column web behaviour with 2D plane and 3D shell elements, and compared the

results with the experiments on the compression zone in fire by Spyrou (2002). It was

shown that the correlation between the 2D models and the tests was very good up to

the point at which plastic buckling of the web occurred. The 3D shell was then used

to predict this out-of-plane effect successfully for the UC 203x203x46 test series.

In the same year, Aribert et al. (2002) used the CASTEM 2000 finite element code to

model the low-cycle fatigue of rolled steel sections subjected to concentrated

transverse loading. They used CU20 elements with 20 nodes and PR15 elements with

15 nodes to model the problem. In the same study, Aribert et al. conducted a number

of fatigue tests which they used to validate the numerical model successfully,

although only a bi-linear stress-strain curve was used and it appears that only one

element through the thickness of flange and web was used.

Later, Zupančič (Vayas et al. 2003, Beg et al. 2004) used the commercial finite

element package ABAQUS to investigate the effects of axial load on the column web

deformation capacity as part of a study into the rotational capacity of endplate joints.

He used 20-noded brick elements in a relatively coarse mesh to simulate the problem

and validated his model against the tests by Kühnemund (2003). In this validation he

assumed the steel grade of the sections used in the experiments as S355, however the

material tests conducted by Kühnemund (2003) showed a yield stress between 246

N/mm2 and 290N/mm2, which means that the models by Zupančič would under

predict the experimental data significantly if the correct material properties are used.

Furthermore, they conducted a parametric study into the effects of axial load onto the

deformation capacity of the compression zone. The resulting empirical equations are

shown in the next chapter.

In the field of patch loading on beams and plate girders, Granath and Lagerqvist

(1999) in Sweden used ABAQUS 5.5 to model plate girders made of high strength

steel (Weldox 700). They used thin shell elements called S9R5 with 9 nodes and

initial imperfections in the form of cosine waves. For the stockiest girder analysed,

with a d/t value of about 40, they showed a large influence of strain hardening on the

patch load resistance. Granath (2000) investigated the elastic behaviour of plate


62

girders under moving patch loads, as found during bridge launching. Again he used

ABAQUS and the S9R5 thin-shell elements for the web, but used beam elements

(B32H) to model the flanges of the girder. Because the flanges were modelled using

beam elements the width of the patch load had to be increased, and for that a value of

ssm = 0.5*(ss+5(tf/2) was used. Comparisons with the results of the numerical analysis

from the study by Granath and Lagerqvist (1999) showed good agreement with the

beam-shell model, although the beam-shell model was found to give slightly stiffer

results.

Around the same time in Norway, Tryland et al. (1999) used LS-DYNA to model the

same high strength steel girders, using shell elements with four nodes and five

integration points through the thickness of the element. Very good comparison

between the slender girders was achieved, but again for the stocky plate girder of a d/t

value of about 40 the analysis under-predicted the capacity found in the experiment

by 11%. Furthermore, the importance of the adopted initial imperfection was

highlighted.

Two years later, Tryland et al. (2001) published a paper concerned with the patch load

resistance of aluminium I-sections and square hollow sections. Having realised that

for stocky sections with a d/t-value of the web below 40, as is normally found in

column sections, shell elements are insufficient as they cannot represent either the

curved form of the root radius properly or the effects of hydrostatic pressure on the

yielding underneath the concentrated load. Therefore, they used LS-DYNA’s 3D

brick elements with eight nodes, and three elements through the thickness of the web

and flange. Although, the brick element models required longer calculation times and

finer meshes, the correlation between the numerical analysis and the experiments

appeared better. However, at the end of the paper Tryland et al. stated that the finite

element models they used in their study became so complex and time consuming that

they saw no real advantage over the use of experimental testing.

What can be seen from the brief literature review above is that there are large

numbers of different ways for modelling the compression zone in a column web.

However, for the column sections it seems most appropriate to use 3D brick elements,

as they can model the very important stress distribution underneath the load better

than shell elements. Nevertheless, such analyses are very time-consuming, and

therefore further alternative ways of numerically representing of the column web


63

component will be investigated. It should be noted further that predictions of the

transverse behaviour of stocky webs gives generally lower results than the

experiments, and that for these webs the plastic part of the stress-strain curve used is

most important.

4.3 2-3D shell element model

As it is the intention of this numerical study to create a model which is suitable for a

parametric study, a way of modelling had to be found which allowed a large number

of analyses to be conducted in a relatively short period of time, within the capacity of

the available computer power and hard-drive space. Therefore, a mixed 2D - 3D shell

model was used, in which the flanges of the column section were very thick shell

elements in-line with the web, so it was possible to model the spread of stress through

the flanges into the web, using shell elements to model the stability and out-of-plane

bending effects in the web.

4.3.1 Modelling of the material for ambient temperatures

In order to achieve a realistic and comparable numerical model of a steel column it is

essential to use the correct non-linear material properties. As was highlighted before,

it is of importance for the modelling of the compression zone that the strain-hardening

part of the stress-strain curve is represented properly. A curve-fit model for this part

of the curve was published by Zheng et al. (2000). The tangent modulus of the strain-

hardening part of the curve can be expressed as

−−=′

y

st

stEEε

εεξexp ...4.1

where E is the elastic modulus; εy is the yield strain; εst is the strain at the onset of

strain hardening, taken as 8εy; Est is the initial strain hardening modulus, taken as

E/40, and E’ is the resulting strain-hardening modulus. The factor ξ is calibrated to

reach the ultimate stress at a strain of 20%.

Because of the differences between the material properties of the flange and web, it

was necessary to use two materials in the finite element model. The material model in

comparison with an ambient-temperature tensile test conducted by Renner (2005) can

be seen in Figure 4.1 below. For this model the factor ξ = -0.0453 was used. As a

comparison, the EC3-1.2 suggestion of a stress-strain curve at ambient temperature is

shown.


64

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 0.20 0.22 0.24 0.26

Strain [-]

Stre

ss [

kN/m

m²]

Tensile test

Material model

EC3-1.2

Figure 4.1: Ambient temperature material model

4.3.2 Modelling of the material for elevated temperatures

To determine the stress-strain curves for high temperatures the stress-strain

relationship and the reduction factors given in EC3: Part 1.2 Figure 3.1 and Table 3.1

are used. Considering the high conductivity of steel, it seems acceptable to assume

the same temperature through the thickness of the column and across the surface.

Therefore, the high temperature stress-strain curves are applied as high-temperature

material properties to the model. This method might be slightly conservative, due to

the non-uniform temperature distribution through the thickness of a column section in

a real fire. In which the core of the flanges and the web will be slightly cooler than

the surfaces, and especially the root radii (important for the local yielding underneath

the concentrated load) will be cooler at their core. This temperature difference may

be small, but the strength and stiffness of the material reduces dramatically in the

range between 400°C and 700°C, which is the critical temperature range for the

compression zone. Therefore, experimental results at elevated temperatures can be

expected to be higher than those for the numerical models.

The stress-strain relationship of steel at elevated temperatures given in EC3: Part 1.2

is formed by two straight lines connected with an elliptical curve. The first linear part

defines the elastic material behaviour until fp,θ , the proportional limit, is reached. The

second linear part starts at a strain of 0.02 and fy,θ , the effective yield strength, is

Stre

ss [

N/m

m2 ]


65

reached and the material becomes plastic. The material model is shown in Figure 4.2

below.

Figure 4.2: Elevated temperature model for S275 mild steel after EC3-1.2

4.3.3 The ANSYS material model

To apply the material properties in the finite element model the Multi-linear Isotropic

Hardening (MISO) option is used. This option allows up to 100 different stress-strain

points, which makes it possible to model non-linear stress-strain relationships,

especially of high temperature curves, with sufficient accuracy. In order to ensure

that the non-linear material is represented adequately in ANSYS the curve is

represented by 42 data points.

The MISO option uses the von Mises yield criterion, which is widely accepted as a

realistic way to calculate the effective stress in multi-axial stress conditions. This

criterion can be applied to many metals, including steel. It uses the principal stresses

to follow the stress-strain curves.

4.3.4 Solution options

ANSYS offers the user a number of different options to solve the large set of non-

linear equations iteratively. The most common ones are described below.

0

50

100

150

200

250

300

350

400

0 0.01 0.02 0.03 0.04 0.05 Strain [-]

Stre

ss [

N/m

m2 ] 100°C

700°C

600°C

500°C

400°C

300°C

200°C

100°C, 200°C,

300°C


66

Newton-Raphson

Because of the plastic behaviour of steel, characterised by non-recoverable strain, it is

necessary to employ a solution method which is able to follow the non-linear stress-

strain curves. Therefore, the well-established Newton-Raphson approach is used.

This method divides the applied loading into steps, called substeps. Before the

solution for each substep is made, the Newton-Raphson method calculates the so-

called out-of-balance force vector, which represents the difference between the

internal forces of the elements and the applied loads. With this vector, the program

performs a linear solution and checks the convergence of the internal forces. If this is

not adequate the stiffness matrix is updated and the next iteration is started. This

procedure is repeated until the solution converges. After this the next substep is

calculated. If the solution does not converge, or other limits such as the maximum

plastic strain increment is reached, the automatic time-stepping feature of ANSYS

performs a bisection and halves the time increment of the substep and before starting

the calculation again.

The Newton-Raphson method, although being very useful, has its limitation. For

example, when the system analysed reaches its buckling load, the system has no

stiffness, and therefore the stiffness matrix is not defined and cannot be solved.

Furthermore, it is not possible for this solution method to follow a ‘down-hill’ path, as

is found in the post-buckling part of the force-displacement curve of the compression

zone. Even though this path is stable there are two possible solutions for the same

applied load. One method to overcome these problems is to use the Newton-Raphson

method in the displacement domain instead of the load domain. However, in the case

of the compression zone this would mean that a large number of substeps are required

in the initial very stiff part of the force-displacement curve. A good alternative to the

Newton-Raphson method is the Arc-Length method, which is described below.

Arc-Length method

The Arc-Length method works like the Newton-Raphson approach, but instead of

searching for convergence with a single criterion, such as force or displacement, it

uses a spherical arc, which represents a combination of both, force and displacement.

The radius of the arc can be changed, in order to achieve a better approximation and

to overcome local instability, by specifying a range of factors used to multiply the

radius of the initial arc. With this method, it is possible to pass over the point where


67

the stiffness of the tangent matrix is zero, to the next point of equilibrium between the

applied forces and the response of the model. Therefore, it is possible to reach the

post-buckling region and its stress distribution. The differences between the Newton-

Raphson method and the Arc-Length method are illustrated for a single-degree-of-

freedom case in Figure 4.3 below.

Figure 4.3: Newton-Raphson and the Arc-Length for a single degree of freedom

(from ANSYS 8.0 User Manual)

4.3.5 Consideration of imperfections

If a perfectly straight steel plate is loaded in pure compression it reaches its buckling

load without any out-of-plane deformations. This effect is called ‘bifurcation’ and

can be described in numerical terms as a singularity of the stiffness matrix. Such a

state is meta-stable, and any magnitude of deformation (positive or negative) can be

assigned to the bifurcation load, if a first order analysis is conducted. Therefore, it is

desirable to prevent such a situation in numerical analysis. However, a real steel

section is never perfectly straight and has always some curvature in the web or the

flanges as well as some ‘out-of-squareness’ meaning that the web is not perpendicular

to the flanges. Therefore, if a column web is loaded in compression these initial

imperfections are amplified by the ‘P-δ’ effect, which creates a smooth way to the

buckling load without bifurcation.

In general, two different forms of imperfections can be used; the first is a small out-

of-plane load applied to the column web, and the second is a change of the geometry

of the column web in the form of a curvature. In an earlier study by the author

(Block, 2002) the former method was used; however it proved difficult to find the

appropriate magnitude of the imperfection force, which should be large enough to

initiate buckling but not large enough to reduce the capacity of the compression zone

significantly. In the present study, the second method was used, by introducing a


68

double sine wave curvature into the web. The equation used to generate this

curvature is:

500

sinsin 21 dz

h

nx

l

ny

=

ππ ...4.2

where n1 and n2 are the half-wave frequencies in the x and z directions respectively; h

and l are respectively the distances between the root radii and the length of the model.

A picture of a typical model is shown in Figure 4.4, in which the amplitude of the

imperfection is scaled by a factor of 10 to make the shape more visible.

x

z

y

d

l

Figure 4.4: Typical imperfection of the finite element model with sine waves

4.3.6 Geometry of the model

Because of the symmetric character of a rolled section and the symmetric loading in

an inner endplate connection, it saves calculation time and computer resource to

utilise symmetry by making longitudinal and transverse cuts through the section and

modelling only one quarter. To include the third dimension, the models are divided

into six areas, and elements with different thicknesses are used. The areas which

represent the flange and the web have the average thickness of the measured

specimens. For the areas of the root radii, an approximation of the thickness was

made as shown in Figure 4.5. These thicknesses are calculated by using equation 4.3,

which is based on a circle with the same radius as the root radius.

( )22, 2 xrrss iroot −−+= ...4.3

where s is the thickness of the web; r is the root radius; x is shown in Figure 4.5.


69

Figure 4.5: Thickness estimation of the 2-3 D model

4.3.7 The finite element type used

As a shell element the ANSYS SHELL181 – Finite Strain Shell was selected. This is

a four-noded element with six degrees of freedom at each node: translation in the x-,

y- and z-directions and rotations about the x-, y- and z-axes. With the SHELL181 it is

possible to include large rotation and large strain effects in the analysis, and non-

linear material relationships can be used. A typical mesh is shown in Figure 4.6

below, both with and without the element thickness shown.

Figure 4.6: Mesh of the 2-3D model (with (a) and without (b) element thickness)

4.3.8 The boundary conditions and the load introduction used

The transverse load was introduced as a point load acting on a node which was

coupled to other nodes along the top flange in order to give a total load introduction

width of 12mm as in the other models. Symmetric boundary conditions were used to

represent the remaining ¾ of the column section. The nodes in the load introduction

Element position

Root radius approximation

Thickness of 2D plane elements

(a) (b)


70

zone were restrained against lateral movement and rotation about the longitudinal axis

of the column.

4.3.9 Mesh study

In order to investigate whether the mesh density was high enough the two meshes

shown in Figure 4.7 below were analysed. The standard mesh was refined in such a

way that each element was split into four, which increased the number of elements

from 1522 to 6088.

Figure 4.7: Element meshes used for convergence study

The resulting force-displacement curves for both meshes are shown in Figure 4.8.

Line of Symmetry


71

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6Displacement [mm]

For

ce [

kN]

Standard mesh

Finer mesh

Figure 4.8: Mesh study on the 2-3D model

As the two lines lie virtually on top of each other an enlarged look is taken at the point

of buckling, and is shown in Figure 4.9.

280

285

290

295

300

305

1 1.25 1.5 1.75 2 2.25 2.5 2.75 3Displacement [mm]

For

ce [

kN]

Standard mesh

Finer mesh

Figure 4.9: Mesh study on the 2-3D model - detail

It can be seen that both meshes behave very similarly but the more finely meshed

model responds in a slightly weaker fashion. However, a difference of 0.1% in the


72

peak load does not justify four times more elements, and therefore it can be concluded

that the standard mesh is sufficient.

4.4 Stress distributions in the 2-3D model

One large advantage of finite element models is that it is possible to investigate the

stress distribution in the modelled material. In the following figures, stress

distributions will be shown for the transverse and the longitudinal direction of the

analysed UC152x152x37 section at ambient temperatures. The figures show the

stress distributions under the peak load of 365kN. It should be noted that the results

of the quarter model have been expanded along the symmetry lines to the full model

in order to illustrate the stress pattern better.

Figure 4.10: Stresses in transverse direction (x) under the peak load

In Figure 4.10, the stresses parallel to the applied load are shown. It can be seen that

most of the web between the loads is still in compression (blue) but a small buckle in

the middle of the web has developed and the membrane stresses have been unloading

the centre of the web at the outside of the buckle. Therefore, small tensile (red)

stresses have developed in the centre of the web. On the back of the section (not

shown) the membrane action has increased the compression stresses. At this stage the

web has almost completely yielded and only stress redistribution and membrane

action can delay the lost of strength. From this point onwards the section has negative

stiffness in the transverse direction.

F

F

Compression Tension


73

Figure 4.11: Stresses in longitudinal direction (z) under the peak load

Figure 4.11, shows the stress pattern perpendicular to the loads. As in the previous

picture, the development of membrane action in the middle of the web can be

observed which is indicated by tensile stresses (red), but more importantly the plastic

hinge mechanism in the flanges can be seen. The first plastic hinge develops

underneath the transverse load. If the load introduction width increases, this inner

plastic hinge will separate into two hinges at the end of the load. On both sides of the

load two further plastic hinges develop at a distance from the inner hinge. This

mechanism has formed the basis of a number of simplified design approaches which

are aimed at predicting the capacity of the compression zone in a column web. The

most accurate of these approaches will be discussed in the following chapters. As the

next step, the 2-3D finite element model developed above is compared with

experimental data.

4.5 Comparison of the numerical model with experimental data

4.5.1 Spyrou’s experiments at elevated temperatures

A comparison between experimental results by Spyrou (2002) at ambient and elevated

temperatures for transverse loading only, and the finite element model described

above is given in Figure 4.12 below.

F

F

Tension Compression


74

0

50

100

150

200

250

300

350

400

450


For

ce [

kN]

Test - 20 C

FEM - 20 C

Test - 400 C

FEM - 400 C

Test - 610 C

FEM - 610 C

Figure 4.12: Comparison of the FEM model and Spyrou’s tests on

UC203x203x46 sections

Good correlation between numerical and experimental results can be seen in the initial

part, but when the load increases the model is weaker and fails at a force about 10%

lower than the tests. This could be due to lower temperatures in the core of the tested

section, and therefore larger residual material strength. In addition, the temperature

reduction factors account for the effects of thermal creep, which are not present in the

steady-state experiments used, due to the relatively fast testing speed. The

significance of these creep effects on the strength and stiffness of steel could be seen

from the material tests presented in the previous chapter. Furthermore, in the EC3-1.2

stress-strain curves no strain hardening effects are considered above 400°C, but strain

hardening has a significant influence on the capacity of the section, because of the

fairly large strains occurring directly under the transverse load and in the plastic

hinges. Furthermore, the displacement of the peak load is under-predicted by the

numerical model, which is due to the limited capability of the model to represent the

stress peaks and the plastic deformations directly underneath the load introduction

plate. It can be seen from the specimens after the tests that the top flange was

indented by about one millimetre on both sides, which correlates with the difference

found in comparing the force-displacement curves.


75

4.5.2 Comparison with the author’s test results

Next, a few of the test results presented in the previous chapter will be compared with

the finite element model. Figure 4.13 shows the results for two ambient-temperature

tests with an axial load ratio of 0% (Test 11) and 30% (Test 3).

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14Displacement [mm]

For

ce [

kN

]

FEM - Test 3

FEM - Test 11

Test 3

Test 11

Figure 4.13: Comparison of the FEM model with the author’s tests at 20°C

A similar effect from the axial load can be observed in the numerical models and in

the experiments. However, as for the models of the tests by Spyrou shown above, the

compression zone capacity is predicted conservatively, and again the peak

displacement is under-predicted.

It was shown at the end of the last chapter that the testing speed has a significant

influence on the strength of steel at elevated temperature. Therefore, the comparisons

between the numerical model and the tests at elevated temperatures have been

conducted with two different stress-strain curves. This has been done firstly using the

material curves recommended in EC3-1.2, and secondly using curves based on the

steady-state material test by Renner (2005).


76

0

50

100

150

200

250

300


For

ce [

kN]

Test 8

FEM -Test 8 - EC3-1.2

FEM - Test 8 - Steady-State

Figure 4.14: Comparison of the FEM model with Test 8 at 558°C

Figure 4.14 shows the comparison between the FEM models and Test 8 with a web

temperature of 558°C and an axial load of 2kN. In the Figure 4.15 below, the force-

displacement curve of Test 9 is compared with the FEM model using the two different

material models.

0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14

Displacement [mm]

For

ce [

kN]

Test 9

FEM - Test - EC3-1.2

FEM - Test - Steady-State

Figure 4.15: Comparison of the FEM model with Test 9 at 591°C


77

In Figure 4.16, the results of Test 13 are compared with the predictions of the 2-3D

model. The test was conducted at a web temperature of 546°C and 266kN axial load.

0

50

100

150

200

250

300


For

ce [

kN

]

Test 13

FEM - Test 13 - EC3-1.2

FEM - Test 13 - Steady State

Figure 4.16: Comparison of the FEM model with Tests 13 at 546°C

Comparing the two types of material model used in the finite element modelling at

elevated temperatures it becomes clear that the results from the EC3 model give

significantly lower values than the models using steady-state material curves. The

reason for this is that the EC3-1.2 approach is based on transient tensile tests, which

implicitly include thermal creep, as was explained in the previous chapter. Therefore,

the two sets of numerical results form an upper (Steady-State) and a lower (EC3-1.2)

bound for the experiments, which have a slower loading speed than standard steady-

state tests and therefore allow some creep effects to happen.

Given the reasonably good comparison between the numerical model and the

experimental data, the real benefit of the simplified shell model, which is calculation

speed, was used in a brief parametric study on the effects of axial load and elevated

temperatures on three different Universal Column sections.

4.6 Parametric study on the effects of axial load

A parametric study on the influence of the axial load in the column section has been

performed in order to investigate a larger variety of section sizes, different


78

temperatures and more axial load ratios. This parametric study was published at the

Structures in Fire Workshop 2004 in Ottawa, Canada (Block et al., 2004a).

The aim of this parametric study was to investigate the influence of the axial load on

the ultimate load and the ductility of the compression zone. In total 50 different

combinations of section type, axial load ratio and temperature were analysed, of

which the details can be seen in Appendix C. In Chapter 6, the results are compared

with design suggestions for the effects of axial load. The study included five different

temperatures: 20°C, 200°C, 450°C, 550°C and 650°C, and axial load ratios between

40% and 80%. Additional models without axial load had to be analysed for

evaluation of the reduction factors. The three different section sizes were

UC254x254x167, UC152x152x37 and UC205x205x46, resulting in a set of d/t-values

for the webs of 10.4 15.5 and 22.3, respectively. These sections represent an

envelope around the slenderness values found in the most common Universal Column

sections. The sections were modelled using the nominal section dimensions and S275

steel. The material models at ambient temperature followed the material model

described above, and at elevated temperatures the EC3-1.2 material model was used.

It was further assumed that the whole cross-section is at the same constant

temperature. A typical set of force-displacement curves is shown in Figure 4.17.

0

50

100

150

200

250

300

350

400

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Displacement [mm]

F [

kN

]

Figure 4.17: Force-displacement curves for UC 203x203x46 at different

temperatures in combination with different axial loads

20°C

450°C

650°C

0 %

50 %

70 %


79

From the force-displacement curves plotted in Figure 4.17, a significant reduction of

the capacity and the ductility of the compression zone is apparent in the presence of

axial load. This reduction increases as the temperature increases, due to the decrease

of the proportional limit of the stress-strain curve and therefore reduced tangent

modulus at low stress levels. Furthermore, the lack of strain hardening at

temperatures above 400°C changes the post-peak strength of the column web.

Whereas at ambient temperature, even for high axial load levels, the web behaviour

recovers slightly due to membrane effects after the peak load, at elevated

temperatures no membrane effects can be developed in the highly strained areas, due

to the lack of strain-hardening. Hence, under high axial loads the post-peak behaviour

decreases rapidly because of the second-order effects in the buckled column web

introduced by the axial load. Furthermore, the numerical results support the

assumption that axial load does not have an influence on the initial stiffness of the

compression zone. From Figure 4.17 it can be seen that initially the results for the

different axial load ratios share the same curve. It is only after the yield strength is

reached that the curves diverge.

4.7 Further FEM study on the compression zone

Towards the end of this project, the results of the finite element study of the

compression zone have been reviewed and further models have been developed in

order to assess the reliability of the 2-3D model used for the parametric study

described above. A generic finite element model of an UC152x152x37 using three-

dimensional solid elements was built and compared with the 2-3D model.

Using 3D solid elements is the most accurate way to model stress distributions in the

section including out-of-plane stresses which are ignored in normal shell elements.

Furthermore, it is possible to model the exact geometry of the section including the

root radius.

4.7.1 Geometry of the 3D model

To be able to compare the different ways of modelling, the same geometrical and

material properties have been used as in the 2-3D model. Due to the symmetry of the

column section and the symmetric loading condition, it is sufficient to model only one

quarter of the section as shown in Figure 4.18 below.


80

Figure 4.18: Typical element mesh of the 3D models (only one quarter is shown)

4.7.2 Load and Boundary Conditions

In a three dimensional environment, it is necessary to constrain a model in six

directions: translation in the x-, y- and z-direction and rotation about the x-, y- and z-

axis. However, by applying symmetrical boundary conditions on two sides of the

quarter model all but one DOF have been restrained. The remaining DOF is

translation in y-direction, which has been fixed at the top of the flange only to allow

the web to buckle laterally.

4.7.3 Pre-deformation in accordance with the eigenvalue buckling shape

In the 2-3D model the web of the modelled column section was pre-deformed using a

series of sine waves. However, this procedure would be too complicated for a 3D

model built of brick elements. Therefore, the feature of ANSYS was used which

allows updating of the geometry of a model with the deformed shape of an earlier

solution. In order to find a deformed shape as close as possible to the buckling shape

of the non-linear analysis, an ‘eigenvalue’ buckling analysis was performed and the

lowest buckling shape found in this analysis was used to update the geometry of the

model of the compression zone. This way proved to be the quickest and most reliable

way of introducing initial imperfections into the numerical analysis of the

compression zone. A typical deformed shape in accordance with the first eigenvalue

buckling mode can be seen in Figure 4.19 below.


81

Figure 4.19: Typical finite element model with eigenvalue buckling imperfections

It should be noted that the deformed shape has been expanded in order to visualise the

model better as it is only necessary to model a quarter of the column section due to

symmetry. Further, the initial imperfection has been scaled by a factor of 20 in the

figure shown above.

4.7.4 Mesh study

The finite element method gives accurate results only when the mesh density is high

enough and a good approximation of the strain and stress has been achieved. A good

way of checking if the mesh is fine enough is to start with a relatively coarse mesh

and then to double the number of elements and compare the calculated displacements

of the model. With increasing numbers of elements, the solution will converge

towards a constant value, which then can be assumed to be the correct result. In the

particular case of the 3D model of the compression zone, it was particularly important

to find the correct number of elements through the thickness of the web and the flange

in order to predict the in-plan bending stresses accurately. Therefore, a model with

two elements through the thickness of the flange and the web has been chosen as a

starting point and then the number of elements through the flange has been increased

until no significant change in peak load and peak displacement could be found. The

three analysed models are shown in Figure 4.20 below.


82

Figure 4.20: 3D model with different numbers of elements through the flange

thickness

The resulting force-displacement graphs of the three different models can be seen in

Figure 4.21 below.

0

50

100

150

200

250

300

0 2 4 6 8 10Displacement [mm]

For

ce [

kN

]

F2W2

F3W2

F4W2

Figure 4.21: Influence of the number of elements through the flange thickness

It can be seen that the difference between the three- and the four-element model is

small and therefore three elements through the flange thickness have been assumed to

be an accurate enough model. In a second step, the number of elements through the

web was varied until convergence was found. The resulting model can be seen in

Figure 4.22 below.


83

Figure 4.22: 3D model with different numbers of elements through the web

thickness

Again, the resulting force-displacement graphs of the three different models can be

seen in Figure 4.23 below.

0

50

100

150

200

250

300


For

ce [

kN

]

F3W2

F3W3

F3W4

Figure 4.23: Influence of the number of elements through the web thickness

Three elements through the thickness give an accurate enough solution considering

the highly increased computing effort of the four elements through the web. In order

to see the conversion of the mesh density better, the peak load and the displacement

under the peak load has been plotted for the number of elements through the thickness

of the flange and the web, shown in Figure 4.24 and Figure 4.25 respectively.


84

200

220

240

260

280

300

1 2 3 4 5Number of elements through thickness

Pea

k lo

ad [

kN]

Flange

Web


thickness on the peak load

0

1

2

3

4

1 2 3 4 5Number of elements through the thickness

Pea

k d

ispl

acem

ent

[mm

]

Flange

Web


thickness on the peak displacement

From the mesh study described above, it can be concluded that three elements through

the thickness of the flange as well as the web gives a good compromise between

accuracy of the results and calculation cost. Another important parameter is the size

of initial imperfection. Therefore, models with three different imperfections between

dw/250 and dw/1000 were analysed and the results can be seen in Figure 4.26 below.


85

0

50

100

150

200

250

300


For

ce [

kN]

F3W3 - dw/250

F3W3 - dw/500

F3W3 - dw/1000

Figure 4.26: Influence of the magnitude of imperfection

A comparison of the displacements found in the experiments with the results of the

imperfection study suggests that an imperfection magnitude of dw/500 is a good

approximation of the real situation.

4.7.5 Deformed shape and stress and strain patterns in the 3D model

As stated above, solid elements give the best representation of the stresses and strains

in a model. They also allow a better visual comparison between the finite element

model and real test specimens than the 2-3D shell element model. Such a comparison

is given in the figures below for one of the test specimens and the 3D model (Figure

4.28) under similar deflections.

Figure 4.27: Deformed shape of test specimen of Test 9


86

Figure 4.28: Deformed shape of the 3D model

As shown for the 2-3D model in Figure 4.10 and Figure 4.11, the stress patterns in

transverse and longitudinal direction under the peak load are shown in Figure 4.29

and Figure 4.30 below.

Figure 4.29: Stresses in transverse direction (x) under the peak load

Very similar stress patterns can be seen from the stress plots above and below in

comparison with the stress distributions predicted by the 2-3D model. Additionally,

the 3D solid model shows the stress distributions in the flanges, which are assumed to

be uniform through the width of the flange in the shell model.


87

Figure 4.30: Stresses in longitudinal direction (z) under the peak load

Furthermore, the von Mises strains are shown in Figure 4.31. It can be seen that

strain levels in the web close to the load is between 3% and 22% which highlights the

importance of the strain-hardening region of the stress-strain curve used to model the

compression zone in a column web.

Figure 4.31: Von Mises strains at peak load

Next the force-displacement curves and the sensitivity of the 2-3D and the 3D models

to axial load will be compared.


88

4.8 Comparison of the axial load sensitivity of the two models

A comparison between the force-displacement curves predicted by the 2-3D and the

3D is shown in Figure 4.32 below.

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10Displacement [mm]

For

ce [

kN

]

3D solid model (F3W3-500)

2-3D shell model - linear

Figure 4.32: Comparison between the force-displacement curves from the 2-3D

and the 3D model

It can be seen that the force-displacement curve predicted by the 2-3D model is

stronger and stiffer than the 3D model. This can be explained by the way the flange is

modelled in the 2-3D model as it is assumed that the flange deformation is uniform

through the width of the flange, whereas from tests and the 3D model it can be seen

that this is not the case. Furthermore, the local deformations and stresses directly

underneath the load are represented better by the 3D solid elements.

After realising the difference between the force-displacement curves of the two types

of models, the influence of axial load on the peak load and displacement was of great

interest. Therefore, the reduction factors of the peak load and the peak displacement

found in the parametric study for the UC 152x152x37 at 450°C have been compared

and the same cases modelled with the 3D solid model. This allowed a first

assessment of the quality of the reduction factors independently of the absolute value.

The force-displacement curves of the 3D models are shown in Figure 4.33 below.


89

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8Displacement [mm]

For

ce [

kN]

N = 0% Npl

N = 40% Npl

N = 60% Npl

Figure 4.33: Effects of axial load on the force-displacement curves (3D model)

The resulting reduction factors for the peak load and the peak displacement are shown

in Figure 4.34 below.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1Axial load ratio N/Npl

For

ce /

Dis

plac

emen

t re

du

ctio

n f

acto

r

Peak Force - 3D Solid

Peak Force - 2-3D Shell

Peak Displacement - 3D Solid

Peak Displacement - 2-3D Shell

Figure 4.34: The axial load reduction factors found in the 2-3D and the 3D model

It can be seen that the reduction factor for the peak load compares accurately with

both models. The reduction of peak displacement however, is predicted higher in the

3D analysis than in the 2-3D model with linear shell elements. This supports the need

for a proper mechanical model for the peak displacement of the column web in

compression.


90


This chapter has summarised the development of a simplified finite element model of

the compression zone. Linear shell elements were used to represent the web and the

flange of the column section. The finite elements in the flange were positioned in line

with the web instead of perpendicular to it. This enabled the modelling of the spread

of the load through the flange into the web. However, very thick shell elements had

to be used and an uniform behaviour of the flange over its entire width had to be

assumed. As a consequence of this, the local stress and strain concentrations

underneath the load were not represented realistically, and therefore the stiffness of

the compression zone was overestimated, which resulted in smaller displacements in

comparison with test results. However, this way of modelling generated very

economic models ideally suited for the use in a parametric study.

During the analyses of the tests results, the effects of thermal creep became obvious

as the models at elevated temperatures under-predicted the capacity of the tests

significantly. The reason for that was the fact that the material model given in EC3-

1.2 was used which is based on transient high temperature material tests and includes

therefore thermal creep strains which make the material weaker. Therefore, the finite

element models were reanalysed using a material model based on isothermal tests by

Kirby and Preston (1988) and by Renner (2005), which resulted in considerably better

correlation between the models and the tests.

Subsequently the developed finite element model was used for a parametric study on

the effects of axial load and temperature on the compression zone. In this study the

axial load was found to have a significant affect on the capacity and peak

deformation. The numerical data generated by this parametric study will be used in

Chapter 6 to validate the analytical model for the compression zone and the derived

axial load reduction factors for the peak displacement.

Towards the end of this project, the quality of the finite element model was improved.

A full 3D model using solid elements was developed using the eigenvalue buckling

shape as geometrical imperfections. The shape and the displacement of the resulting

force-displacement curve compared better with the curves found in the experiments.

An indicative parametric study of the effects of axial load on the peak load and

displacement using the 3D solid element showed that the reduction factors for the

ultimate load found in the parametric study described with the 2-3D element could be


91

confirmed by the more complex model. However, the reduction of the peak

displacement was larger in the 3D model than in the 2-3D model which shows the

sensitivity of this value. Nevertheless, until further research has been conducted, the

results of the parametric study using the 2-3D will be used throughout this project.

Chapter 5: Simplified modelling of the compression zone

92

5 Simplified modelling of the compression zone

5.1 Introduction

After the experimental and numerical investigation of the compression zone

behaviour at elevated temperatures in the previous two chapters, this chapter focuses

on the existing approaches for the force-displacement curve at ambient and elevated

temperatures, which could be used in a connection element. Therefore, a closer look

is taken into existing design approaches for the overall force-displacement behaviour

of the compression zone in the column web of an internal joint. The parameters

commonly used to describe the behaviour of this component will be compared with

test results at ambient and elevated temperatures. In this chapter, the effects of axial

load in the column are not considered, and only tests without axial load are included

in the comparisons. The effects of the axial load will be investigated in the following

chapter.

For column sections with stocky webs the behaviour of the compression zone is

governed by yielding of the web directly underneath the load and formation of a

plastic hinge mechanism in the flanges. After the deflections of the flanges are large

enough it becomes easier for the stocky web to move out-of-plane instead of being

compressed further. At this point, the yielding resistance of the compression zone is

reached and the column web cannot take any further load. For more slender webs,

which can be found in beams or in slender column sections, the behaviour is governed

by the plastic or inelastic buckling resistance of the web. In this mode, failure occurs

due to the reduction of the tangent modulus of the stress-strain curve of the web

material. The boundary between yielding and plastic buckling failure is not fixed. It

depends on the relationship between yield stress and the Young’s modulus as well as

geometrical imperfections of the web and the width of the load introduction zone. If

the slenderness of the web is increased further, buckling might occur before the stress

underneath the load reaches the yield stress, and the web either buckles elastically

over its whole depth or a local instability phenomenon called web crippling occurs.

As slender webs like this are found in plate girders and not in column sections, these

phenomena will not be investigated further.


93

The methodology for this section will be to find the most appropriate ambient-

temperature design approaches for the column web in transverse compression, and to

use these approaches to predict the elevated-temperature response of this component.

The three parameters from which the behaviour can be described are ultimate load, in-

plane deformation capacity and initial stiffness. All of these parameters have been

investigated experimentally in the past, and researchers have developed design

approaches mainly in the development phase of the Eurocodes. The test data will be

used to evaluate the accuracy of the different equations and recommendations.

5.2 Force-displacement curves at ambient temperature

As was highlighted in the second chapter, an accurate model of the force-

displacement behaviour of each connection component is essential to a good

prediction of the overall connection behaviour and also to the development of a

component-based connection element. As part of the development of the component

method a number of approaches at ambient temperature have been developed, which

will be described in this section.

5.2.1 Force-displacement model after Tschemmernegg et al.

Tschemmernegg et al. (1987) conducted a large number of tests on the behaviour of

the compression zone, as part of the development of a mechanical model for semi-

rigid joints, as described in Chapter 2. They defined the elastic stiffness and elastic

capacity of the compression zone, as well the plastic resistance and the displacement

capacity of this component. With these two points, a bi-linear force-displacement

curve can be generated, as seen in Figure 5.1.

Figure 5.1: Force-displacement curve after Tschemmernegg et al.

δ

Fel

kini

δu

Fpl

F


94

5.2.2 Force-displacement model after Eurocode 3–1.8

In EC 3-1.8, an elastic-fully plastic force-displacement curve for each component is

given, forming the foundations of the component method for the prediction of the

stiffness and resistance of a semi-rigid joint. This curve is based on the initial

stiffness kini and the capacity of the component FRd, and can be seen in Figure 5.2

below.

Figure 5.2: Force-displacement curve after EC3-1.8

It should be noticed that there is no displacement limit at component level; a ductility

limit is only introduced after all components are combined to form a moment-rotation

curve. A simplified force-displacement approach like the one in the Eurocode can

only give a crude approximation of the real behaviour of this component, but it is

simple enough to be used in hand calculations.

5.2.3 Force-displacement model after Kühnemund

Kühnemund (2003) developed a multi-linear representation of the compression zone

including the post-buckling stage. The initial behaviour of his approach is based on

modifications to the approach given in EC3-1.8. He used a modified version of the

initial stiffness, as can be seen later in this chapter, and 2/3 of the resistance value

given in EC3-1.8 to define the elastic range. As the next point on the force-

displacement curve, Kühnemund used the component resistance and a displacement

4.5 times larger than the displacement at the end of the elastic range. Furthermore, he

developed an equation to predict the ultimate load of the compression zone, which

will be described in more detail later in this chapter. In order to describe the post-

buckling stage of the compression zone he developed a yield-line model. The

displacement under which the resistance of the compression zone is reached was

δ

F

FRd

kini


95

defined as the point at which the resistance of the yield line model is equal to the

predicted ultimate load. A schematic view of this approach is shown in Figure 5.3.

Figure 5.3: Force-displacement curve after Kühnemund

The approach by Kühnemund is quite sophisticated, but because the rather complex

yield-line model has to be solved iteratively it is not easy to predict the displacement

under the peak load and therefore the end of the stable part of the force-displacement

curve.

5.2.4 Conclusion

Assessing the available models for the force-displacement behaviour of the column

web in compression has shown that the three main characteristics used to generate the

spring behaviour of this component are resistance, initial stiffness and deformation

capacity. In the following sections, all of these parameters will be investigated, first

at ambient temperature and then at elevated temperatures.

5.3 The resistance of the compression zone at ambient temperature

The phenomenon of the behaviour of column webs under concentrated loads has been

the subject of many research projects over the last 45 years. A good summary of the

early research is given by Hendrick and Murray (1984). However, to quote

Lagerqvist and Johansson (1996): “Generally, the resistance to concentrated forces is

considered as being of a very complex nature where it is almost impossible to derive

closed theoretical solutions. Therefore, all studies aiming at predicting the ultimate

resistance of steel girders to concentrated forces end up with more or less empirical

solutions.” This comment has remained true up to the time when this thesis was

written as no purely mechanical solution could be found. Nevertheless, three different

δ

kini

Fpl

4.5 δel

2/3 Fpl

δel

Fu

Yield-line model


96

grades of sophistication of the approaches found in the literature can be categorised.

These different types of approaches for the ultimate load can be summarised as

follows:

• The first group includes purely empirically derived equations based on a large

number of test results or on numerical parametric studies. It is typical for this type

of equation to have a large number of parameters and constant factors.

• In the second group, equations based on the effective width concept are combined.

This type of equation assumes a certain load spread through the flange and the root

radius into the web, which is mainly based on experimental results. In some

instances the effective width is reduced by reduction factors accounting for

stability failure of the web.

• The third and final group includes all approaches using mechanics to predict the

ultimate resistance of the compression zone. In these approaches, plastic

mechanisms in the flanges or the web are combined with plate buckling

considerations. However, all of these approaches include one or more constant

factors in order to calibrate the equations to the available test results.

In the following section, approaches from all groups will be compared with

experimental results collected from different sources in the literature, in order to

identify the most accurate approach at ambient temperature.

5.3.1 Statistical comparison of the resistance approaches at ambient

temperature

The experimental results used for evaluating the different design approaches have

been collected from different sources in the literature (Aribert et al. (1990), Roberts

and Newark (1997), Bailey and Moore (1999), Spyrou (2002), Kühnemund (2003)

and De Mita et al. (2005)). In addition, the ambient-temperature tests by the author

are included. The database includes 106 tests on European, British and American

rolled H- and I-sections with d/t-values between 11.5 and 50, with an average d/t ratio

of about 24. Further, web yield stresses from 213 N/mm2 up to 856 N/mm2 with an

average value of about 350 N/mm2 are included. Therefore, the range of tested

sections exceeds the dimensions and properties commonly used in column sections,

which should give a good indication of the flexibility of the approaches. However, all

the necessary geometrical and material data are not always published. Therefore,

sensible assumptions have been made for the missing data. In the cases where


97

geometrical data is missing, the values given in the appropriate design codes have

been used. If only the material data for the web is given, the same values are assumed

for the flanges. For the Young’s Modulus, a standard value of 205000N/mm2 has

been assumed. For the load-introduction width, based on the tests by Dubas and

Gheri, found in Roberts and Newark (1997), a value of 15mm was assumed. The

detailed experimental data can be found in Appendix D.

In total, ten approaches for the resistance of the compression zone have been

compared with the database. The statistical evaluation has been performed based on

the ratio of the resistance found in the test to the predicted resistance. From this ratio

the mean value, the standard derivation, the coefficient of variation, the lower and the

upper 5% percentile and the correlation coefficient can be calculated. The results of

the evaluation are given in Table 5.1.

Table 5.1: Statistical comparison of ambient-temperature design approaches

No Approach Mean value

Standard deviation

Coefficient of

variation

Lower 5 % percentile

Upper 5% percentile

Correlation coefficient

1 prEN 1993-1-1: 20xx

1.422 0.176 0.124 1.131 1.712 0.988

2 BS EN 1993-1-8: 2005

1.498 0.219 0.146 1.137 1.858 0.982

3 DIN 18800-1:1990 1.332 0.300 0.227 0.830 1.816 0.939

4 BS 5950-1:2000 1.417 0.245 0.171 1.018 1.816 0.971

5 Empirical equation 1.153 0.149 0.129 0.908 1.397 0.988

6 Lagerqvist and Johansson (1996)

1.349 0.161 0.120 1.084 1.614 0.988

7 Kühnemund (2003) 1.219 0.143 0.117 0.984 1.454 0.987

8 Block (2002) 1.195 0.155 0.129 0.941 1.450 0.985

9 Roberts and Newark (1997)

1.520 0.222 0.146 1.154 1.885 0.982

10 Faella et al. (2000) 1.509 0.222 0.147 1.144 1.875 0.939

From the table above it can be seen that all the approaches predict the experimental

results reasonably well, with correlation coefficients of above 0.93. However, some

of the approaches have a lower 5% percentile of below 1.00 which means they over-

predicted the tests and are therefore unconservative (3, 5, 7, 8). Some approaches

have a large coefficient of variation, which means they are not very accurate (3, 4). It

is interesting to note that the approaches in the German and British design codes

predicted the experiments rather inaccurately, and that approach number 1, which


98

gives more accurate results than approach 2, has been removed from the final version

of EC3 Part 1.1 (CEN, 2005c).

In order to reduce the size of this section only the five most accurate approaches, with

correlation coefficients lager than 0.98 and coefficients of variation smaller than 0.13,

will be described further. The approach in the current version of EC3-1.8 is shown, as

well.

5.3.2 Resistance approach after Eurocode 3

The approach in EC 3-1.8 can be classified as a Group 2 approach. It deals with the

behaviour of the column web in transverse compression as part of the component

method. The approach uses a single equation to calculate the yielding resistance,

which may be reduced for relatively slender webs to account for instability effects.

Furthermore, the effects of panel shear and axial load in the column are considered by

reduction factors. These equations are based on the work by Aribert et al. (1990), and

can be seen in Equation 5.1 below.

, , ,, ,

0

wc eff c wc wc y wc

c wc Rd

M

k b t fF

ω

γ= , , ,

21

0.2wc eff c wc wc y wc

M

k b t fω λ

γ λ

−≤

...5.1

where λ gives the slenderness of the web defined as Equation 5.2.

, , , , , ,

3( ) 2

2

0.932

3(1 )

pw eff c wc wc y wc eff c wc wc y wc

w

wccr wc

wc

F b t f b d f

E tF E t

d

λπ

ν

= = =

−

...5.2

in which Fpw is the crushing resistance and Fcr is the elastic buckling resistance of the

web panel in the case of double punching. The notation twc is the column web

thickness, fy,wc is the yield strength of the web and E is the Young’s Modulus. A

reduction factor ω is used to account for shear effects in the column web, which is

taken as 1.0 in the investigated case of an internal joint, γM0 and γM1 are material

safety factors (taken as 1.0) and dwc is the clear depth of the column web. Buckling of

the web becomes critical if the slenderness 0.72λ ≥ , otherwise the capacity of the

column web is limited by yielding of the web over the effective breadth beff given by

Equation 5.3.

( ) ( ), , 2 min ; 2 5eff c wc fb f p f p fcb t a t u a t t s= + + + + + + ...5.3

where tfb is the thickness of the beam flange in compression, af the weld thickness

between the beam flange and the endplate and tp is the endplate thickness. The


99

notation u is the distance between the beam flange and the edge of the endplate, tfc is

the column flange thickness, and finally the parameter s is given as either the root

radius of the column for rolled sections or the weld thickness s = ca2 for welded

columns. The reduction factor kwc for the axial load in the column will be described

in the next chapter. A comparison between the test results and the Eurocode approach

is given in Figure 5.4 below.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.4: Comparison of the resistance after EN 1993-1-8:2005 with tests

The thick red line (used in all the following comparisons between experimental data

and predicted parameters) represents the mean value of the predicted results and the

thin red lines indicate the position of the mean value plus and minus the standard

deviation. These lines shall help to illustrate the accuracy of the different approaches.

From Figure 5.4 it can be seen that the approach in EC 3-1.8 is not very accurate and

underpredicts the test results in most cases quite significantly.

5.3.3 Resistance approach using an empirical equation

An empirical equation probably obtained by statistical methods from some database

has been proposed in Germany and was found in Aribert et al. (1990) with no detailed

reference to the author given. This approach can be seen in equation 5.4 below.

0.28572

23524 2Rd w p w yw

yw

F t t t ff

= +

...5.4

Equation 5.4 considers neither the thickness of the flange, which makes it problematic

to use it with fabricated sections, nor the width of the load. The comparison of the


100

resistance predicted by the empirical equation with the test results is given in Figure

5.5 below.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.5: Comparison of the resistance after the empirical equation with tests

It is surprising how accurately this very simple equation predicts the ultimate load of

the compression zone. However, because not all parameters are considered special

cases with large load introduction width or axial load in the column can not be

assessed properly. It is further questionable how an approach which does not consider

the Young’s modulus explicitly will perform at elevated temperatures, as the

reduction of the Young’s modulus is very significant at elevated temperatures.

5.3.4 Resistance approach after Block

The author also developed an empirical equation based on the available test results

and numerical studies as part of an earlier work (Block, 2002). In this study, a three

dimensional finite element model was used to predict the ultimate load of a rolled

column section similar to the FEM models presented in the previous chapter. This

model of a UC 203x203x46 grade S275 column section was used to vary the web

thickness, the web depth, the flange thickness, the load introduction width and the

yield stress by about ± 50% of the nominal values. The resulting Equation 5.5 is

shown below.

( )

( )

0.42

2 0.35 0.65,

/0.75

1.1

f f y

u w w y w

w y

t r b lt E f

t l c

+ Ρ = +

− ...5.5


101

Where Ew and fy,w are the Young’s Modulus and the yield strength of the column web,

respectively. Furthermore, tw is the thickness of the web and the value tf is the

thickness of the flange, r the root radius and bf the width of the flange. The value c is

the load width and the length ly is equal to distance between the outer plastic hinges,

as specified by Roberts (1981) and shown in Equation 5.6 below.

2

,

,

2 y f f f

y

y w w

f b tl c

f t= + ...5.6

A comparison of the resistance predicted by the equation developed by the author

with the test results is given in Figure 5.6 below.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.6: Comparison of the resistance after Block with tests

This approach correlates well with the experimental results but it is slightly

unconservative. Due to the large number of parameters considered it is likely to be

flexible and to perform well. However, this approach is not based on mechanics and

is therefore less favourable than mechanical models.

5.3.5 Resistance approach after Lagerqvist and Johansson

Lagerqvist and Johansson (1995, 1996) conducted extensive studies into the

behaviour of plate girders subjected to concentrated forces on the flanges, and

developed an analytical approach based on a modification of the plastic hinge model

developed by Roberts and Rockey, summarised in Roberts (1981). As in the original

approach, a length ly is calculated, over which the web is assumed to reach the yield

stress. This is found by equating internal and external energies in the mechanical


102

model shown in Figure 5.7, minimising the load Fw with respect to the distance

between the outer plastic hinges.

Mo Mo

Mi Mi

fyw tw

ly

c+2tf

Fw

Mi Mo

bf bf

tf

0.14dw

Figure 5.7: Assumed plastic mechanism in the column flange after Lagerqvist

The only modification by Lagerqvist and Johansson of the original plastic mechanism

by Roberts and Rockey was that they included a part of the column web in the

calculation of the plastic moment resistance of the outer plastic hinges. This effective

T-section was only assumed for the outer plastic hinges because here an in-plane

stress combination of bi-axial compression exists, which, according to the Von Mises

yield theory, increases the yield resistance of the material. In the inner plastic hinges

a stress combination of tension and compression occurs, which reduces the yield

resistance, and therefore the contribution of the web was ignored. For some

unexplained reason the contribution of the web was ignored in the case of stocky

sections with a plate slenderness factor λF smaller than 0.5. The yielded length of the

web proposed by Lagerqvist and Johansson is given in Equation 5.7 below.

( )1 22 1y fl c t m m= + + + ...5.7

with 1yf f

yw w

f bm

f t= and

2

2 0.02 w

f

dm

t

=

if λF < 0.5 then m2 = 0

The load width c is calculated by using a dispersion angle of 45° through the end-

plate, dw is the clear depth of the web, and tw and tf are the thicknesses of web and

flange respectively. Furthermore, fyw and fyf are the yield stresses of web and flange.

The yield length is reduced if the plate slenderness factor λF exceeds the value 0.53 as

it is assumed that stability effects become important for higher slenderness. This

slenderness factor (Equation 5.8) is defined in the classical way as the square root of

the yield resistance over the elastic buckling resistance of the web.

y y w yw

F

cr cr

F l t f

F Fλ = = ...5.8

where the elastic buckling resistance is given by Equation 5.9 below.


103

( )

3 32

20.9

12 1w w

cr F F

w w

t tEF k k E

d d

π

ν= ≅

− with υ = 0.3 ...5.9

Lagerqvist and Johansson found the buckling coefficient kF by conducting a

numerical parametric study, which resulted in a set of rather complex equations.

These equations were then simplified for design purposes to:

2

3.5 2 w

F

dk

a

= +

for opposite loads i.e. in an internal joint ...5.10

2

6 2 w

F

dk

a

= +

for a single load i.e. in an external joint ...5.11

where a is the distance between web stiffeners. In the case of an unstiffend column

web a becomes infinite and kF becomes 3.5 and 6, respectivley. Following the

classical approach further a χ-function had to be defined which relates the resistance

to the slenderness parameter. To do so, a comparison with a large number of test

results revealed the χ-function shown in Equation 5.12 below.

0.47

0.06 1.0F

χλ

= + ≤ ...5.12

Finally, by using the χ-function, the yield length, the web thickness tw and the yield

stress fyw, the capacity of a web under transverse compression can be calculated as

shown in Equation 5.13.

1

yw w F y

Rd

M

f t lF

χ

γ= ...5.13

Again, this approach has been compared with test results, as can be seen in Figure 5.8.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.8: Comparison of the resistance after Lagerqvist et al. with tests


104

From the figure above a close correlation between the tests and the approach by

Lagerqvist and Johansson can be seen. Furthermore, this approach is mostly based on

mechanics and an extensive numerical parametric study for the buckling coefficient.

It was originally developed for the design of large plate girders with very slender

webs, and the fact that it performs very well with stocky columns shows that the

mechanical approximation represents the real behaviour very well.

5.3.6 Resistance approach after Kühnemund

Kühnemund (2003) based his approach on the work by Ungermann and Sedlacek

(1994) but used the ultimate stress of the web material in his approach rather than the

yield stress to predict the ultimate load of the compression zone. He also derived

correction factors for the ultimate load to fit test results depending on the ratio

between the yield stress and the ultimate stress.

Ungermann’s model starts by assuming an equivalent stress block for the elastic

stresses in the fillet between the root radius and web underneath a point load. This

length is used as the width of the buckling field in the web. He further assumes that

the material in the web is elastic in the direction perpendicular to the load but behaves

plastically with a reduced stiffness parallel to the load. Using the differential equation

of an orthotropic plate, the reduced stiffness at which the assumed buckling field fails

may be found. The equivalent stress block is then calculated by using the reduced

stiffness for the web but the elastic stiffness for the flange, which results in the

effective width shown in Equation 5.14 below.

3 2

,7

, , , 2,

2.359 c fc wc

eff c wc u

u wc

J t Eb

f= ...5.14

Where Jc,fc is the second moment of area of the flange and fu,wc the ultimate stress of

the web material. As this approach is based on point loads, the width of the stiff load

introduction zone has to be added to Equation 5.14. With the effective length beff,c,wc,u

and the correction factor k1, it is then possible to calculate the ultimate resistance as

shown in Equation 5.15.

, , 1 , , , ,c wc u eff c wc u w u wcF k b t f= ...5.15

with

( )( )

, ,

1

, ,

0.23 / 1.05

0.20 / 0.92

u wc y wc

u wc y wc

f f component testsk

f f welded and bolted connections

− +=

− +


105

for , ,1.05 / 2.00u wc y wcf f≤ ≤

A comparison between the approach by Kühnemund and the test database is shown in

Figure 5.9. For the tests where no information about the ratio of the yield and the

ultimate stresses was available, a factor of 1.525 (being the mean value of the range)

was assumed.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.9: Comparison of the resistance after Kühnemund with tests

The approach by Kühnemund predicts the tests results very well and it is easy to use.

However, the approach does not consider the plastic mechanism in the flange and is

based on point loads, which means that it is not certain how the effects of an

increasing load width are predicted.

5.3.7 Conclusion

From the statistical evaluation and the critical discussion of ten design approaches for

the resistance of the compression zone, it can be seen that the approaches by

Lagerqvist and Johansson, and by Kühnemund are the most accurate ones. The

approach by Lagerqvist and Johansson is slightly favourable, as its mechanics are

more transparent and the plastic hinge mechanism in the column flange is considered.

Therefore, this approach will be extended to elevated-temperature conditions.

5.4 Initial stiffness of the compression zone at ambient temperature

The second parameter commonly used to generate a force-displacement curve for the

compression zone is the elastic or initial stiffness. This parameter influences the force

distribution within a connection as well as the distribution of bending moments in a


106

frame with semi-rigid joints. The same approach will be taken as for the ultimate

resistance, and the initial stiffness measured from ambient temperature test results will

be compared with different design approaches. The most accurate approach will then

be used to predict the initial stiffness at elevated temperatures.

5.4.1 Statistical comparison of existing approaches for the initial stiffness

The initial stiffness is a parameter which is very difficult to measure due to the small

displacements in the elastic stage of the tests. Therefore, in most experiments found

in the literature this parameter has not been reported. However, in the tests reported

by Kuhlmann and Kühnemund (2000) and by De Mita et al. (2005) this parameter

was given. Furthermore, the author could extract the initial stiffness values from the

test results of Spyrou (2002) as well as the initial stiffness values from the

experiments conducted within this study. All together, this results in 38 tests. In total

six approaches for the initial stiffness of the compression zone have been compared.

As before, the statistical evaluation has been performed based on the ratio of the

stiffness found in the tests for the predicted stiffness of one side of the compression

zone. From this ratio the mean value, the standard deviation, the coefficient of

variation, the lower and upper 5% percentiles and the correlation coefficient can be

calculated. The results of the evaluation are given in Table 5.2 below.

Table 5.2: Statistical comparison of ambient-temperature stiffness approaches

No. Approach Mean value

Standard deviation

Coefficient of variation


Upper 5% percentile


1 BS EN 1993-1-8: 2005

0.284 0.077 0.270 0.157 0.410 0.632

2 Huber and Tschemmernegg (1996)

0.993 0.268 0.270 0.551 1.434 0.632

3 Faella et al. (2000)

0.426 0.111 0.260 0.244 0.609 0.611

4 Kühnemund (2003)

1.182 0.320 0.270 0.656 1.708 0.632

5 Aribert et al. (2002)

0.633 0.117 0.185 0.441 0.825 0.686

6 Aribert and Younes

0.635 0.114 0.180 0.447 0.823 0.640

From the statistical evaluation of the initial stiffness, it was found that none of the

assessed approaches predict the experiments very accurately, which can be seen


107

clearly by correlation coefficients lower than 0.7. Further, apart from the approaches

by Kühnemund (2003) all equations overestimated the initial stiffness, especially the

approach in the Eurocode that predicts a mean stiffness value over three times that

found in the experiments. The approach by Huber and Tschemmernegg (1996) yields

the most accurate mean value. It is of interest that the approaches by Kühnemund and

by Huber and Tschemmernegg give the same accuracy and correlation as the

Eurocode approach, as they are all based on the same principles apart form empirical

correction factors of the mean value. The two approaches by Aribert et al. gave the

smallest coefficients of variation and the best linear correlation between the measured

and the predicted results, expressed by the correlation coefficient. The Eurocode

approach and the best alternative approaches will be discussed in detail below.

5.4.2 Initial stiffness approach after Eurocode 3-1.8

The Eurocode approach originates from the PhD Thesis of Jaspart (1991), and is

based on the assumption that the column flange acts like a beam on an elastic

foundation with elastic effective width equal to 0.7 of the effective width used to

calculate the plastic resistance, which can be seen in Equation 5.16. Because only one

side of an internal joint is investigated in this study, the depth of the column web dwc

is halved.

, ,, ,

0.7

2

eff c wc wc

c wc ini wcwc

b tk E

d= ...5.16

with ( ) ( )2 min ; 2 5eff fb f p f p fc cb t a t u a t t r= + + + + + +

A comparison between the stiffness approach in the Eurocode and the test results can

be seen in Figure 5.10 below.


108

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000 4500kc,wc,ini,k [kN/mm]

kc,

wc,

ini,t

est [k

N/m

m]

Figure 5.10: Comparison of the initial stiffness after EN 1993-1-8 with tests

It is clear from the comparison above that the approach for the initial stiffness in

Eurocode 3 gives very unconservative results compared with the stiffness found in

component tests.

5.4.3 Initial stiffness approach after Aribert et al.

Aribert et al. (2002) conducted an experimental and numerical simulation of the

column web component in tension and compression to establish rules for low-cycle

fatigue. As part of this study, they developed an empirical equation for the initial

stiffness based on a comparison with finite element models. The approach is shown

below in Equation 5.17.

3

40.45 fc fc wc

wc

b t tk E

d= ...5.17

How this equation predicts the initial stiffness found in the tests can be seen in Figure

5.14 below.


109

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200kc,wc,ini,test [kN/mm]

kc,

wc,

ini,t

est [k

N/m

m]

Figure 5.11: Comparison of the initial stiffness after Aribert et al. with tests

According to De Mita et al. (2005) Equation 5.18 was refined by Aribert and Younes

to the following Equation 5.18.

3 2

40.95 fc fc wc

eff wc

b t tk E

b d= ...5.18

Where beff is the effective length according to Equation 5.3. Again, the prediction of

the equation can be seen below.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200kc,wc,ini,test [kN/mm]

k c,w

c,in

i,tes

t [kN

/mm

]

Figure 5.12: Comparison of the initial stiffness after Aribert and Younes with

tests

Both equations still yield unconservative predictions of the test results but the

accuracy is increased significantly.


110

5.4.4 Conclusion

From the statistical evaluation and the discussion of six design approaches for the

initial stiffness of the compression zone it can be seen that the two approaches by

Aribert are the most accurate approaches. Nevertheless, both are still unconservative,

and so a further modification of the constant factor should be considered. However,

both approaches are based on comparisons with numerical studies at ambient

temperatures and not on mechanics. Therefore, it is not clear if the approaches will be

equally good at elevated temperatures and the same comparison has to be made with

all approaches at elevated temperatures.

5.5 The deformation capacity of the compression zone

The third parameter needed to specify the force-displacement curve, and therefore the

spring characteristics of the column web in compression, is the deformation capacity.

This parameter becomes quite important for plastic design of a frame where rotational

capacity of the joint is required, as the compression zone has limited ductility due to

stability failure of the web in compression. Conservatively, the deformation limit of

this component is defined as the point when the maximum load is reached, as shown

in Figure 5.13.

Figure 5.13: Definition of the deformation limit of the compression zone

The component method in EC 3-1.8 does not consider the ductility of each component

individually. It rather gives limits for the overall rotation of the joint assuming that

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Displacement [mm]

For

ce [

kN

]

unstable stable

deformation limit


111

components with a large ductility, like the column flange or the endplate in tension,

fail under a smaller load than the components with limited ductility, for example the

bolts in tension or the column web in compression. However, if the component

method is used in a non-linear finite element, each component needs a ductility

criterion, otherwise the real behaviour of the connection cannot be predicted.

Therefore, the same procedure used for the resistance and the initial stiffness will be

performed for the ductility of the column web.

5.5.1 Statistical comparison of existing approaches for the deformation

capacity

As with the initial stiffness, the deformation capacity of the compression zone has not

been reported in the early experiments because only the strength of the compression

zone, and not the whole force-displacement relationship, was the research priority at

the time. However, in the tests reported by Kuhlmann and Kühnemund (2000) and

the experiments by De Mita et al. (2005) this parameter was given. Furthermore, the

author could extract the displacement under maximum load from the test results of

Spyrou (2002) as well as the ductility found in the experiments conducted within this

study. Altogether, this results in 24 tests. Only three different approaches for the

deformation capacity of the compression zone could be found in the literature. This is

an indication of the relatively early stage of research into the detailed calculation of

the ductility of joints. These three approaches will be compared statistically in the

same way as the resistance and the initial stiffness approaches previously, and the

results of the evaluation are given in Table 5.3 below.

Table 5.3: Statistical comparison of the displacement approaches


Standard deviation

Coefficient of

variation


Upper 5% percentile



1.003 0.242 0.241 0.605 1.401 0.710

2 Vayas et al. (2003)

0.655 0.185 0.282 0.352 0.959 0.800

3 Block 1.213 0.303 0.250 0.715 1.712 0.882

From the statistical comparison above, it can be seen that approach 1 gives the most

accurate results. However, as this approach can only be used for European sections it


112

was compared with only 16 test results instead of the 24 tests of the whole database.

From the remaining two approaches, the one by the author compares more closely

with experimental data. All three approaches will be described in the next section in

more detail.

5.5.2 Deformation capacity approach after Huber and Tschemmernegg

Huber and Tschemmernegg (1996) suggested limits for the displacement of the

column web at its failure load based on component tests similar to the ones carried out

in this project, but at ambient temperatures and without axial load in the specimens.

The studies were conducted on European column sections of type IPE, HEA, HEB

and HEM. Unfortunately, no information about the applicability of these limits to

British or American sections was given. Therefore, the comparison could only be

conducted sensibly for the tests on European sections. The proposed ductility limits

are summarised in Table 5.4 below. It is assumed that these values are for one side of

the compression zone.

Table 5.4: Deformation capacity after Tschemmernegg and Huber

IPE HEA HEB HEM

Deformation capacity [mm]

1.5 3.0 5.0 7.5

How these deformation limits compare with the test results can be seen in Figure 5.14.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16δδδδu,Rk [mm]

δδ δδu,

Tes

t [m

m]

Figure 5.14: Comparison of the displacement capacity after Huber and

Tschemmernegg with tests


113

It is interesting to see how good these simple deformation limits are compared with

displacements found in the tests on European sections. No limits for British or other

international section types or built up sections are given and therefore, this way of

predicting the deformation capacity is not very acceptable and cannot be used for the

connection element.

5.5.3 Deformation capacity approach after Vayas et al.

Vayas et al. (2003) investigated the deformation capacity of the column web

numerically and derived empirical equations for the ultimate equivalent strain εu in

the column web based on a parametric study on a wide range of European HEB

sections. Axial loads up to 50% of the squash load of the column section were also

considered but only a small influence for stocky sections was found. The finite

element models used in the parametric study were not validated properly, which was

highlighted by Coelho (2004). The resulting equation is:

u u dδ ε= ...5.19

where δu is the deformation under ultimate load and d the clear depth of the web. For

no axial load in the column the following Equations 5.20 for different slenderness

values are recommended by these researchers.

[ ]

18.5 0.75 20

% 5.7 0.11 20 33

2.07 33

w w

u

w w

w

d d

t t

d d

t t

d

t

ε ε

εε ε

ε

− <

= − ≤ <

≤

...5.20

For axial loads larger than 10% of the squash load of the section Vayas et al.

suggested the following Equations (5.21).

[ ]

( )9.4 0.34 15 0.75 0.5 20

% 4.8 0.11 20 33

1.17 33

w w w

u

w w

w

d d dn

t t t

d d

t t

d

t

ε ε ε

εε ε

ε

− + − − <

= − ≤ <

≤

...5.21

With 235

yf

ε = and pl

Nn

N=


114

In the publication of this approach, it was not stated if it predicts the deformation of

both sides of the flange or only of one. Nevertheless, the calibration of the finite

element model, on which the approach is based, has been conducted on the force-

displacement curve of one of the tests by Kuhlmann and Kühnemund (2000), which is

known to be only the displacement of one side of the compression zone. Therefore, it

was assumed that the approach of Vayas et al. predicts the deformation of only one

side. This approach has been compared with the available test results in Figure 5.15.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16δδδδu,Rk [mm]

δδ δδu,

Tes

t [m

m]

Figure 5.15: Comparison of the displacement capacity after Vayas et al. with

tests

From Figure 5.15 it can be seen that the approach by Vayas et al. generally

overestimates the displacements found in the experimental studies. The approach is

based on only three parameters, namely the web thickness, the depth of the web and

the yield stress. Therefore, it relies on the geometrical ratios within the rolled HEB

sections it was derived from. This could lead to errors if the approach is used with

built-up sections.

5.5.4 Deformation capacity approach after Block

In an earlier study the author (Block, 2002, 2004b) also derived an empirical equation

based on a numerical parameter study for the deformation capacity of the column

web. Instead of using different section sizes the author decided to choose a base

section (UC 203x203x46), for which the numerical model was validated against test

results, and varied the web thickness, the web depth, the flange thickness, the load


115

introduction width and the yield stress by about ± 50%. A good correlation was found

between the numerical studies and Equation 5.22 below.

1.5 0.6

2u w w

u

f yw f

P t d

b f t cδ

=

...5.22

Where Pu is the ultimate load, bf and fy,w are the flange width and the yield strength of

the column web respectively, tw is the thickness of the web and tf is the thickness of

the flange. The value c is the load width and dw is the depth of the column web

between the root radii. Subsequently, the equation was simplified further and linked

with the distance of the outer plastic hinges in the column flange ly calculated by the

approach of Lagerqvist and Johansson (1996) described previously. As it was

recognised that the slenderness of the section has an influence on the deformation

capacity, the slenderness parameter χ from the approach of Lagerqvist and Johansson

(see Equation 5.12) was used to increase the deformation capacity for stocky webs

and to reduce it for slender webs. The modified approach is shown in Equation 5.23.

2

2y w w w

u

f f f

l t t d

b t t cδ χ= ...5.23

The deflection δu describes the total deflection of both column flanges in the case of

an internal joint. To calculate the deflection of only one flange δu / 2 should be used.

The approach is compared with test results in Figure 5.16 below.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16δδδδu,Rk [mm]

δδ δδu,

Tes

t [m

m]

Figure 5.16: Comparison of the displacement capacity after Block with tests

The approach predicts the deformation capacity of the compression zone reasonably

well over the whole range of tests.


116

5.5.5 Conclusion

The comparison of the few existing approaches for the ductility of the compression

zone in a column web has shown that the approaches predict the experimental results

reasonable accurately, although the amount of available data is still very limited.

However, the proposed approach by the author predicts the existing test results better

than the two other existing approaches. The approach by Vayas et al. does not

include enough parameters and has too many empirical factors to be a general

equation usable for all I- and H-sections. The approaches by both Vayas et al. and the

author will be compared with the available test results at elevated temperatures in the

following section.

5.6 Design approaches at elevated temperatures

The behaviour of the column web in compression changes at elevated temperatures

due to the reduced material strength and stiffness and the overall change of the shape

of the stress-strain curve. Therefore, the force-displacement curve will be more

rounded, with a smaller elastic range caused by the reduction of the proportional limit

in the stress-strain curve. Furthermore, the amount of reserve strength due to strain

hardening is reduced significantly at temperatures over 400°C. For sections with

more slender webs, instability failures might become more important as the tangent

modulus of the stress-strain curves at elevated temperatures start reducing from about

200°C. In this section, it will be investigated how well the approaches discussed at

ambient temperature predict the different parameters of the force-displacement curve

at elevated temperature. Furthermore, the only other existing approach predicting the

compression zone behaviour in fire will be discussed.

5.6.1 Spyrou’s approach to the compression zone at elevated temperature

Spyrou (2002, 2004b) developed a multi-linear force-displacement model based on

different simplified mechanical models representing the sequential yielding of the

compression zone. For the ultimate load Spyrou used an empirical equation similar to

the one by Drdacky and Novotny (1977). In Figure 5.17 below the yield sequence

can be seen.


117

Figure 5.17: Sequential yielding of the compression zone after Spyrou

As this approach is rather lengthy, and is well documented in the literature (Spyrou et

al., 2004b), it will not be described in detail here. However, a closer look at the

approach has shown that in the original formulation the displacement at third yield is

too low, because the additional displacement from the cantilever model (see Figure

5.17) has to be doubled as the model calculates the total deformation of both sides of

the compression zone in an internal joint.

The approach by Spyrou, as well as the best approaches found in the ambient

temperature comparison, will be compared in accordance with the same methodology

as the ambient temperature test and prediction comparisons.

5.6.2 Statistical comparison of existing approaches at elevated temperatures

The basis of this statistical comparison are the elevated-temperature tests by Spyrou

and the author. In total 25 tests have been performed on small British UC sections, at

temperatures ranging from 280°C to 765°C. For the calculation of all parameters, the

ambient-temperature material properties have been modified to the elevated-

temperature condition by applying the temperature reduction factors given by EC3-

1.2 for the Young’s Modulus and the yield stress.

5.6.3 Resistance of the compression zone at elevated temperatures

Firstly, the different resistance approaches identified at ambient temperature and the

approach by Spyrou will be compared statistically with the high-temperature results.

M

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9 10Displacement [mm]

For

ce [

kN

]

1. Yield

2. Yield

3. Yield

Ultimate load

& displacement


118

The results of the evaluation are given in Table 5.5 below.

Table 5.5: Statistical comparison of elevated-temperature resistance approaches


Standard deviation

Coefficient of

variation


Upper 5% percentile


1 Spyrou (2002) 1.006 0.103 0.102 0.836 1.175 0.985

2 Lagerqvist and Johansson (1996)

1.198 0.129 0.108 0.985 1.410 0.986

3 Kühnemund (2003)

1.250 0.139 0.111 1.020 1.479 0.989

From the table above it can be seen that all three approaches predict the high-

temperature experiments very well. However, the approach by Spyrou is clearly

intended to predict the mean resistance of the compression zone accurately instead of

giving a lower-bound solution, as are the other two approaches. However, a

comparison of this approach at ambient temperature has shown less good comparison

than the other two approaches considered here. The approach by Lagerqvist and

Johansson gives slightly better results, and because of the previously mentioned

advantages of this approach it will be used as a basis for the force-displacement

approach in the connection element. A comparison between the chosen approach and

the high-temperature test results can be seen in Figure 5.18 below. In this figure, the

diamonds represent the tests of Spyrou and the circles the tests by the author.

0

100

200

300

400

500

600

0 100 200 300 400 500 600Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 5.18: Comparison of the resistance after Lagerqvist and Johansson with

tests at elevated temperatures

This approach has been modified by the author to account for the effects of axial load

in the section. The derivation of the new approach and the comparison with tests

Spyrou

Block


119

including axial load will be shown in the next chapter. However, as the new approach

can predict the test results without axial load more accurately than the original

approach by Lagerqvist and Johansson, it will be used for the comparison with the

results of elevated-temperature tests without axial load at the end of this chapter.

5.6.4 Initial stiffness of the compression zone at elevated temperature

The next parameter to be investigated is the initial stiffness of the compression zone at

elevated temperatures. If it is difficult to measure the stiffness at ambient

temperatures, it becomes even more difficult to measure it inside a furnace at high

temperatures. However, based on the available test data a statistical comparison has

been performed. The tests with axial load in the specimen are also included in this

comparison, as the initial stiffness of the material is the same in the uni-axial stress

cases as in the bi-axial one. For the approach of Spyrou, the initial stiffness is taken

as the quotient of the force under first yield and the displacement under this force.

Table 5.6: Statistical comparison of elevated-temperature stiffness approaches


Standard deviation



Upper 5% percentile


1 Spyrou (2002) 1.144 0.318 0.278 0.622 1.666 0.863

2 BS EN 1993-1-8: 2005

0.421 0.119 0.282 0.225 0.616 0.858


1.472 0.416 0.282 0.788 2.156 0.858

4 Faella et al. (2000)

0.633 0.187 0.295 0.326 0.940 0.842

5 Kühnemund (2003)

1.752 0.495 0.282 0.938 2.566 0.858

6 Aribert et al. (2002)

0.760 0.186 0.265 0.429 1.092 0.893

7 Aribert and Younes

0.704 0.186 0.264 0.398 1.010 0.890

Similar to what could be seen at ambient temperature, none of the approaches predicts

the initial stiffness very accurately, but it should be kept in mind that it is difficult to

measure stiffness accurately at elevated temperatures. As was the case at ambient

temperatures the approaches by Aribert et al. (2002) and Aribert and Younes give the

most accurate results at elevated temperatures. The approach by Huber and

Tschemmernegg, which gave the most accurate mean value at ambient temperature, is

now conservative. It appears that the stiffness of the compression zone does not


120

reduce as much as the reduction factor of the Young’s Modulus. Therefore, a

comparison between the normalised initial stiffness of the tests at elevated

temperature and the reduction factors for Young’s modulus and yield stress given by

the BS EN 1993-1-2:2005 is given in Figure 5.19 below.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 100 200 300 400 500 600 700 800 900 1000 1100 1200Temperature [°C]

Nor

mal

ised

init

ial s

tiff

ness

[-]

Tests by Spyrou

Tests by Block

Young's modulusreduction factor

Yield stress reductionfactor

Figure 5.19: Reduction of the initial stiffness with temperature

It can be seen that for the tests by Spyrou the reduction factor for the yield stress gives

a good representation of the mean. However, for the tests by the author the reduction

factor for the Young’s modulus gives a better representation. One explanation for this

could be that the displacements in Spyrou’s test were measured between the insides of

the flanges so the deformation of the flange material is not included, which means that

the measured stiffness values are higher than the real ones. Furthermore, in the tests

by the author the stiffness of the load introduction plate had to be excluded, as the

displacement was measured outside the furnace. Therefore, both results can be

questioned, and therefore the reduction factor of the Young’s modulus as a lower

bound solution will be used until more test results are available. The approach by

Aribert and Younes is compared visually in Figure 5.20 below.


121

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000kc,wc,ini,test [kN/mm]

kc,

wc,

ini,k

[k

N/m

m]

Figure 5.20: Elevated-temperature initial stiffness after Aribert and Younes

If now the original factor of 0.95 is modified to 2/3, and the temperature reduction

factor for the Young’s modulus is introduced, the equation by Aribert and Younes can

be extended to elevated temperatures as shown in Equation 5.24.

3 2

4,

2

3fc fc wc

E

eff wc

b t tk k E

b dθ= ...5.24

with ( ) ( )2 min ; 2 5eff fb f p f p fc cb t a t u a t t r= + + + + + +

How the modified equation compares with the tests at elevated temperature can be

seen in Figure 5.21.

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000kc,wc,ini,test [kN/mm]

kc,

wc,

ini,k

[kN

/mm

]

Figure 5.21: Corrected high-temperature initial stiffness after Aribert and

Younes


122

5.6.5 Ductility of the compression zone at elevated temperature

The final parameter which is needed to create a force-displacement model for the

compression zone at elevated temperatures is the displacement capacity of the

component. No significant influence of increasing temperatures on this parameter

could be found. Only a slight increase of the ductility was discovered at high

temperatures. This can be seen in Figure 5.22, showing the normalised displacements

for each test without axial load under the peak transverse load, which is assumed as

the ductility limit, for increasing temperatures.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Temperature [°C]

Nor

mal

ised

dis

pla

cem

ent

[-]

Figure 5.22: Reduction of the ductility with temperature

The ductility is once more compared on a statistical basis with the available test

results, without axial load as this affects the ductility significantly. The approaches

by Spyrou, Vayas et al. and Block are shown in Table 5.7 below. The material

properties used in the calculations have been reduced in accordance with the

temperature reduction factors given in the Eurocode.

Table 5.7: Statistical comparison of elevated-temperature resistance approaches


Standard deviation

Coefficient of

variation


Upper 5% percentile


1 Spyrou (2002) 0.894 0.374 0.419 0.278 1.510 0.565

2 Vayas et al. (2003)

0.908 0.280 0.308 0.448 1.368 0.832

3 Block 1.353 0.308 0.228 0.846 1.860 0.920


123

It can be seen that the approach by Spyrou does not predict the real behaviour very

well. In addition, the approach by Vayas et al., which compared well at ambient

temperatures, is not very accurate in the fire case. This might be due to the empirical

factors and the limited number of parameters which are included in the calculation

procedure of this approach. However, the equation by the author compares similarly

well as at ambient temperatures and will be used to predict the ductility limit of the

compression spring in the connection element. A detailed comparison of the high-

temperature tests and the chosen approach is shown in Figure 5.23.

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14δδδδu,k [mm]

δδ δδu,

Tes

t [m

m]

Figure 5.23: Comparison of the displacement capacity after Block with high

temperature tests

As with the initial stiffness, it is important to predict the ductility as accurately as

possible, because the internal forces in the whole joint depend on these values. For a

beam with flush endplate connections at both ends, if the calculations under-predict

the ductility of the compression zone and this point is reached the hogging moments

in the connections are redistributed into the beam, which starts to act as simply

supported. However, if in reality the compression zone is more ductile, other

components such as the bolts in tension could fail before the compression zone, which

could lead to a more dramatic failure. Therefore, the equation for the ductility will be

corrected so that it predicts the average value of the test results accurately. The

corrected approach can be seen in Figure 5.24 below.


124

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14δδδδu,k [mm]

δδ δδu,

Tes

t [m

m]

Figure 5.24: Comparison of the corrected displacement capacity after Block with

high temperature tests

All parameters necessary for the description of the compression zone at elevated

temperature have now been found, and only an appropriate force-displacement curve

to link the initial stiffness, the resistance and the ductility limit is required.

5.7 Force-displacement curve model for elevated temperatures

As most rolled column sections have rather stocky webs, the failure will be governed

by yielding of the web and plastic hinges in the flange. Therefore, the shape of the

force-displacement curve will be similar to the stress-strain curve of the base material.

Following this logic, the equations for calculating the stress-strain curve for mild steel

at elevated temperatures given in EC3-1.2 have been modified as a force-

displacement curve for the compression zone in the column web. One great

advantage of this function is that it reaches zero stiffness at a specified load and

displacement level, which the Ramberg-Osgood approach used earlier (Block 2002,

2004b) does not. Therefore, if this curve is used in the connection element the

compression zone component will lose its stiffness gradually, and redistribution of the

load can be achieved more easily. The approach is shown below in Figure 5.25.


125

Figure 5.25: Force-displacement curve for the compression zone at high

temperature

The force-displacement curve is divided into two parts. The first part describes the

elastic part of the curve, and ends at half the ultimate load (see Equation 5.25). This

makes the elastic range smaller than what is assumed for the moment-rotation curve

in EC 3-1.8, which goes up to 2/3 of the ultimate moment. This is due to the

reduction of the proportional limit in the stress-strain curves at elevated temperatures.

, , , , ,i i c wc iniF kθ θ θδ= , ,i elθ θδ δ≤ ...5.25

The second part of the curve is given by an elliptical equation, and its end-point is

defined by the ultimate load and the ductility of the compression zone. The equations

necessary for the second part are given as Equation 5.26 below.

( )22

, , , ,i el u i

bF F c a

aθ θ θ θδ δ= − + − − , , ,el i uθ θ θδ δ δ< ≤ ...5.26

with ,,

, , ,

el

el

c wc ini

F

k

θθ

θ

δ = and ,, 2

u

el

FF

θθ =

The parameters necessary to use Equation 5.26, a, b and c, are shown below.

( )2, , , ,

, , ,u el u el

c wc ini

ca

kθ θ θ θ

θ

δ δ δ δ

= − − +

...5.27

( )2 2, , , , ,u el c wc ini

b c k cθ θ θδ δ= − + ...5.28

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8

Fu,θ

Fel,θ

δu,θ δel,θ

Kc,wc,ini,θ


126

( )

( ) ( )

2

, ,

, , , , , , ,2u el

u el c wc ini u el

F Fc

k F F

θ θ

θ θ θ θ θδ δ

−=

− − − ...5.29

For the incorporation of the force-displacement behaviour of the compression zone, it

is essential to provide the tangent stiffness of the curve, which has been done below in

Equations 5.30 and 5.31.

, , ,T c wc inik k θ= , ,i elθ θδ δ≤ ...5.30

( )

( )

, ,

22, ,

el i

T

el i

bk

a a

θ θ

θ θ

δ δ

δ δ

−=

− − , , ,el i uθ θ θδ δ δ< ≤ ...5.31

How the developed approach compares with the tests at elevated temperatures will be

shown in the next section.

5.8 Validation of the simplified model at elevated temperatures

In order to validate the proposed approach for the compression zone in the column

web at elevated temperatures, a comparison of the approach with the high-temperature

experiments of Spyrou (2002, 2004b) and the tests by the author is shown in this

section. As stated above, the modified version of the resistance approach by

Lagerqvist and Johansson is used in this comparison. The detailed derivation of the

new approach for the resistance can be seen in the next chapter.

5.8.1 Comparison between the simplified model and tests by Spyrou

At first, the simplified model developed will be compared with the tests on six

Universal Column sections UC152x152x30 at temperatures between 410°C and

755°C. This comparison can be seen in Figure 5.26 below. In the following figures

the dots represent the test results whereas the solid lines show the prediction of the

simplified model.


127

0

50

100

150

200

250

300


For

ce [

kN

]

410 C500 C600 C610 C710 C755 C

Figure 5.26: Comparison between the simplified model and tests on


A good comparison can be seen at high temperatures between the new simplified

model and the tests. However, at lower temperatures the resistance of the specimen is

predicted conservatively but the initial stiffness, the ductility and the overall shape of

the force-displacement curves are predicted well. It should be noted that in some of

the tests discussed above Spyrou used web stiffeners at a certain distance from the

compression zone, which could explain the larger capacity in these tests. In the next

figure, the tests on five UC203x203x46 sections between 400°C and 765°C are

predicted.

0

50

100

150

200

250

300

350


For

ce [

kN

]

400 C520 C610 C670 C765 C




128

A very good correlation between the new model and the tests can be seen in Figure

5.27. Similarly to the tests with the UC152x152x30 sections, at 400°C the resistance

approach underpredicts the test results. Next, the three high-temperature tests on

UC203x203x71 are compared with the new simplified model.

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14 16 18 20Displacement [mm]

For

ce [

kN]

535 C

635 C

755 C



Again, a very close comparison can be seen in Figure 5.28 between the test results

and the new approach. Finally, the tests on the largest specimens, UC203x203x86

sections, are evaluated.

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Displacement [mm]

For

ce [

kN

]

585 C650 C705 C750 C




129

Once more, a good correlation between the tests and the new approach can be seen in

Figure 5.29 above. For further validation of the new approach, a comparison against

the high-temperature tests without axial load conducted by the author will be shown.

5.8.2 Comparison between the simplified model and tests by the author

It is difficult to define a specific strain at which the compression zone fails, as the

stress and therefore the strain distribution in the web of the specimen is highly non-

linear. However, the simplified model assumes a uniform stress distribution in the

web, therefore the tests will be predicted with different average strain levels equal to

1%, 2%, 5%, and UTS based on the temperature reduction factors found in the

previous section. This should give a good estimation of the envelope with which the

experimental force-displacement curves should lie. Further, it might be possible to

predict an average failure strain of the compression zone. In the following figures all

elevated-temperature tests conducted without axial load are compared with the

simplified model, using stresses at different strain levels based on reduction factors

from Renner (2005) and Kirby and Preston (1988). Additionally, the results of the

simplified model using the yield stress and the appropriate strength reduction factor

given in the EC3-1.2 are shown. The graphs show the displacement of both sides of

the specimen.

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN]

Test 4 - 450°C

1% - Renner

2% - Renner

5% - Renner

UTS - Renner

1% - Kirby

2% - Kirby

5% - Kirby

UTS - Kirby

EC 3-1.2

Figure 5.30: Comparison between the simplified model and Test 4


130

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 8 -550°C

1% - Renner2% - Renner5% - RennerUTS - Renner1% - Kirby2% - Kirby5% - KirbyUTS - KirbyEC 3-1.2


0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 12 - 598°C

1% - Renner2% - Renner5% - RennerUTS - Renner1% - Kirby2% - Kirby5% - KirbyUTS - KirbyEC 3 - 1.2



131

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16Displacement [mm]

For

ce [

kN

]

Test 15 - 444°C1% - Renner2% - Renner5% - RennerUTS - Renner1% - Kirby2% - Kirby5% - KirbyUTS - KirbyEC 3 - 1.2


0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 17 - 545°C

1% - Renner2% - Renner5% - RennerUTS - Renner1% - Kirby2% - Kirby5% - KirbyUTS - KirbyEC 3 - 1.2


From the figures above, it can be seen that the simplified model generally give a good

prediction of the test results at elevated temperatures. In all tests, apart from Test 12,

the force-displacement curve lies within the solution envelope.


132

The initial stiffness is predicted well for Tests 4, 8, 12. However, Tests 15 and 17

have a very low initial stiffness, which can also be found in Tests 16 and 18 and

therefore must be due to material properties different from the rest of the sections

tested, as all of them are from the same length of column section. Again, it should be

highlighted that the approach for the initial stiffness used by Aribert and Younes

(2002) is purely empirical and gave a large scatter compared with test results.

However, it was the best approach that could be found.

The ultimate displacement is predicted accurately for all tests apart from Tests 4 and

8, which failed in the asymmetric failure mode in which the flanges move laterally.

This failure mode seems to have a lower ductility than the symmetric failure mode.

The resistance of the compression zone in the column web is predicted conservatively

if the yield stress of the material in combination with the temperature reduction given

in EC3-1.2 is used. In this case, the predicted force-displacement curve always lies

below the test results, with a difference of 10% to 15% from the tests. However, if

the steady-state reduction factors are used for the different strain levels a good

prediction can be found if the stresses at 2% and 5% after Kirby and Preston are used.

If one compares the test predictions at lower and higher temperatures it can be seen

that the solution envelope becomes smaller with increasing temperature, showing that

the strain-hardening range of the stress-strain curve reduces more quickly than the

yield stress. If the two sets of reduction factors are compared, it seems that the set

derived from the steady-state tests by Kirby and Preston correlate better with the tests,

which suggests that the average strain-rates in the tests might have been higher than

the ones in the material tests by Renner. However, as the new strain-rate-dependent

reduction factors are based only on a single test series and the real behaviour is not

fully understood, a lot of further work is needed. Therefore, the reduction factors

given in EC3-1.2 will be used in the connection element. However, the fact that the

stress-strain curve found at the slowest tested strain-rate was significantly lower than

the EC3-1.2 prediction remains worrying.

5.9 Conclusion

In this chapter a simplified model for the force-displacement behaviour at elevated

temperature of the compression zone of the column web of an internal joint has been

developed. In order to find an accurate solution for this problem, ambient-

temperature approaches for the resistance, the initial stiffness and the ductility have


133

been compared with a large number of test results. The approaches which compared

best with the tests were then extended to elevated-temperatures with the help of

temperature reduction factors for the yield stress and the Young’s modulus and

compared with elevated temperature test results. From this comparison, the

approaches by Lagerqvist and Johansson, Aribert and Younes and the author

compared most favourably with the experiments for the resistance, the initial stiffness

and the ductility, respectively. These parameters were then combined to a full force-

displacement curve with the help of a modification of the equations given for the

stress-strain curve of steel at elevated temperatures in EC3-1.2.

Further, this model was then compared with the results of high-temperature

experiments on the compression zone by Spyrou and the author. The model

compared very well with the tests conducted by Spyrou. For the tests by the author

the simplified model under-predicted the resistance of the compression zone by about

15%, which could be explained by the higher testing speeds in the tests of the author

than in the experiments by Spyrou. With the reduction factors for steel tested under

steady-state conditions at elevated temperatures, derived form the study by Renner

and steady-state tests by Kirby and Preston, the tests by the author could also be

predicted accurately.

With the proposed simplified model, the ambient and elevated-temperature behaviour

of the compression zone in the column is provided, and will be introduced into the

connection element. However, one parameter which has not yet been investigated is

the influence of axial load in the column on the force-displacement curve of the

compression zone. An analytical and numerical investigation of this parameter will

be presented in the next chapter.

Chapter 6: The influence of axial load on the compression zone

134

6 The influence of axial load on the compression zone

6.1 Introduction

In the previous chapter, a simplified model for the force-displacement behaviour of

the compression zone was developed. However, in this model the effects of axial load

in the column were ignored. Initially in this chapter, the existing reduction factors for

the resistance of the compression zone will be reviewed and discussed. Then a new

analytical approach to include the axial load into the resistance calculations directly

will be presented and validated and finally the reduction of the ductility of the

compression zone due to axial load will be examined.

6.2 Previous research

Research projects in the last 35 years around the world investigated the effects of

axial load on the compression zone behaviour, mainly in conjunction with the

development of new design approaches for the resistance of this joint component.

These studies used numerical calculations, component tests such as the ones in this

study, and tests on full joint assemblies. From these tests, it can be seen that the

ductility, as well as the ultimate resistance force of the compression zone, is reduced

by the presence of axial-stress in the column.

An early attempt to describe the influence of axial load was made by Bose et al.

(1972). They used two series of simple finite element-based numerical simulations of

internal and external beam-column configurations. In the models, the forces

introduced by the beams were replaced by a pair of point loads with opposite

orientation and a shear force directly applied to the column flange. In total three

different internal and two different external joint configurations were analysed at axial

load ratios of up to 85%. Based on these results Bose et al. obtained a very

conservative reduction factor for the effects of axial load, which is shown in Equation

6.1 below.

2

, ,

, ,

1.00 0.5 0.5com Ed com Ed

wc

y wc y wc

kf f

σ σ = − −

...6.1

The original notation in this equation has been changed to the Eurocode standard

where σcom,Ed is the longitudinal stress in the column web and fy,wc is the yield stress of

the column web.


135

In 1980, Zoetemeijer (1980) proposed a transverse resistance reduction factor kwc for

cases where the longitudinal stress in the column web is larger than 50% of its yield

stress. The factor kwc can be calculated from Equation 6.2 below.

,

,

1.25 0.5 1.0com Ed

wc

y wc

kf

σ= − ≤ ...6.2

A slightly modified version of this equation has been adopted in EC3-1.8 (CEN,

2005) and is shown in Equation 6.3. This approach assumes that no reduction of the

capacity of the compression zone occurs until the longitudinal stress in the column

web is larger than 70% of the yield stress of the web.

,

,

1,7 1.0com Ed

wc

y wc

kf

σ= − ≤ ...6.3

This approach is calibrated for the resistance approach in the Eurocode, which is quite

conservative, as was seen in the previous chapter; therefore, a high axial stress level is

required to reduce the capacity below the design value.

Djubek and Skaloud (1976) also derived a reduction factor, in the context of the

influence of bending stresses on the transverse load capacity of plate girders. This

factor is based on the equation of a circle and can be seen in Equation 6.4 below.

2

,

,

1 com Ed

wc

y wc

kf

σ = −

...6.4

Ahmed and Nethercot (1998) applied the von Mises yield theory to the problem, and

concluded that no reduction factor is needed for bi-axial compression in the column

web. Bailey and Moore (1999) conducted a number of cruciform tests with column

loading. These tests were designed to fail the column web in compression or shear.

They investigated Class 1 and Class 4 universal sections as columns, and varied the

load configuration in order to generate equal or opposite moments at both sides of the

beam-to-column joint. Based on these tests and analytical calculations they

developed reduction factors for the yielding and the buckling failure modes. For the

yielding failure the von Mises yield theory with an imperfection factor k was used,

and for the buckling failure a linear interaction between the buckling loads in the

transverse and longitudinal directions of the web panel was used. The von Mises yield

criterion including the reduction factor k is given in Equation 6.5.

2 21 1 2 2y kσ σ σ σ σ= − + ...6.5


136

where σy is the yield stress and σ1 and σ2 are the principal stresses. If this yield

criterion is now extended to stress ratios, σy becomes equal to 1.0 and σ2 is replaced

by the axial stress ratio in the column, then σ1 becomes the reduction factor for axial

load in the column, and the whole approach can be expressed as Equation 6.6.

2 2

, , ,2

, , ,

1 14 4

2 2com Ed com Ed com Ed

wc

y wc y wc y wc

k k kf f f

σ σ σ = + − +

...6.6

Bailey and Moore recommended an imperfection factor of k = 0.7 based on two test

results. Due to the nature of the von Mises yield theory, values larger than unity can

be found for axial load ratios smaller 0.7. However, this increase of resistance was

ignored in the proposed reduction factor for the yielding failure mode. For the

buckling failure mode, the reduction of the transverse capacity due to bi-axial stress is

given in Equation 6.7 below.

( )

−=

axialcr

axialbuckwc

F

Fk 1,

...6.7

where Faxial represents the axial force in the column and Fcr(axial) its load carrying

capacity. However, it is unlikely that a web in a column section is slender enough to

fail in elastic buckling, and therefore this reduction factor is only shown for

completeness.

Kühnemund (2003) modified the same reduction factor as Djubek and Skaloud (1976)

to account for the effects of axial stresses in the column by using the ultimate stress

fu,wc instead of the yield stress fy,wc for the calculation of the axial load ratio:

2

,

,

1 com Ed

wc

u wc

kf

σ = −

...6.8

If it is assumed that for normal steel used in construction the ultimate stress is about

1.5 times the yield stress, this approach can be compared with the other reduction

factors and ambient temperature test results found by Kuhlmann and Kühnemund

(2000) in Figure 6.1 below.


137

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Axial load ratio - N / N pl

Red

ucti

on f

acto

r -

kw

c

Bose et al. (1972)

Zoetemeijer (1975)

Djubek and Skaloud (1976)

Bailey and Moore (1999)

Kuehnemund (2003)

BS EN 1993-1-8:2005

Test - HE 240 A

Test - HE 240 B

Figure 6.1: Comparison between tests and different reduction factors kwc

If the reduction factors derived from these tests are compared with the different

reduction factors found in the literature, it can be seen that none of the approaches

describes the reduction accurately. The equation by Bose et al. seems very

conservative, whereas the approaches by Zoetemeijer, Bailey and Moore and the

Eurocode 3 give unsafe results. The approaches by Djubek and Skaloud and

Kühnemund describe the reduction at low axial loads quite well, but at higher axial

load ratios these approaches diverge from the test results. However, at ambient

temperature in multi-storey building design such high load ratios only occur in stocky

columns at the lower floors, which less likely to be affected by overall buckling.

Normally, the compression-zone resistance in such columns is higher than the load

introduced from the beams, and is therefore not critical for the joint design.

Nevertheless, if the fire case is considered, a possible protection regime could protect

the column only up to the level of the bottom flanges of the beams, which would

leave the joint zone of the column unprotected. When the temperature of the steel in

this joint zone increases and the material loses strength locally, the relative load ratio

in the joint region of the column will increase although the overall buckling capacity

of the column remains almost unaffected. Therefore, it is possible that quite high

axial load ratios can be reached in the compression zone during a fire, and a resistance

approach should be accurate over the whole range of axial load ratios. Furthermore, it


138

is questionable if the “one size fits all” type of approaches consider the influence of

the geometrical properties of a section on its sensitivity to axial load correctly. An

indication for such an influence can be seen in Figure 6.1, where the same axial load

ratio in the more slender HEA 240 section caused a larger reduction of the capacity

than in the more stocky HEB 240 section.

Therefore, the need for a more analytical approach for the compression zone

behaviour, including the effects of axial loads in the column, is apparent and the

approach by Lagerqvist and Johansson (1996), which yielded the most accurate

results in the last chapter, has been extended.

6.3 Proposed analytical approach for the ultimate load

The approach by Lagerqvist and Johansson is based on a plastic hinge mechanism in

the column flange (see Figure 5.7) and a slenderness reduction factor accounting for

instability effects in the web. Firstly, the plastic hinge mechanism will be modified to

account for axial stress in the column flange.

6.3.1 Plastic hinge mechanism in the compression zone with axial load

The plastic moment capacity of the flange is based on a T-section for the outer plastic

hinges including parts of the web and only the flange for the inner plastic hinges, as

can be seen in Figure 5.7. If now the axial load in the column flange is considered,

parts of the available cross section are used to resist the axial load, and the material

available for the plastic moment is reduced. For the inner plastic hinges this

modification is straightforward, and can be found in standard plastic theory textbooks;

the approach used was found in Horne (1979). The procedure is shown schematically

in Figure 6.2.

Figure 6.2: Reduced plastic moment of the inner plastic hinge due to axial load

Horne gives the reduced plastic moment capacity for a rectangular cross-section

dependent on the axial stress ratio n as:

( )2, 1

pl inner p yM n Z f′ = − ...6.9

- fy (Compression)

+ fy (Tension) =

- fy

+

- fy

+ fy

bf

tf


139

with 4

f f

p

b tZ = and axial

y

nf

σ= ...6.10

For the outer plastic hinges, the approach is more complicated, as the T-section is

mono-symmetric and therefore the stress distribution is no longer symmetric.

However, Horne provides a solution procedure for this type of section and this will be

followed below.

Figure 6.3: Notations of the T-section in the outer plastic hinge

The outer plastic moments in a column flange under compression are exposed to a

combination of a hogging moment and compression, which simplifies the approach,

as the plastic neutral axis (PNA) remains in the flange of the T-section. The only

instance where this could change is if the flange area is smaller than the reduced web

area and so the equal-area axis moves into the web, and only a little axial load is

applied. However, the approach by Lagerqvist and Johansson (1996), which is

followed here, assumes that only 14% of the web depth is included in the T-section.

Therefore, the PNA would only move into the web in very deep sections with small

flanges, but not in normally used column sections.

Figure 6.4: Reduced plastic moment of the outer plastic hinge due to axial load

To be able to predict the reduced plastic moment capacity, the first step is to calculate

the position of the centroidal axis, here measured from the top of the flange (see

Figure 6.3) denoted as ye:

a

bf

tw

k dw

tf

ye

Centroidal axis Plastic neutral axis

+ fy (Tension)

- fy (Compression)

=

+ fy

+

- fy - fy


140

2 2 21 12 2f f w w f w w

e

f f w w

b t k d t t k d t

yb t t k d

+ +=

+ ...6.11

The PNA is now defined with respect to the centroidal axis, and the distance between

the two is given by a. To find this distance the total axial stress in the T-section has to

be equated with the distribution of compressive and tensile stresses in the section, as

can be seen in Figure 6.4 where the distance a is kept as variable.

( ) ( ), , , , ,( )f f y f w w y w f e y f f f e y f w w y w

n b t f t k d f b y a f b t y a f t k d f+ = − − + − + + ...6.12

If now the equation above is rearranged with respect to a, the distance between the

PNA and the centroidal axis can be given as Equation 6.13.

( ), ,

,

2 1 1

2

ef f y f w w y w

f

f y f

yn t b f n t k d f

ta

b f

+ − + −

= ...6.13

After the position of the PNA has been found, it is easy to calculate the plastic

moment of the T-section, including the effects of axial load using the equilibrium of

moments:

( ) ( )

( )2

, , , ,

,

2 2 2 2

2

f eepl outer f e y f f y f f f e y f

ww w f e y w

t yy a aM b y a f b f b t y f

k dt k d t y f

− ′ = − + − + −

+ − +

...6.14

and simplified:

2

2 2, , ,2 2

f wpl outer f e f e y f w w f e y w

t k dM b y a t y f t k d t y f

′ = − + − + − +

...6.15

Now the effects of axial load in the column flanges are incorporated into the plastic

moment capacity of the inner and outer plastic hinges, it is possible to follow the

original approach by Roberts (1981) to find the distance between the outer plastic

hinges, and therefore the length over which the web is assumed to yield. To do so, the

total energy of the system, shown in Figure 6.5 below, has to be evaluated and then

minimised with respect to β, which is the distance between an inner and an outer

plastic hinge.


141

Figure 6.5: Plastic mechanism in the column flange

If now the angle α is assumed equal to unity, then the internal and external potential

energies of the system can be written as follows:

( )2, ,2 2 2

in pl outer pl inner yw w yw w fM M f t f t ss tβ β′ ′Π = + + + + ...6.16

( )2

Rk f

ex

f

F ss t

b

β+Π = ...6.17

in exΠ = Π ...6.18

If further the internal and external energies are equated and rearranged with respect to

the external force FRk, this gives:

( )( )

( ), ,2 2 2

2

pl outer pl inner yw w f f

Rk

f

M M f t ss t bF

ss t

β β

β

′ ′+ + + +=

+ ...6.19

To find the length β under which the minimum external force causes the described

mechanism, FRk has to be differentiated once with respect to β and set equal to zero:

( )( )

( )( )( )

, ,

2

2 2

2

2 2 20

2

yw w fRk

f

pl outer pl inner yw w f f

f

f t ss tdF

d ss t

M M f t ss t b

ss t

β

β β

β β

β

+ +=

+

′ ′+ + + +− =

+

...6.20

This expression can now be solved with respect to β.

( ), ,2 yw w pl outer pl inner

yw w

f t M M

f tβ

′ ′+= ± ...6.21

The positive solution for β can now be used to calculate the length over which the

column web yields. Lagerqvist and Johansson reported illogical results from their

approach for stocky sections with slenderness factor smaller than λF = 0.5. However,

fyw tw

ly

ss +2 β β

,pl outerM ′ ,pl outer

M ′

,pl innerM ′ ,pl inner

M ′

Rk

f

F

bα α

α α


142

this could not be found in the new approach and therefore, unlike in the original

approach, the beneficial contribution of the column web towards the plastic moment

capacity of the outer plastic hinges is used over the whole slenderness range.

Furthermore, the factor k, which specifies the size of the part of the web contributing

to the outer plastic hinge, is kept as 0.14, as in the original approach. The length ly

can be calculated from Equation 6.22 below.

, 2 2y axial fl ss t β= + + ...6.22

Finally, the yielding resistance Fy of the can be calculated from Equation 6.23.

, ,y axial w y axial ywF t l f= ...6.23

For column sections with stocky webs, this will give the transverse resistance of the

compression zone. However, for more slender sections, stability effects have to be

considered.

6.3.2 Consideration of the stability of the column web with axial load

The original approach by Lagerqvist and Johansson reduces the yield resistance of the

compression zone calculated in the previous section by a stability reduction factor χ

based on the slenderness factor λF of the column web. This slenderness factor further

depends on the elastic buckling load of the web in the transverse direction. If now the

axial load in the column is considered, the elastic buckling resistance of the web

reduces with increasing axial load. To account for this phenomenon the column web

will be assumed as a simply supported thin plate under bi-axial compression. Classic

buckling theory (Bleich, 1952) gives an interaction equation for this kind of problem:

, ,

1.0yx

cr x cr y

FF

F F

+ =

...6.24

where Fx and Fy are the applied loads in the x-and y-directions respectively, and Fcr,x

and Fcr,y are the elastic buckling resistances of the plate if loaded in only one

direction. This interaction equation gives a linear relationship between the buckling

resistances of a bi-axially loaded plate. If the interaction equation is rearranged it can

be used as a reduction factor for the available buckling resistance in the transverse

direction.

, , ,,

1 long

cr trans cr trans red

cr long

FF F

F

− =

...6.25


143

To be able to use this reduction factor the elastic buckling load of the column web in

the longitudinal direction is needed. This value can be found with the help of

classical buckling theory (Bleich, 1952), if the column web is assumed to be an

isolated plate. The width of this plate is assumed equal to the clear depth of the web,

and the length does not need to be specified as long as the web is unstiffened, as the

plate will buckle under its smallest buckling load. For the edge conditions between

the web and the flange, it seems justifiable to assume rotational fixity, because the

flanges are effectively restrained against rotation by the beams in a real joint, or by

the loading device in a component test. However, this contradicts the assumptions of

the interaction Equation 6.24, but a comparison with numerical and experimental data

supports the assumption taken. Therefore, the elastic buckling load of the column

web in longitudinal direction under axial load only can be calculated from:

( )

2 3

, 212 1w w

cr long

w

E tF k

d

π

ν=

− with 6.97k = and υ = 0.3 ...6.26

and so the reduced plate slenderness factor λF,axial can be calculated from the

following expression:

, ,,

, , , ,

y axial y axial w yw

F axial

cr trans red cr trans red

F l t f

F Fλ = = ...6.27

with ( )

2 3

, , 2,

112 1

long F w wcr trans red

cr long w

F k E tF

F d

π

ν

= − −

with υ = 0.3 ...6.28

The buckling coefficient kF is defined in Equation 5.10 and 5.11 as in the original

approach, which is followed from now on. By using the slenderness reduction factor

including the effects of axial load:

,

0.470.06 1.0

axial

F axial

χλ

= + ≤ ...6.29

the resistance of the compression zone in the column web, including the effects of

axial load in the column, can now be calculated by the following expression.

,,

1

yw w axial y axial

Rd axial

M

f t lF

χ

γ= ...6.30

The partial safety factor γM1 is assumed equal to 1.0 for the comparison with test

results. The approach can be extended to elevated temperatures by reducing the yield

stress and the Young’s modulus of the flange and the web material according to their

temperature, following the material reduction factors given in EC3-1.2 (CEN, 2004).


144

In this new approach, it is assumed that the flanges of the column are restrained in

rotation and translation. Therefore, the effects of the compression forces introduced

by the connections have only local effects to the joint and no interaction with the

overall column behaviour has to be considered. This assumption is also used in EC3-

1.8 and is valid for connections used in multi-storey steel and composite frames in

which the floor plate prevents relative movements of the column flanges.

6.3.3 Validation of the new approach

In order to validate the above derived approach, it will be compared with the ambient-

temperature results of Kuhlmann and Kühnemund (2000). Initially, only the effects

of axial load will be investigated. Therefore, the capacities of the compression zone

found in the tests and calculated from the new approach are normalised and

compared. In Figure 6.6 the results of Test Series 1 and 2 on HEA 240 section are

shown.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20Axial load ratio - N/N pl [-]

Red

uct

ion

fac

tor

- k

wc [

-]

Figure 6.6: Comparison of tests on HEA 240 sections with the new approach

From the comparison above a very good correlation can be seen between the new

approach and the test results. Next, the tests on the HEB 240 sections are compared

with the new approach, which can be seen in Figure 6.7.

Series 1

Series 2


145

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20Axial load ratio - N/N pl [-]

Red

ucti

on f

acto

r -

kw

c [

-]

Figure 6.7: Comparison of tests on HEB 240 sections with the new approach

From the comparison above, good correlation can be seen between the new approach

and the test results on the HEB 240 sections. The new approach under-predicts the

reduction due to axial load for the tests of Series 2 in the low axial load region.

However, at high axial load ratios the comparison becomes very good again. The

next step of the validation process is a comparison between the new approach and the

test results with and without axial load at ambient temperature.

Table 6.1: Statistical comparison of the new approach and tests at 20°C

Type of tests No. of tests

Mean value

Standard deviation



Upper 5% percentile


Ambient temperature – no axial load

106 1.291 0.148 0.114 1.048 1.534 0.988

Ambient temperature – inc axial load

15 1.367 0.096 0.070 1.209 1.525 0.961

From the table above a very good statistical comparison can be observed between the

different test results and the new approach. Furthermore, if one compares the

statistical parameters with the values in Table 5.1, it can be seen that the new

simplified model gives more accurate results than all the other approaches discussed

in the last chapter. A visual comparison between the new approach and the capacities

found in the tests including axial load can be made in Figure 6.8.

Series 1

Series 2


146

0

200

400

600

800

1000

0 200 400 600 800 1000Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [kN

]

Figure 6.8: Comparison between ambient temperature test results with axial

load and the new simplified model

Because the material properties of steel change at elevated temperatures, it is likely

that the effects of axial load are also influenced by this. Therefore, a numerical study

of this problem has been conducted and is presented in the next section.

6.4 Parametric study on the effects of axial load at high temperature

In chapter 4, a parametric study on the effect of axial load in column section was

performed using finite element modelling. The results of this study are now

compared with the newly developed approach for the ultimate load of the

compression zone. The numerical data is also used to find axial load reduction factors

for the peak displacement.

6.4.1 Reduction of the ultimate load due to compression in the column

In order to evaluate the effect of the axial load on the resistance of the compression

zone, the peak transverse loads found in the parametric study have been normalised

for each temperature and are plotted against the axial load ratio in the following

figures. The axial load ratios have been calculated with the help of the temperature

reduction factor for the effective yield stress at 2% strain, given in EC3-1.2.


147

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0Axial load ratio N / N pl [-]

Tra

nsve

rse

load

rat

io F

c,w

c /

Fc,

wc,

0%

[-]

New simplified model - 20°C

New simplified model - > 200°C

FEM - 20°C

FEM - 200°C

FEM - 450°C

FEM - 650°C

Figure 6.9: Reduction of the peak load due to axial loads in an UC203x203x46

It is interesting to notice that there is a distinct difference between the results at

ambient and elevated temperatures. This can be explained by the change of the shape

of the stress-strain curve at elevated temperatures, which has an effect on the moment

capacities of the plastic hinges in the flange, and for more slender sections on the

buckling behaviour of the web as it now buckles under a reduced tangent modulus.

Accounting for these effects in detail within the simplified model would make the

calculation procedure much more complicated because the stress distributions in the

plastic hinges are no longer constant but non-linear, and an iterative procedure would

be needed to find the reduced tangent modulus under which the web loses stability.

However, for the sake of simplification, a factor for the reduction of the ultimate load

due to axial load in the column has been developed based on the parametric study. It

is shown in Equation 6.31:

, ,

1 0.2 NNT

y y wc

kk fθ

σ= − ...6.31

where σN is the longitudinal stress in the column web, ky,θ is the temperature-

dependent yield strength reduction factor given in EC3-1.2 and fy,w the yield stress of

the column web. This factor is to be used only at elevated temperatures together with

Equation 6.30; at ambient temperature, the analytical approach as derived at the


148

beginning of the chapter is sufficient. How the approach compares with the results of

the parametric study is shown below.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Axial load ratio N / N pl [-]

Tra

nsv

erse

load

rat

io F

c,w

c /

Fc,

wc,

0%

[-]



FEM - 20°C

FEM - 450°C

FEM - 550°C

FEM - 650°C


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Axial load ratio N / N pl [-]

Tra

nsve

rse

load

rat

io F

c,w

c /

Fc,

wc,

0%

[-]



FEM - 20°C

FEM - 450°C

FEM - 650°C


From the figures above a satisfying correlation between the predicted reduction of the

ultimate load and the results from the numerical analysis can be seen. Next, the new

approach for the resistance of the compression zone will be compared with the


149

elevated temperature test results without axial load, as well as the tests by the author,

combining elevated temperatures and axial load.

Table 6.2: Comparison of the new approach and tests at high temperatures


Mean value

Standard deviation



Upper 5% percentile


Elevated temperature – no axial load

25 1.134 0.107 0.095 0.958 1.311 0.985

Elevated temperature –

axial load 7 1.242 0.066 0.053 1.133 1.351 0.985

Similar to the observation at ambient temperature, the approach becomes more

conservative if it is compared with the test including axial load. However, it is

questionable whether a statistical comparison is representative, as only seven tests on

the same section type can be compared. Therefore, there is a clear need for more

tests. Nevertheless, the results of the comparison are shown graphically in the

following figures.

0

100

200

300

400

500

600

0 100 200 300 400 500 600Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 6.12: Comparison of the proposed approach with tests at elevated

temperatures without axial load


150

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400Fc,wc,Rk [kN]

Fc,

wc,

Tes

t [k

N]

Figure 6.13: Comparison of the proposed approach with tests at elevated

temperatures including axial load

An accurate but conservative comparison can be seen in the figures above, which is

most likely due to the fact that for the comparison the strength and stiffness reduction

factors given in the EC3-1.2 are used, whereas the steel in the experiments was

stronger due to the high loading speed. However, generally it can be said that the new

approach gives very accurate results in comparison with the available experimental

results. The approach also predicted the effects of axial load accurately in the

numerical study over a large range of d/t values. The next study examines the

reduction of the ductility due to axial load.

6.5 Reduction factor for the displacement under ultimate load

Axial load in a column section has not only a reducing effect on the capacity of the

compression zone; it also reduces the ductility of this component as can be seen in

Figure 4.17 above. As mentioned in Chapter 5, Kuhlmann and Kühnemund (2000)

and Kühnemund (2003) studied this effect experimentally and developed a complex

iterative yield-line model for the ‘down hill’ part of the force-displacement curve

including the effects of axial load. This yield-line model also gives the peak

displacement. However, due to the complexity of the model it will not be reproduced

here. Another attempt to predict the reduction of the ductility was made by Vayas et

al. (2003), which has also been discussed in Chapter 5. However, this approach only


151

predicts a reduction of the ductility for stocky sections with a web slenderness dwc / (ε

twc) < 20. For more slender sections no reduction is predicted.

The results of this approach will be compared with the displacement reductions found

in the experiments by Kuhlmann and Kühnemund. For the lighter HEA240 section,

the comparison can be seen in Figure 6.14 and for the HEB240 section in Figure 6.15

below. The results from the tests are quite scattered, which may be explained by a

difference in initial imperfections in the web.

0.0

0.2

0.4

0.6

0.8

1.0


Dis

pla

cem

ent

rati

o δδ δδ u

,wc

/ δδ δδ u

,wc,

0%

[-]

Simplified model

Vayas et. al.

Tests - A1 - A3

Tests - A4 - A8

Figure 6.14: Comparison of test series 1 and 2 on HEA 240 with the approaches

0.0

0.2

0.4

0.6

0.8

1.0


Dis

pla

cem

ent

rati

o δδ δδ u

,wc

/ δδ δδ u

,wc,

0%

[-]

Simplified model

Vayas et. al.

Tests - B1 - B3

Tests - B4 - B8

Figure 6.15: Comparison of test series 1 and 2 on HEB 240 with the approaches


152

In the figures above, one can see that the approach by Vayas et al. gives a reasonable

prediction for the HEB section, but for the more slender HEA section it clearly

underestimates the effect of the axial load.

The other approach, called the Simplified Model, which is compared in the figures

above, has been derived from the parametric study. Like the resistance, the ductility

is more sensitive to axial load at elevated temperatures than at ambient temperatures.

However, the change in the shape of the stress-strain curve has significantly more

influence on the reduction in ductility than the absolute component temperature, as

there is a large difference between the reduction at 20°C and 450°C, but almost no

change can be seen between 450°C and 650°C. Therefore, it seems sufficient to use

only two different reduction factors for the ductility of the compression zone. That

for ambient temperature is shown in Equation 6.32

, , 20

, ,

1 0.4o

N

N T Cy y wc

kk fδ

θ

σ=

= − ...6.32

and for elevated temperatures in Equation 6.33.

, , 200

, ,

1 0.6o

N

N T Cy y wc

kk fδ

θ

σ>

= − ...6.33

These reduction factors have to be used together with the ductility prediction

approach (shown in Equation 6.30) and the simplified model for the capacity of the

compression zone, which was derived at the beginning of this chapter.

As was the case for resistance, the reduction of the ductility can be extracted from the

numerical study and has been compared with the above-proposed reduction factors

and the approach of Vayas et al.. For the different cross-sections, the results of the

numerical study at ambient and elevated temperatures are shown in the following

figures.


153

0.0

0.2

0.4

0.6

0.8

1.0


Dis

pla

cem

ent

rati

o δδ δδ u

,wc /

δδ δδ u,w

c,0

% [

-]

Simplified model - 20°C

Simplified model - > 200°C

Vayas et. al. - 20°C

FEM - 20°C

FEM - 200°C

FEM - 450°C

FEM - 650°C

Figure 6.16: Reduction of the ductility due to axial loads in an UC203x203x46

0.0

0.2

0.4

0.6

0.8

1.0


Dis

plac

emen

t ra

tio

δδ δδ u,w

c /

δδ δδ u,w

c,0

% [

-]




FEM - 20°C

FEM - 450°C

FEM - 550°C

FEM - 650°C



154

0.0

0.2

0.4

0.6

0.8

1.0


Dis

plac

emen

t ra

tio

δδ δδ u,w

c /

δδ δδ u,w

c,0

% [

-]




FEM - 20°C

FEM - 450°C

FEM - 650°C


Again, a reasonable correlation between the proposed model and the finite element

analysis was achieved. The approach of Vayas et al. however, is over-conservative

for the stocky UC254x254x167 section and it ignores the reduction due to axial load

completely for the more slender section type UC203x203x46. Therefore, this

approach cannot be recommended for general use. Next, the proposed approach will

be compared statistically with the available test data, including axial load at ambient

and elevated temperatures.

Table 6.3: Statistical comparison of the new approach and tests with axial load


Mean value

Standard deviation



Upper 5% percentile


Ambient temperature –

axial load 15 1.296 0.350 0.270 0.720 1.872 0.841

Elevated temperature –

axial load 7 1.158 0.082 0.071 1.023 1.293 0.821

Following the trend in the comparison of the displacement approach without axial

load, the scatter is fairly large and the prediction is not very accurate. However, at

elevated temperatures a good correlation between the tests and the proposed approach

was found. Nevertheless, this might be because only one type of section was tested,


155

and more tests would be necessary to confirm the results. The following figures show

the comparison visually.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8δδδδu,Rk [mm]

δδ δδu,

Tes

t [m

m]

0

1

2

3

4

5

0 1 2 3 4 5δδδδu,Rk [mm]

δδ δδu,

Tes

t [m

m]

Figure 6.19: Comparison of the new approach with tests at 20°C (a) and over

450°C (b) with axial load

After the new approach has been validated, it can be compared with the full force-

displacement curve found in the experiments with axial load.

6.6 Comparison between the simplified model and the experiments

Finally, the proposed simplified model for the compression zone at elevated

temperatures can be compared with the tests conducted in this project including axial

load. As was described in the last chapter, the material strength of the specimens was

increased due to the different testing methods used in the experiments and the

material tests on which the temperature reduction factors in the EC3-1.2 are based.

Therefore, the stress at different strain levels has been used to predict the failure

envelope within which the experimental force-displacement curve should fall. In the

next five figures, this comparison will be shown for Tests 7, 9, 13, 16 and 18. It

should be noted that in the experiments no significant reduction due to axial load

could be observed, which may be due to the fact that it was not possible to maintain

the same temperature and temperature distribution from one test to the other.

(a) (b)


156

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 7 - 464°C

1% - Renner

2% - Renner5% - RennerUTS - Renner

1% - Kirby

2% - Kirby5% - Kirby

UTS - KirbyEC 3-1.2

Figure 6.20: Comparison of the simplified model and Test 7 at a LR of 0.36

0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 9 - 592°C

1% - Renner2% - Renner

5% - RennerUTS - Renner1% - Kirby


UTS - KirbyEC 3-1.2


0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN

] Test 13 - 547°C1% - Renner2% - Renner5% - Renner

UTS - Renner1% - Kirby2% - Kirby

5% - KirbyUTS - KirbyEC 3-1.2



157

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN] Test 16 - 451°C

1% - Renner2% - Renner5% - RennerUTS - Renner1% - Kirby2% - Kirby5% - KirbyUTS - KirbyEC 3-1.2


0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16

Displacement [mm]

For

ce [

kN

] Test 18 - 595°C1% - Renner

2% - Renner5% - Renner

UTS - Renner1% - Kirby


UTS - KirbyEC 3-1.2


Similarly to the comparison between the simplified model and tests without axial

load, it can be seen that when the EC3-1.2 material strength is used the prediction is

quite conservative. However, when the steady-state reduction factors based on the

work by Kirby and Preston or Renner are used, a much better comparison of the

resistance can be achieved.

6.7 Conclusion and discussion

In this chapter, the development of an analytical approach for consideration of the

effects of axial load in a column at ambient and elevated temperatures was presented.

Initially, it was shown that none of the existing simple reduction factors given in

design codes and publications were accurate enough. Most approaches were


158

unconservative, which may be because they have been calibrated in combination with

a conservative design approach for the resistance. Recognising this lack of accuracy

and the ‘one size fits all’ methodology of the simple reduction factors, an analytical

model based on the approach by Lagerqvist and Johansson was derived accounting for

the reduced moment resistance of the plastic hinges in the flange. Subsequently, this

new approach has been validated against ambient temperature test results and a

numerical parametric study including the whole range of d/t-values found in the

British UC sections and temperatures up to 650°C. In the second part of the chapter,

the reduction of the ductility of the compression zone due to axial load has been

investigated and reduction factors for ambient and elevated temperatures have been

derived. Finally, the new simplified model has been compared with the experiments

at elevated temperatures including axial load, and similarly good comparison with the

tests without axial load could be achieved if the strength-temperature reduction

factors for steady-state tests are used.

This is the end of the investigation of the compression zone, and the proposed and

validated simplified model will be introduced into the connection element.

Chapter 7: The component based connection element

159

7 The component based connection element

7.1 Introduction

Since an accurate simplified model for the compression zone in a column web has

been derived, the approaches for the main components necessary for the development

of a component-based joint element have been developed. Now the need for

consideration of the effects of interaction within frames on connections, and vice

versa, at elevated temperatures, as highlighted in the first chapter, can be addressed.

In the second chapter it was concluded that a finite connection element based on the

component method would be a feasible way to introduce detailed connection design

into global frame behaviour. Such an element will be developed in this chapter.

The work described in this chapter has been presented by the author at SiF’06 in

Aveiro, Portugal (Block et al., 2006).

7.2 Selected previous connection elements at ambient temperature

At ambient temperature, researchers have developed more or less complex connection

elements and incorporated them into finite element programs. An early attempt was

made by Poggi (1988). He developed a two-dimensional non-linear line element,

which consisted of a beam element, six springs and a rigid bar with a length equal to

half the depth of the column at either side of the beam element. The springs were

used to simulate the connections at the beam-ends in a piecewise-linear

approximation of the rotational and translational continuity. However, in this study it

was necessary to know the moment-rotation-, normal force-axial displacement- and

the shear force-vertical displacement-relationship of each connection a priori. In

addition, the axial springs were not coupled with the rotational springs, which means

that the effects of axial force in the beam on the moment-rotation behaviour of the

connections were not considered. Later, Atamaz-Sibai and Frey (1993) developed a

joint element as a combination of a two-noded Mindlin beam element representing the

shear panel in the column web and either two or one spring blocks at the beam to

column flange intersection for either internal or external joints. Again, the spring

blocks consisted of three independent springs, two for translation and one for rotation.

Linear linkage equations were used to connect the shear beam element with the spring

blocks. However, like Poggi, Atamaz-Sibai and Frey relied on predefined moment-


160

rotation and force-displacement curves for the spring blocks as well as for the shear

beam. The first to use only translational springs to represent the rotational as well as

both translational degrees of freedom was Li et al. (1995). In this study, the stiffness

matrix of a two-noded connection element was derived, based on two parallel

translational springs in the vertical direction and two in the horizontal direction. In

both pairs, the springs were separated by a certain lever-arm. With this technique, the

coupling effects between axial force and moment, and shear and moment could be

considered. Furthermore, the connection element derived by Li et al. was capable of

simulating the real length of a connection. Although this connection element was

quite complex, it could not represent a connection in detail (i.e. individual tension

springs for each bolt row). Actually, in the example analyses presented by Li et al.

(1995) the connection element was simplified to what was effectively a rotational

spring generated by two translational springs with a lever-arm, by assuming the axial

and shear degrees of freedom as rigid.

The development of the Component Method and its publication in EC3 – Annex J and

later in EC3-1.8 provided the research community with simplified equations for the

resistance and initial stiffness of each component. This allowed Bayo et al. (2005) to

develop a four-noded connection element for internal steel joints based on the

simplified equations of EC3. This element consisted of a diagonal shear spring and a

translational spring for the column web, in compression at the position of the column

web as well as an equivalent translational spring for the tension components at both

sides of the element. Subsequently, the element was used to investigate the effects of

shear panel behaviour on the bending moment distribution within frames. However,

if such an element is used to model frame structures in fire it would need to be able

also to predict the unloading behaviour of the joint.

Although quite complex connection elements have been developed for ambient

conditions, none of the proposed elements in the literature is sophisticated enough to

simulate the behaviour of a connection in a frame structure exposed to heating and

cooling in a natural fire.

What is presently used to solve this problem in Vulcan will be described, and then a

new more comprehensive finite connection element will be derived.


161

7.3 The existing spring models in Vulcan

Vulcan, like most other structural analysis programs can model the connection

between two beam-column elements as rigid or pinned. Additionally, it is possible to

introduce semi-rigid connections by specifying a constant rotational and axial

stiffness. To do this, a simple spring element with two nodes was introduced by

Bailey (1995), which could either have constant temperature-independent stiffnesses

for the different degrees of freedom or could model a certain extended endplate

connection in loading and unloading by using a development of the modified

Ramberg-Osgood relationship by El-Rimawi (1989), which is shown below,

. = +

nM M

0 01A B

φ ...7.1

where M is the moment and φ the rotation between the two nodes of the element.

The parameter A is used to influence the initial stiffness of the curve, the parameter B

the moment resistance and n the general shape of the curve. These three parameters

are temperature-dependent.

In subsequent research projects by Leston-Jones (1997) and Al-Jabri (1999), the

spring element has been extended to incorporate generic moment-rotation-temperature

curves for extended and flush endplate connections based on experimental data.

However, the above described approach to model the moment-rotation-temperature

behaviour of connections has a number of significant limitations, which are similar to

those found in the ambient-temperature attempts:

1. The curves are based on experimental data, but each connection behaves

differently depending on its geometrical and material properties.

2. No failure point of the connection is specified with respect to the real

connection behaviour.

3. The significant effects of axial load on the connection response are not

considered.

Therefore, the existing two-noded spring element in Vulcan has been extended to be

based on the component method.

7.4 The proposed connection element

Like most other connection elements developed so far, the proposed element has no

physical length. This assumption seems acceptable, as the element does not include

the shear panel behaviour in the column web, and therefore the real length of the


162

modelled connection is small in comparison to the length of the attached beams,

which is the governing factor for the influence of connection length on frame

behaviour according to Li et al. (1995). The proposed element, as shown for a flush

endplate connection in Figure 7.1, considers the following parts of a real joint:

1. Endplate in bending

2. Column flange in bending

3. Bolts in tension

4. Column web in compression

The first three components form the tension zone of the connection and are combined

as two T-stubs in series. An additional shear spring (5) had to be included in order to

transfer the vertical load from one node to the other. However, this shear spring is

assumed to be rigid at present although the formulation of the element allows the

implementation of slip and shear failure of the bolts. It was important to position this

shear spring vertically in order to uncouple the vertical and horizontal stiffnesses of

the element. As mentioned above, the shear zone in the column web is not yet

included, which limits the use of the element to internal joints with fairly equal

moments, in which the column web does not experience shear deformations. The

assumed position of the connection element can be seen in Figure 7.1 below.

Figure 7.1: Assumed position and components of the new connection element

In order to include the element into Vulcan, the component method had to be

formulated following the principles of the finite element method. Therefore, the

behaviour of the connection has to be represented in the following form:

4 1, 3 2, 3

5

u

w

φ

lT,1

lT,3

lT,2

l C,1

l C

,2

0 mm


163

= CKF u ...7.2

However, due to the non-linear behaviour of the individual connection components in

fire, and also to the highly non-linear behaviour of the connected structural members,

it is necessary to solve Equation 7.2 iteratively using the tangent stiffness ′CK and

incremental force vector ∆F and the displacements vector ∆u . Therefore, Equation

7.2 should be written as:

′∆ = ∆CKF u ...7.3

with

x i y i z i x i y i z i x j y j z j x j y j z jN V V M M M N V V M M M ∆ = ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ , , , , , , , , , , , ,

TF

and

φ φ φ φ φ φ ∆ = ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ , , , , , ,i i i x i y i z i j j j x j y j z ju v w u v w

Tu

During the iterative process Vulcan assumes ∆u based on the previous step’s stiffness,

and the connection element has to recalculate its stiffness matrix in accordance with

the proposed displacements and also the updated incremental force vector ∆F. Both

are then returned to the main routines of the program and a convergence check based

on the out-of-balance forces is performed. If convergence is reached, the next load or

temperature is calculated, otherwise the incremental displacement is varied until

equilibrium is reached. How the tangent stiffness matrix of the connection element

was derived is shown in the next section.

7.5 Derivation of the stiffness matrix of the connection element

7.5.1 The stiffness matrix

The stiffness matrix of the connection element will be derived based on its simplest

possible form in two dimensions with two horizontal and one vertical springs, which

can be seen in Figure 7.2. In this figure, the orientation of the forces and

displacements at the two nodes of the element is shown. In general, a right-hand

coordinate system was followed.


164

Figure 7.2: The forces and displacement on the 2D connection element

In order to derive the stiffness matrix for the connection element each degree of

freedom (DOF) is moved individually. The three deformation modes for node i of the

element are shown in

Figure 7.3.

Figure 7.3: Deformation modes of node i of the connection element

In the next section, the detailed calculation of each field in the stiffness matrix of the

basic connection element is shown.

Node i - Mode 1

If now node i is translated by ui = 1 and all other DOFs are fixed the following spring

forces are developed:

1 1 iF k u=

3 3 iF k u=

k3

k1

k2

j i

Nj , uj

Vj , wj

Mj , φj

Ni , ui

Vi , wi

Mi , φi

δ1,i

δ3,i

F1 = k1 δ1,i

F3 = k3 δ3,i

δ2,i

F2 = k2 δ2,i

wi = 1 l1

l3

δ1,i

δ3,i

F1 = k1 δ1,i

F3 = k3 δ3,i

φi = 1 ui = 1

Mode 1 Mode 2 Mode 3


165

Spring forces are reaction forces and therefore they always act in opposite direction to

the applied displacement. If now further the force equilibrium on the left hand side of

the element is formed the first field on the main diagonal of the stiffness matrix can be

calculated:

i 1 3N - F - F = 0

i 1 3N = F + F

( )i 1 3 iN = k +k u

1,1K 1 3= k + k

If the stiffness of the two springs is not equal, a reaction moment Mi is generated by

the translation ui.

i 1 1 3 3M - F l + F l = 0

i 1 1 3 3M = F l - F l

( )i 1 1 3 3 iM = k l - k l u

1,3K 1 1 3 3= k l - k l

Node i - Mode 2

If now node i is translated by wi = 1 and a zero element length is assumed only a force

in the shear spring is developed.

2 2 iF = k w

Vertical force equilibrium reveals:

i 2V - F = 0

i 2V = F

i 2 iV = k w

2,2K 2= k

Node i - Mode 3

If now node i is rotated by φi = 1 the following spring forces are developed:

1 1 1 iF =k l φ

3 3 3 iF =k l φ

Moment equilibrium gives the resulting reaction moment Mi.

i 1 1 3 3M - F l - F l = 0


166

i 1 1 3 3M = F l + F l

( )2 2

i 1 1 3 3 iM = k l +k l φ

3,3K 2 2

1 1 3 3= k l + k l

If again the stiffness of the two springs is not equal, a reaction normal force Ni is

generated by the rotation φi.

i 1 3N - F + F = 0

i 1 3N = F - F

( )i 1 1 3 3 iN = k l - k l φ

3,1K 1 1 3 3= k l - k l

Node j

The same procedure can be repeated on node j resulting in the following stiffness

matrix components:

4,4K 1 3k +k=

4,6K 1 1 3 3k l - k l=

5,5K 2-k=

6,6K 2 2

1 1 3 3k l +k l=

6,4K 1 1 3 3k l - k l=

By solving the global force and moment equilibrium on the whole element the

influences of a DOF of node i on the reaction forces on node j can be calculated.

Horizontal equilibrium:

i jN N 0+ =

i jN N= −

( ) ( )i 1 3 j 1 1 3 3 jN - k +k u - k l - k l φ=

( )1,4K 1 3- k +k=

( )1,6K 1 1 3 3- k l - k l=

Vertical equilibrium:

i jV V 0+ =

i jV V= −


167

i 2 jV -k w=

2,5K 2-k=

Moment equilibrium:

i jM M 0+ =

i jM M= −

( ) ( )2 2

i 1 1 3 3 j 1 1 3 3 jM k l k l u k l k l φ= − − − +

( )4,3K 1 1 3 3- k l - k l=

( )6,3K 2 2

1 1 3 3- k l + k l=

The same procedure can be conducted by displacing node j and calculating the

reaction forces on node i. This results in the symmetric stiffness matrix of the

connection element in two dimensions.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

K

1 3 1 1 3 3 1 3 1 1 3 3

2 2

2 2 2 2

1 1 3 3 1 1 3 3 1 1 3 3 1 1 3 3

C

1 3 1 1 3 3 1 3 1 1 3 3

2 2

2 2 2 2

1 1 3 3 1 1 3 3 1 1 3 3 1 1 3 3

k k l k l k k k l k l k0 0

0 k 0 0 k 0

l k l k l k l k l k l k l k l k0 0=

k k l k l k k k l k l k0 0

0 k 0 0 k 0

l k l k l k l k l k l k l k l k0 0

+ − + − − − − − + − +− − + − + −− − − − + − +− −

...7.4

However, if one wants to predict the behaviour of a connection realistically, at least

four horizontal springs are required, namely the upper compression zone, upper bolt

row, lower bolt row and lower compression zone. It is necessary to separate the

tension and compression springs because their lines of action are different. The

compression force is assumed to be transferred at the centreline of the beam flanges

whereas the tension force of each bolt row is assumed to be transferred at the

centreline of the bolts. Therefore, the connection element has to be extended to

include additional horizontal springs. Following the law for parallel springs, it is

clear that the stiffness of each spring can be simply added together. However, for the

mixed terms of the stiffness matrix the position of the springs relative to the centre of

rotation become important and a direction for the lever arm of each spring has to be

introduced. This is defined to be positive from the centre of the beam upwards.

Using this it is possible to express the stiffness matrix for a connection element with


168

multiple bolt rows using summation signs. Furthermore, for the connection element

to be able to be used in Vulcan the third dimension has to be introduced. However,

the out-of-plane DOFs and the torsional DOF are assumed to be of minor importance

in a steel or composite frame building, and therefore these DOF are connected rigidly

and no interaction between them is assumed. So the final tangent stiffness matrix of

the connection element can be shown.

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

0 0 0 0

11 15 11 15

33 33

51 55 51 55

C11 15 11 15

33 33

51 55

K K K K

K K

K K K K

=K K K K

K K

K K

′ ′ ′ ′− −

∞ −∞′ ′−

∞ −∞′ ′ ′ ′− −

∞ −∞′

′ ′ ′ ′− −

−∞ ∞′ ′−

−∞ ∞′ ′ ′− −

K

0 0 0 00 0 0 0 0 0 0 0 0 0

51 55K K

′ −∞ ∞

...7.5

where

, ,

n 2

11 T i C i

i 1 i 1

K k k= =

′ ′ ′= +∑ ∑ ...7.6

, , , ,

n 2

15 51 T i T i C i C i

i 1 i 1

K K l k l k= =

′ ′ ′ ′= = +∑ ∑ ...7.7

33 sK k′ ′= ...7.8

, , , ,

n 22 2

55 T i T i C i C i

i 1 i 1

K l k l k= =

′ ′ ′= +∑ ∑ ...7.9

Where n is the number of bolt rows, and the indices T and C indicate springs acting in

tension and compression only, respectively. The index S indicates a shear spring.

7.5.2 Incorporation of the stiffness matrix into Vulcan

This stiffness matrix has been incorporated into Vulcan. The new connection element

uses the same infrastructure as the existing spring element in Vulcan specified in the

subroutine SEMIJO. SEMIJO provides the necessary incremental displacement

vector for the connection element and returns the tangent stiffness matrix and the

force vector. In order to check that the element has been incorporated correctly, a

number of simple tests using the three-spring model described above have been

conducted.


169

The first test concerns the deformations of the element under a tension force. To test

the off-diagonal terms of the stiffness matrix the two horizontal springs have different

stiffnesses. It has been assumed that the node i is fully fixed so that all the

deformations occur at node j.

Figure 7.4: Stiffness matrix test – normal force

The problem described above is still simple enough to be solved by hand using the

stiffness matrix. Because a constant stiffness for the springs has been assumed it is

possible to apply the full load all at once in the hand calculation. However, Vulcan

will use the full non-linear solution process to solve the connection element, and the

total load will be divided into 25 steps.

Because node i is fully fixed and the shear DOF is uncoupled from the other DOF the

problem described above reduces to the following set of simulations equations.

( ) ( )j 1 3 j 1 1 3 3 jN k k u k l k l φ= + + + ...7.10

( ) ( )2 2

j 1 1 3 3 j 1 1 3 3 jM k l k l u k l k l φ= + + + ...7.11

Solving the above equations the deformations of node j can be calculated directly.

( )( )( ) ( )

( )( )( )( )

j

j 2 2

1 1 3 3

1 3 2 2 221 1 3 3

N 200000u

k l + k l 10000* 200+5000* -200k + k 10000+5000 -

k l +k l 10000* 200 +5000* -200

15 mm

= =

−

=

( )

( )( )( )

( )( )1 1 3 3 j

j 2 2 221 1 3 3

10000* 200+5000* -200 * 15k l +k l u= - = = -0.025 rad

k l + k l 10000* 200 +5000* -200φ

These deformations can now be compared with the results of the connection element

taken from the standard output files of Vulcan.

k3 = 5000 N/mm

k1 = 10000 N/mm

k2

j i Nj = 200 kN 200

mm

20

0 m

m


170

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16Translation [mm]

Nor

mal

for

ce [

kN]

Connection element

Theory

0

50

100

150

200

250

-30 -25 -20 -15 -10 -5 0Rotation [mrad]

Nor

mal

for

ce [

kN]

Connection element

Theory

Figure 7.5: Comparison between the new connection element and theory

If now, additional to the axial load, a moment on node j of M = 100 kNm is applied,

the deformations can again be calculated using the simultaneous equations 7.10 and

7.11.

( ) ( )

( )

( )( ) ( )( )

( )( )

2 2

j 1 1 3 3 j 1 1 3 3

j 2

1 3 1 3

22

2

N k l +k l - M k l +k lu =

k k l - l

2.0e5 1.0e4* 200 +5000* -200 -1.0e8 1.0e4* 200+5000* -200=

1.0e4* 5000* 200 - -200

= 2.5 mm

( )

( )( )( )

( )( )

1 1 3 3 j

j 2 2

1 1 3 3

22

k l +k l u= - M

k l + k l

10000* 200+5000* -200 * 2.5= 1.0e8

10000* 200 +5000* -200

= 0.1625 rad

φ −

−

Again, these deformations are compared with the results of the connection element as

calculated by Vulcan.

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3Translation [mm]

Nor

mal

for

ce [

kN]

Connection element

Theory

0

20

40

60

80

100

120

0 20 40 60 80 100 120 140 160 180Rotation [mrad]

Mom

ent

[kN

m]

Connection element

Theory



171

Again, the connection element calculates the deformations under the applied loads

exactly. From the tests performed, it can be said that the stiffness matrix of the

connection element has been incorporated accurately into Vulcan.

7.6 Relocation of the reference axis

So far, it has been assumed that the nodes of the connection element are situated at the

centre line of the beam. However, in Vulcan it is common practice when modelling

composite construction to position the nodes of the slab and the beam elements in the

middle of the slab element and to use an offset in the beam element formulation to

create the correct internal lever arm of a composite beam. Therefore, the nodes of the

connection element have to be able to be placed at the centre line of the slab, and still

give the same results as if the nodes were situated in the middle of the beam cross

section. One way of achieving this is using the same equations used in the beam

element formulation (Bailey, 1995). In this approach the forces in the new position

(indicated with *) can be calculated from the equations below:

*N N= ...7.12

*V V= ...7.13

*M M N l= + ...7.14

And the displacements in the new position from the equation below.

*u u l sinφ= − ...7.15

*v v= ...7.16

*φ φ= ...7.17

Figure 7.7 shows the general arrangement of the slab, beam and connection elements

most commonly used in Vulcan.


172

Figure 7.7: Offset arrangement of the connection element

The other way of introducing an offset in to the connection element is by including

the offset into the lever arms of each spring. This way is much simpler as no

conversion of the displacement, or the forces has to be conducted. Therefore, this

way has been programmed into the connection element in Vulcan. In order to show

the equivalence of the two approaches the same example spring model as above under

a hogging moment and axial tension will be calculated with both approaches.

However, it will be assumed that the connection element used is in a composite

construction, where the nodes are assumed to be in the centre of the slab and the depth

of the slab is D = 150mm, the height of the beam H = 400mm and hence the offset l is

equal to 275mm. In this example a moment of 100kNm and a normal force of 200kN

is applied to the connection. The stiffness of the springs is specified in Figure 7.4.

The first step is to calculate M* and then to calculate the displacements of the

connection element in the new position. The final step is to convert the displacements

to the original position of the element.

* . . * .M M N l 1 0e8 2 0e5 275 1 55e8 Nmm= + = + =

* .N N 2 0e5 N= =

( ) ( )

( )

( )( ) ( )( )

( )( )

*

.

2 2

j 1 1 3 3 j 1 1 3 3

j 2

1 3 1 3

22

2

N k l +k l - M k l +k lu =

k k l - l

2.0e5 1.0e4* 200 +5000* -200 -1.55e8 1.0e4* 200+5000* -200=

1.0e4* 5000* 200 - -200

= 4 375 mm−

Off

set l

u

u*

Connection element

D

H

Slab element and beam element

Assumed position of beam element

Assumed position of connection element


173

( )

( )( )( ) ( )

( )( )

*

.

1 1 3 3 j

j 2 2

1 1 3 3

22

k l +k l u= - M

k l + k l

10000* 200+5000* -200 * 4 375= 1.55e8

10000* 200 +5000* -200

= 0.2656 rad

φ −

−−

* . * . .u u l sin 4 375 275 sin0 2656 67 816 mmφ= + = − + =

* .0 2656 radφ φ= =

These deformations can now be compared with the results of the connection element

taken from the standard output files of Vulcan.

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80Translation [mm]

Nor

mal

for

ce [

kN]

Connection element

Theory

0

20

40

60

80

100

120

0 50 100 150 200 250 300Rotation [mrad]

Mom

ent

[kN

m]

Connection element

Theory


In general, a very good correlation between the two approaches is achieved.

However, the connection element returned a maximum displacement of u = 68.672

mm, which is about 1.25% larger than what was calculated above. This difference is

due to use of trigonometric functions in the hand calculations, whereas the connection

element assumes small displacements where sinφ φ= . If the hand calculations above

are repeated assuming small displacement theory, a maximum displacement u =

68.665 mm can be calculated, which is very close to the result of the connection

element. Nevertheless, the rotations in the example are considerably larger than what

can be expected in real connections in fire and the difference at these large rotations is

still very small. It can therefore be said that the implemented approach for the offset

in the connection element is sufficiently accurate.

7.7 Spring Component model used

In the previous section, the connection behaviour was very simplified, assuming only

two springs with linear force-displacement characteristics. This simplification was


174

made to be able to validate easily the implementation of the stiffness matrix of the

connection element. In real connections, however, the response of each component is

non-linear and depends on the geometry, the material properties and the temperatures

of the various parts of the connection. In the following section, it will be shown how

these component characteristics are included into the new element.

7.7.1 Tension zone

The tension zone of a connection can be represented as an equivalent T-stub

consisting of either the endplate or the column flange and normally one row of bolts.

Spyrou (2002, 2004a) conducted a large number of experiments studying the

elevated-temperature behaviour of all three typical failure modes occurring in a T-

stub, which can be seen in Figure 7.9. Based on these tests and additional numerical

studies, he developed analytical models based on classical beam theory to predict the

force-displacement-temperature behaviour of the tension zone. This approach has

been well documented and therefore it will not be repeated here. As the approach by

Spyrou is the only existing approach for the tension zone at elevated temperature, it

will be used in the connection element.

Figure 7.9: The three failure modes of a T-stub

Spyrou considered in his model a symmetric combination of two identical T-stubs

with four bolts. However, to be able to model each bolt row in a connection

individually it was necessary to change his formulations accordingly. Furthermore, it

is very unlikely that the endplate and the column flange will have the same

dimensions and therefore each side of the T-stub has to be simulated individually.

This has been done with two springs in series, one representing a T-stub in the

endplate and the other a T-stub in the column flange. However, this creates the need

to distribute the deformation of the bolts to either T-stub. The approach by Spyrou

assumes that in the displacement calculations of both T-stubs the full effective bolt

length, from the centre of the nut to the centre of the bolt head, can be used. More

1st failure mode 2nd failure mode 3rd failure mode


175

correctly, the bolt length should be split equally between the two T-stubs, as is done

in the EC3-1.8, or in accordance with the stiffness of the T-stubs, as suggested by

Kühnemund (2003). Nevertheless, the approach by Spyrou appears to work well and

will be used anyway, as it is beyond the scope of this study to revise the tension zone

models.

As could be seen in the derivation of the stiffness matrix, only the total displacement

of a spring or a system of springs at a certain position in the connection can be

calculated from the translation and the rotation of the nodes of the connection

element. Therefore, it is necessary to combine the two springs, one for the endplate

and one for the column flange, as one effective spring. To be able to do this, the

forces at the change-points which form the multi-linear force-displacement curves of

the individual springs, are sorted in increasing order. These force levels form the

points where the stiffness of the effective spring changes. In order to find the

displacements of the effective spring at these force levels, the displacements of the

individual springs have to be calculated and added together. However, as is typical

for springs in series, the capacity is defined by the weakest spring and therefore the

highest force level has to be ignored. This procedure has to be repeated at each

temperature step and is schematically shown in Figure 7.10 below.

Displacement

For

ce

Figure 7.10: Assembly of the individual springs to the final tension zone spring

Individual springs

Effective spring

Force levels


176

The force-displacement curve of the effective spring is then used in the stiffness

matrix of the connection element. After the analysis has converged to a state of stable

equilibrium and the force in the tension spring is established, the displacement of the

endplate and the column flange side are calculated and output in order to help the user

to evaluate the state of the connection.

In order to validate the implementation of the spring characteristic model in the

connection element some of the experiments conducted by Spyrou will be modelled.

As the tests were done at constant temperatures up to 800°C, which is high enough to

change the material behaviour considerably, the temperature reduction factors for

mild steel found in the EC3-1.2 (CEN 2005) had to be used, and for the bolt material

the reduction factors developed by Kirby (1995) were used, as recommended by

Spyrou. The comparisons between the connection element and a number of T-stub

experiments of Phase C by Spyrou are shown below. At first two tests of the CA

series have been modelled, which have been designed to fail in the first failure mode,

which is plastic hinges in the centre line of the T-stub and bolt failure. As a

comparison, the experimental results of the column web deformations are shown as

dots in the figures below.

0

50

100

150

200

250


Axi

al F

orce

[k

N]

CA1 - Column Flange

CA1 - Endplate

CA1 - Total

0

50

100

150

200

250

300

350


Axi

al F

orce

[kN

]

CA4 - Column Flange

CA4 - Endplate

CA4 - Total

Figure 7.11: Tension zone model and tests CA1 and CA4 at 660°C and 530°C

A good comparison can be seen from the figures above for the first failure mode.

However, it was not part of this project to develop or refine a model for the tension

zone in fire, therefore this validation is only for the correct implementation of the


177

approach by Spyrou. If one compares the results of the connection element with what

has been shown by Spyrou (2002, 2004a) it can be seen that the results for the first

failure mode correlate very well. Next, the connection element has been used to

model two experiments on T-stubs designed to fail in the second failure mode, which

includes plastic hinges at the centre and the bolt line of the T-stub, and final bolt

failure.

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30 35Displacement [mm]

Axi

al F

orce

[k

N]

CB1 - Column Flange

CB1 - Endplate

0

50

100

150

200

250

300

350

400


Axi

al F

orce

[kN

]

CB5 - Column Flange

CB5 - Endplate

Figure 7.12: Tension zone model and tests CB1 and CB5 at 650°C and 505°C

In terms of the capacity, the connection element predicts the T-stub behaviour

reasonably well for the second failure mode. However, the displacement at which the

yielding of the bolts is predicted is significantly different to what can been found in

the publications by Spyrou et al. (2004a), which was about 13mm and 9mm,

respectively. A careful check of the implementation of the approach into the

connection element has shown that the equations for the second failure mode are

implemented as specified in the publications (Spyrou et al., 2004a). Therefore,

further investigation into this matter appears to be necessary, which however is

beyond the scope of this project.

Finally, two experiments designed for failure in the third failure mode. (bolt failure

with the T-stub remaining elastic), have been modelled using the connection element.

Yielding

of bolts

Yielding

of bolts


178

0

10

20

30

40

50

60

70

80

90

100


For

ce [

kN]

CE1 - Endplate

CE1 - Column Flange

0

25

50

75

100

125

150

175

200

225

250

275

0 1 2 3 4Displacement [mm]

For

ce [

kN]

CE4 - Endplate

CE4 - Column Flange

Figure 7.13: Tension zone model and tests CE1 and CE4 at 610°C and 410°C

From the figures above one can see that the approach for the third failure mode

compares well with the test results. After the good comparison with the experimental

results by Spyrou, it can be said that the tension zone approach has been incorporated

into the connection element successfully. However, the T-stub is only a tool to

predict the resistance of bolt rows in real connections. To reduce a real connection

into equivalent T-stubs, it is necessary to find the effective length of the T-stub which

gives the same resistance as the bolt row in the real connection. This step will be

explained in the next section.

7.7.2 Yield line approach for the effective length

As mentioned above, the T-stub is only a simplification of the real behaviour of a bolt

row in a real connection. Therefore, the width of the T-stub has to be specified in

such a way that it ensures that the isolated T-stub behaves in the same ways as the

represented bolt row in either the endplate or the column flange. Zoetemeijer (1974)

specified a number of yield line patterns which can be used to calculate this

equivalent width. These yield line patterns have been adopted into the so-called

‘Green Book’ for moment connections published by the British Steel Construction

Institute (SCI, 1997), and by EC3-1.8. However, in the former document the

approach is presented in a more structured way, and it will therefore be referred to

here. In Table 2.4 of the Green Book, in total eleven yield line patterns are described,


179

six for bolt rows separated by a web, either in a column flange or in an endplate, and

five patterns for a bolt row in a plate extension. From the first set, only the first three

patterns are relevant for unstiffened connections in multi-storey buildings, and these

are shown in Figure 7.14.

Figure 7.14: Considered yield line patterns in a column flange or an endplate

The factor α in the third pattern can be calculated from a set of empirical equations

based on m1, m2 and e, which are given in Appendix III of the Green Book, and have

been implemented into the connection element. From the second set, all patterns have

been included into the connection element, and are shown in Figure 7.15 below.

Figure 7.15: Yield line patterns in the plate extension of an extend endplate

Group end yielding (viii) Double curvature (vii) Corner yielding (ix)

Leff = 0.5 bp

bp

Leff = 2mx + 0.625ex + 0.5g Leff = 2mx + 0.625ex + e

Individual end yielding (x) Circular yielding (xi)

Leff = 4mx + 1.25ex Leff = 2πmx

g

ex mx

ex mx

ex mx mx

e e

Circular yielding (i) Side yielding (ii)

Side yielding near

a beam flange (iii)

m e m e m1

m2

Leff = 2 π m Leff = 4 m + 1.25 e Leff = α m1


180

Further, the Green Book gives in Table 2.5 eight rules for finding the minimum

effective length, and therefore the yield line pattern with the lowest resistance in

different situations. Out of these eight, only four rules apply to the investigated

unstiffened connection in multi-storey buildings (the roman numerals refer to the

diagrams in Figure 7.14 and Figure 7.15) :

1. For a bolt row not influenced by a stiffener or a free end use:

Min{i, ii}

2. For a bolt row below the beam flange of an extended endplate use:

Min{Max{ii, iii}, i}

3. For a bolt row below the beam flange of a flush endplate use:

If g > 0.7 Bp or Tb < 0.8 tp then use: Min{Max{0.5(ii + iii), ii}, i}

otherwise use: Min{Max{ii, iii}, i}

4. For a bolt row in a plate extension use:

Min {vii, viii, ix, x, xi}

These rules have been programmed into the proposed element. However, if the

distance between the bolt rows is below a certain limit, and if the bolt rows are not

separated by stiffeners or the beam flange, it is possible that two or more bolt rows

fail together in a common yield line pattern, which reduces the strength and stiffness

of the tension components. However, due to the complexity of the calculation

procedure, and the considerable programming effort, group effects have not been

included in an automated way into the connection element, but it is possible to

‘overwrite’ manually the effective widths of the bolt rows failing in a group.

7.7.3 Compression zone

For the force-displacement curves of the compression springs, the approach described

in Chapters 5 and 6 has been incorporated in the proposed element. The axial load in

the column, which is necessary for the calculation of the compression zone

characteristics, is conservatively taken from the beam-column element directly

underneath the connection element.


181

7.8 Ambient temperature behaviour of the connection element

7.8.1 Comparison of the connection element with Eurocode 3-1-Annex J

After all the features mentioned above had been implemented into Vulcan, it was

possible to predict the behaviour of endplate connections at ambient temperature. As

an example, the moment-rotation response of five typical endplate connections have

been calculated and compared with the results given by the connection software CoP,

which follows the rules of the EC3-1.8 (2005). In the calculations, all partial safety

factors have been set to 1.0 and nominal material properties have been used. For all

steel parts a steel grade of S275 has been assumed and all bolts are M20 grade 8.8; a

throat thickness of a = 6 mm has been used for welds between the endplate and the

beam. As connected sections, a cruciform arrangement, with two UC 356x171x51 as

beams and a UC 203x203x60 for the column, has been assumed. For the geometry of

the endplate and the position of the bolts, the recommendations in the Green Book

have been followed. The three generic connection types can be seen in Figure 7.16.

The endplate thickness is varied for connection type A in order to generate all three

failure modes of a T-stub.

Figure 7.16: Example endplate connections

A summary of the results, and a comparison of the moment capacity and the initial

stiffness of the moment-rotation curve, is shown in Table 7.1 below. Unfortunately, a

comparison of the predicted rotation capacity of the connections is not possible, as the

Eurocode only states if a connection is ductile enough for plastic design.

200

90 55 55

15 60

60 15

235

200

90 55 55

15 60

60 15

145

90

200

90 55 55

40 60

60 15

235

50

A B C


182

Table 7.1: Summary of the connections behaviour at ambient temperature

Type Endplate thickness

CoP – Eurocode 3-1-Annex J

Vulcan – Connection Element

- tp MRk, EC Kini, EC Rot. MRk, Vul Kini, Vul Rot. MRk, Vul / MRk, EC

Kini, Vul / Kini, EC

- [mm] [kNm] [kNm / mrad]

- [kNm] [kNm / mrad]

[mrad] - -

A 10 74.0 25.2 Ok 79 20.5 23.6 1.07 0.81

A 12 83.2 28.9 Ok 96 22.4 29.9 1.15 0.78

A 25 89.7 34.1 Ok 103 25.7 22.2 1.18 0.75

B 12 109.8 29.0 Ok 118 26.5 21.6 1.07 0.91

C 12 135.1 55.5 Ok 157 44.4 17.4 1.16 0.80

From the comparison above a good correlation between the results calculating the

Eurocode and the response of the connection element can be seen. The connection

element gives a larger moment capacity, which is to be expected as in the component

models used the ultimate capacity rather than a design resistance is calculated.

Further, the initial stiffness is generally lower than the prediction of the Eurocode,

which aligns with the general opinion that the stiffness predictions in EC3-1.8 are

unconservative in comparison with experimental data.

In Figure 7.17 below, the full moment-rotation curves of the three connections of

Type A with different endplate thicknesses are shown. As a comparison, the non-

linear moment-rotation predictions calculated with the help of CoP after EC3-1.8 are

shown as dotted lines.


183

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35Rotation [mrad]

Mom

ent

[kN

m]

Endplate = 25 mm - Failure mode 3



Figure 7.17: Comparison of the connection element and CoP for Type A

In general, the two ways of predicting moment-rotation curves for endplate

connections correlate reasonably well; however for failure mode 2 the connection

element appears a little too soft, which is caused by the above-mentioned problem

with the displacement part of the approach by Spyrou. The comparison between

connection types A, B and C, all with an endplate thickness of 12 mm, is shown in

Figure 7.18. Again, the dotted lines are the EC3-1.8 prediction by CoP.

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35Rotation [mrad]

Mom

ent

[kN

m]

Type A

Type B

Type C

Figure 7.18: Comparison of the connection element and CoP for Type A, B, C


184

From the figure above, a very good correlation between the connection element and

the Eurocode prediction can be seen. It should be noted that the moment-rotation

curve of the proposed element is not multi-linear any more for connections of Types

B and C, because with the extra bolt row the tension zone of the connection becomes

stronger than the compression zone and the connection fails in the column web, as

was also predicted by CoP.

7.8.2 Comparison of the connection element with test results

In order to validate the ambient-temperature behaviour of the connection element two

different types of test have been used. The first test series, by Girão Coelho (2004),

was designed to investigate the rotational capacity of the endplate side of extended

endplate connections. Therefore a short beam (IPE300) was connected to a heavy

column section (HE340M) by three bolt rows (M20-8.8) and an endplate of varying

thickness. In the first (FS1a), the second (FS2a) and the third (FS3a) test endplate

thicknesses tp of 10mm, 15mm and 20mm were used, respectively. In the next three

figures, the predictions of the connection element are compared with the experimental

results. Due to large difference in thickness between the endplate and the column

flange (tf = 40.2mm) the datum of the T-stub displacement in the endplate can be

assumed to be at the centre of the nut on the column flange side of the connection.

Therefore, two different analyses have been conducted, the first denoted as (a) in the

figures, with equal effective bolt length for both T-stubs, and the second with double

the bolt length on the endplate side, denoted as (b) in the figures.


185

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40 45 50Rotation [mrad]

Mom

ent

[kN

m]

Girao Coelho - FS1a

Connection element - a

Connection element - b

Figure 7.19: Comparison of the new element with test FS1a by Girão Coelho

A very good correlation between the predictions by the connection element and the

experimental result was found. As expected, analysis (b) correlates better with the

tests, as a larger part of the bolt contributes to the endplate deformation.

0

20

40

60

80

100

120

140

160

180

200

220

0 5 10 15 20 25 30 35 40 45 50Rotation [mrad]

Mom

ent

[kN

m]

Girao Coelho - FS2a





186

0

20

40

60

80

100

120

140

160

180

200

220

0 5 10 15 20 25 30 35 40 45 50Rotation [mrad]

Mom

ent

[kN

m]

Girao Coelho - FS3a




A comparison of Figure 7.20 and Figure 7.21 shows that the rotational stiffness

predictions of the connection element become less accurate with increasing endplate

thickness. However, the resistance prediction is still very accurate. Furthermore, the

rotation capacity of the connection is under-predicted, which can be explained by the

fact that the analysis stops when the ultimate stress in any bolt is reached. In a real

connection however, the load will be redistributed to adjacent bolt rows until all bolts

are fully yielded or failure of the weld between the beam and the endplate occurs

(Girão Coelho, 2004).

The second group of experiments used for validation was designed to fail in the

compression zone of the column web. First, the ambient-temperature cruciform test

by Leston-Jones (1997) on a flush endplate connection (tp = 12 mm) with three bolt

rows (M16-8.8) connecting two small beams (UB 254x102x22) to a small column

section (UC152x152x23) was modelled. The test failed by plastic buckling of the

column web and large deformations of the column flange in tension. Due to the small

distance between the upper two bolt rows (only 50mm), the assumption that each bolt

row acts individually is unrealistic. Therefore, the effective lengths of the T-stubs

have been calculated by hand, in accordance with the group yield line pattern given in

the ‘Green Book’ (SCI, 1997), assuming the full effective width for the upper bolt

row and the remaining length of the group’s yield line pattern for the lower bolt row.


187

Both analyses are compared with the experiment in Figure 7.22, and a generally good

correlation between the response of the connection element and the experimental M-Φ

data can be seen.

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90 100Rotation [mrad]

Mom

ent

[kN

m]

Test - Leston-Jones

Bolt rows individually

Bolt rows as a group

Figure 7.22: Comparison of the new element with the 20°C test by Leston-Jones

A second example is again a cruciform test, conducted by Bailey and Moore (1999),

but this time using a more representative beam (UB457x191x74) and column

(UC254x254x107) sections, connected by a thick extended endplate (tp = 30mm) and

four bolt rows (M30-8.8). This test was designed to investigate the influence of axial

column load on the compression zone. Two different tests, with different axial

column load ratios, were conducted. As the connection element is able to account for

this effect two runs were conducted, the first without consideration of the axial load

and the second with the axial load considered. From Figure 7.23 and Figure 7.24,

excellent comparison between the proposed element and the experiments can be seen,

if the reduction due to the axial column load is included.


188

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14 16 18 20Rotation [mrad]

Mom

ent

[kN

m]

Bailey and Moore - Test 1 - N = 0.8 Npl

Connection element - with axial load effect

Connection element - without axial load effect

Figure 7.23: Comparison of the new element with Test 1 by Bailey and Moore

0

50

100

150

200

250

300

350

400

450

500

550

0 2 4 6 8 10 12 14 16 18 20Rotation [mrad]

Mom

ent

[kN

m]

Bailey and Moore - Test 2 -N = 0.6 Npl

Connection element - with axial load effect

Connection element -without axial load effect

Figure 7.24: Comparison of the new element with Test 2 by Bailey and Moore

Given the good comparison between the Eurocode approach and the proposed

element, it can be said that Vulcan is able to predict the moment-rotation response at

least as well as predictions by the Eurocode at ambient temperature. Predictions of

the connection element are probably even better, as they include the ductility of a

connection based on the mechanical behaviour of its individual components. From


189

the comparisons with test results, it can be said that the proposed connection element

in Vulcan compares accurately with experimental data at ambient temperature.

However, the rotational capacity is always predicted conservatively. This suggests

that further refinement of the tension zone behaviour is required, as the predicted

rotation was limited in most cases by the fracture of the bolts, which did not always

occur in the tests.

7.9 Elevated Temperature behaviour of the connection element

Connections tend to remain at a lower temperature than the attached beam, due to the

lower volume-to-surface ratio in the joint region, although the connection will still

reach temperatures high enough to reduce significantly the strength and stiffness of

the bolts, the endplate and the column. Therefore, the degeneration of the strength

and stiffness of the connection material with increasing temperatures has to be

included in the connection element. This can be done by using the temperature

reduction factors for mild steel given in EC3-1.2 for the column and the endplate. For

the bolts however, the temperature reduction factors derived by Kirby (1995) have

been used as a comparison against the T-stub experiments by Spyrou, because the

connection element has shown that the EC3-1.2 reduction factors for bolts gave over-

conservative results. However, both sets of reduction factors are included in the

element. The Young’s modulus of the bolts is reduced in accordance with the

temperature reduction factors for mild steel, as determined by Spyrou.

7.9.1 Temperature distribution

There are a number of possible ways of predicting the temperature distribution within

a connection based on either experiments, heat transfer calculations or design codes

like the Eurocode. Whichever method is used, it is common in all of them that the

temperature at the bottom of the connection is higher than that at the top if a floor slab

is present. Furthermore, the column web will have a higher temperature than the rest

of the connection if it is unprotected. Additionally, the time-temperature history of a

connection will be different from that of the connected beam, as experiments at

Cardington have shown by Wald et al. (2004). The connection will heat up more

slowly than the beam, but during cooling it will remain at a higher temperature,

whereas the beam cools down relatively quickly. For these reasons it does not seem

acceptable to assume the connection temperature to be at about 70% of the bottom


190

flange at the mid-span of the beam, as assumed by Lawson (1989, 1990). However, it

was not part of this project to investigate the temperatures of a connection during a

fire, and therefore a generic approach to the consideration of the temperatures in the

connection element has been programmed. To be able to account for the individual

temperature development over time of the connection, a separate time-temperature

curve can be used for each connection element. In order to account for the non-

uniform temperature distribution within the connection, a temperature pattern can be

specified for each element. This pattern consists of temperature multipliers, allowing

the specification of the temperature of the column flange, the bolts and the endplate,

for each bolt row individually, and also for the column flange and the column web in

the two compression zones. This technique gives the user the required flexibility to

consider any temperature distribution across the connection taken from experiments,

analysis or design codes.

7.9.2 Comparison of the connection element with high temperature tests

As part of the validation process of the new connection element, the elevated

temperature connection tests by Leston-Jones have been modelled. The size of the

connected sections and the connections themselves are the same as in the ambient

temperature test discussed in the previous section. However, instead of loading the

cruciform assembly until failure of the connections occurred, a constant moment was

applied to the connection and then the temperature was increased by ~10°C/min, until

runaway failure occurred or the test had to be terminated due to spatial constraints.

As a temperature distribution in the connection, the average temperature multipliers

for each component over the duration of the whole test were used. In total, four tests

with applied connection moments ranging from 5kNm to 20kNm have been compared

with the response of the proposed element. The results of the comparisons can be

seen in Figure 7.25 and Figure 7.26 below.


191

0

100

200

300

400

500

600

700

800

900

0 10 20 30 40 50 60 70 80 90 100Connection rotation [mrad]

Ste

el t

empe

ratu

re [

°C]

Leston-Jones - BFEP 5 - 5 kNm

Connection element - bolt rows individually

Connection element - bolt rows as a group

Leston-Jones - BFEP 15 -15 kNm



Figure 7.25: Comparison of the new element with test BFEP 5 and BFEP 15

In a similar manner to the ambient temperature test shown in Figure 7.22, the bolt

rows yield in a group, and the effects of this can be seen in the comparisons with tests.

0

100

200

300

400

500

600

700

800

900

0 10 20 30 40 50 60 70 80 90 100Connection rotation [mrad]

Stee

l tem

pera

ture

[°C

]







Figure 7.26: Comparison of the new element with test BFEP 10 and BFEP 20

In general, a very good comparison was found between the tests and prediction using

the element. However, the new element under-predicts the rotations found in the test

slightly, which is due to termination of the analysis when any connection component


192

reaches its ultimate load, which ignores the redistribution of internal forces between

the bolt rows.

7.9.3 Anisothermal connection responses of the example connections

After the good correlation between the connection element and the tests has been

obtained, it is now possible to assess the same example connections which have been

investigated at ambient temperature in Section 7.8.1 above, under the influence of

increasing temperature. To test the full range of connection behaviour, four different

load ratios of 0.2, 0.4, 0.6 and 0.8 have been applied, and then the temperature was

increased until the resistance of any of the components had reduced so far that the

applied load could not be supported any more. This temperature is then called the

failure temperature of the connection. First, the results of connection Type A with a

12 mm thick endplate are shown in Figure 7.27 below.

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30 35Rotation [mrad]

Tem

pera

ture

[°C

]

LR = 0.2

LR = 0.4

LR = 0.6

LR = 0.8

Figure 7.27: High temperature behaviour of connection Type A – tp = 12 mm

It can be seen that with increasing load ratio, the failure temperature of the connection

reduces. At a load ratio of 0.8, the connection has plastified at ambient temperature,

which explains the large initial rotation.


193

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30 35

Rotation [mrad]

Tem

pera

ture

[°C

]

LR = 0.2

LR = 0.4

LR = 0.6

LR = 0.8

Figure 7.28: High temperature behaviour of connection Type B

For connection Types B and C it can be observed that, the failure mode changed from

compression failure in the column web at ambient temperature to bolt failure at

elevated temperatures. This is due to the larger strength reduction of the bolt material

with respect to that of mild steel with increasing temperatures. The same change in

failure mode was observed by Lou and Li (2006) in their high-temperature

experiments on extended endplate joints. This contradicts the design approach

proposed by Simões da Silva et al. (2001b), which suggests that the ambient-

temperature resistance of a connection can be multiplied by the strength reduction

factor given in EC3-1.2 to find the high-temperature behaviour of the connection.


194

0

100

200

300

400

500

600

700

800

0 5 10 15 20 25 30 35Rotation [mrad]

Tem

pera

ture

[°C

]

LR = 0.2

LR = 0.4

LR = 0.6

LR = 0.8

Figure 7.29: High-temperature behaviour of connection type C

In Figure 7.30 below, the failure temperatures of the different connections with

respect to the load ratio are summarised. As a comparison, the reduction factor for

2% strain of mild steel given in EC3-1.2 and the strength reduction factors for bolts

derived by Kirby (1995) are plotted.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

250 300 350 400 450 500 550 600 650 700 750 800 850Failure temperature [°C]

Loa

d r

atio

Type A

Type B

Type C

Yield stress reductionfactor - EC3-1.2Bolt strength reductionfactor - Kirby (1995)

Figure 7.30: Summary of the failure temperature – load ratio relationships

From Figure 7.30, it can be seen that the results from the example connection are

situated between the two reduction factor curves. This was expected, as the failure of


195

the connections is governed by the bolts, in combination with yielding of either the

endplate or the column flange. The failure temperatures of connections B and C are

higher than those of connection A, which can be explained by the change of failure

mode from the compression zone at ambient temperature to the tension zone at

elevated temperatures. In connections B and C, the utilisation of the tension zones at

ambient temperature is smaller than in connection A, due to the compression zone

failure. This means that the load ratios of the tension zones in connections B and C

are actually lower than that of connection A, which leads to a higher failure

temperature.


In this chapter, the development of a component-based connection element capable of

predicting the ambient- and elevated-temperature behaviour of flush and extended

endplate connections has been shown. Firstly, the generic stiffness matrix of the

element was derived and included into Vulcan. Secondly, the force-displacement

models used for the individual connection components were described. The element

was then used to predict the moment-rotation response of several ambient-temperature

experiments, and finally the way in which high temperatures are incorporated into the

element is described, and comparisons with elevated temperature experiments were

made.

The proposed element compares well with the tests, although certain limitations still

exist, such as automatic consideration of group effects between bolt rows in the

tension zone, the consideration of the shear deformations in the column web and the

beam-end and bottom-flange buckling of the beams. In theory, it is possible to

consider these additional effects in a similar way to that described above, but to date

no validated high-temperature models for these components have been published.

However, the studies described in this chapter assume that the forces on the

connection remain the same at elevated temperatures as at ambient temperatures,

which may not be the case in a real building. Therefore, all studies on individual

connections are somewhat academic. In the next chapter, the very important effects

of unloading and cooling of connections in frames at high temperatures are

investigated.

Chapter 8: Unloading and cooling of the connection element

196

8 Unloading and cooling of the connection element

In the previous chapter, mainly individual connections exposed to bending moments

at ambient and elevated temperatures have been investigated. In a framed structure

during a fire, however, a beam-to-column connection experiences a changing

combination of axial forces and bending moments. These loads range from pure

bending, to bending and compression, to pure tension and, during the cooling phase of

a fire, to higher tension and a reversed bending moment. In order to respond correctly

to such a changing combination of loads, a connection element for elevated

temperatures needs a robust loading-unloading-reloading approach for constant as

well as changing temperatures.

8.1 Unloading of the connection element at constant temperatures

At ambient temperatures, the classic Massing rule with memory effects has been

included into the new element. This concept is widely used for the consideration of

unloading and cyclic behaviour of metallic materials, and compares very well with

experimental behaviour. Even though the concept has been developed on a material

basis, Gerstle (1988) showed that it could be applied to the cyclic behaviour of semi-

rigid connections too. The concept states that to get the hysteresis (unloading) curve

the skeleton (initial loading) curve has to be doubled. This ensures that the hysteresis

curve meets the skeleton curve in the opposite quadrant at the same load and

deflection level as that where the unloading started. In general, if it is assumed that

the skeleton curve is described as

( )f Fδ = ...8.1

then the hysteresis curve can be described as

( ) ( )A Af F F

2 2

δ δ− −= ...8.2

where δA and FA are respectively the displacement and the force at which unloading

occurred. This concept is shown in Figure 8.1 for force-displacement curves for a

tension zone and a compression zone.


197

Displacement

For

ce

Displacement

For

ce

Figure 8.1: Hysteresis behaviour of the tension (a) and the compression zone (b)

However, it is unrealistic to model a compression spring as transferring tension

forces, as the compressive force in a connection is introduced via direct contact.

Therefore, it is assumed that the compression springs are only active under

compressive forces, shown as the solid lines in Figure 8.1(b). The tension springs,

however, are assumed to be able to act in tension and in compression until they reach

their initial position. This can be imagined as if a T-stub on a rigid plate is deformed

plastically in tension, and is then pushed back until the centre line of the T-stub

touches the base plate again. A good summary of experimental studies on T-stubs

showing this behaviour can be found in Faella et al. (2000). In Figure 8.1(a), the

active range of a tension spring is shown as a solid line.

A complication of this approach originates from the fact that the tensile and

compressive forces in the connection do not share the same line of action. Therefore,

it is necessary for the prediction of the correct internal forces of the connection, that

the internal forces are transferred between tension and compression components. For

the compression springs, it is assumed that if they are plastically deformed and

unloaded until the endplate loses contact with the column (i.e. the compressive force

has reduced to zero), the tension spring next to it has to start taking load from this

deformed position. This is practically done by allowing the compression spring to

reach a small tension force of +500 N. If a force value between 0 N and +500 N is

present in a compression spring, the datum of all tension springs is set to their current

(a) (b)

Skeleton

curve

Hysteresis

curve

δA

FA


198

positions and the compression spring is deactivated. If further loading occurs, the

tension springs are activated and start loading from their new datum.

If, however, a tension spring is deformed plastically and unloads to its initial position,

it is assumed that all subsequent compression force in the tension spring is taken by

the much stiffer compression spring adjacent to the unloading tension spring.

In order to illustrate the force transfer between the different springs better, the

example connection Type A (Figure 7.16) from the previous chapter has been loaded

in pure tension and compression. At ambient temperature, connection Type A has a

tension resistance of 318.46 kN per bolt row calculated after the approach by Spyrou

(2004a) using nominal un-factored material values, which results in a total tensile

resistance of 636.92 kN. Therefore, a tensile force of 600 kN was used in this

example. The compressive resistance of both compression zones were calculated

following the approach derived in Chapters 5 and 6. A total compression resistance

of 940.56 kN was calculated, and therefore the connection in this example was loaded

up to a compressive force of 900 kN. Figure 8.2 shows the results of the example

calculations.

-1000-800-600-400-200

0200400600800

-4 -3 -2 -1 0 1 2 3 4 5 6Displacement [mm]

For

ce [

kN

]

-1000-800-600-400-200

0200400600800

-4 -3 -2 -1 0 1 2 3 4 5 6Displacement [mm]

For

ce [

kN

]

Figure 8.2: Force transfer between the different components

In analysis (a), the connection was initially loaded up to a tension force of 600 kN,

then reverse-loaded to a compression force of 900 kN, and then unloaded to zero. It

can be seen how the load from the tension springs is transferred to the compression

springs when the tension springs reach their starting displacement. In analysis (b) the

connection was loaded initially up to a compression force of 900 kN, then reverse-

loaded up to a tension force of 600 kN, unloaded and then re-loaded until the

(a) (b)


199

compression springs failed at a load of 940.56 kN. From this analysis, it is possible to

see how the compression springs transfer the load to the tension springs at a negative

displacement of -1.92 mm. If the tension springs unload back to this point, the load is

then transferred back to the compression springs, which reload following the

unloading path and returning to the original loading path if the point of initial

unloading is reached.

8.2 Unloading of the connection element at changing temperatures

The concept of loading and unloading in accordance with the Massing rule is

relativity straightforward at a constant temperature. However, if the temperatures of

the components change during loading and unloading, this process becomes a lot

more complicated, as the stiffnesses of the force-displacement curves of the

connection components are temperature-dependent. This can lead to situations in

which the unloading curve crosses the loading curve, which is a mechanical

impossibility. Therefore, the concept of the Reference Point at the position of

permanent displacement has been used to predict the unloading curves at any

temperature.

8.2.1 The Reference Point concept

The Reference Point forms the basis of the whole loading and unloading procedure at

changing temperatures. The concept was originally developed for unloading of

materials at elevated temperatures and assumes that ‘plastic strain is not affected by a

temperature variation’. The concept was used to describe the unloading of composite

beams and columns by Franssen (1990). El-Rimawi et al. (1996) used the concept to

describe the cooling behaviour of steel beam-columns. Bailey (1995, 1996) used the

concept for the incorporation of unloading into the simple moment-rotation spring

element used in an early version of Vulcan, and for the strain reversal in beam-column

elements during cooling. For the connection element presented here, this concept had

to be applied to each individual spring, as each has to be able to load and unload

individually. An example of such a situation can be found by looking at a semi-rigid

beam-to-column connection in a frame structure. At ambient temperature the

connection is loaded with a negative bending moment, causing tension at the top and

compression at the bottom. If the frame is then exposed to a fire, the restrained

thermal expansion of the beam will introduce a compressive force to the connection,


200

which will increase the compression at the bottom of the connection further, but will

reduce the tension force at the top of the connection, generating a situation in which

one spring unloads whereas another loads.

The main underlying assumption to the concept of the Reference Point is that the

permanent displacement of a spring is unaffected by the change of temperature, if the

connection is unloading. This makes it necessary that each force-displacement curve

at a different temperature unloads to the same Reference Point. This unloading

approach has been incorporated both into the compression and the tension springs.

However, the principle remains the same and shall therefore be explained only once

on a tension zone, illustrated in Figure 8.3 below.

Displacement

For

ce

Figure 8.3: Definition of the Reference Point and the Intersection Point

In Figure 8.3, the unloading of a tension zone component with coincident temperature

increase is shown. At temperature T1 the tension zone component is loaded with a

tensile force F1 causing a permanent displacement δpl,1. This permanent displacement

is found firstly by calculating the displacement of the unloading curve (i.e. the

doubled loading curve at temperature T1 under load F1). The displacement found is

then subtracted from the current displacement of the loading curve and the permanent

displacement is found.

δpl,1 - Reference Point

Intersection Point (δinter,

Finter)

T1

T2

F1

F2

Loading curves

( ), , 1load Tf Fδ and

( ), , 2load Tf Fδ

Unloading curves

( ), , 1unload Tf Fδ and

( ), , 2unload Tf Fδ

T1 < T2

F1 > F2

δpl,2

δ1 δ2


201

( ) ( ), , , , ,1 1pl 1 load T 1 unload T 1f F f Fδ δδ = − ...8.3

The position of this displacement on the δ-axis is called the Reference Point and is

used to identify whether unloading occurs in the next load or temperature step, (i.e. at

temperature T2), by comparing the permanent displacement of the new step δpl,2 with

the Reference Point of the previous step.

, ,pl 1 pl 2δ δ< - loading

, ,pl 1 pl 2δ δ> - unloading

If the new permanent displacement is larger than the previous one, the component is

‘loading’ and follows the loading curve at the new temperature, and the Reference

Point is updated. If the new permanent displacement is smaller than the Reference

Point, ‘unloading’ of the component occurs and the unloading curve at the new

temperature has to be used to calculate the response of the component. In order to

define the unloading curve at the increased temperature the principle of the constant

permanent displacement during unloading of a component has to be observed.

Therefore, the unloading curve at temperature T2 has to go through the Reference

Point. To be able to define the unloading curve the Intersection Point between the

unloading and the loading curves at temperature T2 can be found by solving Equation

8.4 with respect to Finter.

( ) ( ), , , , ,2 2 2 2load T inter,T unload T inter,T pl 1f F f Fδ δ δ= + ...8.4

This equation can only be solved iteratively due to the nonlinear character of the

loading and the unloading curves. If the force at the Intersection Point Finter is found,

the displacement of the Intersection Point δinter can be calculated by using the loading

curve at temperature T2.

( ), , ,2 2 2inter T load T inter,Tf Fδδ = ...8.5

Having defined the Intersection Point, the displacement δ2 on the unloading curve at a

force F2 and a temperature of T2 can be calculated relative to the Intersection Point.

( ), ,2 2 22 inter,T unload T inter,T 2f F Fδδ δ= − − ...8.6

In the connection element however, the above described calculation procedure had to

be inverted, due to the fact that during the non-linear solution process in Vulcan a trial

displacement rather than a force is given to the connection element, and a force is

returned to the solver. However, the approach remains the same. It should be noted


202

that if this approach is used for a joint component with a force-displacement model

which includes thermal expansion, this expansion has to be subtracted from the

displacement before the unloading can be calculated. However, for the models used

in this study this is not necessary.

8.2.2 Unloading and heating in tension

In Figure 8.4 to Figure 8.6 connection Type A in tension in combination with heating

and unloading, is shown. The connection was loaded at ambient temperature to +600

kN and then heated and unloaded with a force-temperature ratio of -1.25 kN/°C.

0

100

200

300

400

500

600

700

-200 0 200 400 600Force [kN]

Tem

per

atur

e [°

C]

0

100

200

300

400

500

600

700


Tem

per

atur

e [°

C]

Figure 8.4: Behaviour of connection Type A under tension-heating-unloading

Figure 8.5 shows a comparison of the constant temperature-unloading example

described in Figure 8.2 (a) with the heating-unloading example. It can be seen that

initially the differences between the two curves are relatively small and that both

curves share the same permanent displacement, in accordance with the Reference

Point concept. At temperatures above 500°C, which is coincidental with the change

between tension and compression, the response of the connection element is

considerably weaker. This is due to the dramatic reduction of the yield strength

between temperatures of 500°C and 700°C.


203

-600

-400

-200

0

200

400

600

800


For

ce [

kN]

Figure 8.5: Force-displacement curves for tension zone in heating and unloading

From Figure 8.6 the unloading curves can be seen in more detail. It can be observed

that only if the load drops below 500 kN is there a difference between the two curves.

From Figure 8.4 it can be seen that 500 kN is equivalent to a temperature of 100°C, at

which temperature the Young’s Modulus begins to reduce.

-200

-100

0

100

200

300

400

500

600

700

4.5 4.6 4.7 4.8 4.9 5 5.1 5.2Displacement [mm]

For

ce [

kN]

Figure 8.6: F-δδδδ curves for the tension zone heating-unloading example - detail

Ambient temperature + Unloading

Heating + Unloading

Ambient temperature + Unloading

Heating + Unloading


204

Further, it can be seen that the heating – unloading curve deflects more than the

ambient-temperature curve, which is due to the reduced unloading stiffness at

increasing temperatures and the requirement to unload towards the Reference Point.

8.2.3 Unloading and heating in compression

In a second example, the unloading – heating behaviour of the compression zone will

be investigated. Again, connection Type A is used, which is loaded this time to -900

kN (compression), and is then heated and unloaded with a force-temperature ratio of

+1.25 kN/°C. The force-temperature and the displacement-temperature responses of

the connection element for this example are shown in Figure 8.7 below.

0100200300400500600700800

-1000 -750 -500 -250 0Force [kN]

Tem

per

atur

e [°

C]

0100200300400500600700800

-3 -2.5 -2 -1.5 -1 -0.5 0Displacement [mm]

Tem

pera

ture

[°C

]

Figure 8.7: Behaviour of connection Type A in compression-heating-unloading

Figure 8.8 shows once more the comparison between the constant temperature-

unloading curve from Figure 8.2 (b) and the heating-unloading example. Again, it

can be seen that both curves are coincident up to a load of -800 kN, which is

equivalent to 100°C. At higher temperatures, however, the difference between the

two curves becomes quite significant, especially from a load of -300 kN onwards,

which is equivalent to 500°C. However, if the displacement-temperature plot in

Figure 8.7 is investigated more closely, distinct kinks in the curve can be seen at

500°C, 600°C and 700°C, which is due to the change of rate of the strength reduction

factors given in EC3-1.2 which are used in this study. The difference between the

two curves is so large that a more detailed investigation seems justifiable.


205

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

-2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0Displacement [mm]

Force [kN

]

Figure 8.8: F-δδδδ curves for connection Type A in compression-heating-unloading

In Figure 8.9, the loading and unloading curves of the compression zone at different

temperatures are shown. It can be seen how the displacement of the Intersection Point

increases with increasing temperature, and also how the unloading stiffness reduces.

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Displacement [mm]

Force [k

N]

Figure 8.9: F-δδδδ curves for compression zone heating-unloading example - detail

Furthermore, the recorded force – displacement values at different temperatures are

plotted on the graph and also the heating unloading curves as dotted lines. One can

700°C

600°C

500°C

400°C

200°C

20°C

Ambient temperature - Unloading

Heating - Unloading


206

see that the increase of the displacement at temperatures above 500°C is indeed due to

the large reduction of the strength and stiffness of the compression zone component.

8.3 Cooling behaviour of the connection element

The effects of cooling on a beam-to-column connection can be critical for the survival

of a structure, due to the large tensile forces in the beam, which are developed when

the thermal strains in the beam disappear leaving the post-fire beam considerably

shorter than it was originally due to permanent compressive deformation (Bailey et

al., 1996). This effect has been accounted for in the beam element in Vulcan, which

makes it possible to expose the connection element to the correct forces during

cooling of the structure.

Cooling may be considered to be the same as unloading, but in the temperature

domain instead of the load domain. If a connection is loaded to a certain load level

and the material weakens due to increasing temperature, the connection components

will eventually become plastic and the weakest component will reach its failure point,

at which the connection is deemed to have failed. However, if the connection

temperature is reduced before the failure temperature is reached, a plastically

deformed connection keeps its permanent displacement, and only the elastic

deformations recover. Therefore, the above-developed approach for the unloading of

connection components is also applicable to model the cooling response of a

connection.

8.3.1 Assumed material behaviour for bolts

When high strength bolts have been heated above their annealing temperature of

around 600°C and cooled down naturally, the strength-giving effects of quenching

and tempering during the production of the bolts vanish, and the bolt material returns

to its base material, which is considerably weaker. This has been shown

experimentally by Kirby (1995) and Sakumoto et al. (1993). Unfortunately, to date

there is no experimental data available on bolts which are loaded during cooling, and

therefore this effect has not yet been included into the connection element. It is

assumed that the bolts, as well as the material of the endplate and the column, regain

their strength when cooled down to ambient temperature.


207

8.3.2 Example of a connection under cooling

In the previous chapter, a limiting temperature of the Type A flush endplate

connection with two bolt rows was calculated as 597°C for a load ratio of 0.4.

Therefore, to investigate the cooling response of the connection, it is now uniformly

heated to 580°C and then uniformly cooled to 20°C. As was stated above, it is

assumed that all connection components regain full strength when cooled back to

ambient temperature, a reasonable assumption for sections and plates but dubious for

bolts. In Figure 8.10 the temperature – rotation plot of the example connection in

heating and cooling can be seen. Up to a temperature of about 500°C only a moderate

increase in rotations can seen, but at higher temperatures the resistance of the upper

bolts has reduced so far that the yield strength of the upper bolt row is reached and the

stiffness of this component is reduced significantly. With increasing temperature, the

displacement of the upper bolt row increases further, and the centre of rotation of the

connection moves below the lower bolt row at about 540°C, which activates this

component. This explains the increase of stiffness around this temperature. The

connection rotates further until the maximum temperature, in this example of 580°C,

is reached. From this point, the connection is cooled down to 20°C and rotates back

in accordance with its unloading stiffness. The kink in the cooling curve at 500°C can

be explained by the rapid change of slope of the temperature reduction factor for the

E-Modulus as given in EC3-1.2. A permanent rotation of 16.9 mrad remains.

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14 16 18 20 22Rotation [mrad]

Tem

per

atur

e [°

C]

Figure 8.10: Temperature-rotation plot of a connection in heating and cooling


208

A closer look is now taken at the individual component behaviour. Figure 8.11 shows

the temperature - component displacement response. It should be noted that the lower

bolt row is not decreasing its displacement in cooling, as are the other two springs.

This is caused by the much stiffer unloading of the compression zone and the close

spacing of the two springs.

0

100

200

300

400

500

600

-2 -1 0 1 2 3 4 5 6Spring Displacement [mm]

Tem

per

atu

re [

°C]

Upper bolt row

Lower bolt row

Lower compression zone

Figure 8.11: Temperature-spring displacement plot of a connection in cooling

This means that the cooling compression spring ‘pulls’ on the lower bolt row spring,

which causes additional internal forces in the connection, as can be seen below.

-300

-250

-200

-150

-100

-50

0

50

100

150

200

0 50 100 150 200 250 300 350 400 450 500 550 600

Temperature [°C]

Spri

ng F

orce

[kN

]

Upper bolt row

Lower bolt row

Lower compression zone

Figure 8.12: Spring force-temperature plot of a connection in cooling


209

It can be seen that these additional internal forces are quite large, and after the

connection element has reached ambient temperature again, the force in the

compression zone component has almost doubled. However, as the additional forces

are caused by differential plastic displacements of the component, and not by external

loading, the forces cannot break the connection. If a component starts yielding under

the additional forces, the differential plastic deformation reduces and therefore the

maximum component force which can be reached is the yield force of each

component. The effect discussed here can be significantly increased by non-uniform

cooling of the components. An example of a similar effect caused by non-uniform

cooling is the generation of residual stresses in a hot-rolled section.

8.4 Discussion and conclusion

In this chapter, the approaches used for the unloading and cooling behaviour of the

connection element have been described. For a constant connection temperature ( e.g.

ambient temperature), the classic Massing rule was used to describe the unloading

behaviour of the connection, as is common in seismic engineering. However, if the

connection temperature changes during loading and unloading, a more complex

approach has to be followed, ensuring on one hand that the unloading stiffness is

correct for the current temperature but on the other hand that the plastic deformations

remain unaffected by the changing temperatures. The same concept can also be

applied to describe the cooling behaviour of a connection.

Unfortunately, there is no experimental data available for the detailed forces and

deformations produced during cooling of endplate connections. Therefore, the study

has to remain purely theoretical for the present. However, the approaches adopted

seem logical, and the predicted large residual forces within a connection after cooling

point to the need for further research in this area.

In the next chapter, it will be shown how the connection element can be used in

combination with steel beams and frames in fire, exposing the connection element to a

realistic combination of forces and deformations.

Chapter 9: Preliminary application of the connection element

210

9 Preliminary application of the connection element

After the development of the connection element which has been described in the last

two chapters, a brief application study of the proposed element will be conducted.

Firstly, an isolated beam with three different connections is investigated and the

results are compared with limiting temperatures given in BS5950 Part 8 (BSI, 2003).

Secondly, a small restrained sub-frame is analysed in order to investigate the effect of

axial loads in the beam on the connection. Finally, the same sub-frame is assessed,

including the cooling phase.

9.1 Connection element together with an isolated beam

As a first application of the new element, the effects of different connections on the

high-temperature behaviour of a two-dimensional beam without axial restraint will be

investigated. As an example beam, a 5.50m long small British universal beam section

UB254x102x22 has been used. To be able to specify all connection parameters a

British universal column section UC203x203x71 has been chosen. The endplate

thickness is 12mm. It is further assumed that beam and connections are heated

uniformly along its depth. As upper and lower bounds the beam has been analysed

assuming pinned and fixed end supports. To be able to study the pure bending

behaviour of the beam and the connections, the beam was allowed to expand and

contract freely in the longitudinal direction. A uniformly distributed load was applied

to the beam generating a load ratio of 0.6 with respect to simply supported conditions;

this approach is common in practical design. The beam would be designed as simply

supported and the additional hogging moments at the ends of the beam would be used

as an implicit factor of safety. The other, and probably the more correct way, of using

the concept of load ratios as outlined in BS5950 Part 8 (2003), is to assess each

structural system at ambient temperature in order to find the collapse load, and so be

able to calculate the ‘real’ load ratio of 0.6. Both approaches have been analysed.

The assumed isolated beam is shown in Figure 9.1 below.


211

Figure 9.1: Isolated beam with connection elements

Three different types of connections have been analysed, one with a bending

resistance lower that that of the beam, one with the same bending resistance as the

beam and the last one with a larger resistance than the beam.

Figure 9.2: Analysed endplate connections

In the previous chapter it was found that the connection element underpredicts the

available rotation capacity of a connection because of the conservative failure criteria

used, assuming failure when the maximum stress in any part of any component is

reached. However, the tests by Renner (2005) and Theodorou (2001) have shown that

both mild steel plate and bolts at elevated temperatures can withstand considerably

larger strains than at ambient temperature. Therefore, an option for ductile

component behaviour has been included into the connection element. This has been

done in such a way that when the deformation limit of a component is reached, a very

w = 11.3 kN/m – 22.6 kN/m

UB 254x102x22

Fixed, Pinned or

Connection element

5500 mm

50

130

76 27 27 5 50

55 5

150

130

76 27 27 5 50

55 5

50

100

130

76 27 27

40

55 5

150

40

Endplate thickness = 12 mm - Steel grade: S275 – Bolts: M16 – 8.8

A C B


212

small stiffness of 50 N/mm is assumed and the analysis continues. In the beam study

presented here, the analysis has been continued until a connection rotation of 100

mrad was reached. This limit was chosen because most high-temperature tests on

endplate connections were not extended beyond this limit. Furthermore, very few

tests have been continued until separation of the beam and the column occurs, so the

real rotational capacity remains unknown until further research has been carried out.

9.1.1 Flush endplate with two bolt rows

The first connection analysed, is a flush endplate connection with two bolt rows and a

moment resistance of 47 kNm, which is equal to 66 % to the plastic moment capacity

of the beam.

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800Temperature [°C]

Dis

pla

cmen

t [m

m]

Figure 9.3: Flush endplate (two bolt rows) at a LR = 0.6 (simple beam)

One can see from Figure 9.3 that for both beam-connection temperature ratios the

failure temperature of the beam-connection system is larger than the limiting

temperature given in BS5950 Part 8 for a load ratio of 0.6. However, the mid-span

deflections of the beams are very small at the points of predicted connection failure in

accordance with the failure criteria outlined in the previous chapter. If the ductility of

the connection is increased to 100 mrad, larger deflections are possible.

Span / 10

Span / 20

Span / 30

BS5950 Part 8 limiting temperature: 550°C

Pinned

Rigid

TC = 1.0 TB

TC = 0.8 TB


213

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800Temperature [°C]

Dis

plac

men

t [m

m]

Figure 9.4: Flush endplate (two bolt rows) at the ‘real’ LR = 0.6

If each beam system is loaded to its own load ratio of 0.6, the difference between the

pinned and the rigid case becomes much smaller, as would be expected. However, in

the cases where the flush endplate connections are considered, the limiting

temperatures after BS5950 Part 8 are only just reached for the cases where the

connection temperature is assumed to be 80% of the beam temperature. For the case

with equal temperatures, the connection fails considerably below the limiting

temperature of the beam.

In conclusion, it can be said that, due to the relatively low resistance of the connection

in comparison with the beam, the plastic hinges at the ends of the beam form in the

connection, which demands a large rotational capacity from the connection to form

the final plastic hinge in the mid-span of the beam.

9.1.2 Flush endplate connection with three bolt rows

The second connection assessed was that used by Leston-Jones (1997), a flush

endplate connection with three bolt rows. This connection has a moment resistance of

71 kNm, equal to the plastic bending resistance of the beam.

Span / 10

Span / 20

Span / 30


Pinned

Rigid

TC = 1.0 TB

TC = 0.8 TB


214

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800Temperature [°C]

Dis

pla

cem

ent

[mm

]

Figure 9.5: Flush endplate (three bolt rows) at a LR = 0.6 (simple beam)

In Figure 9.5, the results from the analyses with a load ratio of 0.6 of the simply

supported beam are shown. For the case with TC = 0.8 TB it can be seen that the

connection is strong enough to ensure that the plastic hinges form in the beam, and the

connection survives to very large beam deflection. For the case of equal temperatures

between the beam and the connections, the limiting temperature of the beam is well

exceeded but the connection still fails at small deflections of the beam.

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800

Temperature [°C]

Dis

plac

emen

t [m

m]

Figure 9.6: Flush endplate (three bolt rows) at the ‘real’ LR = 0.6

Span / 10

Span / 20

Span / 30


Pinned

Rigid

TC = 1.0 TB

TC = 0.8 TB

Span / 10

Span / 20

Span / 30


Pinned

Rigid

TC = 1.0 TB TC = 0.8 TB


215

If the ‘real’ load ratios are considered the flush endplate connection with three bolt

rows behaves better than the one with only two bolt rows, but in the case with equal

temperatures the connection is still predicted to fail before the limiting temperature of

the beam is reached.

9.1.3 Extended endplate

The third connection was an extended endplate connection with three bolt rows and a

moment resistance of 86 kNm, or 120% of the beam’s bending resistance.

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800

Temperature [°C]

Dis

pla

cem

ent

[mm

]

Figure 9.7: Extended endplate (three bolt rows) at LR = 0.6 (simple beam)

The results from the analysis of the extended endplate connection are shown in Figure

9.7 above. It can be seen that the beam behaviour is even closer to the rigid case, as

might be expected because extended endplate connections are normally assumed to

behave rigidly, especially if the moment resistance of the connection is larger than

that of the beam. However, for the equal-temperature case the connection still fails

before the beam reaches large deflections, which highlights the need for an accurate

prediction of the connection temperature distribution.

Span / 10

Span / 20

Span / 30


Pinned

Rigid

TC = 1.0 TB

TC = 0.8 TB


216

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800

Temperature [°C]

Dis

plac

emen

t [m

m]

Figure 9.8: Extended endplate (three bolt rows) at the ‘real’ LR = 0.6

In Figure 9.8, it can be seen that the connections with temperature equal to 80 % of

the beam temperature are sufficiently strong for the beam to plastify before the

connection fails. For the equal-temperature case, the beam-connection system did not

reach the limiting temperature of the beam, even though the connection had a 20%

higher bending resistance than the beam at ambient temperature.

From the beam-connection study presented above it can be seen that the relationship

between the resistances of the beam and the connection is a major factor for the

survival of a connection, as it dictates whether the plastic hinge is formed in the beam

or the connection. If the plastic hinge forms in the beam no further moment increase

is possible in the connection, and the beam can deflect further and develop its plastic

mechanism. However, if the beam does not form a hinge at its end, all beam-end

rotation has to be resisted by the connection, and much smaller beam deflections are

possible without breaking the connection. The other important factor governing the

beam-connection interaction is the temperature of the connection relative to the beam

temperature, because it changes the resistance ratio between the beam and the

connection. It appears that if a connection is not sufficiently stronger or colder than

the beam it needs a large rotational capacity to allow the beam to form a plastic

mechanism without separating the connection.

Span / 10

Span / 20

Span / 30


Pinned

Rigid

TC = 1.0 TB TC = 0.8 TB


217

However, the beam study is somewhat academic, as beams are normally used as part

of a frame structure where, in addition to the bending moments axial forces are also

present, which influence the beam and connection behaviour. Therefore, a limited

frame study using the same beam and the flush endplate connection with three bolt

rows is presented in the next section.

9.2 2D Sub-frame

Since a beam with realistic connections has been analysed in isolation, the same beam

should now be analysed as part of a frame providing a high degree of axial and

rotational restraint to the beam. For this study, a so-called ‘restrained rugby goal post

frame’ has been analysed. Similar frame models have been used before for a number

of parametric studies by Bailey (1995), Leston-Jones (1997) and Al-Jabri (1999), but

in all these studies the effects of axial load on the connection could not be assessed, as

the spring element used was only able to predict moment-rotation-temperature

behaviour derived from test data. With the connection element, however, it is

possible to asses the sub-frame more realistically for a wider range of connections and

it is also possible to assess the cooling behaviour of the frame.

9.2.1 Geometry and Loading

The geometry of the restrained sub-frame can be seen in Figure 9.9 below. Using

symmetry, it was possible to model only half of the frame in order to save computing

time. As beams the same small British beam section as in the previous section was

used, a UB 254x102x22. Further, the same column section, a UC 203x203x71, was

used to be able to compare the behaviour of the beam, and especially the connection,

in a frame incorporating the isolated beam analysed before. The beams were loaded

uniformly with a line load of 11.3 kN/m, generating a load ratio of 0.6 with respect to

the simply supported beam. Point loads of 1324 kN were placed on top of the

columns, which were large enough to generate a load ratio in the column of 0.6

together with the beam loads.


218

Figure 9.9: Restraint sub-frame

It was assumed that the fire heats the middle beam uniformly. For the connections

between the middle beam and column two different temperature regimes have been

considered, one assuming a uniform temperature equal to 80% of the beam

temperature and the other at uniform temperature equal to the beam temperature. The

bottom-storey columns are assumed to be protected, and therefore they only reach

50% of the beam temperature. The column protection is only extended to the bottom

flange of the beam, leaving the joint zone exposed. The external beams, the upper-

storey columns and the connections to them are assumed to stay at 20°C.

The frame was initially analysed using pinned and rigid connections to form a

solution envelope for the further studies using the connection element. As a

connection detail, the same flush endplate connection with three bolt rows was used

as in the beam study.

9.2.2 Results

As was seen in the beam study, the endplate connection considered is very stiff, and

almost no difference can be seen between the behaviour of the rigidly connected sub-

frame and that with the connection elements, shown in Figure 9.10 below. However,

20°C

0.5 T

1.0 T

0.8 T

5500 mm 2750 mm 2750 mm

3500

mm

35

00 m

m


219

if the mid-span displacement results from the isolated beam study presented above are

compared with deflections found in the restrained sub-frame study a very different

behaviour can be seen. Whereas the deflections of the isolated beams remain small

until about 600°C, and then increase very rapidly, the deflections of the beam in the

frame start increasing from about 200°C, due to major-axis buckling of the beam

caused by restrained thermal expansion. Then the deflection increase gradually until

the frame loses stability at a temperature about 200°C higher than the isolated

member does (Figure 9.5). If now the two different temperature cases are compared,

it can be seen that the one with equal temperatures of the beam and the connections

stopped at a temperature of 520°C, due to failure in the compression zone. However,

this temperature does not represent the real failure temperature of the frame, because

if the ‘down-hill’ part of the force-displacement curve of the compression zone

component were considered, the bending moment at the connection would reduce and

the load would be shed towards the middle of the beam. However, this has not yet

been included in the connection element, and will have to wait until further research

has been conducted.

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800 900 1000Temperature [°C]

Dis

pla

cem

ent

[mm

]

Rigid

Pinned

Connection Element - Tc = 0.8 Tb


Isolated beam - Tc = 0.8 Tb

Isolated beam - Tc = 1.0 Tb

Figure 9.10: Vertical displacement at mid-span of the heated beam

In Figure 9.11, the rotation-temperature curves for both cases are shown. One can see

how ‘run away’ failure of the connections occurs at 520°C and 920°C for equal

connection temperature and the cooler connection temperature cases, respectively. In

the equal-temperature case, the compression zone in the column web fails due to the


220

combination of large compressive forces in the beam and the material strength

reduction due to the high temperatures. In the case where the connections are cooler,

the compression zone is still strong enough at the beam temperature at which the other

connection failed, and so this connection fails by bolt failure at about 920°C. The

increase of connection stiffness at 860°C, which is equal to a connection temperature

of 700°C, which can be seen in Figure 9.11, is due to the change in slope of the

material strength reduction factors suggested in EC3-1.2. Above 700°C, steel

strength is assumed to reduce less significantly, and therefore a relative increase of

connection stiffness can be seen.

0

5

10

15

20

25

30

35

0 100 200 300 400 500 600 700 800 900 1000Temperature [°C]

Rot

atio

n [m

rad

]



Figure 9.11: Temperature (Tb) - rotation curve of the connection element

The difference between the axial force–temperature and the bending moment–

temperature curves in the two different temperature cases are minimal and therefore

only the curves for the case in which the connection temperature is 80% of the

temperature of the beam will be shown in the figures below. From Figure 9.12, the

large axial forces introduced by the thermal expansion and catenary action of the

beam can be seen. Again, the connection element case behaves very similarly to the

case in which rigid connections are considered. It should be noted that the axial loads

in this sub-frame study might be larger than the forces which would be found in a real

structure, due to the high level of axial restraint. This high restraint is caused by the

symmetric boundary conditions, imposed at the mid-span of the beams in the

neighbouring spans preventing any axial movement, whereas in a real structure the


221

horizontal restraint originates from column and beam bending. In the case of a

continuous frame, the floor slab and the bracing systems would also contribute to

causing lower restraint than has been modelled in this study.

-600

-500

-400

-300

-200

-100

0

100

200

0 100 200 300 400 500 600 700 800 900 1000Temperature [°C]

Axi

al f

orce

[kN

]

Rigid

Pinned

Connection Element

Figure 9.12: Axial force in the connections and the heated beam

The bending moments at the connections and at mid-span of the heated beam are

shown in Figure 9.13. This shows clearly the difference between the bending moment

distribution in the pinned case and the rigid case at ambient temperature. The mid-

span moment of the pinned case is double that in the rigid case, as structural analysis

predicts. By comparing the ambient-temperature moments of the case in which the

flush endplate connections were modelled with the rigid case, the mid-span moment is

slightly larger in the connection element case, which is caused by the lower initial

stiffness of the real connections than the infinitely stiff rigid connections.

With increasing temperature, the bending moment increases in all three cases due to

the second-order effects caused by the thermal expansion of the beam. At

temperatures above 500°C the bending moments reduce to almost zero at the mid-

span, and to a small value at the connections, because the beam loses its ability to

withstand moments, with degradation of the material becoming significant above

400°C.


222

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600 700 800 900 1000Temperature [°C]

Mom

ent

[kN

m]

Rigid

Pinned

Connection Element

Figure 9.13: Moment at the connections and at mid-span of the heated beam

If now a closer look is taken at the connection, from Figure 9.14, the force

distributions in the two compression zones (‘Compression zone 1’ and ‘Compression

zone 2’are located in the column web at the same level as the upper and the lower

beam flanges, repectivley) and the three bolt rows can be seen. It can be seen that the

connection moment at ambient temperature is split up into a compressive force in the

lower compression zone and two tensile forces in the upper two bolt rows. With

increasing compressive force in the beam, the two bolt rows unload, until at 90°C the

connection is fully compressed and the upper compression zone becomes active. The

connection continues loading in compression until the beam buckles at about 200°C,

and the beam deflection starts to increase as well as the beam-end rotation, causing

the upper compression zone to unload whereas the lower compression zone keeps

loading. At 440°C, the force in the upper compression zone has reduced to zero, and

the bolt rows in tension are activated and carry the developing tension load. From this

stage, the lower compression zone keeps unloading as the beam starts to behave in

catenary action.

Connection moments

Beam moments at mid-span


223

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

0 100 200 300 400 500 600 700 800 900 1000Temperature [°C]

Com

pone

nt f

orce

[kN

]

Bolt row 1Bolt row 2Bolt row 3Compression zone 1Compression zone 2

Figure 9.14: Connection component forces

From the sub-frame study with real connections presented above, it can be seen how

the proposed connection element behaves in a frame structure providing restraint at

increasing temperatures. The response appears logical and it should be possible to use

the element in larger frame structures confidently. In the next section, the cooling

behaviour of the above sub-frame will be investigated.

9.2.3 Cooling response of the sub-frame

In order to investigate the cooling behaviour of connections in a frame structure the

above sub-frame has been used. The frame has been heated up 700°C and then cooled

back down to 20°C assuming a uniform temperature distribution in the connection

equal to 80% of the beam temperature. Again, the frame has been analysed assuming

pinned and rigid connections, as well as using the connection element representing the

flush endplate connection with three bolt rows. In Figure 9.15 the vertical

displacements at mid-span of the heated beam are shown. The figure shows how the

deflections of the structure recover during cooling, reducing almost to half the

maximum value.


224

-500

-400

-300

-200

-100

0

0 100 200 300 400 500 600 700 800Temperature [°C]

Dis

plac

emen

t [m

m]

Rigid

Pinned

Connection Element

Figure 9.15: Vertical deflection at mid span of the beam

The connection element predicts failure of the flush endplate connection during

cooling, at a beam temperature of 207.7°C. In Figure 9.16, the temperature-rotation

curve of the connection element is shown. During the cooling phase the connection is

rotated in the opposite direction, which loads the connection in its weak direction, and

the lower bolt row finally reaches its resistance at a rotation of 21.4 mrad.

-25

-20

-15

-10

-5

0

5

10

0 100 200 300 400 500 600 700 800Temperature [°C]

Rot

atio

n [m

rad

]

Figure 9.16: Temperature-rotation curve including the cooling phase


225

Figure 9.17 shows the axial force in the connections and the heated beam. One can

see how the initial compression due to thermal expansion develops until the beam

buckles at about 200°C, and from this point the axial force reduces until just under

700°C when the compression changes into tension. The cooling phase starts at

700°C, and from here considerable tension forces are developed.

-600

-400

-200

0

200

400

600

0 100 200 300 400 500 600 700 800Temperature [°C]

Axi

al f

orce

[kN

]

Rigid

Pinned

Connection Element

Figure 9.17: Axial force in the connections and the heated beam

The bending moment at the mid-span of the heated beam is shown in Figure 9.18

below. In addition to the increase in axial tension force during cooling, a reversal of

the bending moment in the beam can be observed. Shortly before the predicted

failure of the connections occurs, an increase of the moment in the beam can be seen.

This increase is caused by the increasing beam-end rotation due to the plastification of

the connection.


226

-60

-40

-20

0

20

40

60

80

0 100 200 300 400 500 600 700 800Temperature [°C]

Mom

ent

[kN

m]

Rigid

Pinned

Connection Element

Figure 9.18: Mid-span bending moment in the heated beam

In the next Figure 9.19, the beam-end moments are plotted against temperature for the

rigid and the connection element cases. As at the mid-span of the beam, the bending

moment increases during the expansion phase due to second-order effects, and then

reverses during cooling. It is this moment, together with the high axial tensile force,

which causes the connections to fail.

-50

-40

-30

-20

-10

0

10

20

30

0 100 200 300 400 500 600 700 800Temperature [°C]

Mom

ent

[kN

m]

Rigid

Connection Element

Figure 9.19: Connection and beam-end moment


227

From this preliminary analysis, of a restrained frame with realistic connections at

elevated temperatures including the cooling phase, it can be seen how the forces in the

beam and the connection are developed with increasing and decreasing temperature.

In the full-scale fire tests at Cardington the connections were found to be likely to

break during the cooling phase of a fire. The analysis presented verifies that this may

occur. However, it should be kept in mind that the proposed connection element does

not include a realistic bolt material model for the cooling phase, due to a relative lack

of research in this field.

9.3 Conclusion

From the preliminary studies presented in this chapter, it can be seen how the new

connection element can be used in modelling both isolated beams and frame

structures. The connection element appears to behave logically if exposed to a

combination of bending, axial force, heating, cooling and unloading. The proposed

element has the capability to predict realistic connection behaviour in global frame

analysis up to the peak resistance of the connection. However, if the post peak

behaviour of the connections is required, for example to calculate the load

redistribution in a frame, the ‘down-hill’ parts of the force-displacement curves of the

components have to be included. However, only very limited research on this matter

is available at ambient-temperature, and none at elevated temperature.

Chapter 10: Discussion, conclusion and further recommendations

228

10 Discussion, conclusion and further recommendations

10.1 Summary and discussion

In the last decade, the development and use of finite element programs for the

modelling of steel and composite structures has become more and more popular, not

only for research purposes but also as an important tool in structural fire engineering.

The numerical calculations are often used to justify the omission of fire protection on

steel beams. Commonly, these value-engineering exercises are performed during

RIBA Stage C and D, after the general layout of the structural frame has been agreed

between the architect and the structural engineer. At this stage in the design process,

the connections are normally not designed and the structural engineer will have

assumed either pinned or fixed end conditions during the frame analysis. Because of

the general perception that connections are not critical to the fire performance of steel

and composite frames, gained from high temperature experiments on connections in

isolation, the detailed connection behaviour has generally been ignored in finite

element analyses of frames in fire.

However, from the full-scale fire tests in the BRE laboratory in Cardington and from

finite element analyses it has become clear that the loading on a connection during a

fire is a complex combination of axial force and moment, changing with the

temperature of the structure. This loading is very different from the design loads used

during the ambient-temperature design of the connection. This fact, and the large

variety of possible connection details, make it necessary to include detailed

connection behaviour into the global finite element analysis of steel and composite

frames in fire.

A feasible way of doing this has been developed in this thesis. The process that has

been followed comprised of four steps:

1. Identify a feasible method to represent connections in numerical frame

analysis programs (Chapter 2).

2. Extend and further develop the identified method to elevated temperatures

using experimental, numerical, statistically and analytical approaches (Chapter

3 to 6).


229

3. Implement the method into a finite element program in the form of a two-

noded complex spring element that is able to predict the structural behaviour

in fire, and validate it (Chapter 7 and 8).

4. Test the method in a simple beam analysis and a frame analysis (Chapter 9).

In the following section the main findings in each of the steps is summarised and

discussed.

10.1.1 Step 1 of the development process of the connection element

From the initial comparison between curve-fit approaches, mechanical models and

finite element models it became clear that the mechanical models, standardised in

EC3-1.8 as the Component Method, appear most suitable for the inclusion of detailed

modelling of joints in frames. Therefore, the principles and the calculation procedure

of the Component Method have been described, including recent studies on the

moment-normal force interaction. Previous attempts to apply the component method

to elevated temperatures were also summarised. A number of important points can be

highlighted from this chapter:

• The Component Method seems most suitable to model high-temperature joint

behaviour.

• The spring models, together with simplified approaches for the individual

components of the Component Method, can be used to form a finite macro

element assembled on a component basis, to include joint behaviour into

frame analysis.

• In order to model the effects of axial load in the beam on the joint, which is

very important in the fire case, all bolt rows and both compression zones (i.e.

the column web in-line with the top flange of the beam as well as the

compression zone adjacent to the beam bottom flange) have to be included.

• The high-temperature model for the compression zone in the column web

developed by Spyrou does not include the effects of axial load in the column.


In the previous step, a lack of knowledge and therefore the need for further research

was identified on the effects of axial load on the elevated-temperature behaviour of

the compression zone component in a column web. In order to investigate this

phenomenon and to develop a component model for the compression zone, a


230

combination of experimental, numerical, statistical and analytical analyses has been

carried out.

Step 2.1 – Experimental study

The key points of the experimental analysis are:

• The experiments showed the expected reduction in ultimate load with

increasing temperature due to the loss of strength and stiffness of the steel.

• No significant reduction due to axial load was observed at lower temperatures,

and only a slight reduction in resistance and displacement were found at high

temperatures, which can be explained by the increased relative axial load ratio.

• An investigation, associated with the project presented in this thesis, by

Renner (2005) showed a strong influence of testing speed, and therefore

strain-rate, on the strength of steel at elevated temperatures.

Due to the limited amount of time and resources for the experimental study, not

enough tests at high temperatures in combination with high axial load ratios could be

conducted. If it is decided to continue the research, the following points should be

considered:

• It is very difficult to repeat high-temperature experiments, changing only the

axial load and keeping the temperature the same in each test, without a larger

furnace.

• The load introduction plates should be at the same temperature as the

specimen to prevent uncontrollable heat losses in the most important area

underneath the plates.

• The test-rig has to be very robust especially if axial column loads close to the

buckling load of the specimen are tested.

• A constant testing speed, controlled by a programmable actuator-controller,

should be used, as the resulting force-displacement curves are much smoother

then when the load is applied manually in small steps.

Step 2.2 – Numerical study

The second part of the investigation conducted in Step 2 was a numerical study of the

compression zone in a column web, using the finite element method.

• The model was designed to give a fast and economic answer for the overall

force-displacement behaviour of this component rather than an accurate


231

prediction of the experimental results in order to be able to do a more

extensive parametric study.

• The parametric study on the effect of different axial load ratios on different

column sections at elevated temperatures has shown a significant reduction of

the capacity and the ductility of the compression zone.

• The initial stiffness seemed unaffected by the amount of axial load in the

column.

• Further numerical modelling using 3D solid elements to represent the

compression zone, conducted well after the parametric study, confirmed the

reduction of the peak load found in the parametric study but predicted larger

reductions for the peak displacement than the initial modelling suggested.

This highlights the need for a mechanical model describing the peak

displacement.

Step 2.3 – Statistical study

In this part of Step 2, a simplified model for the force-displacement behaviour at

elevated temperature of the compression zone of an internal joint has been developed.

In order to find an accurate solution for this problem, ambient-temperature approaches

for the resistance, initial stiffness and ductility have been compared statistically with a

large number of test results. The approaches which compared best with the tests have

then been extended to elevated temperatures with the help of temperature reduction

factors for the yield stress and the Young’s modulus, and compared with elevated-

temperature test results. The key findings from the statistical comparisons are:

• For the ultimate resistance of the compression zone the approach by

Lagerqvist and Johansson (1996) seemed most appropriate for extension to

elevated temperatures.

• For the initial stiffness of the compression zone, the approach by Aribert and

Younes (De Mita et al., 2005) compared best with the experiments, although

the correlation of all approaches compared was not very good.

• For the displacement of the compression zone under the ultimate load, which

is conservatively assumed as the ductility limit, the approach developed by the

author in a previous study (Block, 2002) compared most favourably.


232

• These parameters were then combined to generate a full force-displacement

curve with the help of a modification of the equations given for the stress-

strain curves of steel at elevated temperatures in EC3-1.2 (CEN, 2005).

• The model compared very well with the tests conducted by Spyrou. For the

tests by the author the simplified model under-predicted the resistance of the

compression zone by about 15%, which could be explained by the higher

testing speed in the author’s tests than in the experiments by Spyrou and

therefore the higher material strength.

• With the reduction factors for steel tested under steady-state conditions at

elevated temperatures, derived form the study by Renner (2005) and steady-

state tests by Kirby and Preston (1988), the tests by the author could also be

predicted accurately.

Step 2 – Analytical study

In this part of Step 2, an analytical approach for the consideration of the effects of

axial load on the column web behaviour at ambient and elevated temperatures has

been developed.

• A literature search revealed that none of the existing simple reduction factors

given in design codes and publications was accurate enough.

• An analytical model based on the approach by Lagerqvist and Johansson

(1996) was derived, accounting for the reduced moment resistance of the

plastic hinges in the flange mathematically.

• Subsequently, this new approach has been validated against ambient-

temperature test results and the numerical parametric study, presented in

Chapter 4, including the whole range of d/t-values found in the British UC

sections and temperatures up to 650°C.

• In the second part of the chapter, the reduction of the ductility of the

compression zone due to axial load has been investigated, and reduction

factors for ambient and elevated temperatures have been derived.

• Finally, the new simplified model has been compared with the experiments at

elevated temperatures including axial load, and similarly good comparison to

the tests without axial load could be reached if the strength-temperature

reduction factors for steady-state tests were used.


233

This is the end of Step 2, as now the most important components of an endplate beam-

to-column connections can be represented by simplified force-displacement models.


The third step of the development process of the connection element followed in this

thesis was the implementation of the Component Method into the finite element

program Vulcan and validation of the new element against test results. The

component-based connection element has been developed to predict the ambient- and

elevated-temperature behaviour of flush and extended endplate connections. The

main points during Step 3 were the following.

• The generic stiffness matrix of the element has been derived and has been

included into Vulcan and validated against simple hand calculations.

• The force-displacement models for the tension zone, developed by Spyrou,

and the one for the compression zone, developed in this thesis, are

implemented into Vulcan in order to populate the stiffness matrix.

• The element is used successfully to predict the moment-rotation response of

several ambient temperature experiments including axial column load.

• The way high temperatures are incorporated into the element is described, and

good comparisons with elevated-temperature experiments are made.

• An initial parametric study on three different endplate connections showed

that the temperature reduction factor for bolts could be used to predict the

lower bound of the high temperature capacity of the connection if the axial

load in the beam can be neglected.

In order to predict the connection response during the full course of a fire, the effects

of cooling and unloading had to be implemented into the new element and the primary

points found during this development are:

• For a constant connection temperature, (e.g. ambient temperature) the classic

Massing rule can be used to describe the unloading behaviour of the

connection, as is common in seismic engineering.

• If the connection temperature changes during loading and unloading, a more

complex approach has to be followed. This approach assumes that plastic

deformations are not affected by changing temperature and is based on the

Development of Bailey et al. (1996). This method ensures that the unloading


234

stiffness is correct for the current temperature. The same concept can also be

applied to describe the cooling behaviour of a connection.

• Unfortunately, there is no available experimental data for the detailed forces

and deformations during cooling of endplate connections. Therefore, the

study has to remain purely theoretical for the present. However, the adopted

approaches seem logical, and the predicted large residual forces within a

connection after cooling point to the need for further research in this area.

The proposed element compares well with the tests, although certain limitations still

exist, such as the automatic consideration of group effects in tension between bolt

rows, the consideration of shear deformations in the column web and the beam-end,

and bottom-flange buckling of the beams. In theory, it is possible to consider these

additional components in the same way as the tension and the compression zone, but

up to the present no validated high-temperature models for these components have

been available. However, once the shear component at the end of the beam, which is

currently being developed by Qian and Tan in Singapore, is published, this

component can be incorporated into the newly-developed element.


In the fourth step of the development, a preliminary study into the application of the

element has been conducted and it was shown how the new connection element could

be used together with isolated beams and frame structures. The connection element

appears to behave logically if exposed to a combination of bending, axial force,

heating, cooling and unloading. It has all the necessary capabilities to predict realistic

connection behaviour in global frame analysis up to the peak resistance of the

connection. However, if the post-peak behaviour of the connections is required (for

example, to calculate the load redistribution in a frame) the ‘down-hill’ parts of the

force-displacement curves of the components have to be included. However, only

very limited research on this matter is available at ambient temperature and none at

elevated temperatures.

10.2 Recommendations for the usage of the connection element

At present, the connection element has only been validated for major axis flush and

extended endplate connections in internal joints. However, it is possible to use the

element to model composite connections, if the following points are considered.


235

• Shear connector elements have to be used which are able to represent the slip

between the steel beam and the concrete.

• Any additional reinforcement in the concrete around the column should be

modelled.

• The slab element mesh should be refined around the connection element to

pick up the steep increase of hogging moment at the column location. If the

slab elements are too large, the moment peak is averaged out and the concrete

does not crack realistically.

These points, if ignored, would increase the stiffness and the strength of a composite

connection significantly.

10.3 Recommendations for further work

During the course of the research leading to this thesis a number of gaps and shortfalls

have been identified in the knowledge of connections in fire in general and the

developed connection element in particular, which could not be addressed due to

limited time and resources. These points are highlighted below.

10.3.1 Extension to compression zone model

Although, the equation describing the capacity of the compression zone is very

accurate, approaches for the initial stiffness and the ductility should be improved

further, and proper mechanical models should be developed for these two parameters.

More experimental and numerical data is necessary to validate the proposed approach

over a larger range of section sizes, load introduction widths, axial load ratios and

temperatures. Furthermore, the influence of shear in the column web at elevated

temperatures has not been investigated so far.

10.3.2 Further development to the connection element

At present, there are still a number of limitations to the use of the connection element,

which restrict its use to internal endplate joints, due to a lack of available simplified

models for the remaining components in fire. However, in theory it is possible to

include all the missing components:

• Improved and corrected tension zone model,

• Group effects in the bolt rows,

• Inclusion of the post-buckling stage for tension and compression zones,


236

• Bolt behaviour during cooling.

• Shear deformation in the column-web,

• Shear deformation in the beam-end zone,

• Local buckling of the bottom flange of the beam.

10.4 Concluding remark

The advantages of the connection element in its present form are evident, as it opens

up the possibility to combine the Component Method, and therefore detailed

connection behaviour, with the overall frame action of a structure at ambient and

elevated temperatures. However, there are also a number of disadvantages associated

with the connection element. The most important is that the accuracy of response of

the element depends on the accuracy of force-displacement-temperature models for

the individual components, which do not exist for all components at present. A more

practical problem is that the required connection details will most likely have not been

designed by the time the structural fire design of a building has to be performed,

which would add another iteration to the building design process, unless standardised

connection details (i.e. SCI, 1997) are agreed with the steelwork fabricator before the

structural fire assessment is conducted. However, in most cases, a steelwork

fabricator will not have been appointed by this time and connection details would

have to be prescribed, which means the responsibility for the connection design lies

by the structural engineer, which is uncommon.

Nevertheless, time and experience with the connection element will show the

significance of the connection behaviour for the behaviour of frame structures in fire,

and in the end, it will be a commercial decision as to which degree the connection

element can be used in practice. However, the benefits for research in terms of

complementing experimental work or by developing a ‘fire connection’ are

indubitable. Furthermore, the safety and economy of steel and composite framed

buildings could be significantly increased if the new element is used to give more

confidence in structural fire engineering solutions involving partially protected

structures. Finally, the usage of the newly developed connection element would allow

more accurately assessments of the robustness of buildings against structural collapse,

which is particularly important if buildings have to be designed against extreme

events like fire, terror and natural catastrophes.

List of references

237

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Appendix A

250

A. Temperature distribution during the test

The temperatures for all elevated temperature tests conducted are given in this

appendix. The values shown are for the beginning of the transverse loading phase, the

time at which the maximum transverse load was recorded and the end of the test. The

average has been calculated form all temperature readings of the particular

thermocouple from start to maximum load.

Table A.1: Temperature distribution of the experiments 1-8

Middle East West Flange Web Flange Web Web Upper Upper Middle Lower Lower Middle Middle

Test #

°C °C °C °C °C °C °C

Start 438.8 441.0 447.6 444.0 435.8 416.3 473.0

Max 455.2 454.1 460.8 443.8 425.8 450.2 503.0

Ave 443.8 444.6 451.3 442.5 429.5 - - 1

End 453.7 452.2 457.3 441.4 425.1 448.7 501.1

Start 519.7 531.8 536.0 529.2 521.8 546.5 562.4

Max 480.6 507.0 531.5 508.8 483.9 562.9 583.1

Ave 495.9 518.7 534.7 519.6 501.6 - - 2

End 484.0 507.1 531.0 507.1 484.0 568.5 591.2

Start 442.4 451.4 450.4 438.2 428.5 424.2 467.3

Max 430.7 449.5 462.3 439.0 418.2 454.6 504.5

Ave 430.0 447.8 456.6 437.1 418.2 - - 4

End 435.7 451.6 462.2 440.7 421.6 456.3 507.9

Start 566.9 573.4 572.5 563.8 553.6 591.3 547.5

Max 516.3 529.2 537.4 528.4 516.1 561.1 512.3

Ave 537.3 545.6 549.6 543.8 536.2 - - 5

End 508.5 530.3 542.1 528.7 507.3 578.3 529.4

Start 466.3 465.9 462.9 458.6 453.5 478.3 471.2

Max 439.4 459.2 475.5 457.5 418.0 524.4 500.0

Ave 435.5 450.2 461.2 449.9 424.5 - - 7

End 456.3 467.9 478.0 467.6 442.9 530.6 511.5

Start 554.8 562.1 557.7 548.1 540.3 521.8 580.8

Max 517.0 553.3 562.2 533.4 508.8 533.1 598.9

Ave 527.4 556.8 561.9 540.2 520.0 - - 8

End 510.6 548.9 558.8 528.9 502.7 534.1 602.2

Appendix A

251

Table A.2: Temperature distribution of the experiments 9-18

Middle East West

Flange Web Flange Web Web

Upper Upper Middle Lower Lower Middle Middle

Test #

°C °C °C °C °C °C °C

Start 606.2 610.2 607.0 598.6 587.1 636.1 557.2

Max 576.4 595.5 598.4 580.7 548.5 636.1 561.9

Ave 585.7 599.1 598.9 586.8 561.9 - - 9

End 586.4 604.7 609.0 593.5 565.9 662.2 579.0

Start 609.5 609.4 604.8 585.7 573.7 593.9 631.6

Max 601.7 606.3 607.6 580.0 563.2 608.7 646.3

Ave 606.7 610.3 609.3 583.4 566.7 - - 12

End 599.2 603.1 609.3 584.5 571.0 611.2 653.6

Start 569.0 564.7 559.1 547.0 531.7 568.4 571.1

Max 539.7 550.2 558.8 530.9 500.8 593.6 596.6

Ave 541.1 550.3 554.6 532.1 504.9 - - 13

End 551.1 555.3 559.6 539.2 519.4 600.7 604.1

Start 440.6 447.3 449.8 439.1 426.4 - -

Max 436.2 443.2 453.0 435.2 420.3 - -

Ave 434.6 445.8 458.3 440.4 420.8 - - 15

End 439.4 444.9 455.5 435.8 421.7 - -

Start 442.5 445.5 448.1 446.3 444.6 459.3 439.0

Max 448.0 453.9 458.5 440.2 429.3 496.2 476.2

Ave 441.6 448.2 453.4 441.2 432.1 - - 16

End 449.0 447.5 448.5 434.9 429.9 487.9 470.8

Start 562.7 565.6 564.5 552.3 549.1 573.6 552.8

Max 547.2 551.3 554.9 528.4 531.1 581.9 543.3

Ave 548.4 554.0 557.0 535.6 538.0 - - 17

End 551.6 551.8 555.6 526.9 533.7 588.4 542.7

Start 612.7 613.4 614.9 601.8 593.1 628.0 591.7

Max 590.3 598.1 604.3 583.0 571.4 628.5 591.9

Ave 601.8 605.1 606.6 590.0 580.5 - - 18

End 600.9 603.9 608.7 593.6 585.4 642.3 608.1

Appendix B

252

B. Force-displacement curves

In this appendix, the recoded deflections from the compression zone experiments are

shown.

0

50

100

150

200

250

300

350

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Displacement [mm]

For

ce [

kN

]

Actuator

LDVT - Average

LDVT - South

LDVT - North

Web

Load introduction plates

Figure B.1: Force-Displacement curve of Test 1 at 446°C and 265 kN axial load

0

50

100

150

200

250

300

0 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

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Average

LVDT - South

LVDT - North



Appendix B

253

0

50

100

150

200

250

300

350

400

450

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Constructed

LVDT - South

LVDT - North

Web


0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Average

LVDT - South

LVDT - North

Web



Appendix B

254

0

50

100

150

200

250

300

-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Displacement [mm]

For

ce [

kN

]

Actuator

LVDT - Average

LVDT - South

LVDT - North

Web

Load introduction plate


0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Average

LVDT - South

LVDT - North

Web


Appendix B

255

0

50

100

150

200

250

300

350

-6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Average

LVDT - South

LVDT - North

Web



0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Constructed

LVDT - South

LVDT - North

Web



Appendix B

256

0

20

40

60

80

100

120

140

160

180

200

-6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Displacement [mm]

For

ce [

kN

]

Acuator LVDT - Average

LVDT - South LVDT - North

Web Load introduction plates


0

50

100

150

200

250

300

350

400

450

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Displacement [mm]

For

ce [

kN

]

Actuator

LVDT - constructed

LVDT - South

LVDT - North

Web


Appendix B

257

0

50

100

150

200

250

300

350

400

450

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Displacement [mm]

For

ce [

kN

]

Actuator

LVDT - constructed

LVDT - South

LVDT - North

Web


0

20

40

60

80

100

120

140

160

180

200

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26

Displacement [mm]

For

ce [

kN

]

Actuator LVDT - Average

Web LVDT - South

LVDT - North Load introduction plate


Appendix B

258

0

50

100

150

200

250

0 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

Displacement [mm]

For

ce [

kN

]

Actuator LVDT - Average

Web LVDT - South

LVDT - North Load introduction plates

Figure B.13: Force-Displacement curve of Test 13 at 546°C and 266kN axial load

0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Displacement [mm]

For

ce [

kN]

Actuator

LVDT - Average

LVDT - South

LVDT - North



Appendix B

259

0

50

100

150

200

250

300

350

400

0 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

Displacement [mm]

For

ce [

kN

]

Acturator

LVDT - Average

Web

LVDT - South

LVDT - North


Figure B.15: Force-Displacement curve of Test 16 at 448°C and 274kN axial load

0

50

100

150

200

250

300

350

400

0 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

Displacement [mm]

For

ce [

kN]

Acturator

LVDT - Average

Web

LVDT - South

LVDT - North



Appendix C

260

C. Results from the parametric study

In this appendix, the results from the numerical parametric study on the compression

zone are shown.

Table C.1: Results from the parametric study on the compression zone

UC Section d / t Temp. Axial Load

Level

Peak Load Peak Load

Reduction

Displ. at

Peak Load

Displ.

Reduction

[-] [°C] N / Npl [kN] [-] [mm] [-]

203x203x46 22.30 20 C 0.00 354.89 1.00 2.17 1.00 0.40 330.35 0.93 1.85 0.85 0.50 315.58 0.89 1.71 0.79 0.60 291.54 0.82 1.34 0.62 0.70 263.54 0.74 1.19 0.55 450 C 0.00 243.56 1.00 1.85 1.00 0.40 209.01 0.86 1.43 0.77 0.50 193.96 0.80 1.27 0.69 0.60 177.67 0.73 1.17 0.63 0.71 160.69 0.66 0.93 0.50 650 C 0.00 92.97 1.00 1.84 1.00 0.35 82.01 0.88 1.51 0.82 0.50 74.17 0.80 1.26 0.68 0.60 68.75 0.74 1.12 0.61 0.71 62.56 0.67 0.91 0.49

152x152x37 15.50 20 C 0.00 345.84 1.00 3.06 1.00 0.41 309.96 0.90 2.30 0.75 0.50 299.12 0.86 2.02 0.66 0.60 279.24 0.81 1.79 0.59 0.70 252.16 0.73 1.79 0.58 450 C 0.00 238.02 1.00 2.20 1.00 0.40 206.32 0.87 1.61 0.73 0.51 190.48 0.80 1.45 0.66 0.60 175.16 0.74 1.31 0.60 0.70 157.20 0.66 1.12 0.51 550 C 0.00 165.75 1.00 2.15 1.00 0.40 143.72 0.87 1.61 0.75 0.50 133.92 0.81 1.50 0.70 0.61 120.80 0.73 1.33 0.62 0.70 109.64 0.66 1.11 0.52 650 C 0.00 91.22 1.00 2.37 1.00 0.40 78.84 0.86 1.84 0.77 0.50 73.60 0.81 1.70 0.72 0.60 67.56 0.74 1.43 0.60 0.70 60.96 0.67 1.19 0.50

254x254x167 10.40 20 C 0.00 2117.56 1.00 10.29 1.00 0.40 1955.08 0.92 8.29 0.81 0.60 1778.94 0.84 7.31 0.71 0.80 1536.58 0.73 6.65 0.65 450 C 0.00 1341.85 1.00 10.65 1.00 0.40 1189.47 0.89 6.59 0.62 0.53 1099.03 0.82 5.68 0.53 0.70 968.93 0.72 4.37 0.41 650 C 0.00 519.99 1.00 10.06 1.00 0.40 461.84 0.89 6.53 0.65

Appendix D

261

D. Experimental data used in the statistical analysis of the compression zone

In this appendix the experimental data from various sources for the ultimate resistance, the peak displacement and the initial stiffness of the

compression zone is given. This data has been used in Chapter 5 and 6 for the statistical analyses.

Table D.1: Ambient temperature tests data without axial load

Web Flange

Index Source Section Type Thickness Height

Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate

stress Thickness Width

Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width

Extra

Plate

Ultimate

Load

Peak

Displ.

Initial

Stiffness

tw H dw Ew fyw fuw tf bf Ef fyf fuf r c tp c+2*tp Fu δu Kini

mm mm mm N/mm² N/mm² N/mm² mm mm N/mm² N/mm² N/mm² mm mm mm mm kN mm kN/mm

1 1 W8x17 5.8 206.4 190.8 205000 344 - 7.8 134.4 205000 344 - 8.1 7.8 - 7.8 265.0 - -

2 3 W12x14 5.7 303.7 292.3 205000 371 - 5.7 101.6 205000 371 - 5.5 5.7 - 5.7 178.0 - -

3 5 W14x22 6.5 351.5 335.3 205000 357 - 8.1 128.5 205000 357 - 10.7 8.1 - 8.1 274.0 - -

4 De Mita HE200A 9.5 200 170 205958 335.771 - 15 200 205958.05 278.262 - 18 25 - 25 812.0 4.44 712

5 De Mita HE200B 10 203.5 174.5 205982 284.309 - 14.5 199 205982.48 272.763 - 18 25 - 25 768.5 4.31 824

6 De Mita HE220A 7.5 215 192 206019 335.712 - 11.5 220 206019.05 321.291 - 18 25 - 25 454.0 2.68 568

7 De Mita HE220B 10.5 220 187.2 205923 283.06 - 16.4 220 205923.13 267.262 - 18 25 - 25 795.5 4.35 912

8 De Mita HE240A 8.3 234 210 206000 396.656 - 12 240 206000.00 341.403 - 21 25 - 25 546.5 3.31 502

9 De Mita HE260A 8.1 253 226.8 206135 318.143 - 13.1 261 206134.55 280.458 - 24 25 - 25 620.5 3.03 700

10 De Mita HE260B 10.9 262 227.4 205975 265.156 - 17.3 260 205975.16 269.039 - 24 25 - 25 998.5 4.12 1100

11 De Mita HE280A 9.3 274 249 205936 319.201 - 12.5 280 205936.37 331.583 - 24 25 - 25 684.5 2.81 790

12 De Mita HE280B 10.7 285 248.4 205900 270.667 - 18.3 283 205899.90 262.293 - 24 25 - 25 1020.5 4.51 930

13 De Mita HE300A 10 295 266.2 205928 306.824 - 14.4 299 205928.08 313.452 - 27 25 - 25 777.5 3.26 1056

14 De Mita HE300B 11.5 340 301.2 205995 323.563 - 19.4 340 205995.02 275.768 - 27 25 - 25 1065.5 4.34 1152

15 Delft 1.1 IPE 240 6.20 240.00 220.4 205000 367 - 9.8 120 205000 367 - 15 40 - 40 380.0 - -

16 Delft 2.1 IPE 240 6.20 240.00 220.4 205000 425 - 9.8 120 205000 425 - 15 40 - 40 381.0 - -

17 Delft 3.1 HEA 240 7.50 230.00 206 205000 317 - 12 240 205000 317 - 21 40 - 40 483.0 - -

18 Delft 4.1 HEA 300 8.50 290.00 262 205000 357 - 14 300 205000 357 - 27 40 - 40 630.0 - -

19 Delft 5.1 HEA 500 12.00 490.00 444 205000 286 - 23 300 205000 286 - 27 40 - 40 980.0 - -

20 Dubas & Gheri 1 13 - 222 205000 254 - 20.5 209 205000 254 - 20 15 - 15 919.0 - -

21 Dubas & Gheri 2 13 - 261 205000 246 - 20.5 258 205000 246 - 20 15 - 15 862.0 - -

22 Dubas & Gheri 3 10.3 - 216 205000 242 - 17.3 206 205000 242 - 20 15 - 15 621.0 - -

23 Dubas & Gheri 4 11.6 - 264 205000 281 - 19 257 205000 281 - 20 15 - 15 796.0 - -

24 Dubas & Gheri 5 12.6 - 318 205000 266 - 20.2 307 205000 266 - 20 15 - 15 1122.0 - -

Appendix D

262

Web Flange


Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate


Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width

Extra

Plate

Ultimate

Load

Peak

Displ.

Initial

Stiffness



25 Dubas & Gheri 6 12.6 - 362 205000 270 - 20.6 370 205000 270 - 20 15 - 15 1134.0 - -

26 Dubas & Gheri 7 9.9 - 308 205000 261 - 15.4 305 205000 261 - 20 15 - 15 648.0 - -

27 Dubas & Gheri 8 11.5 - 360 205000 276 - 19.8 305 205000 276 - 20 15 - 15 1002.0 - -

28 Dubas & Gheri 9 10.6 - 357 205000 269 - 18.2 255 205000 269 - 20 15 - 15 744.0 - -

29 Dubas & Gheri 10 9.6 - 353 205000 254 - 16.3 254 205000 254 - 20 15 - 15 624.0 - -

30 Dubas & Gheri 11 7.5 - 303 205000 282 - 13.1 203 205000 282 - 20 15 - 15 465.0 - -

31 Dubas & Gheri 12 14.6 - 229 205000 217 - 23.7 210 205000 217 - 20 15 - 15 1134.0 - -

32 Dubas & Gheri 13 18 - 333 205000 687 - 28.1 313 205000 687 - 20 15 - 15 4450.0 - -

33 Dubas & Gheri 14 9.3 - 257 205000 406 - 15.7 255 205000 406 - 20 15 - 15 975.0 - -

34 Dubas & Gheri 15 8.1 - 252 205000 856 - 13.4 203 205000 856 - 20 15 - 15 1148.0 - -

35 Dubas & Gheri 16 9.4 - 306 205000 831 - 14.6 204 205000 831 - 20 15 - 15 1180.0 - -

36 Dubas & Gheri 17 7.3 - 260 205000 292 - 12.7 147 205000 292 - 20 15 - 15 408.0 - -

37 Dubas & Gheri 18 6.7 - 307 205000 280 - 11.8 166 205000 280 - 20 15 - 15 277.0 - -

38 Dubas & Gheri 19 4.1 - 100 205000 392 - 5.7 55 205000 392 - 20 15 - 15 151.0 - -

39 Dubas & Gheri 20 4.1 - 100 205000 392 - 5.7 55 205000 392 - 20 15 - 15 172.0 - -

40 Dubas & Gheri 21 4.1 - 100 205000 392 - 5.7 55 205000 392 - 20 15 - 15 175.0 - -

41 Dubas & Gheri 22 6.2 - 240 205000 353 - 9.8 120 205000 353 - 20 15 - 15 334.0 - -

42 Dubas & Gheri 23 6.6 - 270 205000 353 - 10.2 135 205000 353 - 20 15 - 15 394.0 - -

43 Dubas & Gheri 24 6 - 100 205000 300 - 10 100 205000 300 - 20 15 - 15 294.0 - -

44 Dubas & Gheri 25 5.5 - 133 205000 324 - 8.5 140 205000 324 - 20 15 - 15 305.0 - -

45 Dubas & Gheri 26 7 - 140 205000 257 - 12 140 205000 257 - 20 15 - 15 402.0 - -

46 E1 W12x40 7.5 303 276.8 205000 277 - 13.1 203 205000 277 - 15.5 12.7 - 12.7 456.0 - -

47 E14 W8x48 10.3 216 181 205000 237 - 17.5 206 205000 237 - 9.5 12.7 - 12.7 610.0 - -

48 E15 W8x58 12.9 222 181 205000 250 - 20.5 209 205000 250 - 9.7 12.7 - 12.7 901.0 - -

49 E16 W10x66 11.6 264 226 205000 276 - 19 257 205000 276 - 12.8 12.7 - 12.7 782.0 - -

50 E17 W10x72 12.9 267 226 205000 241 - 20.5 258 205000 241 - 12.8 12.7 - 12.7 845.0 - -

51 E18 W12x65 9.9 308 277.2 205000 257 - 15.4 305 205000 257 - 14.8 12.7 - 12.7 636.0 - -

52 E19 W12x85 12.6 318 277.6 205000 261 - 20.2 307 205000 261 - 14.7 12.7 - 12.7 1101.0 - -

53 E20 W14x61 9.6 353 320.4 205000 250 - 16.3 254 205000 250 - 15.4 12.7 - 12.7 612.0 - -

54 E21 W14x68 10.6 357 320.6 205000 264 - 18.2 255 205000 264 - 15.1 12.7 - 12.7 730.0 - -

55 E23 W14x103 12.6 362 320.6 205000 266 - 20.7 370 205000 266 - 15.9 12.7 - 12.7 1112.0 - -

56 INSA L1 HEB 140 7.00 140.00 116 205000 320 - 12 140 205000 280 - 12 10 - 10 365.0 - -

57 INSA L2 HEB 200 9.00 200.00 170 205000 320 - 15 200 205000 280 - 18 10 - 10 770.0 - -

58 INSA L3 HEB 260 10.00 260.00 225 205000 320 - 17.5 260 205000 280 - 24 10 - 10 870.0 - -

59 INSA L4 HEB 140 7.00 140.00 116 205000 320 - 12 140 205000 280 - 12 20 - 20 375.0 - -

Appendix D

263

Web Flange


Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate


Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width

Extra

Plate

Ultimate

Load

Peak

Displ.

Initial

Stiffness



60 INSA L5 HEB 200 9.00 200.00 170 205000 320 - 15 200 205000 280 - 18 15 - 15 780.0 - -

61 INSA L6 HEB 200 9.00 200.00 170 205000 320 - 15 200 205000 280 - 18 20 - 20 825.0 - -

62 INSA L7 HEB 260 10.00 260.00 225 205000 320 - 17.5 260 205000 280 - 24 20 - 20 880.0 - -

63 INSA M1 IPE 140 5.10 141.90 129.1 205000 303 416 6.4 74.2 205000 282 399 7.8 10 - 10 175.0 - -

64 INSA M2 HEA 260 7.80 255.30 231.7 205000 335 459 11.8 259.8 205000 300 456 23.8 15 - 15 608.0 - -

65 INSA M3 IPE 220 6.20 220.00 202.8 205000 284 468 8.6 111.4 205000 305 463 13.1 10 - 10 300.0 - -

66 INSA M4 IPE 360 8.30 359.50 334.5 205000 326 435 12.5 170.8 205000 273 423 16.8 15 - 15 530.0 - -

67 INSA MH1 HEA 140 5.70 135.50 117.9 205000 484 578 8.8 140.2 205000 477 571 11.8 10 - 10 365.0 - -

68 INSA MH10 HEA 160 6.7 154.50 134.3 205000 481 608 10.1 160.7 205000 477 589 13.6 10 10 30 580.0 - -

69 INSA MH11 HEA 160 6.7 154.50 134.5 205000 481 608 10 160.6 205000 477 589 13.7 10 15 40 620.0 - -

70 INSA MH12 HEA 160 6.7 154.20 134.2 205000 481 608 10 160.6 205000 477 589 13.7 10 20 50 664.0 - -

71 INSA MH2 HEA 160 6.70 154.30 134.3 205000 481 608 10 160.6 205000 477 589 13.4 10 - 10 530.0 - -

72 INSA MH3 HEA 160 6.60 154.30 134.3 205000 475 603 10 160.2 205000 441 551 13.6 10 - 10 522.0 - -

73 INSA MH4 HEA 200 7.70 195.90 173.5 205000 542 640 11.2 203.6 205000 512 613 18.2 10 - 10 760.0 - -

74 INSA MH5 HEA 200 7.80 196.20 173.6 205000 542 640 11.3 203.6 205000 512 613 18.6 10 - 10 740.0 - -

75 INSA MH6 HEAA 200 5.70 189.50 172.7 205000 610 697 8.4 202.9 205000 580 660 19.8 10 - 10 402.0 - -

76 INSA MH7 HEAA 300 6.80 285.00 261.6 205000 544 656 11.7 300.1 205000 459 574 26.7 10 - 10 588.0 - -

77 INSA MH8 IPE 240 6.10 238.60 217.4 205000 566 671 10.6 120.2 205000 488 584 14.3 10 - 10 454.0 - -

78 INSA MH9 IPEA 360 6.60 357.50 334.5 205000 524 635 11.5 167.2 205000 463 568 15.6 15 - 15 490.0 - -

79 INSA N1 HEB 160 8.00 160.00 134 205000 275 397 13 160 205000 275 397 15 15 - 15 550.0 - -

80 INSA T1 HEB 200 9.43 200.00 170 205000 265 411 15 200 205000 265 411 18 15 10 35 760.0 - -

81 INSA T2 HEB 200 9.3 200.00 170 205000 265 411 15 200 205000 265 411 18 15 15 45 800.0 - -

82 INSA T3 HEB 200 9.39 200.00 170 205000 265 411 15 200 205000 265 411 18 15 20 55 840.0 - -

83 INSA T4 HEB 200 9.55 200.00 170 205000 265 411 15 200 205000 265 411 18 15 30 75 940.0 - -

84 Kühnemund A3 HE240A 7.5 230 206 198000 286 524 12 240 201000 287 512 21 20 - 20 532.0 2.83 622.4

85 Kühnemund A8 HE240A 7.5 230 206 206000 275 502 12 240 210000 252 497 21 20 - 20 493.0 2.31 757.5

86 Kühnemund B1 HE240B 10 240 206 205000 290 471 17 240 204000 246 440 21 20 - 20 755.0 4.73 724.5

87 Kühnemund B8 HE240B 10 240 206 209000 277 490 17 240 204000 248 476 21 20 - 20 953.0 8.30 1041.6

88 Spyrou UC 152x152x30 6.41 159 140.84 233000 293 474 9.08 151.74 227000 274 467 7.6 12 - 12 320.0 3.75 534

89 Spyrou UC 203x203x107 12.87 267.37 225.47 189300 288 488 20.95 260.34 189300 280 - 12.7 12 - 12 1158.0 8 -

90 Spyrou UC 203x203x46 7.28 203.85 181.47 234000 301 450 11.19 203.5 230000 275 445 10.2 12 - 12 405.0 3.75 766

91 Spyrou UC 203x203x60 9.95 212.3 184.86 204000 304 527 13.72 212.3 204000 304 - 10.2 12 - 12 736.0 6 -

92 Spyrou UC 203x203x86 12.7 223.7 182.7 201000 285 501 20.5 209.1 194970 228 - 10.2 12 - 12 1125.0 8 804

93 Spyrou UC 203x203x86 12.7 223.7 182.7 225000 312 492 20.5 209.1 220000 265 492 10.2 12 - 12 1278.0 10.75 930

94 Spyrou UC 203x203x86 12.7 223.7 182.7 225000 312 492 20.5 209.1 220000 265 492 10.2 12 - 12 1286.0 12.5 940

Appendix D

264

Web Flange


Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate


Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width

Extra

Plate

Ultimate

Load

Peak

Displ.

Initial

Stiffness



95 W12 W12x45 9.8 305.3 276.1 205000 372 - 14.6 204.2 205000 372 - 10.8 12.7 - 12.7 739.0 - -

96 W15 W12x36 8.2 314.5 287.1 205000 763 - 13.7 166.7 205000 763 - 7.1 12.7 - 12.7 1046.0 - -

97 W17 W10x29 7.9 263.4 238 205000 291 - 12.7 147.3 205000 291 - 5.8 12.7 - 12.7 423.0 - -

98 W20 W12x27 6.8 304.8 284.4 205000 281 - 10.2 165 205000 281 - 7.4 12.7 - 12.7 285.0 - -

99 W21 W12x45 9.8 305.3 276.1 205000 392 - 14.6 204.2 205000 392 - 10.8 12.7 - 12.7 748.0 - -

100 W3 W10x39 8.7 253 229.6 205000 841 - 11.7 203 205000 841 - 11.4 12.7 - 12.7 1126.0 - -

101 W4 W12x45 8.7 307 277.6 205000 816 - 14.7 204 205000 816 - 13.5 12.7 - 12.7 1157.0 - -

102 W5 W12x31 6.9 304.5 284.9 205000 275 - 9.8 166 205000 275 - 8 12.7 - 12.7 271.0 - -

103 W6 W10x29 7.8 263 236.4 205000 287 - 13.3 147 205000 287 - 5.2 12.7 - 12.7 401.0 - -

104 W7 W10x54 9.7 256 224.8 205000 399 - 15.6 255 205000 399 - 10.3 12.7 - 12.7 957.0 - -

105 W8 W8x67 14.6 229.6 183.2 205000 213 - 23.2 210.5 205000 213 - 7.8 12.7 - 12.7 1112.0 - -

106 W9 W12x120 17.8 332.5 278.3 205000 674 - 27.1 313 205000 674 - 12.8 12.7 - 12.7 4360.0 - -

Appendix D

265

Table D.2: Ambient temperature tests data with axial load

Web Flange

Index Source Section Type Thick. Height

Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate

Stress Thick. Width

Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width Area

Axial

load

Axial

Stress

Axial

Load

Ratio

Ultimate

Load

Peak

Displ.

Initial

Stiffness

tw H dw Ew fyw fuw tf bf Ef fyf fuf r c A N sN Fu δu Kini

mm mm mm N/mm² mm N/mm² mm mm N/mm² N/mm² N/mm² mm mm mm2 kN N/mm2 - kN mm kN/mm

1 Block 10 UC 152x152x37 7.55 161.4 139.5 205000 289.5 486 10.95 154.7 205000 288 495 7.6 18 4491 265 59.01 0.20 421.7 3.30 698

2 Block 3 UC 152x152x37 7.37 161.2 139.3 205000 289.5 486 10.95 154.05 205000 288 495 7.6 18 4450 394 88.54 0.31 413.7 3.05 659

3 Block 6 UC 152x152x37 7.5 161.5 139.75 205000 289.5 486 10.875 154.1 205000 288 495 7.6 18 4449 398 89.45 0.31 423.4 3.60 689

4 Kühnemund A1 HE240A 7.5 230 206 198000 286 524 12 240 201000 287 512 21 20 7684 1475 191.96 0.67 465.0 2.12 694

5 Kühnemund A2 HE240A 7.5 230 206 198000 286 524 12 240 201000 287 512 21 20 7684 1321 171.90 0.6 453.0 2.21 523

6 Kühnemund A4 HE240A 7.5 230 206 206000 275 502 12 240 210000 252 497 21 20 7684 223 28.99 0.11 481.0 1.94 878

7 Kühnemund A5 HE240A 7.5 230 206 206000 275 502 12 240 210000 252 497 21 20 7684 466 60.61 0.23 473.0 1.36 716

8 Kühnemund A6 HE240A 7.5 230 206 206000 275 502 12 240 210000 252 497 21 20 7684 688 89.59 0.34 467.0 1.86 671

9 Kühnemund A7 HE240A 7.5 230 206 206000 275 502 12 240 210000 252 497 21 20 7684 992 129.12 0.49 455.0 1.84 743

10 Kühnemund B2 HE240B 10 240 206 205000 290 471 17 240 204000 246 440 21 20 10599 1335 125.96 0.47 678.0 3.26 685

11 Kühnemund B3 HE240B 10 240 206 205000 290 471 17 240 204000 246 440 21 20 10599 1875 176.88 0.66 629.0 2.94 753

12 Kühnemund B4 HE240B 10 240 206 209000 277 490 17 240 204000 248 476 21 20 10599 334 31.50 0.12 910.0 7.01 1140

13 Kühnemund B5 HE240B 10 240 206 209000 277 490 17 240 204000 248 476 21 20 10599 723 68.25 0.26 874.0 5.35 994

14 Kühnemund B6 HE240B 10 240 206 209000 277 490 17 240 204000 248 476 21 20 10599 1057 99.75 0.38 842.0 5.05 920

15 Kühnemund B7 HE240B 10 240 206 209000 277 490 17 240 204000 248 476 21 20 10599 1697 160.13 0.61 788.0 4.95 1016

Appendix D

266

Table D.3: Elevated temperature tests data without axial load

Web Flange

Index Source Section Type Temperature Thickness Height

Clear

Depth

Young's

Modulus

Yield

Stress

Ultimate

Stress Thickness Width

Young's

Modulus

Yield

Stress

Ultimate

Stress

Root

Radius

Load

Width

Ultimate

Load

Peak

Displ.

Initial

Stiffness

tw H dw Ew fyw fuw tf bf Ef fyf fuf r c Fu δu Kini

°C mm mm mm N/mm² N/mm² N/mm² mm mm N/mm² N/mm² N/mm² mm mm kN mm kN/mm

1 Spyrou UC 152x152x30 410 6.35 160.7 142.52 160770 286.55 429.83 9.09 151.6 156630 267.97 401.96 7.6 12 280 2.50 634.00

2 Spyrou UC 152x152x30 500 6.1 156.9 138.7 139800 228.54 285.68 9.1 153.75 136200 213.72 267.15 7.6 12 240 3.25 532.00

3 Spyrou UC 152x152x30 600 6.6 158.2 140.4 72230 137.71 151.48 8.9 152.05 70370 128.78 141.66 7.6 12 150 5.00 212.00

4 Spyrou UC 152x152x30 610 6.75 158.55 140.35 68036 130.68 143.75 9.1 152.01 66284 122.20 134.42 7.6 12 129 4.50 182.00

5 Spyrou UC 152x152x30 710 6.4 159.9 141.6 29358 63.87 63.87 9.15 151.6 28602 59.73 59.73 7.6 12 54 5.00 106.00

6 Spyrou UC 152x152x30 755 6.33 160.7 142.68 25164 48.05 48.05 9.01 151.75 24516 44.94 44.94 7.6 12 46 4.75 52.00

7 Spyrou UC 203x203x46 280 7.22 204.03 181.63 191880 301.00 526.75 11.2 203.54 186140 275.00 481.25 10.2 12 376.5 3.25 726.00

8 Spyrou UC 203x203x46 400 7.26 203.95 181.63 163800 301.00 451.50 11.16 203.39 158900 275.00 412.50 10.2 12 333.4 3.00 590.00

9 Spyrou UC 203x203x46 520 7.29 202.2 179.74 126828 216.12 270.15 11.23 203.41 123034 197.45 246.81 10.2 12 241 3.25 396.00

10 Spyrou UC 203x203x46 610 7.27 203.05 180.85 68328 134.25 147.67 11.1 203.66 66284 122.65 134.92 10.2 12 142 3.00 356.00

11 Spyrou UC 203x203x46 670 7.56 202.75 180.65 43056 90.90 99.99 11.05 203.5 41768 83.05 91.36 10.2 12 96.5 4.75 166.00

12 Spyrou UC 203x203x46 765 7.18 203.9 181.66 24336 45.75 45.75 11.12 203.51 23608 41.80 41.80 10.2 12 55.5 4.00 72.00

13 Spyrou UC 203x203x52 610 7.65 210.77 186.37 59013.2 162.79 179.07 12.2 206.02 59013.2 162.79 179.07 10.2 12 210 - -

14 Spyrou UC 203x203x71 535 9.95 215.65 180.73 114655 197.42 246.78 17.46 205.24 109670 183.32 229.15 10.2 12 417 5.00 592.00

15 Spyrou UC 203x203x71 635 10.13 215.73 180.91 56810 113.48 124.83 17.41 205.21 54340 105.38 115.92 10.2 12 220 4.50 494.00

16 Spyrou UC 203x203x71 755 9.86 215.84 181.68 24840 48.22 48.22 17.08 204.74 23760 44.77 44.77 10.2 12 108 7.50 110.00

17 Spyrou UC 203x203x86 585 12.6 223.86 183.8 79537.5 161.15 201.44 20.03 208.52 77770 136.87 171.09 10.2 12 477 9.00 614.00

18 Spyrou UC 203x203x86 650 12.7 223.44 183.3 49500 109.20 120.12 20.07 208.52 48400 92.75 102.03 10.2 12 303 10.50 374.00

19 Spyrou UC 203x203x86 705 12.63 223.26 183 28800 69.89 69.89 20.13 208.46 28160 59.36 59.36 10.2 12 210 9.50 270.00

20 Spyrou UC 203x203x86 750 12.65 223.59 183.55 24750 53.04 53.04 20.02 208.43 24200 45.05 45.05 10.2 12 182 12.00 216.00

21 Block 4 UC 152x152x37 447 7.49 161.40 139.60 133189 257.91 386.86 10.90 154.35 138478 273.42 410 7.6 18 332 2.73 348.40

22 Block 8 UC 152x152x37 553 7.50 161.40 139.60 93513 181.61 227.01 10.90 153.95 115331 213.86 267 7.6 18 257 3.01 259.59

23 Block 12 UC 152x152x37 601 7.48 161.20 139.40 64739 138.10 151.91 10.90 154.20 73954 151.51 167 7.6 18 183 3.53 224.85

24 Block 15 UC 152x152x37 448 7.92 161.30 138.90 134521 276.51 414.77 11.20 152.95 137699 276.63 415 7.6 18 378 4.06 292.66

25 Block 17 UC 152x152x37 549 8.00 161.30 138.50 96307 196.09 245.11 11.40 153.05 99696 194.25 243 7.6 18 272 4.20 178.93

Note: The highlighted fields indicate that the values are either assumed or back calculated as they were not published.

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