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A Thesis Submitted for the Degree of PhD at the University of Warwick

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Lateral-torsional buckling resistance of pultruded
fibre reinforced polymer shapes
by
Tien Thuy Nguyen

A thesis presented for the degree of
Doctor of Philosophy in Engineering
University of Warwick

May 2014
CONTENTS

LIST OF TABLES ............................................................................................................ v
LIST OF FIGURES ....................................................................................................... viii
NOTATIONS .................................................................................................................. xx
ACKNOWLEDGEMENTS ....................................................................................... xxxiii
DECLARATION ........................................................................................................ xxxv
ABSTRACT .............................................................................................................. xxxvii
CHAPTER 1 INTRODUCTION ........................................................................... 1
CHAPTER 2 LITERATURE REVIEW ................................................................ 8
2.1. Introduction ...................................................................................... 8
2.2. Lateral-torsional buckling failure ..................................................... 8
2.3. Lateral-torsional buckling of simply supported beam ................... 14
2.4. Lateral-torsional buckling of cantilever beam ............................... 19
2.5. Aims and objectives ....................................................................... 20
CHAPTER 3 MATERIAL CHARACTERIZATION.......................................... 22
3.1. Introduction .................................................................................... 22
3.2. Full-section properties.................................................................... 23
3.3. Longitudinal elastic modulus ......................................................... 28
3.3.1. Tensile test method ............................................................... 28
3.3.2. Test specimens ...................................................................... 29
i
3.3.3. Test procedure and results .................................................... 33
3.4. Transverse elastic modulus ............................................................ 44
3.4.1. Specimens and test procedure............................................... 44
3.4.2. Test results ............................................................................ 45
3.5. In-plane shear modulus .................................................................. 48
3.5.1. 10o off-axis tensile test method ............................................ 50
3.5.2. Test procedure and results .................................................... 56
3.6. Discussion ...................................................................................... 61
3.7. Local buckling stress ...................................................................... 65
3.7.1. Analytical and numerical predictions ................................... 66
3.7.2. Test specimens ...................................................................... 74
3.7.3. Test arrangements and results ............................................... 75
3.8. Concluding remarks ....................................................................... 83
CHAPTER 4 LATERAL-TORSIONAL BUCKLING TESTS ........................... 85
4.1. Introduction .................................................................................... 85
4.2. Test rig design ................................................................................ 86
4.2.1. Loading fixture ..................................................................... 86
4.2.2. End fixture ............................................................................ 91
4.2.3. Load application method ...................................................... 94
4.3. Test specimens and geometric properties ...................................... 96
4.4. Initial geometric imperfection ........................................................ 98
4.5. Test instruments and procedure ................................................... 103
4.6. Onset of buckling and data reduction method ............................. 108
4.7. Theoretical predictions ................................................................. 111
4.8. Test results and discussion ........................................................... 118
4.8.1. I beam test results ............................................................... 118
4.8.2. Channel beam test results ................................................... 127
4.9. Concluding remarks ..................................................................... 134
CHAPTER 5 NUMERICAL INVESTIGATION .............................................. 137
5.1. Introduction .................................................................................. 137
5.2. Modelling methodology ............................................................... 138
5.2.1. Input of material properties ................................................ 139
ii
5.2.2. Element types and mesh sizes ............................................ 141
5.2.3. Geometrical modelling ....................................................... 144
5.2.4. Simulation of displacement boundary conditions .............. 146
5.2.5. Modelling of vertical load for channel section ................... 149
5.2.6. Analysis methods ................................................................ 150
5.3. Test results vs. eigenvalue analyses ............................................. 154
5.4. Test results vs. nonlinear analyses for I-section .......................... 158
5.5. Sensitivity analyses ...................................................................... 168
5.5.1. Sensitivity of P with the geometrical modelling ........ 168
cr,FEA
5.5.2. Sensitivity of P with elastic constant ......................... 170
cr,FEA
5.5.3. Sensitivity of P with geometric imperfection ......... 173
Limit,FEA
5.5.4. Sensitivity of P with change of lateral load position
Limit,FEA
176
5.5.5. Sensitivity of P with change of vertical load height
Limit,FEA
180
5.5.6. Sensitivity of P with overhang length ........................ 183
cr,FEA
5.6. Concluding remarks ..................................................................... 184
CHAPTER 6 DESIGN PROPOSAL FOR BEAM IN BENDING .................... 188
6.1. Introduction to design proposal.................................................... 188
6.2. Lateral buckling checking methods in standards ......................... 189
6.2.1. Procedure in American Standard AISC 360-10 ................. 189
6.2.2. Procedure in Eurocode BS EN 1993-1-1:2005 ................... 191
6.3. Design proposal and the Eurocode 3 approach ............................ 193
6.3.1. Plateau length ..................................................................... 193
6.3.2. Imperfection factor ............................................................. 195
6.3.3. Partial factor for lateral-torsional buckling ........................ 197
6.4. The lateral-torsional buckling curves ........................................... 211
6.5. Concluding remarks ..................................................................... 215
CHAPTER 7 CONCLUSIONS AND FURTHER WORKs .............................. 217
7.1. Research summary ....................................................................... 217
7.2. Conclusions .................................................................................. 220
iii
7.3. Further work ................................................................................. 224
REFERENCES .............................................................................................................. 227
APPENDIX A PLOTS FOR COUPON TESTS .................................................. 236
A. 1 Plots for longitudinal tensile coupon tests ................................... 237
A. 2 Plots for transverse tensile coupon tests ...................................... 246
A. 3 Plots for 10o off-axis tensile coupon tests .................................... 250
APPENDIX B PLOTS FOR LATERAL BUCKLING TESTS ........................... 258
B. 1 Measurements of out-of-straightness initial imperfection ........... 259
B. 2 Lateral-torsional buckling tests .................................................... 263

iv
LIST OF TABLES

Table 3.1 Geometric properties and critical span lengths of the test sections ................ 25
Table 3.2 w/PL data to obtain the full elastic constants ............................................... 26
Table 3.3 Prediction of section moduli of I- and channel sections ................................. 27
Table 3.4 Measurements of specimen dimensions of I- and C1-sections ....................... 32
Table 3.5 Measurements of specimen dimensions of C2- and C3-sections .................... 32
Table 3.6 Approximation of maximum stress and strain of beam experiences LTB
failure .............................................................................................................................. 35
Table 3.7 Elastic modulus of I-section (120606 mm) .............................................. 40
Table 3.8 Elastic modulus of C1-section (120506 mm) .......................................... 41
Table 3.9 Elastic modulus of C2-section (100506 mm) .......................................... 42
Table 3.10 Elastic modulus of C3-section (100306 mm) ........................................ 43
Table 3.11 Measurements and tests results for transverse (web) coupons of I, C1, C2
and C3 ............................................................................................................................. 45
Table 3.12 Approximation of maximum average shear strain found in LTB testing ..... 55
v
Table 3.13 Measurements and tests results for 10o off-axis specimens of I, C1, C2 and
C3 .................................................................................................................................... 60
Table 3.14 Elastic constants of all sections ..................................................................... 61
Table 3.15 Elastic modulus E,E ,E and shear modulus G,G from design manuals
L T LT
and previous reseaches .................................................................................................... 63
Table 3.16 Properties to predict ............................................................................ 72
Loc,a
Table 3.17 Approximation of  for I, C1 and C2 shapes ........................................ 73
Loc,a
Table 3.18 Mean measured dimensions for I, C1 and C2 ............................................... 75
Table 4.1 Measured geometric properties of the test beams ........................................... 97
Table 4.2 Initial out-of-straightness imperfections of test beams ................................. 102
Table 4.3 Section properties for I- and C-sections, ignored the fillet radius areas ....... 114
Table 4.4 Section properties for I- and C-sections, accounted for the fillet radius areas
....................................................................................................................................... 116
Table 4.5 Buckling results for beams with EC1 and vertical load applied at Top Flange
(TF), Shear Centre (SC) and Bottom Flange (BF) ........................................................ 119
Table 4.6 Buckling results for beams with EC2 and vertical load applied at Top Flange
(TF), Shear Centre (SC) and Bottom Flange (BF) ........................................................ 120
Table 5.1 Material properties for flange and web panels of I- and C1-sections ........... 140
Table 5.2 Elastic LTB loads for different shell elements with mesh refinements ........ 143
Table 5.3 Maximum out-of-straightness imperfection for I and C1 beams .................. 152
vi
Table 5.4 FE results for I-section with EC1 .................................................................. 155
Table 5.5 FE and test results for EC1 and EC2 with I- and C1-sections ...................... 156
Table 5.6 Limiting buckling loads for FEA (with different ) and tests for EC1 .... 163
max
Table 5.7 Limiting buckling loads for FEA (with different ) and tests for EC2 .... 163
max
Table 6.1 Calculation of generalized imperfection factor  ..................................... 196
LT
Table 6.2 Basic variables and their CVs ....................................................................... 199
Table 6.3 re and rt for the I-section ............................................................................. 201
Table 6.4 re and rt for the C1-section .......................................................................... 201
Table 6.5 i and ifor I-section .................................................................................. 205
Table 6.6 iand ifor C1-section ................................................................................ 206
Table 6.7 Calculation of safety partial factor  for I-section .................................... 208
M
Table 6.8 Calculation of safety partial factor  for C1-section .................................. 209
M

vii
LIST OF FIGURES

Figure 1.1 PFRP shapes: (a) conventional shape; (b) bespoke shape ............................... 1
Figure 1.2 PFRP frame structure (http://www.strongwell.com/markets/building-and-
construction/)..................................................................................................................... 2
Figure 1.3 Multistorey building (http://www.fiberline.com/structures/case-stories-other-
structures/-eyecatcher-building/eyecatcher-building) ....................................................... 2
Figure 1.4 Startlink test house (Zafari, 2012) ................................................................... 3
Figure 1.5 Railway pedestrian bridge (www.apatech.ru/chertanovo_eng.html) .............. 4
Figure 1.6 Road bridge over highway (http://www.fiberline.com) .................................. 4
Figure 2.1 LTB of simply supported I-beam under pure bending .................................... 9
Figure 2.2 Standard structural shapes ............................................................................... 9
Figure 2.3 LTB of cantilever I beam under point load at free end .................................. 10
Figure 2.4 Flexural static equilibrium and LTB instability deformation (not to scale) .. 11
Figure 2.5 Test configuration in Mottram (1992a) ......................................................... 14
Figure 2.6 Loading configuration in Razzaq et al. (1996) .............................................. 16
Figure 2.7 Arrangement of LVDTs at mid-span in Davalos et al. (1997) ...................... 16
viii