Cross-sectional instability of aluminium extrusions with
complex cross-sectional shapes
Citation for published version (APA):
Kutanova, N. (2009). Cross-sectional instability of aluminium extrusions with complex cross-sectional shapes. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR652818
DOI:
10.6100/IR652818
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Cross-sectional instability of
aluminium extrusions with complex
cross-sectional shapes
Cross-sectional instability of
aluminium extrusions with complex
cross-sectional shapes
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op dinsdag 23 juni 2009 om 16.00 uur
door
Natalia Kutanova
Dit proefschrift is goedgekeurd door de promotoren:
prof.ir. F. Soetens en
Samenstelling van de Promotiecommissie:
prof. ir. J. Westra TU Eindhoven (voorzitter) prof. ir. F. Soetens TU Eindhoven
prof. ir. H.H. Snijder TU Eindhoven prof. ir. F. van Herwijnen TU Eindhoven prof. ir. F.S.K. Bijlaard TU Delft
prof. dr. T. Pek ¨oz Cornell University, USA
dr. ir. J. Mennink TNO
ir. B.W.E.M. van Hove TU Eindhoven
ISBN 978-90-77172-47-6 First printing May 2009
Keywords: cross-sectional stability, aluminium, local buckling, distortional buckling
This thesis was prepared in LATEX by the author and printed by Print Partners Ipskamp, Enschede
Cover design: N.Kutanova Cover photography: Eva&Ed
Copyright c°2009 by N.Kutanova, The Hague, The Netherlands
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the copyright holder.
This research was carried out under the project number MC1.02146 in the framework of the Research Program of the Materials innovation institute (M2i)
Acknowledgments
This thesis is the result of an extensive work, which often seemed to me as a never-ending period of time. However, the work is finished and that would not have been possible without people around me who have been very helpful during the time it took me to write this thesis.
First, I would like to show my utmost gratitude to my first promotor, Frans Soetens for inviting me to the Netherlands and giving me the opportunity to be-come a PhD researcher. Frans, thank you very much for sharing your expertise, for the support at any moment and for immense kindness. I especially appreciate your patience to correct hundreds of missing articles in my English writing. My grati-tude also extends to my second promotor Bert Snijder for careful reading any piece of my work, for critical suggestions and challenging questions. I am deeply grateful to Jeroen Mennink without whom this work would not have started and, moreover, successfully ended. My thanks and appreciation goes to my thesis committee mem-bers, Frans van Herwijnen, Frans Bijlaard, Dianne van Hove and Theoman Pek ¨oz. I had very fruitful discussions with my reading committee and the questions I have received were a great help.
This thesis would not have been accomplished without funds of the Materials In-novation Institute. I would like to acknowledge the help and organization support of people at the M2i head office. The research was carried out at the University of Eindhoven and also at TNO. I am indebted to many of my TNO and Eindhoven col-leagues who supported me and provided me assistance in numerous ways. Many thanks to Sander, Paul, Roel, Ernst and Edwin for informal chats and for the great working atmosphere. My closest colleague Johan Maljaars was a great source of help especially during the first year of my research. He has made available his support with respect to the DIANA program. Johan also inspired me to bike everyday from the Hague to Delft and was able to bear my company on every Thursday train trip from Delft to Eindhoven. It would have been an impossible task for me to execute experiments without the help of people in Pieter van Musschenbroek laboratory: Theo van de Loo, Erik Wijen, Hans Lamers and Martien Ceelen. It is a pleasure to thank my student Jeroen Berkmortel for performing a part of the experimental program.
Further, I would like to thank those closest to me, whose direct or indirect pres-ence helped making the completion of my work possible. I have been fortunate to
meet many good friends without whom my life would be bleak. I thank my friends in the Netherlands and France. I would like to thank Eva for helping me with a cover design. I would love to name all of your here, but the list might be too long. However, some of my Russian friends who are more like a family to me, I have to mention here: thank you Daria, Andrey, Denis, Andrew, Alexey, Katya and last but not least, Gena.
It is difficult to express how grateful am I to my mother for her everyday emails helping me feeling that I am not away from my country and my family. Special thanks to my sister Katya and my brother Syoma for visiting me in the Netherlands and making me laugh even in my thoughts about you. Many thanks go to my French family in law for their kindness and affection.
Finally, I would like to thank my husband Ugo Lafont for an enormous amount of faith in me, for mental support and lots of wise advices. He encouraged me to concentrate on completing this thesis more than anyone, while keeping me away from other responsibilities. Nothing would have been possible without him.
Natalia Kutanova Eindhoven May 14, 2009
Summary
Aluminium extrusions are of great interest for different industrial fields such as structural applications and transport. The extrusion process allows one to optimize structural elements according to the design requirements with a relative ease. Op-timization of the shape of the aluminium elements often results in the use of thin-walled cross-sectional shapes, which increases the complexity of the cross-section. As a result of thin-walled elements cross-sectional instability - in particular local and distortional buckling - has a substantial effect on the structural behaviour.
In classification of cross-sectional instability local buckling implies changes in geometry without any lateral displacement or twist, while for distortional buck-ling lateral displacement and twist take place with changes in the cross-sectional geometry. The current design rules used by engineers are limited to local buckling of simple and symmetrical cross-sections. Hence, these design rules do not pro-vide an accurate description of distortional buckling behaviour and can not be used for more complex shapes. Extensive research into cross-sectional instability of alu-minium structural elements concerning distortional buckling is required, which is the main subject of this thesis.
An experimental program has been executed in order to predict the ultimate resistance of aluminium structural elements due to cross-sectional instability. This program consists of extruded and cold-formed aluminium specimens with Z-shaped, Angle-shaped and C-shaped sections. Test specimens were subjected to uniform axial compression. Extensive measurements have been performed on initial im-perfections of extruded and cold-formed specimens. The influence of a gradual increase of the complexity of the geometry and material influence on the buckling behaviour have been investigated. The experimental program has resulted in a set of test data on the cross-sectional instability of aluminium structural members with various cross-sectional shapes, including local and distortional buckling, as well as interaction of modes. These data have been used for the numerical model valida-tion.
A finite element (FE) model has been developed and validated based on the re-sults of the experimental program. In this respect, experiments are simulated using the actual geometry, imperfections and material. The comparison between the
FE-model and test results indicates the accuracy of the numerical prediction. It has been shown that the FE-model is a useful tool for the prediction of structural behaviour of uniformly compressed aluminium members with various cross-sectional shapes. The validated FE-model has been used for a detailed investigation of the actual dis-tortional buckling behaviour and local-disdis-tortional (disdis-tortional-local) interaction.
A substantial number of analyses have been carried out to study the distortional buckling effect using the validated FE-model. Based on the actual distortional buck-ling behaviour of C-shaped sections, a prediction model is proposed for the calcula-tion of the ultimate resistance of compressed C-seccalcula-tions subjected to cross-seccalcula-tional instability. As a result, the existing model for local buckling prediction and the newly developed model for distortional buckling prediction including mode inter-action are able to describe the cross-sectional instability behaviour of aluminium C-sections. Furthermore, these models are an important step in the investigation of cross-sectional instability of complex cross-sectional shapes.
List of Figures
1.1 Aluminium profiles application . . . 1
1.2 Instability types . . . 2
1.3 Research approach . . . 4
2.1 Plate geometry . . . 8
2.2 Buckling coefficient for a simply supported plate. . . 9
2.3 Buckling coefficients kcrof uniformly compressed rectangular plates with various boundary conditions. . . 10
2.4 Stages of stress distribution in simply supported compressed plates. 10 2.5 Effective width . . . 11
2.6 Stress-strain diagram based on the Ramberg-Osgood law. . . 13
2.7 Relationship between slenderness parameter and axial resistance. . . 16
2.8 Classification of cross-sections . . . 17
2.9 Buckling modes of lipped channel in compression. . . 20
3.1 Examples of cross-sections selected for investigation in the current project . . . 25
3.2 1st subprogram . . . 26
3.3 2nd subprogram . . . 26
3.4 Specimen notations . . . 28
3.5 Z-shaped profile cut to measure radiuses. . . 29
3.6 Measured heart-to-heart values of cross-sectional dimensions. . . 29
3.7 Schematization of the measuring process. . . 30
3.8 Imperfection test set-up. . . 31
3.9 Imperfection measurements for web and flange. . . 31
3.10 Test set-up for long profile. . . 32
3.11 Fitting plane for measured points, using the Least Square method. . . 32
3.12 Schematization of the measured points. . . 33
3.13 Imperfection amplitude calculation (case 1). . . 33
3.14 Imperfection amplitude calculation (case 2). . . 34
3.16 Tensile coupon . . . 36
3.17 Tensile coupons for cold-formed program. . . 37
3.18 Test set-up for tensile test in 250 kN test bench. . . 37
3.19 Tensile test results: stress-strain relation. . . 38
3.20 Tensile test results: stiffness-strain relation. . . 39
3.21 Compression test set-up for specimens . . . 40
3.22 Gap between the specimen and support. . . 41
3.23 CUFSM result for the selected Z-shaped profile. . . 41
3.24 Experimental result for selected Z-shaped profile. . . 42
3.25 Compression test results ”Extruded” subprogram. . . 43
3.26 Compression test results ”Cold-formed” subprogram. . . 44
3.27 Compression test results: gradually increased complexity of the cross-sectional geometry . . . 45
3.28 Comparison of local and distortional buckling tests for the same profile. 46 3.29 Load-displacement curves for the batch of specimens . . . 47
3.30 Compression test results: material influence. . . 47
4.1 Material stress-strain curves. . . 51
4.2 Mesh applied for the FE-model. . . 52
4.3 Material models for applied springs. . . 53
4.4 Gap during the test and modelled gap. . . 54
4.5 FEM results ”Extruded” subprogram. . . 55
4.6 Comparison of FEM results with and without gap modelling (3C80E7) 55 4.7 Load-displacement and load-deflection curves (specimen 3C80E7). . 56
4.8 Comparison of tangential stiffness of the experiment (LVDT’s and strain gauges), tensile test and numerical analysis (3C80E7). . . 57
4.9 FEM results ”Cold-formed” subprogram. . . 58
4.10 Comparison of test and FEM results for the specimen 2Z100F11A. . . 59
4.11 Mesh density. . . 61
4.12 Load-displacement diagram varying mesh density. . . 62
4.13 Tensile test results for two coupons in rolling and perpendicular to rolling directions. . . 63
4.14 Load-displacement diagram varying material characteristics. . . 64
4.15 Comparison FE-model and test results, applying enhanced material properties for the rounded corners. . . 65
4.16 Euler buckling analysis results for specimen 3C80E7. . . 66
5.1 Z-shaped specimen dimensions. . . 70
5.2 Buckling behaviour of cross-sectional plates and critical limits. . . 71
5.3 Cross-sectional deformation at different load limits. . . 71
5.4 Initial and secondary buckling stress determination for local buckling. 73 5.5 Proposed inelastic buckling approximation. . . 77
5.6 Illustration of the local and distortional buckling behaviour of the cross-section, according to the model. . . 80
5.7 Considered cross-sectional shapes. . . 83
6.1 Cross-section selected for parameter study. . . 88
6.2 CUFSM results for selected C-profile with defined buckling shapes for local and distortional modes. . . 89
6.3 CUFSM results for selected C-profile with defined buckling curves and critical points for local and distortional modes. . . 89
6.4 Applied springs for C-shaped profile. . . 90
6.5 CUFSM result for C-shaped profile with additional springs. . . 90
6.6 Spring application at two points on a critical length distance from both edges. . . 92
6.7 CUFSM results for specimen 5(6)C5. . . 95
6.8 Initial deformed shape according to Euler analysis (mode 1). . . 95
6.9 FEM deformed shapes for specimen 5(6)C5 according to non-linear analysis. . . 96
6.10 Load-displacement (left) and load-deflection (right) plots for speci-men 5(6)C5. . . 96
6.11 FE-results for plate elements of specimen 5(6)C5. . . 97
6.12 Tangential stiffness for plate elements of specimen 5(6)C5. . . 97
6.13 CUFSM results for specimen 2.5(3)C5. . . 98
6.14 FEM deformed shapes for specimen 2.5(3)C5 according to non-linear analysis. . . 99
6.15 Load-displacement (left) and load-deflection (right) plots for speci-men 2.5(3)C5. . . 99
6.16 FE-results for plate elements of specimen 2.5(3)C5. . . 100
6.17 Tangential stiffness for plate elements of specimen 2.5(3)C5. . . 100
6.18 CUFSM results for specimen 1(2)C10. . . 101
6.19 FEM deformed shapes for specimen 1(2)C10 according to non-linear analysis. . . 102
6.20 Load-displacement (left) and load-deflection (right) plots for speci-men 1(2)C10. . . 102
6.21 FE-results for plate elements of specimen 1(2)C10. . . 103
6.22 Tangential stiffness for plate elements of specimen 1(2)C10. . . 103
6.23 FEM deformed shapes for specimen 1(2)C5 according to non-linear analysis. . . 104
6.24 Load-displacement (left) and load-deflection (right) plots for speci-men 1(2)C5. . . 105
6.25 FE-results for plate elements of specimen 1(2)C5. . . 105
6.26 Tangential stiffness for plate elements of specimen 1(2)C5. . . 106
6.27 Initial buckling (outstanding plates group I) and secondary buckling (internal plates group II). . . 107
6.28 Initial buckling (outstanding plates group I) and secondary buckling (internal plates group II). . . 108
6.30 Local-distortional interaction. . . 109
6.31 Investigation of the scatter in distortional buckling behaviour of the outstanding plate elements. . . 110
6.32 Investigation of the scatter in distortional buckling behaviour of the internal plate elements. . . 111
6.33 Comparison of the inelastic distortional buckling (6082–T6 alloy) with the predicted elastic model at εp (squares) and inelastic model σcr;T (dots). . . 115
6.34 Comparison of the inelastic distortional buckling (6060–T66 alloy) with the predicted elastic model at εp(squares) and inelastic model σcr;T (dots). . . 116
6.35 Comparison of the inelastic distortional buckling (5083–H111 alloy) with the predicted elastic model at εp(squares) and inelastic model σcr;T (dots). . . 117
C.1 Buckling curves for three aluminium alloys. . . 138
C.2 Piecewise idealization of the curves. . . 140
D.1 Compressed specimens 6082 aluminium alloy. . . 145
D.2 Compressed specimens 6082 aluminium alloy. . . 145
D.3 Compressed specimens 6060 aluminium alloy. . . 146
D.4 Compressed specimens 6060 aluminium alloy. . . 146
D.5 Compressed specimens 5083 aluminium alloy. . . 147
List of Tables
2.1 Values of kcrfor determining the critical buckling stress . . . 10
3.1 Specification for ”Extruded” subprogram. . . 27
3.2 Specification for ”Cold-formed” subprogram 6082-T6 aluminium alloy 27 3.3 Specification for ”Cold-formed” subprogram 5083-H111 aluminium alloy . . . 28
3.4 Compression test results for ”Extruded” subprogram. . . 43
3.5 Compression test results for ”Cold-formed” subprogram. . . 44
3.6 Repeatability of tests. . . 46
4.1 Comparison of experimental and FE-results for extruded profiles. . . 58
4.2 Comparison of experimental and FE-results for cold-formed profiles. 60 4.3 Sensitivity analysis results for the specimen 3C80E7. . . 66
5.1 Material characteristics used for parameter study . . . 76
5.2 Comparison of the FE-results with Mennink’s prediction model re-sults. ”Extruded” profiles, 6060-T66 aluminium alloy. . . 83
5.3 Comparison of the FE-results with Mennink’s prediction model re-sults. ”Cold-formed” profiles, 6082-T6 aluminium alloy. . . 83
5.4 Comparison of the FE-results with Mennink’s prediction model re-sults. ”Cold-formed profiles”, 5083-H111 aluminium alloy. . . 83
6.1 Specimen specification for parameter study. . . 88
6.2 Parameter study C–shaped specimens specification. . . 94
6.3 CUFSM results for C–shaped specimens of the parameter study (6082-T6 aluminium alloy). . . 94
6.4 Equations for the average post-buckling stiffness of outstanding com-ponents. . . 110
6.5 Equations for the average post-buckling stiffness of internal compo-nents. . . 112
6.7 FEM results for C-specimens (6060-T66). . . 116
6.8 FEM results for C-shaped specimens (5083-H111). . . 117
A.1 Measured dimensions and characteristics, ”Extruded” subprogram. . 129
A.2 Measured dimensions and characteristics, ”Cold-formed” subprogram130 A.3 Measured imperfections, ”Extruded” subprogram. . . 131
A.4 Measured imperfections, ”Cold-formed” subprogram . . . 132
A.5 Material characteristics for aluminium alloy EN AW-6060 T66. . . 133
A.6 Material characteristics for aluminium alloy EN AW-6060 T66. . . 133
A.7 Material characteristics for aluminium alloy EN AW-6060 T66. . . 133
A.8 Material characteristics for aluminium alloy EN AW-6082 T6. . . 134
A.9 Material characteristics for aluminium alloy EN AW-6082 T6. . . 134
A.10 Material characteristics for aluminium alloy EN AW-5083 H111. . . . 134
A.11 Material characteristics for aluminium alloy EN AW-5083 H111. . . . 134
B.1 Dimensions used in FE simulations for extruded specimens. . . 135
B.2 Dimensions used in FE simulations (in mm) for cold-formed specimens.135 B.3 Imperfections used in FE simulations (in mm) for extruded specimens.136 B.4 Imperfections used in FE simulations (in mm) for cold-formed speci-mens. . . 136
D.1 Comparison of the FSM and Eigenvalue analysis results for C-shaped specimens of the parameter study. . . 143
D.2 FEM results for C-shaped specimens of the parameter study (6082-T6). 143 D.3 FEM results for C-shaped specimens of the parameter study (6060-T66).144 D.4 FEM results for C-shaped specimens of the parameter study (5083-H111). . . 144
E.1 Prediction model results for C-profiles of the parameter study (6082-T6).149 E.2 Prediction model results for C-profiles of the parameter study (6060-T66). . . 149
E.3 Prediction model results for C-profiles of the parameter study (5083-H111). . . 150
List of symbols
Ncr;E Euler load of a column, [N] . . . 7
E Modulus of elasticity / Young’s modulus, [N/mm2] . . . 7
I Second moment of inertia, [mm4] . . . 7
L Axial length of the specimen, [mm] . . . .7
t Plate thickness, [mm] . . . 9
σcr Elastic critical stress, [N/mm2] . . . 8
D Plate flexural rigidity, [N/mm2] . . . 8
kcr Plate buckling coefficient, [-] . . . 9
ϕ Plate slenderness parameter, [-] . . . 9
b Plate width, [mm] . . . 9
fy Yield stress, [N/mm2] . . . 10
bef f Effective plate width, [mm] . . . 11
σmax Maximum axial stress at the unloaded plate edges, [N/mm2] . . . 11
C Effective width parameter, [-] . . . 11
f0.1 0.1% proof strength of the material characteristic, [N/mm2] . . . 13
f0.2 0.2% proof strength of the material characteristic, [N/mm2] . . . 12
fu Ultimate stress of the material characteristic, [N/mm2] . . . 13
n Ramberg-Osgood strain-hardening parameter, [-] . . . 13
σ Axial stress, [N/mm2] . . . 12 ε Axial strain, [-] . . . 12 ν Poisson’s ratio, [-] . . . 16 λ Slenderness parameter, [-] . . . 15 N Axial resistance, [N] . . . 16 A Cross-sectional area [mm2] . . . 16
N0.2 Axial squash load, [N] . . . 16
Ncr Critical axial load at initial buckling, [N] . . . 16
Nu Ultimate or failure load, [N] . . . 16
u Axial displacement of the specimen, [mm] . . . 42
w Out-of-plane deflection, [mm] . . . 56
σpl Plastic stress, [N/mm2] . . . 50
εel Elastic strain, [-] . . . .51
σ∗ Non-dimensionalised axial stress at initial buckling , [-] . . . 70
ε∗ Non-dimensionalised axial strain with respect to initial buckling , [-] . . . 70
fp Proportional limit of the material, [kN] . . . 56
Lcr Critical buckling length, [mm] . . . 73
εp Elastic strain according to the proportional limit of the material, [-] . . . 72
σcr;2 Elastic critical stress at secondary buckling, [N/mm2] . . . 75
εcr;2 Elastic critical strain at secondary buckling, [-] . . . 75
Nu;pm Predicted ultimate load, [kN] . . . 75
σcr;T Inelastic critical axial stress at initial buckling, [N/mm2] . . . 76
χT Inelastic buckling coefficient, [-] . . . 137
¯ λ Relative slenderness, [-] . . . 137
Contents
Acknowledgements i Summary iii List of figures v List of tables ix List of symbols xiTable of contents xiii
1 Introduction 1
1.1 Complex cross-sectional shapes . . . 1
1.2 Problem statement and research goals . . . 3
1.3 Research approach and thesis outline . . . 3
2 State of the art: cross-sectional instability 7 2.1 Column Buckling . . . 7
2.2 Buckling Theory of Plates . . . 7
2.2.1 Elastic critical buckling . . . 7
2.2.2 Postbuckling Strength . . . 10
2.2.3 Effective Width Approach . . . 10
2.2.4 Effective Thickness Approach . . . 12
2.3 Stability: influence of material and imperfections . . . 12
2.3.1 Material characteristics . . . 12 2.3.2 Geometrical Imperfections . . . 14 2.4 Local Buckling . . . 15 2.4.1 Slenderness parameter . . . 15 2.4.2 Classification of cross-sections . . . 16 2.5 Distortional buckling . . . 18 2.6 Numerical Investigations . . . 19
2.6.1 Finite Element Method . . . 19
2.6.2 Finite Strip Method . . . 19
2.6.3 Generalized Beam Theory . . . 21
2.7 Direct Strength Method . . . 21
2.8 Mennink’s prediction model . . . 23
2.9 Literature Study Evaluation . . . 24
3 Experimental Investigation 25 3.1 Program Description . . . 25 3.2 Measuring Dimensions . . . 28 3.3 Measuring Imperfections . . . 29 3.3.1 Introduction . . . 29 3.3.2 Measuring concept . . . 30 3.3.3 Test set-up . . . 31
3.3.4 Test results processing . . . 32
3.3.5 Imperfections results . . . 34
3.4 Material characteristics . . . 36
3.4.1 Introduction . . . 36
3.4.2 Tensile coupon . . . 36
3.4.3 Test set-up . . . 37
3.4.4 Tensile test results . . . 38
3.4.5 Summary . . . 39
3.5 Test set-up for compression test . . . 40
3.6 Compression test results . . . 41
3.6.1 Agreement with FSM-prediction . . . 41
3.6.2 Extruded experimental subprogram . . . 42
3.6.3 Cold-formed experimental subprogram . . . 44
3.6.4 Results of the geometry influence . . . 44
3.6.5 Local vs distortional buckling . . . 46
3.6.6 Test repeatability . . . 46
3.6.7 Results of the material study . . . 47
3.7 Conclusions and summary of observations . . . 48
4 Finite Element Model Validation 49 4.1 Scope of the chapter . . . 49
4.2 Numerical program . . . 49 4.2.1 Applied imperfections . . . 50 4.2.2 Material input . . . 50 4.3 FE-model . . . 51 4.3.1 Element types . . . 51 4.3.2 Mesh . . . 52
4.3.3 Supports, loading and modelled gap . . . 52
4.3.4 Analyses types . . . 54
4.4.1 Extruded subprogram . . . 54
4.4.2 Cold-formed subprogram . . . 58
4.5 Influence of the modelling parameters . . . 60
4.5.1 General . . . 60
4.5.2 Mesh density . . . 61
4.5.3 Material anisotropy . . . 62
4.5.4 Material model with enhanced corner properties . . . 64
4.5.5 Imperfection sensitivity . . . 65
4.6 Conclusions . . . 67
5 Assessment of the existing model with respect to distortional buckling 69
5.1 Scope of the chapter . . . 69
5.2 Specimen example for model evaluation . . . 69
5.3 Mennink’s model for local buckling . . . 72
5.3.1 General . . . 72
5.3.2 Initial and secondary buckling determination . . . 72
5.3.3 Elastic local buckling prediction: (σcr< fp) . . . 74 5.3.4 Inelastic buckling prediction: (fp≤ σcr) . . . 76 5.3.5 Inelastic buckling coefficient . . . 76
5.4 Model extension for distortional buckling . . . 79
5.4.1 Description of the behaviour . . . 79
5.4.2 Post-buckling resistance . . . 80
5.4.3 Model proposals . . . 81
5.5 Application of the extended prediction model . . . 81
5.6 Summary and conclusions . . . 85
6 Prediction model for distortional buckling 87
6.1 Scope of the chapter . . . 87
6.2 Selection of the cross-sectional shape for distortional buckling . . . . 88
6.3 FEM set-up for the parameter study . . . 91
6.4 Parameter study definition . . . 93
6.5 Parameter study results: elastic buckling . . . 94
6.5.1 Pure distortional buckling . . . 94
6.5.2 Secondary buckling and distortional-local interaction . . . 98
6.5.3 Secondary buckling and local-distortional interaction . . . 101
6.5.4 Elastic buckling of the reference C-shape . . . 104
6.5.5 Evaluation of elastic buckling results . . . 106
6.6 Development of a prediction model for distortional buckling of C-shaped specimens . . . 109
6.6.1 Initial buckling . . . 110
6.6.2 Secondary buckling . . . 111
6.7 Prediction model for distortional buckling behaviour of C-shaped alu-minium structural elements . . . 112
6.9 Chapter conclusions . . . 118
7 Conclusions and recommendations 119
7.1 Conclusions . . . 119
7.1.1 Distortional buckling in the current design rules . . . 119
7.1.2 Experimental research . . . 119
7.1.3 Numerical research . . . 120
7.1.4 Prediction model for distortional buckling . . . 120
7.2 Recommendations . . . 122
Bibliography 123
Appendixes 129
A Experimental results 129
A.1 Measured dimensions . . . 129
A.2 Measured imperfections . . . 131
A.3 Measured material characteristics . . . 133
B Numerical analysis 135
B.1 Applied dimensions . . . 135
B.2 Applied imperfections . . . 136
C Inelasticity coefficient derivation 137
C.1 Theory . . . 137
C.2 Buckling curves for three aluminium alloys . . . 138
C.3 Mennink’s approximation . . . 140
D Parameter Study Results 143 E Prediction model results 149
Samenvatting 151
Chapter 1
Introduction
Abstract
This chapter outlines the major aspects of the thesis. A general survey of the research background is given and the objectives and the structure of the thesis are introduced. Cross-sectional instability is one of the important design issues for aluminium struc-tural members with complex cross-sectional shapes. Therefore, the current investigation is focused on cross-sectional instability regarding various cross-sectional shapes and re-sulting in the development of a prediction model.
1.1 Complex cross-sectional shapes
T
he last decades structural applications of aluminium have experienced a fast growth due to the outstanding properties of aluminium, namely high strength to weight ratios, good corrosion resistance and ease of maintenance Mazzolani [2002]. Aluminium’s flexibility is an important quality for structural design, as it allows one to meet the requirements of different industrial situations (see Figure 1.1).Figure 1.1: Application of aluminium structural elements with complex cross-sectional shapes in greenhouses.
Extrusion allows one to optimize the cross-section of structural elements accord-ing to the design requirements with relative ease. Extrusion is a technological pro-cess used to form sections with different shapes by pushing hot metal through an
opening called a die. The resulted cross-section corresponds to the shape of such a die. So, even when the cross-sectional shape is complex, the desired aluminium structural element can be simply manufactured by preparing the die accordingly. Aluminium characteristics enable sections of incredible complexity to be produced for a certain application.
One of the important aspects of the design process is to satisfy the practical needs while optimally utilizing the material. This often results in thin-walled sections. En-gineers are searching for new structural forms of aluminium extrusions to enhance practical efficiency. Further optimizing is possible if some additional functions are added to the cross-section like stiffeners or weld backings, all of which leads to the application of complex cross-sectional shapes (see Figure 1.1). Thus, improvements in functionality correspond to increasing the complexity of the cross-section.
Figure 1.2: Buckling modes, instability types.
It is well-known that the modulus of elasticity of aluminium is one third of steel which makes aluminium more susceptible to instability. Two types of insta-bility are usually considered: overall instainsta-bility and cross-sectional instainsta-bility (see Figure 1.2). The stability problem of the entire structure (translation and rotation) is called overall instability, e.g. flexural, torsional or flexural-torsional buckling. Cross-sectional instability concerns stability of the cross-section, like local buckling and distortional buckling. In classification of cross-sectional instability, local buck-ling implies changes in geometry with only rotation occurring at the fold lines of the section. For distortional buckling rotation and also translation take place at the fold lines of the section with changes in the cross-sectional geometry, Yu and Schafer [2006]. Previous research has shown that cross-sectional instability has a consider-able influence on the stability of the whole structural element (Mennink [2002]).
When using aluminium elements such as beams and columns, the design re-sistance should be determined. Local instability effects often determine the struc-tural resistance because optimization of the material (weight) results in the applica-tion of thin-walled secapplica-tions with complex cross-secapplica-tional shapes. The complexity of thin-walled sectional shapes makes the prediction or determination of cross-sectional instability one of the decisive design issues for aluminium extrusions.
1.2 Problem statement and research goals
For building structures, analytical design rules are used to calculate the structural resistance. As mentioned before, cross-sectional instability influences the stability of the whole structure. There are several codes on aluminium structures which deal with the aspect of local buckling. The commonly used approach in Europe is given in Eurocode 9 CEN [2007], described by Mazzolani [1985]. The cross-section is usually seen as a number of plates connected by nodes. Design standards consider the cross-sectional instability as the buckling of individual plate elements which compose the section and do not consider the interaction of the cross-sectional plates. Unfortunately, such an approach is only sufficient for traditional cross-sectional shapes, like rectangular or so-called I-sections.
Hence, the current design rules used by engineers are limited with respect to cross-sectional instability: only a limited range of cross-sections is covered and only with a limited accuracy. Extensive research into cross-sectional instability of alu-minium structural elements concerning variety of shapes is required.
Due to the lack of knowledge in designing aluminium elements, the following research goals are stated:
• To study cross-sectional instability effects (local and distortional buckling) for aluminium structural elements with various cross-sectional shapes;
• To extend the current analytical formulas for calculating the ultimate load of aluminium sections regarding local buckling to a practical design recommen-dation predicting the local and, moreover, distortional buckling resistance of aluminium profiles.
Thus, the main objective of the current research is to result in an improved pre-diction model, which is not only applicable to traditional cross-sectional shapes (as covered by the design rules of the current codes) but could be also applied to more complex cross-sectional shapes. Development of the prediction model is based on the existing model of Mennink [2002], see section 2.8. In the current research Men-nink’s model is extended by taking distortional buckling mode into account.
1.3 Research approach and thesis outline
In general, the objectives of the current research can be achieved by using experi-mental data and numerical methods, such as the finite strip method (FSM) and the finite element method (FEM).
A scheme of the research approach is shown in Figure 1.3. The contents of the thesis correspond to the presented research approach. First two chapters with in-troduction and literature survey are given. These are followed by the description and results of the research steps indicated in the presented layout. Brief commen-taries for the scheme are given below; respective chapter numbers are included in the figure.
Figure 1.3: Research approach scheme.
• Based on the results of finite strip analyses, various cross-sectional shapes are selected for an experimental program. The results of the finite strip analyses are also compared with those of the Euler buckling analyses according to the finite element program DIANA (Hendriks and Wolters [2007]). The outcome of the selection procedure according to the finite strip analyses results is pre-sented in Chapter 3.
• The experimental investigation includes 85 compression tests for aluminium columns with L-shaped, Z-shaped and C-shaped sections by varying the di-mensions of the cross-section and also the length of the specimen. The exper-iments are expected to show a wide range of instability modes, such as local buckling, distortional buckling, torsional and flexural buckling. Complete de-scription of the experimental program is given in Chapter 3.
• The aim of the experiments is mainly to use the data as input for the finite ele-ment (FE) model and also for validation of the FE-model by comparing the re-sults of the finite element (FE) calculations with the test data. Therefore, at the beginning, the FE-model is built by varying, for example, loading conditions, mesh fineness and load step size. Then, the model is adjusted by comparison with the test results. When a suitable accuracy and validity of the numeri-cal model is achieved, the FE-model is assumed to be valid to investigate the buckling resistance of various cross-sections (see Chapter 4).
• The purpose of Chapter 5 is to validate Mennink’s prediction model regard-ing cross-sectional shapes included in the current investigation. Description of Mennink’s model for local buckling prediction is introduced and possible extension of the model for the distortional buckling phenomenon is discussed.
• A parameter study is executed in Chapter 6, using the validated FE-model. The purpose of the parameter study is to estimate the influence of various pa-rameters and properties (e.g. cross-sectional geometry papa-rameters, mechani-cal properties) on distortional buckling resistance. Hence, the ultimate buck-ling resistance of C-shaped sections with various dimensions and geometry is determined. Based on results of the parameter study and Mennink’s model validation, an improved prediction model for distortional buckling behaviour of aluminium C-shaped members is developed.
The thesis finalizes with a last chapter (Chapter 7), which contains conclusions and recommendations for further research.
Chapter 2
State of the art: cross-sectional instability
Abstract
In this chapter the stability theory for columns and plates is briefly described based on previous research of Timoshenko and Gere [1961], Walker [1975], Galambos [1998] and Yu [2000]. The concept of effective width is one of the common design approaches. The description of aluminium properties and characteristics is given. The definition of buckling modes is introduced in the context of aluminium plate buckling. Thus, the current chapter provides an overview of the existing knowledge on the cross-sectional instability problem of aluminium structural members.
2.1 Column Buckling
T
he buckling theory of a column loaded by a compressive force is based on the work of Euler that is described by many authors, e.g. Timoshenko and Gere [1961]. An understanding of the strength of centrally loaded columns is essential to the development of design criteria for compression members in general. Consid-ering the Euler column as a mathematically straight, pin-ended, perfectly centrally loaded column, the buckling load or critical load or bifurcation load is equal to:Ncr;E =
π2EI
L2 (2.1)
where EI is the elastic stiffness and L is the axial length. The Euler load Ncr;Eis the reference value to which the strength of actual columns is usually compared. If end conditions differ from perfectly frictionless pins, the critical load is expressed by:
Ncr;EK=
π2EI
(KL)2 (2.2)
where KL is an effective length defining the portion of the deflected shape between points of zero bending moment. In other words, KL is the length of an equivalent pin-ended column buckling at the same load as the end-restrained column. The general approach for column buckling is to determine the failure or design load as a function of the critical and squash loads.
2.2 Buckling Theory of Plates
2.2.1 Elastic critical buckling
This section regards buckling of rectangular thin plates subjected to uniform pression. Figure 2.1 shows a possible buckling shape in the case of uniform
com-pression. An important contribution to the plate-buckling theory has been made by von K´arm´an et al. [1932] and Winter [1947]. The plate-buckling phenomenon is summarized in many books, see for example: Timoshenko and Gere [1961], Walker [1975], Kirby and Nethercot [1979], Murray [1984], Narayanan [1987], Rhodes and Harvey [1971], Galambos [1998], and Yu [2000]. Earlier investigations established that plates exhibit a stable equilibrium with a substantial amount of post-buckling strength which is not the case for a uniformly compressed column.
Figure 2.1: Rectangular plate loaded by uniform pressure.
The theoretical critical load for a plate is not necessarily a satisfactory basis for design, since the ultimate strength can be much greater than the critical buckling load. For example, a plate loaded in uniaxial compression, with both longitudinal edges supported, provides postbuckling support which means the plate will un-dergo stress redistribution as well as develop transverse tensile membrane stresses after buckling. Initial imperfections in such a plate may cause bending to begin below the buckling load, yet unlike an initially imperfect column, the plate may sustain loads greater than the theoretical buckling load.
The rectangular plate of Figure 2.1 allows various boundary and loading condi-tions. The buckling problem of uniformly compressed plate with no lateral loading was also formulated by Saint-Venant [1883]. This problem was solved first by Bryan [1890]. All four plate-edges are supported, i.e. in-plane translations are restricted whereas rotations are allowed, while one pair of opposite edges is subjected to a uniform compressive stress σ . The elastic critical stress has been defined as σcr:
σcr= Dπ 2 tb2 · m µ b a ¶ +n 2 m ³ a b ´¸2 (2.3) where D = Et 3 12 (1 − ν2) (2.4)
The critical value of σcr, i.e. the smallest value, can be obtained by taking n equal to 1. The physical meaning of this is that a plate buckles in such a way that there can be several half-waves in the direction of compression but only one half-wave
in the perpendicular direction. m is the number of half-waves into which the plate buckles.
For the case of a simply supported rectangular plate under uniform compression the buckling coefficient kcrequals:
kcr= m 2
ϕ2 +
ϕ2
m2 + 2 (2.5)
By introducing the plate-slenderness parameter (ϕ = a/b) in equation 2.5, equa-tion 2.3 gives the critical stress as:
σcr= kcr π 2Et2
12(1 − ν2)b2 (2.6)
Equation 2.6 is similar to Equation C.2 except that the plate width b is squared rather than column length L and the Poisson’s ration ν is included in the flexural rigidity D for plates. The plate buckling coefficient kcrallows for different boundary conditions.
Figure 2.2: Buckling coefficient kcrof a simply supported plate.
Figure 2.2 presents kcr, for a simply supported plate, as a function of the plate slenderness ϕ. In structural engineering long plates have a relatively large a/b ratio. Such plates can be used to describe the behaviour of plates in cross-sections, for example the webs of square hollow sections. In those cases, kcrreduces to the well-known value of 4.0.
Various boundary conditions have been studied by different authors. Figure 2.3 by Gerard and Becker [1957] summarizes five common cases of supports, where
c stands for clamped, ss for simply supported, and free for an unsupported edge.
Values of the buckling coefficient for different boundary conditions are given in Fig-ure 2.1. The buckling coefficient, kcr, for a long rectangular plate simply supported along three sides, with one unloaded free edge equals: kcr= 0.425. However, when the restraining effect of the web is considered, kcr may be taken as 0.5 for the de-sign of the unstiffened compression flange. The supporting effect of edges, which defines kcr, is therefore highly important.
Figure 2.3: Buckling coefficients kcr of
uniformly compressed rectangular plates with various boundary conditions.
Table 2.1: Values of kcr for determining
the critical buckling stress.
2.2.2 Postbuckling Strength
Unlike one-dimensional structural members such as columns, stiffened compres-sion elements will not collapse when the buckling strength is reached. In the plate, the stress distribution is uniform prior to its buckling, as is shown in Figure 2.4(a). After buckling, a portion of the prebuckling load of the center transfers to the edge zone of the plate. As a result, a non-uniform stress distribution is developed, as shown in Figure 2.4(b). The redistribution of stress continues until the stress at the edge reaches the yield point fyand the plate begins to fail (Figure 2.4(c)). The post buckling behaviour of a plate can be analyzed by using large deflection theory.
Figure 2.4: Stages of stress distribution in simply supported compressed plates, according to Yu [2000].
2.2.3 Effective Width Approach
The concept of effective width was introduced by von K´arm´an et al. in 1932. In this approach, instead of considering the non-uniform distribution over the entire width of the plate b, it is assumed that the total load is carried by a fictitious effective
width bef f, subject to a uniformly distributed stress equal to the edge stress σmax, as shown in Figure 2.5.
Figure 2.5: Definition of effective width.
The width b is selected so that the area under the curve of the actual non-uniform stress distribution is equal to the sum of the two parts of the equivalent rectangular shaded area with a total width bef fand an intensity of stress equal to the edge stress
σmax, that is:
b Z 0
σ dx = beffσmax (2.7)
It may also be considered that the effective width represents a particular width of the plate that just buckles when the compressive stress reaches the yield point of material. This results in the Von K´arm´an formula:
beff = Ct s
E
fy (2.8)
where C equals for µ = 0.3:
C =p π
3(1 − µ2)= 1.9 (2.9)
Based on his extensive investigation on light-gauge cold-formed steel sections, Win-ter [1947] indicated that the equation is equally applicable to the element in which the maximum stress σmax is below the yield point fy. This results in the Winter formula: beff = Ct r E σmax (2.10)
In addition, results of tests previously conducted by Sechler [1968] and Winter led to an improvement of the term C:
C = 1.9 " 1 − 0.475 µ t b ¶ r E σmax # (2.11) which led consequently to the modified Winter formula Winter [1947] and Winter [1948] for plates simply supported along both longitudinal edges:
beff = 1.9t r E σmax " 1 − 0.475 µ t b ¶ r E σmax # (2.12) It should be noted that the effective width not only depends on the edge stress but also on the b/t ratio.
2.2.4 Effective Thickness Approach
Design rules on local buckling for steel are based on the effective width approach. In some recently adopted national standards the effective thickness is used since it leads to simple calculations. In Eurocode 9 CEN [2007], the effective thickness is used for calculation of resistance. However, in many cases an effective width concept is used for calculation of stiffness (see Hoglund [1972]). A description of the effective thickness approach can be found in section 2.4.
2.3 Stability: influence of material and imperfections
2.3.1 Material characteristics
The structural analysis for aluminium is based on a material with a generalized in-elastic behaviour. Several authors have formulated various models for inin-elastic be-haviour of material, which are described in Kutanova and Soetens [2006]. The most common way is to separate the stress-strain relationship diagram into three regions. The first region corresponds to elastic behaviour, the second region to inelastic be-haviour and the third region identifies strain-hardening bebe-haviour. With respect to these three regions, models proposed by Baehre [1966] and Mazzolani [1972] define the limits and non-dimensional relations for each region. The stress-strain curve of aluminium alloys is often described as a ”round-house” type DeMartino et al. [1990], which can be modeled accurately using the Ramberg-Osgood expression.
A generalized stress (σ) - strain (ε) law ε = f (σ) has been proposed by Ramberg and Osgood [1943] for aluminium alloys:
ε = σ E + 0.002 µ σ f0.2 ¶n , (2.13)
where E = Young’s modulus at the origin, f0.2 = 0.2% proof stress, ε = variable strain, σ = variable stress.
The exponent n of the Ramberg-Osgood law is a characteristic of the strain-hardening rate of the inelastic portion of the σ − ε diagram and can be expressed as: n = ln 2 ln ³ f0.2 f0.1 ´ (2.14)
where f0.1= 0.1% proof stress , which has to be determined by experiment.
Figure 2.6: Stress-strain diagram based on the Ramberg-Osgood law.
The higher the value of the exponent, n, is, the sharper the knee of the stress-strain curve, and the slower the slope of the stress-strain-hardening portion of the curve. The exponent n is a function of f0.2 and f0.1, and hence this form of the law is in terms of parameters that can all be determined experimentally from a tensile test. The Ramberg-Osgood law is now widely used because its predicted behaviour is very close to the actual behaviour of aluminium alloys, in particular for the first part of the stress-strain diagram.
Sometimes it is not possible to get the value of f0.1and specifications usually do not provide this value with the other mechanical properties. Thus, there are other methods to determine the value of the exponent n. Two proposals of this type are explained below.
The Steinhardt’s proposal [1971] is very simple and concise: E and f0.2 values are assumed to be the minimum required by specifications, and it is also assumed that:
10n = f0.2 (2.15)
This simple expression is based on experimental results that showed a connec-tion between the 0.2% tensile proof stress and the exponent of the Ramberg-Osgood law, n. Non-treated alloys have an exponent, n, in the range of 10-20, and heat-treated alloys have an exponent, n, in the range of 20-40. If the analysis concerns the range of elastic deformations or so-called elastic range (CEN [2007]), Steinhardt’s proposal provides a sufficient prediction of the value of n.
If the analysis concerns the range of plastic deformations or so-called plastic range, Mazzolani’s expression can be used where three values of the material spec-ifications are taken into account [1974]. Starting from the minimum values required by specification for mechanical properties: f0.2- elastic limit, σmax- ultimate strength,
εmax- elongation at rupture, the following approximate expression for the exponent ´ n is proposed: ´ n = ln 2 ln(1 + kχ) (2.16) where χ = σmax− f0.2 10εmax σmax f0.2 [N/mm2] (2.17)
and k is a dimensional constant. This expression has been verified using the statis-tical results of tests carried out at Li`ege University by Bernard et al. [1973]. On this basis it has been assumed that k = 0.028 mm2N−1.
2.3.2 Geometrical Imperfections
Imperfections affect the load-bearing capacity of the members and cannot be ig-nored (Mazzolani [1995]). Unavoidable imperfections are produced during the fab-rication process of structural members, e.g. initial out-of-straightness, thickness de-viation, initial curvature, and load eccentricities. In this section aluminium proper-ties and initial geometrical imperfections are discussed.
By means of geometrical imperfections the differences between the nominal and the actual shape of the structural element, or so-called initial out-of-straightness is considered. Extensive studies for initial out-of-planeness definition have been car-ried out for steel columns: Beer and Schultz [1970], Fukumoto et al. and Essa and Kennedy [1993]. The most common approaches to define plate deflections is to mea-sure the largest out-of-plane deflection of the plate from initial straightness, or to describe the whole deflection pattern in sinusoidal shape by the use of Fourier anal-yses. The post welding shape of welded steel plating in ships has been determined by Antoniou et al. [1984], Antoniou [1980], Carlsen and Czujko [1978] and Kmiecik et al. [1995]. Carlsen and Czujko [1978] found the following expression for the max-imum out-of-plane deflection of a plate:
wmax= 0.016b − 0.36t (2.18)
Corresponding results found by Antoniou [1980] can be expressed as:
wmax= 0.014b − 0.32t (2.19)
Similar studies are known in aluminium, e.g. Clarke [1987] found that the initial imperfection amplitude varies from 0.0066b to 0.021b. It was concluded that slender plates show larger initial imperfection amplitudes than stocky plates.
Due to the production process of aluminium plates, variation in thickness might occur. The plates will have different values of the thickness in different parts of
the plates, and the thickness will differ from nominal thickness. According to Maz-zolani [1995] the different national specifications for extruded profiles allow a de-viation from nominal thickness equal to 5 percent. This value can reach 10 percent in the case of thin profiles with a thickness less than 5 mm. From the results of im-perfections measurements, it was concluded that out-of-straightness of aluminium extruded bars is usually less than in steel. The amplitude of the geometrical imper-fections for extruded bars is limited by the value of about L/2000, for welded bars this limit increases to L/1300.
2.4 Local Buckling
It was mentioned in Chapter 1 that applied thin-walled aluminium structural ele-ments (beams and columns) often consist of complex cross-sectional shapes that are composed of connected plate elements. When a plate is subjected to compression, bending or shear, or a combination of loads, the plate may buckle locally before the whole structural member becomes unstable.
Local buckling is characterized by distortion of the cross-section with only rotation at the fold lines of the section. Local buckling is usually prevalent and occurs with repeated short buckling waves of individual plate elements.
2.4.1 Slenderness parameter
The prediction of the ultimate resistance of the structural member due to the oc-currence of local buckling is rather complex even for simple symmetrical cross-sectional shapes. One of the design methods for buckling analysis makes use of a dimensional strength design curve for a column, which is based on a non-dimensional slenderness.
The buckling load depends on the slenderness of the cross-section element, which is normally determined by the ratio of the width divided by the thickness (β = b/t). In many cases the more general parameter for slenderness, λ is used
λ =
r
f0.2
σcr
, (2.20)
where f0.2is the 0.2% proof stress and σcris the elastic buckling stress for a perfect flat plate without initial buckles or residual stresses. λ is proportional to b/t and p
f0.2/E and depends on the loading conditions and the boundary conditions, e.g. the connection to other cross-sectional elements.
For a plate simply supported along the edges, the critical buckling stress is ex-pressed by Equation 2.6. For an axially loaded plate with simply supported edges,
kcr = 4 (see Figure 2.1). The slenderness parameter can now be determined by inserting expression 2.6 into 2.20.
λ = s f0.2 kcr12(1−νπ2E2)t 2 b2 = 0.526b t r f0.2 E , for kcr= 4 and ν = 0.3. (2.21)
A principal relationship between axial resistance N (N = σA, where A is the cross-sectional area ) and the slenderness parameter λ for a slender plate is given in Figure 2.7. Failure of the plate elements is represented by the axial squash load
N0.2 = f0.2A , critical axial load Ncr = σcrA and ultimate load Nu = σmaxA . It can be shown that the load exceeds the buckling load, which is called post-buckling strength. The resistance in the post-buckling stage is important for slender plates.
Figure 2.7: Relationship between slenderness parameter λ, buckling load Ncrand ultimate
load Nu.
2.4.2 Classification of cross-sections
The edge restraining conditions of the plate elements of the cross-section influence the occurrence of local buckling (see Figure 2.3 and Table 2.1). From this point of view, two groups of plate elements are recognized: flat internal and flat outstanding elements. Cross-sectional shapes composed by different plate elements (e.g. a rect-angular hollow section composed by only internal flat elements) are investigated by many authors. An extensive experimental program for aluminium square hol-low sections, rectangular holhol-low sections, channels and angles has been developed to calibrate a prediction approach for local buckling ( Mazzolani et al. [1996], Maz-zolani et al. [1997] and MazMaz-zolani et al. [2001]). The aim of the experiments was to provide Eurocode 9 CEN [2007] with the necessary data for classification of alu-minium members.
Design methods control the capability of members to reach a given limit state. Referring to a relationship between a relative force F/Fmaxand a relative displace-ment u/umaxin Figure 2.8, the above classes correspond to the following behaviour according to Eurocode 9 (CEN [2007]):
Figure 2.8:Classification of cross-sections (from CEN [2007]).
1. Class 1 ductile cross-sections can develop the ultimate resistance without the
occurrence of buckling, reaching the ultimate value of deformation with - de-pending on the type of alloy (collapse limit state) - considerable plastic strains;
2. Class 2 compact cross-sections are capable of developing ultimate plastic
resis-tance but have limited deformation capacity, which is prevented by plastic instability phenomena (plastic limit state α1);
3. Class 3 semi-compact cross-sections are able to reach the proof strength f0.2 of the material, but the development of the important plastic deformations is prevented. Only small plastic deformations are allowed, sometimes giving rise to a brittle behavior ( elastic limit state α2);
4. Class 4 slender cross-sections, whose strength is governed by the occurrence of
local buckling phenomena, are not able to reach the proof strength f0.2. No plastic deformation is allowed within the section (elastic buckling limit state
α3).
Therefore, cross-sections of Class 4 need to be checked for local buckling. A slender cross-section must be designed according to resistance formulae calculated with reference to the effective thickness of the element. The effective thickness is determined by the reduction factor for local buckling.
Three different types of elements are recognized in a cross-section: flat outstand, flat internal and curved internal elements. The basic parameter for the classification criterion is the width-to-thickness ratio (b/t) of each element. Experiments have been executed to identify the influence of the width-to-thickness ratios on the local buckling phenomenon (Mazzolani et al. [1996]). On the basis of tests, the classi-fication of cross-sectional internal and outstand elements linked to the values of
the slenderness parameter has been introduced in Eurocode 9 CEN [2007]. Clas-sification in Eurocode 9 does not account for the interaction between flat internal elements and flat outstand elements. Thus, another classification criterion has been developed by Faella et al. [2000], taking into account the interaction between the slenderness parameters of the plate elements. Updating the classification method provided by Eurocode 9 was advised.
2.5 Distortional buckling
The distortional type buckling mode was first detected by Lundquist and Stow-ell [1942] and later also by Gallaher and Boughan [1971] in their investigation on sta-bility of panels with Z-stiffeners. Van der Maas [1954] studied the buckling behavior of hat-section columns and noticed another kind of local buckling phenomenon in-volving flange-stiffener rotations. Later, Sharp [1966] called the attention to ”flange-stiffener” instability and developed a simple formula for the buckling load of lipped channel columns failing in this mode. The influence of edge and intermediate stiff-eners on the buckling behaviour of lipped channel columns and beams has been studied further by Desmond, Pekoz and Winter [1977], [1981] at Cornell University. A significant amount of experimental and analytical investigations for ”flange-stiffener” buckling has been carried out at the University of Sydney by Hancock et al. [1990] and Hancock [1997]. Finally, distortional buckling received its current name. As a result of this research, an analytical prediction model for the distortional buckling stress of thin-walled cold-formed carbon steel sections has been developed by Lau and Hancock [1986]. This analytical method has been implemented in the Australian/New Zealand codes [2005].
A clear definition for distortional buckling was given in the work of Yu and Schafer [2003]:
Distortional buckling involves changes in geometry when both rotation and trans-lation occur at the flange/lip fold lines of the section. The wavelength of dis-tortional buckling is in between that of local buckling and torsional-flexural buckling, while torsional-flexural buckling is characterized by one half-wave over the length of the structural member.
In general, for distortional buckling a reduction is applied in addition to local ling. Existing design rules do not distinguish between local and distortional buck-ling. One of the reasons of the lack of knowledge of distortional buckling, is that it is hard to perform experiments for distortional buckling.
Distortional buckling tests on cold-formed steel beams with C and Z sections have been executed by Yu and Schafer [2006]. The results of the distortional buck-ling tests have been compared with the results of local buckbuck-ling tests carried out by Yu and Schafer [2003]. In both tests steel beams with identical geometry and material have been used and identical test set-up, except the restraint conditions for the compression flange. In the distortional buckling test the compression flange was
not restrained by attachment to paneling, while for local buckling test the compres-sion flange was restrained by a special fastener configuration to be able to prevent the distortional buckling mode. It was discovered, that the elastic stiffness for local buckling and distortional buckling tests was the same, but failure in case of dis-tortional buckling was initiated before failure according to local buckling occurred. The results of the local and distortional buckling tests have been compared with design specifications and with a direct strength method developed by Schafer and Pek ¨oz [1998]. It was concluded that the so-called direct strength method provides an adequate prediction of the distortional buckling capacity. A description of this method can be found in section 2.7.
2.6 Numerical Investigations
2.6.1 Finite Element Method
Various numerical methods can be used to analyse the buckling behaviour of alu-minium thin-walled members. The finite element method (FEM) is one of the most popular, as it successfully handles almost any buckling problem. In the current investigation, the finite element program DIANA is used. General aspects are de-scribed here, detailed information can be found in the DIANA manual, Hendriks and Wolters [2007].
Finite element calculations of the thesis are performed by three analyses: linear elastic, stability and non-linear analysis. Linear elastic analysis assumes the first order analysis with elastic material characteristics. In most large deformation prob-lems, linear elastic assumptions are not satisfactory. Therefore, material and geo-metrical nonlinear analysis is considered for the best approximate solutions. How-ever, the result of the linear elastic analysis is an essential input for the non-linear analysis. The stability analysis according to the finite element code (Euler Eigen-value analysis) results in several buckling modes. Each mode represents a Eigen-value of the buckling load and a buckling deformation pattern. The deformation pattern of the first Euler buckling mode is applied as imperfection pattern for the geometri-cal non-linear analysis. The Newton-Raphson iterative method is used to solve the nonlinear problem. For material hardening behaviour, the von Mises yield criterion and a work-hardening stress-strain relation is applied.
2.6.2 Finite Strip Method
FSM uses a discretization of a cross-section into a series of longitudinal elements, so-called strips. Based on these strips elastic and geometric stiffness matrices can be formulated. Hancock [1978] presented the formulation of a semi-analytical fi-nite strip method (FSM) to perform buckling analyses of thin-walled members. This paper introduced first the concept of the ”signature curve”: a curve representing the relation of the buckling stress with the member length, which exhibited a local minimum, identified with local buckling, and a descending branch, identified with
lateral-torsional buckling. Based on numerical study of columns with edge stiff-ened cross-sections, Hancock [1985] has shown later that the ”signature curve” has two local minima, identified with local buckling for short columns and distortional buckling for intermediate columns. Furthermore, is has been shown that distor-tional buckling may be critical.
Figure 2.9: Buckling modes of lipped channel in compression (from Hancock [2008]).
A result of the finite strip buckling analysis of the lipped channel section, sub-jected to a uniform compression, is shown in Figure 2.9 as the buckling load versus the half-wavelength. Point A corresponds to the local buckling mode and the buck-ling shape includes deformation of the web without translation at the flange/lip junction. Point B corresponds to the distortional buckling mode and the buckling shape includes deformation of the web and translation at the flange/lip junction. Points C, D and E correspond to torsional-flexural and flexural buckling.
The constrained finite strip method (cFSM) is a new extension to the FSM. The constrained method is developed by Adany and Schafer [2006] and is able to pro-vide both modal decomposition and modal identification. It means that local, dis-tortional and global deformation modes can be recognized and decomposed, and participation of each buckling mode can be defined. The conventional finite strip method combined with the constrained finite strip method is implemented in an open source program CUFSM Schafer [2006].
