Zainal Abidin-AR-2013-PhD-Thesis

281 Pages·2013·26.12 MB·English

M L E B
ODELLING OF OCAL LASTIC UCKLING FOR
S B W O
TEEL EAMS WITH EB PENINGS

by

Ahmad Razin Zainal Abidin
BEng, MSc

January 2013

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

Imperial College London
Department of Civil and Environmental Engineering
Imperial College London
London SW7 2AZ

1
DECLARATION

I confirm that this thesis is my own work and that any quotation from,
or description of, published or unpublished work of others
is acknowledged herein by reference to the sources.

Ahmad Razin Zainal Abidin

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ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my supervisor Professor Bassam A. Izzuddin
for his patience, careful guidance and continuous encouragement throughout the duration of this study.
His constructive comments and unfailing support in many parts concerning this work are invaluable
and gratefully acknowledged.
The financial support provided by Universiti Teknologi Malaysia (UTM) throughout the
course of the research is also gratefully acknowledged.
Special thanks goes to the closest person to me, Nafisah, whose love and support has always
been an enormous source of inspiration and strength for me. Not to mention my beloved kids, Ismail
and Ilyas, to whom this thesis is dedicated.
Finally, I wish to thank my parents, who patiently and constantly supported me in the best
possible way during the whole period of this study. I largely owe to them all I have achieved in my life
thus far. Thanks is also due to my colleagues and friends in room 317A for their unforgettable help
and company.

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ABSTRACT

This work is mainly concerned with the development of sophisticated yet efficient methods for
assessing the elastic buckling of steel beams with web openings, focussing on local buckling effects in
the web region. A new computational method is proposed which extends the use of the Element Free
Galerkin (EFG) method for the numerical discretisation combined with a simplified buckling
assessment approach based on the Rotational Spring Analogy (RSA). The new approach considers
several potential simplifications offering a balance between computational efficiency and accuracy in
local buckling analysis.
In the present EFG/RSA method, considerable advantage is established by separating the
planar and out-of-plane responses. Planar analysis is further enhanced using modular concepts, where
the beam is divided into unit cells, each of which resembles a super-element with a reduced number of
freedoms, and solved using a standard discrete procedure. As for the out-of-plane analysis, the
application of a ‘local region’ is adopted to significantly reduce the size of the original buckling
problem. Finally, local buckling assessment is conducted using an effective approach that utilises an
iterative procedure based on a rank 2 reduced eigenvalue problem along with a shifting local region.
Several illustrative examples are provided which highlight the efficiency and accuracy of the
developed approach in comparison with detailed nonlinear finite element analysis performed using
ADAPTIC, and which demonstrate general applicability to local buckling analysis of steel beams with
web openings of various shapes and sizes.
This work also considers the development of a simplified design-oriented method which is
presented particularly for web-post buckling. Towards this end, a simplified analytical model is
proposed based on an analogy with equivalent rectangular thin beams (RTB), where a semi-empirical
approach is used to calibrate the suggested formulation against the results obtained from the numerical
work performed earlier.

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TABLE OF CONTENTS

DECLARATION 2
ACKNOWLEDGMENTS 3
ABSTRACT 4
TABLE OF CONTENTS 5
LIST OF TABLES 10
LIST OF FIGURES 11
NOTATION 19
CHAPTER 1 INTRODUCTION 24
1.1 Introduction 24
1.1.1 Beams with web openings as important structures 25
1.1.2 Structural response characteristics of steel beams with web openings 26
1.1.3 Current methods of assessment 27
1.1.4 Need for advanced modelling of beams with web openings 29
1.2 Objectives and Scope of the Present Research 30
1.3 Organization of the Thesis 31
CHAPTER 2 LITERATURE REVIEW 33
2.1 Introduction 33
2.2 Behaviour of Steel Beams with Web Openings 33
2.2.1 Complexity of behaviour 37
2.2.2 Potential failure modes 42
2.3 Design Guidance 47
2.3.1 Design-Oriented Models 49
2.4 Applied Research 50
2.5 Finite Element Modelling 53
2.5.1 FE analysis for perforated beams 53

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2.5.2 Mesh and element types 58
2.5.3 Boundary conditions and load application 59
2.5.4 FE models for the present work 59
2.6 Meshless Methods for Plates Analysis 63
2.6.1 Recent developments 64
2.6.2 Application issues 65
CHAPTER 3 NUMERICAL MODELLING OF PLANAR RESPONSE 70
3.1 Overview of Proposed Model 70
3.2 Discretisation with EFG Method 70
3.2.1 Formulation for planar response 71
3.2.2 Shape function by MLS approximation 72
3.2.3 Essential boundary conditions 78
3.2.4 Background cells for integration 80
3.2.5 Computational implementation of the EFG planar analysis 81
3.3 Numerical Examples and Discussion 85
3.3.1 Rectangular solid panel 86
3.3.2 Panel with irregular shape 96
3.4 Planar Response of Beams with Web Openings 109
3.4.1 Unit cell level 110
3.4.2 Local analysis and representative actions 111
3.4.3 Global analysis 114
3.4.4 Recovery of planar stresses over unit cell 115
3.4.5 Extended scheme for perforated beams 115
3.5 Verification 117
3.5.1 Simply supported cellular beam under UDL 117
3.5.2 Cellular beams with different support conditions 120
3.5.3 Computational benefits of EFG method for increasing number of holes 124
CHAPTER 4 NUMERICAL MODELLING OF OUT-OF-PLANE RESPONSE 126
4.1 Introduction 126
4.2 EFG Method for Thin Plates 126
4.2.1 Problem formulation 128
4.2.2 Higher order derivatives of shape function 129
4.2.3 Computational implementation for the out-of-plane analysis 132
4.3 Numerical Examples and Discussion 134

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4.3.1 Rectangular thin plates 134
4.3.2 Plates with irregular shape 139
4.4 Out-of-plane Response of Beams with Web Holes 145
4.4.1 Local region 145
4.4.2 Torsional stiffness of flanges 146
4.4.3 Extended scheme for perforated beams 147
4.5 Verification 149
4.5.1 Cellular beam with a point load at mid-span 150
4.5.2 Cellular beam subject to web-post twisting load 152
CHAPTER 5 NUMERICAL MODELLING OF LOCAL BUCKLING 155
5.1 Introduction 155
5.2 Buckling Analysis of Structural Systems 156
5.2.1 Rotational Spring Analogy 157
5.3 Proposed EFG/RSA Method 159
5.3.1 General formulation 159
5.3.2 Geometric stiffness matrix, K 160
G
5.3.3 Probing process 161
5.4 Numerical Investigations and Discussion 165
5.4.1 Buckling of rectangular plate 166
5.4.2 Buckling of irregular plates 177
5.5 Buckling of Beams with Regular Web Openings 183
5.5.1 Additional geometric stiffness from flanges 184
5.5.2 Buckling analysis strategy 185
5.5.3 Flowchart for the EFG/RSA method 185
5.5.4 Verification 186
5.6 Advanced Application to Beams with Irregular Holes 197
5.6.1 Issues in modelling irregular holes 197
5.6.2 Application examples and discussion 200
CHAPTER 6 PARAMETRIC STUDIES 208
6.1 Introduction 208
6.2 Geometric Configuration 208
6.2.1 Effect of hole size 209
6.2.2 Effect of beam depth 214
6.2.3 Effect of hole spacing 216

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6.3 Support Conditions 219
6.4 Irregular Beam Profile 223
6.4.1 Offset holes 223
6.4.2 In-filled holes 225
6.4.3 Elongated holes 226
CHAPTER 7 SIMPLIFIED MODEL FOR WEB-POST BUCKLING 228
7.1 Introduction 228
7.2 Web-post Buckling 228
7.2.1 Empirical fit of numerical finite element simulations 229
7.2.2 Simplified strut model under compression 230
7.3 Rectangular Thin Beam Analogy 232
7.3.1 Concept and formulation 233
7.3.2 Verification of RTBA 236
7.4 Web-post Buckling Model by RTBA 238
7.4.1 Basic assumptions 238
7.4.2 Model calibration 239
7.4.3 Comparison with existing models 243
7.5 RTB Buckling Model for Combined Actions 245
7.5.1 Co-existing axial force 245
7.5.2 Co-existing bending moment 248
CHAPTER 8 SUMMARY AND CONCLUSIONS 250
8.1 Meshless Method for Local Buckling of Beams with Openings 250
8.1.1 Planar analysis 251
8.1.2 Out-of-plane analysis 252
8.1.3 Buckling assessment 252
8.2 Simplified Model for Web-Post Buckling 253
8.3 Suggestions for Future Research 254
REFERENCES 256
APPENDIX A GRAPHICAL MESH GENERATION TOOL 265
A.1 General Procedure 265
A.2 Examples of Graphical Mesh Generation Tool 267
APPENDIX B ADAPTIC DATA FILES 271

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B.1 Monolithic analysis 271
B.2 Partitioned analysis using HPC system 273
B.2.1 Script file 274
B.2.2 Parent data file 274
B.2.3 Partition data files 278

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LIST OF TABLES

Table 2-1 – Relevant design guidelines in Eurocode 3 and BS5950:Part1:2000 48
Table 3-1 – Displacement of point A (u ) and the total strain energy (e) resulting from different
A
quadrature rules (Units mm and N.mm) 92
Table 4-1 – Parameter  for a centrally loaded square plate under different quadrature rules 137
Table 4-2 – Parameter  for simply supported rectangular plates 138
Table 4-3 – Parameter  for fully clamped rectangular plates 139
Table 5-1 – Material stiffness energy (e ), geometric stiffness energy (e ), and buckling load
E G
factor (λ ) for different quadrature rules 168
cr
Table 5-2 – Performance of different approaches for establishing  170
cr
Table 5-3 – Material stiffness energy (e ), geometric stiffness energy (e ), and buckling load
E G
factor (λ ) for different quadrature rules 175
cr
Table 5-4 – Performance of different approaches in probing for the lowest buckling load factor 176
Table 5-5 – Convergence of probing approaches to lowest buckling load factor 182

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