Modern Structural Analysis Modelling Process and Guidance

Acknowledgements iii
Foreword iv

1 Introduction 1
1.1 Scope and definitions 1
1.2 Why “modern” structural analysis? 1
1.3 Issues for practice 2
1.4 Issues for education 2
1.4.1 The wider context 3
1.5 Finite elements 3
1.6 Accuracy of the information provided in the text 4
1.7 Website 4

2 Basic principles 5
2.1 Managing the analysis process 5
2.1.1 Quality management systems 5
2.1.2 Use the modelling process 5
2.1.3 Competence 5
2.2 Modelling principles 5
2.2.1 Use the simplest practical model 5
2.2.2 Estimate results before you analyse 6
2.2.3 Increment the complexity 6
2.2.4 When you get results, assume that they may be errors 6
2.2.5 Troubleshooting for errors 6
2.2.6 Relationship between the analysis model and the design code of
Practice 7
2.2.7 Case study – the Ronan Point collapse 8
2.3 Principles in the use of structural mechanics 8
2.3.1 Local and resultant stresses – the St Venant principle 8
2.3.2 Principle of superposition 9
2.3.3 Lower bound theorem in plasticity 10
2.4 Understanding structural behaviour 11
2.4.1 General 11
2.4.2 Model validation 11
2.4.3 Results verification and checking models 11
2.4.4 Sensitivity analysis 11
2.4.5 Solution comparisons 13
2.4.6 Convergence analysis 14
2.4.7 Identify patterns 14
2.4.8 Mathematics 14
2.4.9 Physical modelling and testing 14

3 The modelling process 15
3.1 Overview of the modelling process 15
3.1.1 General 15
3.1.2 Representations of the modelling process 15
3.1.3 Validation and verification 17
3.1.4 Error and uncertainty 17
3.2 Defining the system to be modelled 18
3.3 The model development process 18
3.3.1 Conceptual and computational models 18
3.3.2 Model options 19
3.4 Validation of the analysis model 19
3.4.1 Validation process 19
3.4.2 Validation the conceptual model 20
3.4.3 Validation the computational model 20
3.5 The solution process 21
3.5.1 Selecting software 21
3.5.2 Software validation and verification 21
3.5.3 Truncation error, ill-conditioning 22
3.6 Verifying the results 22
3.6.1 Acceptance criteria for results 22
3.6.2 Verification process 22
3.6.3 Checking models 23
3.6.4 Checking loadcase 25
3.7 The modelling review 25
3.7.1 Sensitivity analysis 25
3.7.2 Overall acceptance of the results 25
3.7.3 The modelling review document 25
3.8 Case studies 26
3.8.1 The Tay Bridge disaster 26
3.8.2 The Hartford Civil Center roof collapse 27
3.8.3 The Sleipner platform collapse 27

4 Modelling with finite elements 29
4.1 Introduction 29
4.2 Elements 29
4.2.1 Constitutive relationships 29
4.2.2 Line elements 30
4.2.3 Surface elements 30
4.2.4 Volume elements 32
4.2.5 Joint elements 33
4.2.6 Basic principles for the derivation of finite element stiffness matrices 34
4.3 Mesh refinement 36
4.3.1 Discretisation error 36
4.3.2 Convergence 36
4.3.3 Singularities 37
4.3.4 Benchmark tests 38
4.3.5 Case study – mesh layouts for a cantilever bracket 38
4.3.6 Meshing principles 39
4.4 Case study – convergence analysis of a plane stress cantilever beam model 41
4.4.1 General 41
4.4.2 The context 41
4.4.3 Elements used in the convergence analysis 41
4.4.4 Reference solution 42
4.4.5 Convergence parameters 43
4.4.6 Meshes 44
4.4.7 Results 44
4.4.8 Overview 45
4.5 Constraints 46
4.5.1 General 46
4.5.2 Rigid Constraint conditions 47
4.5.3 Constraint equations 47
4.6 Symmetry 48
4.6.1 General 48
4.6.2 Mirror symmetry 48
4.6.3 Symmetry checking 49

5. Skeletal frames – modelling with line elements 51
5.1 Introduction 51
5.1.1 Members and elements 52
5.2 Bending 52
5.2.1 Background 52
5.2.2 Behaviour 53
5.2.3 Basic relationships for bending 53
5.2.4 Symmetric and asymmetric bending 53
5.2.5 Shear in bending 54
5.2.6 Combined bending and shear 56
5.2.7 Validation information for the engineers’ theory of bending 56
5.3 Axial effects 58
5.3.1 Behaviour 58
5.3.2 Basic relationships 59
5.3.3 Validation information 59
5.4 Torsion 60
5.4.1 Behaviour 60
5.4.2 Basic relationships for shear torsion 61
5.4.3 Basic relat6ionships for bending torsion 62
5.4.4 Combined torsion 63
5.4.5 Validation information for torsion 63
5.5 Bar elements and beam elements 64
5.5.1 Bar elements 64
5.5.2 Engineering beam elements 64
5.5.3 Higher-order beam elements 66
5.6 Connections 66
5.6.1 Basic connection types 66
5.6.2 Treatment of the finite depth of a beam using rigid links 68
5.6.3 Modelling beam-to-column connections in steelwork 68
5.6.4 Connections in concrete 71
5.6.5 Eccentricity of members at a joint 72
5.7 Distribution of load in skeletal frames 74
5.7.1 Vertical load in beam systems 74
5.7.2 Distribution of lateral load 75
5.8 5.8.1 Curved members 75
5.8.2 Case study – modelling of curved beams 75
5.8.3 Modelling members with non-uniform cross section 77
5.8.4 Case study – tapered cantilever 77
5.8.5 Cantilever with a tapered soffit 79
5.8.6 Haunched beams 79
5.9 Triangulated frames 79
5.9.1 Modelling issues 79
5.9.2 Euler buckling effect of members 80
5.10 Parallel chord trusses 80
5.10.1 General 80
5.10.2 Definitions 81
5.10.3 Behaviour 81
5.10.4 Equivalent beam model 82
5.11 Vierendeel frames 85
5.11.1 Definitions 85
5.11.2 Behaviour 86
5.11.3 Equivalent beam model 86
5.12 Grillage models 87
5.13 3D models 88
5.14 Plastic collapse of frames 88
5.14.1 Prediction of collapse loads – limit analysis 88
5.14.2 Prediction of plastic collapse using an iterated elastic analysis 88
5.14.3 Prediction of plastic collapse using a finite element solution 89
5.14.4 Validation information 89

6 Plates in bending and slabs 91
6.1 Introduction 91
6.2 Plate bending elements 91
6.2.1 Plate bending elements 91
6.2.2 Validation information for biaxial plate bending 92
6.2.3 Output stresses and moments 92
6.2.4 Checking models for plates in bending 94
6.3 Concrete slabs 94
6.3.1 General 94
6.3.2 Element models for slab analysis 94
6.3.3 Reinforcing moments and forces for concrete slabs 95
6.3.4 Plate bending and shell element models 95
6.3.5 Shear lag effect 97
6.3.6 Plate grillage models for concrete slabs 98
6.3.7 Ribbed slabs 100
6.3.8 Plastic collapse of concrete slabs 0 the yield line method 101

7 Material models 103 7.1 Introduction 103
7.2 Linear elastic behaviour 103
7.2.1 General 103
7.2.2 Types of elastic behaviour 104
7.2.3 Values of elastic constants 104
7.2.4 Validation information for linear elastic materials 105
7.3 Non-linear material behaviour 106
7.3.1 Plasticity 106
7.3.2 Other non-linear constitutive relationships 108

8 Support models 109
8.1 Introduction 109 8.2 Modelling support fixity 109
8.2.1 General 109
8.2.2 Support requirements 109
8.2.3 Roller supports 110
8.2.4 Pin supports 112
8.2.5 Rotational restraint at a cantilever support 112
8.2.6 Rotational restraints at column bases 113
8.2.7 Slab support 114
8.3 Modelling the ground 114
8.3.1 General 114
8.3.2 The Winkler model for soil behaviour 115
8.3.3 Half space models 116
8.3.4 Finite element models 117
8.4 Foundation structures 118
8.4.1 Ground beams 118
8.4.2 Raft foundations 118
8.4.3 Piles 118

9 Loading 119
9.1 Introduction 119
9.2 Dead loading 119
9.3 Live loading 119
9.4 Wind loading 119
9.5 Earthquake loading 119
9.6 Fire 121
9.7 Temperature 121
9.7.1 General 121
9.7.2 Basic relationships 121
9.8 Influence lines for moving loads 121
9.8.1 General 121
9.8.2 Basic concept 121
9.8.3 Using influence lines 122
9.8.4 Defining influence lines 123
9.8.5 Validation information for the use of the Mueller-Breslau method
For defining influence lines 123
9.9 Prestressing 123
9.10 Impact loading 124
9.10.1 Gravity impact 124

10 Non-linear geometry 125
10.1 Introduction 125
10.1.1 Basic behaviour 125
10.1.2 Cantilever strut example – the P-∆effect 125
10.2 Modelling for geometric non-linearity 126
10.2.1 Using the non-linear geometry option in finite element packages 126
10.2.2 Use of the critical load ratio magnification factor 126
10.2.3 Case study – non-linear geometry analysis of a cantilever 127
10.2.4 Validation information for non-linear geometry effects 128
10.3 Critical load analysis of skeletal frames 129
10.3.1 The Euler critical load for single members 129
10.3.2 Non-sway instability of a column in a frame 130
10.3.3 The critical load ratio for an axially loaded member of a frame 130
10.3.4 Estimation of critical loads using eigenvalue extraction 131
10.3.5 Case study – eigenvalue analysis of a cantilever strut 131
10.4 Global critical load analysis of building structures 132

11 Dynamic behaviour 134
11.1 Introduction 134
11.2 Dynamic behaviour of a single mass and spring system 134
11.2.1 Governing equation 134
11.2.2 Validation information for equation (11.1) 135
11.2.3 Free undamped vibration 136
11.2.4 Damping 136
11.3 Multi-degree of freedom systems 137
11.3.1 Basic behaviour 137
11.3.2 Governing equation for multi-degree of freedom systems 138
11.3.3 Modelling for dynamic eigenvalue extraction 139
11.3.4 Verification of output for dynamic models 139
11.4 Resonance 139
11.4.1 Description 139
11.4.2 Systems subject to vibratory loading 140
11.5 Transient load 141
11.6 Checking models for natural frequencies 141
11.6.1 Single-span beams 141
11.6.2 The maximum deflection formula 141
11.6.3 Case study – use of equation (11.12) 142
11.6.4 Single mass and spring 142
11.6.5 Combinations of frequencies 143

12 Case studies 144
12.1 Case study 1 – vierendeel frame 144
12.1.1 General 144
12.1.2 Definition of the system to be modelled – the engineering model 144
12.1.3 Model development 144
12.1.4 The analysis model 146
12.1.5 Model validation 147
12.1.6 Results verification 147
12.1.7 Sensitivity analysis 153
12.1.8 Overall acceptance 155
12.1.9 Modelling review document 155
12.2 Case study 2 – four-storey building 155
12.2.1 General 155
12.2.2 Definition of the system to be modelled – the engineering model 155
12.2.3 Model development 157
12.2.4 Model validation 160
12.2.5 Results verification 162
12.2.6 Sensitivity analysis 169
12.2.7 Model review 170

Appendix – Tables of material and geometric properties 171

Bibliography 176

References 180

Index 183