Modern structural analysis

Modelling process and guidance

Iain A. MacLeod

Published by Thomas Telford Publishing, Thomas Telford Ltd, 1 Heron Quay, London E14 4JD.

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First published 2005

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ISBN: 0 7277 3279 X

# Thomas Telford Limited 2005

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Typeset by Academic Technical, BristolPrinted and bound in Great Britain by MPG Books, Bodmin

Acknowledgements

This book follows on from Analytical modelling of structural systems published in1990. I was involved in a working group of the Institution of Structural Engineerswhich resulted in the 2002 publication of the booklet The use of computers forengineering calculations. A number of ideas about modelling process which Ihave used in the book arose from the work of the group and I acknowledgewith thanks the contributions of Andrew Bond, Peter Gardner, Peter Harris,Bill Harvey, Nigel Knowles and Brain Neale to these ideas.I am specially grateful to Sam Thorburn, Yaqub Raq and Steven McKerlie

who read a draft of the book and provided me with many useful suggestions.I record my thanks to the following people for advice and information on the

production of this book: Kamal Badrah, Callum Bennett, Prabakhara Bhatt,Roy Cairns, Andrew Clark, Graeme Harley, Paul Lyons, John Morrison,Matthew Petticrew, Ian Salisbury, David Scott, Richard Wood, Howard Wrightand Karoly Zalka.Finally my thanks to Barbara, Mairi, Alastair and Iseabail for their love and

support.

Foreword

This interesting book promotes a new way of looking at structural analysis. Itsuggests that the ability to work with the model (as distinct from the solutionprocess) is a primary issue which should be formally addressed in practice andin education. The content is focused on modelling issues and I know of no othertext which does this so comprehensively.The early chapters contain much advice necessary to help the reader establish

how to formulate a numerical model that might be capable of simulating theperformance of the actual structural system under investigation. The later chaptersinclude a good outline of the issues involved in modelling of structures using niteelements. The two case studies given at the end of the book are a good device to putthe excellent advice given in the earlier sections into some perspective for thereader.I found it most useful to have in the same book a reminder of the theoretical

basis of the full range of nite element types and a sound method as to how toemploy analysis as a reective tool towards a better understanding of structuralbehaviour. The rigorous treatment for the process of validation of a model ismost enlightening as is that outlined for verication of the results. After all, theiterative process of model validation and output verication are the main activitiesfor gaining a true understanding of structural behaviour.My own experience working with Buro Happold tells me that robust structural

design requires the willingness to develop an understanding of structural behaviourwith a questioning mind. In most consulting oces, current practice is to under-take this using nite element models of increasing complexity as understandingof the problem at hand grows. Iain MacLeod describes clearly how to build upthis understanding using sensitivity analysis and simplied loadings to test validityagainst expectations from parallel calculation and modelling experiences. It isargued that risk will be reduced in practice if there is a rigorous analytical processthat reects the realities of current engineering practice in most oces.Most structures are of a reasonably conventional type and use well tried framing

systems. Substantial experience already exists on their likely performance so handcalculations based on structural theory can be done to initiate formulation of themodel or to act as a check on the results. However, even advanced classicalmethods struggle to model the sophistication of load paths in redundant or non-linear structures where individual stiness, material response and denition ofrestraint determines structural performance. In this case, I have found thatcomparison of the output of simplied analytical results with physical modelsvery useful as an addition to classical calculation as advocated in the secondchapter.

The book is thus both a useful reference for the practitioner and a comprehen-sive learning guide for the student. It builds on the publication by the Institution ofStructural Engineers Guidelines for the Use of Computers for Engineering Calcula-tion published in 2002. Its carefully constructed content successfully redresses theimbalance in risk between the nite element process based around generallydeterminate calculation output that has itself been derived from a possibly non-determinate understanding of the actual modelling process. In the Introduction,the author suggests that all structural engineers and all civil engineers who usestructural analysis will nd the contents of the book to be useful. I think that heis right.

Michael Dickson FIStructEDirector, Design and Technology Board, Buro HappoldPresident, Institution of Structural Engineers 200506

FOREWORD v

Contents

Acknowledgements iiiForeword iv

1 Introduction 11.1 Scope and denitions 11.2 Why modern structural analysis? 11.3 Issues for practice 21.4 Issues for education 2

1.4.1 The wider context 31.5 Finite elements 31.6 Accuracy of the information provided in the text 41.7 Website 4

2 Basic principles 52.1 Managing the analysis process 5

2.1.1 Quality management system 52.1.2 Use the modelling process 52.1.3 Competence 5

2.2 Modelling principles 52.2.1 Use the simplest practical model 52.2.2 Estimate results before you analyse 62.2.3 Increment the complexity 62.2.4 When you get results, assume that they may be errors 62.2.5 Troubleshooting for errors 62.2.6 Relationship between the analysis model and the design

code of practice 72.2.7 Case study the Ronan Point collapse 8

2.3 Principles in the use of structural mechanics 82.3.1 Local and resultant stresses the St Venant principle 82.3.2 Principle of superposition 92.3.3 Lower bound theorem in plasticity 10

2.4 Understanding structural behaviour 112.4.1 General 112.4.2 Model validation 112.4.3 Results verication and checking models 112.4.4 Sensitivity analysis 11

2.4.5 Solution comparisons 132.4.6 Convergence analysis 142.4.7 Identify patterns 142.4.8 Mathematics 142.4.9 Physical modelling and testing 14

3 The modelling process 153.1 Overview of the modelling process 15

3.1.1 General 153.1.2 Representations of the modelling process 153.1.3 Validation and verication 173.1.4 Error and uncertainty 17

3.2 Dening the system to be modelled 183.3 The model development process 18

3.3.1 Conceptual and computational models 183.3.2 Model options 19

3.4 Validation of the analysis model 193.4.1 Validation process 193.4.2 Validating the conceptual model 203.4.3 Validating the computational model 20

3.5 The solution process 213.5.1 Selecting software 213.5.2 Software validation and verication 213.5.3 Truncation error, ill-conditioning 22

3.6 Verifying the results 223.6.1 Acceptance criteria for results 223.6.2 Verication process 223.6.3 Checking models 233.6.4 Checking loadcase 25

3.7 The modelling review 253.7.1 Sensitivity analysis 253.7.2 Overall acceptance of the results 253.7.3 The modelling review document 25

3.8 Case studies 263.8.1 The Tay Bridge disaster 263.8.2 The Hartford Civic Center roof collapse 273.8.3 The Sleipner platform collapse 27

4 Modelling with nite elements 294.1 Introduction 294.2 Elements 29

4.2.1 Constitutive relationships 294.2.2 Line elements 304.2.3 Surface elements 30

viii MODERN STRUCTURAL ANALYSIS

4.2.4 Volume elements 324.2.5 Joint elements 334.2.6 Basic principles for the derivation of nite element

stiness matrices 344.3 Mesh renement 36

4.3.1 Discretisation error 364.3.2 Convergence 364.3.3 Singularities 374.3.4 Benchmark tests 384.3.5 Case study mesh layouts for a cantilever bracket 384.3.6 Meshing principles 39

4.4 Case study convergence analysis of a plane stress cantileverbeam model 414.4.1 General 414.4.2 The context 414.4.3 Elements used in the convergence analysis 414.4.4 Reference solution 424.4.5 Convergence parameters 434.4.6 Meshes 444.4.7 Results 444.4.8 Overview 45

4.5 Constraints 464.5.1 General 464.5.2 Rigid constraint conditions 464.5.3 Constraint equations 47

4.6 Symmetry 484.6.1 General 484.6.2 Mirror symmetry 484.6.3 Symmetry checking 50

5 Skeletal frames modelling with line elements 515.1 Introduction 51

5.1.1 Members and elements 525.2 Bending 52

5.2.1 Background 525.2.2 Behaviour 525.2.3 Basic relationships for bending 535.2.4 Symmetric and asymmetric bending 535.2.5 Shear in bending 545.2.6 Combined bending and shear 565.2.7 Validation information for the engineers theory of

bending 565.3 Axial eects 58

5.3.1 Behaviour 585.3.2 Basic relationships 595.3.3 Validation information 59

CONTENTS ix

5.4 Torsion 605.4.1 Behaviour 605.4.2 Basic relationships for shear torsion 615.4.3 Basic relationships for bending torsion 625.4.4 Combined torsion 635.4.5 Validation information for torsion 63

5.5 Bar elements and beam elements 645.5.1 Bar elements 645.5.2 Engineering beam elements 645.5.3 Higher-order beam elements 66

5.6 Connections 665.6.1 Basic connection types 665.6.2 Treatment of the nite depth of a beam using rigid links 685.6.3 Modelling beam-to-column connections in steelwork 685.6.4 Connections in concrete 715.6.5 Eccentricity of members at a joint 72

5.7 Distribution of load in skeletal frames 745.7.1 Vertical load in beam systems 745.7.2 Distribution of lateral load 75

5.8 Modelling curved and non-uniform members 755.8.1 Curved members 755.8.2 Case study modelling of curved beams 755.8.3 Modelling members with non-uniform cross section 775.8.4 Case study tapered cantilever 775.8.5 Cantilever with a tapered sot 795.8.6 Haunched beams 79

5.9 Triangulated frames 795.9.1 Modelling issues 795.9.2 Euler buckling eect of members 80

5.10 Parallel chord trusses 805.10.1 General 805.10.2 Denitions 815.10.3 Behaviour 815.10.4 Equivalent beam model 82

5.11 Vierendeel frames 855.11.1 Denitions 855.11.2 Behaviour 865.11.3 Equivalent beam model 86

5.12 Grillage models 875.13 3D models 885.14 Plastic collapse of frames 88

5.14.1 Prediction of collapse loads limit analysis 885.14.2 Prediction of plastic collapse using an iterated elastic

analysis 885.14.3 Prediction of plastic collapse using a nite element

solution 895.14.4 Validation information 89

x MODERN STRUCTURAL ANALYSIS

6 Plates in bending and slabs 916.1 Introduction 916.2 Plate bending elements 91

6.2.1 Plate bending element basics 916.2.2 Validation information for biaxial plate bending 926.2.3 Output stresses and moments 926.2.4 Checking models for plates in bending 94

6.3 Concrete slabs 946.3.1 General 946.3.2 Element models for slab analysis 946.3.3 Reinforcing moments and forces for concrete slabs 956.3.4 Plate bending and shell element models 956.3.5 Shear lag eect 976.3.6 Plate grillage models for concrete slabs 986.3.7 Ribbed slabs 1006.3.8 Plastic collapse of concrete slabs the yield line method 101

7 Material models 1037.1 Introduction 1037.2 Linear elastic behaviour 103

7.2.1 General 1037.2.2 Types of elastic behaviour 1047.2.3 Values of elastic constants 1047.2.4 Validation information for linear elastic materials 105

7.3 Non-linear material behaviour 1067.3.1 Plasticity 1067.3.2 Other non-linear constitutive relationships 108

8 Support models 1098.1 Introduction 1098.2 Modelling support xity 109

8.2.1 General 1098.2.2 Support requirements 1098.2.3 Roller supports 1108.2.4 Pin supports 1128.2.5 Rotational restraint at a cantilever support 1128.2.6 Rotational restraints at column bases 1138.2.7 Slab supports 114

8.3 Modelling the ground 1148.3.1 General 1148.3.2 The Winkler model for soil behaviour 1158.3.3 Half space models 1168.3.4 Finite element models 117

8.4 Foundation structures 1188.4.1 Ground beams 118

CONTENTS xi

8.4.2 Raft foundations 1188.4.3 Piles 118

9 Loading 1199.1 Introduction 1199.2 Dead loading 1199.3 Live loading 1199.4 Wind loading 1199.5 Earthquake loading 1199.6 Fire 1219.7 Temperature 121

9.7.1 General 1219.7.2 Basic relationships 121

9.8 Inuence lines for moving loads 1219.8.1 General 1219.8.2 Basic concept 1229.8.3 Using inuence lines 1229.8.4 Dening inuence lines 1239.8.5 Validation information for the use of the MuellerBreslau

method for dening inuence lines 1239.9 Prestressing 1239.10 Impact loading 124

9.10.1 Gravity impact 124

10 Non-linear geometry 12510.1 Introduction 125

10.1.1 Basic behaviour 12510.1.2 Cantilever strut example the P- eect 125

10.2 Modelling for geometric non-linearity 12610.2.1 Using the non-linear geometry option in nite element

packages 12610.2.2 Use of the critical load ratio magnication factor 12610.2.3 Case study non-linear geometry analysis of a

cantilever 12710.2.4 Validation information for non-linear geometry eects 128

10.3 Critical load analysis of skeletal frames 12910.3.1 The Euler critical load for single members 12910.3.2 Non-sway instability of a column in a frame 13010.3.3 The critical load ratio for an axially loaded member of a

frame 13010.3.4 Estimation of critical loads using eigenvalue extraction 13110.3.5 Case study eigenvalue analysis of a cantilever strut 131

10.4 Global critical load analysis of building structures 132

xii MODERN STRUCTURAL ANALYSIS

11 Dynamic behaviour 13411.1 Introduction 13411.2 Dynamic behaviour of a single mass and spring system 134

11.2.1 Governing equation 13411.2.2 Validation information for equation (11.1) 13511.2.3 Free undamped vibration 13611.2.4 Damping 136

11.3 Multi-degree of freedom systems 13711.3.1 Basic behaviour 13711.3.2 Governing equation for multi-degree of freedom

systems 13811.3.3 Modelling for dynamic eigenvalue extraction 13911.3.4 Verication of output for dynamic models 139

11.4 Resonance 13911.4.1 Description 13911.4.2 Systems subject to vibratory loading 140

11.5 Transient load 14111.6 Checking models for natural frequencies 141

11.6.1 Single-span beams 14111.6.2 The maximum deection formula 14111.6.3 Case study use of equation (11.12) 14211.6.4 Single mass and spring 14211.6.5 Combinations of frequencies 143

12 Case studies 14412.1 Case study 1 vierendeel frame 144

12.1.1 General 14412.1.2 Denition of the system to be modelled the engineering

model 14412.1.3 Model development 14412.1.4 The analysis model 14612.1.5 Model validation 14712.1.6 Results verication 14712.1.7 Sensitivity analysis 15312.1.8 Overall acceptance 15512.1.9 Modelling review document 155

12.2 Case study 2 four-storey building 15512.2.1 General 15512.2.2 Denition of the system to be modelled the engineering

model 15512.2.3 Model development 15712.2.4 Model validation 16012.2.5 Results verication 16212.2.6 Sensitivity analysis 16912.2.7 Model review 170

CONTENTS xiii

Appendix Tables of material and geometric properties 171

Bibliography 176

References 180

Index 183

xiv MODERN STRUCTURAL ANALYSIS

Title PageAcknowledgementsForewordContents