Optimal structural analysis

Optimal Structural Analysis

Optimal Structural Analysis

Second edition

A. Kaveh

Iran University of Science and Technology, Iran

John Wiley & Sons, Ltd Research Studies Press Limited

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Library of Congress Cataloging-in-Publication Data

Kaveh, A. (Ali), 1948-

Optimal structural analysis / A. Kaveh. -- 2nd ed.

p. cm. -- (RSP bird)

Includes bibliographical references and index.

ISBN-13: 978-0-470-03015-8 (cloth : alk. paper)

ISBN-10: 0-470-03015-1 (cloth : alk. paper)

1. Structural analysis (Engineering) 2. Structural optimization. I. Title. II. Series.

TA645.K38 2006

624.1'71--dc22

2006014662

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-03015-8 (HB)

ISBN-10 0-470-03015-1 (HB)

Typeset in 10/12pt Times New Roman by Laserwords Private Limited, Chennai, India

Printed and bound in Great Britain by TJ International, Padstow, Cornwall

This book is printed on acid-free paper responsibly manufactured from sustainable forestry

in which at least two trees are planted for each one used for paper production.

CONTENTS

Foreword of the first edition xvi

Preface xvii

List of Abbreviations xix

1. Basic Concepts and Theorems of Structural Analysis 1

1.1 Introduction 1

1.1.1 Definitions 1

1.1.2 Structural Analysis and Design 4

1.2 General Concepts of Structural Analysis 4

1.2.1 Main Steps of Structural Analysis 4

1.2.2 Member Force and Displacements 6

1.2.3 Member Flexibility and Stiffness Matrices 8

1.3 Important Structural Theorems 11

1.3.1 Work and Energy 11

1.3.2 Castigliano’s Theorem 14

1.3.3 Principle of Virtual Work 15

1.3.4 Contragradient Principle 18

1.3.5 Reciprocal Work Theorem 19

Exercises 20

2. Static Indeterminacy and Rigidity of Skeletal Structures 23

2.1 Introduction 23

2.2 Mathematical Model of a Skeletal Structure 25

2.3 Expansion Process for Determining the Degree of Statical Indeter￾minacy

27

2.3.1 Classical Formulae 27

2.3.2 A Unifying Function 28

CONTENTS vi

2.3.3 An Expansion Process 28

2.3.4 An Intersection Theorem 29

2.3.5 A Method for Determining the DSI of Structures 30

2.4 The DSI of Structures: Special Methods 33

2.5 Space Structures and their Planar Drawings 35

2.5.1 Admissible Drawing of a Space Structure 35

2.5.2 The DSI of Frames 37

2.5.3 The DSI of Space Trusses 38

2.5.4 A Mixed Planar drawing - Expansion Method 39

2.6 Rigidity of Structures 41

2.7 Rigidity of Planar Trusses 45

2.7.1 Complete Matching Method 45

2.7.2 Decomposition Method 47

2.7.3 Grid-form Trusses with Bracings 48

2.8 Connectivity and Rigidity 50

Exercises 50

3. Optimal Force Method of Structural Analysis 53

3.1 Introduction 53

3.2 Formulation of the Force Method 54

3.2.1 Equilibrium Equations 54

3.2.2 Member Flexibility Matrices 57

3.2.3 Explicit Method for Imposing Compatibility 60

3.2.4 Implicit Approach for Imposing Compatibility 62

3.2.5 Structural Flexibility Matrices 64

3.2.6 Computational Procedure 64

3.2.7 Optimal Force Method 69

3.3 Force Method for the Analysis of Frame Structures 70

3.3.1 Minimal and Optimal Cycle Bases 71

3.3.2 Selection of Minimal and Subminimal Cycle Bases 72

CONTENTS vii

3.3.3 Examples 79

3.3.4 Optimal and Suboptimal Cycle Bases 81

3.3.5 Examples 84

3.3.6 An Improved Turn-Back Method for the Formation of

Cycle Bases

87

3.3.7 Examples 88

3.3.8 An Algebraic Graph-Theoretical Method for Cycle Basis

Selection

91

3.3.9 Examples 93

3.4 Conditioning of the Flexibility Matrices 97

3.4.1 Condition Number 98

3.4.2 Weighted Graph and an Admissible Member 101

3.4.3 Optimally Conditioned Cycle Bases 101

3.4.4 Formulation of the Conditioning Problem 103

3.4.5 Suboptimally Conditioned Cycle Bases 104

3.4.6 Examples 107

3.4.7 Formation of B0 and B1 matrices 109

3.5 Generalised Cycle Bases of a Graph 115

3.5.1 Definitions 115

3.5.2 Minimal and Optimal Generalized Cycle Bases 118

3.6 Force Method for the Analysis of Pin-jointed Planar Trusses 119

3.6.1 Associate Graphs for Selection of a Suboptimal GCB 119

3.6.2 Minimal GCB of a Graph 122

3.6.3 Selection of a Subminimal GCB: Practical Methods 123

3.7 Force Method of Analysis for General Structures 125

3.7.1 Flexibility Matrices of Finite Elements 125

3.7.2 Algebraic Methods 131

Exercises 139

CONTENTS viii

4. Optimal Displacement Method of Structural Analysis 141

4.1 Introduction 141

4.2 Formulation 142

4.2.1 Coordinate Systems Transformation 142

4.2.2 Element Stiffness Matrix using Unit Displacement Method 146

4.2.3 Element Stiffness Matrix using Castigliano’s Theorem 150

4.2.4 Stiffness Matrix of a Structure 153

4.2.5 Stiffness Matrix of a Structure: An Algorithmic Approach 158

4.3 Transformation of Stiffness Matrices 160

4.3.1 Stiffness Matrix of a Bar Element 161

4.3.2 Stiffness Matrix of a Beam Element 163

4.4 Displacement Method of Analysis 166

4.4.1 Boundary Conditions 168

4.4.2 General Loading 169

4.5 Stiffness Matrix of a Finite Element 173

4.5.1 Stiffness Matrix of a Triangular Element 173

4.6 Computational Aspects of the Matrix Displacement Method 176

4.6.1 Algorithm 176

4.6.2 Example 178

4.7 Optimally Conditioned Cutset Bases 180

4.7.1 Mathematical Formulation of the Problem 181

4.7.2 Suboptimally Conditioned Cutset Bases 182

4.7.3 Algorithms 183

4.7.4 Example 184

Exercises 186

5. Ordering for Optimal Patterns of Structural Matrices: Graph

Theory Methods

191

5.1 Introduction 191

5.2 Bandwidth Optimisation 192

CONTENTS ix

5.3 Preliminaries 194

5.4 A Shortest Route Tree and its Properties 196

5.5 Nodal Ordering for Bandwidth Reduction 197

5.5.1 A Good Starting Node 198

5.5.2 Primary Nodal Decomposition 201

5.5.3 Transversal P of an SRT 201

5.5.4 Nodal Ordering 202

5.5.5 Example 202

5.6 Finite Element Nodal Ordering for Bandwidth Optimisation 203

5.6.1 Element Clique Graph Method (ECGM) 204

5.6.2 Skeleton Graph Method (SGM) 205

5.6.3 Element Star Graph Method (ESGM) 208

5.6.4 Element Wheel Graph Method (EWGM) 209

5.6.5 Partially Triangulated Graph Method (PTGM) 211

5.6.6 Triangulated Graph Method (TGM) 212

5.6.7 Natural Associate Graph Method (NAGM) 214

5.6.8 Incidence Graph Method (IGM) 217

5.6.9 Representative Graph Method (RGM) 218

5.6.10 Discussion of the Analysis of Algorithms 220

5.6.11 Computational Results 221

5.6.12 Discussions 223

5.7 Finite Element Nodal Ordering for Profile Optimisation 224

5.7.1 Introduction 224

5.7.2 Graph Nodal Numbering for Profile Reduction 226

5.7.3 Nodal Ordering with Element Clique Graph (NOECG) 230

5.7.4 Nodal Ordering with Skeleton Graph (NOSG) 230

5.7.5 Nodal Ordering with Element Star Graph (NOESG) 232

5.7.6 Nodal Ordering with Element Wheel Graph (NOEWG) 232

5.7.7 Nodal Ordering with Partially Triangulated Graph

(NOPTG)

232

CONTENTS x

5.7.8 Nodal Ordering with Triangulated Graph (NOTG) 233

5.7.9 Nodal Ordering with Natural Associate Graph

(NONAG)

233

5.7.10 Nodal Ordering with Incidence Graph (NOIG) 234

5.7.11 Nodal Ordering with Representative Graph (NORG) 234

5.7.12 Nodal Ordering with Element Clique Representative

Graph (NOECRG)

236

5.7.13 Computational Results 236

5.7.14 Discussions 240

5.8 Element Ordering for Frontwidth Reduction 241

5.8.1 Definitions 242

5.8.2 Different Strategies for Frontwidth Reduction 244

5.8.3 Efficient Root Selection 246

5.8.4 Algorithm for Frontwidth Reduction 249

5.8.5 Complexity of the Algorithm 252

5.8.6 Computational Results 253

5.8.7 Discussions 256

5.9 Element Ordering for Bandwidth Optimisation of Flexibility

Matrices

256

5.9.1 An Associate Graph 257

5.9.2 Distance Number of an Element 257

5.9.3 Element Ordering Algorithms 258

5.10 Bandwidth Reduction for Rectangular Matrices 260

5.10.1 Definitions 260

5.10.2 Algorithms 262

5.10.3 Examples 262

5.10.4 Bandwidth Reduction of Finite Element Models 264

5.11 Graph-Theoretical interpretation of Gaussian Elimination 266

Exercises 269

CONTENTS xi

6. Ordering for Optimal Patterns of Structural Matrices:

Algebraic Graph Theory Methods

273

6.1 Introduction 273

6.2 Adjacency Matrix of a Graph for Nodal Ordering 273

6.2.1 Basic Concepts and Definition 273

6.2.2 A Good Starting Node 277

6.2.3 Primary Nodal Decomposition 277

6.2.4 Transversal P of an SRT 277

6.2.5 Nodal Ordering 278

6.2.6 Example 278

6.3 Laplacian Matrix of a Graph for Nodal Ordering 279

6.3.1 Basic Concepts and Definitions 279

6.3.2 Nodal Numbering Algorithm 282

6.3.3 Example 283

6.4 A Hybrid Method for Ordering 284

6.4.1 Development of the Method 284

6.4.2 Numerical Results 285

6.4.3 Discussions 290

Exercises 291

7. Decomposition for Parallel Computing: Graph Theory Methods 293

7.1 Introduction 293

7.2 Earlier Works on Partitioning 294

7.2.1 Nested Dissection 294

7.2.2 A modified Level-Tree Separator Algorithm 294

7.3 Substructuring for Parallel Analysis of Skeletal Structures 295

7.3.1 Introduction 295

7.3.2 Substructuring Displacement Method 296

7.3.3 Methods of Substructuring 298

7.3.4 Main Algorithm for Substructuring 300

CONTENTS xii

7.3.5 Examples 301

7.3.6 Simplified Algorithm for Substructuring 304

7.3.7 Greedy Type Algorithm 305

7.4 Domain Decomposition for Finite Element Analysis 305

7.4.1 Introduction 306

7.4.2 A Graph-Based Method for Subdomaining 307

7.4.3 Renumbering of Decomposed Finite Element Models 309

7.4.4 Complexity Analysis of the Graph-Based Method 310

7.4.5 Computational Results of the Graph-Based Method 312

7.4.6 Discussions on the Graph-Based Method 315

7.4.7 Engineering-Based Method for Subdomaining 316

7.4.8 Genre Structure Algorithm 317

7.4.9 Example 320

7.4.10 Complexity Analysis of the Engineering-Based Method 323

7.4.11 Computational Results of the Engineering-Based Method 325

7.4.12 Discussions 328

7.5 Substructuring: Force Method 330

7.5.1 Algorithm for the Force Method Substructuring 330

7.5.2 Examples 333

7.6 Substructuring for Dynamic Analysis 336

7.6.1 Modal Analysis of a Substructure 336

7.6.2 Partitioning of the Transfer Matrix H(w) 338

7.6.3 Dynamic Equation of the Entire Structure 338

7.6.4 Examples 342

Exercises 346

CONTENTS xiii

8. Decomposition for Parallel Computing: Algebraic Graph

Theory Methods

349

8.1 Introduction 349

8.2 Algebraic Graph Theory for Subdomaining 350

8.2.1 Basic Definitions and Concepts 350

8.2.2 Lanczos Method 354

8.2.3 Recursive Spectral Bisection Partitioning Algorithm 359

8.2.4 Recursive Spectral Sequential-Cut Partitioning Algorithm 362

8.2.5 Recursive Spectral Two-way Partitioning Algorithm 362

8.3 Mixed Method for Subdomaining 363

8.3.1 Introduction 363

8.3.2 Mixed Method for Graph Bisection 364

8.3.3 Examples 369

8.3.4 Discussions 371

8.4 Spectral Bisection for Adaptive FEM; Weighted Graphs 371

8.4.1 Basic Concepts 372

8.4.2 Partitioning of Adaptive FE Meshes 374

8.4.3 Computational Results 376

8.5 Spectral Trisection of Finite Element Models 378

8.5.1 Criteria for Partitioning 378

8.5.2 Weighted Incidence Graphs for Finite Element Models 380

8.5.3 Graph Trisection Algorithm 381

8.5.4 Numerical Results 387

8.5.5 Discussions 389

8.6 Bisection of Finite Element Meshes using Ritz and Fiedler Vectors 389

8.6.1 Definitions and Algorithms 390

8.6.2 Graph Partitioning 390

8.6.3 Determination of Pseudo-Peripheral Nodes 391

8.6.4 Formation of an Approximate Fiedler Vector 391

8.6.5 Graph Coarsening 392