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Foreword of the first edition xvi
Preface xvii
List of Abbreviations xix
Basic Concepts and Theorems of Structural Analysis 1
Introduction 1
Definitions 1
Structural Analysis and Design 4
General Concepts of Structural Analysis 4
Main Steps of Structural Analysis 4
Member Force and Displacements 6
Member Flexibility and Stiffness Matrices 8
Important Structural Theorems 11
Work and Energy 11
Castigliano's Theorem 14
Principle of Virtual Work 15
Contragradient Principle 18
Reciprocal Work Theorem 19
Exercises 20
Static Indeterminacy and Rigidity of Skeletal Structures 23
Introduction 23
Mathematical Model of a Skeletal Structure 25
Expansion Process for Determining the Degree of Statical Indeterminacy 27
Classical Formulae 27
A Unifying Function 28
An Expansion Process 28
An Intersection Theorem 29
A Method for Determining the DSI of Structures 30
The DSI of Structures: Special Methods 33
Space Structures and their Planar Drawings 35
Admissible Drawing of a Space Structure 35
The DSI of Frames 37
The DSI of Space Trusses 38
A Mixed Planar drawing - Expansion Method 39
Rigidity of Structures 41
Rigidity of Planar Trusses 45
Complete Matching Method 45
Decomposition Method 47
Grid-form Trusses with Bracings 48
Connectivity and Rigidity 50
Exercises 50
Optimal Force Method of Structural Analysis 53
Introduction 53
Formulation of the Force Method 54
Equilibrium Equations 54
Member Flexibility Matrices 57
Explicit Method for Imposing Compatibility 60
Implicit Approach for Imposing Compatibility 62
Structural Flexibility Matrices 64
Computational Procedure 64
Optimal Force Method 69
Force Method for the Analysis of Frame Structures 70
Minimal and Optimal Cycle Bases 71
Selection of Minimal and Subminimal Cycle Bases 72
Examples 79
Optimal and Suboptimal Cycle Bases 81
Examples 84
An Improved Turn-Back Method for the Formation of Cycle Bases 87
Examples 88
An Algebraic Graph-Theoretical Method for Cycle Basis Selection 91
Examples 93
Conditioning of the Flexibility Matrices 97
Condition Number 98
Weighted Graph and an Admissible Member 101
Optimally Conditioned Cycle Bases 101
Formulation of the Conditioning Problem 103
Suboptimally Conditioned Cycle Bases 104
Examples 107
Formation of B[subscript 0] and B[subscript 1] matrices 109
Generalised Cycle Bases of a Graph 115
Definitions 115
Minimal and Optimal Generalized Cycle Bases 118
Force Method for the Analysis of Pin-jointed Planar Trusses 119
Associate Graphs for Selection of a Suboptimal GCB 119
Minimal GCB of a Graph 122
Selection of a Subminimal GCB: Practical Methods 123
Force Method of Analysis for General Structures 125
Flexibility Matrices of Finite Elements 125
Algebraic Methods 131
Exercises 139
Optimal Displacement Method of Structural Analysis 141
Introduction 141
Formulation 142
Coordinate Systems Transformation 142
Element Stiffness Matrix using Unit Displacement Method 146
Element Stiffness Matrix using Castigliano's Theorem 150
Stiffness Matrix of a Structure 153
Stiffness Matrix of a Structure: An Algorithmic Approach 158
Transformation of Stiffness Matrices 160
Stiffness Matrix of a Bar Element 161
Stiffness Matrix of a Beam Element 163
Displacement Method of Analysis 166
Boundary Conditions 168
General Loading 169
Stiffness Matrix of a Finite Element 173
Stiffness Matrix of a Triangular Element 173
Computational Aspects of the Matrix Displacement Method 176
Algorithm 176
Example 178
Optimally Conditioned Cutset Bases 180
Mathematical Formulation of the Problem 181
Suboptimally Conditioned Cutset Bases 182
Algorithms 183
Example 184
Exercises 186
Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods 191
Introduction 191
Bandwidth Optimisation 192
Preliminaries 194
A Shortest Route Tree and its Properties 196
Nodal Ordering for Bandwidth Reduction 197
A Good Starting Node 198
Primary Nodal Decomposition 201
Transversal P of an SRT 201
Nodal Ordering 202
Example 202
Finite Element Nodal Ordering for Bandwidth Optimisation 203
Element Clique Graph Method (ECGM) 204
Skeleton Graph Method (SGM) 205
Element Star Graph Method (ESGM) 208
Element Wheel Graph Method (EWGM) 209
Partially Triangulated Graph Method (PTGM) 211
Triangulated Graph Method (TGM) 212
Natural Associate Graph Method (NAGM) 214
Incidence Graph Method (IGM) 217
Representative Graph Method (RGM) 218
Discussion of the Analysis of Algorithms 220
Computational Results 221
Discussions 223
Finite Element Nodal Ordering for Profile Optimisation 224
Introduction 224
Graph Nodal Numbering for Profile Reduction 226
Nodal Ordering with Element Clique Graph (NOECG) 230
Nodal Ordering with Skeleton Graph (NOSG) 230
Nodal Ordering with Element Star Graph (NOESG) 232
Nodal Ordering with Element Wheel Graph (NOEWG) 232
Nodal Ordering with Partially Triangulated Graph (NOPTG) 232
Nodal Ordering with Triangulated Graph (NOTG) 233
Nodal Ordering with Natural Associate Graph (NONAG) 233
Nodal Ordering with Incidence Graph (NOIG) 234
Nodal Ordering with Representative Graph (NORG) 234
Nodal Ordering with Element Clique Representative Graph (NOECRG) 236
Computational Results 236
Discussions 240
Element Ordering for Frontwidth Reduction 241
Definitions 242
Different Strategies for Frontwidth Reduction 244
Efficient Root Selection 246
Algorithm for Frontwidth Reduction 249
Complexity of the Algorithm 252
Computational Results 253
Discussions 256
Element Ordering for Bandwidth Optimisation of Flexibility Matrices 256
An Associate Graph 257
Distance Number of an Element 257
Element Ordering Algorithms 258
Bandwidth Reduction for Rectangular Matrices 260
Definitions 260
Algorithms 262
Examples 262
Bandwidth Reduction of Finite Element Models 264
Graph-Theoretical interpretation of Gaussian Elimination 266
Exercises 269
Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory Methods 273
Introduction 273
Adjacency Matrix of a Graph for Nodal Ordering 273
Basic Concepts and Definition 273
A Good Starting Node 277
Primary Nodal Decomposition 277
Transversal P of an SRT 277
Nodal Ordering 278
Example 278
Laplacian Matrix of a Graph for Nodal Ordering 279
Basic Concepts and Definitions 279
Nodal Numbering Algorithm 282
Example 283
A Hybrid Method for Ordering 284
Development of the Method 284
Numerical Results 285
Discussions 290
Exercises 291
Decomposition for Parallel Computing: Graph Theory Methods 293
Introduction 293
Earlier Works on Partitioning 294
Nested Dissection 294
A modified Level-Tree Separator Algorithm 294
Substructuring for Parallel Analysis of Skeletal Structures 295
Introduction 295
Substructuring Displacement Method 296
Methods of Substructuring 298
Main Algorithm for Substructuring 300
Examples 301
Simplified Algorithm for Substructuring 304
Greedy Type Algorithm 305
Domain Decomposition for Finite Element Analysis 305
Introduction 306
A Graph-Based Method for Subdomaining 307
Renumbering of Decomposed Finite Element Models 309
Complexity Analysis of the Graph-Based Method 310
Computational Results of the Graph-Based Method 312
Discussions on the Graph-Based Method 315
Engineering-Based Method for Subdomaining 316
Genre Structure Algorithm 317
Example 320
Complexity Analysis of the Engineering-Based Method 323
Computational Results of the Engineering-Based Method 325
Discussions 328
Substructuring: Force Method 330
Algorithm for the Force Method Substructuring 330
Examples 333
Substructuring for Dynamic Analysis 336
Modal Analysis of a Substructure 336
Partitioning of the Transfer Matrix H(w) 338
Dynamic Equation of the Entire Structure 338
Examples 342
Exercises 346
Decomposition for Parallel Computing: Algebraic Graph Theory Methods 349
Introduction 349
Algebraic Graph Theory for Subdomaining 350
Basic Definitions and Concepts 350
Lanczos Method 354
Recursive Spectral Bisection Partitioning Algorithm 359
Recursive Spectral Sequential-Cut Partitioning Algorithm 362
Recursive Spectral Two-way Partitioning Algorithm 362
Mixed Method for Subdomaining 363
Introduction 363
Mixed Method for Graph Bisection 364
Examples 369
Discussions 371
Spectral Bisection for Adaptive FEM; Weighted Graphs 371
Basic Concepts 372
Partitioning of Adaptive FE Meshes 374
Computational Results 376
Spectral Trisection of Finite Element Models 378
Criteria for Partitioning 378
Weighted Incidence Graphs for Finite Element Models 380
Graph Trisection Algorithm 381
Numerical Results 387
Discussions 389
Bisection of Finite Element Meshes using Ritz and Fiedler Vectors 389
Definitions and Algorithms 390
Graph Partitioning 390
Determination of Pseudo-Peripheral Nodes 391
Formation of an Approximate Fiedler Vector 391
Graph Coarsening 392
Domain Decomposition using Ritz and Fiedler Vectors 393
Illustrative Example 393
Numerical Results 397
Discussions 401
Exercises 401
Decomposition and Nodal Ordering of Regular Structures 403
Introduction 403
Definitions of Different Graph Products 404
Boolean Operations on Graphs 404
Cartesian Product of Two Graphs 404
Strong Cartesian Product of Two Graphs 407
Direct Product of Two Graphs 409
Eigenvalues of Graphs Matrices for Different Products 410
Kronecker Product 151
Cartesian Product 411
Strong Cartesian Product 414
Direct Product 417
Second Eigenvalues for Different Graph Products 419
Eigenvalues of A and L Matrices for Cycles and Paths 421
Computing [lambda][subscript 2] for Laplacian of Regular Models 424
Algorithm 425
Numerical Examples 425
Examples for Cartesian Product 426
Examples for Strong Cartesian Product 430
Examples for Direct Product 431
Spectral Method for Profile Reduction 433
Algorithm 433
Examples 433
Non-Compact Extended p-Sum 435
Exercises 436
Basic Concepts and Definitions of Graph Theory 437
Introduction 437
Basic Definitions 437
Vector Spaces Associated with a Graph 445
Matrices Associated with a Graph 448
Directed Graphs and their Matrices 456
Graphs Associated with Matrices 458
Planar Graphs: Euler's Polyhedron Formula 459
Maximal Matching in Bipartite Graphs 462
Greedy Algorithm and its Applications 465
Axiom System for a Matroid 465
Matroids Applied to Structural Mechanics 467
Cocycle Matroid of a Graph 470
Matroid for Null Basis of a Matrix 471
Combinatorial Optimisation: the Greedy Algorithm 472
Application of the Greedy Algorithm 473
Formation of Sparse Null Bases 474
References 477
Index 495
Index of Symbols 505
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