Computational Structural Analysis and Finite Element Methods
Computational Structural Analysis
and Finite Element Methods
Contents
Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs 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 5
1.2.1 Main Steps of Structural Analysis 5
1.2.2 Member Forces and Displacements . 6
1.2.3 Member Flexibility and Stiffness Matrices 7
1.3 Important Structural Theorems . 11
1.3.1 Work and Energy 11
1.3.2 Castigliano’s Theorems . 13
1.3.3 Principle of Virtual Work 13
1.3.4 Contragradient Principle 16
1.3.5 Reciprocal Work Theorem 17
1.4 Basic Concepts and Definitions of Graph Theory 18
1.4.1 Basic Definitions 19
1.4.2 Definition of a Graph 19
1.4.3 Adjacency and Incidence 20
1.4.4 Graph Operations 20
1.4.5 Walks, Trails and Paths . 21
1.4.6 Cycles and Cutsets . 22
1.4.7 Trees, Spanning Trees and Shortest Route Trees 23
1.4.8 Different Types of Graphs 23
1.5 Vector Spaces Associated with a Graph . 25
1.5.1 Cycle Space 26
1.5.2 Cutset Space 26
1.5.3 Orthogonality Property . 26
1.5.4 Fundamental Cycle Bases
1.6 Matrices Associated with a Graph 28
1.6.1 Matrix Representation of a Graph 29
1.6.2 Cycle Bases Matrices 32
1.6.3 Special Patterns for Fundamental Cycle Bases 33
1.6.4 Cutset Bases Matrices 34
1.6.5 Special Patterns for Fundamental Cutset Bases 34
1.7 Directed Graphs and Their Matrices 35
2 Optimal Force Method: Analysis of Skeletal Structures 39
2.1 Introduction 39
2.2 Static Indeterminacy of Structures 40
2.2.1 Mathematical Model of a Skeletal Structure . 41
2.2.2 Expansion Process for Determining the Degree of Static Indeterminacy . 42
2.3 Formulation of the Force Method 46
2.3.1 Equilibrium Equations 46
2.3.2 Member Flexibility Matrices 49
2.3.3 Explicit Method for Imposing Compatibility . 52
2.3.4 Implicit Approach for Imposing Compatibility 53
2.3.5 Structural Flexibility Matrices 55
2.3.6 Computational Procedure 55
2.3.7 Optimal Force Method . 60
2.4 Force Method for the Analysis of Frame Structures . 60
2.4.1 Minimal and Optimal Cycle Bases 61
2.4.2 Selection of Minimal and Subminimal Cycle Bases 62
2.4.3 Examples . 67
2.4.4 Optimal and Suboptimal Cycle Bases 69
2.4.5 Examples . 72
2.4.6 An Improved Turn Back Method for the Formation of Cycle Bases 75
2.4.7 Examples . 76
2.4.8 Formation of B 0 and B 1 Matrices 78
2.5 Generalized Cycle Bases of a Graph 82
2.5.1 Definitions 83
2.5.2 Minimal and Optimal Generalized Cycle Bases 85
2.6 Force Method for the Analysis of Pin-Jointed Planar Trusses . 86
2.6.1 Associate Graphs for Selection of a Suboptimal GCB . 86
2.6.2 Minimal GCB of a Graph 89
2.6.3 Selection of a Subminimal GCB: Practical Methods 89
2.7 Algebraic Force Methods of Analysis 91
2.7.1 Algebraic Methods . 91
3 Optimal Displacement Method of Structural Analysis 101
3.1 Introduction 101
3.2 Formulation 101
3.2.1 Coordinate Systems Transformation . 102
3.2.2 Element Stiffness Matrix Using Unit Displacement Method 105
3.2.3 Element Stiffness Matrix Using Castigliano’s Theorem 109
3.2.4 The Stiffness Matrix of a Structure 111
3.2.5 Stiffness Matrix of a Structure;an Algorithmic Approach 116
3.3 Transformation of Stiffness Matrices 118
3.3.1 Stiffness Matrix of a Bar Element 118
3.3.2 Stiffness Matrix of a Beam Element . 120
3.4 Displacement Method of Analysis 122
3.4.1 Boundary Conditions 124
3.4.2 General Loading 125
3.5 Stiffness Matrix of a Finite Element 128
3.5.1 Stiffness Matrix of a Triangular Element 129
3.6 Computational Aspects of the Matrix Displacement Method 132
4 Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods . 137
4.1 Introduction 137
4.2 Bandwidth Optimisation . 138
4.3 Preliminaries 140
4.4 A Shortest Route Tree and Its Properties . 142
4.5 Nodal Ordering for Bandwidth Reduction 142
4.5.1 A Good Starting Node 143
4.5.2 Primary Nodal Decomposition 145
4.5.3 Transversal P of an SRT 146
4.5.4 Nodal Ordering . 146
4.5.5 Example 147
4.6 Finite Element Nodal Ordering for Bandwidth Optimisation 147
4.6.1 Element Clique Graph Method (ECGM) 149
4.6.2 Skeleton Graph Method (SkGM) 149
4.6.3 Element Star Graph Method (EStGM) 150
4.6.4 Element Wheel Graph Method (EWGM) 151
4.6.5 Partially Triangulated Graph Method (PTGM) 152
4.6.6 Triangulated Graph Method (TGM) 153
4.6.7 Natural Associate Graph Method (NAGM) 153
4.6.8 Incidence Graph Method (IGM) 155
4.6.9 Representative Graph Method (RGM) 156
4.6.10 Computational Results . 157
4.6.11 Discussions 158
4.7 Finite Element Nodal Ordering for Profile Optimisation 160
4.7.1 Introduction 160
4.7.2 Graph Nodal Numbering for Profile Reduction 162
4.7.3 Nodal Ordering with Element Clique Graph (NOECG) 164
4.7.4 Nodal Ordering with Skeleton Graph (NOSG) 165
4.7.5 Nodal Ordering with Element Star Graph (NOESG) 166
4.7.6 Nodal Ordering with Element Wheel Graph (NOEWG) 166
4.7.7 Nodal Ordering with Partially Triangulated Graph (NOPTG) 167
4.7.8 Nodal Ordering with Triangulated Graph (NOTG) 167
4.7.9 Nodal Ordering with Natural Associate Graph (NONAG) 168
4.7.10 Nodal Ordering with Incidence Graph (NOIG) 168
4.7.11 Nodal Ordering with Representative Graph (NORG) 168
4.7.12 Nodal Ordering with Element Clique Representative Graph (NOECRG) . 170
4.7.13 Computational Results . 170
4.7.14 Discussions 170
4.8 Element Ordering for Frontwidth Reduction 171
4.9 Element Ordering for Bandwidth Optimisation of Flexibility Matrices 174
4.9.1 An Associate Graph . 174
4.9.2 Distance Number of an Element . 175
4.9.3 Element Ordering Algorithms 175
4.10 Bandwidth Reduction for Rectangular Matrices 177
4.10.1 Definitions 177
4.10.2 Algorithms . 178
4.10.3 Examples 179
4.10.4 Bandwidth Reduction of Finite Element Models 181
4.11 Graph-Theoretical Interpretation of Gaussian Elimination 182
5 Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory and Meta-heuristic Based Methods 187
5.1 Introduction 187
5.2 Adjacency Matrix of a Graph for Nodal Ordering 187
5.2.1 Basic Concepts and Definitions . 187
5.2.2 A Good Starting Node 190
5.2.3 Primary Nodal Decomposition 190
5.2.4 Transversal P of an SRT 191
5.2.5 Nodal Ordering 191
5.2.6 Example 192
5.3 Laplacian Matrix of a Graph for Nodal Ordering 192
5.3.1 Basic Concepts and Definitions . 192
5.3.2 Nodal Numbering Algorithm 196
5.3.3 Example 196
5.4 A Hybrid Method for Ordering . 196
5.4.1 Development of the Method . 197
5.4.2 Numerical Results 198
5.4.3 Discussions 199
5.5 Ordering via Charged System Search Algorithm 203
5.5.1 Charged System Search . 203
5.5.2 The CSS Algorithm for Nodal Ordering . 208
5.5.3 Numerical Examples 211
6 Optimal Force Method for FEMs: Low Order Elements 215
6.1 Introduction 215
6.2 Force Method for Finite Element Models: Rectangular and Triangular Plane Stress and Plane Strain Elements 215
6.2.1 Member Flexibility Matrices 216
6.2.2 Graphs Associated with FEMs 220
6.2.3 Pattern Corresponding to the Self Stress Systems . 221
6.2.4 Selection of Optimal γ -Cycles Corresponding
to Type II Self Stress Systems 224
6.2.5 Selection of Optimal Lists 225
6.2.6 Numerical Examples 227
6.3 Finite Element Analysis Force Method: Triangular and Rectangular Plate Bending Elements 230
6.3.1 Graphs Associated with Finite Element Models 233
6.3.2 Subgraphs Corresponding to Self-Equilibrating Systems 233
6.3.3 Numerical Examples 240
6.4 Force Method for Three Dimensional Finite Element Analysis 244
6.4.1 Graphs Associated with Finite Element Model 244
6.4.2 The Pattern Corresponding to the Self Stress Systems . 245
6.4.3 Relationship Between γ (S) and b 1 (A(S)) 248
6.4.4 Selection of Optimal γ -Cycles Corresponding to Type II Self Stress Systems 251
6.4.5 Selection of Optimal Lists 252
6.4.6 Numerical Examples 254
6.5 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Brick Element 257
6.5.1 Definition of the Independent Element Forces 258
6.5.2 Flexibility Matrix of an Element 259
6.5.3 Graphs Associated with Finite Element Model 261
6.5.4 Topological Interpretation of Static Indeterminacy 263
6.5.5 Models Including Internal Node . 270
6.5.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems 271
6.5.7 Numerical Examples 272
7 Optimal Force Method for FEMS: Higher Order Elements 281
7.1 Introduction 281
7.2 Finite Element Analysis of Models Comprised of Higher Order Triangular Elements 281
7.2.1 Definition of the Element Force System . 282
7.2.2 Flexibility Matrix of the Element 282
7.2.3 Graphs Associated with Finite Element Model 282
7.2.4 Topological Interpretation of Static Indeterminacies 284
7.2.5 Models Including Opening 287
7.2.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems 290
7.2.7 Numerical Examples 291
7.3 Finite Element Analysis of Models Comprised of Higher Order Rectangular Elements . 297
7.3.1 Definition of Element Force System . 298
7.3.2 Flexibility Matrix of the Element 300
7.3.3 Graphs Associated with Finite Element Model 301
7.3.4 Topological Interpretation of Static Indeterminacies 303
7.3.5 Selection of Generators for SESs of Type II and Type III 307
7.3.6 Algorithm . 308
7.3.7 Numerical Examples 309
7.4 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Hexa-Hedron Elements . 316
7.4.1 Independent Element Forces and Flexibility Matrix of Hexahedron Elements 317
7.4.2 Graphs Associated with Finite Element Models 321
7.4.3 Negative Incidence Number . 325
7.4.4 Pattern Corresponding to Self-Equilibrating Systems 325
7.4.5 Selection of Generators for SESs of Type II and
7.4.6 Numerical Examples 334
8 Decomposition for Parallel Computing: Graph Theory Methods . 341
8.1 Introduction 341
8.2 Earlier Works on Partitioning 342
8.2.1 Nested Dissection 342
8.2.2 A Modified Level-Tree Separator Algorithm . 342
8.3 Substructuring for Parallel Analysis of Skeletal Structures 343
8.3.1 Introduction 343
8.3.2 Substructuring Displacement Method 344
8.3.3 Methods of Substructuring 346
8.3.4 Main Algorithm for Substructuring . 348
8.3.5 Examples . 348
8.3.6 Simplified Algorithm for Substructuring . 350
8.3.7 Greedy Type Algorithm . 352
8.4 Domain Decomposition for Finite Element Analysis 352
8.4.1 Introduction . 353
8.4.2 A Graph Based Method for Subdomaining . 354
8.4.3 Renumbering of Decomposed Finite Element
Models 356
8.4.4 Computational Results of the Graph BasedMethod 356
8.4.5 Discussions on the Graph Based Method 359
8.4.6 Engineering Based Method for Subdomaining 360
8.4.7 Genre Structure Algorithm . 361
8.4.8 Example . 364
8.4.9 Computational Results of the EngineeringBased Method 367
8.4.10 Discussions 367
8.5 Substructuring: Force Method 370
8.5.1 Algorithm for the Force Method Substructuring 370
8.5.2 Examples . 373
9 Analysis of Regular Structures Using Graph Products 377
9.1 Introduction 377
9.2 Definitions of Different Graph Products . 377
9.2.1 Boolean Operation on Graphs 377
9.2.2 Cartesian Product of Two Graphs 378
9.2.3 Strong Cartesian Product of Two Graphs 380
9.2.4 Direct Product of Two Graphs 381
9.3 Analysis of Near-Regular Structures Using Force Method 383
9.3.1 Formulation of the Flexibility Matrix 385
9.3.2 A Simple Method for the Formation of theMatrix A T . 388
9.4 Analysis of Regular Structures with Excessive Members . 389
9.4.1 Summary of the Algorithm . 390
9.4.2 Investigation of a Simple Example 390
9.5 Analysis of Regular Structures with Some Missing Members . 393
9.5.1 Investigation of an Illustrative Simple Example 393
9.6 Practical Examples 396
10 Simultaneous Analysis, Design and Optimization of StructuresUsing Force Method and Supervised Charged System Search . 407
10.1 Introduction 407
10.2 Supervised Charged System Search Algorithm 408
10.3 Analysis by Force Method and Charged System Search 409
10.4 Procedure of Structural Design Using Force Methodand the CSS 414
10.4.1 Pre-selected Stress Ratio 415
10.5 Minimum Weight 420
1.5.5 Fundamental Cutset Bases 27
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