## 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
1.3.5 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
1.4.4 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
3.5 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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