Optimal Structural Analysis -- Copyright -- Contents -- Foreword of the first edition -- Preface -- List of Abbreviations -- Chapter 1 Basic Concepts and Theorems of Structural Analysis -- 1.1 Introduction -- 1.1.1 Definitions -- 1.1.2 Structural Analysis and Design -- 1.2 General Concepts of Structural Analysis -- 1.2.1 Main Steps of Structural Analysis -- 1.2.2 Member Force and Displacements -- 1.2.3 Member Flexibility and Stiffness Matrices -- 1.3 Important Structural Theorems -- 1.3.1 Work and Energy -- 1.3.2 Castigliano's Theorem -- 1.3.3 Principle of Virtual Work -- 1.3.4 Contragradient Principle -- 1.3.5 Reciprocal Work Theorem -- Exercises -- Chapter 2 Static Indeterminacy and Rigidity of Skeletal Structures -- 2.1 Introduction -- 2.2 Mathematical Model of a Skeletal Structure -- 2.3 Expansion Process for Determining the Degree of Statical Indeterminacy -- 2.3.1 Classical Formulae -- 2.3.2 A Unifying Function -- 2.3.3 An Expansion Process -- 2.3.4 An Intersection Theorem -- 2.3.5 A Method for Determining the DSI of Structures -- 2.4 The DSI of Structures: Special Methods -- 2.5 Space Structures and their Planar Drawings -- 2.5.1 Admissible Drawing of a Space Structure -- 2.5.2 The DSI of Frames -- 2.5.3 The DSI of Space Trusses -- 2.5.4 A Mixed Planar drawing - Expansion Method -- 2.6 Rigidity of Structures -- 2.7 Rigidity of Planar Trusses -- 2.7.1 Complete Matching Method -- 2.7.2 Decomposition Method -- 2.7.3 Grid-form Trusses with Bracings -- 2.8 Connectivity and Rigidity -- Exercises -- Chapter 3 Optimal Force Method of Structural Analysis -- 3.1 Introduction -- 3.2 Formulation of the Force Method -- 3.2.1 Equilibrium Equations -- 3.2.2 Member Flexibility Matrices -- 3.2.3 Explicit Method for Imposing Compatibility -- 3.2.4 Implicit Approach for Imposing Compatibility -- 3.2.5 Structural Flexibility Matrices.

3.2.6 Computational Procedure -- 3.2.7 Optimal Force Method -- 3.3 Force Method for the Analysis of Frame Structures -- 3.3.1 Minimal and Optimal Cycle Bases -- 3.3.2 Selection of Minimal and Subminimal Cycle Bases -- 3.3.3 Examples -- 3.3.4 Optimal and Suboptimal Cycle Bases -- 3.3.5 Examples -- 3.3.6 An Improved Turn-Back Method for the Formation of Cycle Bases -- 3.3.7 Examples -- 3.3.8 An Algebraic Graph-Theoretical Method for Cycle Basis Selection -- 3.3.9 Examples -- 3.4 Conditioning of the Flexibility Matrices -- 3.4.1 Condition Number -- 3.4.2 Weighted Graph and an Admissible Member -- 3.4.3 Optimally Conditioned Cycle Bases -- 3.4.4 Formulation of the Conditioning Problem -- 3.4.5 Suboptimally Conditioned Cycle Bases -- 3.4.6 Examples -- 3.4.7 Formation of B0 and B1 matrices -- 3.5 Generalised Cycle Bases of a Graph -- 3.5.1 Definitions -- 3.5.2 Minimal and Optimal Generalized Cycle Bases -- 3.6 Force Method for the Analysis of Pin-jointed Planar Trusses -- 3.6.1 Associate Graphs for Selection of a Suboptimal GCB -- 3.6.2 Minimal GCB of a Graph -- 3.6.3 Selection of a Subminimal GCB: Practical Methods -- 3.7 Force Method of Analysis for General Structures -- 3.7.1 Flexibility Matrices of Finite Elements -- 3.7.2 Algebraic Methods -- Exercises -- Chapter 4 Optimal Displacement Method of Structural Analysis -- 4.1 Introduction -- 4.2 Formulation -- 4.2.1 Coordinate Systems Transformation -- 4.2.2 Element Stiffness Matrix using Unit Displacement Method -- 4.2.3 Element Stiffness Matrix using Castigliano's Theorem -- 4.2.4 Stiffness Matrix of a Structure -- 4.2.5 Stiffness Matrix of a Structure: An Algorithmic Approach -- 4.3 Transformation of Stiffness Matrices -- 4.3.1 Stiffness Matrix of a Bar Element -- 4.3.2 Stiffness Matrix of a Beam Element -- 4.4 Displacement Method of Analysis -- 4.4.1 Boundary Conditions -- 4.4.2 General Loading.

4.5 Stiffness Matrix of a Finite Element -- 4.5.1 Stiffness Matrix of a Triangular Element -- 4.6 Computational Aspects of the Matrix Displacement Method -- 4.6.1 Algorithm -- 4.6.2 Example -- 4.7 Optimally Conditioned Cutset Bases -- 4.7.1 Mathematical Formulation of the Problem -- 4.7.2 Suboptimally Conditioned Cutset Bases -- 4.7.3 Algorithms -- 4.7.4 Example -- Exercises -- Chapter 5 Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods -- 5.1 Introduction -- 5.2 Bandwidth Optimisation -- 5.3 Preliminaries -- 5.4 A Shortest Route Tree and its Properties -- 5.5 Nodal Ordering for Bandwidth Reduction -- 5.5.1 A Good Starting Node -- 5.5.2 Primary Nodal Decomposition -- 5.5.3 Transversal P of an SRT -- 5.5.4 Nodal Ordering -- 5.5.5 Example -- 5.6 Finite Element Nodal Ordering for Bandwidth Optimisation -- 5.6.1 Element Clique Graph Method (ECGM) -- 5.6.2 Skeleton Graph Method (SGM) -- 5.6.3 Element Star Graph Method (ESGM) -- 5.6.4 Element Wheel Graph Method (EWGM) -- 5.6.5 Partially Triangulated Graph Method (PTGM) -- 5.6.6 Triangulated Graph Method (TGM) -- 5.6.7 Natural Associate Graph Method (NAGM) -- 5.6.8 Incidence Graph Method (IGM) -- 5.6.9 Representative Graph Method (RGM) -- 5.6.10 Discussion of the Analysis of Algorithms -- 5.6.11 Computational Results -- 5.6.12 Discussions -- 5.7 Finite Element Nodal Ordering for Profile Optimisation -- 5.7.1 Introduction -- 5.7.2 Graph Nodal Numbering for Profile Reduction -- 5.7.3 Nodal Ordering with Element Clique Graph (NOECG) -- 5.7.4 Nodal Ordering with Skeleton Graph (NOSG) -- 5.7.5 Nodal Ordering with Element Star Graph (NOESG) -- 5.7.6 Nodal Ordering with Element Wheel Graph (NOEWG) -- 5.7.7 Nodal Ordering with Partially Triangulated Graph (NOPTG) -- 5.7.8 Nodal Ordering with Triangulated Graph (NOTG) -- 5.7.9 Nodal Ordering with Natural Associate Graph (NONAG).

5.7.10 Nodal Ordering with Incidence Graph (NOIG) -- 5.7.11 Nodal Ordering with Representative Graph (NORG) -- 5.7.12 Nodal Ordering with Element Clique Representative Graph (NOECRG) -- 5.7.13 Computational Results -- 5.7.14 Discussions -- 5.8 Element Ordering for Frontwidth Reduction -- 5.8.1 Definitions -- 5.8.2 Different Strategies for Frontwidth Reduction -- 5.8.3 Efficient Root Selection -- 5.8.4 Algorithm for Frontwidth Reduction -- 5.8.5 Complexity of the Algorithm -- 5.8.6 Computational Results -- 5.8.7 Discussions -- 5.9 Element Ordering for Bandwidth Optimisation of Flexibility Matrices -- 5.9.1 An Associate Graph -- 5.9.2 Distance Number of an Element -- 5.9.3 Element Ordering Algorithms -- 5.10 Bandwidth Reduction for Rectangular Matrices -- 5.10.1 Definitions -- 5.10.2 Algorithms -- 5.10.3 Examples -- 5.10.4 Bandwidth Reduction of Finite Element Models -- 5.11 Graph-Theoretical interpretation of Gaussian Elimination -- Exercises -- Chapter 6 Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory Methods -- 6.1 Introduction -- 6.2 Adjacency Matrix of a Graph for Nodal Ordering -- 6.2.1 Basic Concepts and Definition -- 6.2.2 A Good Starting Node -- 6.2.3 Primary Nodal Decomposition -- 6.2.4 Transversal P of an SRT -- 6.2.5 Nodal Ordering -- 6.2.6 Example -- 6.3 Laplacian Matrix of a Graph for Nodal Ordering -- 6.3.1 Basic Concepts and Definitions -- 6.3.2 Nodal Numbering Algorithm -- 6.3.3 Example -- 6.4 A Hybrid Method for Ordering -- 6.4.1 Development of the Method -- 6.4.2 Numerical Results -- 6.4.3 Discussions -- Exercises -- Chapter 7 Decomposition for Parallel Computing: Graph Theory Methods -- 7.1 Introduction -- 7.2 Earlier Works on Partitioning -- 7.2.1 Nested Dissection -- 7.2.2 A modified Level-Tree Separator Algorithm -- 7.3 Substructuring for Parallel Analysis of Skeletal Structures -- 7.3.1 Introduction.

7.3.2 Substructuring Displacement Method -- 7.3.3 Methods of Substructuring -- 7.3.4 Main Algorithm for Substructuring -- 7.3.5 Examples -- 7.3.6 Simplified Algorithm for Substructuring -- 7.3.7 Greedy Type Algorithm -- 7.4 Domain Decomposition for Finite Element Analysis -- 7.4.1 Introduction -- 7.4.2 A Graph-Based Method for Subdomaining -- 7.4.3 Renumbering of Decomposed Finite Element Models -- 7.4.4 Complexity Analysis of the Graph-Based Method -- 7.4.5 Computational Results of the Graph-Based Method -- 7.4.6 Discussions on the Graph-Based Method -- 7.4.7 Engineering-Based Method for Subdomaining -- 7.4.8 Genre Structure Algorithm -- 7.4.9 Example -- 7.4.10 Complexity Analysis of the Engineering-Based Method -- 7.4.11 Computational Results of the Engineering-Based Method -- 7.4.12 Discussions -- 7.5 Substructuring: Force Method -- 7.5.1 Algorithm for the Force Method Substructuring -- 7.5.2 Examples -- 7.6 Substructuring for Dynamic Analysis -- 7.6.1 Modal Analysis of a Substructure -- 7.6.2 Partitioning of the Transfer Matrix H(w) -- 7.6.3 Dynamic Equation of the Entire Structure -- 7.6.4 Examples -- Exercises -- Chapter 8 Decomposition for Parallel Computing: Algebraic Graph Theory Methods -- 8.1 Introduction -- 8.2 Algebraic Graph Theory for Subdomaining -- 8.2.1 Basic Definitions and Concepts -- 8.2.2 Lanczos Method -- 8.2.3 Recursive Spectral Bisection Partitioning Algorithm -- 8.2.4 Recursive Spectral Sequential-Cut Partitioning Algorithm -- 8.2.5 Recursive Spectral Two-way Partitioning Algorithm -- 8.3 Mixed Method for Subdomaining -- 8.3.1 Introduction -- 8.3.2 Mixed Method for Graph Bisection -- 8.3.3 Examples -- 8.3.4 Discussions -- 8.4 Spectral Bisection for Adaptive FEM -- Weighted Graphs -- 8.4.1 Basic Concepts -- 8.4.2 Partitioning of Adaptive FE Meshes -- 8.4.3 Computational Results.

8.5 Spectral Trisection of Finite Element Models.