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9780470033326

Optimal Structural Analysis, 2nd Edition

by
  • ISBN13:

    9780470033326

  • ISBN10:

    0470033320

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2011-06-01
  • Publisher: WILEY

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Summary

This second edition of the highly acclaimed and successful first edition, deals primarily with the analysis of structural engineering systems, with applicable methods to other types of structures. The concepts presented in the book are not only relevant to skeletal structures but can equally be used for the analysis of other systems such as hydraulic and electrical networks. The book has been substantially revised to include recent developments and applications of the algebraic graph theory and matroids.

Author Biography

Ali Kaveh is Professor of Structural Engineering at Iran University of Science & Technology, Tehran. He has had over 200 papers published in international journals and conferences. He has held the position of Chief editor of the Asian Journal of Structural Engineering and was a member of the editorial board for 5 international journals and 3 national journals. His research interests include structural mechanics: graph and matrix methods, strength of materials, stability, finite elements and comptuer methods of structural analysis. He is the recipient of various awards, including: Press Media Prize; Educational Gold Medal; Kharuzmi Research Prize and the Alborz Prize; and his previous book “Structural Mechanics: Graph and Matrix Methods, 2nd Edition, 1995” won an award for the best engineering book of its year in Iran.

Table of Contents

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 Castigiliano’s Theorem.
1.3.3 Principle of Virtual Work.
1.3.4 Contragradient Principle.
1.3.5 Reciprocal Work Theorem.
Exercises.
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.
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 Suboptimal 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 Generalized Cycle Basis of a Graph .
1.2.1 Definitions.
1.2.2 Minimal and optimal Generalized Cycle Bases.
3.6 Force Method for the Analysis of Pin-jointed Trusses.
3.6.1 Associate Graph for the Selection of a Suboptimal GCB.
3.6.2 Minimal Generalized Cycle Bases of a Graph.
3.6.3 Selection of a Subminimal GCB; Practical Method.
3.7 Force Method Analysis of General Structures.
3.7.1 Flexibility Matrices of Finite Elements.
3.7.2 Algebraic Methods.
Exercises.
4. Optimal Displacement Method of Structural Analysis.
4.1 Introduction.
4.2 Formulation.
4.2.1 Coordinate System Transformation.
4.2.2 Element Stiffness Matrix Using Unit Displacement Method.
4.2.3 Element Stiffness Matrix Using Castigiliano’s Theorem.
4.2.4 The 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.
4.4.2 Example.
4.7 Optimal 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 Examples.
Exercises.
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; Graph Theory Methods.
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 the Algorithms.
5.6.11 Computational Results.
5.7.12 Discussions.
5.8 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 Route 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.
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 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.
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.
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 Structural 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 Subdimaining.
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.
8.5.1 Criteria for Partitioning.
8.5.2 Weighted Incidence Graphs for Finite Element Models.
8.5.3 Graph Trisection Algorithm.
8.5.4 Numerical Results.
8.5.5 Discussions.
8.6 Bisection of Finite Element Meshes Using Ritz and Fiedler Vectors.
8.6.1 Definitions and Algorithms.
8.6.2 Graph Partitioning.
8.6.3 Determination of Pseudo-Peripheral Nodes.
8.6.4 Formation of an Approximate Fiedler Vector.
8.6.5 Graph Coarsening.
8.6.6 Domain Decomposition Using Ritz and Fiedler Vectors.
8.6.7 Illustrative Example.
8.6.8 Numerical Results.
8.6.9 Discussions.
Exercises.
9. Decomposition and Nodal Ordering of Regular Structures.
9.1 Introduction.
9.2 Definitions of Different Graph Products.
9.2.1 Boolean Operations on Graphs.
9.2.2 Cartesian Product of Two Graphs.
9.2.3 Strong Cartesian Product of Two Graphs.
9.2.4 Direct Product of Two Graphs.
9.3 Eigenvalues of Graphs Matrices for Different Products.
9.3.1 Kronecker Product.
9.3.2 Cartesian Product.
9.3.3 Strong Cartesian Product.
9.3.4 Direct Product.
9.3.5 Second Eigenvalues for Different Products.
9.4 Eigenvalues of A and L Matrices for Cycles and Paths.
9.4.1 Computing l2 for Laplacian of Regular Models.
9.4.2 Algorithm.
9.5 Numerical Examples.
9.5.1 Examples for Cartesian Product.
9.5.2 Examples for Strong Cartesian Product.
9.5.3 Examples for Direct Product.
9.6 Spectral Method for Profile Reduction.
9.6.1 Algorithm.
9.6.2 Examples.
9.7 Non-Compact Extended p-Sum.
Exercises.
References.
Appendix A Basic Concepts and Definitions of Graph Theory.
A.1 Introduction.
A.2 Basic Definitions.
A.3 Vector Spaces Associated with a Graph.
A.4 Matrices Associated with a Graph.
A.5 Directed Graphs and Their Matrices.
A.6 Graphs Associated with Matrices.
A.7 Planar Graphs: Euler’s Polyhedron Formula.
A.8 Maximal Matching in Bipartite Graphs.
Appendix B Greedy Algorithm and its Applications.
B.1 Axiom System for a Matroid.
B.2 Matroids Applied to Structural Mechanics.
B.3 Cocycle Matroid of a Graph.
B.4 Matroid for Null Bases of a Matrix.
B.5 Combinatorial Optimisation: the Greedy Algorithm.
B.6 Application of the Greedy Algorithm.
B.7 Formation of Sparse Null Bases.
Index.
Index of Symbols.

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