Optimal Structural Analysis (2^nd Ed.) RSP Series

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.

Foreword of the First Edition.

Preface.

List of Abbreviations.

1. Basic Concepts and Theorems of Structural Analysis.

1.1 Introduction.

1.2 General Concepts of Structural Analysis.

1.3 Important Structural Theorems.

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.

2.4 The DSI of Structures: Special Methods.

2.5 Space Structures and Their Planar Drawings.

2.6 Rigidity of Structures.

2.7 Rigidity of Planar Trusses.

2.8 Connectivity and Rigidity.

Exercises.

3. Optimal Force Method of Structural Analysis.

3.1 Introduction.

3.2 Formulation of the Force Method.

3.3 Force Method for the Analysis of Frame Structures.

3.4 Conditioning of the Flexibility Matrices.

3.5 Generalized Cycle Basis of a Graph .

3.6 Force Method for the Analysis of Pin-jointed Trusses.

3.7 Force Method Analysis of General Structures.

Exercises.

4. Optimal Displacement Method of Structural Analysis.

4.1 Introduction.

4.2 Formulation.

4.3 Transformation of Stiffness Matrices.

4.4 Displacement Method of Analysis.

4.5 Stiffness Matrix of a Finite Element. 

4.6 Computational Aspects of the Matrix Displacement Method.

4.7 Optimal Conditioned Cutset Bases.

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.

5.6 Finite Element Nodal Ordering For Bandwidth Optimisation.

5.7 Finite Element Nodal Ordering for Profile Optimisation.

5.8 Element Ordering for Frontwidth Reduction.

5.9 Element Ordering for Bandwidth Optimisation of Flexibility Matrices.

5.10 Bandwidth Reduction for Rectangular Matrices.

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.

6.3 Laplacian Matrix of a Graph for Nodal Ordering.

6.4 A Hybrid Method for Ordering.

Exercises.

7. Decomposition for Parallel Computing: Graph Theory Methods.

7.1 Introduction.

7.2 Earlier Works on Partitioning.

7.3 Substructuring for Parallel Analysis of Skeletal Structures.

7.4 Domain Decomposition for Finite Element Analysis.

7.5 Substructuring; Force Method.

7.6 Substructuring for Dynamic Analysis.

Exercises.

8. Decomposition for Parallel Computing: Algebraic Graph Theory Methods.

8.1 Introduction.

8.2 Algebraic Graph theory for Subdomaining.

8.3 Mixed Method for Subdimaining.

8.4 Spectral Bisection for Adaptive FEM; Weighted Graphs.

8.5 Spectral Trisection of Finite Element Models.

8.6 Bisection of Finite Element Meshes Using Ritz and Fiedler Vectors.

Exercises.

9. Decomposition and Nodal Ordering of Regular Structures.

9.1 Introduction.

9.2 Definitions of Different Graph Products.

9.3 Eigenvalues of Graphs Matrices for Different Products. 

9.4 Eigenvalues of A and L Matrices for Cycles and Paths.

9.5 Numerical Examples.

9.6 Spectral Method for Profile Reduction.

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.

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.