Optimal Structural Analysis

by Kaveh, Ali

• ISBN13: 9780470030158
• ISBN10: 0470030151
• Edition: 2nd
• Format: Hardcover
• Copyright: 2006-08-25
• Publisher: WILEY

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

Foreword of the first edition	xvi
Preface	xvii
List of Abbreviations	xix
1. Basic Concepts and Theorems of Structural Analysis	1	(22)
1.1 Introduction	1	(3)
1.1.1 Definitions	1	(3)
1.1.2 Structural Analysis and Design	4	(1)
1.2 General Concepts of Structural Analysis	4	(7)
1.2.1 Main Steps of Structural Analysis	4	(2)
1.2.2 Member Force and Displacements	6	(2)
1.2.3 Member Flexibility and Stiffness Matrices	8	(3)
1.3 Important Structural Theorems	11	(9)
1.3.1 Work and Energy	11	(3)
1.3.2 Castigliano's Theorem	14	(1)
1.3.3 Principle of Virtual Work	15	(3)
1.3.4 Contragradient Principle	18	(1)
1.3.5 Reciprocal Work Theorem	19	(1)
Exercises	20	(3)
2. Static indeterminacy and Rigidity of Skeletal Structures	23	(30)
2.1 Introduction	23	(2)
2.2 Mathematical Model of a Skeletal Structure	25	(2)
2.3 Expansion Process for Determining the Degree of Statical Indeterminacy	27	(6)
2.3.1 Classical Formulae	27	(1)
2.3.2 A Unifying Function	28	(1)
2.3.3 An Expansion Process	28	(1)
2.3.4 An Intersection Theorem	29	(1)
2.3.5 A Method for Determining the DSI of Structures	30	(3)
2.4 The DSI of Structures: Special Methods	33	(2)
2.5 Space Structures and their Planar Drawings	35	(6)
2.5.1 Admissible Drawing of a Space Structure	35	(2)
2.5.2 The DSI of Frames	37	(1)
2.5.3 The DSI of Space Trusses	38	(1)
2.5.4 A Mixed Planar drawing - Expansion Method	39	(2)
2.6 Rigidity of Structures	41	(4)
2.7 Rigidity of Planar Trusses	45	(5)
2.7.1 Complete Matching Method	45	(2)
2.7.2 Decomposition Method	47	(1)
2.7.3 Grid-form Trusses with Bracings	48	(2)
2.8 Connectivity and Rigidity	50	(1)
Exercises	50	(3)
3. Optimal Force Method of Structural Analysis	53	(88)
3.1 Introduction	53	(1)
3.2 Formulation of the Force Method	54	(16)
3.2.1 Equilibrium Equations	54	(3)
3.2.2 Member Flexibility Matrices	57	(3)
3.2.3 Explicit Method for Imposing Compatibility	60	(2)
3.2.4 Implicit Approach for Imposing Compatibility	62	(2)
3.2.5 Structural Flexibility Matrices	64	(1)
3.2.6 Computational Procedure	64	(5)
3.2.7 Optimal Force Method	69	(1)
3.3 Force Method for the Analysis of Frame Structures	70	(27)
3.3.1 Minimal and Optimal Cycle Bases	71	(1)
3.3.2 Selection of Minimal and Subminimal Cycle Bases	72	(7)
3.3.3 Examples	79	(2)
3.3.4 Optimal and Suboptimal Cycle Bases	81	(3)
3.3.5 Examples	84	(3)
3.3.6 An Improved Turn-Back Method for the Formation of Cycle Bases	87	(1)
3.3.7 Examples	88	(3)
3.3.8 An Algebraic Graph-Theoretical Method for Cycle Basis Selection	91	(2)
3.3.9 Examples	93	(4)
3.4 Conditioning of the Flexibility Matrices	97	(18)
3.4.1 Condition Number	98	(3)
3.4.2 Weighted Graph and an Admissible Member	101	(1)
3.4.3 Optimally Conditioned Cycle Bases	101	(2)
3.4.4 Formulation of the Conditioning Problem	103	(1)
3.4.5 Suboptimally Conditioned Cycle Bases	104	(3)
3.4.6 Examples	107	(2)
3.4.7 Formation of B0 and B1 matrices	109	(6)
3.5 Generalised Cycle Bases of a Graph	115	(4)
3.5.1 Definitions	115	(3)
3.5.2 Minimal and Optimal Generalized Cycle Bases	118	(1)
3.6 Force Method for the Analysis of Pin-jointed Planar Trusses	119	(6)
3.6.1 Associate Graphs for Selection of a Suboptimal GCB	119	(3)
3.6.2 Minimal GCB of a Graph	122	(1)
3.6.3 Selection of a Subminimal GCB: Practical Methods	123	(2)
3.7 Force Method of Analysis for General Structures	125	(14)
3.7.1 Flexibility Matrices of Finite Elements	125	(6)
3.7.2 Algebraic Methods	131	(8)
Exercises	139	(2)
4. Optimal Displacement Method of Structural Analysis	141	(50)
4.1 Introduction	141	(1)
4.2 Formulation	142	(18)
4.2.1 Coordinate Systems Transformation	142	(4)
4.2.2 Element Stiffness Matrix using Unit Displacement Method	146	(4)
4.2.3 Element Stiffness Matrix using Castigliano's Theorem	150	(3)
4.2.4 Stiffness Matrix of a Structure	153	(5)
4.2.5 Stiffness Matrix of a Structure: An Algorithmic Approach	158	(2)
4.3 Transformation of Stiffness Matrices	160	(6)
4.3.1 Stiffness Matrix of a Bar Element	161	(2)
4.3.2 Stiffness Matrix of a Beam Element	163	(3)
4.4 Displacement Method of Analysis	166	(7)
4.4.1 Boundary Conditions	168	(1)
4.4.2 General Loading	169	(4)
4.5 Stiffness Matrix of a Finite Element	173	(3)
4.5.1 Stiffness Matrix of a Triangular Element	173	(3)
4.6 Computational Aspects of the Matrix Displacement Method	176	(4)
4.6.1 Algorithm	176	(2)
4.6.2 Example	178	(2)
4.7 Optimally Conditioned Cutset Bases	180	(6)
4.7.1 Mathematical Formulation of the Problem	181	(1)
4.7.2 Suboptimally Conditioned Cutset Bases	182	(1)
4.7.3 Algorithms	183	(1)
4.7.4 Example	184	(2)
Exercises	186	(5)
5. Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods	191	(82)
5.1 Introduction	191	(1)
5.2 Bandwidth Optimisation	192	(2)
5.3 Preliminaries	194	(2)
5.4 A Shortest Route Tree and its Properties	196	(1)
5.5 Nodal Ordering for Bandwidth Reduction	197	(6)
5.5.1 A Good Starting Node	198	(3)
5.5.2 Primary Nodal Decomposition	201	(1)
5.5.3 Transversal P of an SRT	201	(1)
5.5.4 Nodal Ordering	202	(1)
5.5.5 Example	202	(1)
5.6 Finite Element Nodal Ordering for Bandwidth Optimisation	203	(21)
5.6.1 Element Clique Graph Method (ECGM)	204	(1)
5.6.2 Skeleton Graph Method (SGM)	205	(3)
5.6.3 Element Star Graph Method (ESGM)	208	(1)
5.6.4 Element Wheel Graph Method (EWGM)	209	(2)
5.6.5 Partially Triangulated Graph Method (PTGM)	211	(1)
5.6.6 Triangulated Graph Method (TGM)	212	(2)
5.6.7 Natural Associate Graph Method (NAGM)	214	(3)
5.6.8 Incidence Graph Method (IGM)	217	(1)
5.6.9 Representative Graph Method (RGM)	218	(2)
5.6.10 Discussion of the Analysis of Algorithms	220	(1)
5.6.11 Computational Results	221	(502)
5.6.12 Discussions	723
5.7 Finite Element Nodal Ordering for Profile Optimisation	224	(17)
5.7.1 Introduction	224	(2)
5.7.2 Graph Nodal Numbering for Profile Reduction	226	(4)
5.7.3 Nodal Ordering with Element Clique Graph (NOECG)	230	(1)
5.7.4 Nodal Ordering with Skeleton Graph (NOSG)	230	(2)
5.7.5 Nodal Ordering with Element Star Graph (NOESO)	232	(1)
5.7.6 Nodal Ordering with Element Wheel Graph (NOEWG)	232	(1)
5.7.7 Nodal Ordering with Partially Triangulated Graph (NOPTG)	232	(1)
5.7.8 Nodal Ordering with Triangulated Graph (NOTG)	233	(1)
5.7.9 Nodal Ordering with Natural Associate Graph (NONAG)	233	(1)
5.7.10 Nodal Ordering with Incidence Graph (NOIG)	234	(1)
5.7.11 Nodal Ordering with Representative Graph (NORG)	234	(2)
5.7.12 Nodal Ordering with Element Clique Representative Graph (NOECRG)	236	(1)
5.7.13 Computational Results	236	(4)
5.7.14 Discussions	240	(1)
5.8 Element Ordering for Frontwidth Reduction	241	(15)
5.8.1 Definitions	242	(2)
5.8.2 Different Strategies for Frontwidth Reduction	244	(2)
5.8.3 Efficient Root Selection	246	(3)
5.8.4 Algorithm for Frontwidth Reduction	249	(3)
5.8.5 Complexity of the Algorithm	252	(1)
5.8.6 Computational Results	253	(3)
5.8.7 Discussions	256	(1)
5.9 Element Ordering for Bandwidth Optimisation of Flexibility Matrices	256	(4)
5.9.1 An Associate Graph	257	(1)
5.9.2 Distance Number of an Element	257	(1)
5.9.3 Element Ordering Algorithms	258	(2)
5.10 Bandwidth Reduction for Rectangular Matrices	260	(6)
5.10.1 Definitions	260	(2)
5.10.2 Algorithms	262	(1)
5.10.3 Examples	262	(2)
5.10.4 Bandwidth Reduction of Finite Element Models	264	(2)
5.11 Graph-Theoretical interpretation of Gaussian Elimination	266	(3)
Exercises	269	(4)
6. Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory Methods	273	(20)
6.1 Introduction	273	(1)
6.2 Adjacency Matrix of a Graph for Nodal Ordering	273	(6)
6.2.1 Basic Concepts and Definition	273	(4)
6.2.2 A Good Starting Node	277	(1)
6.2.3 Primary Nodal Decomposition	277	(1)
6.2.4 Transversal P of an SRT	277	(1)
6.2.5 Nodal Ordering	278	(1)
6.2.6 Example	278	(1)
6.3 Laplacian Matrix of a Graph for Nodal Ordering	279	(12)
6.3.1 Basic Concepts and Definitions	279	(3)
6.3.2 Nodal Numbering Algorithm	282	(1)
6.3.3 Example	283	(1)
6.4 A Hybrid Method for Ordering	284	(1)
6.4.1 Development of the Method	284	(1)
6.4.2 Numerical Results	285	(5)
6.4.3 Discussions	290	(1)
Exercises	291	(2)
7. Decomposition for Parallel Computing: Graph Theory Methods	293	(56)
7.1 Introduction	293	(1)
7.2 Earlier Works on Partitioning	294	(1)
7.2.1 Nested Dissection	294	(1)
7.2.2 A modified Level-Tree Separator Algorithm	294	(1)
7.3 Substructuring for Parallel Analysis of Skeletal Structures	295	(10)
7.3.1 Introduction	295	(1)
7.3.2 Substructuring Displacement Method	296	(2)
7.3.3 Methods of Substructuring	298	(2)
7.3.4 Main Algorithm for Substructuring	300	(1)
7.3.5 Examples	301	(3)
7.3.6 Simplified Algorithm for Substructuring	304	(1)
7.3.7 Greedy Type Algorithm	305	(1)
7.4 Domain Decomposition for Finite Element Analysis	305	(25)
7.4.1 Introduction	306	(1)
7.4.2 A Graph-Based Method for Subdomaining	307	(2)
7.4.3 Renumbering of Decomposed Finite Element Models	309	(1)
7.4.4 Complexity Analysis of the Graph-Based Method	310	(2)
7.4.5 Computational Results of the Graph-Based Method	312	(3)
7.4.6 Discussions on the Graph-Based Method	315	(1)
7.4.7 Engineering-Based Method for Subdomaining	316	(1)
7.4.8 Genre Structure Algorithm	317	(3)
7.4.9 Example	320	(3)
7.4.10 Complexity Analysis of the Engineering-Based Method	323	(2)
7.4.11 Computational Results of the Engineering-Based Method	325	(3)
7.4.12 Discussions	328	(2)
7.5 Substructuring: Force Method	330	(6)
7.5.1 Algorithm for the Force Method Substructuring	330	(3)
7.5.2 Examples	333	(3)
7.6 Substructuring for Dynamic Analysis	336	(10)
7.6.1 Modal Analysis of a Substructure	336	(2)
7.6.2 Partitioning of the Transfer Matrix H(w)	338	(1)
7.6.3 Dynamic Equation of the Entire Structure	338	(4)
7.6.4 Examples	342	(4)
Exercises	346	(3)
8. Decomposition for Parallel Computing: Algebraic Graph Theory Methods	349	(54)
8.1 Introduction	349	(1)
8.2 Algebraic Graph Theory for Subdomaining	350	(13)
8.2.1 Basic Definitions and Concepts	350	(4)
8.2.2 Lanczos Method	354	(5)
8.2.3 Recursive Spectral Bisection Partitioning Algorithm	359	(3)
8.2.4 Recursive Spectral Sequential-Cut Partitioning Algorithm	362	(1)
8.2.5 Recursive Spectral Two-way Partitioning Algorithm	362	(1)
8.3 Mixed Method for Subdomaining	363	(8)
8.3.1 Introduction	363	(1)
8.3.2 Mixed Method for Graph Bisection	364	(5)
8.3.3 Examples	369	(2)
8.3.4 Discussions	371	(1)
8.4 Spectral Bisection for Adaptive FEM; Weighted Graphs	371	(7)
8.4.1 Basic Concepts	372	(2)
8.4.2 Partitioning of Adaptive FE Meshes	374	(2)
8.4.3 Computational Results	376	(2)
8.5 Spectral Trisection of Finite Element Models	378	(11)
8.5.1 Criteria for Partitioning	378	(2)
8.5.2 Weighted Incidence Graphs for Finite Element Models	380	(1)
8.5.3 Graph Trisection Algorithm	381	(6)
8.5.4 Numerical Results	387	(2)
8.5.5 Discussions	389	(1)
8.6 Bisection of Finite Element Meshes using Ritz and Fiedler Vectors	389	(12)
8.6.1 Definitions and Algorithms	390	(1)
8.6.2 Graph Partitioning	390	(1)
8.6.3 Determination of Pseudo-Peripheral Nodes	391	(1)
8.6.4 Formation of an Approximate Fiedler Vector	391	(1)
8.6.5 Graph Coarsening	392	(1)
8.6.6 Domain Decomposition using Ritz and Fiedler Vectors	393	(1)
8.6.7 Illustrative Example	393	(4)
8.6.8 Numerical Results	397	(4)
8.6.9 Discussions	401	(1)
Exercises	401	(2)
9. Decomposition and Nodal Ordering of Regular Structures	403	(34)
9.1 Introduction	403	(1)
9.2 Definitions of Different Graph Products	404	(6)
9.2.1 Boolean Operations on Graphs	404	(1)
9.2.2 Cartesian Product of Two Graphs	404	(3)
9.2.3 Strong Cartesian Product of Two Graphs	407	(2)
9.2.4 Direct Product of Two Graphs	409	(1)
9.3 Eigenvalues of Graphs Matrices for Different Products	410	(11)
9.3.1 Kronecker Product	410	(1)
9.3.2 Cartesian Product	411	(3)
9.3.3 Strong Cartesian Product	414	(3)
9.3.4 Direct Product	417	(2)
9.3.5 Second Eigenvalues for Different Graph Products	419	(2)
9.4 Eigenvalues of A and L Matrices for Cycles and Paths	421	(5)
9.4.1 Computing λ2 for Laplacian of Regular Models	424	(1)
9.4.2 Algorithm	425	(1)
9.5 Numerical Examples	426	(7)
9.5.1 Examples for Cartesian Product	426	(4)
9.5.2 Examples for Strong Cartesian Product	430	(1)
9.5.3 Examples for Direct Product	431	(2)
9.6 Spectral Method for Profile Reduction	433	(2)
9.6.1 Algorithm	433	(1)
9.6.2 Examples	433	(2)
9.7 Non-Compact Extended p-Sum	435	(1)
Exercises	436	(1)
Appendix A Basic Concepts and Definitions of Graph Theory	437	(28)
A.1 Introduction	437	(1)
A.2 Basic Definitions	437	(8)
A.3 Vector Spaces Associated with a Graph	445	(3)
A.4 Matrices Associated with a Graph	448	(8)
A.5 Directed Graphs and their Matrices	456	(2)
A.6 Graphs Associated with Matrices	458	(1)
A.7 Planar Graphs: Euler's Polyhedron Formula	459	(3)
A.8 Maximal Matching in Bipartite Graphs	462	(3)
Appendix B Greedy Algorithm and its Applications	465	(12)
B.1 Axiom System for a Matroid	465	(2)
B.2 Matroids Applied to Structural Mechanics	467	(3)
B.3 Cocycle Matroid of a Graph	470	(1)
B.4 Matroid for Null Basis of a Matrix	471	(1)
B.5 Combinatorial Optimisation: the Greedy Algorithm	472	(1)
B.6 Application of the Greedy Algorithm	473	(1)
B.7 Formation of Sparse Null Bases	474	(3)
References	477	(18)
Index	495	(10)
Index of Symbols	505