Optimal Structural Analysis

Optimal Structural Analysis

by A. Kaveh




The increasing need to demonstrate structural safety has driven many recent advances in structural technology that require greater accuracy, efficiency and speed in the analysis of their systems. These new methods of analysis have to be sufficiently accurate to cope with complex and large-scale structures. In addition, there is also a growing need to achieve more efficient and optimal use of materials.

Following on from the highly acclaimed and successful first edition, Optimal Structural Analysis now deals primarily with the analysis of structural engineering systems, with applicable methods to other types of structures.

  • Presents efficient and practical methods for optimal analysis of structures.
  • Provides a complete reference for many applications of graph theory, algebraic graph theory and matroids in computational structural mechanics.
  • Substantially revised to include recent developments and applications of the algebraic graph theory and matroids, which are ideally suited for modern computational techniques.
  • Describes recent developments in the matrix force methods of structural analysis.
  • Presents novel applications of graph products in structural mechanics.

Optimal Structural Analysis will be of interest to post-graduate students in the fields of structures and mechanics, and applied mathematics particularly discrete mathematics. It will also appeal to practitioners developing programs for structures and finite element analysis.

Product Details

ISBN-13: 9780471975199
Publisher: Taylor & Francis, Inc.
Publication date: 05/01/1997
Series: Applied and Engineering Mathematics Series
Pages: 568

About the Author

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
Basic Concepts and Theorems of Structural Analysis     1
Introduction     1
Definitions     1
Structural Analysis and Design     4
General Concepts of Structural Analysis     4
Main Steps of Structural Analysis     4
Member Force and Displacements     6
Member Flexibility and Stiffness Matrices     8
Important Structural Theorems     11
Work and Energy     11
Castigliano's Theorem     14
Principle of Virtual Work     15
Contragradient Principle     18
Reciprocal Work Theorem     19
Exercises     20
Static Indeterminacy and Rigidity of Skeletal Structures     23
Introduction     23
Mathematical Model of a Skeletal Structure     25
Expansion Process for Determining the Degree of Statical Indeterminacy     27
Classical Formulae     27
A Unifying Function     28
An Expansion Process     28
An Intersection Theorem     29
A Method for Determining the DSI of Structures     30
The DSI of Structures: Special Methods     33
Space Structures and their Planar Drawings     35
Admissible Drawing of a Space Structure     35
The DSI of Frames     37
The DSI of Space Trusses     38
A Mixed Planar drawing - Expansion Method     39
Rigidity of Structures     41
Rigidity of Planar Trusses     45
Complete Matching Method     45
Decomposition Method     47
Grid-form Trusses with Bracings     48
Connectivity and Rigidity     50
Exercises     50
Optimal Force Method of Structural Analysis     53
Introduction     53
Formulation of the Force Method     54
Equilibrium Equations     54
Member Flexibility Matrices     57
Explicit Method for Imposing Compatibility     60
Implicit Approach for Imposing Compatibility     62
Structural Flexibility Matrices     64
Computational Procedure     64
Optimal Force Method     69
Force Method for the Analysis of Frame Structures     70
Minimal and Optimal Cycle Bases     71
Selection of Minimal and Subminimal Cycle Bases      72
Examples     79
Optimal and Suboptimal Cycle Bases     81
Examples     84
An Improved Turn-Back Method for the Formation of Cycle Bases     87
Examples     88
An Algebraic Graph-Theoretical Method for Cycle Basis Selection     91
Examples     93
Conditioning of the Flexibility Matrices     97
Condition Number     98
Weighted Graph and an Admissible Member     101
Optimally Conditioned Cycle Bases     101
Formulation of the Conditioning Problem     103
Suboptimally Conditioned Cycle Bases     104
Examples     107
Formation of B[subscript 0] and B[subscript 1] matrices     109
Generalised Cycle Bases of a Graph     115
Definitions     115
Minimal and Optimal Generalized Cycle Bases     118
Force Method for the Analysis of Pin-jointed Planar Trusses     119
Associate Graphs for Selection of a Suboptimal GCB     119
Minimal GCB of a Graph     122
Selection of a Subminimal GCB: Practical Methods     123
Force Method of Analysis for General Structures     125
Flexibility Matrices of Finite Elements     125
Algebraic Methods      131
Exercises     139
Optimal Displacement Method of Structural Analysis     141
Introduction     141
Formulation     142
Coordinate Systems Transformation     142
Element Stiffness Matrix using Unit Displacement Method     146
Element Stiffness Matrix using Castigliano's Theorem     150
Stiffness Matrix of a Structure     153
Stiffness Matrix of a Structure: An Algorithmic Approach     158
Transformation of Stiffness Matrices     160
Stiffness Matrix of a Bar Element     161
Stiffness Matrix of a Beam Element     163
Displacement Method of Analysis     166
Boundary Conditions     168
General Loading     169
Stiffness Matrix of a Finite Element     173
Stiffness Matrix of a Triangular Element     173
Computational Aspects of the Matrix Displacement Method     176
Algorithm     176
Example     178
Optimally Conditioned Cutset Bases     180
Mathematical Formulation of the Problem     181
Suboptimally Conditioned Cutset Bases     182
Algorithms     183
Example     184
Exercises      186
Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods     191
Introduction     191
Bandwidth Optimisation     192
Preliminaries     194
A Shortest Route Tree and its Properties     196
Nodal Ordering for Bandwidth Reduction     197
A Good Starting Node     198
Primary Nodal Decomposition     201
Transversal P of an SRT     201
Nodal Ordering     202
Example     202
Finite Element Nodal Ordering for Bandwidth Optimisation     203
Element Clique Graph Method (ECGM)     204
Skeleton Graph Method (SGM)     205
Element Star Graph Method (ESGM)     208
Element Wheel Graph Method (EWGM)     209
Partially Triangulated Graph Method (PTGM)     211
Triangulated Graph Method (TGM)     212
Natural Associate Graph Method (NAGM)     214
Incidence Graph Method (IGM)     217
Representative Graph Method (RGM)     218
Discussion of the Analysis of Algorithms     220
Computational Results     221
Discussions     223
Finite Element Nodal Ordering for Profile Optimisation      224
Introduction     224
Graph Nodal Numbering for Profile Reduction     226
Nodal Ordering with Element Clique Graph (NOECG)     230
Nodal Ordering with Skeleton Graph (NOSG)     230
Nodal Ordering with Element Star Graph (NOESG)     232
Nodal Ordering with Element Wheel Graph (NOEWG)     232
Nodal Ordering with Partially Triangulated Graph (NOPTG)     232
Nodal Ordering with Triangulated Graph (NOTG)     233
Nodal Ordering with Natural Associate Graph (NONAG)     233
Nodal Ordering with Incidence Graph (NOIG)     234
Nodal Ordering with Representative Graph (NORG)     234
Nodal Ordering with Element Clique Representative Graph (NOECRG)     236
Computational Results     236
Discussions     240
Element Ordering for Frontwidth Reduction     241
Definitions     242
Different Strategies for Frontwidth Reduction     244
Efficient Root Selection     246
Algorithm for Frontwidth Reduction     249
Complexity of the Algorithm     252
Computational Results     253
Discussions     256
Element Ordering for Bandwidth Optimisation of Flexibility Matrices     256
An Associate Graph     257
Distance Number of an Element     257
Element Ordering Algorithms     258
Bandwidth Reduction for Rectangular Matrices     260
Definitions     260
Algorithms     262
Examples     262
Bandwidth Reduction of Finite Element Models     264
Graph-Theoretical interpretation of Gaussian Elimination     266
Exercises     269
Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory Methods     273
Introduction     273
Adjacency Matrix of a Graph for Nodal Ordering     273
Basic Concepts and Definition     273
A Good Starting Node     277
Primary Nodal Decomposition     277
Transversal P of an SRT     277
Nodal Ordering     278
Example     278
Laplacian Matrix of a Graph for Nodal Ordering     279
Basic Concepts and Definitions     279
Nodal Numbering Algorithm     282
Example     283
A Hybrid Method for Ordering     284
Development of the Method     284
Numerical Results     285
Discussions     290
Exercises      291
Decomposition for Parallel Computing: Graph Theory Methods     293
Introduction     293
Earlier Works on Partitioning     294
Nested Dissection     294
A modified Level-Tree Separator Algorithm     294
Substructuring for Parallel Analysis of Skeletal Structures     295
Introduction     295
Substructuring Displacement Method     296
Methods of Substructuring     298
Main Algorithm for Substructuring     300
Examples     301
Simplified Algorithm for Substructuring     304
Greedy Type Algorithm     305
Domain Decomposition for Finite Element Analysis     305
Introduction     306
A Graph-Based Method for Subdomaining     307
Renumbering of Decomposed Finite Element Models     309
Complexity Analysis of the Graph-Based Method     310
Computational Results of the Graph-Based Method     312
Discussions on the Graph-Based Method     315
Engineering-Based Method for Subdomaining     316
Genre Structure Algorithm     317
Example     320
Complexity Analysis of the Engineering-Based Method     323
Computational Results of the Engineering-Based Method     325
Discussions     328
Substructuring: Force Method     330
Algorithm for the Force Method Substructuring     330
Examples     333
Substructuring for Dynamic Analysis     336
Modal Analysis of a Substructure     336
Partitioning of the Transfer Matrix H(w)     338
Dynamic Equation of the Entire Structure     338
Examples     342
Exercises     346
Decomposition for Parallel Computing: Algebraic Graph Theory Methods     349
Introduction     349
Algebraic Graph Theory for Subdomaining     350
Basic Definitions and Concepts     350
Lanczos Method     354
Recursive Spectral Bisection Partitioning Algorithm     359
Recursive Spectral Sequential-Cut Partitioning Algorithm     362
Recursive Spectral Two-way Partitioning Algorithm     362
Mixed Method for Subdomaining     363
Introduction     363
Mixed Method for Graph Bisection     364
Examples     369
Discussions     371
Spectral Bisection for Adaptive FEM; Weighted Graphs     371
Basic Concepts      372
Partitioning of Adaptive FE Meshes     374
Computational Results     376
Spectral Trisection of Finite Element Models     378
Criteria for Partitioning     378
Weighted Incidence Graphs for Finite Element Models     380
Graph Trisection Algorithm     381
Numerical Results     387
Discussions     389
Bisection of Finite Element Meshes using Ritz and Fiedler Vectors     389
Definitions and Algorithms     390
Graph Partitioning     390
Determination of Pseudo-Peripheral Nodes     391
Formation of an Approximate Fiedler Vector     391
Graph Coarsening     392
Domain Decomposition using Ritz and Fiedler Vectors     393
Illustrative Example     393
Numerical Results     397
Discussions     401
Exercises     401
Decomposition and Nodal Ordering of Regular Structures     403
Introduction     403
Definitions of Different Graph Products     404
Boolean Operations on Graphs     404
Cartesian Product of Two Graphs     404
Strong Cartesian Product of Two Graphs     407
Direct Product of Two Graphs      409
Eigenvalues of Graphs Matrices for Different Products     410
Kronecker Product     151
Cartesian Product     411
Strong Cartesian Product     414
Direct Product     417
Second Eigenvalues for Different Graph Products     419
Eigenvalues of A and L Matrices for Cycles and Paths     421
Computing [lambda][subscript 2] for Laplacian of Regular Models     424
Algorithm     425
Numerical Examples     425
Examples for Cartesian Product     426
Examples for Strong Cartesian Product     430
Examples for Direct Product     431
Spectral Method for Profile Reduction     433
Algorithm     433
Examples     433
Non-Compact Extended p-Sum     435
Exercises     436
Basic Concepts and Definitions of Graph Theory     437
Introduction     437
Basic Definitions     437
Vector Spaces Associated with a Graph     445
Matrices Associated with a Graph     448
Directed Graphs and their Matrices     456
Graphs Associated with Matrices     458
Planar Graphs: Euler's Polyhedron Formula      459
Maximal Matching in Bipartite Graphs     462
Greedy Algorithm and its Applications     465
Axiom System for a Matroid     465
Matroids Applied to Structural Mechanics     467
Cocycle Matroid of a Graph     470
Matroid for Null Basis of a Matrix     471
Combinatorial Optimisation: the Greedy Algorithm     472
Application of the Greedy Algorithm     473
Formation of Sparse Null Bases     474
References     477
Index     495
Index of Symbols     505

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews