Structural Mechanics: Graph and Matrix Methods

Transcription

1 Structural Mechanics: Graph and Matrix Methods A. Kaveh Department of Civil Engineering Technical University of Vienna Austria RESEARCH STUDIES PRESS LTD. Taunton, Somerset, England 0 JOHN WILEY & SONS INC. New York Chichester Toronto Brisbane Singapore

2 Contents Page number BASIC CONCEPTS AND DEFINITIONS OF GRAPH THEORY Introduction Basic Definitions Definition of a Graph Adjacency and Incidence Lsomorphic Graphs Graph Operations Walks, Trials and Paths Connectedness Cycles and Cut Sets Trees, Spanning Trees and Shortest Route Trees Different Types of Graphs Vector Spaces Associated with a Graph Cycle Space Cut Set Space Orthogonality Property Fundamental Cycle Bases Fundamental Cut Set Bases 1.5 Matrices Associated with a Graph Matrix Representation of a Graph Cycle Bases Matrices Cut Set Bases Matrices Directed Graphs and Their Matrices PlanarGraphs-Euler's Polyhedra Formula Planar Graphs ' Theorems for Planarity Maximal Matching in Bipartite Graphs Definitions Theorems on Matching Maximum Matching 28 Exercises TOPOLOGICAL PROPERTIES OF SKELETAL STRUCTURES Introduction Mathematical Model of a Skeletal Structure Union-intersection Method A Unifying Function An Expansion Process An Intersection Theorem 42 (xi)

3 (xi.) CONTENTS A Method for Determining the DKI and DSI of Structures Modifications on a Structure Identification Method The DSI of Structures; Special Methods Space Structures and Their Planar Drawings Admissible Drawing of a Space Structure The Degree of Statical Indeterminacy of Frames TheDegreeof Statical Indeterminacy of Space Trusses Suboptimal Drawing ofa Space Structure 68 Exercises 74 RIGIDITY OF SKELETAL STRUCTURES Introduction Definitions Complete Matching for the Recognition of Generic Independence A Decomposition Approach for the Recognition of Generic Independence Special Methods for Rigidity of Planar Trusses Simple Trusses Trusses in the Form of 2-trees A y-tree and its Rigidity Grid-form Trusses with Bracings Connectivity and Rigidity 93 Exercises 95 NETWORK FORMULATION OF STRUCTURAL ANALYSIS Introduction Theory of Networks Basic Concepts of Network Theory Topological Properties of Networks Algebraic Properties of Networks Formulation of Network Analysis Formulation of Structural Analysis 111 Exercises 115 MATRIX DISPLACEMENT METHOD Introduction Formulation Element Stiffness Matrices Stiffness Matrix of a General Element Stiffness Matrix ofa Bar Element Stiffness Matrix ofabeam Element Overall Stiffness Matrix of a Structure 139

4 CONTENTS (xiii) 5.5 General Loading Computational Aspects of the Matrix Displacement Method 143 Exercises 145 MATRIX FORCE METHOD Introduction Formulation Generalized Cycle Bases of a Graph Minimal and Optimal Generalized Cycle Bases Pattern Equivalence of Flexibility and Cycle Adjacency Matrices Minimal GCB of a Graph Selection of a Subminimal GCB: Practical Methods Method Method Method Force Method for the Analysis of Rigid-jointed Skeletal Structures Cycle Bases Selection: Topological Methods Cycle Bases Selection: Graph-theoretical Methods Formation of B o and B i Matrices Force Method for the Analysis of Pin-jointed Planar Trasses Associate Graphs for Selection of a Subminimal GCB Analysis of General Structures by the Force Method Algebraic Methods Algebraic-topological Methods 199 Exercises 202 ORDERING FOR BANDWIDTH AND PROFILE OPTIMIZATION Introduction 205 J 7.2 Preliminaries Pattern Equivalence of Stiffness and Cut Set Adjacency Matrices A Shortest Route Tree and its Properties Nodal Ordering for Bandwidth Optimization A Good Starting Node Primary Nodal Decomposition Transversal P of an SRT Nodal Ordering Examples A Connectivity Coordinate System for Nodal Ordering A Connectivity Coordinate System for Planar Graphs A Connectivity Coordinate System for Space Graphs Nodal Numbering for Profile Reduction Graph-theoretical Interpretation of Gaussian Elimination Element Ordering for Bandwidth Optimization of Flexibility Matrices 230

5 (xiv) CONTENTS An Associate Graph Distance Number of an Element Element Ordering Algorithms Example '" Nodal Numbering for Finite Element Models Bandwidth Reduction for Rectangular Matrices Definitions 236 Exercises CONDITIONING OF STRUCTURAL MATRICES Introduction Condition Numbers The Ratio of Extreme Eigenvalues Determinant of a Row Normalized Matrix The Ratio of Determinants Weighted Graph and an Admissible Member Optimally Conditioned Cycle Bases Formulation of the Problem r Suboptimally Conditioned Cycle Bases AlgorithmA ' AlgorithmB AlgorithmC Examples Optimally Conditioned Cut Set Bases Mathematical Formulation of the Problem Suboptimally Conditioned Cut Set Bases Algorithm Example 264 Exercises MATROIDS AND SKELETAL STRUCTURES Introduction Axiom Systems for a Matroid Definition in terms of Independence Definition in terms of Bases Definition in terms of Circuits Definition in terms of Rank Matroids Relevant to Structural Mechanics A Basis for a Finite Vector Space A Basis for Cycle Space of a Graph A Basis for Cut Set Space ofa Graph Cycle Matroid ofa Graph Cocycle Matroid of a Graph Rigidity Matroid ofa Graph Matroid for Null Basis ofa Matrix 275

6 CONTENTS (xv) 9.4 Combinatorial Optimization: the Greedy Algorithm Application of the Greedy Algorithm to the Force Method, A Combinatorial Approach Problems with Application of the Greedy Algorithm Concluding Remarks 282 Exercises A GRAPH THEORETICAL APPROACH FOR CONFIGURATION PROCESSING Introduction Mathematical Elements of Configuration Processing Operations on Graphs and Their Representations Special Graphs Some UsefuI Functions for Configuration Processing Translation Functions Rotation Functions Reflection Functions Projection Functions Geometry of Structures Extension to Hypergraphs 304 Exercises 306 REFERENCES 309 APPENDIX A 325 SOLUTIONS TO SELECTED EXERCISES 327 INDEX 337 INDEX OF SYMBOLS 343