Computational structural analysis and finite element methods

However, as the author makes clear, the early relationshipbetween graph theory and skeletal structures and finite element models is nowobvious: the structure of the mathematics is well s

Computational Structural Analysis and Finite Element Methods

A Kaveh

Tai ngay!!! Ban co the xoa dong chu nay!!!

Computational Structural Analysis and Finite Element Methods

.

A Kaveh

Computational Structural

Analysis and Finite Element Methods

A Kaveh

Centre of Excellence for Fundamental Studies in

Structural Engineering

School of Civil Engineering

Iran University of Science and Technology

Tehran

Iran

ISBN 978-3-319-02963-4 ISBN 978-3-319-02964-1 (eBook)

DOI 10.1007/978-3-319-02964-1

Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013956541

© Springer International Publishing Switzerland 2014

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts

in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication

of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law.

The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Recent advances in structural technology require greater accuracy, efficiency andspeed in the analysis of structural systems It is therefore not surprising that newmethods have been developed for the analysis of structures with complex config-urations and large number of elements

The requirement of accuracy in analysis has been brought about by the need fordemonstrating structural safety Consequently, accurate methods of analysis had to

be developed, since conventional methods, although perfectly satisfactory whenused on simple structures, have been found inadequate when applied to complexand large-scale structures Another reason why higher speed is required results fromthe need to have optimal design, where analysis is repeated hundred or eventhousands of times

This book can be considered as an application of discrete mathematics ratherthan the more usual calculus-based methods of analysis of structures and finiteelement methods The subject of graph theory has become important in science andengineering through its strong links with matrix algebra and computer science

At first glance, it seems extraordinary that such abstract material should have quitepractical applications However, as the author makes clear, the early relationshipbetween graph theory and skeletal structures and finite element models is nowobvious: the structure of the mathematics is well suited to the structure of thephysical problem In fact, could there be any other way of dealing with thisstructural problem? The engineer studying these applications of structural analysishas either to apply the computer programs as a black box, or to become involved ingraph theory, matrix algebra and sparse matrix technology This book is addressed

to those scientists and engineers, and their students, who wish to understand thetheory

The methods of analysis in this book employ matrix algebra and graph theory,which are ideally suited for modern computational mechanics Although this textdeals primarily with analysis of structural engineering systems, it should berecognised that these methods are also applicable to other types of systems such

as hydraulic and electrical networks

v

The author has been involved in various developments and applications of graphtheory in the last four decades The present book contains part of this researchsuitable for various aspects of matrix structural analysis and finite element methods,with particular attention to the finite element force method.

determining the degree of static indeterminacy of structures and provides systematicmethods for studying the connectivity properties of structural models In this chapter,force method of analysis for skeletal structures is described mostly based on the

of stiffness matrices These methods are especially suitable for the formation of

methods are investigated Efficient methods are presented for both node and elementordering Many new graphs are introduced for transforming the connectivity proper-

graph theory and algebraic graph theory methods for the force method of finiteelement meshes of low order and high order, respectively These new methods use

partitioning algorithms are developed for solution of multi-member systems, whichcan be categorized as graph theory methods and algebraic graph theory approaches

which are obtained by addition or removal of some members to regular structural

new optimization algorithm called SCSS is applied to the analysis procedure Then,using the SCSS and prescribed stress ratios, structures are analyzed and designed Inall the chapters, many examples are included to make the text easier to be understood

I would like to take this opportunity to acknowledge a deep sense of gratitude to

a number of colleagues and friends who in different ways have helped in thepreparation of this book Mr J C de C Henderson, formerly of Imperial College

of Science and Technology, first introduced me to the subject with most stimulatingdiscussions on various aspects of topology and combinatorial mathematics Profes-sor F Ziegler and Prof Ch Bucher encouraged and supported me to write thisbook My special thanks are due to Mrs Silvia Schilgerius, the senior editor of theApplied Sciences of Springer, for her constructive comments, editing and unfailingkindness in the course of the preparation of this book My sincere appreciation isextended to our Springer colleagues Ms Beate Siek and Ms G Ramya Prakash

I would like to thank my former Ph.D and M.Sc students, Dr H Rahami,

Dr M S Massoudi, Dr K Koohestani, Dr P Sharafi, Mr M J Tolou Kian,

Dr A Mokhtar-zadeh, Mr G R Roosta, Ms E Ebrahimi, Mr M Ardalan, and

Mr B Ahmadi for using our joint papers and for their help in various stages ofwriting this book I would like to thank the publishers who permitted some of ourpapers to be utilized in the preparation of this book, consisting of Springer-Verlag,John Wiley and Sons, and Elsevier

My warmest gratitude is due to my family and in particular my wife, Mrs.Leopoldine Kaveh, for her continued support in the course of preparing this book

Every effort has been made to render the book error free However, the authorwould appreciate any remaining errors being brought to his attention through hisemail-address: alikaveh@iust.ac.ir.

December 2013

.

of Graphs 1

1.1 Introduction 1

1.1.1 Definitions 1

1.1.2 Structural Analysis and Design 4

1.2 General Concepts of Structural Analysis 5

1.2.1 Main Steps of Structural Analysis 5

1.2.2 Member Forces and Displacements 6

1.2.3 Member Flexibility and Stiffness Matrices 7

1.3 Important Structural Theorems 11

1.3.1 Work and Energy 11

1.3.2 Castigliano’s Theorems 13

1.3.3 Principle of Virtual Work 13

1.3.4 Contragradient Principle 16

1.3.5 Reciprocal Work Theorem 17

1.4 Basic Concepts and Definitions of Graph Theory 18

1.4.1 Basic Definitions 19

1.4.2 Definition of a Graph 19

1.4.3 Adjacency and Incidence 20

1.4.4 Graph Operations 20

1.4.5 Walks, Trails and Paths 21

1.4.6 Cycles and Cutsets 22

1.4.7 Trees, Spanning Trees and Shortest Route Trees 23

1.4.8 Different Types of Graphs 23

1.5 Vector Spaces Associated with a Graph 25

1.5.1 Cycle Space 26

1.5.2 Cutset Space 26

1.5.3 Orthogonality Property 26

1.5.4 Fundamental Cycle Bases 27

1.5.5 Fundamental Cutset Bases 27

ix

1.6 Matrices Associated with a Graph 28

1.6.1 Matrix Representation of a Graph 29

1.6.2 Cycle Bases Matrices 32

1.6.3 Special Patterns for Fundamental Cycle Bases 33

1.6.4 Cutset Bases Matrices 34

1.6.5 Special Patterns for Fundamental Cutset Bases 34

1.7 Directed Graphs and Their Matrices 35

References 37

2 Optimal Force Method: Analysis of Skeletal Structures 39

2.1 Introduction 39

2.2 Static Indeterminacy of Structures 40

2.2.1 Mathematical Model of a Skeletal Structure 41

2.2.2 Expansion Process for Determining the Degree of Static Indeterminacy 42

2.3 Formulation of the Force Method 46

2.3.1 Equilibrium Equations 46

2.3.2 Member Flexibility Matrices 49

2.3.3 Explicit Method for Imposing Compatibility 52

2.3.4 Implicit Approach for Imposing Compatibility 53

2.3.5 Structural Flexibility Matrices 55

2.3.6 Computational Procedure 55

2.3.7 Optimal Force Method 60

2.4 Force Method for the Analysis of Frame Structures 60

2.4.1 Minimal and Optimal Cycle Bases 61

2.4.2 Selection of Minimal and Subminimal Cycle Bases 62

2.4.3 Examples 67

2.4.4 Optimal and Suboptimal Cycle Bases 69

2.4.5 Examples 72

2.4.6 An Improved Turn Back Method for the Formation of Cycle Bases 75

2.4.7 Examples 76

2.4.8 Formation of B0and B1Matrices 78

2.5 Generalized Cycle Bases of a Graph 82

2.5.1 Definitions 83

2.5.2 Minimal and Optimal Generalized Cycle Bases 85

2.6 Force Method for the Analysis of Pin-Jointed Planar Trusses 86

2.6.1 Associate Graphs for Selection of a Suboptimal GCB 86

2.6.2 Minimal GCB of a Graph 89

2.6.3 Selection of a Subminimal GCB: Practical Methods 89

2.7 Algebraic Force Methods of Analysis 91

2.7.1 Algebraic Methods 91

References 98

3 Optimal Displacement Method of Structural Analysis 101

3.1 Introduction 101

3.2 Formulation 101

3.2.1 Coordinate Systems Transformation 102

3.2.2 Element Stiffness Matrix Using Unit Displacement Method 105

3.2.3 Element Stiffness Matrix Using Castigliano’s Theorem 109

3.2.4 The Stiffness Matrix of a Structure 111

3.2.5 Stiffness Matrix of a Structure; an Algorithmic Approach 116

3.3 Transformation of Stiffness Matrices 118

3.3.1 Stiffness Matrix of a Bar Element 118

3.3.2 Stiffness Matrix of a Beam Element 120

3.4 Displacement Method of Analysis 122

3.4.1 Boundary Conditions 124

3.4.2 General Loading 125

3.5 Stiffness Matrix of a Finite Element 128

3.5.1 Stiffness Matrix of a Triangular Element 129

3.6 Computational Aspects of the Matrix Displacement Method 132

References 135

4 Ordering for Optimal Patterns of Structural Matrices: Graph Theory Methods 137

4.1 Introduction 137

4.2 Bandwidth Optimisation 138

4.3 Preliminaries 140

4.4 A Shortest Route Tree and Its Properties 142

4.5 Nodal Ordering for Bandwidth Reduction 142

4.5.1 A Good Starting Node 143

4.5.2 Primary Nodal Decomposition 145

4.5.3 Transversal P of an SRT 146

4.5.4 Nodal Ordering 146

4.5.5 Example 147

4.6 Finite Element Nodal Ordering for Bandwidth Optimisation 147

4.6.1 Element Clique Graph Method (ECGM) 149

4.6.2 Skeleton Graph Method (SkGM) 149

4.6.3 Element Star Graph Method (EStGM) 150

4.6.4 Element Wheel Graph Method (EWGM) 151

4.6.5 Partially Triangulated Graph Method (PTGM) 152

4.6.6 Triangulated Graph Method (TGM) 153

4.6.7 Natural Associate Graph Method (NAGM) 153

4.6.8 Incidence Graph Method (IGM) 155

4.6.9 Representative Graph Method (RGM) 156

4.6.10 Computational Results 157

4.6.11 Discussions 158

4.7 Finite Element Nodal Ordering for Profile Optimisation 160

4.7.1 Introduction 160

4.7.2 Graph Nodal Numbering for Profile Reduction 162

4.7.3 Nodal Ordering with Element Clique Graph (NOECG) 164

4.7.4 Nodal Ordering with Skeleton Graph (NOSG) 165

4.7.5 Nodal Ordering with Element Star Graph (NOESG) 166

4.7.6 Nodal Ordering with Element Wheel Graph (NOEWG) 166

4.7.7 Nodal Ordering with Partially Triangulated Graph (NOPTG) 167

4.7.8 Nodal Ordering with Triangulated Graph (NOTG) 167

4.7.9 Nodal Ordering with Natural Associate Graph (NONAG) 168

4.7.10 Nodal Ordering with Incidence Graph (NOIG) 168

4.7.11 Nodal Ordering with Representative Graph (NORG) 168

4.7.12 Nodal Ordering with Element Clique Representative Graph (NOECRG) 170

4.7.13 Computational Results 170

4.7.14 Discussions 170

4.8 Element Ordering for Frontwidth Reduction 171

4.9 Element Ordering for Bandwidth Optimisation of Flexibility Matrices 174

4.9.1 An Associate Graph 174

4.9.2 Distance Number of an Element 175

4.9.3 Element Ordering Algorithms 175

4.10 Bandwidth Reduction for Rectangular Matrices 177

4.10.1 Definitions 177

4.10.2 Algorithms 178

4.10.3 Examples 179

4.10.4 Bandwidth Reduction of Finite Element Models 181

4.11 Graph-Theoretical Interpretation of Gaussian Elimination 182

References 185

5 Ordering for Optimal Patterns of Structural Matrices: Algebraic Graph Theory and Meta-heuristic Based Methods 187

5.1 Introduction 187

5.2 Adjacency Matrix of a Graph for Nodal Ordering 187

5.2.1 Basic Concepts and Definitions 187

5.2.2 A Good Starting Node 190

5.2.3 Primary Nodal Decomposition 190

5.2.4 Transversal P of an SRT 191

5.2.5 Nodal Ordering 191

5.2.6 Example 192

5.3 Laplacian Matrix of a Graph for Nodal Ordering 192

5.3.1 Basic Concepts and Definitions 192

5.3.2 Nodal Numbering Algorithm 196

5.3.3 Example 196

5.4 A Hybrid Method for Ordering 196

5.4.1 Development of the Method 197

5.4.2 Numerical Results 198

5.4.3 Discussions 199

5.5 Ordering via Charged System Search Algorithm 203

5.5.1 Charged System Search 203

5.5.2 The CSS Algorithm for Nodal Ordering 208

5.5.3 Numerical Examples 211

References 213

6 Optimal Force Method for FEMs: Low Order Elements 215

6.1 Introduction 215

6.2 Force Method for Finite Element Models: Rectangular and Triangular Plane Stress and Plane Strain Elements 215

6.2.1 Member Flexibility Matrices 216

6.2.2 Graphs Associated with FEMs 220

6.2.3 Pattern Corresponding to the Self Stress Systems 221

6.2.4 Selection of Optimalγ-Cycles Corresponding to Type II Self Stress Systems 224

6.2.5 Selection of Optimal Lists 225

6.2.6 Numerical Examples 227

6.3 Finite Element Analysis Force Method: Triangular and Rectangular Plate Bending Elements 230

6.3.1 Graphs Associated with Finite Element Models 233

6.3.2 Subgraphs Corresponding to Self-Equilibrating Systems 233

6.3.3 Numerical Examples 240

6.4 Force Method for Three Dimensional Finite Element Analysis 244

6.4.1 Graphs Associated with Finite Element Model 244

6.4.2 The Pattern Corresponding to the Self Stress Systems 245

6.4.3 Relationship Betweenγ(S) and b1(A(S)) 248

6.4.4 Selection of Optimalγ-Cycles Corresponding to Type II Self Stress Systems 251

6.4.5 Selection of Optimal Lists 252

6.4.6 Numerical Examples 254

6.5 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Brick Element 257

6.5.1 Definition of the Independent Element Forces 258

6.5.2 Flexibility Matrix of an Element 259

6.5.3 Graphs Associated with Finite Element Model 261

6.5.4 Topological Interpretation of Static Indeterminacy 263

6.5.5 Models Including Internal Node 270

6.5.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems 271

6.5.7 Numerical Examples 272

References 279

7 Optimal Force Method for FEMS: Higher Order Elements 281

7.1 Introduction 281

7.2 Finite Element Analysis of Models Comprised of Higher Order Triangular Elements 281

7.2.1 Definition of the Element Force System 282

7.2.2 Flexibility Matrix of the Element 282

7.2.3 Graphs Associated with Finite Element Model 282

7.2.4 Topological Interpretation of Static Indeterminacies 284

7.2.5 Models Including Opening 287

7.2.6 Selection of an Optimal List Corresponding to Minimal Self-Equilibrating Stress Systems 290

7.2.7 Numerical Examples 291

7.3 Finite Element Analysis of Models Comprised of Higher Order Rectangular Elements 297

7.3.1 Definition of Element Force System 298

7.3.2 Flexibility Matrix of the Element 300

7.3.3 Graphs Associated with Finite Element Model 301

7.3.4 Topological Interpretation of Static Indeterminacies 303

7.3.5 Selection of Generators for SESs of Type II and Type III 307

7.3.6 Algorithm 308

7.3.7 Numerical Examples 309

7.4 Efficient Finite Element Analysis Using Graph-Theoretical Force Method: Hexa-Hedron Elements 316

7.4.1 Independent Element Forces and Flexibility Matrix of Hexahedron Elements 317

7.4.2 Graphs Associated with Finite Element Models 321

7.4.3 Negative Incidence Number 325

7.4.4 Pattern Corresponding to Self-Equilibrating Systems 325

7.4.5 Selection of Generators for SESs of Type II and Type III 331

7.4.6 Numerical Examples 334

References 338

8 Decomposition for Parallel Computing: Graph Theory Methods 341

8.1 Introduction 341

8.2 Earlier Works on Partitioning 342

8.2.1 Nested Dissection 342

8.2.2 A Modified Level-Tree Separator Algorithm 342

8.3 Substructuring for Parallel Analysis of Skeletal Structures 343

8.3.1 Introduction 343

8.3.2 Substructuring Displacement Method 344

8.3.3 Methods of Substructuring 346

8.3.4 Main Algorithm for Substructuring 348

8.3.5 Examples 348

8.3.6 Simplified Algorithm for Substructuring 350

8.3.7 Greedy Type Algorithm 352

8.4 Domain Decomposition for Finite Element Analysis 352

8.4.1 Introduction 353

8.4.2 A Graph Based Method for Subdomaining 354

8.4.3 Renumbering of Decomposed Finite Element Models 356

8.4.4 Computational Results of the Graph Based Method 356

8.4.5 Discussions on the Graph Based Method 359

8.4.6 Engineering Based Method for Subdomaining 360

8.4.7 Genre Structure Algorithm 361

8.4.8 Example 364

8.4.9 Computational Results of the Engineering Based Method 367

8.4.10 Discussions 367

8.5 Substructuring: Force Method 370

8.5.1 Algorithm for the Force Method Substructuring 370

8.5.2 Examples 373

References 376

9 Analysis of Regular Structures Using Graph Products 377

9.1 Introduction 377

9.2 Definitions of Different Graph Products 377

9.2.1 Boolean Operation on Graphs 377

9.2.2 Cartesian Product of Two Graphs 378

9.2.3 Strong Cartesian Product of Two Graphs 380

9.2.4 Direct Product of Two Graphs 381

9.3 Analysis of Near-Regular Structures Using Force Method 383

9.3.1 Formulation of the Flexibility Matrix 385

9.3.2 A Simple Method for the Formation of the Matrix AT 388

9.4 Analysis of Regular Structures with Excessive Members 389

9.4.1 Summary of the Algorithm 390

9.4.2 Investigation of a Simple Example 390

9.5 Analysis of Regular Structures with Some Missing Members 393

9.5.1 Investigation of an Illustrative Simple Example 393

9.6 Practical Examples 396

References 406

10 Simultaneous Analysis, Design and Optimization of Structures

Using Force Method and Supervised Charged System Search 407

10.1 Introduction 407

10.2 Supervised Charged System Search Algorithm 408

10.3 Analysis by Force Method and Charged System Search 409

10.4 Procedure of Structural Design Using Force Method and the CSS 414

10.4.1 Pre-selected Stress Ratio 415

10.5 Minimum Weight 420

References 432

be obtained, satisfying the boundary conditions In the matrix methods of structuralanalysis, one must also use these basic equations In order to provide a readyreference for the development of the general theory of matrix structural analysis,the most important basic theorems are introduced in this chapter, and illustratedthrough simple examples.

In the second part, basic concepts and definitions of graph theory are presented.Since some of the readers may be unfamiliar with the theory of graphs, simpleexamples are included to make it easier to understand the presented concepts

temperature changes, and support settlements, without undue deformation ing frames, industrial building, bridges, halls, towers, dams, reservoirs, tanks,retaining walls, channels, pavements are typical structures of interest to civilengineers

Build-A structure can be considered as an assemblage of members and nodes

space trusses, planar and space frames, single and double-layer grids are examples

A Kaveh, Computational Structural Analysis and Finite Element Methods,

DOI 10.1007/978-3-319-02964-1_1, © Springer International Publishing Switzerland 2014 1

Structures which must artificially be divided into members (elements) are called

The underlying principles for the analysis of other structures are more or less thesame Airplane, missile and satellite structures are of interest to the aviationengineer The analysis and design of a ship is interesting for a naval architect Amachine engineer should be able to design machine parts However, in this bookonly structures of interest to structural engineers are studied

Structural analysis is the determination of the response of a structure to external

design is the selection of a suitable arrangement of members, and a selection ofmaterials and member sections, to withstand the stress resultants (internal forces)

by a specified set of loads, and satisfy the stress and displacement constraints, andother requirements specified by the utilized code of practice The diagram shown in

In optimal design of structures this cycle should be repeated hundred andsometime thousands of times to reduce the weight or cost of the structure.Structural theories may be classified from different points of view as follows:Static versus dynamic;

Planar versus space;

Structure

Structural

Analysis

Structural Design

Stress Analysis

Fig 1.3 The cycle of analysis and design of a structure

Fig 1.2 Examples of continua (a) A plate (b) A dam

4 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

Linear versus non-linear;

Skeletal versus continua;

Statically determinate versus statically indeterminate

In this book, static analyses of linear structures are mainly discussed for thestatically determinate and indeterminate cases Here, both planar and space skeletalstructures and continua models are of interest

A correct solution of a structure should satisfy the following requirements:

1 Equilibrium: The external forces applied to a structure and the internal forcesinduced in its members should be in equilibrium at each node

2 Compatibility: The members should deform so that they all fit together

3 Force-displacement relationship: The internal forces and deformations satisfythe stress–strain relationships of the members

For structural analysis two basic methods are in use:

Force method: In this method, some of the internal forces and/or reactions aretaken as primary unknowns, called redundants Then the stress–strain relation-ship is used to express the deformations of the members in terms of external andredundant forces Finally, by applying the compatibility conditions that thedeformed members must fit together, a set of linear equations yield the values

of the redundant forces The stress resultants in the members are then calculatedand the displacements at the nodes in the direction of external forces are found

Displacement method: In this method, the displacements of the nodes necessary todescribe the deformed state of the structure are taken as unknowns The deforma-tions of the members are then calculated in terms of these displacements, and byuse of the stress–strain relationship, the internal forces are related to them Finally,

by applying the equilibrium equations at each node, a set of linear equations isobtained, the solution of which results in the unknown nodal displacements This

For choosing the most suitable method for a particular structure, the number ofunknowns is one of the main criteria A comparison for the force and displacementmethods can be made, by calculating the degree of static indeterminacy andkinematic indeterminacy As an example, for the truss structure shown in

number of unknowns for the force and displacement methods is 4 and 2, tively Efficient methods for calculating the indeterminacies are discussed in

for choosing the most suitable method is the conditioning of the flexibility and

A structure can be considered as an assembly of its members, subjected to externaleffects These effects will be considered as external loads applied at nodes, sinceany other effect can be reduced to such equivalent nodal loads The state of stress in

a member (internal forces) is defined by a vector,

The relationship between member forces and displacements can be written as:

Flexibility matrices can be written only for members supported in a stable manner,because rigid body motion of the undefined amplitude would otherwise result fromapplication of applied loads These matrices can be written in as many ways as thereare stable and statically determinate support conditions

Fig 1.4 Some simple structures (a) A planar truss (b) A planar frame (c) A simple planar truss

6 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

The stiffness and flexibility matrices can be derived using different approaches.For simple members like bar elements and beam elements, methods based on theprinciples of strength of materials or classical theory of structures will be sufficient.However, for more complicated elements the principle of virtual work or alterna-tively variational methods can be employed In this section, only simple members

two components of member forces From the equilibrium,

then only one end force need be specified in order to determine the state of stressthroughout the member The corresponding deformation of the member is simplythe elongation, and hence:

From Hooke’s law NR

Now consider a prismatic beam of a planar frame with length L and bending

related by the following two equilibrium equations:

Therefore, only two end-force components should be specified as internal forces

Using classical formulae, such as those of the strength of materials or deflection equations of the theory of structures, the force-displacement relation-ships can be established As an example, the flexibility matrix for a prismatic beamsupported as a cantilever is obtained using the differential equation of the elasticdeformation curve as follows:

B B

d d

L

L

m + d Fig 1.6 Internal forces and deformation of a bar element

8 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

3775

Using a similar method, for a simply supported beam with two moments acting

at the two ends, we have:

266

37

266664

377775

L

4EIL

266664

377775

Fig 1.8 Two sets of end

forces and displacements

for a beam element

10 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

It should be mentioned that both flexibility and stiffness matrices are symmetric,

on account of the Maxwell-Betti reciprocal work theorem proven in the nextsection More general methods for the derivation of member flexibility and stiffness

direction of that force is the product rdu

area under this curve represents the work done, denoted by W The area above thiscurve is the complementary work designated by W*

W*

u r

i

i

O d

this curve represents the density of the strain energy, and when integrated over thevolume of the member (or structure) results in the strain energy U The area to theleft of the stress–strain curve is the density of the complementary strain energy, and

by integration over the member (or structure) the complementary energy U* is

Since the work done by external actions on an elastic system is equal to the strainenergy stored internally in the system (work-energy law), therefore:

s

s

e e

1.3.2 Castigliano’s Theorems

The partial derivative of the strain energy with respect to a displacement, is equal to the force applied at the point and along the considered displacement.

∂U

The work done by a set of external forces P acting on a structure, in moving through the associated displacements v, is equal to the work done by some other set of forces R, which

is statically equivalent to P, moving through associated displacements u, which is ible with v Associated forces and displacements have the same lines of actions.

Using a statically admissible set of forces and the work equation, the bility relations between the deformations and displacements can be derived Alter-natively, employing a compatible set of displacements and the work equation, oneobtains the equations of equilibrium between the forces These approaches areelegant and practical.

compati-Dummy Load Theorem This theorem can be used to determine the conditions ofcompatibility Suppose that the deformed shape of each member of a structure isknown, then it is possible to find the deflection of the structure at any point by usingthe principle of virtual work For this purpose a dummy load (usually unit load) isapplied at the point and in the direction of required displacement, which is why it is

C 1

1

c

Fig 1.12 Three different systems capable of supporting the dummy load

14 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

It should be noted that the dummy load theorem is a condition on the geometry

of the structure In fact, once the deformations of elements are known, one can drawthe deflected shape of the structure, and the results obtained for the deflections willagree with those of the dummy load theorem

vertical deflection at node C, when the structure is subjected to a certain loading

A unit load is applied at C, and a set of internal forces statically equivalent to theunit load is chosen However, for such equivalent internal forces, there exists a widechoice of systems, since there are several numbers of structural possibilities whichcan sustain the load at C Three examples of such systems are shown inFig.1.12a–c

Obviously, system (a) will need more calculation because of being staticallyindeterminate

System (c) is used here, since it has a smaller number of members than (b), andsymmetry is also preserved Internal forces of the members in this system shown in

p

ffiffiffi2

p

ffiffiffi2

ffiffiffi2p

ffiffiffi2p

þ

ffiffiffi2p

dummy displacement applied

in the direction of unknownsactual external forces

Example 2 For the truss studied in Example 1, it is required to find the magnitude

of P by measuring the internal forces in the members of the truss

In these systems, the internal forces to be measured are shown in bold lines Due tothe symmetry, in both cases only two measurements are needed Applying thedummy-displacement theorem to system (a) yields:

ffiffiffi 2 p

2 þ r2dþ r1d

ffiffiffi 2 p

Fig 1.14 Element deformations equivalent to unit dummy displacement

16 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

In a general structure, if member forces R are related to external nodal loads P,

distortions v and nodal displacement u will be related by an equation similar to

The work done by {P1, P2, , P m } through displacements { δ 1 , δ 2 , , δ m } produced by {Q1, Q2, , Q n }, is the same as the work done by {Q1, Q2, , Q n } through displacements { Δ 1 , Δ 2 , , Δ n } produced by {P1, P2, , P m }; i.e.

The deflection at point i due to a load at point j is the same as deflection at j when the same load is applied at i.

The proof of the reciprocal work theorem is constructed by equating the strain

both sets of loads are applied simultaneously, while in the second sequence, loads

Some of the uses of the theory of graphs in the context of civil engineering are asfollows: A graph can be a model of a structure, a hydraulic network, a trafficnetwork, a transportation system, a construction system, or a resource allocation

D D D D

d d d P

P

P

Q Q Q Q

1

2

m

1 2 2 1

1 2 3

Fig 1.15 A structure subjected to two sets of loads

18 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

system, for example In this book, the theory of graphs is used as the model of askeletal structure, and it is employed also as a way of transforming the connectivityproperties of finite element meshes to those of graphs Many such graphs are

performed for optimal analysis of skeletal structures and finite element models.This part of the chapter will also enable the readers to develop their own ideas andmethods in the light of the principles of graph theory For further definitions and

as important as the mechanical properties of its members Hence, it is important torepresent a structure so that its topology can be understood clearly The graphmodel of a structure provides a powerful means for this purpose

Two or more members joining the same pair of nodes are collectively known as amultiple member, and a member joining a node to itself is called a loop A graph

The above definitions correspond to abstract graphs; however, a graph may bevisualised as a set of points connected by line segments in Euclidean space; thenodes of a graph are identified with points, and its members are identified as line

1.4.3 Adjacency and Incidence

members incident with that node Since each member has two end nodes, the sum ofnode-degrees of a graph is twice the number of its members

node-sets and member-sets of the two subgraphs The intersection of two subgraphsdoes not need to consist only of nodes, but it is usually considered to do so in the

L

suppressing or inserting nodes of degree 2 in the members

Fig 1.17 A graph, two of

its subgraphs, their union,

intersection and ring sum.

Fig 1.16 Simple and

non-simple graphs (a) A

simple graph (b) A graph

with loop and multiple

members

20 1 Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

1.4.5 Walks, Trails and Paths

of the members of a shortest path between these nodes

is a path of length 3 in which no node and no member is repeated

component of a graph S is a maximal connected subgraph, i.e it is not a subgraph ofany other connected subgraph of S

n n

1

3 n 2

n n

1

a

Fig 1.18 A walk, a trail

and a path in S (a) A walk

w in S (b) A trail t in S.

(c) A path P in S

1.4.6 Cycles and Cutsets

1.4.7 Trees, Spanning Trees and Shortest Route Trees

can be generated by the following simple algorithm:

the members incident to “0” as tree members Repeat the process of labelling with

“2” the unnumbered ends of all the members incident with nodes labelled as “1”,again recording the tree members This process terminates when each node of S islabelled and all the tree members are recorded This algorithm has many applica-

It is easy to prove that, for a tree T,

where M(T) and N(T) are the numbers of members and nodes of T, respectively

the number of chords is given by:

In order to simplify the study of the properties of graphs, different types of graphshave been defined Some important ones are as follows: