Matrix Methods ofStructural Analysis

Matrix Methods ofStructural Analysis

S.S. BhavikattiEmeritus Professor

BVBCET, HubliFormerly Professor and Dean

NITK, Surathkaland

SDMLET, Dharwad andPrincipal, RYMEC, Bellary

NEW DELHI • BANGALORE

Published by

I.K. International Publishing House Pvt. Ltd.S-25, Green Park ExtensionUphaar Cinema MarketNew Delhi – 110 016 (India)

E-mail: info@ikinternational.com

Website: www.ikbooks.com

ISBN 978-81141-35-9

© 2014 I.K. International Publishing House Pvt. Ltd.

10 9 8 7 6 5 4 3 2 1

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or any means: electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from the publisher.

Published by Krishan Makhijani for I.K. International Publishing House Pvt Ltd., S-25, Green Park Extension, Uphaar Cinema Market, New Delhi – 110 016 and Printed by Rekha Printers Pvt. Ltd., Okhla Industrial Area, Phase II, New Delhi – 110 020

Preface

Systematic development of consistent deformation and slope deflection methods to suit computer programming led to flexibility and stiffness matrix methods. To get better understanding of the methods structure approach is explained first. Later stiff-ness matrix method by element approach is dealt with. However, flexibility matrix method by element approach is not dealt with in this book since it has not found favour with the program developers. Stiffness matrix method by element approach is more important since it is this method which has led to most modern method of structural analysis, namely, finite element method. Hence, I have preferred to deal with stiffness matrix method in detail. A chapter is given at the end to give guidelines for developing computer program.

I hope that the book will find acceptance among students and teachers since it gives systematic approach to Matrix Method of Structural Analysis.

Author

Contents

Preface v

1. Introduction 1 1.1 Static Indeterminacy 2

1.2 Degree of Static Indeterminacy 4

1.3 Kinematic Indeterminacy 11

1.4 Generalized Coordinates 14

1.5 Generalized Force and Deformation 14

1.6 Flexibility Matrix 15

1.7 Flexibility Equation 17

1.8 Stiffness Matrix 17

1.9 Stiffness Equation 18

1.10 Relationship between Flexibility and Stiffness Matrices 18

2. Review of Consistent Deformation and SlopeDefl ection Methods 21

2.1 Consistent Deformation Method 21

2.2 Methods of Finding Displacements 23

2.3 Slope Defl ection Method 30

3. Flexibility Matrix Method: Structures Approach 38 3.1 Flexibility Equation 38

3.2 The Method 39

3.3 Analysis of Beams 40

3.4 Analysis of Plane Rigid Frames 57

3.5 Analysis of Pin-jointed Plane Frames 63

4. Stiffness Matrix Method: Structures Approach 70 4.1 Stiffness Equation 70

4.2 The Method 71

4.3 Analysis of Beams 72

4.4 Analysis of Frames 81

4.5 Analysis of Pin-jointed Plane Frames 92

4.6 Comparison of Methods 96

viii Contents

5. Introduction to Element Approach 98 5.1 New Terminology 98

5.2 Relationship between Global and Local Stiffness Matrices 103

5.3 Analysis of Pin-jointed Plane Frames 103

5.4 Analysis of Pin-jointed Space Frames 114

5.5 Continuous Beams 119

5.6 Analysis of Plane Frames 135

5.7 Analysis of Rigid Jointed Three-dimensional Frames 143

5.8 Analysis of Grids 148

5.9 General Procedure of Analysis by Stiffness Matrix-Element 156Approach

6. Computer Programming Preliminaries 160 6.1 Input 160

6.2 Processing 165

6.3 Output 166

6.4 Techniques of Saving Computer Memory Requirements 166

6.5 Software Developers to Note 168

6.6 Software Users to Note 169

Appendix 171

Index 173

1 Introduction

The various types of structures we come across may be classified into the following groups:

1. One dimensional (skeleton)

2. Two dimensional (plates)

3. Three dimensional

One dimensional, also known as skeleton or framed structures consist of one or more number of members which have one dimension (length) considerably larger compared to any dimension in its cross-section. Examples of such structures are pin-jointed frames, rigid jointed frames and grids.

Two-dimensional structures are also known as plates. Common civil engineering examples are slabs and walls. These structures have two dimensions considerably larger compared to their third dimension (thickness).

Three-dimensional structures have their dimension in any direction considerably large. Examples of such structures are thick plates, soil mass under footing.

Matrix method of structured analysis is suitable only for skeleton structures. Hence, in this book only skeleton structures are dealt with. In fact, in late 1950s the term matrix method was used even for the analysis of two-dimensional structures.

In 1960, R.W. Clough introduced the term finite element analysis when he used energy concept for the analysis of plates.

A structural engineer comes across the design of indeterminate structures. The first step to analyse indeterminate structures is that he should know the analysis of sim-ple determinate structures. Then he should know the relationships among forces and deformations (deflection and rotations) at the ends of members.

There are basically two methods of finding force and deformation relations:

1. Based on equilibrium compatibility and superposition: Examples of such methods are double integration (Mecaulay’s) method, moment area method and conjugate beam method.

2. Based on energy concepts: Examples are strain energy method, Castigliano’s method and virtual work method. When loads considered are virtual it leads to unit load method.

After gaining the above knowledge one can analyse indeterminate structures by consistent deformation method or by slope deflection method. These methods were

2 Matrix Method of Structural Analysis

considered suitable if the degree of indeterminacy is less, since they involve solution of simultaneous equations.

In the period 1930s, the problem of solving large size simultaneous equations led to analysis by distribution procedures. Hardy Cross’s moment distribution and Kani’s rotation contribution method became popular methods of structural analysis in the pre-computer era. Three to five-storey building frames were analysed conveniently by Kani’s method.

As the need for the analysis of very large structures developed, structural engineers started looking at the use of approximate methods like

1. Substitute frame method for vertical loads.

2. Portal method, cantilever method and factor method for wind load analysis.

During 1950s, development of high-speed computers gave rise to numerical meth-ods for solving simultaneous equations. Hence, it renewed the interest of analysts again in consistent deformation and slope deflection methods. The systematic devel-opment of these methods to suit computer application gave rise to matrix method of structural analysis. The development of consistent deformation method has led to flex-ibility matrix method while the development of slope deflection method has led to stiffness matrix method.

In this chapter the new technical terms used in matrix method of structure analysis are defined and explained.

1.1 STATIC INDETERMINACY

A structure which can be analysed with the use of equations of equilibrium of forces only is called statically determinate structure. Simply supported beams, beams with one end hinged and another on roller and cantilever beams are determinate beams. Similarly, pin-jointed plane frames with only three reaction components and num-ber of members equal to 2j-3 where ‘j’ is number of joints and single bay, single storey rigid frames with only three reaction components are determinate structures. Figure 1.1 shows some of the determinate structures.

Fig. 1.1 Statically determinate structures.

Introduction 3

A structure which cannot be analysed with equations of equilibrium only is a stati-cally indeterminate structure. Figure 1.2 shows some of the indeterminate beams and frames, where support reactions are more than three. Figure 1.3 shows indeterminate frames where indeterminacy is due to more number of member forces than the number of available independent equations.

1

2 42 4

1

3

3

1

2 4

31

2 5

3 64

2 32 4

1 3 1

4

Fig. 1.2 Statically indeterminate structures.

(a) (b)

Fig. 1.3 Indeterminate due to additional members.

continue

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z #

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