Integrated Matrix Analysis of Structures - Theory and Computation (Kluwer International Series in Engineering & Computer Science)

334 Pages·2001·4.47 MB·English

by  Mario Paz

INTEGRATED MATRIX ANALYSIS
OF STRUCTURES
Theory and Computation
by
Mario Paz
Speed Scientific School
University of Louisville
Louisville, Kentucky
and
William Leigh
University of Central Florida
Orlando, Florida
00
KLUWER ACADEMIC PUBLISHERS
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Library of Congress Cataloging-in-Publication Data
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from the Library of Congress.
Copyright © 2001 by Kluwer Academic Publishers
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo-
copying, recording, or otherwise, without the prior written permission of the
publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell,
Massachusetts 02061.
Printed on acid free paper.
Printed in the United States of America.
To Annis

Contents
PREFACE xi
ACKNOWLEDGMENTS xiii
I BEAMS: ANALYTICAL METHODOLOGY
1.1 Introduction 1
1.2 Elements, Nodes and Nodal Coordinates 2
1.3 Shape Functions and Stiffness Coefficients 3
1.4 Element Stiffness Matrix 6
1.5 Assembly of the System Stiffness Matrix 7
1.6 System Force Vector 10
1.7 System Nodal Displacements and Support Reactions 15
1.8 Element End Forces 17
1.9 End Releases in Beam Elements 20
1.10 Support Displacements 25
1.11 Element Displacement Functions 30
1.12 Shear Force and Bending Moment Functions 32
1.13 The Temperature Effect 33
1.14 Elastic Supports 37
1.15 Analytical Problems 42
1.16 Practice Problems 49
2 BEAMS: COMPUTER APPLICATIONS
2.1 Computer Program 55
2.2 Problems 86
3 PLANE FRAMES
3.1 Introduction 93
3.2 Stiffness Coefficients for Axial Forces 93
3.3 Displacement Functions for Axially Loaded Beams 95
3.4 Element Stiffness Matrix for Plane Frame Elements 96
3.5 Coordinate Transformation 97
3.6 Equivalent Nodal Force for an Axially Loaded Beam
Element 99
3.7 Element End Forces 99
3.8 Inclined Roller Supports 109
3.9 Analytical Problems 133
3.10 Practice Problems 137
4 GRID FRAMES
4.1 Introduction 145
4.2 Torsional Effects 145
4.3 Stiffness Matrix for a Grid Element 147
4.4 Transformation of Coordinates 148
4.5 Analysis of Grid Frames 150
4.6 Problems 167
5 SPACE FRAMES
5.1 Introduction 171
5.2 Element Stiffness Matrix 171
5.3 Transformation of Coordinates 173
5.4 Reduced System Stiffness Equation 180
5.5 Analytical Problems 193
5.6 Practice Problems 200
6 PLANE TRUSSES
6.1 Introduction 203
6.2 Element Stiffness Matrix in Local Coordinates 203
6.3 Transformation of Coordinates 204
6.4 Element Stiffness Matrix in Global Coordinates 205
6.5 Assemblage of System Stiffness Matrix 206
6.6 End-Forces for an Element of a Truss 206
6.7 Problems 217
viii
7 SPACE TRUSSES
7.1 Introduction 221
7.2 Element Stiffness Matrix of a Space Truss
Local Coordinates 221
7.3 Transformation of the Element Stiffness Matrix 223
7.4 Element Axial Force 224
7.5 Assemblage of the System Stiffness Matrix 225
7.6 Problems 236
STATIC CONDENSATION AND SUBSTRUCTURING
8.1 Introduction 239
8.2 Static Condensation 239
8.3 Substructuring 244
8.4 Problems 259
9 INTRODUCTION TO FINITE ELEMENT METHOD
9.1 Introduction 261
9.2 Plane Elasticity Problems 262
9.3 Plate Bending 285
9.4 Rectangular Finite Element for Plate Bending 285
9.5 Problems 298
APPENDIX I
Equivalent Nodal Forces 301
APPENDIX II
Displacement Functions for Fixed-End Beams 305
GLOSSARY 309
SELECTED BIBLIOGRAPHY 317
INDEX 319
ix

Preface
This is the first volume of a series of integrated textbooks for the analysis and
design of structures. The series is projected to include a first volume in Matrix
Structural Analysis to be followed by volumes in Structural Dynamics and
Earthquake Engineering as well as other volumes dealing with specialized or
advanced topics in the analysis and design of structures. An important objective in
the preparation of these volumes is to integrate and unify the presentation using
common notation, symbols and general format. Furthermore, all of these volumes
will be using the same structural computer program, SAP2000, developed and
maintained by Computers and Structures, Inc., Berkeley, California. SAP2000 is a
commercially available program with a most powerful capability for the analysis
and computer aided design of structures under static or dynamic loads that may
result in elastic or inelastic behavior of the structure. The CD-ROM included in this
volume contains the student version of SAP2000 with limitations in both the size
and behavior of the structure. Even with these limitations, the student version of
SAP2000 is extremely useful in introducing the student to a powerful structural
program used by professional engineers world-wide. In particular, for this first
volume on Matrix Structural Analysis, the only perceptible limitation of the student
version is the size of the structure. The CD-ROM accompanying this book
contains, in addition to the student version of SAP2000, various tutorials, example
problems and a complete set of user manuals for SAP2000.
In order to maintain the continuity of the text, we present in the various
chapters the development of mathematical formulas in a section on Analytical
Problems, which is followed by another section containing Practice Problems.
Throughout the book numerous Illustrative Examples are given with detailed
numerical hand solutions as well as solutions obtained with SAP2000. In using
SAP2000, the student is guided (as he or she becomes acquainted with the program)