Analysis of Geometrically Nonlinear Structures

276 Pages·2003·8.415 MB·English

by  Robert Levy, William R. Spillers (auth.)

Analysis of Geometrically Nonlinear Structures
Analysis of Geometrically
Nonlinear Structures
Second Edition
by
Robert Levy
Technion - Israel Institute of Technology,
Haifa, Israel
and
William R. Spillers
New Jersey Institute of Technology,
Newark, New Jersey, U.S.A.
Springer-Science+Business Media, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6438-7 ISBN 978-94-017-0243-0 (eBook)
DOI 10.1007/978-94-017-0243-0
Printed on acid-free paper
All Rights Reserved
© 2003 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2003.
Softcover reprint of the hardcover 1s t edition 2003
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permission from the Publisher, with the
exception of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
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To the late Jewell Garrelts of
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VII
BOOK CONTENTS
Preface XIII
Using the CD xv
Chapter 1 Overview 1
1.1 Introduction 1
1.2 Newton's Method 2
1.3 Restrictions of Small Strain 4
1.4 Stress Stiffening 5
1.5 Buckling 6
1.5.1 Snap Through 7
1.5.2 Thermal Buckling 8
1.5.3 Euler's Column 8
1.5.4 Moment Distribution 9
1.5.5 Eigenvalue Approach 12
1.5.6 An Exact Approach 15
1.6 Prestress 17
1.6.1 A Fundamental Theorem 18
1.7 Problems 21
Chapter 2 Linear Structural Analysis 23
2.1 Introduction 23
2.2 The Truss Problem 25
2.3 Computer Programs 29
2.3.1 Programs PI-TR3D.FOR and P2-TR2D.FOR 30
2.4 Examples 30
2.4.1 Example 2.1. A 3-Bar Truss 30
2.4.2 Example 2.2. A 24-Bar Space Truss Dome 32
2.4.3 Example 2.3. A 72-Bar Double Layered Grid 35
2.5 Problems 39
Chapter 3 "Exact" Analysis of Trusses 41
3.1 Introduction 41
3.2 Linearization of the Joint Equilibrium Equations 42
3.3 The Geometric Stiffness Matrix 43
3.4 Overall Buckling 47
VIII
3.5 Computer Programs 49
3.5.1 Programs P3-TR3DNL.FOR and P4-TR2DNL.FOR 49
3.5.1.1 Nonlinear Analysis and Newton's Method 51
3.5.2 Programs P5-BUCK3D.FOR and P6-BUCK2D.FOR 51
3.6 Examples 52
3.6.1 Example 3.1. Biot's 2-Bar Prestressed Truss 52
3.6.2 Example 3.2. A Prestressed Cablenet 54
3.6.3 Example 3.3. Plane Truss Buckling 57
3.6.4 Example 3.4. Buckling of a Symmetric 2-Bar Truss 61
3.6.5 Example 3.5. Buckling of a 4-Bar Shallow Space Truss 63
3.6.6 Example 3.6. Buckling of the 24-Bar Dome 66
3.7 Problems 68
Chapter 4 Nonlinear Analysis of Plane Frames 71
4.1 Linear Analysis 71
4.2 Computer Program P7-FR2D.FOR 75
4.2.1 Example 4.1 A Simple Plane Frame 76
4.3 The Geometric Stiffness Matrix 78
4.4 Computer Program P8-FR2DNL.FOR 81
4.4.1 Examples 82
4.4.1.1 Example 4.2. Buckling of a Portal Frame 82
4.4.1.2 Example 4.3. Buckling of a Plane Frame 85
4.4.1.3 Example 4.4. Large Rotations of a Circular Cantilever
Beam 87
4.5 Problems 90
Chapter 5 Nonlinear Analysis of Space Frames 93
5.1 Introduction 93
5.2 Linear Analysis 93
5.3 Computer Program P9-FR3D.FOR 98
5.3.1 Example 5.1. A 2-Storey Simple Space Frame 99
5.3.2 Example 5.2. A More Complex Space Frame 102
5.4 Nonlinear Effects 105
5.5 The Geometric Stiffeness Matrix 107
5.6 Computer Programs PI0-FR3DNL2.FOR, PII-FR3DNL3.FOR and
P12-FR3DNLSR 113
5.6.1 Example 5.3. Lateral Torsional Buckling 114
5.7 Problems 119
Chapter 6 Nonlinear Analysis of Membranes 121
6.1 Introduction 121
6.2 The Geometric Stiffness Matrix of the Plane Stress Triangular Finite
Element 121
IX
6.3 Three Dimensional Members 127
6.4 A Direct Alternative Derivation of the Geometric Stiffness Matrix of
Three-Dimensional Membranes 131
6.5 Computer Programs 132
6.5.1 Program P13-FEMPS.FOR 132
6.5.2 Program PI4-MEMBR.FOR 132
6.5.3 Program PI5-MEMNL.FOR 133
6.6 Examples 133
6.6.1 Example 6.1. A Deep Beam 133
6.6.2 Example 6.2. A Spherical Cap 136
6.6.3 Example 6.3. A Flat Stretched Membrane 139
6.7 Problems 148
Chapter 7 Cablenets and Fabric Structures 151
7.1 Introduction 151
7.2 Basic Methods of Shape Finding 152
7.2.1 Deformed Shape 152
7.2.2 Force Density Method 153
7.2.3 Grid Method 154
7.2.4 Smoothing 155
7.3 The Grid Method 155
7.3.1 Example 7.1. A Piece of a Cross Arched Skylight 156
7.3.2 Example 7.2. A Squared Base Skylight 158
7.4 Smoothing 163
7.5 A More Complex Example 166
7.6 Membrane Finite Element Model 177
7.7 Patterning 180
7.8 Computer Programs For Cable Nets And Fabric Structures 182
7.8.1 LAYOUT.FOR (The grid method) 182
7.8.2 LPLOT1.FOR 183
7.8.3 LAYOUTPLOT.FOR 183
7.8.4 PATTERN. FOR 184
7.9 Problems 186
Chapter 8 Three-Dimensional Beam-Columns 187
8.1 Introduction 187
8.2 The Equation of Three-Dimensional Beam-Columns 188
8.3 The Member Stiffness Matrix 191
8.4 Numerical Solution 194
8.5 Special Cases 196
8.5.1 The Elastic Beam 196
8.5.2 Two-Dimensional Beam-Column 196
x
8.5.3 Lateral Buckling 197
8.5.4 A More Complex Case 197
8.5.5 The Effect ofInitial Torsion 199
8.6 Problems 203
Chapter 9 Nonlinear Analysis of Shells 205
9.1 Introduction 205
9.2 The Geometric Stiffness Matrix of Triangular Eelement Shells 206
9.2.1 In-Plane Contribution of the Triangular Membrane
Element 207
9.2.2 In-Plane Contribution of the Triangular Plate Bending
Element 207
9.2.3 Out-Of-Plane Contribution to the Shell Geometric Stiffness
Matrix 210
9.3 Element Pure Deformational Rotations and Translations 214
9.3.1 Stress Retrieval in the Membrane Finite Element 214
9.3.2 Stress Retrieval in the Plate Finite Element 215
9.4 Computer Program PI6-SHELLNL.FOR 218
9.5 Examples 219
9.5.1 Example 9.1. Bending of a Cantilever Plate 220
9.5.2 Example 9.2. Simply Supported Plate 223
9.5.3 Example 9.3. Analysis of a Shallow Cylindrical Shell 226
9.5.4 Example 9.4. Leicester's Shallow Spherical Shell 233
9.5.5 Example 9.5. Mescall's Shallow Spherical Shell 234
9.5.6 Example 9.6. Open Hemispherical Shell 235
9.5.7 Example 9.7. Lateral Buckling of an L-Frame 236
9.6 Some Remarks 237
9.7 Problems 237
References 239
Appendix 1 Member Stiffness When Beam-Column Effects are
Included 243
Appendix 2 Determinants 247
Appendix 3 The Rotation Matrix 249
Appendix 4 Perturbation Methods Applied to Plane Beams 257
Appendix 5 Introduction to Computer Programs 259
A5.1 Introduction 259
XI
AS.2 Space Trusses 259
AS.3 Plane Frames 260
AS.4 Listing for TR3D.FOR 261
AS.S Listing for FR2D.FOR 263
Appendix 6 Graphics on a PC 267
A6.1 Introduction 267
A6.2 Plotting in 2-D 268
A6.3 Drawing Lines in 2-D 269
Index 271
XIII
PREFACE
The availability of computers has, in real terms, moved forward the practice
of structural engineering. Where it was once enough to have any analysis
given a complex configuration, the profession today is much more
demanding. How engineers should be more demanding is the subject of this
book.
In terms of the theory of structures, the importance of geometric
nonlinearities is explained by the theorem which states that
"In the presence ofp restress, geometric nonlinearities are of the same order
of magnitude as linear elastic effects in structures. "
This theorem implies that in most cases (in all cases of incremental
analysis) geometric nonlinearities should be considered. And it is well
known that problems of buckling, cable nets, fabric structures, ... REQUIRE
the inclusion of geometric nonlinearities.
What is offered in the book which follows is a unified approach (for
both discrete and continuous systems) to geometric nonlinearities which
incidentally does not require a discussion of large strain. What makes this all
work is perturbation theory. Let the equations of equilibrium for a system be
written as
where P represents the applied loads, F represents the member forces or
stresses, and N represents the operator which describes system equilibrium.
(This equation can also be thought of as the matrix equation of node
equilibrium for a discrete system.) Under a load perturbation dP this system
responds as
dN TF + NT dF = dP
It is the first term in the above equation which describes so-called geometric
nonlinearity and it is the second term which returns linear theory. For a
discrete system it turns out to be a relatively simple matter to convert this
equation into the usual