Computers and Structures, Inc.
Berkeley, California, USA
Third Edition
Reprint January 2002
Three-Dimensional
Static and Dynamic
Analysis of Structures
A Physical Approach
With Emphasis on Earthquake Engineering
Edward L. Wilson
Professor Emeritus of Structural Engineering
University of California at Berkeley
Copyright

by Computers and Structures, Inc. No part of this publication may be
reproduced or distributed in any form or by any means, without the prior written
permission of Computers and Structures, Inc.
Copies of this publication may be obtained from:
Computers and Structures, Inc.
1995 University Avenue
Berkeley, California 94704 USA
Phone: (510) 845-2177
FAX: (510) 845-4096
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
Copyright Computers and Structures, Inc., 1996-2001
The CSI Logo is a trademark of Computers and Structures, Inc.
SAP90, SAP2000, SAFE, FLOOR and ETABS are trademarks of
Computers and Structures, Inc.
ISBN 0-923907-00-9
STRUCTURAL ENGINEERING IS

effects of modal damping on the results of a dynamic analysis.
Appendix H, on the speed of modern personal computers, has been updated. It is now
possible to purchase a personal computer for approximately $1,500 that is 25 times
faster than a $10,000,000 CRAY computer produced in 1974.
Several other additions and modifications have been made in this printing. Please send
your comments and questions
to

.
Edward L. Wilson
April 2000
Personal Remarks
My freshman Physics instructor dogmatically warned the class “do not use an equation
you cannot derive.” The same instructor once stated that “if a person had five minutes to
solve a problem, that their life depended upon, the individual should spend three
minutes reading and clearly understanding the problem." For the past forty years these
simple, practical remarks have guided my work and I hope that the same philosophy has
been passed along to my students. With respect to modern structural engineering, one
can restate these remarks as “do not use a structural analysis program unless you fully
understand the theory and approximations used within the program” and “do not create
a computer model until the loading, material properties and boundary conditions are
clearly defined.”
Therefore, the major purpose of this book is to present the essential theoretical
background so that the users of computer programs for structural analysis can
understand the basic approximations used within the program, verify the results of all
analyses and assume professional responsibility for the results. It is assumed that the
reader has an understanding of statics, mechanics of solids, and elementary structural
analysis. The level of knowledge expected is equal to that of an individual with an
undergraduate degree in Civil or Mechanical Engineering. Elementary matrix and
vector notations are defined in the Appendices and are used extensively. A background

former students and professional colleagues. Their contributions are acknowledged.
Ashraf Habibullah, Iqbal Suharwardy, Robert Morris, Syed Hasanain, Dolly Gurrola,
Marilyn Wilkes and Randy Corson of Computers and Structures, Inc., deserve special
recognition. In addition, I would like to thank the large number of structural engineers
who have used the TABS and SAP series of programs. They have provided the
motivation for this publication.
The material presented in the first edition of

Three Dimensional Dynamic Analysis of
Structures

is included and updated in this book. I am looking forward to additional
comments and questions from the readers in order to expand the material in future
editions of the book.
Edward L. Wilson
July 1998
CONTENTS
1. Material Properties
1.1 Introduction 1-1
1.2 Anisotropic Materials 1-1
1.3 Use of Material Properties within Computer Programs 1-4
1.4 Orthotropic Materials 1-5
1.5 Isotropic Materials 1-5
1.6 Plane Strain Isotropic Materials 1-6
1.7 Plane Stress Isotropic Materials 1-7
1.8 Properties of Fluid-Like Materials 1-8
1.9 Shear and Compression Wave Velocities 1-9
1.1 Axisymmetric Material Properties 1-10
1.11 Force-Deformation Relationships 1-11
1.12 Summary 1-12

4. One-Dimensional Elements
4.1 Introduction 4-1
4.2 Analysis of an Axial Element 4-2
4.3 Two-Dimensional Frame Element 4-4
4.4 Three-Dimensional Frame Element 4-8
4.5 Member End-Releases 4-12
4.6 Summary 4-13
5. Isoparametric Elements
5.1 Introduction 5-1
5.2 A Simple One-Dimensional Example 5-2
5.3 One-Dimensional Integration Formulas 5-4
5.4 Restriction on Locations of Mid-Side Nodes 5-6
5.5 Two-Dimensional Shape Functions 5-6
5.6 Numerical Integration in Two Dimensions 5-10
5.7 Three-Dimensional Shape Functions 5-12
5.8 Triangular and Tetrahedral Elements 5-14
5.9 Summary 5-15
CONTENTS iii
5.1 References 5-16
6. Incompatible Elements
6.1 Introduction 6-1
6.2 Elements With Shear Locking 6-2
6.3 Addition of Incompatible Modes 6-3
6.4 Formation of Element Stiffness Matrix 6-4
6.5 Incompatible Two-Dimensional Elements 6-5
6.6 Example Using Incompatible Displacements 6-6
6.7 Three-Dimensional Incompatible Elements 6-7
6.8 Summary 6-8
6.9 References 6-9
7. Boundary Conditions and General Constraints

9.1 Introduction 9-1
9.2 Basic Assumptions 9-2
9.3 Displacement Approximation 9-3
9.4 Introduction of Node Rotation 9-4
9.5 Strain-Displacement Equations 9-5
9.6 Stress-Strain Relationship 9-6
9.7 Transform Relative to Absolute Rotations 9-6
9.8 Triangular Membrane Element 9-8
9.9 Numerical Example 9-8
9.1 Summary 9-9
9.11 References 9-10
10. Shell Elements
10.1 Introduction 10-1
10.2 A Simple Quadrilateral Shell Element 10-2
10.3 Modeling Curved Shells with Flat Elements 10-3
10.4 Triangular Shell Elements 10-4
10.5 Use of Solid Elements for Shell Analysis 10-5
10.6 Analysis of The Scordelis-Lo Barrel Vault 10-5
10.7 Hemispherical Shell Example 10-7
10.8 Summary 10-8
10.9 References 10-8
CONTENTS v
11. Geometric Stiffness and P-Delta Effects
11.1 Definition of Geometric Stiffness 11-1
11.2 Approximate Buckling Analysis 11-3
11.3 P-Delta Analysis of Buildings 11-5
11.4 Equations for Three-Dimensional Buildings 11-8
11.5 The Magnitude of P-Delta Effects 11-9
11.6 P-Delta Analysis without Computer Program Modification 11-10
11.7 Effective Length - K Factors 11-11

14.4 Inverse Iteration 14-3
14.5 Gram-Schmidt Orthogonalization 14-4
14.6 Block Subspace Iteration 14-5
14.7 Solution of Singular Systems 14-6
14.8 Generation of Load-Dependent Ritz Vectors 14-7
14.9 A Physical Explanation of the LDR Algorithm 14-9
14.1 Comparison of Solutions Using Eigen And Ritz Vectors 14-11
14.11 Correction for Higher Mode Truncation 14-13
14.12 Vertical Direction Seismic Response 14-15
14.13 Summary 14-18
14.14 References 14-19
15. Dynamic Analysis Using Response Spectrum Seismic Loading
15.1 Introduction 15-1
15.2 Definition of a Response Spectrum 15-2
15.3 Calculation of Modal Response 15-4
15.4 Typical Response Spectrum Curves 15-4
15.5 The CQC Method of Modal Combination 15-8
15.6 Numerical Example of Modal Combination 15-9
15.7 Design Spectra 15-12
15.8 Orthogonal Effects in Spectral Analysis 15-13
15.8.1 Basic Equations for Calculation of Spectral Forces 15-14
15.8.2 The General CQC3 Method 15-16
15.8.3 Examples of Three-Dimensional Spectra Analyses 15-17
15.8.4 Recommendations on Orthogonal Effects 15-21
15.9 Limitations of the Response Spectrum Method 15-21
15.9.1 Story Drift Calculations 15-21
15.9.2 Estimation of Spectra Stresses in Beams 15-22
CONTENTS vii
15.9.3 Design Checks for Steel and Concrete Beams 15-22
15.9.4 Calculation of Shear Force in Bolts 15-23

18.1 Introduction 18-1
CONTENTS viii
18.2 Structures with a Limited Number of Nonlinear Elements 18-2
18.3 Fundamental Equilibrium Equations 18-3
18.4 Calculation of Nonlinear Forces 18-4
18.5 Transformation to Modal Coordinates 18-5
18.6 Solution of Nonlinear Modal Equations 18-7
18.7 Static Nonlinear Analysis of Frame Structure 18-9
18.8 Dynamic Nonlinear Analysis of Frame Structure 18-12
18.9 Seismic Analysis of Elevated Water Tank 18-14
18.1 Summary 18-15
19. Linear Viscous Damping
19.1 Introduction 19-1
19.2 Energy Dissipation in Real Structures 19-2
19.3 Physical Interpretation of Viscous Damping 19-4
19.4 Modal Damping Violates Dynamic Equilibrium 19-4
19.5 Numerical Example 19-5
19.6 Stiffness and Mass Proportional Damping 19-6
19.7 Calculation of Orthogonal Damping Matrices 19-7
19.8 Structures with Non-Classical Damping 19-9
19.9 Nonlinear Energy Dissipation 19-9
19.1 Summary 19-10
19.11 References 19-10
20. Dynamic Analysis Using Numerical Integration
20.1 Introduction 20-1
20.2 Newmark Family of Methods 20-2
20.3 Stability of Newmark’s Method 20-4
20.4 The Average Acceleration Method 20-5
20.5 Wilson’s Factor 20-6
20.6 The Use of Stiffness Proportional Damping 20-7

Appendix A Vector Notation
A.1 Introduction A-1
A.2 Vector Cross Product A-2
A.3 Vectors to Define a Local Reference System A-4
A.4 Fortran Subroutines for Vector Operations A-5
CONTENTS x
Appendix B Matrix Notation
B.1 Introduction B-1
B.2 Definition of Matrix Notation B-2
B.3 Matrix Transpose and Scalar Multiplication B-4
B.4 Definition of a Numerical Operation B-6
B.5 Programming Matrix Multiplication B-6
B.6 Order of Matrix Multiplication B-7
B.7 Summary B-7
Appendix C Solution or Inversion of Linear Equations
C.1 Introduction C-1
C.2 Numerical Example C-2
C.3 The Gauss Elimination Algorithm C-3
C.4 Solution of a General Set of Linear Equations C-6
C.5 Alternative to Pivoting C-6
C.6 Matrix Inversion C-9
C.7 Physical Interpretation of Matrix Inversion C-11
C.8 Partial Gauss Elimination, Static Condensation and Substructure
Analysis C-13
C.9 Equations Stored in Banded or Profile Form C-15
C.10 LDL Factorization C-16
C10.1 Triangularization or Factorization of the A Matrix C-17
C10.2 Forward Reduction of the b Matrix C-18
C10.3 Calculation of x by Backsubstitution C-19
C.11 Diagonal Cancellation and Numerical Accuracy C-20

H.3 Speed of Different Computer Systems H-2
H.4 Speed of Personal Computer Systems H-3
H.5 Paging Operating Systems H-3
H.6 Summary H-4
Appendix I Method of Least Square
I.1 Simple Example I-1
I.2 General Formulation I-3
I.3 Calculation Of Stresses Within Finite Elements I-4
CONTENTS xii
Appendix J Consistent Earthquake Acceleration and Displacement
Records
J.1 Introduction J-1
J.2 Ground Acceleration Records J-2
J.3 Calculation of Acceleration Record From Displacement Record J-3
J.4 Creating Consistent Acceleration Record J-5
J.5 Summary J-8
Index
1.
MATERIAL PROPERTIES
Material Properties Must Be Evaluated
By Laboratory or Field Tests
1.1

INTRODUCTION
The fundamental equations of structural mechanics can be placed in three
categories[1]. First, the stress-strain relationship contains the material property
information that must be evaluated by laboratory or field experiments. Second,
the total structure, each element, and each infinitesimal particle within each
element must be in force equilibrium in their deformed position. Third,
displacement compatibility conditions must be satisfied.

exists. Hence, it is reasonable to start with a definition of anisotropic materials,
which may be different in every element in a structure.
The positive definition of stresses, in reference to an orthogonal 1-2-3 system, is
shown in Figure 1.1.
Figure 1.1 Definition of Positive Stresses
All stresses are by definition in units of force-per-unit-area. In matrix notation,
the six independent stresses can be defined by:
[]
233121321
τττσσσ
=
T
f
(1.1)
1
2
3
2
σ
3
σ
1
σ
21
τ
23
τ
31
τ
12















∆+

























−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
−−−−−
=















54
54
3
53
2
52
1
51
6
46
5
45
43
43
2
42
1
41
6
36
4
35
4
34
32
32
1
31
6
26

1
α
α
α
α
α
α
τ
τ
τ
σ
σ
σ
ννννν
ννννν
ννν
νν
ννννν
ννννν
νν
ν
ν
ν
γ
γ
γ
ε
ε
ε
T

j
ij
EE
νν
=
(1.5)
However, because of experimental error or small nonlinear behavior of the
material, this condition is not identically satisfied for most materials. Therefore,
these experimental values are normally averaged so that symmetrical values can
be used in the analyses.
1.3

USE OF MATERIAL PROPERTIES WITHIN COMPUTER
PROGRAMS
Most of the modern computer programs for finite element analysis require that
the stresses be expressed in terms of the strains and temperature change.
Therefore, an equation of the following form is required within the program:
0
fEdf
+=
(1.6)
in which
C
= E
-1
. Therefore, the zero-strain thermal stresses are defined by:
Eaf
0
-
T
















∆+

























−−
−−
−−
=

















6
5
4
32
32
1
31
3
23
21
21
3
13
2
12
1
23
31
21
3
2
1
α
α
α
τ
τ
τ
σ
σ

ISOTROPIC MATERIALS
An isotropic material has equal properties in all directions and is the most
commonly used approximation to predict the behavior of linear elastic materials.
For isotropic materials, Equation (1.3) is of the following form:
1-
6
STATIC AND DYNAMIC ANALYSIS




















∆+





















−−
−−
−−
=









000
1
23
31
21
3
2
1
23
31
21
3
2
1
T
G
G
G
EEE
EEE
EEE
α
τ
τ
τ
σ
σ
σ
νν
νν

γ
γ
ε
131
are zero, the structure is in a state of plane strain. For
this case the compliance matrix is reduced to a 3 x 3 array. The cross-sections of
many dams, tunnels, and solids with a near infinite dimension along the 3-axis
can be considered in a state of plane strain for constant loading in the 1-2 plane.
For plane strain and isotropic materials, the stress-strain relationship is:
MATERIAL PROPERTIES 1-7










∆−










2
21
00
01
01
12
2
1
12
2
1
ETE
α
γ
ε
ε
ν
νν
νν
τ
σ
σ
(1.11)
where
)21)(1(
νν
−+
=
E
E








∆−























ε
ν
ν
ν
τ
σ
σ
(1.14)
where