Three Dimensional Static and Dynamic Analysis Of Structures

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 Author and The Book


Personal Remarks

Table of Contents

Chapter 1.  Material Properties - click and wait to download an PDF file

Chapter 2.   Equilibrium and Compatibility 

New Chapter 2. - 2008

Chapter 3.    Energy and Work

Chapter 4.   One- Dimensional Elements

Chapter 5.   Isoparametric Elements

Chapter 6.    Incompatible Elements

Chapter 7.    Boundary Conditions and General Constraints

Chapter 8.   Plate Bending Elements

Chapter 9.  Membrane Element with Normal Rotations

Chapter 10. Shell Elements

Chapter 11. Geometric Stiffness and P-Delta Effects

Chapter 12. Dynamic Analysis

Chapter 13.  Dynamic Analysis using Mode Superpositione

Chapter 14   Calculation of Stiffness and Mass Orthogonal Vectors

Chapter 15  Dynamic Response using the the response Spectrum Method

Chapter 16.  Soil-Structure Interaction

Chapter 17. Seismic Analysis Modeling to Satisfy Building Codes

Chapter 18  The Fast Nonlinear Analysis Method

Chapter 19. Linear Viscous Damping

Chapter 20  Dynamic Analysis by Direct Integration

Chapter 21.  Nonlinear Elements

Chapter 22.  Seismic Analysis using Displacement Loading

Chapter 23  Fluid-Structure Interaction

Chapter 24  Natural Mode Shapes   Work in Progress


Appendix A  Vector Notation

Appendix BMatrix Notation

Appendix C  Solution or Inversion of Linear Equations

Appendix D  The Classical Eigenvalue Problem

Appendix E  Transformation of Material Properties

Appendix F  A Displacement Based Beam Element with Shear Deformation

Appendix G  Numerical Integration

Appendix H  Speed of Computers

Appendix I   Method of Least Square

Appendix J  Consistent Earthquake Acceleration and Displacement Records

Appendix K  Wall-Slab Interface


The Author and The Book

Professor Wilson has over sixty years of professional experience in Civil, Mechanical and Aerospace Engineering. He was a Professor of Structural Engineering at the University of California at Berkeley during the period 1965 to 1991 and has published over 180 papers and books. His research and development contributions have earned him many awards including the election to the National Academy of Engineering in 1985.

Professor Wilson wrote the first automated finite element analysis computer program in 1961 and was the original developer of the CAL, SAP and ETABS series of computer programs. These programs are noted for their accuracy, speed, use of very efficient numerical algorithms and accurate finite elements. During the past 40 years, Ed Wilson has worked as a Senior Consultant to CSI on the programming and documentation of these new methods of computational structural analysis.

The major purpose of this book is to summarize the theoretical development of the finite elements and numerical methods used in the latest versions of the SAP and ETABS programs. Most of the elements and numerical methods used in these programs are new and are not presented in current textbooks on structural analysis. In addition, the book summarizes the fundamental equations of mechanics.

A minimum mathematical background is required in order to completely understand the material presented in the book. However, an understanding of the physical behavior of real structures is essential. A computer programming background is not required.

A new three-dimensional quadrilateral SHELL element, with normal rotational degrees-of-freedom, is presented that is accurate for both thin and thick plates and shells. Therefore, shell elements can be easily connected to classical FRAME elements. The three-dimensional SOLID element can be used to model both fluids and solids.

Dynamic analysis is presented as a logical extension of static analysis in which inertia and damping forces are added to satisfy equilibrium at every point in time. The use of Load Dependent Ritz, LDR, vectors in a dynamic analysis produce far more accurate results than if the exact dynamic eigenvectors are used.

The use of LDR vectors allows the classical mode superposition method to be extended to nonlinear dynamic analysis by the use of the Fast Nonlinear Analysis, FNA, method. This new method of nonlinear, dynamic analysis allows structures, with a limited number of nonlinear elements, to be analyzed with almost the same computational time as required for a linear dynamic analysis of the same structure.

This is a “must read” book for all researchers and professionals working in the field of modern structural engineering




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 (c) 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



This edition of the book contains corrections and additions to the July 1998 edition. Most of the new material that has been added is in response to questions and comments from the users of SAP2000, ETABS and SAFE.

Chapter 22 has been written on the direct use of absolute earthquake displacement loading acting at the base of the structure. Several new types of numerical errors, for absolute displacement loading, are identified. First, the fundamental nature of displacement loading is significantly different from the base acceleration loading traditionally used in earthquake engineering. Second, a smaller integration time step is required to define the earthquake displacement and to solve the dynamic equilibrium equations. Third, a large number of modes are required for absolute displacement loading in order to obtain the same accuracy as produced when base acceleration is used as the loading. Fourth, the 90 percent mass participation rule, intended to assure accuracy of the analysis, does not apply for absolute displacement loading. Finally, the effective modal damping for displacement loading is larger than when acceleration loading is used.

In order to reduce theses errors associated with displacement loading a higher order integration method, based on a cubic variation of loads within a time step, is introduced in Chapter 13. In addition, static and dynamic participation factors have been defined which allow the structural engineer to minimize the errors associated with displacement type of loading. In addition, Chapter 19 on viscous damping has been expanded in order to illustrate the physical 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 in order 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 in tensor notation and complex variables is not required.

All equations are developed using a physical approach, since this book is written for the student and professional engineer and not for my academic colleagues. Three dimensional structural analysis is relatively simple due to the high speed of the modern computer. Therefore, all equations are presented in three dimensional form and anisotropic material properties are automatically included. A computer programming background is not necessary in order to use a computer program intelligently. However, detailed numerical algorithms are given in order that the readers completely understand the computational methods that are summarized in this book. The Appendices contain an elementary summary of the numerical methods used; therefore, it should not be necessary to spend additional time reading theoretical research papers in order to understand the theory presented in this book.

The author has developed and published many computational techniques for the static and dynamic analysis of structures. It has been personally satisfying that many members of the engineering profession have found these computational methods useful. Therefore, one reason for compiling this theoretical and application book is to consolidate in one publication this research and development. In addition, the recently developed Fast Nonlinear Analysis (FNA) method and other numerical methods are presented in detail for the first time.

The fundamental physical laws that are the basis of the static and dynamic analysis of structures are over 100 years old. Therefore, anyone who believes they have discovered a new fundamental principle of mechanics is a victim of their own ignorance. This book contains computational tricks that the author has found to be effective for the development of structural analysis programs.

The static and dynamic analysis of structures has been automated to a large degree due to the existence of inexpensive personal computers. However, the field of structural engineering, in my opinion, will never be automated. The idea that an expert-system computer program, with artificial intelligence, will replace a creative human is an insult to all structural engineers.

The material presented in the first edition, 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



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

1.13      References   1-12

2.         Equilibrium and Compatibility

2.1        Introduction   2-1

2.2        Fundamental Equilibrium Equations   2-2

2.3        Stress Resultants - Forces And Moments   2-2

2.4        Compatibility Requirements   2-3

2.5        Strain Displacement Equations   2-4

2.6        Definition of Rotation   2-4

2.7        Equations at Material Interfaces   2-5

2.8        Interface Equations in Finite Element Systems   2-7

2.9        Statically Determinate Structures   2-7

2.1        Displacement Transformation Matrix   2-9

2.11      Element Stiffness and Flexibility Matrices   2-11

2.12      Solution of Statically Determinate System   2-11

2.13      General Solution of Structural Systems   2-12

2.14      Summary   2-13

2.15      References   2-14

3.         Energy and Work

3.1        Introduction   3-1

3.2        Virtual and Real Work   3-2

3.3        Potential Energy and Kinetic Energy   3-4

3.4        Strain Energy   3-6

3.5        External Work   3-7

3.6        Stationary Energy Principle   3-9

3.7        The Force Method   3-10

3.8        Lagrange’s Equation of Motion   3-12

3.9        Conservation of Momentum   3-13

3.1        Summary   3-15

3.11      References   3-16

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

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

7.1        Introduction   7-1

7.2        Displacement Boundary Conditions   7-2

7.3        Numerical Problems in Structural Analysis   7-3

7.4        General Theory Associated With Constraints   7-4

7.5        Floor Diaphragm Constraints   7-6

7.6        Rigid Constraints   7-11

7.7        Use of Constraints in Beam-Shell Analysis   7-12

7.8        Use of Constraints in Shear Wall Analysis   7-13

7.9        Use of Constraints for Mesh Transitions   7-14

7.1        Lagrange Multipliers and Penalty Functions   7-16

7.11      Summary   7-17

8.         Plate Bending Elements

8.1        Introduction   8-1

8.2        The Quadrilateral Element   8-3

8.3        Strain-Displacement Equations   8-7

8.4        The Quadrilateral Element Stiffness   8-8

8.5        Satisfying the Patch Test   8-9

8.6        Static Condensation   8-10

8.7        Triangular Plate Bending Element   8-10

8.8        Other Plate Bending Elements   8-10

8.9        Numerical Examples   8-11

8.9.1     One Element Beam   8-12

8.9.2     Point Load on Simply Supported Square Plate   8-13

8.9.3     Uniform Load on Simply Supported Square Plate   8-14

8.9.4     Evaluation of Triangular Plate Bending Elements   8-15

8.9.5     Use of Plate Element to Model Torsion in Beams   8-16

8.1        Summary   8-17

8.11      References   8-17

9.         Membrane Element with Normal Rotations

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

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

11.8      General Formulation of Geometry Stiffness   11-11

11.9      Summary   11-13

11.1      References   11-14

12.       Dynamic Analysis

12.1      Introduction   12-1

12.2      Dynamic Equilibrium   12-2

12.3      Step-By-Step Solution Method   12-4

12.4      Mode Superposition Method   12-5

12.5      Response Spectra Analysis   12-5

12.6      Solution in the Frequency Domain   12-6

12.7      Solution of Linear Equations   12-7

12.8      Undamped Harmonic Response   12-7

12.9      Undamped Free Vibrations   12-8

12.1      Summary   12-9

12.11    References   12-10

13.       Dynamic Analysis Using Mode Superposition

13.1      Equations to be Solved   13-1

13.2      Transformation to Modal Equations   13-2

13.3      Response Due to Initial Conditions Only   13-4

13.4      General Solution Due to Arbitrary Loading   13-5

13.5      Solution for Periodic Loading   13-10

13.6      Participating Mass Ratios   13-11

13.7      Static Load Participation Ratios   13-13

13.8      Dynamic Load Participation Ratios   13-14

13.9      Summary   13-16

14.       Calculation of Stiffness and Mass Orthogonal Vectors

14.1      Introduction   14-1

14.2      Determinate Search Method   14-2

14.3      Sturm Sequence Check   14-3

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

15.9.3   Design Checks for Steel and Concrete Beams   15-22

15.9.4   Calculation of Shear Force in Bolts   15-23

15.1      Summary   15-23

15.11    References   15-24

16.       Soil Structure Interaction

16.1      Introduction   16-1

16.2      Site Response Analysis   16-2

16.3      Kinematic or Soil Structure Interaction   16-2

16.4      Response Due to Multi-Support Input Motions   16-6

16.5      Analysis of Gravity Dam and Foundation   16-9

16.6      The Massless Foundation Approximation   16-11

16.7      Approximate Radiation Boundary Conditions   16-11

16.8      Use of Springs at the Base of a Structure   16-14

16.9      Summary   16-15

16.1      References   16-15

17.       Seismic Analysis Modeling to Satisfy Building Codes

17.1      Introduction   17-1

17.2      Three-Dimensional Computer Model   17-3

17.3      Three-Dimensional Mode Shapes and Frequencies   17-4

17.4      Three-Dimensional Dynamic Analysis   17-8

17.4.1   Dynamic Design Base Shear   17-9

17.4.2   Definition of Principal Directions   17-10

17.4.3   Directional and Orthogonal Effects   17-10

17.4.4   Basic Method of Seismic Analysis   17-11

17.4.5   Scaling of Results   17-11

17.4.6   Dynamic Displacements and Member Forces   17-11

17.4.7   Torsional Effects   17-12

17.5      Numerical Example   17-12

17.6      Dynamic Analysis Method Summary   17-15

17.7      Summary   17-16

17.8      References   17-18

18.       Fast Nonlinear Analysis

18.1      Introduction   18-1

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

20.7      The Hilber, Hughes and Taylor Method   20-8

20.8      Selection of a Direct Integration Method   20-9

20.9      Nonlinear Analysis   20-9

20.1      Summary   20-10

20.11    References   20-10

21.       Nonlinear Elements

21.1      Introduction   21-1

21.2      General Three-Dimensional Two-Node Element   21-2

21.3      General Plasticity Element   21-3

21.4      Different Positive and Negative Properties   21-5

21.5      The Bilinear Tension-Gap-Yield Element   21-6

21.6      Nonlinear Gap-Crush Element   21-7

21.7      Viscous Damping Elements   21-8

21.8      Three-Dimensional Friction-Gap Element   21-10

21.9      Summary   21-12

22.       Seismic Analysis Using Displacement Loading

22.1      Introduction   22-1

22.2      Equilibrium Equations for Displacement Input   22-3

22.3      Use of Pseudo-Static Displacements   22-5

22.4      Solution of Dynamic Equilibrium Equations   22-6

22.5      Numerical Example   22-7

22.5.1   Example Structure   22-7

22.5.2   Earthquake Loading   22-9

22.5.3   Effect of Time Step Size for Zero Damping   22-9

22.5.4   Earthquake Analysis with Finite Damping   22-12

22.5.5   The Effect of Mode Truncation   22-15

22.6      Use of Load Dependent Ritz Vectors   22-17

22.7      Solution Using Step-By-Step Integration   22-18

22.8      Summary   22-20

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

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

C.12     Summary   C-20

C.13     References   C-21

Appendix D  The Eigenvalue Problem

D.1       Introduction   D-1

D.2       The Jacobi Method   D-2

D.3       Calculation of 3d Principal Stresses   D-4

D.4       Solution of the General Eigenvalue Problem   D-5

D.5       Summary   D-6

Appendix E  Transformation of Material Properties

E.1       Introduction   E-1

E.2       Summary   E-4

Appendix F  A Displacement-Based Beam Element With Shear Deformations

F.1       Introduction   F-1

F.2       Basic Assumptions   F-2

F.3       Effective Shear Area   F-5

Appendix G  Numerical Integration

G.1       Introduction   G-1

G.2       One-Dimensional Gauss Quadrature   G-2

G.3       Numerical Integration in Two Dimensions   G-4

G.4       An Eight-Point Two-Dimensional Rule   G-5

G.5       An Eight-Point Lower Order Rule   G-6

G.6       A Five-Point Integration Rule   G-7

G.7       Three-Dimensional Integration Rules   G-8

G.8       Selective Integration   G-11

G.9       Summary   G-11

Appendix H  Speed of Computer Systems

H.1       Introduction   H-1

H.2       Definition of One Numerical Operation   H-1

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

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