Three-Dimensional Static andDynamic Analysis

ofStructuresA Physical Approach

With Emphasis on Earthquake Engineering

Edward L. WilsonProfessor Emeritus of Structural Engineering

University of California at Berkeley

Copyright by Computers and Structures, Inc. No part of this publication may bereproduced 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 AvenueBerkeley, California 94704 USA

Phone: (510) 845-2177FAX: (510) 845-4096

e-mail: info@csiberkeley.com

Copyright Computers and Structures, Inc., 1996-2001The CSI Logo is a trademark of Computers and Structures, Inc.SAP90, SAP2000, SAFE, FLOOR and ETABS are trademarks ofComputers and Structures, Inc.ISBN 0-923907-00-9

STRUCTURAL ENGINEERING IS

THE ART OF USING MATERIALSThat Have Properties Which Can Only Be Estimated

TO BUILD REAL STRUCTURESThat Can Only Be Approximately Analyzed

TO WITHSTAND FORCESThat Are Not Accurately Known

SO THAT OUR RESPONSIBILITY WITH RESPECT TOPUBLIC SAFETY IS SATISFIED.

Adapted From An Unknown Author

Preface To Third Edition

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 commentsfrom the users of SAP2000, ETABS and SAFE.

Chapter?? has been written on the direct use of absolute earthquake displacementloading acting at the base of the structure. Several new types of numerical errors forabsolute displacement loading have been identified. First, the fundamental nature ofdisplacement loading is significantly different from the base acceleration loading tra-ditionally used in earthquake engineering. Second, a smaller integration time step isrequired to define the earthquake displacement and to solve the dynamic equilibriumequations. Third, a large number of modes are required for absolute displacement load-ing to obtain the same accuracy as produced when base acceleration is used as theloading. Fourth, the 90 percent mass participation rule, intended to assure accuracy ofthe analysis, does not apply for absolute displacement loading. Finally, the effectivemodal damping for displacement loading is larger than when acceleration loading isused.

To reduce those errors associated with displacement loading, a higher order inte-gration method based on a cubic variation of loads within a time step is introduced inChapter??. In addition, static and dynamic participation factors have been defined thatallow the structural engineer to minimize the errors associated with displacement typeloading. In addition, Chapter?? on viscous damping has been expanded to illustratethe physical effects of modal damping on the results of a dynamic analysis.

Appendix??, on the speed of modern personal computers, has been updated. It isnow possible to purchase a personal computer for approximately $1,500 that is 25 timesfaster than a $10,000,000 CRAY computer produced in 1974.

Several other additions and modifications have been made in this printing. Pleasesend your comments and questionsto ed@csiberkeley.com.

Edward L. WilsonApril 2000

Personal Remarks

My freshman Physics instructor dogmatically warned the class “do not use an equa-tion you cannot derive.” The same instructor once stated that “if a person had five min-utes to solve a problem, that their life depended upon, the individual should spend threeminutes reading and clearly understanding the problem.” For the past forty years thesesimple, practical remarks have guided my work and I hope that the same philosophyhas 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 youfully understand the theory and approximations used within the program” and “do notcreate a computer model until the loading, material properties and boundary conditionsare clearly defined.”

Therefore, the major purpose of this book is to present the essential theoretical back-ground so that the users of computer programs for structural analysis can understandthe basic approximations used within the program, verify the results of all analysis andassume professional responsibility for the results. It is assumed that the reader has anunderstanding of statics, mechanics of solids, and elementary structural analysis. Thelevel of knowledge expected is equal to that of an individual with an undergraduate de-gree in Civil or Mechanical Engineering. Elementary matrix and vector notations aredefined in the Appendices and are used extensively. A background in tensor notationand complex variables is not required.

All equations are developed using a physical approach, because this book is writtenfor the student and professional engineer and not for my academic colleagues. Three-dimensional structural analysis is relatively simple because of the high speed of themodern computer. Therefore, all equations are presented in three-dimensional formand anisotropic material properties are automatically included. A computer program-ming background is not necessary to use a computer program intelligently. However,detailed numerical algorithms are given so that the readers completely understand thecomputational methods that are summarized in this book. The Appendices contain anelementary summary of the numerical methods used; therefore, it should not be nec-essary to spend additional time reading theoretical research papers to understand thetheory presented in this book.

The author has developed and published many computational techniques for thestatic and dynamic analysis of structures. It has been personally satisfying that manymembers of the engineering profession have found these computational methods use-ful. Therefore, one reason for compiling this theoretical and application book is toconsolidate in one publication this research and development. In addition, the recentlydeveloped Fast Nonlinear Analysis (FNA) method and other numerical methods arepresented in detail for the first time.

The fundamental physical laws that are the basis of the static and dynamic analysisof structures are over 100 years old. Therefore, anyone who believes they have dis-covered 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 thedevelopment of structural analysis programs.

viii Three-Dimensional Static and Dynamic Analysis of Structures

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

The material in this book has evolved over the past thirty-five years with the helpof my former students and professional colleagues. Their contributions are acknowl-edged. Ashraf Habibullah, Iqbal Suharwardy, Robert Morris, Syed Hasanain, DollyGurrola, Marilyn Wilkes and Randy Corson of Computers and Structures, Inc., deservespecial recognition. In addition, I would like to thank the large number of structuralengineers who have used the EABS and SAP series of programs. They have providedthe motivation for this publication.

The material presented in the first edition ofThree Dimensional Dynamic Analysisof Structuresis included and updated in this book. I am looking forward to additionalcomments and questions from the readers in order to expand the material in futureeditions of the book.

Edward L. WilsonJuly 1998

Contents

Contents ix

List of Tables xi

List of Figures xiii

1 EQUILIBRIUM AND COMPATIBILITY 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fundamental Equilibrium Equations. . . . . . . . . . . . . . . . . . . 21.3 Stress Resultants - Forces And Moments. . . . . . . . . . . . . . . . . 21.4 Compatibility Requirements. . . . . . . . . . . . . . . . . . . . . . . 21.5 Strain Displacement Equations. . . . . . . . . . . . . . . . . . . . . . 31.6 Definition Of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Equations At Material Interfaces. . . . . . . . . . . . . . . . . . . . . 41.8 Interface Equations In Finite Element Systems. . . . . . . . . . . . . . 51.9 Statically Determinate Structures. . . . . . . . . . . . . . . . . . . . . 61.10 Displacement Transformation Matrix. . . . . . . . . . . . . . . . . . . 71.11 Element Stiffness And Flexibility Matrices. . . . . . . . . . . . . . . . 91.12 Solution Of Statically Determinate System. . . . . . . . . . . . . . . . 91.13 General Solution Of Structural Systems. . . . . . . . . . . . . . . . . 91.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

BIBLIOGRAPHY 13

INDEX 17

ix

x Three-Dimensional Static and Dynamic Analysis of Structures

List of Tables

xi

xii Three-Dimensional Static and Dynamic Analysis of Structures

List of Figures

1.1 Material Interface Properties. . . . . . . . . . . . . . . . . . . . . . . 41.2 Simple Truss Structure. . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Definition of Positive Joint Forces and Node Displacements. . . . . . . 71.4 Typical Two-Dimension Truss Element. . . . . . . . . . . . . . . . . 8

xiii

xiv Three-Dimensional Static and Dynamic Analysis of Structures

CHAPTER

1

EQUILIBRIUM AND COMPATIBILITY

Equilibrium Is Essential - Compatibility Is Optional

1.1 Introduction

Equilibrium equations set the externally applied loads equal to the sum of the inter-nal element forces at all joints or node points of a structural system; they are the mostfundamental equations in structural analysis and design. The exact solution for a prob-lem in solid mechanics requires that the differential equations of equilibrium for allinfinitesimal elements within the solid must be satisfied.Equilibrium is a fundamentallaw of physics and cannot be violated within a “real” structural system.Therefore, itis critical that the mathematical model, which is used to simulate the behavior of a realstructure, also satisfies those basic equilibrium equations.

It is important to note that within a finite element, which is based on a formal dis-placement formulation, the differential stress-equilibrium equations are not always sat-isfied. However, inter-element force-equilibrium equations are identically satisfied atall node points (joints). The computer program user who does not understand the ap-proximations used to develop a finite element can obtain results that are in significanterror if the element mesh is not sufficiently fine in areas of stress concentrationCook,Malkus, and Plesha[1989].

Compatibility requirements should be satisfied. However, if one has a choice be-tween satisfying equilibrium or compatibility, one should use the equilibrium- basedsolution. For real nonlinear structures, equilibrium is always satisfied in the deformedposition. Many real structures do not satisfy compatibility caused by creep, joint slip-page, incremental construction and directional yielding.

1

2 Three-Dimensional Static and Dynamic Analysis of Structures

1.2 Fundamental Equilibrium Equations

The three-dimensional equilibrium of an infinitesimal element, shown in Figure??, isgiven by the following equilibrium equationsP. [1985] :

∂σ1

∂x1+

∂τ12

∂x2+

∂τ13

∂x3+ β1 = 0

∂τ21

∂x1+

∂σ2

∂x2+

∂τ23

∂x3+ β2 = 0

∂τ31

∂x1+

∂τ32

∂x2+

∂σ3

∂x3+ β3 = 0

(1.1)

The body force,βi, is per unit of volume in the i-direction and represents gravitationalforces or pore pressure gradients. Becauseτij = τji, the infinitesimal element is au-tomatically in rotational equilibrium. Of course for this equation to be valid for largedisplacements, it must be satisfied in the deformed position, and all stresses must bedefined as force per unit of deformed area.

1.3 Stress Resultants - Forces And Moments

In structural analysis it is standard practice to write equilibrium equations in terms ofstress resultants rather than in terms of stresses. Force stress resultants are calculated bythe integration of normal or shear stresses acting on a surface. Moment stress resultantsare the integration of stresses on a surface times a distance from an axis.

A point load, which is a stress resultant, is by definition an infinite stress times aninfinitesimal area and is physically impossible on all real structures. Also, a point mo-ment is a mathematical definition and does not have a unique stress field as a physicalinterpretation. Clearly, the use of forces and moments is fundamental in structural anal-ysis and design. However, a clear understanding of their use in finite element analysisis absolutely necessary if stress results are to be physically evaluated.

For a finite size element or joint, a substructure, or a complete structural systemthefollowing six equilibrium equations must be satisfied:

∑Fx = 0

∑Fy = 0

∑Fz = 0

∑Mx = 0

∑My = 0

∑Mz = 0

(1.2)

For two dimensional structures only three of these equations need to be satisfied.

1.4 Compatibility Requirements

For continuous solids we have defined strains as displacements per unit length. Tocalculate absolute displacements at a point, we must integrate the strains with respectto a fixed boundary condition. This integration can be conducted over many differentpaths. A solution is compatible if the displacement at all points is not a function of

CHAPTER 1. EQUILIBRIUM AND COMPATIBILITY 3

the path. Therefore, a displacement compatible solution involves the existence of auniquely defined displacement field.

In the analysis of a structural system of discrete elements, all elements connected toa joint or node point must have the same absolute displacement. If the node displace-ments are given, all element deformations can be calculated from the basic equations ofgeometry. In a displacement-based finite element analysis, node displacement compat-ibility is satisfied. However, it is not necessary that the displacements along the sidesof the elements be compatible if the element passes the “patch test.”

A finite element passes the patch test “if a group (or patch) of elements, of arbitraryshape, is subjected to node displacements associated with constant strain; and the resultsof a finite element analysis of the patch of elements yield constant strain.” In the caseof plate bending elements, the application of a constant curvature displacement patternat the nodes must produce constant curvature within a patch of elements. If an elementdoes not pass the patch test, it may not converge to the exact solution. Also, in the caseof a coarse mesh, elements that do not pass the patch test may produce results withsignificant errors.

1.5 Strain Displacement Equations

If the small displacement fieldsu1, u2, andu3 are specified, assumed or calculated,the consistent strains can be calculated directly from the following well-known strain-displacement equationsP. [1985]:

ε1 =∂u1

∂x1(1.3a)

ε2 =∂u2

∂x2(1.3b)

ε3 =∂u3

∂x3(1.3c)

γ12 =∂u1

∂x2+

∂u2

∂x1(1.3d)

γ13 =∂u1

∂x3+

∂u3

∂x1(1.3e)

γ23 =∂u2

∂x3+

∂u3

∂x2(1.3f)

1.6 Definition Of Rotation

A unique rotation at a point in a real structure does not exist. A rotation of a horizontalline may be different from the rotation of a vertical line. However, in many theoreticalbooks on continuum mechanics the following mathematical equations are used to define

4 Three-Dimensional Static and Dynamic Analysis of Structures

rotation of the three axes:

θ3 ≡ 12

[∂u1

∂x2− ∂u2

∂x1

](1.4a)

θ2 ≡ 12

[∂u3

∂x1− ∂u1

∂x3

](1.4b)

θ1 ≡ 12

[∂u2

∂x3− ∂u3

∂x2

](1.4c)

It is of interest to note that this definition of rotation is the average rotation of twonormal lines. It is important to recognize that these definitions are not the same asused in beam theory when shearing deformations are included. When beam sectionsare connected, the absolute rotation of the end sections must be equal.

1.7 Equations At Material Interfaces

One can clearly understand the fundamental equilibrium and compatibility require-ments from an examination of the stresses and strains at the interface between twomaterials. A typical interface for a two-dimensional continuum is shown in Figure1.1.By definition, the displacements at the interface are equal. Or,us(s, n) = us(s, n) andun(s, n) = un(s, n).

s, us(s,n)n, un(s,n)

Figure 1.1:Material Interface Properties

Normal equilibrium at the interface requires that the normal stresses be equal. Or:

σn = σn (1.5a)

Also, the shear stresses at the interface are equal. Or:

τns = τns (1.5b)

CHAPTER 1. EQUILIBRIUM AND COMPATIBILITY 5

Because displacementus andus must be equal and continuous at the interface:

εs = εs (1.5c)

Because the material properties that relate stress to strain are not equal for the twomaterials, it can be concluded that:

σs 6= σs (1.5d)

εn 6= εn (1.5e)

γns 6= γns (1.5f)

For a three-dimensional material interface on a s-t surface, it is apparent that the fol-lowing 12 equilibrium and compatibility equations exist:

σn = σn εn 6= εn (1.6a)

σs 6= σs εs = εs (1.6b)

σt 6= σt εt = εt (1.6c)

τns = τns γns 6= γns (1.6d)

τnt = τnt γnt 6= γnt (1.6e)

τst 6= τst γst = γst (1.6f)

These 12 equations cannot be derived because they are fundamental physical laws ofequilibrium and compatibility. It is important to note that if a stress is continuous, thecorresponding strain, derivative of the displacement, is discontinuous. Also, if a stressis discontinuous, the corresponding strain, derivative of the displacement, is continuous.

The continuity of displacements between elements and at material interfaces is de-fined asC0 displacement fields. Elements with continuities of the derivatives of thedisplacements are defined byC1 continuous elements. It is apparent that elements withC1 displacement compatibility cannot be used at material interfaces.

1.8 Interface Equations In Finite Element Systems

In the case of a finite element system in which the equilibrium and compatibility equa-tions are satisfied only at node points along the interface, the fundamental equilibriumequations can be written as:

∑Fn +

∑Fn = 0 (1.7a)

∑Fs +

∑F s = 0 (1.7b)

∑Ft +

∑F t = 0 (1.7c)

Each node on the interface between elements has a unique set of displacements; there-fore, compatibility at the interface is satisfied at a finite number of points. As the finiteelement mesh is refined, the element stresses and strains approach the equilibrium andcompatibility requirements given by Equations (1.6a) to (1.6f). Therefore, each elementin the structure may have different material properties.

6 Three-Dimensional Static and Dynamic Analysis of Structures

1.9 Statically Determinate Structures

The internal forces of some structures can be determined directly from the equations ofequilibrium only. For example, the truss structure shown in Figure1.2will be analyzedto illustrate that the classical “method of joints” is nothing more than solving a set ofequilibrium equations.

Figure 1.2:Simple Truss Structure

Positive external node loads and node displacements are shown in Figure1.3. Mem-ber forcesfi and deformationsdi are positive in tension.

Equating two external loads,Rj , at each joint to the sum of the internal memberforces,fi, (see Appendix?? for details) yields the following seven equilibrium equa-tions written as one matrix equation:

R1

R2

R3

R4

R5

R6

R7

=

−1.0 −0.6 0 0 0 0 0

0 −0.8 0 0 0 0 0

1.0 0 0 0 −0.6 0 0

0 0 −1.0 0 −0.8 −1.0 0

0 0.6 0 −1.0 0 0 0

0 0.8 1.0 0 0 0 0

0 0 0 0 0 0 −1.0

f1

f2

f3

f4

f5

f6

f7

(1.8)

Or, symbolically

R = Af (1.9)

CHAPTER 1. EQUILIBRIUM AND COMPATIBILITY 7

2,2uR

1,1uR

6,6uR

5,5uR

4,4uR 3,3

uR7,7

uR

1,

1df

2,

2df

5,

5df

4,

4df

7,

7df

6,

6df

3,

3df

Figure 1.3:Definition of Positive Joint Forces and Node Displacements

whereA is a load-force transformation matrix and is a function of the geometry ofthe structure only. For thisstatically determinatestructure,we have seven unknown ele-ment forces and seven joint equilibrium equations; therefore, the above set of equationscan be solved directly for any number of joint load conditions. If the structure hadone additional diagonal member, there would be eight unknown member forces, anda direct solution would not be possible because the structure would bestatically inde-terminate.The major purpose of this example is to express the well-known traditionalmethod of analysis(“method of joints”) in matrix notation.

1.10 Displacement Transformation Matrix

After the member forces have been calculated, there are many different traditional meth-ods to calculate joint displacements. Again, to illustrate the use of matrix notation, themember deformationsdi will be expressed in terms of joint displacementsuj . Considera typical truss element as shown in Figure1.4.

The axial deformation of the element can be expressed as the sum of the axial de-formations resulting from the four displacements at the two ends of the element. Thetotal axial deformation written in matrix form is:

d =[−Lx

L−Ly

L

Lx

L

Ly

L

]

v1

v2

v3

v4

(1.10)

Application of Equation (1.10) to all members of the truss shown in Figure1.3 yields

8 Three-Dimensional Static and Dynamic Analysis of Structures

Lx

x

y

Ly L

Deformed Position

Initial Position

v1

v2

v3 v4

Figure 1.4:Typical Two-Dimension Truss Element

the following matrix equation:

d1

d2

d3

d4

d5

d6

d7

=

−1.0 0 1.0 0 0 0 0

−0.6 −0.8 0 0 0.6 0.8 0

0 0 0 −1.0 0 1.0 0

0 0 0 0 −1.0 0 0

0 0 −0.6 −0.8 0 0 0

0 0 −1.0 0 0 0 0

0 0 0 0 0 0 −1.0

u1

u2

u3

u4

u5

u6

u7

(1.11)

Or, symbolically:

d = Bu (1.12)

The element deformation-displacement transformation matrix,B, is a function of thegeometry of the structure. Of greater significance, however, is the fact that the matrixBis the transpose of the matrixA defined by the joint equilibrium Equation (1.8). There-fore, given the element deformations within this statically determinate truss structure,we can solve Equation (1.11) for the joint displacements.

CHAPTER 1. EQUILIBRIUM AND COMPATIBILITY 9

1.11 Element Stiffness And Flexibility Matrices

The forces in the elements can be expressed in terms of the deformations in the elementsusing the following matrix equations:

f = kd or, d = k−1f (1.13)

The element stiffness matrixk is diagonal for this truss structure, where the diagonalterms arekii = AiEi

Liand all other terms are zero. The element flexibility matrix is the

inverse of the stiffness matrix, where the diagonal terms areLi

AiEi. It is important to note

that the element stiffness and flexibility matrices are only a function of the mechanicalproperties of the elements.

1.12 Solution Of Statically Determinate System

The three fundamental equations of structural analysis for this simple truss structureare equilibrium, Equation (1.8); compatibility, Equation!(1.11); and force-deformation,Equation (1.13). For each load condition R, the solution steps can be summarized asfollows:

1. Calculate the element forces from Equation (1.8).

2. Calculate element deformations from Equation (1.13).

3. Solve for joint displacements using Equation (1.11).

All traditional methods of structural analysis use these basic equations. However, beforethe availability of inexpensive digital computers that can solve over 100 equations inless than one second, many special techniques were developed to minimize the numberof hand calculations. Therefore, at this point in time, there is little value to summarizethose methods in this book on the static and dynamic analysis of structures.

1.13 General Solution Of Structural Systems

In structural analysis using digital computers, the same equations used in classical struc-tural analysis are applied. The starting point is always joint equilibrium. Or,R = A f .From the element force-deformation equation,f = k d, the joint equilibrium equationcan be written asR = A k d. From the compatibility equation,d = B u, joint equi-librium can be written in terms of joint displacements asR = A k B u. Therefore, thegeneral joint equilibrium can be written as:

R = Ku (1.14)

The global stiffness matrixK is given by one of the following matrix equations:

K = AkB or K = AkAT or K = BT kB (1.15)

10 Three-Dimensional Static and Dynamic Analysis of Structures

It is of interest to note that the equations of equilibrium or the equations of compatibilitycan be used to calculate the global stiffness matrixK.

The standard approach is to solve Equation (1.14) for the joint displacements andthen calculate the member forces from:

f = kBu or f = kAT u (1.16)

It should be noted that within a computer program, the sparse matricesA, B, k andK are never formed because of their large storage requirements. The symmetric globalstiffness matrixK is formed and solved in condensed form.

1.14 Summary

Internal member forces and stresses must be in equilibrium with the applied loads anddisplacements. All real structures satisfy this fundamental law of physics. Hence, ourcomputer models must satisfy the same law.

At material interfaces, all stresses and strains are not continuous. Computer pro-grams that average node stresses at material interfaces produce plot stress contours thatare continuous; however, the results will not converge and significant errors can beintroduced by this approximation.

Compatibility conditions, which require that all elements attached to a rigid jointhave the same displacement, are fundamental requirements in structural analysis andcan be physically understood. Satisfying displacement compatibility involves the use ofsimple equations of geometry. However, the compatibility equations have many forms,and most engineering students and many practicing engineers can have difficulty inunderstanding the displacement compatibility requirement. Some of the reasons wehave difficulty in the enforcement of the compatibility equations are the following:

1. The displacements that exist in most linear structural systems are small comparedto the dimensions of the structure. Therefore, deflected shape drawing must begrossly exaggerated to write equations of geometry.

2. For structural systems that are statically determinate, the internal member forcesand stresses can be calculated exactly without the use of the compatibility equa-tions.

3. Many popular (approximate) methods of analysis exist that do not satisfy the dis-placement compatibility equations. For example, for rectangular frames, both thecantilever and portal methods of analysis assume the inflection points to exist at apredetermined location within the beams or columns; therefore, the displacementcompatibility equations are not satisfied.

4. Many materials, such as soils and fluids, do not satisfy the compatibility equa-tions. Also, locked in construction stresses, creep and slippage within joints arereal violations of displacement compatibility. Therefore, approximate methodsthat satisfy statics may produce more realistic results for the purpose of design.

CHAPTER 1. EQUILIBRIUM AND COMPATIBILITY 11

5. In addition, engineering students are not normally required to take a course ingeometry; whereas, all students take a course in statics. Hence, there has notbeen an emphasis on the application of the equations of geometry.

The relaxation of the displacement compatibility requirement has been justified for handcalculation to minimize computational time. Also, if one must make a choice betweensatisfying the equations of statics or the equations of geometry, in general, we shouldsatisfy the equations of statics for the reasons previously stated.

However, because of the existence of inexpensive powerful computers and efficientmodern computer programs, it is not necessary to approximate the compatibility re-quirements. For many structures, such approximations can produce significant errors inthe force distribution in the structure in addition to incorrect displacements.

BIBLIOGRAPHY

K. J. Bathe and E. L. Wilson. Large eigenvalue problems in dynamic analysis.Pro-ceedings American Society of Civil Engineers Journal of the Engineering MechanicsDivision EM6, pages 1471–1485, December 1972.

E. Bayo and E. L. Wilson. Use of ritz vectors in wave propagation and foundationresponse.Earthquake Engineering and Structural Dynamics, 12:499–505, 1984.

A. Chopra.Dynamics of Structures.Prentice-Hall Inc., Englewood Cliffs, New Jersey07632., 1995. ISBN 0-13-855214-2.

R. Clough and J. Penzien.Dynamics of Structures Second Edition. McGraw-Hill,1993. ISBN 0-07-011394-7.

R. W. Clough, editor.The Finite Element Method in Plane Stress Analysis, Pittsburg,PA, September 1960. Conf. On Electronic Computations.

Computers and Structures Inc.SAP2000-Integrated Structural Analysis and DesignSoftware. Berkeley, California, 1997.

R. D. Cook, D. S. Malkus, and M. E. Plesha.Concepts and Applications of FiniteElement Analysis. John Wiley & Sons Inc., third edition, 1989. ISBN 0-471-84788-7.

A. Habibullah. ETABS-Three Dimensional Analysis of Building Systems. Computersand Structures Inc., Berkeley California, 1997.

M. Hoit and E. L. Wilson. An equation numbering algorithm based on a minimumfront criterion.J. Computers and Structures, 16(1–4):225–239, 1983.

T. J. Hughes and R.Hughes Thomas.The Finite Element Method - Linear Static andDynamic Finite Element Analysis. Prentice Hall, Inc, 1987.

Adnan Ibrahimbegovic. Quadrilateral elements for analysis of thick and thin plates.Computer Methods in Applied Mechanics and Engineering, 110:195–209, 1993.

Adnan Ibrahimbegovic, R. L. Taylor, and E. L. Wilson. A robust membrane quadri-lateral element with drilling degrees of freedom.International Journal for NumericalMethods in Engineering, 1990.

13

14 Three-Dimensional Static and Dynamic Analysis of Structures

American Concrete Institute.Building Code Requirements for Reinforced Concrete(ACI 318-95) and Commentary (ACI 318R-95). American Concrete Institute, Farm-ington Hills, Michigan, 1995.

B. M. Irons and A. Razzaque.Experience with the Patch Test in The MathematicalFoundations of the Finite Element Method. Academic Press, 1972. 557–87.

B. M. Irons and O. C. Zienkiewicz.The Isoparametric Finite Element System - A NewConcept in Finite Element Analysis. Proc. Conf. Recent Advances in Stress Analysis.Royal Aeronautical Society, London, 1968.

Wilson E. L. and Joseph Penzien. Evaluation of orthogonal matrices.InternationalJournal for Numerical Methods in Engineering, 4:5–10, 1972.

R. H. MacNeal and R. C. Harder. A proposed standard set to test element accuracy.Finite Elements in Analysis and Design, 1:3–20, 1985.

C. Menun and A. Der Kiureghian. A replacement for the 30Earthquake Spectra, 13(1), Febuary 1998.

N. M. Newmark. A method of computation for structural dynamics.ASCE Journal ofthe Engineering Mechanics Division, 85(EM3), 1959.

American Institute of Steel Construction Inc.Load and Resistance Factor DesignSpecification for Structural Steel Buildings. American Institute of Steel ConstructionInc., December 1993.

Boresi A. P.Advanced Mechanics of Materials. John Wiley & Sons Inc., 1985. ISBN0-471-88392-1.

J. Penzien and M. Watabe. Characteristics of 3-d earthquake ground motions.Earth-quake Engineering and Structural Dynamics, 3:365–373, 1975.

E. P. Popov.Engineering Mechanics of Solids. Prentice-Hall Inc., 1990. ISBN 0-13-279258-3.

A. Rutenberg. Simplified p-delta analysis for asymmetric structures.ASCE Journal ofthe Structural Division, 108(9), September 1982.

A. C. Scordelis and K. S. Lo. Computer analysis of cylinder shells.Journal of Amer-ican Concrete Institute, 61, May 1964.

J. C. Simo and M. S. Rafai. A class of assumed strain method and incompatible modes.J. Numerical Methods in Engineering, 29:1595–1638, 1990.

G. Strang. Variational crimes in the finite element method.The Mathematical Foun-dations of the Finite Element Method, pages 689–710, 1972.

R. L. Taylor, P. J. Beresford, and E. L. Wilson.A Non-Conforming Element for StressAnalysis. 1976. 1211–20.

BIBLIOGRAPHY 15

E. L. Wilson. Dynamic Response by Step-By-Step Matrix Analysis Proceedings Sym-posium On The Use of Computers in Civil Engineering. Labortorio Nacional de En-genharia Civil, Lisbon, Portugal, October 1962. 1–5.

E. L. Wilson. Structural analysis of axisymmetric solids.AIAA Journal, 3, 1965.

E. L. Wilson. The static condensation algorithm.International Journal for NumericalMethods in Engineering, 8:198–203, January 1974.

E. L. Wilson and K. Bathe. Stability and accuracy analysis of direct integration meth-ods.Earthquake Engineering and Structural Dynamics, 1(283–291), 1973.

E. L. Wilson and A. Habibullah. Static and dynamic analysis of multi-story buildingsincluding p-delta effects.Earthquake Engineering Research Institute, Vol. 3(No.3),May 1987.

E. L. Wilson and Adnan Ibrahimbegovic. Use of incompatible displacement modesfor the calculation of element stiffnesses and stresses.Finite Elements in Analysis andDesign, Vol. 7:229–247, 1990.

E. L. Wilson and T. Itoh. An eigensolution strategy for large systems.Computers andStructures, 16(1–4):259–265, 1983.

E. L. Wilson, A. Der Kiureghian, and E. R. Bayo. A replacement for the srss methodin seismic analysis.Earthquake Engineering and Structural Dynamics, 9:187–192,1981.

E. L. Wilson, R. L. Taylor, W. Doherty, and J. Ghaboussi.Incompatible Displace-ment Models Proceedings ONR Symposium on Numerical and Computer Methods inStructural Mechanics. University of Illinois, Urbana, September 1971.

E. L. Wilson, M. Yuan, and J. Dickens. Dynamic analysis by direct superposition ofritz vectors.Earthquake Engineering and Structural Dynamics, 10:813–823, 1982.

E.L. Wilson. Finite Element Analysis of Two-Dimensional Structures. PhD thesis,University of California at Berkeley, 1963.

E.L. Wilson, K. J. Bathe, and W. P. Doherty. Direct solution of large systems of linearequations.Computers and Structures, 4:363–372, January 1974.

16 Three-Dimensional Static and Dynamic Analysis of Structures

INDEX

Bathe and Wilson [1972],13Bayo and Wilson [1984],13Chopra [1995],13Clough and Penzien [1993],13Clough [1960],13Computers and Inc. [1997],13Cook et al. [1989],1, 13Habibullah [1997],13Hoit and Wilson [1983],13Hughes and Thomas [1987],13Ibrahimbegovic et al. [1990],13Ibrahimbegovic [1993],13Institute [1995],13Irons and Razzaque [1972],14Irons and Zienkiewicz [1968],14L. and Penzien [1972],14MacNeal and Harder [1985],14Menun and Kiureghian [1998],14Newmark [1959],14P. [1985],2, 3, 14Penzien and Watabe [1975],14Popov [1990],14Rutenberg [1982],14Scordelis and Lo [1964],14Simo and Rafai [1990],14Strang [1972],14Taylor et al. [1976],14Wilson and Bathe [1973],15Wilson and Habibullah [1987],15Wilson and Ibrahimbegovic [1990],15Wilson and Itoh [1983],15Wilson et al. [1971],15Wilson et al. [1974],15

Wilson et al. [1981],15Wilson et al. [1982],15Wilson [1962],14Wilson [1963],15Wilson [1965],15Wilson [1974],15of Steel Construction Inc. [1993],14

Axial Deformations,7

Compatibility,2, 10

Deformed Position,2, 7Displacement Compatibility,3, 5, 10Displacement Transformation Matrix,7, 8

Element Flexibility,9Element Stiffness,9Equilibrium,1, 2

Force Transformation Matrix,7

Initial Position,7

Material Interface,4Method of Joints,6

Patch Test,3Point Loads,2

Rotation Definition,3

Solution of Equations,9Statically Determinate Structures,6Strain Compatibility,2

17

18 Three-Dimensional Static and Dynamic Analysis of Structures

Strain Displacement Equations3D Linear Solids,3

Stress Continuity,4Stress Resultant,2

Truss Element,6