02 three dimensional static and dynamic analysis of structure

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 Computers and Structures, Inc Berkeley, California, USA Third Edition Reprint January 2002 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 e-mail: info@csiberkeley.com  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 THE ART OF USING MATERIALS That Have Properties Which Can Only Be Estimated TO BUILD REAL STRUCTURES That Can Only Be Approximately Analyzed TO WITHSTAND FORCES That Are Not Accurately Known SO THAT OUR RESPONSIBILITY WITH RESPECT TO PUBLIC 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 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 have been 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 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 To reduce those 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 that allow the structural engineer to minimize the errors associated with displacement type loading In addition, Chapter 19 on viscous damping has been expanded 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 ed@csiberkeley.com 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 in tensor notation and complex variables is not required All equations are developed using a physical approach, because this book is written for the student and professional engineer and not for my academic colleagues Threedimensional structural analysis is relatively simple because of 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 to use a computer program intelligently However, detailed numerical algorithms are given so 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 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 because of the existence of inexpensive personal computers However, the field of structural engineering, in my opinion, will never be automated The idea that an expertsystem computer program, with artificial intelligence, will replace a creative human is an insult to all structural engineers The material in this book has evolved over the past thirty-five years with the help of my 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 Material Properties 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.1 1.11 1.12 1.13 Introduction 1-1 Anisotropic Materials 1-1 Use of Material Properties within Computer Programs 1-4 Orthotropic Materials 1-5 Isotropic Materials 1-5 Plane Strain Isotropic Materials 1-6 Plane Stress Isotropic Materials 1-7 Properties of Fluid-Like Materials 1-8 Shear and Compression Wave Velocities 1-9 Axisymmetric Material Properties 1-10 Force-Deformation Relationships 1-11 Summary 1-12 References 1-12 Equilibrium and Compatibility 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.1 2.11 2.12 2.13 Introduction 2-1 Fundamental Equilibrium Equations 2-2 Stress Resultants - Forces And Moments 2-2 Compatibility Requirements 2-3 Strain Displacement Equations 2-4 Definition of Rotation 2-4 Equations at Material Interfaces 2-5 Interface Equations in Finite Element Systems 2-7 Statically Determinate Structures 2-7 Displacement Transformation Matrix 2-9 Element Stiffness and Flexibility Matrices 2-11 Solution of Statically Determinate System 2-11 General Solution of Structural Systems 2-12 CONTENTS 2.14 2.15 Introduction 3-1 Virtual and Real Work 3-2 Potential Energy and Kinetic Energy 3-4 Strain Energy 3-6 External Work 3-7 Stationary Energy Principle 3-9 The Force Method 3-10 Lagrange’s Equation of Motion 3-12 Conservation of Momentum 3-13 Summary 3-15 References 3-16 One-Dimensional Elements 4.1 4.2 4.3 4.4 4.5 4.6 Summary 2-13 References 2-14 Energy and Work 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.1 3.11 ii Introduction 4-1 Analysis of an Axial Element 4-2 Two-Dimensional Frame Element 4-4 Three-Dimensional Frame Element 4-8 Member End-Releases 4-12 Summary 4-13 Isoparametric Elements 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Introduction 5-1 A Simple One-Dimensional Example 5-2 One-Dimensional Integration Formulas 5-4 Restriction on Locations of Mid-Side Nodes 5-6 Two-Dimensional Shape Functions 5-6 Numerical Integration in Two Dimensions 5-10 Three-Dimensional Shape Functions 5-12 Triangular and Tetrahedral Elements 5-14 Summary 5-15 CONTENTS 5.1 Introduction 6-1 Elements With Shear Locking 6-2 Addition of Incompatible Modes 6-3 Formation of Element Stiffness Matrix 6-4 Incompatible Two-Dimensional Elements 6-5 Example Using Incompatible Displacements 6-6 Three-Dimensional Incompatible Elements 6-7 Summary 6-8 References 6-9 Boundary Conditions and General Constraints 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.1 7.11 References 5-16 Incompatible Elements 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 iii Introduction 7-1 Displacement Boundary Conditions 7-2 Numerical Problems in Structural Analysis 7-3 General Theory Associated With Constraints 7-4 Floor Diaphragm Constraints 7-6 Rigid Constraints 7-11 Use of Constraints in Beam-Shell Analysis 7-12 Use of Constraints in Shear Wall Analysis 7-13 Use of Constraints for Mesh Transitions 7-14 Lagrange Multipliers and Penalty Functions 7-16 Summary 7-17 Plate Bending Elements 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Introduction 8-1 The Quadrilateral Element 8-3 Strain-Displacement Equations 8-7 The Quadrilateral Element Stiffness 8-8 Satisfying the Patch Test 8-9 Static Condensation 8-10 Triangular Plate Bending Element 8-10 CONTENTS 8.8 8.9 8.1 8.11 Other Plate Bending Elements 8-10 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 Summary 8-17 References 8-17 Membrane Element with Normal Rotations 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.1 9.11 10 iv Introduction 9-1 Basic Assumptions 9-2 Displacement Approximation 9-3 Introduction of Node Rotation 9-4 Strain-Displacement Equations 9-5 Stress-Strain Relationship 9-6 Transform Relative to Absolute Rotations 9-6 Triangular Membrane Element 9-8 Numerical Example 9-8 Summary 9-9 References 9-10 Shell Elements 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 Introduction 10-1 A Simple Quadrilateral Shell Element 10-2 Modeling Curved Shells with Flat Elements 10-3 Triangular Shell Elements 10-4 Use of Solid Elements for Shell Analysis 10-5 Analysis of The Scordelis-Lo Barrel Vault 10-5 Hemispherical Shell Example 10-7 Summary 10-8 References 10-8 APPENDIX I-6 STATIC AND DYNAMIC ANALYSIS where P is a by 21 array that is a function of the global x, y and z reference system The element node forces can be expressed in terms of the assumed stress distribution by the direct application of the principle of virtual work in which the virtual displacements d are of the same form as the basic displacement approximation Or, from Equation (6.3), the virtual displacements, including incompatible modes, are: d = [BC u BI ]  α  (I.16) If the virtual and incompatible displacements are all set to one, the following equation can be used to calculate node forces for an eight-node solid element: T  BC   fi  T f =   = d s dV =   T P dV  c = Qc   Vol Vol BI     ∫ ∫ (I.17) The 33 by 21 matrix Q is calculated using standard numerical integration The forces associated with the nine incompatible modes are zero The system of equations is approximately solved by the least square method, which involves the solution of: QT Qc = QT f or Hc = B (I.18) After c is evaluated for each load condition, the six components of stress at any point (x,y,z) within the element can be evaluated from Equation (I.15) APPENDIX J CONSISTENT EARTHQUAKE ACCELERATION AND DISPLACEMENT RECORDS Earthqua ke Ac cel erations can b e Measur ed Ho wever, Stru ctures ar e Su bjected to Earthq uake Displacemen ts J.1 INTRODUCTION At the present time most earthquake motions are approximately recorded by accelerometers at equal time intervals After correcting the acceleration record, as a result of the dynamic properties of the instrument, the record may still contain recording errors Assuming a linear acceleration within each time interval, a direct integration of the accelerations generally produces a velocity record with a non-zero velocity at the end of the record that should be zero And an exact integration of the velocity record does not produce a zero displacement at the end of the record One method currently used to mathematically produce a zero displacement at the end of the record is to introduce a small initial velocity so that the displacement at the end of the record is zero However, this initial condition is not taken into account in the dynamic analysis of the computer model of the structure In addition, those displacement records cannot be used directly in multi-support earthquake response analysis The purpose of this appendix is to summarize the fundamental equations associated with time history records It will be demonstrated that the recovery of accelerations from displacements is an unstable numerical operation A new APPENDIX J-2 STATIC AND DYNAMIC ANALYSIS numerical method is presented for the modification of an acceleration record, or part of an acceleration record, so that it satisfies the fundamental laws of physics in which the displacement, velocity and acceleration records are consistent J.2 GROUND ACCELERATION RECORDS Normally, 200 points per second are used to define an acceleration record, and it is assumed that the acceleration function is linear within each time increment, as shown in Figure J.1 u ui ui −1 t ∆t TIME Figure J.1 Typical Earthquake Acceleration Record Ground velocities and displacements can then be calculated from the integration of the accelerations and velocities within each time step Or: (u i − u i − ) ∆t u(t) = ui −1 + t u u= t2 u t2 t3 + u i −1 + u u(t) = ui −1 + t ui −1 + u(t) = ui −1 + t ui −1 (J.1) The evaluation of those equations at t = ∆t produces the following set of recursive equations: CONSISTENT EARTHQUAKE RECORDS APPENDIX J-3 ( u i − u i −1 ) ∆t ui = ui −1 + ∆t u u= i=1,2,3 ∆t u ∆t ∆t u i −1 + u + ui = ui −1 + ∆t ui −1 + ui = ui −1 + ∆t ui −1 (J.2) The integration of ground acceleration records should produce zero velocity at the end of the record In addition, except for near fault earthquake records, zero displacements should be obtained at the end of the record Real earthquake accelerations are normally corrected to satisfy those requirements Note that the displacements are cubic functions within each time increment Therefore, if displacements are used as the specified seismic loading, smaller time steps or a higher order solution method, based on cubic displacements, must be used for the dynamic structural analysis On the other hand, if accelerations are used as the basic loading, a lower order solution method, based on linear functions, may be used to solve the dynamic response problem J.3 CALCULATION OF ACCELERATION RECORD FROM DISPLACEMENT RECORD Rewriting Equation (J.2), it should be possible, given the displacement record, to calculate the velocity and acceleration records from the following equations: ∆t u i −1 ] [ui − ui −1 − ∆t ui −1 − ∆t ∆t ui = ui −1 + ∆t ui −1 + u ui = ui −1 + ∆t u u= (J.3) On the basis of linear acceleration within each time step, Equations (J.2) and (J.3) are theoretically exact, given the same initial conditions However, computer round off introduces errors in the velocities and accelerations and the recurrence Equation (J.3) is unstable and cannot be used to recover the input acceleration APPENDIX J-4 STATIC AND DYNAMIC ANALYSIS record This instability can be illustrated by rewriting the equations in the following form: ui = −2 ui −1 − ui = − u i −1 ∆t ∆t u i −1 + ( u i − u i −1 ) ∆t − u i − + ( u i − u i −1 ) ∆t (J.4) If the displacements are constant, the recurrence equation written in matrix form is:  2 u u = −   i   ∆t ∆t   u    u i −1  (J.5) Or, if a small round-off error, ε , is introduced as an initial condition, the following results are produced: ε  u u0 =   =   , u 0   u = −ε  , 6 / ∆t   u2 = ε  24 / ∆t (J.6) It is apparent from Equation (J.6) that the introduction of a small round-off error in the velocity or acceleration at any step will have an opposite sign and be amplified in subsequent time steps Therefore, it is necessary to use an alternate approach to calculate the velocities and accelerations directly from the displacement record It is possible to use cubic spline functions to fit the displacement data and to recover the velocity and acceleration data The application of Taylor’s series at point i produces the following equations for the displacement and velocity: t2 t3 ui + u t u(t) = u i + t ui + u u(t) = u i + t ui + (J.7) Elimination of u from these equations produces an equation for the acceleration at time ti Or: CONSISTENT EARTHQUAKE RECORDS ui = (ui − u(t)) + (u(t) + 2ui ) t t APPENDIX J-5 (J.8) Evaluation of Equation (22.10) at t = ± ∆t (at i + and i-1) produces the following equations: ui = 6 (ui − ui +1 ) + (ui +1 + 2ui ) = (ui − ui −1 ) − (ui −1 + 2ui ) (J.9) ∆t ∆t ∆t ∆t Requiring that u be continuous, the following equation must be satisfied at each point: u i − + 4u i + u i + = (u i + − u i − ) ∆t (J.10) Therefore, there is one unknown velocity per point This well-conditioned tridiagonal set of equations can be solved directly or by iteration Those equations are identical to the moment equilibrium equations for a continuous beam that is subjected to normal displacements After velocities (slopes) are calculated, accelerations (curvatures) and derivatives (shears) are calculated from: (ui − u i +1 ) + (ui + 2ui ) ∆t ∆t u − u i −1 u= i ∆t ui = (J.11) This “spline function” approach eliminates the numerical instability problems associated with the direct application of Equations (J.4) However, it is difficult to physically justify how the displacements at a future time point i + can affect the velocities and accelerations at time point i J.4 CREATING CONSISTENT ACCELERATION RECORD Earthquake compression, shear and surface waves propagate from a fault rupture at different speeds with the small amplitude compression waves arriving first For example, acceleration records recorded near the San Francisco-Oakland Bay Bridge from the 1989 Loma Prieta earthquake indicate high frequency, small acceleration motions for the first ten seconds The large acceleration phase of the APPENDIX J-6 STATIC AND DYNAMIC ANALYSIS record is between 10 and 15 seconds only However, the official record released covers approximately a 40-second time span Such a long record is not suitable for a nonlinear, time-history response analysis of a structural model because of the large computer storage and execution time required It is possible to select the “large acceleration part of the record” and use it as the basic input for the computer model To satisfy the fundamental laws of physics, the truncated acceleration record must produce zero velocity and displacement at the beginning and end of the earthquake This can be accomplished by applying a correction to the truncated acceleration record that is based on the fact that any earthquake acceleration record is a sum of acceleration pluses, as shown in Figure J.2 ui Area = Ai = u i ∆t i −1 i +1 ∆t TIME ∆t t I − ti ti tI Figure J.2 Typical Earthquake Acceleration Pulse From spline theory, the exact displacement at the end of the record is given by the following equation: uI = I ∑ (t I − ti ) ui ∆t = ∆U (J.12) i =1 A correction to the acceleration record may now be calculated so that the displacement at the end of the record, Equation (J.12), is identically equal to zero Rather then apply an initial velocity, the first second or two of the acceleration record can be modified to obtain zero displacement at the end of the APPENDIX J-7 CONSISTENT EARTHQUAKE RECORDS record Let us assume that all of the correction is to be applied to the first “L” values of the acceleration record To avoid a discontinuity in the acceleration record, the correction will be weighted by a linear function, from α at time zero to zero at time t L Therefore, the displacement resulting from the correction function at the end of the record is of the following form: L ∑α i =1 L−i (t I − ti ) ui ∆t = α pU pos + α nU neg = −∆U L (J.13) For Equation (J.13) the positive and negative terms are calculated separately If it is assumed that the correction is equal for the positive and negative terms, the amplitudes of the correction constants are given by: αp = − U pos ∆U and α n = − U neg ∆U (J.14a and J.14b) Therefore, the correction function can be added to the first “L” values of the acceleration record to obtain zero displacement at the end of the record This simple correction algorithm is summarized in Table J.1 If the correction period is less that one second, this very simple algorithm, presented in Table J.1, produces almost identical maximum and minimum displacements and velocities as the mathematical method of selecting an initial velocity However, this simple one-step method produces physically consistent displacement, velocity and acceleration records This method does not filter important frequencies from the record and the maximum peak acceleration is maintained The velocity at the end of the record can be set to zero if a similar correction is applied to the final few seconds of the acceleration record Iteration would be required to satisfy both the zero displacement and velocity at the end of the record APPENDIX J-8 STATIC AND DYNAMIC ANALYSIS Table J.1 Algorithm to Set Displacement at End of Records to Zero GIVEN UNCORRECTED ACCELERATION RECORD 0, u1 , u , u , u , u I −1 ,0 and L COMPUTE CORRECTION FUNCTION ∆U = I ∑ (t I − ti ) ui ∆t i =1 L−i (t I − ti ) ui ∆t = U pos + U neg L i =0 ∆U ∆U αp = − and α n = − 2U pos 2U neg L ∑ CORRECT ACCELERATION RECORD L−i ) ui L L−i if ui < then ui = (1 + α n ) ui L if ui > then ui = (1 + α p J.5 i = 1, L SUMMARY Acceleration records can be accurately defined by 200 points per second and with the assumption that the acceleration is a linear function within each time step However, the resulting displacements are cubic functions within each time step and smaller time steps must be user-define displacement records The direct calculation of an acceleration record from a displacement record is a numerically unstable problem, and special numerical procedures must be used to solve this problem The mathematical method of using an initial velocity to force the displacement at the end of the record to zero produces an inconsistent displacement record that should not be directly used in a dynamic analysis A simple algorithm for the correction of the acceleration record has been proposed that produces physically acceptable displacement, velocity and acceleration records INDEX Automated Computer Program, 5-1 Axial Deformations, 2-10, 7-5 Axial Element, 4-2 Axial Flexibility, 1-11 Axial Stiffness, 1-11 Axisymmetric Material Properties, 1-10 A Acceleration Records, 15-4, J-1 Accidental Torsion, 17-12 Algorithms for Bilinear Plasticity Element, 21-5 Correction of Acceleration Records, J-5 Damping Element, 21-8 Evaluation of Damping, 19-3 Fast Nonlinear Analysis, 18-8 Friction-Gap Element, 21-12 Gap-Crush Element, 21-8 Gauss Elimination, C-4 Gram-Schmidt, 14-6 Inverse Iteration, 14-5 Jacobi Method, D-2 LDL Factorization, C-16 Load Dependent Ritz Vectors, 14-17 Matrix Inversion, C-9 Newmark Integration Method, 20-4 Nonlinear Damping, 21-10 Partial Gauss Elimination, C-13 Solution of General Set of Equations, C-6 Solution of Modal Equations, 13-9 Static Condensation, C-13 Subspace Iteration, 14-6 Tension-Gap-Yield Element, 21-7 Viscous Element, 21-8 Anisotropic Material, 1-1 Arbitrary Dynamic Loading, 13-5 Arbitrary Frame Element, 4-1 Arbitrary Shells, 10-3 Arch Dam, 10-1 Assumed Displacement, 4-3 Assumed Stress, 4-3 B B Bar Method, 6-2 Banded Equations, C-15 Base Shear, 17-2 Bathe, K J., 12-10, 14-19 Bayo, E., 14-19, 15-24 Beams, 4-1 Beam-Shell Analysis, 7-12 Body Forces, 3-8 Boeing Airplane Company, 5-1 Boresi, 1-12, 2-14 Boundary Conditions, 7-1 Bridge Analysis, 22-2 Buckling Analysis, 11-3, 18-2 Building Codes, 17-1 Bulk Modulus, 1-8 C Cable Element, 11-1 CDC-6400, H-2 Center of Mass, 7-7 Cholesky, C-1 Chopra, A., 12-10 Clough, Ray W., 3-16, 5-1, 16-15 Coefficient of Thermal Expansion, 1-1 Compatibility, 1-1, 2-3, 2-13 Compliance Matrix, 1-3, 1-6, 1-10 Compression Wave Velocity, 1-9 Consistent Mass, 3-10, 3-13 II STATIC AND DYNAMIC ANALYSIS Constraints, 7-4, 7-14 Cook, R., 2-14, 5-15, 5-16 Correction for Higher Mode Truncation, 14-13 Correction for Static Loads, 14-13 Correction Matrix, 6-4, 9-5 Cramer's Rule, C-2 Cubic Displacement Functions, 22-4, J-3 Direct Integration, 12-4, 20-1 Frequency Domain, 12-6 Mode Superposition, 12-5 Response Spectrum, 12-5 Dynamic Equilibrium Equations, 3-13 Dynamic Participation Ratios, 13-14 Dynamic Response Equations, 13-4, 13-5 E D Damping Classical Damping, 13-3, 19-3 Decay Ratio, 19-3 Energy Loss Per Cycle, 19-4 Equilibrium Violation, 19-4 Experimental Evaluation, 19-4 Mass Proportional, 19-7 Nonlinear, 18-3, 19-9 Rayleigh, 19-1, 19-6 Stiffness Proportional, 19-7 Damping Matrix, 13-3 Deformed Position, 2-2, 2-10 Der Kiureghian, A., 15-24 Determinate Search Method, 14-2 Diagonal Cancellation, C-20 Diagonal Mass Matrix, 7-7, 7-11 Dickens, J., 14-19 Direct Flexibility Method, 4-6 Direct Stiffness Method, 3-7 Displacement Boundary Conditions, 7-2 Displacement Compatibility, 2-3, 2-7, 2-13, 6-1 Displacement Seismic Loading, 22-1 Displacement Transformation Matrix, 2-9, 2-11 Distorted Elements, 9-8 Doherty, W P., C-21 Double Precision, 7-4 Dovey, H H., C-21 Duhamel Integral, 13-8 Dynamic Analysis by Earthquake Excitation Factors, 13-3 Earthquake Loading, 12-3 Effective Length, 11-11 Effective Shear Area, F-5 Effective Stiffness, 18-4, 18-15 Eigenvalue Problem, D-1, D-6 Eigenvalues, 14-1, D-1 Eigenvalues of Singular System, D-1 Eigenvectors, 14-1, D-1 Element Flexibility, 2-11 Element Stiffness, 2-11 Energy, 3-1 Complementary Energy, 3-11 Energy Dissipation Elements, 21-1 Energy Pump, 3-5 External Work, 3-7 Kinetic Energy, 3-4, 3-12, 3-15 Mechanical Energy, 12-9 Minimum Potential Energy, 3-9 Potential Energy, 3-4 Stationary Energy, 3-9 Strain Energy, 3-6, 3-7 Zero Strain Energy, 12-9 Equilibrium, 1-1, 2-1, 2-2 Exponent Range, 7-4 External Work, 3-1, 3-7 F Fast Nonlinear Analysis Method, 18-1 Finite Element Method, 5-1 Floor Diaphragm Constraints, 7-4 FLOOR Program, 8-17 Fluid Properties, 1-8 INDEX III FNA Method, 18-1 Force Method, 3-10 Force Transformation Matrix, 2-9 Frame Element, 4-1 Absolute Reference System, 4-11 Displacement Transformation, 4-10 Geometric Stiffness, 11-2 Local Reference System, 4-9 Member End Releases, 4-12 Member Loading, 4-12 Properties, 4-7 Properties, 4-7 Free-Field Displacements, 16-4 Friction-Gap Element, 21-10 G Gap Element, 18-3 Gap-Crush Element, 19-9 Gauss Integration, 5-4 Gauss Points, 5-4 Generalized Mass, 13-2 Geometric Stiffness, 11-1, 11-11 Givens, 14-18 Gram-Schmidt Orthogonalization, 14-4, 14-18 Incremental Solution, 7-4 Infinitesimal Displacements, 3-2 Initial Conditions, 13-4 Initial Position, 2-10 Initial Stresses, 1-4 Inverse Iteration, 14-3 Irons, Bruce M., 5-1, 6-1 Isoparametric Elements, 5-1 Area, 5-9 Definition, 5-3 Mid-Side Nodes, 5-6 Shape Functions, 5-3, 5-6 Isotropic Materials, 1-5 Iteration, 7-4 Itoh, T., 14-19 J Jacobi Method, 14-18, D-2 Jacobian Matrix, 5-9, 6-1 K K Factor, 11-11 Kinetic Energy, 1-1 Kirchhoff Approximation, 8-2, 10-7 L H Half-Space Equations, 16-14 Harmonic Loading, 12-7 Hart, J., 16-15 Hierarchical Functions, 8-4 Higher Mode Damping, 22-5 Hilber, 20-8 Hoit, M., C-21 Hourglass Displacement Mode, 5-10 Householder, 14-18 Hughes, T J R., 20-11 I Ibrahimbegovic, Adnan, 8-17, 9-10 Impact, 3-14 Incompatible Displacements, 5-1, 6-2, 6-6 Lagrange Multipliers, 7-16 Lagrange's Equations of Motion, 3-12 Lame's Constants, 1-8 Lanczos Method, 14-11 Large Strains, 18-1 LDL Factorization, C-16 LDR Vectors, 14-1, 14-7, 14-18, 16-9 Lysmer, J., 16-15 M Mass, Generalized, 13-2 Mass Density, 1-10 Mass Participation Ratios, 13-11 Mass Participation Rule, 13-12 Massless Foundation Approximation, 16-11 IV STATIC AND DYNAMIC ANALYSIS Master Node, 7-7 Material Interface, 2-5 Material Properties Summary, 1-9 Material Property Transformation, E-1 Matrix Inversion, C-1, C-9 Matrix Multiplication, B-1 Matrix Notation, B-1 Matrix Transpose, B-4 Mechanical Energy, 12-9 Membrane Element, 9-1 Mesh Transitions, 7-14 Method of Joints, 2-7 Modal Damping, 13-3 Modal Participation Factors, 13-3 Mode Shapes, 13-1 Mode Superposition Analysis, 13-1 Mode Truncation, 22-15 Modulus of Elasticity, 1-1 Moment Curvature, 1-12 Momentum, Conservation, 3-13 Multi-Support Earthquake Motions, 16-6, 22-1 N Newmark Integration Method, 20-2 Alpha Modification, The, 20-9 Average Acceleration Method, 20-5 Stability, 20-4 Summary of Methods, 20-9 Wilson's Modification, 20-6 Newton's Second Law, 3-5, 12-1 Nonlinear Elements, 21-1 Nonlinear Equilibrium Equations, 18-3 Nonlinear Stress-Strain, 18-2 Non-Prismatic Element, 4-1 Normal Rotations, 9-1 Numerical Accuracy, C-20 Numerical Damping, 13-5 Numerical Integration Rules, G-1 Gauss 1D Rule, 5-4 Point 2D Rule, G-1 Point 2D Rule, G-1 Point 3D Rule, G-10 14 Point 3D Rule, G-8 Numerical Operation, Definition, B-6, H-1 Numerical Problems, 7-3 Numerical Truncation, 7-3 O Orthogonal Damping Matrices, 19-7 Orthogonality Conditions, 13-2 Orthotropic Materials, 1-5 P Paging Operating System, H-3 Partial Gauss Elimination, 4-13, C-13 Participating Mass Ratios, 13-11 Patch Test, 2-3, 6-1, 8-9 P-Delta Effects, 11-1, 17-3, 18-1 Penalty Functions, 7-16 Penzien, J., 15-24, 16-15 Periodic Dynamic Loading, 13-10 Piece-Wise Linear Loading, 13-1 Pivoting, C-6 Plane Strain, 1-6 Plane Stress, 1-7 Plasticity Element, 21-3 Plate Bending Elements, 8-1 Constant Moment, 8-9 Convergence, 8-11 DKE, 8-12 DSE, 8-12, 10-6 Examples, 8-11 Patch Test, 8-9 Point Load, 8-13 Positive Displacements, 8-5 PQ2, 8-11 Properties, 8-8 Reference Surface, 8-1 Shearing Deformations, 8-2 Strain-Displacement Equations, 8-7 Thin Plates, 8-1 INDEX Torsion, 8-16 Triangular Element, 8-10 Point Loads, 2-2 Poisson's Ratio, 1-1, 6-3 Popov, Egor P., 1-12 Principal Directions, 17-10 Principal Stresses, D-4 Profile Storage of Stiffness Matrix, C-15 Q Quadrature Rules, G-1, G-2 Quadrilateral Element, 5-7, 8-3, 9-2, 10-2 R Radiation Boundary Conditions, 16-11 Rafai, M.S., 6-2 Rank Deficient Matrix, 5-11 Rayleigh Damping, 19-1, 19-6 Recurrence Solution for Arbitrary Dynamic Loading, 13-10 Relative Displacements, 22-2 Relative Rotations, 9-6, 9-7 Response Spectrum Analyses, 15-1 CQC Modal Combination, 15-8 CQC3 Direction Combination, 15-15 Definitions, 15-2 Design Spectra, 15-12 Limitations, 15-21 Numerical Evaluation, 13-8 Principal Stresses, 15-22 SRSS Directional Combination, 15-17 SRSS Modal Combinations, 15-8 Story Drift, 15-21 Stress Calculations, 15-22 Typical Curves, 15-4 100/30 and 100/40 Directional Combination Rules, 15-17 Rigid Body Constraints, 7-11 V Rigid Body Displacements, 4-4, 7-11 Rigid Body Rotation, 7-8 Rigid Elements, 7-3 Rigid Zones, 7-14 Ritz Method, 14-1 Rotation Definition, 2-4 S SAFE Program, 8-17 Scaling of Results, 17-11 Scordelis-Lo Barrel Vault, 10-5 Section Forces, 4-6 Section Properties, 4-6 Selective Integration, G-11 SHAKE Program, 16-2 Shape Functions, 5-3 Shear Locking, 6-2, 9-9 Shear Modulus, 1-6, 1-7, 1-8, 1-9 Shear Wall Analysis, 7-13 Shear Wave Velocity, 1-9 Shearing Deformations, 4-8, 8-2, F-1 Shell Elements, 10-1 Shifting of Eigenvalues, 14-6 Significant Figures, 7-4 Simo, J C., 6-2 Simpson's Rule, 5-4 Site Response Analysis, 16-2 Slave Nodes, 7-9 Soil-Structure Interactions, 16-1 Solution of Equations, 2-11, 12-7, C-1 Sparse Matrix, 7-10 Speed of Computers, H-1 Spline Functions, J-4 Static Condensation, 4-13, 6-5, 8-10, 13-1, C-13 Static Load Participation Ratios, 13-13 Statically Determinate Structures, 2-7 Statically Indeterminate Structures, 3-1, 3-10 Step by Step Integration, 20-1 Strain Compatibility, 2-3 VI STATIC AND DYNAMIC ANALYSIS Strain Displacement Equations 2D Plane Elements, 9-5 3D Linear Solids, 2-4 3D Nonlinear Solids, 11-11 Plate Bending, 8-7 Strain Energy, 1-1 Strang, G., 6-1 Stress Continuity, 2-5 Stress Definition, 1-2 Stress Resultant, 2-2 Stress-Strain Relationship, 1-1, 9-6 Structural Design, 7-3 Sturm Sequence Check, 14-3 Subspace Iteration, 14-5 Substructure Analysis, C-13 Truss Element, 2-7 U Undamped Free Vibration, 12-8 Uplifting Frame, 18-9, 18-13 V Vector Cross Product, A-2 Vector Notation, A-1 Vectors Used to Define Local Reference System, A-4 Vertical Earthquake Response, 14-15 Virtual Displacements, 3-3 Virtual Work, 3-1 Viscous Damping, 19-1 Volume Change, 1-8 T Tapered Rod Analysis, 5-5 Taylor, R L., 6-1, 20-8 Tension Only Element, 18-3 Tension-Gap-Yield Element, 21-6 Tetrahedral Elements, 5-14 Thermal Strains, 1-3 Thermal Stresses, 1-4 Time Increment, 13-5, 13-8 Time Step Size, 22-9 Torsional Effects on Buildings, 17-12 Torsional Flexibility, 4-8 Torsional Force, 1-11 Torsional Moment of Inertia, 1-11 Torsional Stiffness, 1-11, 4-8 Triangular Elements, 5-14, 9-8, 10-4 Truncation Errors, 7-3 W Watabe, M., 15-24 Wave Loading, 13-10 Wave Propagation, 1-10, 16-11 WAVES Program, 16-2, 16-15 Wind Loading, 13-10 Y Young's Modulus, 1-6, 1-8 Yuan, M., 14-19 Z Zero Energy Mode, 9-7 ... and are defined in terms of three numbers: modulus of elasticity E , Poisson’s ratio 1-2 STATIC AND DYNAMIC ANALYSIS ν and coefficient of thermal expansion α In addition, the unit weight w and. .. 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... effective for the development of structural analysis programs The static and dynamic analysis of structures has been automated to a large degree because of the existence of inexpensive personal computers