Matrix structural dynamics

Matrix Analysis of Structural Dynamics Matrix Analysis of Structural Dynamics Applications and Earthquake Engineering Franklin Y Cheng University of Missouri, Rolla Rolla, Missouri MARCEL DEKKER, INC NEW YORK • BASEL Library of Congress Cataloging-in-Publication Data Cheng, Franklin Y Matrix analysis of structural dynamics: applications and earthquake engineering/ Franklin Y Cheng p cm - (Civil and environmental engineering; 4) Includes index ISBN 0-8427-0387-1 (alk paper) Structural dynamics Earthquake engineering Matrices I Title II Series TA654.C515 2000 624.1'7-dc21 00-031595 This book printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http: / / www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above Copyright © 2001 by Marcel Dekker, Inc AH Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit): 10 PRINTED IN THE UNITED STATES OF AMERICA Civil and Environmental Engineering A Series of Reference Books and Textbooks Editor Michael D Meyer Department of Civil and Environmental Engineering Georgia Institute of Technology Atlanta, Georgia Preliminary Design of Bridges for Architects and Engineers Michele Melaragno Concrete Formwork Systems Awad S Hanna Multilayered Aquifer Systems: Fundamentals and Applications Alexander H.-D Cheng Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering Franklin Y Cheng Hazardous Gases Underground: Applications to Tunnel Engineering Barry R Doyle Cold-Formed Steel Structures to the AISI Specification Gregory J Hancock, Thomas M Murray, Duane S Ellifritt Fundamentals of Infrastructure Engineering: Civil Engineering Systems: Second Edition, Revised and Expanded Patrick H McDonald Handbook of Pollution Control and Waste Minimization Abbas Ghassemi Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods Victor N Kaliakin 10 Geotechnical Engineering: Principles and Practices of Soil Mechanics and Foundation Engineering V N S Murthy Additional Volumes in Production Chemical Grouting and Soil Stabilization: Third Edition, Revised and Expanded Reuben H Karol Estimating Building Costs Calin M Popescu, Kan Phaobunjong, Nuntapong Ovararin PREFACE OBJECTIVES AND ORGANIZATION This book covers several related topics: the displacement method with matrix formulation, theory and analysis of structural dynamics as well as application to earthquake engineering, and seismic building codes As computer technology rapidly advances and buildings become taller and more slender, dynamic behavior of such structures must be studied using state-of-the-art methodology with matrix formulation Analytical accuracy and computational efficiency of dynamic structural problems depends on several key features: structural modeling, material property idealization, loading assumptions, and numerical techniques The features of this book can be summarized as follows Three structural models are studied: lumped mass, consistent mass, and distributed mass Material properties are presented in two categories: damping and hysteretic behavior Damping is formulated in two types: proportional and nonproportional Hysteretic behavior is studied with eight models suited to different construction materials such as steel and reinforced concrete Loading comprises a range of time-dependent excitations, for example, steady-state vibration, impact loading, free and transient vibration, and earthquake ground motion Numerical techniques emphasize two areas: eigensolution and numerical integration The former covers fundamental as well as advanced techniques for five predominant methods; the latter covers five well-known integration techniques Structural dynamics theory is used to substantiate seismic building-code provisions Representative codes are discussed to illustrate their similarities and differences This book is intended for graduate students as well as advanced senior undergraduates in civil, mechanical, and aeronautical engineering It is also intended as a reference tool for practitioners In the preparation of this text, six organizing principles served as guidelines The book functions as a self-study unit Its technical detail requires the reader to be knowledgeable only in strength of materials, fundamental static structural analysis, calculus, and linear algebra Essential information on algebraic matrix formulation, ordiIII iv PREFACE nary and partial differential equations, vector analysis, and complex variables is reviewed where necessary Step-by-step numerical examples are provided This serves to illustrate mathematical formulations and to interpret physical representations, enabling the reader to understand the formulae vis-a-vis their associated engineering applications Each chapter discusses a specific topic There is a progression in every chapter from fundamental to more advanced levels; for instance, eigensolution methods are grouped accordingly in Chapter 2, numerical integration techniques in Chapter 7, and hysteresis models in Chapter This approach may help the reader to follow the subject matter and the instructor to select material for classroom presentation Topic areas are covered comprehensively For example, three structural models are studied for uncoupling and coupling vibrations with longitudinal, flexural, and torsional motions Flexural vibration extends from bending deformation to bending and shear deformation, rotatory inertia, P-A effect, and elastic media support The reader can attain greater understanding from this integrative approach 3-D building structures are treated in one chapter Comprehensive formulations are developed for member, joint, and global coordinate transformation for general 3-D structures Building systems in particular are extensively analyzed with consideration of floor diaphragms, bracings, beams, columns, shear walls, and the rigid zone at connecting joints These elements are not collectively covered in a structural dynamics text or a static structural analysis text; this book can supplement the latter Examples are designed to help the reader grasp the concepts presented Contained in the book are 114 examples and a set of problems with solutions for each chapter A detailed solutions manual is available Computer programs are included that further clarify the numerical procedures presented in the text SCOPE OF TEXT AND TEACHING SUGGESTIONS The text can be used for two semesters of coursework, and the sequence of 10 chapters is organized accordingly Chapters 1-6 compose the first semester, and Chapters 7-10 the second Fundamental and advanced topics within chapters are marked as Part A and Part B, respectively If the book is used for one semester, Part B can be omitted at the instructor's discretion The scope of the text is summarized as follows Chapter presents single degree- of-freedom (d.o.f.) systems Various response behaviors are shown for different types of time-dependent excitations Well-known solution techniques are elaborated Chapter is devoted to response behavior of multiple d.o.f systems without damping The significance of individual modes contributing to this behavior is the focus, and comprehensive understanding of modal matrix is the goal of this chapter As a function of computational accuracy and efficiency, eigensolution methods are examined These methods include determinant, iteration, Jacobian, Choleski decomposition, and Sturm sequence Response analysis extends from general problems with symmetric matrix and distinct frequencies to unsymmetric matrix as well as zero and repeated eigenvalues for various fields of engineering Chapter examines the characteristics of proportional and nonproportional damping Numerical methods for eigenvalues and for response considering both types of damping are included, and solutions are compared Chapter presents the fundamentals of distributed mass systems Emphasis is placed on dynamic stiffness formulation, steady-state vibration for undamped harmonic excitation, and transient vibration for general forcing function including earthquake excitation with and without damping Chapter continues the topic of distributed mass systems to include longitudinal, flexural, and torsional coupling vibration Also included are bending and shear deformation, rotatory inertia, and P-A effect with and without elastic media support Vibrations of trusses, elastic frames, and plane grid systems are discussed PREFACE v Chapter introduces consistent mass model for finite elements Frameworks and plates are studied with emphasis on isoparametric finite element formulation Advanced topics include tapered members with Timoshenko theory and P-A effect Note that the structural model of a distributed mass system in Chapter yields the lower bound of an eigensolution while the model in Chapter yields a solution between a lumped mass and a distributed mass model Solutions are thus compared Chapter covers structural analysis and aseismic design as well as earthquake characteristics and ground rotational movement Well-known numerical integration methods such as Newmark's, Wilson-0, and Runge-Kutta fourth-order are presented with solution criteria for error and stability behavior Procedures for constructing elastic and inelastic response spectra are established, followed by design spectra This chapter introduces six components of ground motion: three translational and three rotational Response spectra are then established to reveal the effect of those components on structural response Modal combination techniques such as CQC (Complete Quadratic Combination) are presented in detail Computer program listings are appended for the numerical integration and modal combination methods so that they can be used without sophisticated testing for possible bugs Chapter focuses on 3-D build structural systems composed of various steel and reinforced concrete (RC) members The formulations and numerical procedures outlined here are essential for tall building analysis with P-A effect, static load, seismic excitation, or dynamic force Chapter presents inelastic response analysis and hysteresis models such as elasto-plastic, bilinear, curvilinear, and Ramberg-Osgood Additional models for steel bracings, RC beams and columns, coupling bending shear and axial deformations of low-rise shear walls, and axial hysteresis of walls are provided with computer program listings to show calculation procedures in detail These programs have been thoroughly tested and can be easily implemented for structural analysis Also included are nonlinear geometric analysis and large deformation formulae Chapter 10 examines three seismic building codes: the Uniform Building Codes of 1994 and 1997 and the International Building Code of 2000 IBC-2000 creates uniformity among the US seismic building codes, and replaces them This chapter relates code provisions to the analytical derivations of previous chapters It explains individual specifications and compares them across the codes Since the IBC departed from the UBC format in organization of sections, figures, tables and equations, the chapter concludes with summary comparisons of the codes Numerical examples in parallel form delineate the similarities and differences ACKNOWLEDGMENTS This book consolidates results from my years as a teacher and researcher Teaching consists of classes at University of Missouri-Rolla (UMR) and the UMR Engineering Education Center in St Louis, lectures for receiving honorary professorships in China at Harbin University of Architecture and Engineering, Xian University of Architecture and Technology, Taiyuan University of Technology, and Yunnan Polytechnic University as well as UMR Continuing Education short courses Distinguished guest speakers at the short courses—the late Professor Nathan M Newmark, Professor N Khachaturian, and Dr V B Venkayya—have my wholehearted appreciation for their contribution UMR has my continued thanks for bestowing on me the distinguished Curators' professorship to enhance my research and teaching My deep gratitude goes to the National Science Foundation, particularly Dr S C Liu, for sustained guidance and support of my research I am grateful to my former graduate students, especially Drs J F Ger, K Z Truman, G E Mertz, D S Juang, D Li, K Y Lou, H P Jiang, and Z Q Wang as well as Misses Y Wang and C Y Luo, for their endeavors to improve the manuscript and solutions manual Also my thanks go to Dr O R Mitchell, Dean of School of Engineering, for his enthusiasm in my career development, and to departmental staff members C Ousley and E Farrell who gracefully rendered their valuable assistance over a long period of time I extend special appreciation to Brian Black, Technical Coordinator of book editorial, and B J Clark, executive acquisitions editor with Marcel Dekker Inc Mr darks' vision of engineering education and pub- vi PREFACE lication motivated accomplishment of this project My mentors, Professors C K Wang and T C Huang, have my continued appreciation for their early influence and inspiration Everlasting thanks go to my family, including my wife, brothers Jefrey and Ji-Yu, son George, daughter Deborah, daughter-in-law Annie, and grandson Alex Haur-Yih I dedicate this book to my wife Beatrice (Pi-Yu) for her care and encouragement throughout my academic career Franklin Y Cheng CONTENTS Preface iii Characteristics of Free and Forced Vibrations of Elementary Systems 1.1 Introduction 1.2 Free Undamped Vibration 1.2.1 Motion Equation and Solution 1.2.2 Initial Conditions, Phase Angle and Natural Frequency 1.2.3 Periodic and Harmonic Motion 1.3 Free Damped Vibration 1.3.1 Motion Equation and Viscous Damping 1.3.2 Critical Damping, Overdamping and Underdamping 1.3.3 Logarithmic Decrement and Evaluation of Viscous Damping Coefficient 11 1.4 Forced Undamped Vibration 14 1.4.1 Harmonic Forces 14 1.4.2 Steady-State Vibration and Resonance 15 1.4.3 Impulses and Shock Spectra 19 1.4.4 General Loading—Step Forcing Function Method vs Duhamel's Integral 24 1.5 Forced Damped Vibration 29 1.5.1 Harmonic Forces 29 1.5.2 Steady-State Vibration for Damped Vibration, Resonant and Peak Amplitude 30 1.5.3 General Loading—Step-Forcing Function Method vs Duhamel's Integral 32 1.5.4 Transmissibility and Response to Foundation Motion 36 VII 984 NOTATION adjusted maximum considered earthquake spectral response acceleration for short and sec periods, respectively [Tn elastic pseudo-displacement, pseudo-velocity and pseudo-acceleration, respectively Tr constraint matrix between the forces {-fjm} and {Fjs} acting at master and slave joint, respectively [7](r) transmissibility, A[/F orthogonal transformation matrix at the rth iteration defined by Eq (2.141) velocity response spectrum of the rth mode f torsional spectrum in the rth earthquake principal direction time time at maximum response U ay; Svy; S^ inelastic pseudo-displacement, pseudo-velocity and pseudo-acceleration, respectively SMO; SMY; SVY displacement functions of element in x and y directions, respectively ux; uy; uz flexural dynamic stiffness coefficients composite translational spectrum stiffness coefficient matrix [Sg] geometric stiffness coefficient matrix [SNU] dynamic axial stiffness coefficient matrix accelerations of main, intermediate and minor seismic principal components, respectively ux\; ux2; composite torsional spectrum IS) strain (potential) energy; longitudinal displacement ^vertical accelerations of seismic components in structural reference axes, x\, X2, xj, respectively acceleration records in E-W, N-S and vertical directions, respectively displacement matrix ua; ub axial deformation at top and bottom of axial spring, respectively kinetic energy V structural base shear approximate fundamental period Va, Vb shears at top and bottom of shear spring, respectively period in E-W and N-S directions, respectively KD design base shear V base shear calculated by elastic response spectrum analysis [57V] dynamic torsional stiffness coefficient matrix T period; torsional moment; rigid-zone transfer matrix NOTATION 985 Ks base shear calculated by equivalent static procedure va; vb lateral shear deformation at top and bottom of shear spring, respectively vu xe; xm; xy equivalent elastic, maximum and yield displacements, respectively x^; xp homogeneous and particular solution, respectively relative unit shear deformation Vy; Vz unit vectors in element's Ye and Ze axes, respectively Vxi vector at wall top Vxb vector at wall battom V'y vector along with the average longitudinal axis of wall displacement of the rth mode *;•;*:• X sti uncoupled displacement and acceleration of the rth mode, respectively pseudo-static displacement of rth mode static deflection, F/K Vi, Vj shears at end-/ and end-j, respectively Kp primal wave Vs secondary wave V(x,i) time-dependent shear force W;w weight; work done due to external forces; wind load WEI effective weight at the rth mode denned by Eq (10.14) 3cg earthquake record in terms of g (acceleration of gravity) x displacement at l(m)ih dynamic d.o.f due to force applied at the m(€)th dynamic d.o.f t,mi xm,t X 'it0'x'ito uncoupled initial displacement and velocity of the rth mode at t = to respectively xfotal displacement of the kih d.o.f due to translational and rotational seismic components xn +1 displacement at time (n+l)h in which h is the time interval Xji normal mode displacement of yth floor at rth mode Wx\ wx weight at level x Wf, wi weight factor w(x,t) time-dependent load Xm location of mass center [X] eigenvector matrix XR Location of rigidity center [X\q eigenvector for the qih mode x'; x"; x^ r 1st, 2nd and mih differentials of x [X\ eigenvector at the rth iteration x; x; x displacement, velocity and acceleration, respectively f , -r, XQ; XQ initial displacement and velocity at t = Q, respectively XIQ; XK, initial displacement and velocity at t = t0, respectively 3x/d£, and dx/dr] {x}; {x}; {x} displacement, velocity and acceleration vectors, respectively {x'}; {x1} uncoupled displacement and acceleration vectors, respectively 986 NOTATION displacement due to elastic and rigid-body motion, respectively GREEK SYMBOLS a,-; y.j displacement and acceleration of foundation, respectively cross-correlation coefficient between mode ; and mode y, defined by Eq (7.209) displacement and acceleration of super-structure, respectively a fraction of the wall's height at top and bottom, respectively nth normal mode vector logarithmic decrement, In (Xn/Xn + i); shear slope uncoupled initial displacement and velocity, respectively {.x}e(r) elastic displacements due to initial condition and applied force, respectively WavgJ {-*}avg average velocity and acceleration, respectively {Ajc} incremental velocity {Ai^}; {Axg} incremental velocity from t to ? + C and from t to t + OAt, respectively plastic angles at end-; and end-y', respectively participation factor of the y'th mode in the structural rth reference axis y(x) torsional displacement function deflection; variation symbol flexibility matrix Kronnecker delta c5w; A K "avg virtual work and virtual displacement, respectively average displacement at extreme points of structure at level x Yf, YJ deflections at end-; and end-y Ym location of mass center FR location of rigidity center maximum displacement at level x Fr yield reduction factor element deformation Yslfc pseudo-static displacement at kih mode deformation of joint-center Y(x) shape function y(x,t) time-dependent deflection y',; y'n dy/3£, and 3y/3rj {z} state vector, { x } / { x } , defined by Eq (3.35a) Z seismic zone factor displacement of master joint displacement of slave joint deflection by elastic analysis deflection at level x deflection of level x in the mth mode at mass center by elastic analysis NOTATION 987 $x', dy', Oz displacement at master joint Oya; Oyi, rotations in Vy direction 5xf, byf, Oyj displacement at jth node Oza; Oz/, rotations in V^ direction increment symbol; dummy 0U relative unit rotation global displacement Ocr critical seismic input angle {A}, displacement at the iih story 0' angle from ME-W component to wx component Aa allowable drift AM maximum inelastic response A variable in integral; story drift; eigenvalue displacement or story drift ).q eigenvalue of the qih mode Am modal drift in a story /,/ spectral distribution factor corresponding to ground As elastic response displacement or story drift Ax AM(t) story drift at story level x principal translational component n Poisson's ratio; ductility factor p viscous damping factor defined by Eq (1.31); redundancy /reliability factor incremental bending moment prior to time t {At/} unbalanced force {AZ>},- incremental displacement of member /' pg damping factor of the yth mode at the ;th d.o.f incremental force of member ; [O] modal matrix incremental joint displacement {}, modal displacement of the zth mode (®}u> {^ modal displacement of the wth and vth modes, respectively 4>x; 4>y; < components of rotational deformations of member /' summation operation n multiplication operation a; {a} stress and stress matrix e;{g} strain and strain matrix t; natural coordinate C forced period parameter in Wilson's method 0^, Oj rotations at end /' and end j, respectively ground rotational acceleration in the rth earthquake principal direction Oa; O/, torsional frequency parameter; bending slope \jii (i) total rotation of oscillator in the iih earthquake principal direction co angular frequency of excitations flexural rotations at top and bottom of bending spring, respectively 988 n/ (0 NOTATION EPV effective peak velocity direction FFA floor-by-floor assembly seismic force amplification for ft feet GCS global coordinate system GSA general system assembly IBC International Building Code relative rotation in the rth earthquake principal overstrength factor reduction factor MATRIX SYMBOLS { ) column matrix in inch [0] zero matrix JCS joint coordinate system unit matrix k 1000 pounds inverse of matrix kg kilogram transpose of matrix kN 1000 Newtons [\-\\ diagonal matrix ksi 1000 lb/in det F determinant of matrix Ib pound LHS left-hand side N Newton, a unit of force; axial force NEHRP National Earthquake Hazard ABBREVIATIONS BOCA Basic Building Code BT backtracking cm centimeter OS overshooting CQC complete-quadraticcombination PGA peak ground acceleration cps cycle per second rad radian det determinant RHS right-hand side d.o.f degree of freedom SBC Standard Building Code ECS element coordinate system sec; s second EPA effective peak acceleration SRSS square root of the sum of the squares EPGA effective peak ground acceleration UBC Uniform Building Code Reduction Program Index Abcissa, 187 ABSSUM (summation of individual absolute value of individual modes) (see Modal combination method) Acceleration(s), 280, 281 effective, 609 pseudo, 367 spectral value, 394 Acceleration method: average, 329, 358 linear, 330-332, 358, 560 incremental, 542, 546, 596 Amplification factor, deflection, 17, 20, 22, 64, 207, 210, 639 for accidental torsion, 662 force, 626 spectral, 367, 372 Amplitude, 3, 51, 157 decay, 32, maximum, 20 method, 43 peak, 30 resonant, 30 of transmitting force, 38 Assembly, floor by floor, 490, 498 general system, 490 ATC-3-06, 607, 682, 705 Average annual risk, 607 Axial displacement transformation (see Force transformation) Base shear, 610, 641, 650, 693 maximum, 394 Backtracking (see Yield limit) Bandwidth method, 43 Bauschinger effect, 528, 534, 564 Bernoulli-Euler equation, 163 compared to Timoshenko theory, 252 with elastic media, 237 with elastic media and P-A effect, 238 Binomial expansion, 44 Bisection procedure, 97 Branch curve (see Stiffness formulation) Building Code: IBC-2000, 607, 633, 641, 648, 675, 689 UBC-94, 123, 607, 612, 641, 648, 675, ( UBC-97, 123, 607, 624, 641, 648, 676, ( 989 990 Castigliano's first theorem, 266 Center, geometric, 295-296 mass, 621, 657 rigidity, 621, 657, 672, 673 Centroid, inertia force, 174, 280 Chain rule for differentiation, 298 Choleski's decomposition, 47, 81-83, 98 Coding method, 490 Coefficient, cross correlation, 394 damping, evaluation: all modes required, 125 two modes required, 121 equivalent damping, 42, 43 mass, 271, 275 numerical, 626 stability, 640 stiffness, 166, 214, 230, 244, 267, 269, 288 coupling flexural and longitudinal, 225 coupling flexural and torsional, 234 matrix, 54, 178, 220, 266, 435, 535 with static stability (two cases), 240 with torsional vibration, 229, 230, 233, 235-237 viscous damping, 8, 13, 42 Cofactor method, 298 Collapse mechanism formation, delay of 628 Compatibility condition, 304-305 Composite response spectrum (see Spectrum) Composite spectral modal analysis (see Modal analysis) Composite torsional spectral quantity (see Spectra, composite torsional) Composite translational spectral value (see Spectra, composite translational) Conservation of energy method, 375 Constraint matrix (see Matrix) Coordinate(s), generalized: in finite element formulation, 262, 286 global, 218, 222, 262 local (see also d.o.f, local and global), 221, 262, 264 natural, in finite element formulation, 262, 286, 291 system, global, 418 element, 418 joint, 418 CQC (see Modal combination method) Curve-fitting technique, 122 D'Alembert's method, 45 Damping, 7, 283 absolute, 120 coefficient (see Coefficient, damping) coulumb, 117 critical, INDEX [Damping] equivalent, 42, 43 evaluation of, 42 factor (see also Factor), calculated from damping coefficient, 125, 126 matrix, nonproportional vs proportional, 126-128 nonproportional, 152, 156-157 nonviscous, 42 over, 10 proportional, 120, 126-128, 156 relative, 120 structural, 117 subcritical, 10 supercritical, 10 under, 10 viscous, 7, 8, 43, 117 Dashpot, 8, 37 Deflection, dynamic, 263 large, 527 small, theory, 162, 527 static, Deformation, axial, 581 bending, 213, 240, 312 boundary, 174 capacity, 377 displacement, relationship (see Matrices, compatibility) end, 166, 223 force, 476 magnitude of elastic curve (see Elastic curve, deformation magnitude) peak, 374 virtual, 178 Degrees of freedom (d.o.f.), for elastic structure, 175 dummy, 455 global, 220, 221, 426 local, 221 linear displacement, 175 linear inertia force, 306 multiple (see Multiple d.o.f system) out-of-plane, 444 rigid frame, 175 rotational, 273, 306 side-sway, 175, 176, 273 single (see Single d.o.f system) rotation, 176 truss, 175 two (see Two d.o.f system) Determinant method, 50, 99, 101, 107 function, 187, 190 solution, 191, 193 zero, 186, 191, 194, 226, 227, 257 991 INDEX Design, allowable stress (ASD), 627, 648 strength (LRFD), 627, 648 Design: ground shaking, 682 response displacement, 632 response spectra, 677 spectral response acceleration coefficient, 633 at 1-second, 636 at short-period, 636 structural categories, 633 Diaphragms, rigid and flexible, 612 Differential equation, 5, 8, 17, 49, 230, 238 for foundation movement, 185 in matrix, 58 nonhomogeneous, 181 ordinary, 181 of uniform load, 182 Direct element formulation, 218, 220, 223, 233, 490 Direct stiffness method, 179 Displacement: axial, 264, 303 deformation, relationship (see Matrices, compatibility) elastic, 109-111 end, 243 force, relationship (see Rayleigh's dynamic reciprocal principle) function, in finite element formulation, 286 linear (see Degrees of freedom, linear) longitudinal, 214, 263 maximum, 20, 23, 210 pseudo, 367 pseudo-static, 64, 207, 248, 250 response, 65, 66, 112-113, 148, 152, 156 spectral value, 393 strain relationship, 288 transverse, 263 steady-state, 171 virtual, 178 Dot (scalar) product (see Product) Drift, 632, 645, 664, 697 allowable story, 623, 640 calculated story, 623 design story, 639 Ductility, 315, 377 allowable, 377 capacity, 377 demand, 377 load displacement, 380 moment curvature, 381 moment energy, 381 moment rotation, 381 structural system, 382 Duhamel's integral, 24, 26, 34, 59, 282, 346, 349, 363 Dynamic stiffness, (see Coefficient, dynamic stiffness): method, 194, 195 vs lumped mass method, 196, 197 Earthquake, epicenter, 322 focal depth, 322 ground motion components rotational, 383, 389 translational, 327, 383 hypocenter, 322 principal components: main, intermediate and minor, 327, 385 Eccentricity, 658 Effective earthquake force, 609, 611 mass, 609 peak acceleration, 682 peak velocity, 682 Eigenpair, 95, 98, 402 Eigensolution, 146, 161, 240, 306, 318 Choleski's decomposition, 47 damped vs undamped, 130-132 determinant, 186 extraction, 47, 80 iteration, 47, 72-74, 101-103 Jocobi method, 47, 87-89 Sturm sequence, 47, 95-96 for symmetric matrix, 47, 107 for unsymmetric matrix, 47, 101-102 Eigenvalue: comparison of lumped mass, dynamic stiffness and consistent mass, 283-285 complex, 128, 130 damped vs undamped, 135-136 multiple, 105 zero and repeating, 105-114 Eigenvector: complex, 128, 139 iteration, 79 polar form, 153 upper triangle, 189 El Centro earthquake, 35, 328 Elastic curve, 281 frame, 226, 265 media, 238, 252, 258 response parameter, 677 structure, 216 support, 252 Elasticity, fundamental theory, 388 Element coordinate system (ECS) (see Coordinate system) 992 Element(s), finite, 261 isoparametric, 262, 286, 291 quadrilateral, 286, 295-300, 299-300 rectangular, 286, 299-300 subparametric, 292 superparametric, 292 triangular, 286, 299-300 Energy, dissipated, 43 kinetic, 203, 264 potential, 264, 303-304 strain, 179, 204, 266-267 Energy method, 117, 203 Energy theorem, 265 Equilibrium: condition, 549, 550 equation, 5, 176, 213, 221 position, 1, Equivalent damping coefficient method, 42 Equivalent lateral force procedure, 514 Error, numerical (or computational), 358 characteristics of, 358 truncation, 340 Essential facilities, 612 Euler buckling load, 257 Factor: allowable stress, 614, 626 damping, 9, 121, 131, 137, 373 design load, 614 ductility, 373-374 ductility reduction, 614 dynamic load (see Amplification factor) force reduction, 373 importance, 612 magnification, 298 near source, 624, 625, 684 overstrength, 614 participation, 65, 207, 249, 393 response modification, 613 seismic zone, 612 spectral distribution, 412 weight, 294, 299 yield reduction, 373 Fault rupture, 625 Finite element (see Element, finite) Fixed-end moments (and shears): derivation of, 180-183 Fixed-end forces, 186, 256, 276, 278 Floor-by-floor assembly (see Assembly) Force(s): axial, 214, 238, 252, 258, 267, 303 compressive, 240, 589 its effect, and use of negative sign, 305 damping, nonlinear, 43 INDEX [Force(s)] deformation, relationship, 166, 183, 255, 476, 566, 581 rotational, 566 shear, 213, 240, 254, 258 bending, 213, 240 displacement, relationship, 267 dynamic, 58, 61 end, 167, 243 equivalent, 18 final, 282 fixed-end (see Fixed-end forces) harmonic, 29 impulsive, 19 inertia, 186, 193, 216, 262, 280 nodal, due to unit global displacement, 290 rectangular, 19 restoring, 42 tensile, use of positive sign, 305 transformation, 479, 480 transverse inertia, 216, 221 triangular, 21, 511 unbalanced (nodal), 193, 220, 223, 550 technique, 546, 550 Foundation movement, 36, 185 Fourier integral, 387 Frames, moment-resisting, 617 Frequency, angular, complex, 129, 132 coupling, 227, 233-237 damped, 11 damping effect on, 119 equation, 50, 134, 136, 165, 225 flexural, 226, 235 forcing, 17, 198, 199 fundamental, 50, 72, 75, 79 highest, 79 longitudinal and flexural, 224-226 natural, parameter, 187, 199, 252 pseudo-coupling, 227, 228 repeating, 56, 107-108, 111 torsional, 234 Function, determinant, 187-189 displacement, 248, 250, 208-209, 304 shape, 49, 163, 204, 264, 266, 271, 297 bending and shear slope in, 308 sine (see Sine function) time, 49, 163, 214 trigonometric, 163 Fundamental structural mechanics, 218, 221, 272 Gauss elimination, 191, 194, 541 Gaussian quadrature, 293, 295, 298 Generalized forces and displacements, 175 993 INDEX General system assembly (see Assembly) Geometric nonlinearity (see Nonlinearity) Global coordinate system (GCS) (see Coordinate system) Global d.o.f (see d.o.f.) Grillage, 231 (see also Plane grid system) Hooke's law, 288 Housner's average design spectra, 369 Hysteresis model (see Model) Impulse, rectangular, 21 triangular, 21 [Lagrange's equation] multiplier form, 402 Laplace transform, 45, 150 Lateral force: horizontal shear distribution, 639, 644 procedure, 624 vertical distribution, 620, 638, 644, 650, 695 Lateral force, dynamic, 633 equivalent, 633, 659 L'Hospital's operations/rule, 173, 561 Load, buckling, 240, 583, 584 Load combination, ASD, 631, 647, 669, 701 LRFD, 627, 646, 671, 703 Local d.o.f (see d.o.f.) Logarithmic decrement, 12, 13, 120 Logarithmic plot, tripartite, 36, 363-365 (see also Force, rectangular and triangular) Inelastic analysis (see Nonlinear analysis, scope of) Inelastic response displacement, maximum, 632 Inertia force (see Force, inertia) Map, microzonation, 682 regionalization, 682 Inertia: rotatory, 213, 240-242, 258, 311, 315, 317 torsional, 229 seismic zone, 682, 683 transverse, 258, 311, 315-316 Mapping concept, 490 Initial condition, 3, 33 Mass, consistent, 68, 70, 269, 318-319 Initial yield moment (see Moment) distributed, 68, 161 Integrals (see also Duhamel's integral) effective, 608 foundation, 67 impulse, 27 Integration by parts, 34 inertia effect, 427 Interaction effect, 589 lumped, (see also Model, mass lumped): Internal moments due to shears, 600-602 method vs dynamic stiffness method, 197 Interpolation function (see also Function, pseudo-dynamic, 250 shape), 291 rotationary (see Inertia, rotary) scheme, 297 superstructure, 67 Inverse transform, 151 Inversion, 270, 279, 298 matrix (see Matrix) Irregularities: extreme torsional, 633 plan structural, 612, 685 torsional, 633 vertical structural, 612, 676, 685 Iteration method/procedure, 74, 76, 79, 82, 101-105 for complex eigensolutions, 137-138, 156 Jacobian, 292, 298 Jacobi method, 47, 87-89, 95 Joint coordinate system (JCS) (see Coordinate system) Joint, start, end, 429 Joint plastification, 553 Kinematics, 276 Kronecker delta, 118, 120, 126 Lagrange's equation, 203, 264, 304 interpolation, 292, 297 Matrix (matrices) band, skyline of, 490 coefficient, 189 mass (see Coefficient) stiffness (see Coefficient) compatibility, 177-179, 276 condensation, 490, 541 constraint, 425 damping, 120, (see also Damping matrix) nonsymmetric, 117 symmetric, 117 diagonal, 81, 505-506 dynamic, equilibrium, 72, 99-100, 108 dynamic load, 179 dynamic, reduced, 72, 74, 76 zero, 77 dynamic system, 184-185, 193, 199 eigensolution (see Eigensolution) equation, 215, 222 equilibrium, 177, 178 transpose of, 179 flexibility, 81, 508 generalized force, 268, 270, 272, 312, 317 geometric, 303, 305-306, 312, 438 994 [Matrix (matrices)] half band, 490 identity, 78 inertial force (see Force, inertia, matrix) inversion, 81, 279 kinematic (see Matrices, compatibility) linear array, 490 load, 193, 201, 278 mass: consistent (see Mass consistent) lumped (see Model lumped) modal, 56, 64, 149 analysis, 281 nonsingular, 77 reduced dynamic, 74, 76 partition, 68 singular, 187 stiffness: consistent mass model, 266, 271 distributed mass model, 178 lumped mass model, 55, 435, 438 string stiffness (see String stiffness) sweeping, 74, 105 system, equation, 180, 186 transformation, 87 triangular, 81, 98 unsymmetric, 99 Maxwell's reciprocal theorem, 60 Mean recurrence interval, 683 Mean-value theorem, 336, 340 Modal combination method: ABSSUM, 514 CQC 67, 394, 404-405, 514, 679 deflection, 681 drift, 681 SRSS,67, 393, 394, 400-402, 514, 676, 678 Modal analysis, 676, 678 composite spectral, 412 Mode, buckling, 240 characteristic, 165 first, 75, 102, 144, 217 fundamental, 72, 101, 137 higher, 73, 101 dynamic stiffness vs lumped mass, 197 natural, 49, 50, 67 normal, 49, 50, 203 shape, in relation to torsional vs flexural vibration, 237 Model(s), bilinear, 528, 534, 560-561 consistent mass (see Mass, consistent; Matrix, mass) coordinate, 100 curvilinear, 528, 555, 560-561 distributed mass (see Mass, distributed) INDEX [Model(s), bilinear] elastic frame (see Elastic frame) elasto-plastic, 375, 532-534, 560 hysteresis, 528 lumped mass, 48, 68, 71 mathematical, 100 physical, 99 Ramberg-Osgood (see Ramberg-Osgood) rigid-body spring, 442 spring mass, 47 string stiffness (see P-A effect) Modified Mercalli Intensity Scale, 322 Moment, bending, 243, 271, 462 initial yield, 528 overturning, 611, 621, 639, 645, 651 primary, 622 restoring, 631 secondary, 622 second-order, 303 and shear equation, 250 ultimate, 532, 604 unbalanced, 226 Moment-curvature relationship, 238 axial load on, 589 Motion (displacement), elastic, 109, 111 equation 25, 49, 264 uncoupled, 58, 150, 206 harmonic, 6, 52, 199 foundation movement due to 185 periodic, 6, 52 rigid-body, 80, 105, 108-110 time-periodic, 11 Miiller-Breslau principle, 173 Multiple-component seismic input, 397 Multiple d.o.f system, 55-60, 117, 390, 393 NEHRP, 607, 705 Newmark elastic design spectra, 371-372 Newmark inelastic design spectra, 372-378 Newmark integration method, 329, 337, 346-347, 356-358 in relation to average acceleration and linear acceleration method, 329-330 in relation to general numerical integration, 334 in relation to numerical stability and error, 350 in relation to trapezoidal rule, 329 Newton's second law, 26 Nonlinear: analysis, scope of, 527 time-history analysis, 632 Nonlinearity, geometric, 579-581 Normalization (see Vector, normalized) 995 INDEX Numerical elimination technique, 507 Numerical error: amplitude decay, 358 periodic elongation, 358 Occupancy categories, 612 Ordinates, spectral, 371 Orthogonality, 55, 73, 101-102, 105-107, 137, 203-205, 392 proof for eigenvectors, 130 Out-of-plane d.o.f (see d.o.f.) stiffness (see Stiffness) Overdamping (see Damping, supercritical) Overshooting (see Yield limit) P-A effect, 100, 238, 252, 303, 314, 438, 579, 585-586, 589, 621, 632, 640, 646, 666 string stiffness model (see also String stiffness), 438 Participation factors (see Factor) Peak ground acceleration, 326-327 Period, elongation (see also Numerical error) damped, 10 forcing, 20 fundamental, equation, 123, 618, 637 return, 683 natural, 5, 20, 361 undamped, 10 Phase angle, 4, 37, 151, 156 Physical interpretation, 179, 193, 218, 220, 223 Pin-connected member, 216 (see also Coefficient, stiffness; Dynamic) Planar constraint, 426 Plane grid system (see also Grillage), 230, 233, 234 Plane strain, 285, 288, 298 Plane stress, 285, 288, 298 Plastic hinge formation, 532, 591 rotation(s), 591 Plastic moment, 591 reduced, 591 Points, infinite, 257 inflectional, 187 sampling, 294, 299 zero, 257 Poisson's ratio, 288 Polar moment of inertia, 269, 659, 672, 673 Polynomial equation, 19, 183 Position: [Pseudo acceleration] dynamic procedure (based on response spectra), 514 static displacement, 64 velocity, 36 Pulse, 20 after, 20 during, 20 Quadratic equation, 88 Radius of gyration, 99 Ratio, amplitude, 12, 40 damping, 38, 45, 368 frequencies, 39, 252 natural period to forcing period, 19 Poisson's (see Poisson's ratio) slenderness, 227, 257 effect on coupling vs pseudo-coupling frequencies in higher modes, 235-236 Ramberg-Osgood: hysteresis model, 528, 562, 576 moment-curvature, 563 stress-strain, 562 Rayleigh's dynamic reciprocal principle, 171, 173 equation, 619 Reciprocal theorem, 60 Reduced plastic moment (see Plastic moment) Redundancy/reliability factor, 628 Region of convergence, 187 of divergence, 187 (see also Inflectional points) Resonance, 16, 168 Response analysis: displacement (see Displacement, response) due to forces, initial distrubance, seismic excitation, 58 maximum (worst-case), 399-400 modal matrix analysis, (see Displacement, response) steady-state 15, 198 Richter Magnitude Scale, 322, 324 Rigid-body spring model (see Model) Rigid zone, 464 Rigid-zone transfer matrix, 465 Root-mean-square (see SSRS) Runge-Kutta fourth-order method, 338-346 in relation to numerical stability, 358, 360 equilibrium, neutral, Principles, d'Alembert's, 45 Hamilton's, 45 virtual displacement, 45 Pseudo acceleration, 36 displacement, 36 Scalar product (see Product) SEAOC, 676 Secant: modulus, 563 yielding stress, 562-563 Second-order moment, 238 996 Seismic: acceleration, 326 coefficient, 617, 624, 625 dead load, 616 design categories, 634, 635 force amplification, 614 intensity, 323 magnitude, 224 use group, 633 Seismic waves, types of, 322 Separation, element, 624 Separation of variables, 163 Shear: deformation, 314 maximum base, 394 modulus, 241, 269 stress, 269 strain, 241, 269 Side-sway, 180, 193 Sine function, Site coefficient, 617 Skeleton curve (see Stiffness formulation) Slip rate, 625 Slope, bending, 240-241, 308 shear, 240-241, 308 Soil: profiles, 380 stiffness, 71 Spectral distribution factor (see Factor) Spectrum (spectra): acceleration, 36 composite response, 410 composite torsional, 411, 412 spectral quantity (value), 412 composite translational, 412 spectral quantity (value), 412 design, 367 elastic, 375 inelastic, 375 (see Housner's average design spectra) (see Newmark elastic and inelastic design spectra) normalized, 369, 371 ordinates (see Ordinates, spectral) principal component, 362 intermediate, 369 main, 369 minor, 369 pseudo, 36, 365-368 response, 21, 36, 362-363, 366-367, 368-369 rock, 625 hard, 625 shock, 21, 35, 67 site dependent, 378 INDEX [Spectrum (spectra)] torsional response, 383, 389, 390 UBC design, 378 velocity response, 36, 64, 393 Spring constant, 237 SSRS (square-root-of-the-sum-of-the-squares method (or root-mean-square method)) (see Modal combination method) Stability criterion, 351 coefficient, 640 Stability, conditional, 357 unconditional, 357 Stability, numerical, 350 State vector, 129, 133, 135, 149 Static deflection, Steel, alloy, yielding stress, 562 structural, stress-strain relationship, 528 Step function method, 32 Stiffness formulation matrix (see also Physical interpretation; fundamental structural mechanics) branch curve, 572-575, 578 direct element, 583 geometric, 583 (see also Matrix, geometric) out-of-plane, 455 skeleton curve, 565-570, 575 unsymmetric, 72 String stiffness, geometric, 583 matrix, 306, 587 model (see P-A effect) Structure, flexible, 632 isolated, 633 nonisolated, 633 Structural joint/node, model, 417 node, 462 optimization, 640 redundancy, 614 Sturm sequence method, 95-98 Substitution, backward, forward, 82 Superposition technique, 28 Sweeping cycle, first, 90-92 second, 93-94 Systems, bearing-wall, building-frame, dual, 617 Tall buildings, 499 Taylor series, 335-341, 358 Time-history response (using numerical integration), 514, 515, 523 analysis, 515, 523, 678, 687 nonlinear, 632 Time-dependence, 131 Timoshenko beam, 240, 244, 309, 313 bending moment and shear, 243 compared to Bernoulli-Euler theory, 252 997 INDEX [Timoshenko beam] equation, 240, 252 fixed-end forces, 246 member's geometric matrix, 305 (see also Matrix, geometric) member's mass matrix, 272 (see also Mass consistent) shear coefficient, 241 tapered and prismatic, 317 Torsion, 621, 632, 654, 658 accidental, 621, 676 primary, 621 for uniform force, 200 Torsional response spectra (see Spectra) Torsional member: consistent mass method, 268-270 dynamic stiffness method, 229-230 Transformation technique, 465 Transformed section method, 444 Transmissibility, 38, 40 Triangular distributed wind forces, 511 Truss(es), 216, 220 dynamic stiffness matrix for, 218-221 vierendeel, 276 Two d.o.f system, 49, 105, 126, 135 Two-force member (see also Truss(es)), 265 Unbalanced force technique, 539 Uncoupled equation, 58, 63-64 Underdamping (see also Damping, subcritical), 10, 130 Vector, displacement, 55, 583 complex, normalization of, 132 force, 276, 581 generalized response, 58 [Vector, displacement] state (see State-vector) sum, Vibration, bending (flexural), 161, 224, 234, 240, 270 coupling, 224, 234 rigid-frame, behavior, 192 forced, 14, 60, 63-64 forced damped, 29 forced undamped, 14 free, 2, 14, 51, 162, 193 free damped, 14 free undamped, 14 longitudinal, 213, 224-227 pseudo-coupling, 234-236 steady-state, 15, 65, 180, 185, 198, 240, 276 steady-state forced damped, 30 torsional (see also Torsional member), 229, 234 transient, 15, 30 uncoupled (uncoupling), 224, 225 with elastic media, 214, 215 Virtual displacement (see Displacement, virtual) Virtual load concept, 581 Virtual work, 276 Wilson-6 method, 332-333, 346, 351, 356-358 in relation to amplitude decay, 358 in relation to general numerical integration, 334 in relation to linear acceleration method, 332-333 in relation to numerical stability and error, 350 in relation to period elongation, 358 Worst-case response analysis (see Response analysis, maximum) incremental joint, 579 Yield limit influence coefficient, 390 length, 56 real trial, 138-139 rotating, 131 backtracking (nonlinear state), 539-545 overshooting (linear state), 539-546 Yield, state(s) of, 536, 538-539, 552 Young's modulus, 562