Matrix Analysis of Structural Dynamics: Applications and Earthquake Engineering (Civil and Environmental Engineering)

1017 Pages·2000·31.39 MB·English

by  Franklin Y. Cheng

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)
1. Structural dynamics. 2. Earthquake engineering. 3. Matrices. I. Title. II. Series
TA654.C515 2000
624.1'7-dc21
00-031595
This book printed on acid-free paper.
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Civil and Environmental Engineering
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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 con-
struction 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. Represen-
tative 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.
1. 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, ordi-
III
iv PREFACE
nary and partial differential equations, vector analysis, and complex variables is reviewed
where necessary.
2. 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.
3. Each chapter discusses a specific topic. There is a progression in every chapter from fun-
damental 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 9. This approach may help the reader to follow the subject matter
and the instructor to select material for classroom presentation.
4. 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.
5. 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 con-
necting joints. These elements are not collectively covered in a structural dynamics text
or a static structural analysis text; this book can supplement the latter.
6. 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. Fundamen-
tal 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 1 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 2 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 accu-
racy 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 3 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 4 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 5 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 6 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 4 yields the lower bound of an eigensolution while the model
in Chapter 6 yields a solution between a lumped mass and a distributed mass model. Solutions are
thus compared.
Chapter 7 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 8 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 9 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 struc-
tural 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 ana-
lytical 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 Uni-
versity 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 enthusi-
asm 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
1 Characteristics of Free and Forced Vibrations of Elementary Systems 1
1.1 Introduction 1
1.2 Free Undamped Vibration 1
1.2.1 Motion Equation and Solution 1
1.2.2 Initial Conditions, Phase Angle and Natural Frequency 3
1.2.3 Periodic and Harmonic Motion 6
1.3 Free Damped Vibration 7
1.3.1 Motion Equation and Viscous Damping 7
1.3.2 Critical Damping, Overdamping and Underdamping 9
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