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ORIGINAL MONOGRAPH SERIES Engineering Monographs on Earthquake Criteria, Structural Design, and Strong Motion Records Coordinating Editor: Mihran S. Agbabian l\.1N0-1 Reading and Interpreting Strong Motion Accelerograms, by Donald E. Hudson, 1979 l\.1N0-2

Dynamics of Structures, A Primer, by Anil K. Chopra, 1982 (out of print)

l\.1N0-3

Earthquake Spectra and Design, by Nathan M. Newmark and William J. Hall, 1982

l\.1N0-4

Earthquake Design Criteria, by George W. Hausner and Paul C. Jennings, 1982

l\.1N0-5

Ground Motions and Soil Liquefaction During Earthquakes, by H. Bolton Seed and I. M. Idriss, 1983

l\.1N0-6

Seismic Design Codes and Procedures, by Glen V Berg, 1983 (out of print)

MN0-7

An Introduction to the Seismicity of the United States, by S. T. Algermissen, 1983 SECOND MONOGRAPH SERIES Engineering Monographs on Miscellaneous Earthquake Engineering Topics

l\.1N0-8

Seismic Design with Supplemental Energy Dissipation Devices, by Robert D. Hanson and Tsu T. Soong, 2001

l\.1N0-9

Fundamentals of Seismic Protection for Bridges, by Mark Yashinsky and M. J. Karshenas, 2003

l\.1N0-1 0

Seismic Hazard and Risk Analysis, by Robin K. McGuire, 2004

l\.1N0-11

SECOND EDITION Earthquake Dynamics of Structures, A Primer, by Ani\ K. Chopra, 2005 (first edition: l\.1N0-2)


EARTHQUAKE DYNAMICS OF STRUCTURES A Primer

by ANIL K. CHOPRA University of California, Berkeley

Second Edition

This monograph was sponsored by the Earthquake Engineering Research Institute with support from FEMAIU.S. Department of Homeland Security

EARTHQUAKE ENGINEERING RESEARCH INSTITUTE MN0-11


Š2005 Earthquake Engineering Research Institute, Oakland, CA 94612-1934. All rights reserved. No part of this book may be reproduced in any form or by any means without the prior written permission of the publisher, Earthquake Engineering Research Institute (EERI) . The publication of this book was supported by FEMA/U.S. Department of Homeland Security under grant #EMW-2004-CA0297. EERI is a nonprofit corporation. The objective ofEERI is toreduce earthquake risk by advancing the science and practice of earthquake engineering; by improving the understanding of the impact of earthquakes on the physical, social, economic, political, and cultural environment; and by advocating comprehensive and realistic measures for reducing the harmful effects of earthquakes. Any opinions, findings , conclusions, or recommendations expressed herein are the author's and do not necessarily reflect the views of FEMA or EERI. Copies of this publication may be ordered from EERI, 499 14th Street, Suite 320, Oakland, California 94612-1934; tel: (510) 451-0905; fax: (510) 451-5411 ; e-mail: eeri@eeri.org; web site: http ://www. eeri.org. This is the second edition of the monograph originally entitled Dynamics ofStructures, A Primer, by Ani! K. Chopra, first published in 1982 by EERI (Publication No. MN0-2) . Printed in the United States of America. ISBN #1 -932884-07-6 EERI Publication No. MN0-11 Technical Editor: Barbara Zeiders Production Coordinator: Eloise Gilland Layout and Production: Interactive Composition Corporation


CONTENTS Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 1: Linearly Elastic Systems 1 Elastic Single-Degree-of-Freedom Systems . . . . . . . . 1.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . 1.2 System Parameters . . . . . . . . . . . . . . . . . . . . 1.3 Earthquake Ground Motion . . . . . . . . . . . . . . 1.4 Response History . . . . . . . . . . . . . . . . . . . . . 1.5 Response Spectrum . . . . . . . . . . . . . . . . . . . 1.5.1 The Concept . . . . . . . . . . . . . . . . . . . . 1.5.2 Deformation Response Spectrum . . . . . . 1.5.3 Pseudo-Velocity Response Spectrum . . . . 1.5.4 Pseudo-Acceleration Response Spectrum . 1.5.5 Combined D- V-A Spectrum . . . . . . . . . . 1.5.6 Response Spectrum . . . . . . . . . . . . . . . 1.6 Spectral Regions . . . . . . . . . . . . . . . . . . . . . 1. 7 Elastic Design Spectrum . . . . . . . . . . . . . . . . 1.8 Peak Structural Response from Spectrum . . . . . 2 Elastic Multi-Degree-of-Freedom Systems ..... . .. 2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . 2.2 Modal Analysis: Basic Concept . . . . . . . . . . . 2.2.1 Modal Expansion of Effective Forces . . . . 2.3 Modal Response History Analysis . . . . . . . . . . 2.3.1 Modal Responses. . . . . . . . . . . . . . . . . 2.3.2 Total Response . . . . . . . . . . . . . . . . . . 2.3.3 Interpretation of Modal Analysis . . . . . . 2.3.4 Modal Static Response . . . . . . . . . . . . . 2.3.5 Effective Modal Mass and Modal Height . 2.3.6 Example. . . . . . . . . . . . . . . . . . . . . . .

v

.... .... . . . . . . . . . . . . . . . . . . . . . . . . . .

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

1x x1

3 3 4 6 8 9 9 10 11 12 14 15 15 19 24 27 28 29 29 30 30 31 32 32 33 34


2.4

Modal Response Spectrum Analysis . . . . . . . . . . . . 38 2.4.1 PeakModalResponses .. . ....... . . .... . 38 2.4.2 Modal Combination Rules . . . . . . . . . . . . . . . 39 2.4.3 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.4 Alternative Interpretations of Response Spectrum Analysis . . . . . . . . . . . . . . . . . . . . 42 2.5

2.6

Higher-Mode Contributions to Earthquake Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5.1 Modal Contribution Factors . . . . . . . . . . . . . 43 2.5.2 Structural Properties . . . . . . . . . . . . . . . . . . 44 2.5.3 Influence ofT, on Higher-Mode Response. . . . 45 2.5.4 Influence of p on Higher-Mode Response . . . . 47 How Many Modes to Include ........ . . . . . .... 47

Part II: Inelastic Systems 3 Inelastic Single-Degree-of-Freedom Systems . . . . . . . . . . 53 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Idealized Inelastic Systems . . . . . . . . . . . . . . Yield-Strength Reduction Factor and Ductility Factor . . . . . . . . . . . . . . . . . . . . . . Equation of Motion and Controlling Parameters Peak Deformation and Ductility Demand . . . . . Response Spectrum for Yield Deformation and Yield Strength . . . . . . . . . . . . . . . . . . . . . . . Ry -Jl- - Tn Relations .. .. . .. . ... .. .. .. . . Constant-Ductility Design Spectrum . . . . . . . . Applications ofthe Inelastic Design Spectrum .

. . . . 53 . . . . 56 . . . . 57 . . . . 57 .. .. .. ..

.. .. .. ..

3.8.1 Structural Design for Allowable Ductility . . . . . 3.8.2 Deformation of an Existing Structure . . . . . . . 3.9 Deformation by Graphical Method . . . . . . . . . . . . . 3.10 Inelastic Deformation Ratio . . . . . . . . . . . . . . . . . .

60 61 63 65 65 67 68 69

3.10.1 Estimating the Deformation of Inelastic Systems . . . . . . . . . . . . . . . . . . . . 73 4 Inelastic Multistory Buildings. . . . . . . . . . . . . . . . . . . . . 75 4.1 4.2 4.3 4.4 4.5 4.6

Nonlinear Response History Analysis . . . . . . Why Modal Analysis Is Not Applicable . . . . . Strength Demands for SDF and MDF Systems Influence oflnelastic Behavior on Story Drifts P-1'3. Effects . . . . . . . . . . . . . . . . . . . . . . . Influence ofModeling Assumptions on Seismic Demands . . . . . . . . . . . . . . . . . . . .

vi

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

75 77 77 80 82

. . . . . 83


4.7 Statistical Variation in Seismic Demands . . . . . . . . . 4.8 Story Drift Demands. . . . . . . . . . . . . . . . . . . . . . . Part III: Building Design Codes and Evaluation Guidelines 5 Structural Dynamics in Building Design Codes . . . . . . . . . 5.1 International Building Code . . . . . . . . . . . . . . . . . . 5.1.1 Base Shear . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Lateral Forces . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Story Forces . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Relation between IBC and Dynamic Analysis . . . . . . 5.2.1 Base Shear . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Design Force Reduction . . . . . . . . . . . . . . . . 5.2.3 Lateral Force Distribution . . .. . .. . .. . . .. . 6 Structural Dynamics in Building Evaluation Guidelines .. 6.1 Nonlinear Response History Analysis ........... 6.2 Current Practice .......................... 6.3 SDF-System Estimate of Roof Displacement ...... 6.4 Estimating Deformation of Inelastic SDF Systems ... 6.4.1 ATC-40 Method ...................... 6.4.2 FEMA-356 Method .... . .. . .. . . .. . .. .. 6.5 FEMA-356 Nonlinear Static Procedure .. .. .. . .. . 6.6 Improved Nonlinear Static Procedures ... ... . .. . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation ........... .... . ... ....... . . .. .. . ... .

vii

85 87 91 91 92 93 94 94 95 98 101 105 105 108 108 110 111

111 112 115 119 125


FOREWORD The original seven EERI monographs were published between 1979 and 1983 and grew out of a seminar on earthquake engineering organized by EERI and presented in several cities. The monographs covered the basic aspects of earthquake engineering in some detail, including seismicity, strong-motion records, earthquake spectra, liquefaction, dynamics, design criteria, and codes. The themes were fundamental and focused, and the content was thorough and generally noncontroversial. These monographs filled a gap in available documents and were highly acclaimed. This is the first monograph to be published that is intended to update a subject area covered in the original series. The first monograph covering linear structural dynamics, Dynamics ofStructures, A Primer, authored by Anil K. Chopra, has become a classic text used around the world. However, the author recommended that EERI not reprint that document without an update to acknowledge the increased use of dynamic analysis in building codes and in structural design as well as the current interest in nonlinear analysis. The update has become a major rewriting of the original material with significant additions covering various aspects of nonlinear analysis. Since techniques to perform simplified nonlinear analysis are evolving, the premise that monographs are basic and conceptual and free of significant material that might become dated created difficult review issues. The reviewers, Robert D. Hanson, Helmut Krawinkler, and Farzad Naeim, and the author are to be congratulated for working through several drafts to achieve an appropriate balance in this regard. EERI intends to continue to capture the essence of the original series by publishing, from time to time, updates or additional monographs covering a wide variety of subjects. The new monographs will still be expositions of narrowly focused aspects of earthquake engineering prepared by especially qualified experts. However, they may provide background and insight on a topic that will be of ix


particular value to readers from different specialties or disciplines than the authors. Depending on the technical characteristics of the topic, the intended audience may include design professionals, researchers, social scientists, or policy makers. The new monographs are not intended to be design guides or detailed and highly technical state-of-the-art papers. The contents will therefore tend to be basic and conceptual, and in general they will not include material that may quickly become outdated. Rather than creating a prioritized list of subjects and rigid publishing schedules, EERI approves monograph subjects individually as the opportunities arise. WILLLAM

T. HoLMES

CH A IR, MONOGRAPH COMMITIEE

July 2005

X


PREFACE Based on seminars given in the late 1970s, the EERI monograph Dynamics ofStructures, A Primer (1980), was written for generations of professional engineers who had not had the opportunity to study structural dynamics formally as students. It was written at a time when building design was based on elastic analysis for lateral forces that included a factor to recognize inelastic effects indirectly. It provided the nonspecialist in structural dynamics with the basic concepts and knowledge needed to understand the response of structures excited by ground shaking. But for three pages, this 126-page monograph was limited to elastic dynamic analysis of buildings. The structural engineering profession and practice, especially in California, has changed greatly since the first monograph was published, in at least two major ways. First, most current practicing engineers have taken a structural dynamics course in college. Second, performance-based guidelines for evaluating existing buildingssuch as FEMA-273; its successor, FEMA-356; and ATC-40-consider inelastic behavior explicitly in estimating seismic demands at low performance levels, such as life safety and collapse prevention. Intended to be responsive to the current needs of the profession, this new monograph has two objectives: 1. To provide the nonspecialist in dynamics of structures with the basic concepts and knowledge needed to understand the response of structures to earthquake excitation. 2. To present structural dynamics concepts and analysis procedures in elastic and inelastic response of structures that in one form or the other are utilized in design codes and seismic evaluations guidelines. This monograph is organized in three parts: I. Linearly Elastic Systems; II. Inelastic Systems; and III. Building Design Codes and Evaluation Guidelines. Part I presents a modern treatment of the dynamics of elastic systems, including many of the topics in the original monograph. Chapter 3 of Part II is devoted to the dynamics xi


of inelastic single-degree-of-freedom systems, response spectra for inelastic systems, inelastic design spectrum and its applications in the design of new structures, and safety evaluation of existing structures. Chapter 4 is an introductory presentation of the vast subject of inelastic analysis and response of multistory buildings. Chapter 5 of Part III presents the lateral forces specified in the 2003 International Building Code, together with their relationship to the structural dynamics concepts in Chapters 1 to 4; pertinent comments on three other building codes are included. In Chapter 6, selected aspects of computing seismic demands according to FEMA and ATC guidelines for evaluating existing buildings are discussed in light of structural dynamics theory presented in Chapters 1 to 4. The list of references, although extensive, is not exhaustive; only those references that are most directly related are included. I have included comments with references to add historical perspective to some of the seminal concepts that have become an integral part of textbooks and engineering practice. They are so well integrated into earthquake engineering that many researchers and engineers using them may not be aware of their origins. I am grateful to several people whose time and insight were instrumental in completing this monograph: Rakesh K. Goel and Chatpan Chintanapakdee developed and executed the computer software necessary to generate numerical results and create figures. Akshay Gupta and Aladdin Nassar provided numerical data from their published research for several figures in Chapter 4, which were prepared by Gabriel Hurtado. Many figures are taken from Dynamics of Structures: Th eory and Applications to Earthquake Engineering by Anil K. Chopra Š 2001 and are reprinted by permission of Pearson Education, Inc., Upper Saddle River, New Jersey. Charles D. James, of the Earthquake Engineering Research Center at the University of California, Berkeley, helped in selecting and collecting the photographs. Technical review of this monograph was provided by Robert D. Hanson, Helmut Krawinkler, and Farzad A. Naeim . Claire Johnson prepared the text and assembled the manuscript, which was copyedited by Barbara Zeiders. This monograph was prepared during a year of appointment as Miller Research Professor at the University of California, Berkeley, an award for which I am grateful to the Miller Institute for Basic Research in Science. A NIL

xii

K.

CHOPRA


PART 1: LINEARLY ELASTIC SYSTEMS


1 ELASTIC SINGLE-DEGREE-OF-FREEDOM SYSTEMS

We begin our study of structural dynamics with simple structures, such as the pergola shown in Fig. 1.1 and the elevated water tank shown in Fig. 1.2. Such simple structures can be idealized as a concentrated or lumped mass supported by a massless structure that provides stiffness. In addition, damping must be included to account for the energy dissipated during vibration of the structure.

1.1 Equation of Motion The system being considered (Fig. 1.3) consists of a mass m concentrated at the top, a massless frame that provides lateral stiffness k to the system, and a viscous damper (also known as a dashpot) defined by a damping coefficient c that dissipates the vibrational energy of the system. We are interested in studying the dynamics of this system subjected to earthquake-induced motion ug(t) at the base of the structure. This chapter is restricted to linearly elastic systems. The equation governing u, the structural deformation or displacement of the mass relative to the ground, is mit+ cit+ ku

=

-milg(t)

(1.1)

Note that the ground acceleration ilg(t) appears on the right side of this differential equation. Because the motion of the system can be described by a single displacement, it is called a single-degree-offreedom (SDF) system. The mass and stiffness can be calculated from the dimensions of the structure and the sizes of the structural elements. However, damping cannot be calculated from these structural properties. Free vibration tests or harmonic vibration generator experiments on actual structures provide data for determining the damping ratio

3


Figure 1.1. This pergola at the Macuto-Sheraton Hotel near Caracas, Venezuela, was damaged by the earthquake of July 29, 1967. The magnitude 6.5 event, which was centered about 15 miles from the hotel, overstrained the steel pipe columns (courtesy ofNational Information Service for Earthquake Engineering, University of California, Berkeley).

at small amplitudes of motion; system identification methods applied to structural motions recorded during earthquakes lead to damping ratio values at higher amplitudes of motion. Dividing Eq. ( 1.1) by m gives the equation of motion in terms of two-system parameters: (1.2) where c

l; = 2mwn

(1.3)

1.2 System Parameters The two system parameters, Wn and l;, that appear in Eq. (1.2) are known as the natural circular frequency of vibration (in units of radians per second) and the damping ratio, a dimensionless measure of damping. Related to Wn is the natural period of vibration of the 4


Figure 1.2. This reinforced concrete tank on a 40-ft-tall single concrete column, located near the Valdivia Airport, was undamaged by the Chilean earthquakes of May 1960. When the tank is full of water, the structure can be idealized as a lumped single-mass system (from K. V Steinbrugge Collection, courtesy ofNational Information Service for Earthquake Engineering, University of California, Berkeley).

5


k

Figure 1.3. Single-degree-offreedom system.

system, defined as the time required for the undamped system to complete one cycle of motion when vibrating freely without external excitation; denoted as Tn (units of seconds), it is related to Wn as follows: (1.4) A system executes 1I Tn cycles in 1 sec of free vibration. This natu1I Tn; its units ral cyclic frequency of vibration is denoted by fn are hertz (Hz; or cycles per second). The term natural frequency of vibration applies to both Wn and fn. The natural vibration properties Wn, Tn, and fn depend only on the mass and stiffness of the structure. The qualifier natural is used in defining Wn, Tn, and fn .to emphasize the fact that these are natural properties of the system when it is vibrating freely without external excitation. These properties are valid for a system vibrating within its linear range of behavior. Given the natural vibration period Tn and damping ratio of the system, we are interested in finding its deformation response u(t) to a specified ground acceleration ug(t).

=

s

1.3 Earthquake Ground Motion For engineering purposes, the time variation of ground acceleration is the most useful way of defining the shaking of the ground during an earthquake; it appears on the right side of differential equation (1.2). Instruments known as strong-motion accelerographs record three components of ground acceleration: two orthogonal components of horizontal motion and the vertical component. Figure 1.4 shows the north-south component of the ground motion recorded at a site in El Centro, California, during the Imperial Valley earthquake of 6


0

5

10

15

20

25

30

Time, sec

Figure 1.4. North-south component of horizontal ground acceleration recorded at the Imperial Valley Irrigation District substation, El Centro, California, during the Imperial Valley earthquake of May 18, 1940. The ground velocity and ground displacement were computed by integrating the ground acceleration.

May 18, 1940. It is apparent that ground acceleration varies with time in a highly irregular manner. No matter how irregular, the ground motion is presumed to be known and independent of the structural response, implying no soil- structure interaction. The earthquake accelerogram is digitized and processed, and the processed record is then defined by numerical values at discrete time instants for dynamic response analysis [ 1,2]. These time instants should be closely spaced to describe accurately the highly irregular variation of acceleration with time. Typically, the time interval is chosen to be 1I 100 to 1I 50 of a second, requiring 1500 to 3000 ordinates to describe the ground motion of Fig. 1.4. In addition to the recorded ground acceleration, Fig. 1.4 shows the ground velocity and ground displacement computed by successive integration of the acceleration-time function; the peak values of all three are identified. Accurate determination of the ground velocity and displacement is difficult because (1) the very long-period 7


components of motion are usually filtered out during record processing; and (2) analog accelerographs do not record the initial part- until the accelerograph is triggered- of the acceleration-time function, and thus the base (zero acceleration) line is unknown. Digital accelerographs overcome the second problem by providing a short memory so that the onset of motion is measured.

1.4 Response History Ground acceleration during earthquakes varies irregularly to such an extent (see Fig. 1.4) that analytical solution of the equation of motion must be ruled out. Therefore, numerical methods for solving Eq. (1.2) are implemented on a computer to determine the structural response. For a given ground acceleration iig(t), the deformation response u(t) of the system depends only on the natural vibration period Tn of the system and the damping ratio £;. Figure 1.5a shows the deformation response of three different systems to ground acceleration at

(a)

Tn = 0.5 sec,

(b)

s= 0.02

Tn = 2 sec,

9.9 1 in.

~

lA II

1001 .......... ••• " fiPf.UUh-hwnn• w• 1

s= 0

AAfi~AfiAAfi8Afil

vuvv~~~~~~vvvv

2.67 in.

- 10

Tn = 2 sec,

s= 0.02

lA 1\ A8AAA8118hOA \fCJQ \TV I[V \fV V\1 vvv A

, - 10

Af\AAAAAAA!\flfi(\A

1 0

I

s

T" = 2 sec, = 0.05

T,. ~ , "''·, ~ 0.02

0

7.47 in .

v Qvvvvvvvvvvvv

s.m". t.f\AAI\AI\of\1\1\1\_-

V\Jl) VlJlJ

v \J\T<Tvv "'"

7.47 in.

10

20

30

Time, sec

0

10

20 Time, sec

Figure 1.5 . Deformation response of SDF systems to El Centro ground motion [3].

8

30


- --

- - Vb(t)

~Mb(t)

Figure 1.6. Equivalent static force .

El Centro. The damping ratio t = 2% is the same for the three systems, so that only the differences in their natural periods are responsible for the large differences in deformation response. Figure 1.5b shows the deformation response of three systems with the same vibration period Tn but different damping. We observe the trend expected: that systems with more damping respond less strongly than do lightly damped systems. Once the deformation response history u (t) has been determined by dynamic analysis of the structure [i .e., solving the equation of motion (1.2)], the internal forces and stresses at each time instant can be determined by static analysis of the structure subjected to the equivalent static force (Fig. 1.6), defined by fs(t)

= ku(t)

= mw~u(t) = mA(t)

(1.5)

where A (t)

= w~u (t)

(1.6)

In particular, the base shear Vb(t) and base-overturning moment Mb(t) are Vb(t)

= mA(t)

(1.7)

Observe that the equivalent static force ism times A(t) , the pseudoacceleration, not m times the total acceleration u1 (t); see Fig. 1.3 for a definition of total displacement, u 1 .

1.5 Response Spectrum 1.5.1 The Concept The complete history r(t) of any response quantity can be determined by the numerical procedures mentioned above. However, for the design of new structures or the evaluation of existing structures, 9


it is generally sufficient to know only the peak value of response, or for brevity, peak response, defined as the maximum of the absolute value of the response quantity: r 0 =max lr(t)l t

(1.8)

By definition, the peak response is positive; the algebraic sign is dropped because it is usually irrelevant for design. A plot of the peak value of a response quantity as a function of the natural vibration period 1;1 of the system, or a related parameter such as circular frequency W 11 or cyclic frequency j 11 , is called the response spectrum for that quantity. Each such plot is for SDF systems having a fixed damping ratio t;, and several such plots for different values of t; are included to cover the range of damping values encountered in actual structures. Hausner was instrumental in the widespread acceptance of the concept of the earthquake response spectrum [4] introduced in 1933 by Biot [5]. The response spectrum is now a central concept in earthquake engineering. In this section, the deformation response spectrum and two related spectra, the pseudo-velocity and pseudo-acceleration response spectra, are defined. As shown in the preceding section, only the deformation u(t) is needed to compute the internal forces at any time instant t. Obviously, then, the deformation response spectrum will provide all the information necessary to compute peak values of deformation and internal forces. The pseudo-velocity and pseudoacceleration response spectra are included, however, because they are useful in studying the characteristics of response spectra, constructing design spectra, and relating response spectra to the base shear equation in building codes. 1.5.2 Deformation Response Spectrum This is a plot of the peak deformation D = u 0 as a function of the natural vibration period T;1 of the system. Figure 1. 7 shows the procedure to compute this spectrum: Part (b) of this figure shows the time variation of the deformation of three SDF systems- with different periods but the same damping ratio- due to the ground motion shown in part (a). For each system, the peak value of deformation D = U 0 is determined from its response history. The D value so determined for each system provides one point on the deformation response spectrum; the three values of D are identified in Fig. 1.7c. 10


(a)

0.4

~0~

-0.4r~~-------~ 0

10

20

30

Time, sec (b)

T,.

(c)

20,--

~ t;0.5~ 2o/i se~.

- -- - - ,

15

T,.

-~

~ I sec1

5

"'

1; ~ 2o/i

T,. ~ 2

secT

1; ~ 2o/i

II C)

10

"I v IJvvvvvvvvvvvv Af\AAAfif\fl(\(\(lfiAA

0

- 10

7.47 in.

0

10

20

30

2

Time, sec

Figure 1.7. (a) Ground acceleration; (b) deformation response of three SDF systems with~ = 2% and T,, = 0.5, 1, and 2 sec; (c) deformation response spectrum for~ = 2% [3].

Repeating such computations for a range of values of Tn while keeping l; constant provides the deformation response spectrum shown in Fig. 1. 7 c. As we show later, the complete response spectrum includes such spectrum curves for several values of the damping ratio.

1.5.3 Pseudo-Velocity Response Spectrum Consider a quantity V for an SDF system with natural frequency w 11 related to its peak deformation D by the equation 2rr

V=w 11 D= - D Tn

(1.9)

The quantity V has units of velocity. Associated with this velocity, the kinetic energy of the structural mass m is equal to the peak value of strain energy, Es 0 , stored in the system during the earthquake: E so

mV 2 2

=- 11

(1.1 0)


The pseudo-velocity response spectrum is a plot of V 1 as a function ofthe natural vibration period T11 , or natural vibration frequency j;,, of the system. For the ground motion of Fig. 1.7a, V for a system with period T11 can be calculated from Eq. (1.9) and the peak deformation D available from Fig. 1.7c, which has been reproduced in Fig. 1.8a. The resulting values of V are plotted in Fig. 1.8b as a function of T11 for a fixed value of the damping ratio, to obtain the pseudo-velocity response spectrum. 2

1.5.4 Pseudo-Acceleration Response Spectrum Consider a quantity A for an SDF system with natural vibration frequency w11 related to its peak deformation D by

A= w~D =

(-2:rr) Tn

2

D

(1.11)

The quantity A has units of acceleration and is related to the peak value of base shear Vbo (or the peak value of the equivalent static force !so [Eqs. (1.5) and (1.7a)]): A Vbo =!so = mA = - w

g

(1.12)

where w is the weight of the system and g the gravitational acceleration. Observe that the peak base shear is equal to the inertia force associated with the mass m undergoing acceleration A, not the true acceleration ii~ . When the base shear is written in the latter form in Eq. (1.12), A/g may be interpreted as the base shear coefficient or lateral force coefficient. It is used in building codes to represent the coefficient by which the structural weight is multiplied to obtain the base shear. The pseudo-acceleration response spectrum is a plot of A as a function of the natural vibration period Tn , or natural vibration frequency fn, of the system. For the ground motion of Fig. 1. 7a, A for a system with period Tn can be calculated from Eq. (1.11) and the peak deformation, D , available from Fig. 1.7c, which has been 1

2

The notation used here is compared with that in Hudson 's monograph [1] ; the latter has been used in some building codes ang evaluation guidelines: V , D , and A here correspond to PSV, SD, and PSA in [1]. The prefix pseudo is used to distinguish this spectrum from the velocity response spectrum, a plot of peak relative velocity u0 as a function ofT,, (see also Section 1.8).

12


20 15

.5

c::l

10

(a)

5 0 50 u

<!)

~

.5 ::,._"

(b)

1.5.-----=~~--------------------------~

c::

(c)

2

3

Figure 1.8. Response spectra (I; = 2%) for El Centro ground motion: (a) deformation response spectrum; (b) pseudo-velocity response spectrum; and (c) pseudo-acceleration response spectrum [3].

reproduced in Fig. 1.8a. The resulting values of A are plotted in Fig. 1.8c as a function of Tn for a fixed value of damping ratio to obtain the pseudo-acceleration response spectrum. 3

3

The prefix pseudo is used to distinguish this spectrum from the acceleration response spectrum, a plot of peak total acceleration ii~ as a function of Tn (see also Section 1.8).

13


1.5.5 Combined D- V- A Spectrum Each of the deformation, pseudo-velocity, and pseudoacceleration response spectra for a given ground motion contains the same information, no more and no less. The three spectra are simply different ways of presenting the same information on structural response. Knowing one of the spectra, the other two can be obtained from Eqs. (1.9) and (1.11). The simple relations among the three- deformation, pseudovelocity, and pseudo-acceleration- response spectra make it possible to combine them into a single plot on four-way logarithmic graph paper from which all three spectral quantities- D , V , and A- can be read. The vertical and horizontal scales for V and Tn are standard logarithmic scales. The two scales for D and A sloping at +45째 and -45째 to the Tn axis are also logarithmic scales but not identical to the vertical scale. The V - Tn data in the linear plot of Fig. 1.8b are replotted in Fig. 1.9 on logarithmic scales. For a given natural period Tn, the D and A values can be read from the diagonal scales. The four-way plot is a compact presentation of the three response spectra,

100 50 1;, = 0.02

/o.

1?

20 <) <l)

10

~

.e

::,._"

5 2

T,,,

sec

Figure 1.9. Combined D- V- A response spectrum for El Centro ground motion; ~ = 2% [3].

14


100

so u

~

20

.5

~

c

10

路c; 0

"6 >

5

"0

"

"' ~

2

0.05

0.1

0.2

0.5

2

5

10

20

50

Natural vibration period Tn , sec

Figure 1.10. Combined D-V-A response spectrum for El Centro ground motion; I; = 0, 2, 5, 10, and 20% [3].

for a single plot of this form replaces the three linear plots ofFig. 1.8. Earthquake response spectra were plotted in this form for the first time in 1960, by Veletsos and Newmark [6]. 1.5.6 Response Spectrum A response spectrum should cover a wide range of natural vibration periods and several damping values so that it provides the peak response of all possible structures. Figure 1.10 shows spectrum curves fort = 0, 2, 5, 10, and 20% over the period range 0.02 to 50 sec. This, then, is the response spectrum for the north-south component of ground motion recorded at one location during the Imperial Valley earthquake of May 18, 1940. The corresponding pseudo-acceleration response spectrum is plotted in Fig. 1.11 and the deformation response spectrum in Fig. 1.12.

1.6 Spectral Regions Figure 1.13 shows one of the spectrum curves of Fig. 1.1 0, the one for 5% damping, together with an idealized version shown by 15


3

"'<: II

;;:

2

1.5

0.5

2

2.5

3

T11 , sec

Figure 1.11. Normalized pseudo-acceleration, or base shear coefficient, response spectrum for El Centro ground motion; t; = 0, 2, 5, 10, and 20% [3]. 50 20 10 5 2

.~ Q.

0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.02

0.05

0.1

0.2

0.5

2

10

20

50

T11 , sec

Figure 1.12. Deformation response spectrum for El Centro ground motion; t; = 0, 2, 5, 10, and 20% [3].


Spectral Region s Acceleration Velocity Displacement 1 sensitive I sensitive sensiti ve

5

T, 1 , sec

Figure 1.13. Response spectrum for El Centro ground motion shown by a solid line together with an idealized version shown by a lighter dashed line, plotted with normalized scales A /ug 0 , V /ug 0 , and D /iig 0 ; t; = 5%. Spectral regions are defined [3].

dashed lines; both are plotted using normalized scales. The A, V, and D axes have been normalized to Ajiig0 , V jitg 0 , and D /ug 0 , where iig 0 , Ug 0 , and Uga are the peak ground acceleration, velocity, and displacement, respectively; they are identified in Fig. 1.4 for the El Centro ground motion. Figure 1.13 will help identify important characteristics of earthquake response spectra and define spectral regions. For systems with very short periods, say Tn < Ta , the peak pseudo-acceleration A approaches ii go; this is also apparent in Fig. 1.11. For systems with very long periods, say Tn > TJ, the peak deformation D approaches Uga; this can also be observed in Fig. 1.12. For short-period systems with Tn between Ta and Tc, A exceeds iig0 , with the amplification depending on T,, and l;. Over a portion of the range, Tb to Tc, A may be idealized as a constant equal to iigo amplified by a factor depending 17


on£;. For long-period systems with Tn between Td and Tj , D generally exceeds U g 0 , with the amplification factor depending on Tn and £;. Over a portion of this period range, Td to Te, D may be idealized as constant at a value equal to U go amplified by a factor depending on £;. For intermediate-period systems with Tn between Tc and Td, V may be idealized as constant at a value equal to itgo amplified by a factor depending on£; . The idealized spectrum in Fig. 1.13, which is a series of straight lines a- b-c-d- e-fin the four-way logarithmic plot, is not necessarily a close approximation to the actual spectrum of an individual ground motion. This may not be apparent visually but becomes obvious when we note that the scales are logarithmic. As we shall see in the next section, the greatest benefit of the idealized spectrum is in constructing a design spectrum representative of many ground motions. Based on these observations, it is logical to divide the spectrum intothreeperiodranges(Fig. 1.13). Thelong-periodregion, Tn > Td , is called the displacement-sensitive region of the spectrum because structural response is related most directly to ground displacement. The short-period region, Tn < Tc, is called the acceleration-sensitive region because structural response is related most directly to ground acceleration. The intermediate-period region, Tc < Tn < Td , is called the velocity-sensitive region because structural response appears better related to ground velocity than any other parameter. The concept of spectral regions was introduced in 1969 by Veletsos [7]. The spectral regions and the periods Ta, Tb , Tc , Td , Te, and T1 separating them depend on the time variation of ground motion: in particular, the relative values of peak ground acceleration, velocity, and displacement, as indicated by their ratios, itg0 / ug0 , and Ug0 / itgo· These ground motion characteristics depend on the earthquake magnitude, fault-to-site distance, source-to-site geology, and the soil conditions at the site. Near-fault ground motions often contain large long-period pulses. Figure 1.14 shows the fault-normal component of ground motion recorded at a station located toward the direction of fault rupture. Evident is a long-period pulse in the acceleration history that appears as a coherent pulse in the velocity and displacement histories. Such a pronounced pulse does not exist in ground motions recorded at locations away from the near-fault region. The ratios itg0 / ugo and Ug0 / ug0 are very different between the two types of ground motions. As a result, the response spectra for near- and far-fault motions are

18


I0~

1 00

200

]

.,'"

-"=

174.78

1\ (\

AO

A

nO

VV\1\('"'~

__l"v....

=~=

0-

0

~ 100 ~200

50

39.81

~50+---------------.----------------.--------------~

0

10

15

Time, sec

Figure 1.14. Fault-normal component of horizontal ground acceleration recorded at the Rinaldi Receiving Station, California, during the Northridge earthquake of January 17, 1994. The ground velocity and ground displacement were computed by integrating the ground acceleration.

very different in shape. Shown in Fig. 1.15 are the idealized versions of response spectra for one far-fault motion and the fault-normal component of three near-fault motions from earthquakes of similar magnitude. Comparing them indicates that the velocity-sensitive region is much narrower for near-fault motions, and their accelerationand displacement-sensitive regions are much wider.

1. 7 Elastic Design Spectrum A design spectrum differs conceptually from a response spectrum in two important ways. First, the jagged response spectrum is a plot of the peak response of all possible SDF systems and hence is a description of the particular ground motion. The smooth design spectrum, however, is a specification of the level of seismic design force, or deformation, as a function of natural vibration period and damping ratio. Second, the design spectrum recognizes that ground

19


NR94rrs

100 :\-\:I

r/]

E (.)

~

.c

¡u 0

v>

10

I

0

"0

;::l

<!) r/]

P-.

0.02

0.1

10

50

Natural vibration period T11 , sec Figure 1.15. Idealized response spectra for fault-normal component of three near-fault ground motions-and of the 1952 Taft record ; ~ = 5% [8]. motions at a site may originate from different earthquake faults , each characterized by the largest-magnitude earthquake that it can generate and its distance from the site. Detailed discussion of the construction of design spectra is beyond the scope of this monograph. The presentation here emphasizes structural dynamics concepts underlying design spectra and focuses on the narrow question of how to develop a design spectrum that is representative of an available ensemble (or set) of recorded ground motions. The design spectrum is based on statistical analysis of response spectra for the ensemble of ground motions. The response spectrum for each ground motion is computed, leading to as many spectral values at each period T11 as the number of ground motions in the ensemble. Statistical analyses of these data provide the probability distribution for the spectral ordinate, its mean value, and its standard derivation at each period ~1 â&#x20AC;˘ The probability distributions are shown schematically in Fig. 1.16 at three selected Tn values. Connecting

20


0.1

Figure 1.16. Mean and mean +I a spectra with probability distributions for Vat Tn = 0.25 , 1, and 4 sec; I; = 5%. Dashed lines show an idealized design spectrum [3]. (Based on numerical data from [9].)

all the mean values gives the mean response spectrum. Similarly, connecting all the mean-plus-one-standard-deviation values gives the mean-plus-one-standard-deviation response spectrum. The quantities u g 0 , it g 0 , and Ugo in the normalized scales ofFig. 1.16 are the mean values of the peak ground displacement, velocity, and accelerationaveraged over the ensemble of ground motions. Observe that these two response spectra are much smoother than the response spectrum for an individual ground motion (Fig. 1.13). As shown in Fig. 1.16, such a smooth spectrum curve lends itself to idealization by a series of straight lines much better than does the spectrum for an individual ground motion (Fig. 1.13 ). This idealized spectrum provides the basis for a procedure to construct design spectra, which is illustrated schematically in Fig. 1.17, wherein the values recommended for T0 , Tb, Te, and TJ are identified, and amplification factors a A, a v, and a o for the three spectral regions used in constructing the spectrum are illustrated. Developed by the preceding statistical analysis for a larger ensemble of ground motions,

21


Elastic design spectrum

c

/

d

r - -tko - - - '.

.;;-'.

/ //v,.,g

<Po

'.

/~ Peak ground acceleration, velocity, and displacement

l/33 sec 33Hz

I 0 sec 1/1 0 Hz

1/8 sec 8Hz

33 sec 1/33 Hz

Natural vibration period (log scale)

Figure 1.1 7. Construction of elastic design spectrum [3] .

numerical values of the amplification factors for two different nonexceedance probabilities, 50% and 84.1 %, are available for several damping values, and a procedure for constructing a design spectrum from the peak values of ground acceleration, velocity, and displacement has been presented by Newmark and Hall [ 10]. Once the pseudovelocity design spectrum has been developed, the pseudo-acceleration design spectrum and deformation design spectrum can be determined using Eqs. (1.9) and (1.11). The 84.1 th percentile pseudo-acceleration design spectrum for ground motions on firm ground with peak ground acceleration Ugo = 1g, peak ground velocity it go = 48 in./ sec, and peak ground displacement Ug o = 36 in. is presented in two formats : logarithmic scales (Fig. 1.18) and linear scales (Fig. 1.19). The latter plot includes spectrum curves for five damping values, whereas the first plot is for r; = 5% only. As expected, A approaches Ugo = lg atTn = 0, and the design spectrwn can be defined completely by numerical values for Ta, Tb, Tc, Ta , Te, and Tf and equations for A ( Tn) for each branch of

22

)


5

,-----------------------------------------~

Oil

""' 0

Accelerationsensiti ve region

.~ ~ ....

region

<!)

03 () ()

ro

6

-o

0.1

;::>

<!)

"'

~

)

() <!)

"'

0.1

10 Natural vibration period Tm sec

Figure 1.18. Pseudo-acceleration design spectrum (84.1 th percentile) for ground motions with Ugo = lg, Ugo = 48 in./sec, and U go = 36 in.; I; = 5% [3].

J

5

s = 1% 4

3 Oil "

"'<:

2 s = 2o% -,l

0 0

Tn, sec

2

3

Figure 1.19. Pseudo-acceleration design spectrum (84.1th percentile) for ground motions with iigo = lg, Ugo = 48 in./sec, and U ga = 36 in.; I;= 5% [3] .


the spectrum (Fig. 1.18). Scaling the spectrum by TJ is the simplest way to obtain a design spectrum for ground motions with ii go = T]g.

1.8 Peak Structural Response from Spectrum The peak response of an SDF system can be determined readily from the earthquake response (or design) spectrum without computing the response history. Corresponding to the natural vibration period 'ÂŁ7 and damping ratio t; of the system, the values of D , V, or A are read from the spectrum, such as Fig. 1.12, 1.1 0, or 1.11 , respectively; D and A can also be read from Fig. 1.1 0. Then all response quantities or seismic demands can be expressed in terms of D, V, or A and the mass or stiffness properties of the system. In particular, the peak deformation of the system is U0

=

D

= 2Tn7T V = ( 2'ÂŁ7T1 )

2

A

(1.13) ")

and the peak value of the equivalent static force fso is [fromEqs. (1 .12) and (1.11)] ! so

A w g

= kD = m A = -

(1.14)

Static analysis of the structure subjected to lateral force fso (Fig. 1.20) provides the internal forces (e.g. , shears and moments in columns and beams) and stresses. In particular, the peak values of the shear and overturning moment at the base of the one-story structure are Vbo =

A

( 1.15)

- w g

Only one of the three response (or design) spectra- deformation, pseudo-velocity, or pseudo-acceleration- is sufficient for computing

J

f J$/NMM##ff#/I ~~

'

- - - - - - v bo

~Mbo

Figure 1.20. Peak value of equivalent static force .

24


the peak deformations and forces required in structural design. For such applications, the velocity or acceleration response spectra mentioned earlier are not needed. Therefore, these spectra or the Fourier spectrum are not included here; readers are referred to other books (Newmark and Rosenblueth [11], Chap. 1; Hudson [1], pp. 55- 64; Chopra [3], Sec. 6.12).

J

25


2 ELASTIC MULTI-DEGREE-OF-FREEDOM SYSTEMS

Structures more complex than the simple structures considered in Section 1.1 should be treated as systems with multiple degrees of freedom (MDF) because their motion cannot be described by a single displacement. The equations governing the motion of an MDF system and procedures to analyze its dynamic response can be developed in a general form such that they apply to all classes of structures (e.g., buildings, bridges, dams, etc.) subjected to spatially varying ground motion. Such generality is typical of presentations in textbooks (Clough and Penzien [12], Chaps. 9-10; Chopra [3], Chaps. 9 and 13). However, a specialized approach was adopted in this monograph for its intended audience, and the presentation is limited to multistory buildings with their planwise distribution of mass and stiffness perfectly symmetric about x andy axes subjected to uniform ground motion (i.e., identical excitation at all supports). Such symmetric-plan systems can be analyzed independently in the two lateral directions; when subjected to ground motion along one of the two axes (say, the x-axis), the system responds only in the same direction. The theory and concepts are developed in a form that facilitates their extension to unsymmetric-plan buildings (not included here). When subjected to the x or y component of ground motion, unsymmetric-plan systems would undergo simultaneously lateral motion in two (x andy) directions and torsion aboutthe vertical (z) axis ([3], Chaps. 9 and 13). Even symmetric-plan buildings may undergo "accidental" torsional motions for two principal reasons: the building as built is usually not perfectly symmetric; and even minor spatial variations in ground motion may induce torsional motion of the building even if its plan is perfectly symmetric. The topic of accidental torsion is also not included here. 27


2.1 Equations of Motion The equations governing the motion of a symmetric-plan multistory building (or multistory frame shown in Fig. 2.1) due to horizontal earthquake ground acceleration iig(t) along an axis of symmetry are as follows:

mii + cit + ku

=

- mtiig(t)

(2.1)

where u is the vector of N lateral floor displacements u1 (see Fig. 2.1) relative to the ground; and m , c, and k are the mass, damping, and lateral stiffness matrices ofthe system; each element of the influence vector t is equal to unity because all floor displacements are in the same direction as the ground motion. The right-hand side of Eq. (2 .1) can be interpreted as effective earthquake forces: Peff(t)

=

- mtiig(t)

(2.2)

The time variation of these effective earthquake forces is defined by ii g(t) and their spatial distribution by the vectors = mt, which is the same as distribution of :floor masses over the height of the building. Given the structural matrices m, c, and k and ground acceleration ii g(t), the displacement response of the structure can be determined by solving the N differential equations represented by the matrix equation [Eq. (2.1)] for the N floor displacements u1 (t) ,

Floor N

j

2

7,0~

7,0~

7-'?.:'

7,0~

Figure 2.1. Dynamic degrees of freedom of a multistory frame : lateral displacements relative to the ground [3].

28


j = 1, 2, ... , N. Each of these equations contains more than one unknown and therefore the set of N equations must be solved simultaneously. Internal forces can subsequently be determined from the displacement response. Whereas the mass and stiffness matrices of a structure can be computed from the dimensions and sizes of structural and nonstructural elements, it is impractical to compute the damping matrix in a similar manner. Damping in a structure is therefore usually specified on a global basis in terms of modal damping ratios, with values obtained from experiments on similar structures serving as a guide. Modal damping ratios can be determined from ambient vibration tests or from harmonic vibration generator experiments on actual structures and from their motion recorded during earthquakes ([3], Chap. 11 ).

2.2 Modal Analysis: Basic Concept The modal analysis procedure is based on the fact that forcertain forms of damping that are reasonable models for many buildings, the response in each natural mode of vibration can be computed independent of the others, and the modal responses can be combined to determine the total response. Each mode responds with its own particular pattern of deformation, the natural mode of vibration </> 11 ; with its own frequency, the natural frequency of vibration w 11 ; and with its own modal damping ratio, /;11 â&#x20AC;˘ Each modal response can be computed by analysis of an SDF system with properties chosen to be representative of the particular mode. The modal analysis procedure avoids simultaneous solution of the coupled equations (2.1 ). This method is strictly valid only for the earthquake analysis of structures responding within their linear elastic range of behavior, but provides good results even if the response is moderately large, as long as it is essentially linear. The method provides the complete history of dynamic response during the earthquake and can be adapted to provide estimates of peak response directly from the earthquake response spectrum.

2.2.1 Modal Expansion of Effective Forces To develop the modal analysis procedure, the force distribution s is expanded as a summation of modal inertia force distributions s11 : N

mt

N

= Lsn = Lrnm<f>n n=l

n= l

29

(2.3)


where if>n is the nth natural vibration mode of the structure, and

1n

= -Ln

Mn

Ln

=

T

if>n mt

Mn

= if>nT mif>n

(2.4)

The effective earthquake forces can then be expressed as Peff(t) =

N

N

n= l

n= l

L Peff,n(t) = L

-Sn Ug(t)

(2.5)

The nth-mode components ofpetTCt) and s are (2 .6) The expansion ofEq. (2.3) has an important and useful property that can be proven analytically ([3], Sec. 12.8): The force vector Peff,11 (t) produces response only in the nth mode but no response in any other mode.

2.3 Modal Response History Analysis The preceding property provides a conceptual basis for the classical modal analysis procedure to determine structural response as a function of time (i.e., response history). Structural response due to individual excitation terms Peff,n (t) is determined first for each n, and these N modal responses are combined algebraically at each time instant to obtain the total response.

2.3.1 Modal Responses Because the response of an MDF system to Peff,n (t) is entirely in the nth mode, with no contributions from other modes, the floor displacements are U 17 (f)

= if>nq

17

(f)

(2.7)

where the modal coordinate q 11 (t) is governed by

qn + 2t;nwniJn + w~qn =

-1 11 ilg(t)

(2 .8)

in which W 17 is the natural vibration frequency and 1;11 is the damping ratio for the nth mode. The solution q 11 (t) ofEq. (2.8) is given by (2.9) where D 11 (t) is governed by the equation of motion for the nth-mode linear SDF system, an SDF system with vibration properties- natural frequency W n and damping ratio sn - Of the nth mode of the MDF

30


system, subjected to ug(t): -

Dn

. + 2{nWnDn + W2 Dn = -ug(t) 11

(2.10)

Substituting Eq. (2.9) into Eq. (2.7) gives the floor displacements, (2.11) Any response quantity r(t)- floor displacements, story drifts, internal element forces , etc.-can be expressed as (2.12) where r~t denotes the modal static response, the static value (indicated by the superscript "st'') of r due to external forces 1 S11 , and (2 .13) is the pseudo-acceleration response of the nth-mode SDF system (Section 1.4). Observe that r~t may be positive or negative and is independent of how the mode is normalized.

2.3.2 Total Response Equations (2.11) and (2 .12) represent the response of the MDF system to Petr,11 (t) [Eq. (2.6a)], where n = 1, 2, . .. , N. These N modal responses are algebraically combined to obtain the response of the system to the total excitation P etrU): N

u(t)

N

= .L:UnU) = L n= l

n= l

N

N

rn4>nDn(t)

r(t) = Lr11 (t) = Lr~ 1 A 11 (t) n=l

(2.14)

(2.15)

n= l

This is the classical modal RHA procedure: Eq. (2 .8) is the standard modal equation governing q11 (t) , Eqs. (2.11) and (2.12) define the contribution of the nth mode to the response, and Eqs. (2.14) and (2.15) reflect combining the response contributions of all modes to obtain the total response.

1

Although we refer loosely to sn as forces, they have units of mass . Thus, r~t does not have the same units as r , but Eq. (2.12) gives the correct units for r 11 â&#x20AC;˘

31


Forces

(a)

Sn

(b)

LA

11

(t)

l "''" --<>

Ug(t)

Figure 2.2. Conceptual explanation of modal response history analysis of elastic MDF systems: (a) static analysis of structure; (b) dynamic analysis of an SDF system [3].

2.3.3 Interpretation of Modal Analysis In the first phase of this dynamic analysis procedure, the vibration properties-natural vibration frequencies and modes- of the structure are computed and the force distribution vector mt is expanded into its modal components S11 â&#x20AC;˘ In the second phase, the contribution of the nth mode to the dynamic response is obtained by multiplying the results of two analyses : (1) static analysis of the structure with applied forces S11 , and (2) dynamic analysis of the nth-mode SDF system excited by iig(t). These two analyses are shown schematically in Fig. 2.2. In the third and final phase, at each time instant the modal responses are combined algebraically to determine the total response of the structure. 2.3.4 Modal Static Response The modal static response r,~ 1 is determined by static analysis of the building subjected to external forces S11 (Fig. 2.2). The modal static response for base shear Vb and base overturning moment Mb are given by st M bn

where

rn

= h*M* n n

(2.16)

and Ln were defined in Eq. (2.4) and (2 .17)

32


= < h 1, h 2 , ... , h N> and h1 is the height of the jth floor above the base. Equations for modal static response for other response quantities- forces in other stories and in structural elements- can be written similarly. The modal static response for jth-floor displacement u1 and jth-story drift b. J are given by hT

st

uj n

=

rn

?

w;;

(2.18)

¢ Jn

2.3.5 Effective Modal Mass and Modal Height The base shear due to the nth mode is obtained by specializing Eq. (2.12) for Vb and using Eq. (2.16a): (2.19) Comparing Eq. (2.19) with Eq. (1.7a) indicates that the total mass of the single-mass (SDF) system is effective in producing base shear, whereas only the portion M; of the mass of a multi mass (MDF) system is effective in producing the base shear due to the nth mode; the portion depends on the distribution of mass over the height and on the shape of the mode. Thus, M; is called the effective modal mass. Equation (2 .17a) will give values of M,7that are independent of how the modes are normalized. It can be proven analytically that the sum of the effective modal masses M; over all the modes is equal to the total mass of the building. The base overturning moment due to the nth mode is obtained by specializing Eq. (2 .12) for Mb and using Eqs. (2.16b) and (2.19): (2 .20) Comparing Eqs. (2.20) and (1.7b) indicates that the total height of a single-mass (SDF) system is effective in producing base overturning moment Mb , whereas the height h~ that appears in the modal contribution Mbn to Mb is less than the total height of a multimass system. Known as effective modal height, h ~ depends on the distribution of mass over the height and on the shape of the mode. For some of the vibration modes higher than the first mode, the effective modal height computed from Eq. (2.17b) may be negative. A negative value of h ~ implies that at any instant of time, base shear Vbn (t) and base overturning moment Mbn(f) have opposite algebraic signs; the Vb l (t) and Mb 1 (t) for the first mode are by definition both positive.

33


9th 8th 7th 6th 0'""' 0

5th

~ 4th

3rd 2nd

I st Ground - 1.5

-I

- 0.5 0.5 0 Mode Shape Component

1.5

Figure 2.3. First three natural-vibration periods and modes of the 9-story building [14].

2.3.6 Example The structure considered is the SAC-Los Angeles 9-story building. 2 The first three vibration modes and periods of the building are shown in Fig. 2.3 ; they are ordered in sequence from longest to shortest period. The floor displacements in the first (or fundamental) mode are all in the same direction, but they reverse direction in higher modes as one moves up the building. The modal expansion of the distribution s = mt of the effective earthquake forces is shown in Fig. 2.4. Observe that the direction of force SJn at the jth-ftoor level is controlled by the algebraic sign of ¢ 1n , the jth-ftoor displacement in mode {jJ 11 â&#x20AC;˘ Hence, these forces for the first mode all act in the same direction, but for the second and 2

SAC commissioned three consulting firms to design 3-, 9-, and 20-story steel-moment-resisting-frame buildings according to local code requirements in three cities: Los Angeles (UBC, 1994), Seattle (UBC, 1994), and Boston (BOCA, 1993). Square in plan, these buildings have identical properties in both lateral directions. Descriptions of their dimensions in plan and elevation, member sizes, and other properties are available in several publications (e.g., [13]). Perimeter frames of the Los Angeles 3-, 9-, and 20-story buildings are used as examples throughout this monograph. Ground motions used include the 20 ground motion records assembled by SAC to represent 2% of probability of exceedance in 50 years (return period of 2475 years).

34


- 0.573111

0.260m

- 0. 125m

0.06 1m

- 0.284m

- 0.033m

O. ll 6m

- 0. 105m

m

- 0.005m

- 0.231m

0. 143m

-0.0 16m

m

0.2 13m

- 0.250m

- 0.0 13m

0. 107111

tl1

0.339m

1.08m

1.475111

Ill

- 0. 11 8m

+

- 0. 137m

+

0.028m

+

+ ooo

Ill

0.395m

0.062m

- 0.123m

- 0.088m

Ill

0.381m

0.202111

0.004m

-0.067m

0.385m

0.3 15m

0.251m

0. 117111

0.039m

0.235m

0.208m

0. 196m

0. 13 1m

0.090m

Ill

l. 02m

ml

Figure 2.4. Modal expansion of the distributions earthquake forces [ 14].

= mt of effective

higher modes they change direction as one moves up the building. The contribution of the first mode to the force distribution s is largest, and the modal contributions decrease progressively for higher modes. The effective modal masses M,~ and effective modal heights h~ are shown schematically in Fig. 2.5; where the h~ are plotted without their algebraic sign. Observe that both M~ and h~ decrease progressively for higher modes. Figure 2.6 shows that when subjected to forces Peff,n (t) defined in Eq. (2.6a), the building responds only in its nth mode; none of the other modes are excited. This provides a numerical confirmation of the earlier assertion (Section 2.2), which is the basis of the modal analysis procedure. 0.83 1M

~

N

i2 6

Mode

0. 109M

2

4

Figure 2.5. Effective modal masses and effective modal heights.

35


:~':IH :,:If-----~-) Model

- 15

- 5 L_~-~-~~--__j

:.':I

1

Mode

21

§ : ::!"<:_:.

I

-MOOd

:~ :51

5

10 15 20 Time, sec

25

••

.

- Is~~--~-~-~~

0

2.226

- s~~--~-~--~

- 15~·-~~-~-~~-~.

:<':11----

.au~llllll!lUIII~aaum;a~ct~:.:..1 ...., , , , , , , " ' ' ' " " ' ' " " ' " ' "

30

0

5

10 15 20 Time, sec

25

30

Figure 2.6. Response to modal component P eff,11 (t) of effective earthquake forces: (a) n = 1; (b) n = 2; P eff,11 (t) = -S11 iig(t), where ii g(t) = 0.25x El Centro ground motion [14].

The earthquake response of the building was determined by the modal analysis procedure implemented in a computer program. A tiny portion of the response results (i.e., the roof displacement and top-story drift) is presented in Fig. 2.7. Shown as a function of time during the earthquake are the contributions of the first three modes to each response quantity, together with the total response (including the contributions of all nine modes). The peak values of all floor displacements and all story drifts, including the contributions of one, two, three, or all modes, are presented in Fig. 2.8. It is apparent that the first mode is dominant in floor displacements but not in story drifts, implying that the significance of higher-mode contributions depends on the response quantity of interest. Observe that the significance of higher-mode response varies over the height of the building, and the first three modes capture essentially the total response- floor displacements as well as story drifts- of this 9-story building. A very attractive feature of modal analysis is that it permits analysis of an MDF system by independent analysis of SDF systems,

36


(a)

(b)

j:~~~~~~ :~:1 • •~iz2

.

~~~

:c:'----------'---1

I ~":1 ~~''I - 1 ~~--~----~--~~

- 5~~--~----~--~~

:~: 1

I :<:

c___________.__...__________.___0.0256 •

3

==0.0292Mode

;. :I • ~ ;._:~-"\-l\-I-10: f-A-iI~

''\f-\:11 moflPdes'JV.

0

5

I 0 15 20 Time, sec

25

30

0

5

I 0 15 20 Time, sec

25

30

Figure 2.7. Response history of (a) roof displacement and (b) roof-story drift: first three modal contributions and total (all modes) response to one of the SAC ground motions. (b)

(a) 9

'

, il

6 ~

~

0

0 0

0

G:

u:

RHA 3

- All Modes • - a ] Mode o- o 2 Modes .,._....3Modes

2

4

6

\0

Displace ment/height,%

G

I

RH A -AI\ Modes • -a I Mode o- o 2 Modes .,._....3 Modes

0

2

4

Q

6

8

\0

12

Story driftL\. RHA'%

Figure 2.8. Peak values of (a) floor displacements and (b) story drifts from RHA, including contributions of one, two, three, or all modes, due to one of the SAC ground motions.

37


one for each natural vibration mode (Fig. 2.2). Also significant is the fact that, in general, the response need be determined only in the first few modes because the contribution of higher modes to earthquake response is usually negligible (e.g., as noted earlier from Fig. 2.8, the first three modes are sufficient for the example 9-story building). Thus, only the first few vibration frequencies and modes need be computed, the response computations need to be repeated only for these modes, and the summations in Eqs. (2.14) and (2.15) can be truncated accordingly. The response history analysis (RHA) procedure described in this section provides the time variation of all response quantities- floor displacements, story drifts, story shears and moments, and internal forces in each structural member- for the structure due to a specified ground acceleration history. From the response history, the peak value of response can readily be determined, as noted in Fig. 2.7.

2.4 Modal Response Spectrum Analysis The complete history of structural response is seldom needed for design or evaluation of structures; the peak value of the forces and deformations usually suffice. A good estimate (but not the exact value) of the peak response can be determined directly from the response spectrum for the ground motion. The spectrum provides exact values of peak modal responses that are combined to obtain an estimate of the peak value of the total response. Such a response spectrum analysis (RSA) procedure is described in this section for structures excited by a single component of ground motion. 2.4.1 Peak Modal Responses Because the contribution of each mode to structural response has been related to the response of an SDF system (Fig. 2.2), the exact value of peak modal response can be determined directly from the response spectrum. The peak value rna of the nth-mode contribution rn(t) to response r(t) is given by

(2.21) where An is the ordinate A(Tn , ~11 ) of the pseudo-acceleration response spectrum, corresponding to the natural vibration period Tn = 2rr / w 11 and the damping ratio ~n of the nth mode. The algebraic sign of rna is the same as that of the modal static response r~ 1 because A 11 38


is positive by definition. 3 Although it has an algebraic sign, rno will be referred to as the peak modal response. The algebraic sign must be retained because it can be important, as will be seen later. 3

2.4.2 Modal Combination Rules We are interested in determining the peak value r 0 of the total response r(t). It will not be possible to determine the exact value of r0 from the peak modal responses r no because, in general, the modal responses rn (t) attain their peaks at different time instants and the combined response r (t) attains its peak at yet a different time instant. This phenomenon can be observed in Fig. 2.7b, where results for the roof-story drift of the 9-story building are presented. The individual modal responses t::,.,.n (t) , n = 1, 2, .. . , 3 are shown together with the total response t::,.,.(t). Assuming that all modal peaks occur at the same time and ignoring their algebraic sign provides an upper bound for the peak value of the total response: N

Yo .:S

L

lrnol

(2.22)

n= l

Because this estimate is usually excessively conservative, this absolute sum modal combination rule is not popular in structural design applications. The peak modal responses are usually combined according to the square-root-of-sum-of-squares (SRSS) rule or the complete quadratic combination (CQC) rule. The SRSS rule provides an estimate of the peak value of the total response according to the equation N

Yo::::::::

(

L:r,;o

) 1/ 2

(2.23)

n= l

The algebraic signs ofr no do not affect the value ofr0 from Eq. (2.23). Valid for structures with well-separated natural vibration frequencies, such as multistory buildings with a symmetric plan, the SRSS rule was developed by Rosenblueth in his Ph.D. thesis (1951) [15], and about the same time (1950) applied in the seismic design of the Latino 3

This notation rna should not be confused with the use of a subscript "o" in Chapter I to denote the maximum (over time) of the absolute value of the response quantity, which is positive by definition.

39


Americana Tower in Mexico City, a project for which N. M. Newmark was a consultant. The CQC rule for modal combination is applicable to a wider class of structures, including those with close-spaced natural vibration frequencies (e.g., unsymmetric-plan buildings). According to the CQC rule, (2.24) Each of the N 2 terms on the right side of this equation is the product of the peak responses in the ith and nth modes and the correlation coefficient, Pin, for these two modes; Pin varies between 0 and 1, and Pin = 1 fori = n. Thus, Eq. (2.24) can be rewritten as 1/2

N ro:::::

~r~o n=l

N

N

+ ~ ~Pinriorno

(2.25)

i=l n=l '--v--'

i=f.n

to show that the first summation on the right side is identical to the SRSS combination rule of Eq. (2.23). Each cross term in the double summation may be positive or negative depending on the algebraic signs ofrio and rna路 Thus the estimate r 0 obtained by the CQC rule may be larger or smaller than the estimate provided by the SRSS rule. The preceding modal combination rule was first derived by Rosenblueth and Elorduy in 1969 [ 16]-and widely disseminated by the Newmark-Rosenblueth book [11]-although they did not give it the CQC name, which came later [ 17]. The correlation coefficient Pin is a function of the ratio f3in = wi / Wn of the modal frequencies and the modal damping ratios Si and sn; Pin approaches 1 as wi and Wn become close to each other, and it diminishes rapidly as wi and Wn move farther apart. For structures with well-separated vibration frequencies, the coefficients Pin vanish; as a result, all cross (i # n) terms in the CQC rule, Eq. (2.25), can be neglected and it reduces to the SRSS rule, Eq. (2.23). Several equations are available for the correlation coefficient, and two of the more important ones are included in textbooks (e.g., [3], Chap. 13). Considering that the SRSS and CQC modal combination rules are based on random vibration theory, r 0 should be interpreted as the mean (or median) of the peak values of response to an ensemble 40

-_L


of earthquake excitations. Thus, the modal combination rules are intended for use when the excitation is characterized by a smooth response (or design) spectrum, based on the response spectra for many earthquake excitations. The smooth spectrum may be the mean (or median) of the individual response spectra, or it may be a more conservative spectrum, such as the 84.1 th percentile spectrum mentioned in Section 1.7. When used in conjunction with, say, the mean spectrum, the CQC or SRSS modal combination rule provides an estimate of the peak response that is reasonably close to the mean of the peak values of response to individual excitations. The error in the estimate of the peak may be on either side, conservative or unconservative. These modal combination rules may be used to estimate the peak structural response due to a single ground motion characterized by a jagged response spectrum, but the errors may be significantly larger in this case. 2.4.3 Example Figure 2.9 presents the peak value of the total response of the 9story building determined by (1) RHA considering all modes (this is the "exact response" reproduced from Fig. 2.8); and (2) RSA combining the contributions of one, two, or three vibration modes according to Eq. (2.23). Although when included in RHA the first three modes provided essentially the exact value of response (Fig. 2.8), the RSA estimate errs significantly because of the approximation inherent in

(a)

(b)

9

9

6

6

00

a

-RHA RSA

00

c:

c:

â&#x20AC;˘ -a ! Mode o- <> 2 Modes ...._.3 Modes

3

RSA a -a I Mode ._ o 2 Modes ...._.3 Modes G

0

2

4 8 6 Disp lacement/height,%

10

G

0

4 6 10 Story drift 6RSA or 6RHA'%

12

Figure 2.9. Peak values of(a) floor displacements and (b) story drifts determined by RSA, combining contributions of one, two, or three modes, and by RHA considering all modes due to one of the SAC ground motions; shading indicates modal combination error.

41


I

6

-

RHA

RSA 1 Mode G- ~ 2 Modes ,...___... 3 Modes

0'""" 0

Iii-

~

3

QL---~----~--~----~----~--~

0

2

4 6 8 10 Story drift L\sA or t.RHA' %

12

Figure 2.10. Median of peak values of story drifts determined by RSA, combining contributions of one, two, or three modes, and by RHA considering all modes due to the SAC ensemble of 20 ground motions; shading indicates modal combination error.

combining the peak modal responses. For a given structure, this error can vary widely with the ground motion. Modal combination rules are more dependable in estimating the mean (or median) of the peak values of response to an ensemble of earthquake excitations. This is illustrated in Fig. 2.1 0, where the median (over 20 ground motions) of story drifts determined by two procedures, RHA and RSA, are presented. Clearly, the errors in the RSA estimate are now significantly smaller than noted earlier in estimating the response to the single excitation selected in Fig. 2.9, but this observation may not hold for another excitation. 2.4.4 Alternative Interpretations of Response Spectrum Analysis The RSA procedure, which is a dynamic analysis procedure, can be interpreted in two ways: as static analysis or pushover analysis. Static analysis of the structure subjected to lateral forces

(2.26) will provide the same value ofrn 0 , the peak nth-mode response, as in Eq. (2.21). 42


Alternatively, this response value can be obtained by linear static analysis of the structure subjected to monotonically increasing lateral forces with an invariant height-wise distribution: (2 .27) pushing the structure up to the roof displacement, Urn o (the subscript r denotes "roof"), the peak value of the roof displacement due to the nth mode, which from Eq. (2.11) is (2.28) where D n = An /w~ is the ordinate D(Tn , sn) of the deformation response spectrum corresponding to the period Tn and damping ratio Sn of the nth mode. The peak modal responses, r11 0 , each determined by one pushover analysis, can be combined according to Eq. (2 .23) or (2.24), as appropriate, to obtain an estimate of the peak value r 0 of the total response. Equivalent to the well-known RSA procedure described earlier, this modal pushover analysis (MPA) procedure offers no advantage for linearly elastic systems. It becomes attractive when extended to estimate seismic demands for inelastic systems (Chapter 6).

2.5 Higher-Mode Contributions to Earthquake Response The response contributions of all the natural vibration modes must be included if the "exact" value of the structural response to earthquake excitation is desired, but the first few modes can usually provide sufficiently accurate results. For example, the first three modes were sufficient to determine the peak earthquake response of a 9-story building (Fig. 2.8). More modes may be necessary to achieve similar accuracy for response to near-fault ground motions with a forward directivity pulse. The relative contributions of the various modes to the total response and the number of modes necessary in the analysis depend on the vibration properties of the system, the shape of the response or design spectrum, and the response quantity of interest. These topics are the subject of this section. 2.5.1 Modal Contribution Factors Equation (2 .21) for the peak value of the nth-mode contribution r 11 (t) to response r(t) can be rewritten as

(2.29) 43


where r 51 is the static value ofr due to external forces s modal contribution factor

=

mt and the

(2.30) The relative contributions of various modes to the total response depend on two factors, modal contribution factor r11 and spectral ordinate A 11 , that enter into modal response equation (2.29). These modal contribution factors have three important properties: (1) by definition they are dimensionless; (2) they are independent of how the modes are normalized; and (3) the sum of modal contribution factors over all modes is unity; that is, (2 .31)

2.5.2 Structural Properties The fundamental natural vibration period T1 and the beam-tocolumn stiffness ratio are two parameters that are especially influential on the earthquake response of a building. Introduced in 1968 by Blume [ 18] as a joint rotation index, the latter parameter is based on the properties of the beams and columns in the story closest to the midheight of the frame: p

=

Lbeams Elb/ Lb

(2.32)

"""' L...co lumns Elc/Lc

where EIb and EIc define the flexural rigidity of beams and columns, Lb and L c are the lengths of beams and columns, and the summations include all the beams and columns in the midheight story. This parameter defines the overall behavior of the frame under lateral forces. For p = 0, the beams impose no restraint on joint rotations and the frame behaves as a flexural beam (Fig. 2.11a). For p = oo, the beams constrain the joint rotation completely, and the structure behaves as a shear beam with double-curvature bending of the columns in each story (Fig. 2.llc). An intermediate value of p represents a frame in which beams and columns undergo double-curvature bending with joint rotation (Fig. 2.11 b). Although defined for a frame, the parameter p is also meaningful for other structural systems (e.g., buildings with a single shear wall as the lateral resisting element would be characterized by p close to zero). The parameter p controls several

44


(a)

(b)

(c)

Figure 2.11. Defl ected shapes: (a) p = 0; (b) p = 1/ 8; (c) p = oo [3].

properties of the structure: the fundamental period T1 , the relative closeness or separation of vibration periods Tn, and the shapes of the vibration modes. The vibration periods of a frame with small p are more separated from each other than if p is large.

2.5.3 Influence of T1 on Higher-Mode Response As T1 increases within the velocity- and displacement-sensitive regions of the spectrum, the higher-mode response generally becomes an increasing percentage of the total response. Figures 2.12 and 2.13 show the results of response spectrum analyses of many framescovering a wide range of T1 and three values of p (0, 1/8, and oo )for ground motion characterized by the design spectrum of Fig. 1.18 multiplied by 0.5. Figure 2.12 presents the normalized base shear Vb/ W!- where Wt = Mig and Mi is the effective modal mass for the first mode [Eq. (2 .17a)]- for each frame determined by two analyses, considering (1) all modes, and (2) only the first mode. The one-mode curves are independent of p and identical to the design spectrum selected. The difference between the two results for base shear is the higher-mode response, which has been expressed as a percentage of total response and presented in Fig. 2.13 ; also included therein are similar results for three other response quantities: base overturning moment Mb , top-story shear Vr , and roof displacement u,. The higher-mode response of buildings is negligible for T 1 in the acceleration-sensitive region of the spectrum and increases with increasing T1 in the velocity- and displacement-sensitive regions. This implies that higher-mode contributions are more significant in the response of taller buildings than in the response of shorter buildings.

45


p=O 0.1

RSA, 1 mode, all p

Fundamental natural period T1, sec

Figure 2.12. Normalized base shear in uniform frames for three values of p. Results were obtained by RSA, including one or five modes [3]. Such results are useful in evaluating the lateral force provisions in building design codes (Chapter 5). The preceding trends on how the significance of higher-mode response depends on Tt can be predicted from Eq. (2.29), the shape of the design spectrum, and where T 1 falls on the spectrum ([3], Chap. 18). Figure 2.13 demonstrates that the higher-mode response is more significant (1) for forces (e.g. , Vt, Vb , and Mb) than for displacements (e.g., ur) ; (2) for base shear than for base overturning moment; and (3) for top-story shear than for base shear. Such results are useful in interpreting and evaluating the lateral force provisions in building codes (Chapter 5). The preceding trends on how the significance of highermode response depends on the response quantity can be predicted

46


100

100

80

80

60

60

E 40

40

:? ,; OJ

0

s:2

"

"0 0

,'.

"

..c

bO

i

20

20

0

0 0.1

10

10

0.1

0.1

10

Fundamental natural period Tt, sec

Figure 2.13. Higher-mode response in base shear Vb, top-story shear fit, base overturning moment Mb , and roof displacement Ur for uniform frames for three values of p: (a) p = 0; (b) p = 1/8; (c) p = oo [3].

from numerical values of modal contribution factors [Eq. (2.30)] for these four response quantities that enter into Eq. (2.29) ([3], Chap. 18). 2.5.4 Influence of p on Higher-Mode Response For buildings with T1 within the velocity- and displacementsensitive regions of the spectrum, as p decreases, the higher-mode response becomes an increasing percentage of the total response (Fig. 2.13). This implies that higher-mode contributions are more significant in systems deforming like flexural beams (e.g., a shearwall building) than in systems deforming like shear beams (e.g., a stiff beam/flexible column frame). The preceding trends on how the significance of higher-mode response depends on p can be predicted from Eq. (2.29), numerical values of the modal contribution factors for various p values (Eq. 2.30), and the p dependence of the ratios of spectral ordinates, An/ A 1, which in turn depend on the period ratios TnfT, ([3], Chap. 18).

2. 6 How Many Modes to Include The number of modes to be included in dynamic analysis depends on two factors: modal contribution factor rn and spectral ordinates An, that enter into the modal response equation (2.29). If only the

47


first J modes are included, the error in the static response is J

e1

=

1-

Lrn

(2.33)

n=l

Thus modal analysis may be truncated when le1 1, the magnitude of e1, becomes sufficiently small for the response quantity of interest. However, e1 by itself is not indicative of the modal-truncation error in earthquake response because the relative values of An depend on the shape of the earthquake design spectrum and where the modal periods Tn fall on the period scale. The results presented in Fig. 2.12 and their earlier discussion suggest that for the same desired accuracy: (I) more modes should be included in the analysis of buildings with longer T1 (e.g. , taller buildings) than the number of modes necessary for shorter-period buildings (e.g., shorter buildings); and (2) more modes should be included in the analysis ofbuildings with smaller p values (e.g., shearwall buildings) than the number of modes necessary in buildings with larger p values (e.g., moment-resisting frames) . These expectations regarding how T, and p influence the number of modes that should be included in earthquake response analysis are confirmed by the results of Fig. 2.14, where for each p value, response curves for base shear are identified by indicating the number of modes included in the analysis. 10

10 (a) p = 0

(b) p = 1/8

(c)

p =~

N umber of modes

all ~-

\

,.~

\ 0. 1

\ \

\

0.1

2

\

0.01 0. 1

10

0. 1

10

0. 1

0.01 10

Fundamental natu ral period T 1, sec

Figure 2.14. Normalized base shear in uniform frames for three values of p : (a) p = 0; (b) p = 1/ 8; (c) p = oo. Results were determined by RSA considering one, two, three, four, or all modes [3].

48


In light of the preceding observations, it is interesting to examine the 90% rule for participating mass specified in some building codes [ 19 ,20] . Because the effective modal mass is equal to the modal static response Vt~ for base shear (see Section 2.3.4), the rule above implies that enough- say, J - modes should be included so that eJ [Eq. (2.33)] for base shear is less than 10%. However, e J varies with the response quantity because the modal contribution factors rn vary, and therefore this error may exceed 10% for other response quantities, such as shears in upper stories of a building and bending moments and shears in structural members. Even if a sufficient number of modes were included so that the error e J in the static response is less than 10% for all response quantities of interest, the error in dynamic response may exceed 10% for buildings with longer T1 and smaller p . Thus, the code criterion to decide the number of modes to include in dynamic analysis should be improved.

49


PART II: INELASTIC SYSTEMS


3 INELASTIC SINGLE-DEGREE-OF-FREEDOM SYSTEMS

Most buildings are designed for base shear smaller than the peak base shear in the structure computed under the assumption that it is linearly elastic. This is evident from Fig. 3.1, in which the base-shear coefficient Ajg [see (Eq. 1.12)] from the design spectrum for 5% damping in Fig. 1.19, scaled by 0.4 to correspond to peak ground acceleration of 0.4g, is seen to be much larger than the base-shear coefficient specified in the International Building Code. This iinplies that buildings designed for code forces would be deformed beyond the limit of linearly elastic behavior (i.e., they would be damaged when subjected to ground motions represented by the 0.4g design spectrum). The challenge to the engineer is to design the structure so that the damage is controlled to an acceptable degree. Structural design is obviously not successful if the damage is too severe to be repaired economically (Fig. 3 .2) or if the structure collapses (Fig. 3.3). Therefore, the response of structures deforming into their inelastic range is of central importance in earthquake engineering.

3.1 Idealized Inelastic Systems Since the 1960s, thousands of laboratory tests have been conducted to determine the force- deformation behavior of structural components for earthquake conditions. The experimental results show that the cyclic force- deformation behavior of structural members, assemblages of members, scaled models of structures, and small fullscale structures depends on the structural material and on the structural system. However, all force- deformation plots show hysteresis loops under cyclic deformations because of inelastic behavior. Idealized versions of these force-deformation relations appropriate for the 53


1.2

Elastic design spectrum

iigo = 0.4g

0

Q)

'G

0.8

!..:::

""'0 Q)

0

.... 0.6

"' Q)

..c:

"' Q)

co"'"'

0.4

0.2

0 0

2 Natural vibration period T11 , sec

Figure 3.1. Comparison ofbase-shear coefficients from an elastic design spectrum and the International Building Code [3].

Figure 3.2. Barrington Medical Building: Reinforced concrete structure at the corner of S. Barrington Ave. and W. Olympic Blvd. in west Los Angeles. This view shows the X-cracking in the columns in the lower stories, resulting from the January 17, 1994, magnitude 6.7, Northridge, California, earthquake. It appears that the panels at the beam levels acted as structural components, leading to captive-column shear failures. This building was demolished within days of the earthquake.

3


Figure 3.3 . Psychiatric Day Care Center before and after the San Fernando, California, earthquake of February 9, 1971. This magnitude 6.4 earthquake caused very strong ground shaking at this site, which disintegrated the first story of this building. structural material and structural system of interest have been used in many computer simulation studies on the earthquake response of single-degree-of-freedom (SDF) systems. For this introductory presentation, the simplest such idealized force- deformation behavior is chosen. Figure 3.4a shows the elasticperfectly plastic (or elastoplastic) force- deformation relation, fs (u , sign it) . The elastic stiffness is k and the post-yield stiffness is zero. The yield strength is /y and the yield deformation is uy . Unloading

55


(b)

(a)

fs

fs 1

fo

J;,

Corresponding linear system

I

Elastoplastic system

J;,

L__--'--'---------..J--- u

Figure 3.4. (a) Elastoplastic force- deformation relation; (b) elastoplastic system and its corresponding linear system [3].

(the algebraic sign of it is negative) and reloading (the algebraic sign of it is positive) of the hysteretic system occur along a path parallel to the initial elastic branch without any deterioration of stiffness or strength. Within the linearly elastic range, the system has a natural vibration period Tn (frequency Wn = 2n j Tn) and damping ratio r;. At larger amplitudes of motion the natural vibration period is not defined for inelastic systems.

3.2 Yield-Strength Reduction Factor and Ductility Factor The yield strength reduction factor, Ry, is defined by

.fo Uo Ry= - = /y

(3.1)

Uy

where .fo and u 0 are the minimum yield strength and yield deformation required for the structure to remain elastic during ground motion. (For brevity, the notation .fo has been used instead of the .fso employed in Chapter 1.) They are also the peak response values for the corresponding linear system, defined to have the same stiffness as the initial stiffness ofthe elastoplastic system (see Fig. 3.4b). The peak deformation or absolute (without regard to algebraic sign) maximum deformation of the inelastic system due to ground motion is denoted by Um and the ductility factor by (3.2)

56


It can be shown that the inelastic deformation ratio, defined as the

ratio of deformations of inelastic and corresponding linear systems, is related to fL and Ry by Um

(3.3)

3.3 Equation of Motion and Controlling Parameters The system considered is the one shown in Fig. 1.3 , with one important difference. The force- deformation relation is no longer linearly elastic but is defined by is (u , sign u) , which is shown in Fig. 3.4a for the elastoplastic systems considered. The equation governing the structural deformation u, defined as the displacement of the mass relative to the ground, is mit

+ cu + Is

= - mug(t)

(3.4a)

Dividing by m gives ..

U

j~ + 2 {W U. +m 11

=

"() t

-U g

(3.4b)

As for linearly elastic systems, numerical methods for solving Eq. (3 .4) are implemented on a computer to determine deformation as a function of time. Special attention must be given in the numerical procedure to detect the time instants accurately enough when the system transitions from an elastic to a yielding branch, or vice versa. For a given excitation ug(t) , u(t) depends on three system parameters: Wn (or Tn = 2n / W 11 ) , { , and u y; and the ductility factor fL depends on w 11 , { , and Ry. The latter statement implies that doubling the ground acceleration ug(t) will produce the same ductility demand for the system as if its yield strength had been halved but the excitation remained as ug(t).

3.4 Peak Deformation and Ductility Demand The deformation response of an inelastic system- defined by its initial elastic vibration period Tn, damping ratio {, and forcedeformation relation fs (u , sign u)-and its corresponding linear system, are determined by numerical solution of Eqs. (3.4b) and (1.2), respectively. The peak deformations U 111 and u 0 of the two systems are determined from their response histories. The ductility demand fL is then computed .from Eq. (3 .2) using the known yield deformation u y for the system or from Eq. (3 .3) using Ry available from Eq. (3.1).

57


(b)

(a) Spectral Regi o n s Acceleration sensitive

'-

~~ 0

sensitive

Acceleration

'

sensiti ve

IVelocity Displacement ''sensiti ve•I• sensiti ve '

0.1

0

';:!.

Velocity Displacement sensiti ve

0.01

--•••.. .

Ry= l Ry = 1.5 Ry= 2 Ry= 4 Ry = 8

0.00 I LLLL~L~:L::::::::IL:::::::'J '---'L~'--'-'__._.,~~-'--'--'-.......J....---'---'-'-' 0.02 O.l 10 O.Q2 0.1 10 50 Period ~" sec

Period T11 , sec

Figure 3.5. Peak deformations U 111 and u 0 of elastoplastic systems and corresponding linear systems due to (a) El Centro ground motion, and (b) LMSR ensemble of ground motions (median values are presented). T,, is varied; Ry = l , 1.5, 2, 4, and 8; t; = 5%.

Figures 3.5a and 3.6a are plots of peak deformations u 111 and u 0 and of ductility demand f.L , respectively, due to El Centro ground motion as a function of Tn for selected values of Ry. [Note that U 111 and u 0 have been divided by the peak ground displacement Ugo = 8.40 in. (noted in Fig. 1.4)]. Figures 3.5b and 3.6b present analogous results for the LMSR 1 ensemble of 20 ground motions. The response to each ground motion is computed, leading to 20 response values at each period Tn. The median (or geometric mean) 2 value of these response data at each period Tn is determined and the median values at all Tn are joined to obtain the curves in Figs. 3.5b and 3.6b. As expected, the median response curves (Figs. 3.5b and 3.6b) are much smoother than the curves for individual ground motions (Figs. 3.5a and 3.6a). The ductility demand and the relationship between peak deformations um and u 0 of inelastic and corresponding linear systems 1

2

Assembled by H. Krawinkler and hi s students, LMSR represents ground motions from large-magnitude earthquakes (M = 6.6-6.9) recorded at short distances (R = 13- 30 km) . Median refers to the exponent of the natural log values of the data set.

58


----------------~-·-

---

(a)

(b) Spectral Reg i ons

Acceleration sensitive

Velocity Displacement sensitive

• Acceleration , .velocity l• Displacement

sensitive

sensitive

1

1

sensitive

:::l..

-

"0

=

"'8 <l)

1

sensitive

Ry= l Ry = 1.5 Rr =2

--- Rr = 4 ••·•·• •• ••• ... Ry = 8

10

"0

&

~

10

0.1

0.02

Period T11 , sec

0.1

10

50

Period T11 , sec

Figure 3.6. Ductility demand for elastoplastic systems due to (a) El Centro ground motion and (b) LMSR ensemble of ground motions (median values are presented); Ry = 1, 1.5, 2, 4, and 8; l; = 5%.

depend on the vibration period Tn and yield strength reduction factor Ry (Figs. 3.5 and 3.6). For systems with Tn in the accelerationsensitive region (defined in Section 1.6), starting from um :::::: u 0 at Tn = Tc, Um exceeds u 0 increasingly for shorter periods, where Um is very sensitive to the yield strength, increasing as the yield strength is reduced. Even if their strength is only slightly smaller than the minimum strength required for the system to remain elastic (e.g., Ry = 1.5), very short-period systems experience deformation much larger than that of the elastic system; the ductility demand p, for these very short-period systems is much larger than Ry. For systems with Tn in the velocity-sensitive region, Um due to an individual excitation may be larger or smaller than u 0 , but in the median (over an excitation ensemble) Um :::::: u 0 , essentially independent of the yield strength, and the ductility demand is close to Ry, unless the system is very weak (e.g., Ry = 8). In the displacement-sensitive region, Um < u 0 for systems in the period range Td to TJ, where um decreases as the strength is reduced, the ductility demand p, is less than Ry . However, for systems with periods longer than TJ, Um :::::: u 0 , essentially independent of strength and the ductility factor p, :::::: Ry; for very long-period systems, both Um and u 0 are equal to Ug 0 , the peak ground displacement, independent of yield strength, and p, = Ry. 59


Figure 3.5 may be viewed as a deformation response spectrum that provides the deformation of an existing structure with known yield strength /y or yield deformation uy . Then Ry can be determined from Eq. (3.1) because fa can be calculated by Eq. (1.12), where A is known from the response spectrum for linearly elastic systems. Corresponding to this Ry and period Tn of the system, the deformation u 111 can be read from Fig. 3.5. Similarly, Fig. 3.6 may be viewed as a ductility demand response spectrum. Corresponding to known R y and T11 , the ductility demand f.L can be read from Fig. 3.6, and Eq. (3.2) provides the peak deformation Um.

3.5 Response Spectrum for Yield Deformation and Yield Strength In designing a structure, one approach is to determine the yield strength /y necessary to limit the ductility demand imposed by the ground motion to a specified allowable value. In 1960, Veletsos and Newmark [6] developed a response spectrum for elastoplastic systems that readily provides the desired information. Response spectra are plotted for the quantities Dy =uy

Vy=

2

nDy

Tn

2 2 Ay=( n) Dy

Tn

(3.5)

Note that Dy is the yield deformation of the elastoplastic system, not its peak deformation u 171 â&#x20AC;˘ A plot of Dy against T,, for fixed values of the ductility factor f.L is called the yield-deformation response spectrum. Following the definitions for linearly elastic systems (Section 1.5), similar plots of Vy and Ay are called the p seudo-velocity response spectrum and pseudo-acceleration response spectrum, respectively. The quantities Dy, Vy, and Ay can be presented in a single four-way logarithmic plot in the same manner as for elastic systems (Section 1.5). The yield strength of an inelastic system necessary to limit its ductility demand to a selected ductility factor f.L is Ay /y=-W g

(3.6)

which is analogous to Eq. (1.12) for linearly elastic systems. The response spectrum for elastoplastic systems with l; = 5% subjected to the El Centro ground motion is presented for f.L = 1, 1.5, 2, 4, and 8 in two different forms : a linear plot of Ay j g versus

60


0.8

0.6 bi)

路,

"'<:

0.4

0.2

2

3

T"' sec

Figure 3.7. Constant-ductility response spectrum for elastoplastic systems and El Centro ground motion; p., = I , 1.5, 2, 4, and 8; I; = 5% [3].

T11 (Fig. 3.7) and a four-way logarithmic plot showing Dy, Vy, and Ay (Fig. 3.8). [The f-L = 1 curve is essentially the same as the elastic response spectrum curve for = 5% (Figs. 1.10 and 1.11) because this value of 1-L implies a linearly elastic system.] An interpolative or iterative procedure is necessary to obtain the yield strength (or A y ) for a specified ductility factor because the response of a system with arbitrarily selected yield strength will seldom correspond to the desired ductility value. Originally developed by Veletsos and Newmark [21 ], such a procedure to construct the inelastic response spectrum has now become a part of textbooks ([3], Sec. 7.5).

s

3. 6 R y- JL - Tn Relations The yield strength /y required of an SDF system permitted to undergo inelastic deformation is less than the minimum strength fa necessary for the structure to remain elastic, implying that the yieldstrength reduction factor R y > 1. Figure 3. 7 shows that the yield strength required is reduced with increasing values of the ductility factor f-L. Even small amounts of inelastic deformation, corresponding to f-L = 1.5, produce a significant reduction in the required strength. 61


50 ,~

20 10 0

Q)

~

5

.5 "

~"' 2

Figure 3.8. Constant-ductility response spectrum for elastoplastic systems and El Centro ground motion; J-L = 1, 1.5 , 2, 4, and 8; ~ = 5% [3].

Additional reductions are achieved with increasing values of M but at a slower rate. Figure 3.9 shows the yield-strength reduction factor Ry for elastoplastic systems as a function of Tn for selected values of f-t . These data for the El Centro ground motion are shown in Fig. 3.9a and the median value over 20 ground motions in Fig. 3.9b. The reduction in yield strength permitted for a specified ductility factor varies with Tn . At the short-period end of the spectrum, Ry tends to 1, implying no reduction. At the long-period end of the spectrum, Ry tends to f-t. In between, Ry varies with Tn in an irregular manner for a single ground motion, but its ensemble median varies relatively smoothly with Tn , generally increasing significantly with 'ÂŁ1 over the acceleration-sensitive spectral region, but only slightly over the velocity-sensitive region and the Td -to-Tf part of the displacementsensitive region; in the period range longer than Tf, Ry decreases as Tn increases and approaches M at very long periods. Based on results similar to those presented in Fig. 3.9b, several researchers have proposed equations for the variation of Ry with 'ÂŁ7 and f-t . The earliest such equation for Ry goes back to the work of

62


(a)

(b)

Spectral Region s Acceleration

Velocity Di splace ment

sensiti ve

sensitive

Acceleration

sensiti ve

sensiti ve

Velocity Displace ment sensiti ve

sensiti ve

20

-

~~

~~

10

<:>::" '

-

... :

5

---

·····-

3

1 1.5

~~2 ~~4

·····.:····

~ ~8

2

d

0.5 0.02

0.1

10

0.02

0.1

Period Tm sec

10

50

Period Tn , sec

Figure 3.9. Yield-strength reduction factor Ry for elastoplastic systems as a function of Tn for f.1, = 1, 1.5, 2, 4, and 8; I; = 5%: (a) El Centro ground motion; (b) LMSR ensemble of ground motions (median values are presented).

Veletsos and Newmark [6]: (3 .7) Tb < Tn < Tc' Tn > Tc where the periods Ta, Tb, . .. , TJ separating the spectral regions were defined in Figs. 1.16 and 1.17, and Tc' will become clear later. The basis for the well-known design spectra developed by Newmark and Hall [10], Eq. (3 .7), is plotted for several values of f..L in a Jog-log format in Fig. 3.10, where sloping straight lines are included to provide transitions among the three constant segments.

3. 7 Constant-Ductility Design Spectrum A design spectrum for an inelastic system is a plot of its pseudoacceleration Ay or pseudo-velocity Vy versus the initial elastic period Tn , for selected values of f..L. This spectrum is constructed by dividing the elastic design spectrum by Ry. This procedure is shown schematically in Fig. 3.11 , where the elastic design spectrum a-b- c-d-e- f (Fig. 1.17) is divided by Ry defined by Eq. (3 .7) and Fig. 3.10 to obtain the inelastic design spectrum a' -b' -c' -d' - e' - f '. Detailed

63


10

11 =8 5 11 = 4 "<"'

2

11 = 2 11 = 1.5 11 = 1

T,., sec

Figure 3.10. Yield-strength reduction factor defined by Eq. (3.7) [3].

Elastic desi n spectrum

V = av~igo d

c

,.-_ <!)

-;;; () <f)

bJl 0

V / 11

0

:::.." 0 :::..

Inelastic design spectrum

cG 0

n>;> 6 -o ;::l

<!) <f)

0.,

/

1/33 sec 33Hz

1/8 sec 8Hz

10 sec 1/10 Hz

Natural vibration period T11 (log scale)

Figure 3.11 . Construction of inelastic design spectrum [3].

64

~

33 sec 1/33 Hz


3.------,------,------,------.------.----~

2

Elastic design spectrum

2

3

T"' sec

Figure 3.12 . Inelastic (pseudo-acceleration) design spectrum (84th percentile) for ground motions with ii go = lg, Ugo = 48 in./sec, Ugo = 36 in., and 1-L = 1, 1.5, 2, 4, and 8; 1; = 5% [3].

descriptions of the procedure and illustrative examples are available in textbooks (e.g., [3], Sec. 7.11). Using this procedure and starting with the elastic design spectrum of Fig. 1.18 (also Fig. 1.19 for 5% damping), inelastic design spectra were constructed for several values of the ductility factor, as shown in Fig. 3.12. Ry-J.-L- T11 equations other than Eq. (3.7) can obviously be used to construct the design spectrum.

3.8 Applications of the Inelastic Design Spectrum Spectra such as those described above and the underlying theory provide the basis to address questions that arise in the design of new structures and the safety evaluation of existing structures. Two such applications, structural design for allowable ductility and evaluation of an existing structure, are summarized next for SDF systems; the application to displacement-based structural design is discussed in Sec. 7.12 of [3]. 3.8.1 Structural Design for Allowable Ductility Given the design spectrum, the structural properties T11 and~ , and an allowable ductility factor f-L , the design yield strength and design 65


deformation for an SDF system can be determined from the inelastic design spectrum. Corresponding to Tn , l;, and f.L, the value of Ay I g is read from the spectrum of Fig. 3.12. The minimum yield strength necessary to limit the ductility demand to the allowable ductility factor is given by Eq. (3.6), repeated here for convenience:

/y

Ay

= -w

(3.8)

g

The peak deformation of the inelastic system can also be determined from A y I g using 2 Urn

=

T11 ) f.L ( 2n

(3.9)

Ay

Alternatively, the peak deformation can be determined directly from A, the pseudo-acceleration design spectrum for elastic systems, using the Ry-t.L - Tn relationship: U 111

1 = f.LRy

(T -

11

2 )

2n

(3 .1 0)

A

To illustrate the use of Eqs. (3. 8) to (3 .1 0), we determine the strength and deformation demands for an SDF system with T11 = 1.0 sec and t; = 5% due to ground motions characterized by the elastic design spectrum of Fig. 1.19 scaled by 0.5, corresponding to a peak ground acceleration of 0.5g. Table 3.1 presents the strength and deformation demands for ductility factors of 8, 4, 2, and 1 (an elastic system). These results indicate that if the structure is permitted to yield, its design force is reduced by the factor f.L but the design deformation is unaffected, a conclusion that is valid for systems with an initial elastic period Tn longer than Tc (i.e., in the velocity- or displacement-sensitive regions of the design spectrum). Table 3.1 Strength and Deformation Demands Strength Demand Ductility Factor /L

Deformation u 11, in.

1

0.9

2

0.45 0.225 0.113

4

8

100

50 25 12.5

66

8.8 8.8 8.8 8.8


The practical implication of these results is that a structure may be designed for earthquake resistance by making it very strong, by making it very ductile, or by designing it for economic combination of both properties. For the design spectrum considered, Table 3.1 indicates that if the system selected (T,1 = 1.0 sec and t; = 5%) is designed for strength Ia = 0.9w or larger, it will not yield; therefore, it need not be ductile. On the other hand, if it can develop a ductility factor of 8, it need be designed for only 12.5% of the minimum strength Ia required for elastic behavior. Alternatively, it may be designed for strength equal to 50% of Ia and a ductility capacity of 2; or strength equal to 25% of Ia and a ductility capacity of 4. For some types of structural systems and materials (e.g., masonry bearing-wall construction), ductility is difficult to achieve, and economy dictates designing for larger lateral forces; for others (e.g., steel moment-resisting frames), providing ductility is much easier than providing lateral strength, and the design practice reflects this. If the combination of strength and ductility provided is inadequate, the structure may be damaged to an extent that repair is not economical (see Fig. 3.2), or it may collapse (Fig. 3.3).

3.8.2 Deformation of an Existing Structure The deformation of an existing structure at which its performance should be evaluated can be estimated from the inelastic design spectrum. This procedure is illustrated here for a structure that can be idealized as an SDF system. The initial elastic vibration period Tn and the yield strength IY of the system is determined first from its properties: dimensions, member sizes, and design details (reinforcement in reinforced-concrete structures, connections in steel structures, etc.). For a system with known values of Tn and t;, A is read from the elastic design spectrum and A y is obtained from the known value of I Y, by inverting Eq. (3.8): A

Y -

ly

wjg

(3 .11)

With A and Ay known, Ry is calculated from A

Ry = Ay

(3.12)

Corresponding to this Ry and Tn , fJv is determined from Fig. 3.10 and is substituted together with Ay and T,1 in Eq. (3 .9) to estimate the peak deformation U 111 â&#x20AC;˘

67


3.9 Deformation by Graphical Method In the preceding section we demonstrated that structural deformation can readily be calculated from the structural properties and design spectrum. Alternatively, deformation can be determined by a graphical method [22- 24], an approach that may be of interest in engineering practice. This graphical procedure is summarized next and illustrated by an example. The inelastic design spectrum of Fig. 3.12, scaled by a factor of 0.5 to represent ground motions with peak ground acceleration = 0.5g, which is in the standard Ay versus Tn format, is converted to the Ay versus D format. The peak deformation of the system D Um is determined by using Eq. (3 .9), and the resulting data pairs (A y, D) are plotted in Fig. 3.13 to obtain the demand diagram for inelastic systems, wherein the period values associated with radial lines are noted. Superimposed on the demand diagram in Fig. 3.14 is the force- deformation curve, or capacity curve, for an elastoplastic SDF system with Tn = 1.0 sec and /y = 0.225w , which implies that A y/ g = 0.225 [see Eq. (3.11)]. The capacity curve intersects the demand diagram curves for various ductility values. Dividing the deformation at each intersection point by the yield deformation

=

D, cm

Figure 3.13. Inelastic demand diagram [24].

68


2.-------, --------.--------.--------,-------, 00

1.5

100

D, cm

Figure 3.14. Capacity-demand-diagram method of determining structural deformation [24] .

of the system gives the corresponding ductility factor; its value at each intersection point is noted. At one of the intersection points, the ductility factor calculated from the capacity curve matches the ductility value associated with the intersecting demand curve. This intersection point (shown circled) provides the deformation demand, Um = 22.32 em (8 .8 in.). This is identical to the value in Table 3.1 determined by using Eq. (3.9).

3.10 Inelastic Deformation Ratio Since the early 1990s there has been increasing emphasis on estimating structural deformations and on displacement-based design, which has been advocated as a more relevant and rational approach than traditional strength-based seismic design of structures. This has led to renewed interest in the relationship between peak deformations u 111 and u 0 of inelastic and corresponding linear SDF systems, respectively, a problem first studied by Veletsos and Newmark [6]. If expressed as a function of the initial elastic vibration period Tn and ductility factor f.L , the inelastic deformation ratio can be used to determine the inelastic deformation of a new or rehabilitated structure where global ductility capacity can be estimated. If expressed as a

69


Spect r a l Regions Acceleration sensitive

Velocity Disp lacement sensitive

sensitive

-

~ = 1.5

-

~ =2

Acceleration sensiti ve

Velocity Displacement sensitive

sensitive

Ry -

~=4

~=6

=

1. 5

Rr= 2 Ry = 4 Ry= 6

10

cJ

0.5 t_u.:c.._-'-":.._.._.u_~-'-"'-~L---"..........J L_..c;_._~...c:....__.__.w.~__._j~---'.....----"c..........J 0.5 0.02 0.1 10 50 0.02 0.1 Period T"' sec Period T"' sec

Figure 3.15. Median inelastic deformation ratios elL and c R for elastoplastic systems subjected to the LMSR ensemble of ground motions [25].

function of Tn and yield-strength reduction factor Ry, the inelastic deformation ratio can be used to determine the deformation of an existing structure with known lateral strength. The inelastic deformation ratio will be denoted by Ci-t = umfu 0 or CR = umfu 0 , where the subscripts fL and R represent systems with known ductility capacity fL or known yield strength defined by the reduction factor Ry, respectively; the peak deformations Um and u 0 are determined by numerical solution ofEqs. (3.4) and (1.2). Figure 3.15(a) and (b) present the median values of CI-t and C R, respectively, as a function of T11 , for elastoplastic systems subjected to the LMSR3 ensemble of 20 ground motions; the spectral regions are noted in the plots. In the acceleration-sensitive region, CI-t and CR ::::::: 1 at T11 = Tc , but they exceed unity increasingly for shorter periods and larger fL or Ry, indicating greater inelastic action. For these short-period systems, CI-t and C R are very sensitive to yield strength, increasing as the yield strength is reduced. For very short-period systems (T11 < Ta), even if their strength is only slightly smaller than the minimwn strength required for the structure 3

Seven ensembles of far-fault ground motions, each with 20 records, are mentioned. Assembled by H. Krawinkler and his students, the first group of ensembles, denoted by LMSR, LMLR, SMSR, and SMLR, represent four combinations of large (M = 6.6- 6.9) or small (M = 5.8- 6.5) magnitude and short (R = 13- 30 km) or large (R = 30-60 km) distance.

70


to remain elastic (e.g., Ry = 1.5) , both elL and eR are much larger than unity. In the velocity-sensitive region, e IL and e R c::::::: 1 and are essentially independent of the ductility factor or yield strength. In the displacement-sensitive region, e IL and e R < 1 for systems in the period range Td to TJ , where these ratios decrease as the ductility factor is increased or strength is reduced; however, for systems with periods longer than T1, eIL and e R c::::::: 1 are essentially independent of ductility factor or strength, and both elL and eR= 1 for very longperiod systems, independent of JL or Ry. Results such as these are the basis for the widely used equal-deformation rule (i.e. , u 111 = u 0 ) , which is reasonable for systems in the velocity- and displacementsensitive regions of the spectrum, but not in the acceleration-sensitive regwn. What is the influence of earthquake magnitude and distance on the inelastic deformation ratio? To answer this question, the median eIL is plotted against Tn in Fig. 3.16a for the LMSR, LMLR, SMSR, and SMLR ground motion ensembles. 3 These results indicate that the inelastic deformation ratio is essentially independent of earthquake magnitude and distance; however, it may be different for near-fault ground motions, as will be shown later. What is the influence of soil conditions at the recording sites on the inelastic deformation ratio? To answer this question, the median eIL is presented in Fig. 3.16b for three ensembles of ground motions

(a) Varying Mand R ; J.t = 4, a = O%

(b) Varying site class; 11 = 4, a = 0%

m

5 ~~~~--;======;l LMSR 路 LMLR 路 SMSR SMLR

3

0.5

c

D

c........~~~~~~~~~......_J

0.02

0. 1

10

0.02

Period T, , sec

0. 1

10 Period T, , sec

Figure 3.16. Comparison of CJl. for elastoplastic systems with fL = 4 subjected to (a) LMSR, LMLR, SMSR, and SMLR ground motion ensembles; and (b) site class B, C, and D ensembles [25] .

71

50


(b)

(a)

·"":'

3

..... ,

'~ .·. .' 1\ ··. ' .'-\..'\ . \ . ···~.\

0.5 ~~~~~-~~--~ 0.02 O.l 10 0.02 Period Tn, sec

0.1

10

50

Figure 3.17. Comparison of CJ.L for far-fault (LMSR) and near-fault (NFFN and NFFP) ground motion ensembles plotted versus (a) initial elastic vibration period T,, and (b) normalized period T,, / Tc with fL = 4 [25].

recorded on NEHRP site classes B, C, and D, 4 all of which are firm soil sites. The median CJ-L versus Tn curves for the three site classes are very similar to each other and to the LMSR result. Thus, the inelastic deformation ratio is essentially independent of local soil conditions as long as they are firm soil sites, but it may be affected by soft soil conditions, such as in parts of Mexico City and around the margins of San Francisco Bay. The median inelastic deformation ratios C J-L (and C R) for nearfault (NF) ground motions 5 are significantly different than those for far-fault (FF) motions, e.g. , LMSR ensemble (Figs. 3.17a and 3.18a). This systematic difference between the values of CJ-L (and C R) for NF and FF ground motions, especially in the acceleration-sensitive spectral region, is due primarily to the differences between the spectral shapes and values of Tc for the two types of excitations (Section 1.6); recall that Tc is the period separating the acceleration- and 4

5

Assembled by E. Miranda, the second group of three ensembles are categorized by NEHRP site classes B, C, or D. These ground motions were recorded during earthquakes with magnitudes ranging from 6.0 to 7.4 at distances ranging from 11 to 118 km. The two ensembles ofNF ground motions, denoted by NFFN and NFFP, are the two horizontal components [fault normal (FN) and fault parallel (FP)] of 15 NF ground motions, recorded during earthquakes of magnitudes ranging from 6.2 to 6.9 at distances ranging from 0 to 9 krn.

72


(b)

(a)

100 \ \ \

I \

"'

\

' II

\..)

= "'

10

I

'0 2

\

<l)

'' \ 0.5 0.02

I

0.1

0.02

10

Period T11 , sec

0.1

10

50

T"/Tc

Figure 3.18. Comparison of C R for far-fault (LMSR) and near-fault (NFFN and NFFP) ground motion ensembles plotted versus (a) initial elastic vibration period T,, and (b) normalized period T,, I Tc with f.L = 4 [25].

velocity-sensitive spectral regions (Fig. 1.13). This assertion is demonstrated by plotting the ensemble median of the individual ground motion data for ett (and e R) as a function of the normalized vibration period Tn j Tc (Figs. 3 .17b and 3 .18b ). Now the inelastic deformation ratio plots for FF ground motions and both (i.e., FN and FP) components ofNF ground motions have become very similar in all spectral regwns. 3.10.1 Estimating the Deformation of Inelastic Systems Simplified equations for inelastic deformation ratios ett and e R can facilitate estimation of the deformation of an inelastic SDF system because deformation of the corresponding linear system is readily known from the elastic design spectrum [Eq. (1.13)]. Such an equation for ett could be used to determine deformation of a new or rehabilitated structure where the global displacement ductility capacity can be estimated. Similarly, an equation for eR could be used to determine the deformation of an existing structure with known lateral strength. Such equations have been developed by several researchers (e.g., [25] and [26]). In particular, two equations have been developed for ett and eRas functions of Tn/Tc, and f..L or Ry, respectively, which are valid for NF ground motions recorded on soil or rock and FF ground motions associated with a wide range of earthquake magnitudes and distances recorded on NEHRP site classes B, C, and D [25].

73


4 INELASTIC MULTISTORY BUILDINGS

As demonstrated in Chapter 2, the complex dynamics of multistory buildings requires that they be idealized as systems with multiple degrees of freedom for analysis of their response to earthquakes. Because most buildings are designed for base shear smaller than the peak base shear computed under the assumption that the structure is Iinearly elastic (Fig. 3.1 ), the earthquake response of multistory buildings deforming into their inelastic range is of central importance in earthquake engineering. This introductory presentation is limited to two aspects of this vast subject. Identified first are the differences in methodologies for analyzing elastic versus inelastic MDF systems and in the response of inelastic SDF versus MDF systems. Next, we demonstrate that the inelastic response of buildings is strongly influenced by assumptions in idealizing or modeling the structure, by second-order P - 11 effects of gravity loads acting on the laterally deformed state of the structure, and by the detailed variation of ground motion with time. These factors are much more influential in the response of structures responding into the inelastic range than in those remaining elastic.

4.1 Nonlinear Response History Analysis The stiffness term in the equations of motion for an elastic MDF system [Eq. (2.1)] is modified to recognize inelastic behavior of a building. The force- deformation relation for each structural member undergoing cyclic deformations is now nonlinear and hysteretic. The initial loading curve is nonlinear at larger amplitudes of deformation, and the unloading and reloading curves differ from the initial loading branch. Based on experimental data, force- deformation relations appropriate for various structural elements (beams, columns, 75


wall, braces, etc.) and structural materials (steel, reinforced concrete, masonry, wood, etc.) have been developed but are not included here. The relationship between lateral forces fs at the N floor levels and resulting lateral floor displacements u is no longer single valued but depends on the history of displacements; thus, fs = fs (u, sign u)

(4.1)

With this generalization for inelastic systems, Eq. (2.1) becomes mii +cia+ fs (u , sign it) = -mtiig(t)

(4.2)

where m, c, and t are as defined in Section 2.1. This matrix equation represents N nonlinear differential equations for the N floor displacements tlj (t) , j = 1, 2, ... , N. Given the structural mass matrix m , damping matrix c, inelastic force- deformation relation fs (u, sign u), and ground acceleration iig(t) , numerical solution ofEq. (4.2) gives the displacement response of the structure, and internal forces can be determined from the displacements. Formulation of the nonlinear differential equations, in particular the fs (u , sign it) term, and their numerical solution pose several challenges. The structural stiffness matrix must be reformulated at each time instant from the element stiffness matrices corresponding to the deformation and state of each structural element- whether it is on the initial loading, unloading, or reloading branches of the element force-deformation relation- and this process must be repeated for thousands of structural elements in a large building. Numerical solution ofEq. (4.2) is computationally demanding for large (number of DOFS) inelastic systems because these coupled differential equations must be solved simultaneously; for inelastic systems they cannot be uncoupled by using modal concepts. Such numerical solutions must be repeated at every time step b..t , which must be very short, short enough to ensure that the numerical procedure converges, remains stable, and gives accurate results. A vast body ofliterature, including entire books, major chapters of several books, and research papers, on numerical methods to solve nonlinear differential equations, such as Eq. (4.2), is available. The literature includes mathematical development of the methods; their accuracy, convergence, and stability properties; and their computer implementation. Selected methods in the context of structural dynamics are presented in several books, including Clough and Penzien [12], Chaps. 7 and 15; Humar [27], Chaps. 8 and 13; and Chopra [3],

76


Chaps. 5 and 15. Several commercial software packages that implement these methods are available to compute the inelastic response of multistory buildings to earthquake ground motions.

4.2 Why Modal Analysis Is Not Applicable The classical modal analysis procedure is based on the property that the force vector P eff,n (f)-the nth-mode component of the effective earthquake forces [Eq. (2.6)]- produces building response only in its nth natural mode of vibration but no response in any other mode. For linearly elastic systems this property can be proven analytically and was confirmed numerically in Fig. 2.6 for the SAC- Los Angeles 9-story building. However, this property is not valid for structures responding beyond their elastic range, as demonstrated by the response of the same building to ground motion intense enough to cause significant yielding of the structure. Figure 4.1 shows that the response of the building to the force vector Peff,n (t) is due primarily to the nth mode but that other modes contribute to the response. The second, third, and fourth modes start responding to excitation P eff, 1 (t) the instant the structure first yields (Fig. 4.1 a). Similarly, the first, third, and fourth modes start responding to excitation Peff,2 (t) the instant the structure first yields (Fig. 4.1 b) . Although the natural vibration modes of the elastic system are no longer uncoupled if the system responds in the inelastic range, modal coupling is weak. In the structural response due to Peff, 1(t), the contributions to roof displacement of the second, third, and fourth modes are only 6, 3, and 2%, respectively, of the first-mode response (Fig. 4.1a). In the structural response to Peff,2 (t), the contributions to roof displacement of the first, third, and fourth modes are 25, 13 , and 2%, respectively, of the second-mode response (Fig. 4.1 b). This weak coupling of modes permitted development of the modal pushover analysis (MPA) procedure to include contributions of modes higher than the first mode in nonlinear static procedures for estimating seismic demands for inelastic systems [ 14].

4.3 Strength Demands for SDF and MDF Systems What base-shear yield strength is required in a multistory building to keep the earthquake-induced ductility demand in every story below a selected value? To address this question, previously considered by Krawinkler and Nassar [28], we examine the story ductility 77


150 ,----

(a) peff,l = - s 1 x LA25 - - - ' , . - - - - -- ----, Mode 1

(b)

p eff,2 =

- s2 X

LA25

E u~ 0~~~~~-~~~~ ;::

- 150

73 .962

'-----~--~--__j

:~':1 4~28 - 150

30.---

Mode 2

E ~

0~~~~~~~~~~

"'

;::-

16.5 16

L---~--~--____J

:~': 1

-30 '-----~--~-------'

3 2393

Mode

3 1 :"

:

I

- 30

Modo

Ll5 l

â&#x20AC;˘

Mode

'-----~--~-------'

'I :! I :

- 30 5 10 Time, sec

15

3!

7

3

- 150 0

0, 2

- 150

:~':I

-------. Mode 2

o.m

Mode

'I

L---~--~--__j

0

Figure 4 .1. Response to modal component Peff,n (t) tive earthquake forces : (a) n = l ; (b) n = 2 [52].

5 10 Time, sec

= -s

11

15

ii g(t) of effec-

demands for a building with base-shear strength, given by

Ay Vby = - W g

(4.3)

where W is the total weight of the building and A y is the pseudoacceleration corresponding to a selected SDF-system ductility factor, JL , and known initial elastic vibration properties of the fundamental mode: natural period T 1 and damping ratio t; 1â&#x20AC;˘ The pseudo-acceleration is determined from the mean inelastic response spectrum for an ensemble of ground motions. Before presenting the ductility demands for the multistory building, we note that the mean ductility demand imposed by the ensemble of ground motions on the corresponding SDF system will be essentially identical to the ductility selected (Chapter 3). The weight of this SDF system is W, its yield base shear is

78


20 16 SDF-system ductility factor

.... 12 0 0

~

8 4

G

0

4

12

8

16

20

Story ducti lity factor

Figure 4.2. Mean story ductility demands for beam-hinge and column-hinge models of a 20-story frame due to an ensemble of 15 ground motions, compared with the SDFsystem ductility factor fh = 8. (Data from [29].)

defined by Eq. (4.3), and its initial elastic period ~1 = T, and damping ratio l; = ;;,. For multistory buildings, however, the ductility demands differ from the selected SDF-system ductility factor f.L , and vary over the height. Figure 4.2 shows the mean values (over an ensemble of ground motions) of story ductility factors for two 20-story building frames (plastic hinges form in beams for the beam-hinge model and in columns for the column-hinge model) with base shear strength defined according to Eq. (4.3) for f.L = 8. It is clear that story ductility demands differ from the SDF-system ductility factor and are not constant over height because of the more complex dynamics ofMDF systems responding in several "modes" of vibration. The first-story ductility demand (often the largest among all stories) increases with the fundamental period, Tt (or the number of stories), and may exceed the SDF -system ductility factor. It is also affected by the plastic hinge mechanism, as distinguished by columnhinge versus beam-hinge models. These trends are shown in Fig. 4.3 , where the median ductility demand in the first story of buildings 2, 5, 10, 20, 30, and 40 stories high is presented as a function offundamental period T1 for f.L = 2 and 8. It is evident from the preceding results (Fig. 4.3) and related observations that unlike SDF systems, the base-shear yield strength determined from Eq. (4.3) is not sufficient to limit the story ductility 79


30.-----~----~------~----~----~

,.... .....,CH,J..l=8 /

/

BH, J..l=8

OL-----~----~------~----~----~

0

0.5 1.5 2 Fundamental vibration period T" sec

2

5

10

20 30 Number of stories

2.5

40

Figure 4.3 . Mean first-story ductility demands for beamhinge and column-hinge models of2-, 5-, 10-, 20-, 30-, and 40-story frames designed for SDF-system ductility factors p., = 2 and 8. (Data from [29] .)

demands in a multistory building below the SDF-system ductility factor. Thus, the base shear yield strength Vby for SDF systems needs to be increased to account for two factors that exist in MDF systems: multimode effects and height-wise variation of ductility demand. The modification factor ( Vby )MDF ...;- ( Vby )sDF, where ( Vby )MDF and ( Vby )sDF are the base-shear yield strengths ofMDF and SDF systems, respectively, varies between 1 and 2.5 for the examples considered, increases with T1 (or number of stories) and the ductility factor, and is also influenced by the plastic hinge mechanism (Fig. 4.4).

4.4 Influence of Inelastic Behavior on Story Drifts The distribution of story drifts over the height of a multistory frame also depends on how far the frame deforms into the inelastic range, as demonstrated in Fig. 4.5. Presented are the median values of story drift demands due to the LMSR ensemble of20 ground motions for five 9-story frames, designed for base shear given by Eq. (4.3), where Ay is chosen to correspond to the SDF-system ductility factor f.L = 1, 1.5, 2, 4, and 6; included are story drifts for the frame assumed to be linearly elastic. The story drift demands and their height-wise variation for inelastic systems differ from those of elastic systems and 80


3 ~

.s""

,--.._"'

2

.....,

-

~

.....

+

~

""

,__2.

--

...,CH, 11=8 ,.....0-

BH, 11=8 -oCH, 11=2 BH, 11=2

.s

~

'--'

0

0

0.5 1.5 2 Fundamental vibration period Tl> sec 2

10

5

20 30 Number of stories

2.5

40

Figure 4.4. Modification factor to obtain the base-shear yield strength of multistory frames from the base-shear yield strength of the corresponding SDF system for beamhinge and column-hinge models. (Data from [29].)

9.-----~----~----~------~----, ~

\

I

8 7 6

Q5

Elastic

0

Vi 4

f..L = 1

3

2

1.5 2 4

1

6

GL-----~----~----~------~--~

0

0.2

0.4

0.6

0.8

Story drift,% Figure 4.5. Variation of story drift demands in 9-story frames designed for different values of the SDF-system ductility factor J.1- [32] .

81


depend significantly on the ductility factor, a measure of the degree of inelastic behavior. The story drifts increase at the upper stories of the elastic frame, where the response contributions from higher vibration modes are known to be significant (Section 2.5). As the ductility factor, f-1,, increases (i.e., the strength of the frame decreases, implying a larger degree of inelastic action), the drifts in upper stories decrease and the largest drift occurs near the base of the structure.

4.5 P-A Effects The second-order effect of the downward gravity loads acting on the laterally deformed state of a structure, known as P-!J. effects, can profoundly influence the pushover curves and earthquake response of buildings in their inelastic range, as demonstrated by Gupta and Krawinkler [30]. With or without these effects, Fig. 4.6 shows pushover curves- plots of the base shear Vb (normalized by the total weight W) against roof displacement (normalized by building height)- for the SAC- Los Angeles 20-story building determined by nonlinear static analysis in which lateral forces with specified heightwise distribution are applied and are gradually increased to push the building to large displacements. P - !J. effects reduce the initial elastic stiffness of a structure slightly and will therefore have little influence on the earthquake response of the structure if it remains elastic during the design ground motion. However, P-!J. effects have a profound influence on the post-yield response, which now displays a 0.14 0.12

.:c

bJl

'C)

0.1

....

~

P-!1 effects excluded

+ 0.08

~

<)

..0 C/)

0.06

<) C/)

"'

CQ

0.04

"

0.02 0

2 4 3 Roof displacement+ building height, %

Figure 4.6. Pushover curves for the SAC- Los Angeles 20-story building with and without P - f>.. effects [30].

82

5


~--------------------------------------------------------------~~~

15 'cf.

.:c 10 OJ)

.ii)

..c

P-l'l effects included

Q

2CJJ +

5

垄::

路c '0

Q

0

0

Ul -5

0

10

20 30 Time, sec

40

50

Figure 4.7. Importance of P - 8. effects on the secondstory drift of the SAC- Los Angeles 20-story building due to LA30 ground motion. (Adapted from [30].)

short constant-strength plateau at a reduced yield strength, followed by a rapid decrease in lateral force resistance represented by the negative slope of the curve, culminating in zero lateral resistance at a roof displacement of 4%; in contrast, the post-yield stiffness remains positive if P - j'j. effects are ignored. These profound differences in the post-yield static behavior of the building suggest that P-j'j. effects should also be important in a building's response to earthquake excitation. This expectation is confirmed in Fig. 4.7, where the response history of interstory drift (normalized by story height) in the second story of the building due to one of the SAC ground motions is presented for two cases: P - j'j. effects included or excluded. When these effects are included, after the first episode of major yielding the story drift grows in one direction without any reversal toward the opposite lateral direction, resulting in dynamic instability. In contrast, analysis excluding these effects predicts oscillatory response that remains bounded. Clearly, it is essential to include P - j'j. effects in predicting the earthquake response of buildings deforming significantly into their inelastic range.

4.6 Influence ofModeling Assumptions on Seismic Demands The earthquake response of a building can be influenced significantly by the assumptions in modeling (or idealizing) the structure for computer analysis. To demonstrate this possibility, three different

83

.


15 ::?. 0

.:c

OJ)

路a;

Model M2

Model Ml

10

..c

>..

'-<

2 "'+

5

<:::::

I

路c "0

>.. .....

r-ModelM2A I ,/\/\/'/~~/\/~ ..1

0

0

U5 -5

0

10

20

30 Time, sec

40

50

Figure 4.8. Influence of modeling assumptions on the second-story drift in SAC-Los Angeles 20-story building due to LA30 ground motion. (Adapted from [30].)

planar idealizations of a perimeter frame of the SAC- Los Angeles 20-story building were considered by Gupta and Krawinkler [31]: (1) model M1, a basic centerline model in which the panel zone size, strength, and stiffness are not represented; (2) model M2, a model that explicitly incorporates the strength, and stiffness properties of panel zones; and (3) model M2A, an enhanced version of model M2, which includes the interior gravity colunms, shear connections, and floor slabs. The earthquake response of a building may be profoundly affected by the differences in these analytical models, as demonstrated by the response history of second-story drift due to one of the SAC ground motions: the LA30 (Tabas) record (Fig. 4.8). The Ml model predicts that after the first large inelastic excursion, the story drift will not reverse direction and will continue to grow rapidly so that the building would become dynamically unstable within the first 20 seconds of the excitation. This early instability does not occur in the M2 model, but the subsequent (after 20 seconds) ground motion, although weaker, causes the drift to grow to near dynamic instability of the frame. When other sources of stiffness and strength are considered (model 2A), the response is radically different. After the first large inelastic excursion, the story drift now recovers partially and oscillates about a shifted position, showing no signs of dynamic instability of the structure. It is evident that dynamic response is extremely sensitive to modeling assumptions once P-6. effects 84


20 16 .... 12 0 0

i:L:

8

4

G

0

2

4

10 6 8 Story drift, %

12

14

16

Figure 4.9. Influence of modeling assumptions on the story drift demands for the SAC- Los Angeles 20-story building due to LA30 ground motion; results are shown for M2 and M2A models, but model Ml predicted collapse of the building. (Adapted from [31].)

become important and a story is deformed into the range of negative post-yield stiffness. Consequently, the story drift demands for a building may be affected profoundly by the modeling assumptions. This is evident in Fig. 4.9, where the peak values of story drifts for the same building due to LA30 ground motion are presented. No results are shown for model M1 because it predicted collapse of the building. Model M2 predicts story drifts approaching 15%, which are so large that performance of the building would not be acceptable. However, the most realistic model (M2A) predicts much smaller story drifts, with the largest drift among all stories near 5%.

4. 7 Statistical Variation in Seismic Demands The story drift demands are also sensitive to the time variation of ground acceleration in addition to assumptions in structural modeling demonstrated in Section 4.6. This is evident from the peak values of story drifts in the M2 model of the same 20-story building due to 20 SAC ground motions (Fig. 4.1 0). The story drift demands imposed by the 20 individual ground motions vary widely, implying that the

85


20 16 ..., 12 0 0

G::

8

4 G

0

2

4

10 8 Story drift, %

6

12

14

16

Figure 4.10. Story drift demands for the SAC- Los Angeles 20-story building due to 20 SAC ground motions. (Data provided by Akshay Gupta.)

response to any one excitation should not be the basis for designing new buildings or evaluating existing buildings. The seismic demands due to a large enough number of ground motions must be determined and their statistical variability considered in selecting demand values for the design and evaluation of structures. The median 1 and 84th percentile2 values of the story drift demands for the M2 model of the structure are presented in Fig. 4.11 for three ensembles of 20 ground motions for Los Angeles representing different probabilities of exceedance: 2% in 50 years (return period of 2475 years), 10% in 50 years (return period of 475 years), and 50% in 50 years (return period of72 years). As the intensity of the ground motion ensembl~ increases, both the median and 84th percentile values of story drift demands become larger, as expected; more important, dispersion of the demand increases. The excitation-to-excitation variability in response is larger for the most intense ensemble of ground

1

2

Median refers to the exponent of the mean of the natural log values of the data set. The 84th p ercentile is the median multiplied by the exponent of the dispersion, where dispersion is determined as the standard deviation of the natural log values of the data set.

86


20 Median 50/50 -A- 10/50 -e- 2/50 84th Percentile -o - 50150 -6. - 10/50 -o- 2/50 ---â&#x201A;Ź--

16

..., 12 0 0

~

8

-o-

4

-

--e

"\ -~

G-

G 2

0

4 Story drift, %

0

6

8

Figure 4.11. Median and 84th percentile values of the story drift demands for the SAC- Los Angeles 20-story buildings for three ensembles of ground motions [31] .

motions because they cause story drifts large enough for P - 11 effects to be important.

4.8 Story Drift Demands The story drift demands and their distribution over height are strongly dependent on structural and ground motion characteristics. Figure 4.12 presents median demands for SAC buildings designed for Los Angeles, Seattle, and Boston, due to ground motion ensembles (a)

(b) 20

9

~LA

16 6

- - SE .....,.... BO

12

0'""" 0

~

8

3

4 G

G 0

2 3 Story drift, %

4

5

0

2 3 Story drift, %

4

5

Figure 4.12. Median story drift demands for the SAC- Los Angeles (LA), Seattle (SE), and Boston (BO) buildings due to the 2/50 ground motion ensemble: (a) 9-story buildings; (b) 20-story buildings. (Adapted from [31].)

87


representing 2% probability of exceedance in 50 years at each location. The story drifts vary considerably among the structures, from essentially within the elastic range for Boston buildings to moderately beyond yielding for Seattle buildings to far into the inelastic range for Los Angeles buildings. The story-drift increases in the upper stories of Boston buildings, is concentrated in the upper stories of Seattle buildings, and in the lower stories of Los Angeles buildings.

88


PART Ill: BUILDING DESIGN CODES AND EVALUATION GUIDELINES


5 STRUCTURAL DYNAMICS IN BUILDING DESIGN CODES

Most seismic building codes require that structures be designed toresist static lateral forces determined from the properties of the building and the seismicity of the region. Based on an estimate of the fundamental natural vibration period of the structure, formulas are specified for the base shear and the distribution of lateral forces over the height of the building. Static analysis of the building for these forces provides the design forces, including shears and overturning moments for each story, with some codes permitting reduction for statically computed overturning moments. The lateral forces specified in the 2003 International Building Code [20] are presented first in this chapter, followed by a discussion of how they are related to structural dynamics concepts presented in Chapters 1 to 4; pertinent comments on three other codes [33- 35] are included. Modern codes also permit linear dynamic analysis proceduresboth response spectrum analysis (RSA) and response history analysis (RHA)- and nonlinear RHA for all structures. They require dynamic analysis- linear or nonlinear- for longer-period or irregular buildings, and nonlinear RHA for base-isolated buildings. The California Building Code requires dynamic analysis of hospital structures. The code versions of these procedures are not included here because they are essentially equivalent to those that were presented earlier in this monograph.

5.1 International Building Code (United States) In the 2003 Edition of the International Building Code (IBC), the earthquake-induced forces for many buildings may be computed by the equivalent lateral force (ELF) analysis procedure described next. 91


5.1.1 Base Shear The 2003 edition of the International Building Codes (IBC) specifies the base shear as 1 (5.1) where W is the total dead load and applicable portions of other loads, and the seismic coefficient (5.2) The coefficient corresponding to R elastic seismic coefficient:

= 1 will

be referred to as the

Ce = IC

(5.3)

where the importance factor I = 1.0, 1.25, or 1.5; I = 1 for most structures, I = 1.25 for structures that have a substantial public hazard due to occupancy or use, and I = 1.5 for "essential facilities" that are required in post-earthquake recovery and for facilities containing hazardous substances. The period-dependent coefficient C depends on the location of the structure and the site class, and is related to ordinates of the pseudo-acceleration design spectrum: A(Tn ,short) , the pseudoacceleration at short periods; and A(Tn = 1.0 sec), the pseudoacceleration at 1.0 sec. Maps of the United States show these two A values for ground motion due to the maximum considered earthquake (MCE), which are m~ltiplied by 2/3 to obtain A values for the design basis earthquake (DBE). Consider, for example, a location representative of coastal California regions not in the near field of known active faults. For site class B, the maps provide A(Tn ,short)-:-g = 1.5 and A(Tn = 1.0 sec)-:- g = 0.6, which are multiplied by 2/3 to obtain 1.0 and 0.4, respectively, for the DBE. The numerical coefficient C is then g1ven as

c --

1.0 { 0.4jT,

r,

< 0.4

Tt :": 0.4

(5.4)

where T 1 is the fundamental natural vibration period of the structure in seconds. 1

To keep it consistent with Chapters 1 to 4, the notation used is not the same as that used in the IBC.

92


l.2 - , - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,

0.8

-'2fl

'"0..."'

I I 0.6 I A/g

c..3' 0.4

0.2

0 0

2

3

4

5

T1 or Tm sec

Figure 5.1. IBC (2003): elastic seismic coefficient Ce and pseudoacceleration A/g for a location in California coastal regions not in the near field of known active faults [3].

Figure 5.1 shows this elastic coefficient for the above-mentioned example location and site, class, and I = 1, which is valid for most structures. The code also specifies (for use in dynamic analysis) the elastic pseudo-acceleration design spectrum that is the basis for defining C. This spectrum, for the example location considered, is also shown in Fig. 5.1, where Ajg is plotted against Tn, the natural vibration period of an SDF system. The response modification factor R depends on several factors, including the ductility capacity and inelastic performance of structural materials and systems during past earthquakes. Specified values of R vary between 1.25 (for certain inverted pendulum systems) and 8 (for special moment-resisting frames, steel eccentrically braced frames, and certain dual systems with special detailing for ductile behavior).

5.1.2 Lateral Forces The distribution of lateral forces over the height of the building is determined from the base shear in accordance with the formula for 93


N

Wj

j

FJ

wz

2

Fz

WJ

I

FI

/

vb Figure 5.2 . IBC lateral forces.

the lateral force at the jth floor (Fig. 5.2):

FJ

=

Vb

WjhY

k (5.5) Li=l w;h; where w; is the weight at the ith floor, h; is the height of the ith floor above the base, and k is a coefficient related to the vibration period r, as follows: N

r, _:: : o.5

r, < 2.5 r, : : : 2.5

0.5 <

(5 .6)

5.1.3 Story Forces According to the ELF analysis procedure, the design values of story shears and story overturning moments are determined by static analysis of the structure subjected to these lateral forces; the effects of gravity and other loads should be included.

5.2 Relation between IBC and Dynamic Analysis The ELF analysis procedure reduces a complex nonlinear dynamic analysis to a simple linear static analysis. By necessity, many compromises are introduced to develop such a simplified approach. However, several of the code provisions are related to the theory of structural dynamics. In this section we discuss this interrelationship 94


and evaluate how well the seismic forces in building codes agree with the results of dynamic analysis presented in Chapters 2 and 4.

5.2.1 Base Shear The elastic seismic coefficient Ce in the IBC is related to the pseudo-acceleration spectrum for linearly elastic systems. In a linear SDF (one-story) system of weight w, the peak value of base shear is [see Eq. (1.12)] A

vb = -w g

(5.7)

and the peak base shear in a multistory building due to its nth mode of vibration [see Eqs. (2.16) and (2.21)] is

Vbn

=

An * -Wn g

(5.8)

w,:

where is the effective weight and A 11 jg the normalized pseudoacceleration, both for the nth mode. For buildings, the IBC gives the design base shear as (5.9) Specializing this equation for R = 1, Ce in the IBC corresponds to A j g, the pseudo-acceleration for linearly elastic systems normalized with respect to gravitational acceleration. The significance of contributions of higher vibration modes in the earthquake response of buildings (Section 2.5) plays a central role in interpreting the code formulas for lateral forces. Based on Fig. 2.12, we had concluded that the base shear for buildings with Tt in the acceleration-sensitive region of the spectrum is essentially all due to the first mode. However, for buildings with T1 in the velocityor displacement-sensitive regions of the spectrum, the higher-mode response can be significant, increasing with increasing T1 and with decreasing beam-to-column stiffness ratio p, for reasons discussed in Section 2.5. For buildings with Tt in the acceleration-sensitive region of the spectrum, these results and Eq. (5.8) indicate that the code formula, Eq. (5.9), would predict the base shear for elastic buildings consistent with structural dynamics theory if the seismic coefficient Ce were defined as A 1 jg and the total weight W were replaced by the first-mode effective weight Wi. If W is used instead of Wi, as in building codes,

95


RSA, 5 modes

p=O 0.1 p = 1/8

p = oo

0.01L_L-LL~UL~~~_L~~~--~~~~~--------~

0.02

0.1

10

Figure 5.3. Base shear Vb (normalized by total weight W) in buildings computed for the design spectrm11 shown; results are presented for three values of p [3].

the base shear is overestimated. This become obvious by normalizing the base shear data of Fig. 2.12 with respect to the total weight, as shown in Fig. 5.3; the overestimation varies with p. For buildings with T 1 in the velocity- or displacement-sensitive spectral regions, however, the increase in base shear by using the total weight of the building may not be sufficient to compensate for the higher-mode response. This is clear from Fig. 5.3 , where we observe that Vb I W exceeds A 1I g for longer T1 and smaller p. Recognizing such results, the IBC does not permit the use of the ELF analysis procedure for buildings with T1 exceeding 3.5Tc, where Tc is the period separating the acceleration- and velocity-sensitive regions of the spectrum; Tc = 0.66 sec for the design spectrum of Fig. 1.18.

96


3 RSA , p = 0

:?

,.... ' NBCC

r:/)

c

.. · · ·. EC

OJ)

---""'

+ 2

/,•

.-<•'

·X· -

~

""

~

IBC

..... 0 ,.....__

.., Ul

- ·- ·- ·- .- .- .- .-

-o 0

\

~ OJ)

---""'

...

/

RSA, p = 1/8 MFDC RSA , p = co IBC

+

u'" 0 0

2

3

4

5

T 1, sec

Figure 5.4. Comparison of Ce --;- A jg in four building codes and V6 j Wj --;- A jg from RSA for three values of p [3]. An alternative to such a restriction would be to specify the seismic coefficient larger than the design spectrum A I g in the velocity- and displacement-sensitive regions of the spectrum. The required increase can be determined from results of dynamic analysis. It is given by the ratio VbI Wj -:- A I g of the two values of base shear, the first including responses of all modes and the other considering only the first mode (Fig. 2.12). The curves plotted in Fig. 5.4 for this ratio represent data obtained from Fig. 2.12 but curve-fitted as described in Section 21 .6 of[3]. The degree to which the spectrum should be raised depends on the fundamental vibration period and the beam-to-column stiffness ratio p ; the spectrum needs to be raised very little for shear buildings (p = oo) but to an increasing degree with increasing frame action or wall-type behavior (i.e. , decreasing p). Multiplying the seismic coefficient from the raised spectrum by the first-mode effective weight Wj instead of the total weight W would give base shear consistent with structural dynamics theory over a wide period range and would avoid overestimation of base shear in the acceleration-sensitive spectral region and underestimation in the velocity- and displacement-sensitive regions. Such an approach

97


would extend the range of applicability of the ELF analysis procedure to structures responding significantly in higher vibration modes. Alternatively, these higher-mode responses can rationally be included by dynamic analysis of the structure as required by the IBC. In contrast to the IBC, where the design spectrum Alg is not raised and the ELF analysis procedure is restricted to short-period buildings, several other codes raise the spectrum, as shown in Fig. 5 .4, where the ratio Ce --:- Alg from four building codes is included as a function of T 1â&#x20AC;˘ Three of these codes- the National Building Code of Canada (NBCC) [33] , the Mexico Federal District Code (MFDC) [34], and Eurocode 8 (EC) [35]- specify that Ce = Alg only at very short periods (less than about 0.5 sec), but that Ce exceeds Algincreasingly as the period gets longer. To recognize the dependence of higher-mode response on the fundamental period T1 and stiffness ratio p , building codes should define Ce in terms of the design spectrum A I g, which in turn should be specified explicitly. Once this transparent format is adopted, Ce can be defined by raising the spectrum in its velocity- and displacement-sensitive spectral regions, based on dynamic response results of the type presented in Fig. 2.12, and multiplied by Wj to estimate the base shear. The inelastic response behavior of multistory buildings and its differences relative to SDF systems should also be considered in specifying the seismic coefficient in building codes. This important concept is illustrated by returning to Fig. 4.4 from nonlinear dynamic analysis, showing the ratio of base-shear yield strengths in multistory buildings and SDF systems necessary to limit the ductility demand to the same allowable value. Superimposed on these results in Fig. 5.5 is the ratio Ce --:- A I g for four building codes; the code ratios are independent of the allowable reduction factor R. In contrast, dynamic response results indicate that the strength increase required to account for MDF effects depends significantly on the ductility factor and failure mechanism.

5.2.2 Design Force Reduction Building design codes specify the design base shear to be smaller than the elastic base shear (determined using the elastic seismic coefficient Ce). The reduction factor R in building codes used to determine the design strength of the building is intended to encompass several factors, including the yield-strength reduction factor Ry (introduced in Section 3.2 in the context of SDF systems), the difference between design strength and yield strength (because of many sources of

98


u..

Cl

~(/)

~ '-' +

2 .------ -----,-----,------ ----,--- ---r--- ----, BH, J..L=8 EC ... 路

路:...;..: 'NBCC BH, J..L=2

路 - MFDC

r-~~~~=-~-------- IBC

\.3'

OL_----~----~----~------~--~

0

0.5 1.5 2 Fundamental vibration period T 1 , sec

2

5

10

20 30 Number of stories

2.5

40

Figure 5.5. Comparison of Ce -:- Ajg in four building codes and (Vby)MoF/(Vby )SDF from nonlinear RHA of beam-hinge and column-hinge models of frames for two values of SDF-system ductility factor. Dynamic analysis data are from [29].

overstrength in actual buildings not explicitly considered in design), and the performance of different structural systems and materials during past earthquakes. Despite these conceptual differences between Rand Ry, the Ry in Fig. 3.1 0, determined from dynamic response analysis of yielding SDF systems, provides useful guidelines for specifying R in codes. As shown in Fig. 5.6, Ry increases from Ry = 1 at T, 1 = 0 to Ry = J-L (selected as 4) in the velocity- and displacement-sensitive spectral regions. However, the R factor in the IBC is independentofT1 (Fig. 5.6), which contradicts dynamic response results for structures with their fundamental period in the acceleration-sensitive region of the design spectrum. In contrast, several other codes, including MFDC and EC8, specify reduction factors that vary with vibration period in a manner generally consistent with structural dynamics theory (Fig. 5.6). The discrepancy in the design spectra resulting from a periodindependent R factor is seen in Fig. 5.7 for two values of the ductility factor J-L . The inelastic design spectra shown are from Fig. 3.12, scaled

99


5,------------------------------------------,

.B (!!?G._an..拢 N~Qg_

_ . -r---------------i

I

I ~"'

0

Ry

Q'(MFDC) 2

'"""'

0.5

2 T1 or T"' sec

5

10

20

50

Figure 5.6. Comparison of yield-strength reduction factor in four building codes and for Ry for elastoplastic SDF systems, f.L = 4 [3].

1.2.-------------------------------------------.

\

'

\

\

bJl

~ ""<:

'

- - Inelastic design spectrum _ - - Elastic design spectrum + J..l

\

0.8

'

J..l = 2\

0.6

'路,

/ Elastic design spectrum 路....: J..l=l

路,

0.4

_____

0.2

..

__ _. __ _ --- . _

o~~~~==~~~~~~~~ 0

2

3

4

5

Figure 5.7. Comparison of an inelastic design spectrum and an elastic design spectrum reduced by the period-independent factor f.L; results are presented for f.L = 2 and 8 [3].

100


by 0.4 so that they correspond to peak ground acceleration ii go = 0.4g. The elastic design spectrum reduced by the period-independent factor f.L is lower in the acceleration-sensitive period region. Thus, by ignoring the period dependence of the yield-strength reduction factor, the IBC gives a design force smaller than that from structural dynamics theory for structures in this period region. However, the code value may not be unconservative because overstrength is generally more significant in short-period buildings. 5.2.3 Lateral Force Distribution Structural dynamics gives the base shear and equivalent static lateral force at floor level j for mode n of a multistory building (Sections 2.3 and 2.4): (5.10) Using the definitions of M~ and f expressed in terms of Vb n:

f jn

11 ,

Eqs. (2.17) and (2.4), f; n can be Wjc/Jjn

= Vbn ------;N ,..,.-'--"--

(5.11)

Li = l Wic/Jin

The lateral force distribution specified in the IBC is intended to represent the inelastic dynamic response of the building but is independent of the R factor or the degree of inelastic action. Although the code forces should not be distributed strictly according to the dynamic response of elastic systems, comparison of the two provides an understanding of how higher-mode response is considered in code force distribution. For this purpose we now compare the force distribution ofEq. (5.11) from structural dynamics with code specifications. The IBC with k = 1 in Eq. (5 .5) gives FJ

=

Vb

w 1 h路1 N Li= l Wihi

(5.12)

This force distribution agrees with Eq. (5.11) if垄J11 were proportional to hJ, that is, if the mode shape varied linearly with height above the base of the building. The linear shape is a reasonable approximation to the fundamental mode of many buildings. In the IBC the height-wise distribution oflateral forces is given by Eq. (5 .5), based on the assumption that lateral displacements are proportional to h J if T1 _::: 0.5 sec, to h] if T 1 ::: 2.5 sec, and to an intermediate power of hJ for T1 between 0.5 and 2.5 sec. These 101


(a) T1 = 0.5 sec

(b) T1 = 3sec

Ratio oflateral force to base shear FJ/Vb (codes) and Jj !Vb (RSA)

Figure 5.8. Comparison of equivalent static force distributions in four building codes and from RSA for three values of p for two values of T1: (a) T1 = 0.5 sec; (b) T1 = 3.0 sec; (Adapted from [3]).

variations in force distributions are intended to recognize the changing fundamental mode and increasing higher-mode contributions to response with increasing T1. Figure 5.8 compares the distribution of lateral forces specified in the IBC and three other building codes with those computed from dynamic analysis (RSA). 2 Included are results for two values of T1 = 0.5 and 3 sec, chosen to be representative of two spectral regions. Note that for easier visualization, the numerical values of the equivalent static forces that are concentrated at floor levels have been joined by straight lines. For buildings with T1 in the acceleration-sensitive region of the spectrum, the height-wise distributions oflateral forces specified by the four codes are essentially identical to each other and fall between the dynamic response curves for p = 0 and oo. With increasing T1 , the code distributions for lateral forces differ among codes and all four codes differ increasingly from dynamic response. These differences are especially significant for the smaller values of 2

The equivalent static forces were computed from the story shears determined by RSA, as the differences between the shears in consecutive stories (equal to the discontinuity in shears at the floor levels).

102

r


p because higher-mode response increases with increasing T 1 and

decreasing p (Section 2.5). This discrepancy is avoided in the IBC by not permitting the ELF analysis procedure for buildings with T 1 longer than 3.5Tc. Story overturning moments determined by static analysis of buildings subjected to lateral forces specified in codes [Eq. (5.5) for the IBC] usually exceed the results of elastic dynamic analysis ([3] , Sec. 21.8), and could therefore be reduced. The 2003 edition of the IBC does not permit any reduction in overturning moments (except at the foundation level), whereas several codes, including the 2000 IBC, NBCC, and MFDC, specify a reduction factor by which statically computed overturning moments may be reduced. The IBC 2003 provision may be reasonable in light of the results of nonlinear dynamic analyses by Medina and Krawinkler [36], which do not show such a reduction in overturning moments in the inelastic range of behavior, and to ensure conservative estimates of design force in columns, obviously critical to the stability of buildings.

103


6 STRUCTURAL DYNAMICS IN BUILDING EVALUATION GUIDELINES

A major challenge for performance-based seismic engineering is to develop simple, yet sufficiently accurate methods for analyzing designed structures and evaluating existing buildings to meet selected performance objectives. As reflected in post-1995 guidelines for evaluating existing buildings, such as the FEMA-273 [37], its successor FEMA-356 [38], and ATC-40 [39] documents, the profession has shifted away from the traditional elastic analysis of structures subjected to seismic forces reduced to recognize indirectly inelastic response; instead, inelastic behavior of structures is considered explicitly in estimating seismic demands at low performance levels, such as life safety and collapse prevention. In this chapter, selected aspects of the above-mentioned guidelines for computing seismic demands are discussed in light of structural dynamics theory presented in Chapters 1 to 4. These guideline documents include linear dynamic analysis procedures that are generally consistent with those described in Chapter 2 and therefore not discussed here.

6.1 Nonlinear Response History Analysis Currently, building evaluation guidelines permit use of two nonlinear analysis methods to estimate seismic demands: the nonlinear static procedure (NSP) and the nonlinear dynamic procedure (NDP); the latter is essentially equivalent to nonlinear response history analysis (RHA) mentioned in Chapter 4. The FEMA-356 specifications for the NDP, which are the same as those in the International Building Code [20] and ASCE-7 -02 [40], state that the seismic demand may be estimated as ( 1) the maximum of demands due to three ground motions, or (2) the mean value of demands due to seven ground motions. 105


(a)

(b)

80

40

s

()

q::"

60

~

~

57.0

53.3

r--

21.4

24.1

-

-

40

if>

~ 20 Q)

p..

0

n

2 Trial No.

12.7

r--

0 3

2 Trial No.

3

Figure 6.1. First-story drift: (a) maximum of demands due to three excitations; and (b) average of demands due to seven excitations. The excitations were selected randomly three times from an ensemble of 17 excitations.

These estimates can vary widely, as demonstrated next for the SAC- Los Angeles 9-story building subjected to an ensemble of 20 SAC ground motions; nonlinear RHA predicted collapse of the building during three of these excitations. The nonlinear RHA results for first-story drift led to a mean value of 20.4 em over 17 excitations (excluding three that caused collapse of the building). 1 The results, shown in Fig. 6.1, demonstrate a large variation in the drift estimated by three implementations of both versions of the FEMA-356 criteria. Such wide variability obviously implies that different engineers following the same criteria could arrive at contradictory conclusions about seismic safety and rehabilitation requirements for an existing building. This observation points to the need for better criteria to select and scale ground motions, better in the sense that variability in structural response should be reduced to permit the use of a smaller number of records to estimate seismic demands. Several efforts are in progress to find improved ground motion intensity measures (e.g., the pseudoacceleration spectral ordinate at fundamental period of the building, inelastic deformation spectral ordinates, combinations of spectral values at modal periods [41 ]). 1

Ignoring the three collapses in computing the mean is strictly incorrect. Working with the median value would be better, but the mean of the data for 17 excitations was used to conform to FEMA-356 guidelines.

106


Seismic demands computed by nonlinear analysis procedures depend significantly on assumptions in preparing an inelastic model of the building and software used in implementing the computation. For example, three different models of the SAC- Los Angeles 20-story building predicted widely different response to the LA30 ground motion: two models predicted story drifts approaching 5 and 15%, and the third model predicted collapse of the building (Fig. 4.9); and the story drifts computed by analyzing the same model by three widely used computer programs differed by up to 30%. Such variability implies that requirements for inelastic modeling valid at large structural deformations should be established and robust computer programs should be developed. With these developments, nonlinear analysis procedures would provide more realistic and reliable estimates of seismic demands. At the present time, nonlinear RHA is an onerous task, for several reasons. First, an ensemble of site-specific ground motions compatible with the seismic hazard spectrum for the site must be simulated. Second, despite increasing computing power, inelastic modeling and nonlinear RHA remains computationally demanding, especially for unsymmetric-plan buildings- which require three-dimensional analysis to account for coupling between lateral and torsional motionssubjected to two horizontal components of motion. Third, such analyses must be repeated for several excitations because of the wide variability in demand due to plausible ground motions (see Fig. 4.10), and the statistics of response must be considered. Fourth, the structural model must be sophisticated enough to represent a building realistically, especially deterioration in strength at large displacements. Fifth, commercial software is so far not robust enough to predict response with high reliability. Sixth, an independent peer review of the results of nonlinear RHA is required by the FEMA-356 guidelines, adding to the project duration and cost. With additional research and software development, most of the preceding issues should be resolved, and nonlinear RHA may eventually become the dominant method in structural engineering practice. Opinions within both the research and professional communities differ on whether nonlinear RHA and the implementing software are ready for practical application. Even if nonlinear RHA is ripe for application, it is unreasonable to require this onerous procedure for every building- no matter how simple- and of every structural engineering office- no matter how small. Therefore, nonlinear static procedures are expected to remain important in structural engineering 107


practice. Such simplified methods must be rooted in structural dynamics theory and their underlying assumptions and range of applicability identified. Nonlinear RHA may be employed for final evaluation of those combinations of buildings and ground motions where a simplified procedure begins to loose its accuracy.

6.2 Current Practice According to the NSP described in FEMA-273/356 and ATC40 guidelines, seismic demands are computed by nonlinear static analysis of a structure subjected to monotonically increasing lateral forces (known as pushover analyses) with a specified, usually invariant height-wise distribution until a predetermined target displacement is reached; supplementary elastic analyses with relaxed acceptance criteria are required for structures with significant higher-mode responses, but the NSP is still permitted in FEMA-356. The target displacement is estimated from the deformation of an inelastic SDF system derived from the pushover curve by either an iterative procedure, requiring analysis of a sequence of equivalent linear SDF systems [39], or by empirical equations based on RHA of a large number of inelastic SDF systems [38]. In light of structural dynamics theory (Chapters 1 to 4), issues related to estimation of target displacement are examined in Section 6.3 and 6.4, and pushover analysis is discussed in Section 6.5.

6.3 SDF-System Estimate of Roof Displacement How well can the roof displacement u,. of a multistory building be determined from the deformation of an SDF system? To address this question, we compare the values of roof displacement determined by two methods: the "exact" value (u,. )MDF, determined by nonlinear RHA of the multistory building treated as an MDF system; and the SDF-system estimate: (u,.)sDF = r,¢r l D I , where r 1 was defined in Eq. (2.4), ¢r 1 is the value at the roof in the first mode lP 1, and D 1 is the peak deformation of the inelastic SDF system with its forcedeformation relation determined from the pushover curve obtained by nonlinear static analysis of the building using the first-mode force distribution: sj = mlP 1 [see Eq. (2.6)]. D , is determined by nonlinear RHA of the SDF system, thus avoiding any of the approximations

108

(


underlying the simplified methods for estimating its deformation (see the FEMA-273/356 or ATC-40 documents). The response of the SAC- Los Angeles 3-, 9-, and 20-story buildings to each of 20 SAC ground motions is computed and the displacement ratio is determined: (u:.)SDF = (ur)sDF-:- (ur)MDF 路 The difference between the median of this displacement ratio and unity indicates the bias in the SDF -system estimate of median roof displacement. The SDF system estimates the median roof displacement of multistory buildings to a useful degree of accuracy. Figure 6.2 shows histograms of the values of the displacement ratio, together with its

(b)

(a) 15

15 CCl

"' 10

"0 :..; 0

(.)

(.)

"'

"'....

:..;

0

0

c

4-<

4-<

z

a,

"' 10

~

"0 .... 0

Range: 0.63- 1.65

"' '6

0

z0

5

0 0

0.5

1.5

::8"'

5

0

2

0

0.5

l.5

2

(u,~) s oF

(u,~)sm

(c) 15

Range: 0.82- 1.49

a,

-

"E"' 10 0 (.)

c

~

"' '6

4-<

0

0

z

::8"'

5

0 0

0.5

1.5

2

(u,~)sm

Figure 6.2. Histograms ofratio (u; )soF for SAC-Los Angeles buildings: (a) 3-story; (b) 9-story; (c) 20-story. The range of values and median value of this ratio are noted [42].

109


range of values and the median value for each of the three SAC- Los Angeles buildings. 2 The SDF system estimates the median roof displacement accurately, within 3% of the "exact" value for the 3-story building, but overestimates it by 19% for the 9- and 20-story buildings. The median roof displacement of taller buildings is not always overestimated by the SDF system; for example, it is underestimated by 14 and 18% for the SAC- Boston 9- and 20-story buildings, respectively, and by 5% for the SAC-Seattle 9- and 20-story buildings [42]. The bias in the SDF-system estimate of median roof displacement depends on the vibration properties of the building and how far it is deformed into the inelastic range. It would be useful to develop correction factors to eliminate this bias. The SDF system may not estimate the roof displacement of multistory buildings due to individual excitations to a useful degree of accuracy. This estimate can be alarmingly small (as low as 63% of the exact value in Fig. 6.2 for the Los Angeles buildings and as low as 31% of the exact value among the nine SAC buildings [42] or surprisingly large (as large as 165% of the exact value in Fig. 6.2 for the Los Angeles buildings and as large as 215% of the exact value among the nine SAC buildings [42]). The errors are actually worse than indicated by Fig. 6.2 because it does not include those cases where nonlinear RHA predicted collapse of the first-"mode" SDF system but not of the building as a whole. This large discrepancy arises because for individual ground motions the SDF system may significantly underestimate or overestimate the yielding-induced permanent drift in the response of the building [42].

6.4 Estimating Deformation of Inelastic SDF Systems As mentioned earlier, seismic demands are estimated in current engineering practice by pushover analysis up to a target displacement of the roof, determined from the deformation D of an inelastic SDF system. The methods described in the ATC-40 and FEMA-356 guidelines are commonly used to determine D.

2

Data for excitations that caused collapse of the SDF system are excluded, reducing the number of data to 17 for the Los Angeles 9-story building, and 14 for the Los Angeles 20-story building; the median values for these buildings are computed by the counting method.

110

(


6.4.1 ATC-40 Method The deformation of an inelastic SDF system is estimated by the capacity-spectrum method, an iterative method requiring analysis of a sequence of equivalent linear systems; the method is typically implemented graphically. Unfortunately, the ATC-40 iterative procedure does not always converge; when it does converge it underestimates by as much as 40 to 50% the deformation over a wide range of periods [43]. The two flaws in the ATC-40 capacity spectrum method-lack of convergence in some cases and large errors in many cases- appear to have been rectified in the FEMA-440 report [44]. It derives the optimal vibration period and damping ratio parameters for the equivalent linear system by minimizing the differences between its response and that of the actual inelastic system. Such an equivalent linear method would obviously give essentially the correct deformation. However, the benefit in making the equivalent linearization detour is unclear when the deformation of an inelastic system can be determined readily from the inelastic design spectrum (Section 3.8) or by using available equations for the inelastic deformation ratio (e.g., [25] and [26]). As mentioned earlier, the capacity-demand-diagram method described in Section 3.9 gives results identical to the theoretical results of Section 3.8. This method is graphically similar to the capacity spectrum method in ATC-40, thus retaining its attractive features , but the two differ in an important sense; the demand diagram used is different: the constant-ductility demand diagram for inelastic systems in the capacity-demand-diagram method instead of the elastic spectrum for equivalent linear systems in ATC-40. 6.4.2 FEMA-356 Method The deformation of an inelastic SDF system is estimated by 2

D

=

11 C 1C2C3 ( Trr ) 2

A

(6.1)

Multiplying the deformation of the elastic system [given by Eq. (1.13)] are three coefficients, C 1, C2 , and C3. The coefficient C 1 represents the inelastic deformation ratio, u 111 ju 0 [see Section 3.10] for inelastic systems without pinching, stiffness degradation, or strength deterioration of their hysteresis loop. The coefficient C 2 accounts for the increase in deformation of the inelastic system due to these effects not considered in C 1, and C3 accounts for P-6. effects.

111


Equations and numerical values for these coefficients specified in the FEMA-356 guidelines are based on research results and on judgment. However, some of the numerical values are not supported by research results; for example, C1 is limited to 1.5, which is much smaller than the inelastic deformation ratio determined from dynamic response analyses for systems in the acceleration-sensitive region of the spectrum (Fig. 3.15); however, the value of C1 = 1.0 at longer periods is theoretically correct (Section 3.10). As part of the FEMA440 project, coefficients C1 and Cz were investigated and improved specifications were developed for these coefficients [44] .

6.5 FEMA-356 Nonlinear Static Procedure The nonlinear static procedure in FEMA-356 requires development of a pushover curve, a plot of base shear versus roof displacement, by nonlinear static analysis of the structure subjected first to gravity loads, followed by monotonically increasing lateral forces with a specified invariant height-wise distribution. At least two force distributions must be considered. The first is to be selected from among the following : first-mode distribution, equivalent lateral force (ELF) distribution, and RSA distribution. The second distribution is either the "uniform" distribution or an adaptive distribution; several options are mentioned for the latter, which varies with change in deflected shape of the structure as it yields. The other four force distributions mentioned above are defined as follows: l. First-mode distribution: sj = m1¢ J 1 , where mJ is the mass and ¢ J 1 is the mode shape value at the jth floor. 2. Equivalent lateral force (ELF) distribution: sj = m 1 where hJ and k are as defined in Section 5 .1. 3. RSA distribution: s* is defined by the lateral forces backcalculated from the story shears determined by response spectrum analysis of the structure, assumed to be linearly elastic (Section 2.4). 4. "Uniform" distribution: sj = m1 . Each of these force distributions pushes the building in the same direction over the height of the building (Fig. 6.3). If the higher modes of vibration contribute significantly, as defined in FEMA-356, to the elastic response of the structure, the NSP must be supplemented by linear dynamic analysis (LDP), and seismic demands computed by the two procedures are evaluated against their

hJ,

112

(


(a)

(b)

(d)

(c)

.------0.367 /

0.119

.177

0.0197 0.00719

Figure 6.3. FEMA-356 force distributions for Los Angeles 9-story building: (a) first mode, (b) ELF, (c) RSA, and (d) "uniform."

respective acceptance criteria. The SAC 9- and 20-story buildings fall into this category [45]. The limitations of the FEMA-356 force distributions are demonstrated in Figs. 6.4 and 6.5, where the resulting estimates of the median story drift and plastic hinge rotation demands imposed on the SAC- Los Angeles buildings by the ensemble of20 SAC ground motions are compared with the "exact" median value determined by nonlinear RHA. The target displacement for FEMA analysis was not determined using Eq. (6.1) but was calculated accurately to ensure a meaningful comparison of the two sets of results. The FEMA-356lateral force distributions provide a good estimate of story drifts for the 3-story building. However, consistent with earlier results (Fig. 2.14) for elastic buildings, the first-mode force distribution grossly underestimates the story drifts, especially in the upper stories of the 9- and 20-story buildings, showing that higher-mode contributions are especially significant in the seismic demands for upper stories . Although the ELF and RSA force distributions are intended to account for higher-mode responses, they do not provide satisfactory estimates of seismic demands for buildings that remain essentially elastic (Boston buildings) or buildings that are deformed far into the inelastic range (Los Angeles buildings) [45] . The "uniform" force distribution seems unnecessary because it grossly underestimates drifts in upper stories and grossly overestimates them in lower stories. Because FEMA-356

113


(a)

(b) 9

3

h

-Nonlinear RHA FEMA $ ~ 1st Mode \ 2 JJ- oELF \ \ \

0 0

~

-

'

~9RSA

\

JJ-

~路~

41\ ~ ~~~

~~..!\,-

6

....

' &..," '.....~

0 0

oUniform

~

.....~

,.......

3

.... . G

'----------'--

0

~

---'---_L_-----'---'--_J

2

3

4

5

"

b

6

Story drift ~NL-RHA or ~FEMA' %

Story drift ~NL-RHA or ~F EMA' % (c)

20

16 .... 12 0 0

~

8 4

G

0

2

3

4

5

6

Story drift ~NL-RHA or ~FEMA' % Figure 6.4. Median story drifts for SAC- Los Angeles bu il dings: (a) 3-story; (b) 9-story; (c) 20-story. Determined by nonlinear RHA and four FEMA-356 force distributions: first mode, ELF, RSA, and "uniform" [45].

requires that seismic demands be estimated as the larger of results from at least two lateral force distributions, it is useful to examine the upper bound of results from the four force distributions considered. This upper bound also significantly underestimates drifts in upper stories of taller buildings but overestimates them in lower stories (Fig. 6.4). The FEMA-356 lateral force distributions provide a good estimate of plastic hinge rotations for the 3-story building, but either fail to identify or significantly underestimate plastic hinge rotations in beams at the upper floors of9- and 20-story buildings (Fig. 6.5). Many

114


(a)

(b)

3

....

9

6

2

....

0 0

0 0

~

~

3

G~----~----~----~

0

0.02

0.04

0.06

Beam plastic rotation, rad

Beam plastic rotation, rad

(c) 20~~----~~7.'--~~

16 ....

12

0

8

-Nonlinear RHA FEMA - 路 lst Mode 11- aELF

~ ~RSA 11-

oUniform

0.02

0.04

4

0.06

Beam plastic rotation, rad

Figure 6.5. Median plastic hinge rotations in interior beams of SAC- Los Angeles buildings: (a) 3-story; (b) 9-story; (c) 20-story. Determined by nonlinear RHA and four FEMA-356 force distributions: first mode, ELF, RSA, and "uniform" [45].

discussions of pushover analysis and its potential and limitations are available in the literature (e.g., [46)-[ 49]).

6.6 Improved Nonlinear Static Procedures It is clear from the preceding discussion that the seismic demand estimated by NSP using the first-mode force distribution (or others in FEMA-356) should be improved. One approach to reducing the discrepancy in this approximate procedure relative to nonlinear RHA is to include the contributions ofhigher modes of vibration to seismic

115


demands. Recall that when higher-mode responses were included in the response spectrum analysis (RSA) procedure, improved results were obtained for linearly elastic systems (Figs. 2.9 and 2.1 0). Although modal analysis theory is strictly not valid for inelastic systems, the fact that elastic modes are coupled only weakly in theresponse of inelastic systems (Fig. 4.1) permitted extension of the modal pushover analysis procedure (MPA), first mentioned in Section 2.4.4, to inelastic systems. In the MPA procedure, the peak response of the building to Peff,n(t) defined in Eq. (2.6)- or the peak "modal" demand r 11 - is determined by a nonlinear static or pushover analysis using the modal force distributions~ = mif> 11 up to the target displacement determined from the deformation of the nth-"mode" inelastic SDF system; and the peak modal demands r11 are then combined by an appropriate modal combination rule (Section 2.4.2). The original version of the MPA procedure [14] has been improved, especially in its treatment of P - 1:1 effects due to gravity loads and calculation of plastic hinge rotations [45] and has been extended to compute member forces [50]. This procedure has been shown to estimate seismic demands better than FEMA-356 procedures for the SAC 9- and 20story buildings [45] and generic frames (vertically "regular" as well as vertically irregular) of height varying from 3 to 18 stories [51]. Modal pushover analysis has also been extended to estimate seismic demands for unsymmetric-plan buildings, which respond in coupled lateral-torsional motions during earthquakes [52]. Higher-mode force distributions, which push some floors and pull others (Fig. 2.4), may reveal plastic hinge mechanisms for the building that are not detected by the first-mode distribution [53] or other FEMA-356 force distributions, which push all floors in the same direction (Fig. 6.3). Other approaches have been developed to account for the contributions of higher modes to response, redistribution of inertia forces because of structural yielding, and the associated changes in the vibration properties of the structure. Adaptive force distributions that attempt to follow more closely the time-variant distributions of inertia forces have been proposed [48,54,55]. An incremental response spectrum analysis procedure (IRSA) has been developed that requires at each incremental loading step a response spectrum analysis of the structure in its current yielded state, treating it as linearly elastic until the next step [56]. Although these procedures may provide good estimates of seismic demands for the examples considered, their accuracy remains to be evaluated for a wide range of buildings and ground motion ensembles. Although these procedures are notable 116


research contributions, they may be too complicated conceptually for implementation in structural engineering practice. Furthermore, the accuracy of most pushover analysis procedures becomes questionable for buildings subjected to very intense ground motions that deform them far into the region of negative post-yield stiffness, with significant deterioration of structural strength. For such cases, pushover procedures may have to be abandoned in favor of nonlinear RHA.

117


REFERENCES [1] Hudson, D. E. (1979). Reading and Interpreting Strong Motion Accelerograms, Proceedings, Earthquake Engineering Research Institute, Berkeley, Calif. [2] Shakal, A. F., Huang, M. J., and Grazier, V. M. (2003). "Strongmotion data processing," in International Handbook of Earthquake and Engineering Seismology, W H. K. Lee, H. Kanamori, P. C. Jennings, and C. Kisslinger, Eds., Academic Press, London. [3] Chopra, A. K. (2001). Dynamics ofStructures: Theory and Applications to Earthquake Engineering, 2nd ed. , Prentice Hall, Upper Saddle River, N.J. , 844 pp. [4] Hausner, G. W (1959). "Behavior of structures during earthquakes," ASCE, J. Engrg. Mech., EM4:109- 129. [5] Biot, M. A. (1933). "Theory of elastic systems under transient loading with an application to earthquake proof buildings," Proceedings, Nat!. Acad. Sci., 19:262-268. [6] Veletsos, A. S., and Newmark, N. M. (1960). "Effect of inelastic behavior on the response of simple systems to earthquake motions," Proceedings, 2nd World Conf. Earthq. Engrg., Japan, 2:895- 912. [7] Veletsos, A. S. (1969). "Maximum deformation of certain nonlinear systems," Proceedings, 4th World Conf. Earthq. Engrg., Santiago, Chile, 1:155- 170. [8] Chopra, A. K., and Chintanapakdee, C. (2001). "Comparing response of SDF systems to near-fault and far-fault earthquake motions in the context of spectral regions ," Earthq. Engrg. Struc. Dyn., 30:1769- 1789. [9] Riddell, R. , and Newmark, N . M . (1979). "Statistical analysis of the response of nonlinear systems subjected to earthquakes," Structural Research Series 468, University ofillinois at UrbanaChampaign, Urbana, Ill.

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J

)

NOTATION Abbreviations

ATC CQC DOF EC ELF FEMA FF FN FP IBC MDF MFDC MPA NBCC NDP NF NSP RHA RSA SDF SRSS UBC Subscripts eff

g

Applied Technology Council complete quadratic combination degree of freedom Eurocode 8 equivalent lateral forces Federal Emergency Management Agency far fault fault normal fault parallel International Building Code multi -degree-of-freedom M exico s Federal District Code modal pushover analysis National Building Code of Canada nonlinear dynamic procedure near fault nonlinear static procedure response history analysis response spectrum analysis single-degree-of-freedom square root of the sum of squares Uniform Building Code

effective ground 125


y

floor number; story number; DOF peak value for inelastic systems; maximum natural; mode number peak value yield

Superscripts st t

static total

i, j

m

n 0

Roman Symbols A A(t)

pseudo-acceleration spectrum ordinate pseudo-acceleration pseudo-acceleration spectrum ordinate A(T,1 , sn) pseudo-acceleration of nth-mode SDF system 2

WnU Y

c c D Dn Dn(t)

Dy Eso eJ

FJ

In fJn

Is

numerical coefficient, UBC elastic seismic coefficient seismic coefficient inelastic deformation ratio, fixed f-t inelastic deformation ratio, fixed R damping coefficient damping matrix deformation spectrum ordinate deformation spectrum ordinate D(T,l> sn) deformation of nth-mode SDF system yield deformation spectrum ordinate maximum strain energy error in static response code lateral force at floor j natural frequency (Hz) lateral force : floor j, mode n elastic or inelastic resisting force ; equivalent static force 126


)

fso, fo

/y fn, fno

hj h~ I k k

Len m m Mb Mbo Mbn(t)

Mz~ M~

N Peff

Peff

qn (t) Ry r(t)

Sjn

s

peak value of fs(t) yield strength peak value offn(t) height of jth floor effective modal height, mode n importance factor, IBC stiffness or spring constant stiffness matrix see Eq. (2.17) mass mass matrix mass at ith floor base overturning moment peak value of Mb(t) Mb(t) due to mode n nth modal static response Mb effective modal mass, mode n number ofDOFs effective earthquake force effective earthquake force vector nth modal coordinate yield-strength reduction factor any response quantity peak value of r n (t) nth modal contribution factor r(t) due to mode n peak value of r n (t) peak value of r (t) static response to forces s nth modal static response jth element of sn spatial distribution of forces defined by Eq. (2.6) defined by Eq. (2.27) 127


Ta, Tb , Tc, Td, Te, TJ Tn u

Ug Ug Ug Ugo Ug o Ug o

u j (t)

time periods that define spectral regions natural period of SDF system; nth natural period ofMDF system displacement; deformation; displacement relative to ground displacement vector total displacement ground displacement ground velocity ground acceleration peak ground displacement peak ground velocity peak ground acceleration relative displacement of floor j max lu(t)l for an inelastic system t

u,.

Uy

v vb Vbn (t) Vbn Vbo Vby v:st

bn Vy

w W*11

w Wi

roof displacement peak or maximum value ofu (t) u(t) due to mode n yield deformation pseudo-velocity response spectrum ordinate base shear Vb(t) due to mode n peak value of Vbn(t) peak value of Vb(t) yield strength value of Vb nth modal static response Vb W 11 U y

total weight of building; total dead load and applicable portion of other loads effective modal weight, mode n weight of SDF system weight at ith floor

128


Greek Symbols

p 6.; " st

Djl1

Pin

spectral amplification factors see Eq. (2.4) damping ratio damping ratio for nth mode influence vector beam-to-column stiffness ratio jth-story deformation or drift nth modal static response 6. J cross-correlation coefficient for modes i and n jth element of ~ 11 roof element of ~ 11 nth natural vibration mode natural frequency of SDF system; nth natural frequency ofMDF system ductility factor

129


Chopra earthquake dynamics of structures