tJ)F/tU+- 780085

PB 283 705A Report on Research Sponsored by

NATIONAL SCIENCE FOUNDATION(RANN)

Grant No. ENV74-14766

STRUCTURAL WALLS IN

EARTHQUAKE-RESISTANT BUILDINGS

Dynamic Analysis of

Isolated Structural Walls

PARAMETRIC STUDIES

by

Arnaldo T. Derecho

Satyendra K. Ghosh

Mohammad Iqbal

George N. Freskakis

Mark Fintel

Submitted byRESEARCH AND DEVELOPMENT

CONSTRUCTION TECHNOLOGY LABORATORIESPORTLAND CEMENT ASSOCIATION

5420 Old Orchard RoadSkokie, Illinois 60077

March 1978

50272 -101

REPORT DOCUMENTATION Il~REPORT NO_

PAGE NSF/RA-7800854. Title and Subtitle 5:' RePort Dat~ ,-

March 1978

7. Author(s)

Portland Cement Association5420 Old Orchard RoadSkokie, Illinois 60077

--- --------------111. Contract{C) or Grant(G) No.

(e) ENV7414766(G)

----------------1

Applications (ASRA)Applied Science and ResearchNational Science Foundation1800 G Street, N.W.WR<hington, 0 C ..2Q5."-5u.O '

15. Supplement"" Note' ----------

1--------------------~----------__+---------------___112. Sponsoring Organization Name and Address 13. Type of Report & Period Covered

(1974-1978)Final--- - ------ ----1

14_

This is the second of a series of four report, on the Analytical Investigation onEarthquake-Resistant Structural Walls and Wall Systems

--------------·16. Abstract (Limit: 200 words)

The results of a parametric study to identify the most significant structural andground motions parameters, as these affect the dynamic inelastic response of isolatedstructural walls, is presented. The parameters examined include intensity, durationand frequency content of the ground motion and the following variables relating to thestructure: fundamental period, yield level in flexure, yield stiffness ratio, unloadingand reloading parameters cliaracterizing the decreasing stiffness hysteretic loopassumecfor the model, damping, stiffness and strength taper along height of structure, and basefixity condition. Results are presented in the form of maximum response envelopes forhorizontal and interstory displacements, bending moment, shear, rotational ductilityand cumulative plastic hinge rotations as well as time-history plots of response.Results of the study indicate that the most important parameters are the fundamentalperiod, yield load, and the earthquake intensity.

17. Document Analysis 3. Descriptors

EarthquakesSeismic wavesCriti ca1 frequenci es

IntensityDynamic structural analysisDynamic response

Earthquake resistant structuresDesign criteria

b. Identifiers/Open-Ended Terms

c. COSATI Field/Group

18. Availability Statement

NTIS.

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CAPiTP,~ SYSTeMS G,'''-!~'', .6110 EXECUTIVE GOULEVAf,1)

SUITE 250ROCKVILLE, MP,RYI.AND 20852

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TABLE OF CONTENTS

SYNOPSIS

BACKGROUND

OBJECTIVES

Dynamic Inelastic Response Analysis

Sectional Analysis Under Combined Flexureand Axial Load

ANALYSIS PROCEDURE - AN OVERVIEW

Structural Model

Input Ground Motions

Response Parameters of Interest

Computer Program for Dynamic Analysis

Preliminary Analysis

Basic Approach to Analysis

Presentation of Results

COMPUTER PROGRAMS

Dynamic Inelastic Analysis of Structures ­Program DRAIN-2D

PRELIMINARY STUDIES

PARAMETRIC STUDIES

DISCUSSION OF RESULTS

SUMMARY AND CONCLUSIONS

ACKNOWLEDGEMENTS

REFERENCES

Page No.

i (l-)

1

5

5

6

8

8

9

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10

11

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12

13

13

19

24

29

62

72

73

SYNOPSIS

Although structural walls have a long history of satisfac­

tory use in stiffening buildings against wind, there is insuf­

ficient information on their behavior under strong earthquakes.

Observations of the performance of buildings during recent

earthquakes have demonstrated excellent behavior of buildings

stiffened by properly proportioned and designed structural

walls. Both safety and damage control can be obtained economi­

cally with structural walls.

The primary objective of the analytical investigation, of

which the work reported here is a part, is the estimation of

maximum forces and deformations that can reasonably be expected

in critical regions of structural walls in buildings subjected

to strong ground motion. The results of the analytical inves­

tigation, when correlated with data from the concurrent experi­

mental program, will form the basis for a design procedure to

be developed as the ultimate objective of the overall inves­

tigation.

This is the second part of a comprehensive report on the

analytical investigation. It discusses the results of para­

metric studies of various structural and ground motion para­

meters. These parameters are examined in terms of their effects

on the dynamic inelastic response of isolated structural walls.

Among the structural parameters considered are fundamental

period, yield level, yield stiffness ratio, character of the

hysteretic force-displacement loop (reloading and unloading

stiffnesses) damping, stiffness and strength taper, and degree

of base fixity. Also considered are the three parameters

characterizing stong-motion accelerograms: duration, intensity,

and frequency content.

-i- (tl

Dynamic Analysis of Isolated Structural Walls

PARAMETRIC STUDIES

by

(I)A. T. Derecho , S. K.

G. N. Freskakis(4)

(2) (3)Ghosh , M. Iqbal ,

and M. Fintel{S)

BACKGROUND

Although structural walls (shear walls)* have a long his­

tory of satisfactory use in stiffening multistory buildings

against wind, not enough information is available on the be­

havior of such elements under strong earthquake conditions.

Observations of the performance of buildings subjected to

earthquakes during the past decade have focused attention on

the need to minimize damage in addition to ensuring the general

safety of buildings during strong earthquakes. The need to

control damage to structural and nonstructural components dur­

ing earthquakes becomes particularly important in hosptials and

other facilities that must continue operation following a major

disaster. Damage control, in addition to life safety, is also

economically desirable in tall buildings designed for residen­

tial and commercial occupancy, since the nonstructural compo­

nents in such buildings usually account for 60 to 80 percent of

the total cost.

There is little doubt that structural walls offer an

cient way to stiffen a building against lateral loads.

effi­

When

{l)Manager, and (3)Structural Engineer, Struc{~fal AnalyticalSection, Engineering (S?eVelopment Department; Senior Struc­tural Engineer, ~~9 Director, Advanced Engineering ServicesDepartment; and Formerly, Senior Structural Engineer, DesignDevelopment Section, Portland Cement Association, Skokie,Illinois.

*In conformity with the nomenclature adopted by the AppliedTechnology Council(l) and in the forthcoming revised edition ofAppendix A to ACI 318-71, "Building Code Requirements for Rein­forced Concrete", the term "structural wall" is used in placeof "shear wall".

-1-

proportioned so that they possess adequate lateral stiffness to

reduce interstory distortions due to earthquake-induced motions,

walls effectively reduce the likelihood of damage to the non­

structural elements in a building. When used with rigid frames,

walls form a structural system that combines the gravity-load­

carrying efficiency of the rigid frame with the lateral-load­

resisting efficiency of the structural wall.

Observations of the comparative performance of rigid frame

buildings and buildings stiffened by structural walls during

recent earthquakes, (3,4,5) have clearly demonstrated the

superior performance of buildings stiffened by properly propor­

tioned structural walls. Performance of structural wall build­

ings was better both from the point of view of safety as well

as damage control.

The need to minimize damage during strong earthquakes, in

addition to the primary requirement of life safety (i.e., no

collapse), clearly imposes more stringent requirements on the

design of structures. This provided the impetus for a closer

examination of the structural wall as an earthquake-resisting

element. Among the more immediate questions to be answered

before a design procedure can be developed are:

1. What magnitude of deformation and associated forces

can reasonably be expected at critical regions of

structural walls corresponding to specific combinations

of structural and ground motion parameters? How many

cycles of large deformations can be expected in criti­

cal regions of walls under earthquakes of average

duration?

2. What stiffness and strength should structural walls,

in representative building configurations, have rela­

tive to the expected ground motion in order to limit

the deformations to acceptable levels?

3. What design and detailing requirements must be met to

provide walls with the strength and deformation capa­

cities indicated by analysis?

-2-

The combined analytical and experimental investigation, of

which this study is a part, was undertaken to provide answers

to the above questions. The objective of the overall investi­

gation is to develop practical and reliable design procedures

for earthquake-resistant structural walls and wall systems.

The analytical program to accomplish part of the desired

objective consists of the following steps:

a. Characterization of input motions in terms of the

significant parameters to enable the calculation of

critical or "near-maximum" response using a minimum

number of input motions(6).

b. Determination of the relative influence of the various

structural and ground motion parameters on dynamic

structural response through parametric studies(7).

The purpose of these studies is to identify the most

significant variables.

c. Calculation of estimates of strength and deformation

demands in critical regions of structural walls as

affected by the significant parameters determined in

Step (b). A number of input accelerograms chosen on

the basis of information developed in Steps (a) and(b) are used(8).

d. Development of procedure for determining design force

levels(8) by correlating the stiffness, strength and

deformation demands obtained in Step (c) with the cor­

responding capacities determined from the concurrent

experimental program(9).

Another important result of the analytical investigation is

the determination of a representative loading history that can

be used in testing laboratory specimens under slowly reversingloads(IO) .

The first phase of this

isolated structural walls.

investigation deals mainly

A detailed consideration of

with

the

dynamic response of frame-wall and coupled wall structures is

planned for subsequent phases of the investigation.

-3-

This is the second part of the report on the analytical

investigation. It discusses the results of parametric studies

of structural and ground motion parameters as they affect the

dynamic inelastic response of isolated structural walls. The

evaluation of the relative influence of these parameters on

dynamic inelastic response represents a major step towards

developing a design procedure for earthquake-resistant struc­

tural walls. By comparing the effects of different variables

on the relevant response quantities, the parametric studies

serve to identify the most significant variables. The subse­

quent development can then be formulated in terms of these major

variables. The material presented here is based on Part B of

Ref. 7 and corresponds to Step (b) listed above.

The first part of the report dealt mainly with the charac­

terization of input motions in terms of duration, intensity and

frequency content, particularly as they related to dynamic in­

elastic response.

The investigation has been supported in major part by Grant

No. ENV74-l4766 from the National Science Foundation, RANN Pro­

gram. Any opinions, findings and conclusions expressed in this

report are those of the authors and do not necessarily reflect

the views of the National Science Foundation.

-4-

OBJECTIVES

Dynamic Inelastic Response Analysis

The major objective of the parametric studies is to evaluate

the relative influence of the various structural and ground

motion parameters on the dynamic inelastic response of struc­

tural walls and wall systems. Effects of selected variables on

response quantities such as maximum total and interstory dis­

placements are examined. Particular attention is placed on the

effects of the parameters studied on the maximum forces (espe­

cially shear) and deformations in critical regions of structural

walls.

The parametric study is necessary in view of the many vari­

ables that affect dynamic structural response to ground motions.

Because of the need to develop a simple design procedure, only

the most significant parameters can be considered in formulating

the design methodology. By identifying the most important vari­

ables, the parametric study allows the subsequent effort, i.e.,

the determination of estimates of force and deformation demands

in critical regions of structural walls, to concentrate on a

manageable few of the most significant parameters. These can

then form the basis for developing the design procedure.

During the first phase of the program, dynamic analyses

were carried out on isolated structural walls, with only

exploratory runs made for frame-wall and coupled wall systems.

Isolated walls were analyzed not only to obtain dynamic response

data on this basic element, but also to establish a reference

with which results for the more complex structural wall systems

can be compared.

In certain cases, the frame in a frame-wall structure or

the coupling beams in a coupled wall structure are relatively

flexible compared to the structural wall. In these cases, the

wall can be considered to act essentially as an isolated struc­

tural element.

-5-

Sectional Analysis Under Combined Flexure and Axial Load

In addition to the parametric study dealing with dynamic

response of isolated walls, a start was made on a parametric

study of wall sections subjected to combined flexure and axial

load. The objective of this analysis was to evaluate the rela­

tive effects of selected design variables on ductility and

other characteristics of the primary moment-curvature (M- ~ )

curve for representative structural wall sections. For the

primary M- ~ curve, the loading applied is an essentially

static, monotonically increasing flexure with constant axial

load.

Among the more important input to the dynamic response

analysis is the force-deformation relationship of members, in

addition to structure dimensions and geometry. The effects of

design variables on dynamic response can thus be examined in

terms of their influence on the primary M-~ curve of critical

sections -- at least to the extent that the primary M-~ curve

affects dynamic response. Among the design variables consi­

dered were:

a. shape of structural wall cross-section

b. percentage of longitudinal reinforcement

c. level of axial load

d. degree of confinement of compression zone concrete

Results of the study of the effects of sectional shape on

the behavior of wall sections under combined bending and axial

load were included in the Progress Report of August 1975(11).

An initial effort also was made toward developing charts

(interaction diagrams) to serve as a basis for proportioning

wall sections in earthquake-resistant structures. Preliminary

charts based on sectional analyses under combined bending and

axial load, covering a wide range of selected parameters, were

presented in Supplement 3(12) to the Progress Report of

August 1975.

of

Because any recommendations on the

structural wall sections will have to

-6-

detailed proportioning

await final reduction

and evaluation of experimental data, it was decided

this work until adequate test data became available.

concern here is the effect of shear and cyclic loading

behavior of structural walls.

The subsequent discussion is concerned mainly

analysis of inelastic dynamic response of structures.

-7-

to defer

Of major

on the

with the

ANALYSIS PROCEDURE - AN OVERVIEW

It was recognized early in the study that if the design

procedure to be developed is to be put in practical form, it

will have to involve only the most important parameters affect­

ing the behavior of structural walls under seismic conditions.

A parametric study considering reasonable variations in what

are thought to be significant structural and ground motion vari­

ables was carried out as a preliminary step to the compilation

of data for the design procedure. Once the most significant

variables are identified, the design procedure can be based on

these few major parameters.

The general procedure followed in the parametric study is

described below.

1. Structural Model. A reference 20-story isolated

structural wall, representing a typical element in a

structural wall or "cross-wall" structure, was

designed on the basis of the 1973 Uniform Building

Code (UBC) (13), Zone 3 requirements.*

A number of other designs were also considered to

determine the practical range of variation of the re­

quired strength (yield level) corresponding to differ­

ent stiffnesses and wall cross sections. Based on

this study, the ranges of variation of the different

structural parameters were established.

Among the more important structural parameters

considered were the fundamental period, Tl , and the

flexural yield level, My' i.e., the force required

to produce first yield in bending. For the purpose of

defining the "first yield", a bilinear idealization

was used to replace the actual curvilinear force-

*It is pointed out that the use of UBC requirements to establishthe dimensions (mainly relating to strength) of the referencestructure has little bearing on the results of the analysis.The principal reason for the use of the UBC provisions (or ofany code provisions for that matter) is to establish the prac­tical range of variation of certain design parameters.

-8-

deformation relationship of structural members. For

the 20-story structure studied, an initial fundamental

period ranging from 0.8 sec. to 2.4 sec., and a yield

level for the base of the wall ranging from 500,000

in.-kips (56,500 kN·m), to 1,500,000 in.-kips (169,500

kN·m) were considered.

2. Input Ground Motions. For input motions, a small num­

ber of accelerograms were selected from among the

recorded and artificially generated accelerograms

catalogued in Ref. 6. These were chosen so that the

ranges of dominant frequencies for the "peaking accel­

erograms,,(6) more or less covered the period range

of interest, i.e., between 0.80 sec. and 3.0 sec.

In studying input motions, the following three

characteristics affecting dynamic structural response

were recognized(6}:

a. intensity - used here as a measure of the ampli­

tude of the large acceleration pulses;

b. duration of the large-amplitude pulses;

c. frequency characteristics.

Except where duration was the parameter of inter­

est, all the structures analyzed were subjected to 10

seconds of ground motion. The input motions were nor­

malized with respect to intensity by multiplying the

as-recorded or as-generated acceleration ordinates by

a factor calculated to yield a "spectrum intensity"*

equal to some percentage of a reference spectrum in­

tensity, SI ref . The reference spectrum intensity

used in this study is that corresponding to the first

10 seconds of the NS component of the 1940 El Centro

record. Factors designed to yield 1.5 SI ref were

used to normalize most of the input accelerograms.

3. Response Parameters of Interest. A major requirement

of the analysis for the dynamic response study was

*Defined here as the area under the 5%-damped relative velocityresponse spectrum between periods of 0.1 and 3.0 seconds.

-9-

duced into theDerecho(15) at

consideration of inelastic deformations at critical

regions of structural members. Economic considerations

in design generally result in structural dimensions

that allow inelasticity to develop when a structure is

subjected to a major earthquake. Consequently, the

magnitude of the inelastic deformations (or the duc­

tility requirement) becomes the principal response

parameter of interest. In addition to the ductility

requirement, especially at the base of the wall, the

maximum horizontal and interstory displacements and

associated maximum forces, i.e., moments and shears,

were noted.

4. Computer Program for Dynamic Analysis. A number of

dynamic inelastic analysis computer programs were

examined and the availability of support for modifica­

tions deemed essential for the planned investigation

were considered. On the basis of these considerations,

the program DRAIN-2D(14) was chosen for this work.

The program has been implemented on the CDC 6400 at

Northwestern University. The present version of the

program, used in the analysis of isolated structural

walls, includes a capability for considering a

'degrading stiffness' model for reinforced concrete

beams, developed by R. W. Litton and G. H. Powell.

Also included is an option to output compact time his­

tories of response. Further modifications were intro-

program by S. K. Ghosh and A. T.

the Portland Cement Association.

These allow plotting of response data. Modifications

to Program DRAIN-2D, done by I. Buckle and G. H. Powell

(August 1976) at the request of PCA, include a model

for a shear-shear slip mechanism at plastic hinges, in

addition to the flexural hinge. An option to consider

a third, descending branch of the primary M-8 and V-y

curves is also included in this latest modification.

-10-

5. Preliminary Analysis. To permit extensive evaluation

of selected parameters, an effort was made to minimize

the cost per analysis without sacrificing accuracy in

the relevant data. To do this, a preliminary series

of analyses was done to examine the possibility of

using a lumped-mass model of an isolated wall with a

lesser number of concentrated masses than floor levels.

Also, the maximum permissible length of time step for

use in numerical integration was evaluated. The ques­

tion of the most appropriate model for the hinging

region at the base of the wall also was considered.

6. Basic Approach to Analysis. The dynamic analyses

assumed structural elements with unlimited deformation

capacity (or ductility). This was done since a major

objective of these analyses is the determination of

the magnitude of deformation requirements correspond­

ing to specific combinations of parameter values.

To isolate the effect of individual variables on

response, only the particular parameter under study

was varied in the reference structure at anyone time.

It is recognized that the process of modifying only

one basic parameter of a structure while suppressing

any change in other parameters* is artificial and may

sometimes result in unrealistic structures. However,

this was found necessary to avoid uncertainties in

evaluating the effect of each variable.

The frequency content of the input motion was

considered first because of the significant effect it

can have on the dynamic response of a structure. It

was necessary to identify early in the investigation

the frequency content distribution that would produce

near-maximum response in structures with specific

*For instance, increasing the stiffness (and hence the fre­quency) of a structural wall by increasing its overall dimen­sions or its depth will ordinarily be accompanied by an in­crease in its strength or yield level.

-11-

periods and yield levels. This would allow use of the

appropriate input motion in subsequent analyses where

other parameters were varied.

7. Presentation of Results. The results of the parameter

study are presented mainly in the form of envelopes of

selected response quantities, i.e., the maximum hori­

zontal displacements, interstory displacements, mo­

ments, shears, ductility ratios and cumUlative plastic

hinge rotations along the height of the structure.

Time history plots of a number of response quan­

tities were also obtained and are presented where they

help in understanding the observed behavior. In addi­

tion, moment-rotation curves are shown for some cases.

plots summarizing the results of the parametric

studies are presented. An attempt is then made to

assess the relative importance of the different struc­

tural parameters. This is done by comparing the ef­

fects of these parameters on selected response quanti­

ties, the effect of each parameter being normalized

with respect to its corresponding practical range of

variation to allow comparison.

-12-

COMPUTER PROGRAMS

General features of the computer programs used

analytical investigation are described briefly below.

these programs have been implemented on the CDC 6400

Vogelback Computing Center of Northwestern University.

in

Most

at

the

of

the

Dynamic Inelastic Analysis of Structures - Program DRAIN-2D

The dynamic response analyses were done using program

DRAIN-2D, developed at the University of California at Berkeley.

A number of modifications, intended to allow more efficient

extraction and plotting of output data, have been introduced

into the program at PCA. Detailed information concerning the

program is given in Refs. 14 and 15.

The essential features of the program, as implemented at

Northwestern University, are as follows:

1. The program considers only plane structures.

2. A structure may consist of a combination of beam or

beam-column elements, truss elements or infill panel

elements. The moment-rotation characteristics for the

beam elements can be defined as a bilinear relation­

ship that may be stable hysteretic or may exhibit the

"degrading stiffness" characteristic for deformation

cycles subsequent to yielding. More recent modifica­

tions to the program relating to force-displacement

characteristics are discussed in a subsequent para­

graph.

3. The mass of the structure is assumed to be concen­

trated at nodal points.

4. Horizontal and vertical components of the input (base)

acceleration may be considered simultaneously. The

input motions are assumed to be applied directly to

the base of the structure (no soil-structure interac­

tion effects are considered).

5. Several types of damping may be specified, including

mass-proportional and stiffness-proportional damping.

-13-

6. Elastic shear deformation and the P- effect in frame

elements can be taken into account.

7. Output options include printouts of response quanti­

ties (e.g., displacement components and forces at node

points and plastic rotations at member ends) corres­

ponding to specified nodes and response envelopes at

given time intervals. Envelopes of basic response

quantities are automatically printed at the end of

each computer run. Time histories of specified re­

sponse quantities presented in compact form also can

be obtained.

For structures consisting of beam elements, (including

columns under constant axial load) plots of a variety of re­

sponse quantities can be obtained during each run.

The structural stiffness matrix is formulated by the direct

stiffness method, with the nodal displacements as unknowns.

Dynamic response is determined using step-by-step integration

by assuming a constant response acceleration during each time

step.

Elements of particular interest in this study are beams and

beam-columns, and especially beams characterized by a progres­

sive decrease in reloading stiffness with cycles of loading

subsequent to yield. In DRAIN-2D, both flexural and axial

stiffnesses of these elements are considered. Variable cross

sections may be taken into account by specifying the appropriate

stiffness coefficients. Inelasticity is allowed in the form of

concentrated plastic hinges at the element ends. For beam­

column elements, interaction between axial force and moment in

causing yielding is taken into account in an approximate manner.

The primary moment-rotation curves for the inelastic hinges

at the ends of beam and beam-column elements are specified in

terms of an initial stiffness, k, and a post-yield stiffness,

ky = ryk as shown in Fig. 1. The ratio r y = ky/k will be re­

ferred to as the yield stiffness ratio.

-14-

Two types of hysteresis loops can be specified for the

moment-rotation curve characterizing the inelastic point hinges

at element ends. The first is the more common stable loop,

shown in Fig. 2a. For this case, the unloading and reloading

stiffnesses are both equal to the initial stiffness. The pro­

gram accounts for the inelastic behavior of this type of element

by assuming an equivalent element consisting of two parallel

components, one elastic and the other elasto-plastic.

The second type of hysteresis loop is one that exhibits a

decreasing stiffness for reloading cycles subsequent to yield.

The basic or primary moment-rotation curve for the beam element

with decreasing stiffness is defined in Fig. 1. After yielding

occurs, however, the reloading branch, and to a minor degree

the unloading branch, of the curve exhibits a decrease in slope

(i.e., stiffness). This decrease in stiffness is assumed to be

a function of the maximum rotation reached during any previous

cycle. The detailed behavior of an inelastic point hinge, such

as unloading and reloading from intermediate points within the

primary envelope, is determined by a set of rules that are an

extended version of those proposed by Takeda and 50zen(16).

A single element is used for this case.

Modifications Introduced into DRAIN-2D: Early runs using

DRAIN-2D indicated the desirability of certain changes in the

program. At the request of the Portland Cement Association, G.

H. Powell and R. W. Litton at the University of California,

Berkeley, introduced changes into the program to enable it to

consider modifications of the basic Takeda model for the de­

creasing stiffness beam element. The unloading and reloading

parameters a and S shown in Fig. 2b allow the basic decreasing

stiffness model to be varied to correspond more closely to

experimental results. These have been incorporated into the

program. Powell and Litton also provided options for printing

and storing on file the time histories of most response quanti­

ties in a compact and convenient form.

-15-

In addition to the above modifications, a major effort was

made at the Portland Cement Association to incorporate plotting

capabilities into the program. Plots of time histories of

forces and deformations, as well as response envelopes, can be

obtained automatically with each run. Options to punch cards

for both envelopes and time histories of response have also

been incorporated. These can be used to plot curves in compar­

ative studies.

as used in the

and are discussed

the program DRAIN-2D

have been documented

All modifications to

study of isolated walls,

in detail in Ref. 15.

A more recent (August 1976) modification to DRAIN-2D,

undertaken by I. Buckle and G. H. Powell at the request of PCA,

allows the modelling of the hinging region in a beam element in

the form of a flexural 'point hinge' and a shear-shear slip

mechanism. The latter is designed to simulate the transverse

relative displacement that can occur between the ends of a

hinging region. The force-displacement relationships for both

of these mechanisms can be specified in terms of a three-segment

curve. The third segment represents the loss in strength ac­

companying displacements beyond the point of maximum resistance.

Behavior of the hysteresis loops under cyclic reversed

loading with the modified program can range from the 'stable

loop' to Clough's model for degrading stiffness(17). Plot­

ting routines developed for earlier versions of the program

have already been adapted to the subroutines for this latest

modification. These latest options added to DRAIN-2D have not

been used in the study of isolated walls reported here. How­

ever, it is intended to use these options in the study of

coupled walls and frame-wall systems.

Sectional Analysis Under Combined Flexure and Axial Load

As a first step in the parametric study of sections under

combined flexure and axial compression, a computer program was

developed for the analysis of typical wall cross sections.

This program allows a wide range of geometric and material pro-

-16-

perties to be specified. Sections of all commonly encountered

shapes, containing a large number of reinforcement layers, can

be considered.

Stress-strain properties of steel and concrete are repre­

sented by realistic analytical relationships built into the

program, or may be in the form of experimental data defined at

discrete points. Output for each section analyzed is obtained

as a set of bending moment and curvature values, corresponding

to a given magnitude of axial load applied to the section.

Detailed documentation for this program is given in Ref. 18.

This program was used also to determine the ranges of vari­

ation of the yield level and yield stiffness ratio corresponding

to practical wall cross sections used in the dynamic response

parametric studies.

the dynamic

linear and

for

this

spectra

with

Other Programs

Two dynamic analysis computer programs, DYMFR(19) and

DYCAN(20), developed at the Portland Cement Association, were

used primarily to determine the undamped natural frequencies

and mode shapes of the buildings considered in the parametric

study. DYMFR is a program for dynamic analysis of plane frame­

wall systems, while DYCAN is designed specifically for dynamic

analysis of inelastic isolated cantilevers. Both these programs

are implemented on the META4-l130 computing system at the Port­

land Cement Association.

Another PCA-developed program, DYSDF(21), for

analysis of single-degree-of-freedom systems (both

nonlinear) was used to calculate and plot response

various input motions considered in connection

investigation.

A number of other independent programs have been developed

for specific purposes. Among these is a program, AREAMR(22),

that utilizes the punched data from DRAIN-2D to compute the

cumulative areas under moment-rotation diagrams, and the cumu­

lative rotational ductilities. Results can be plotted as func­

tions of elapsed time.

-17-

Another useful program is ANYDATA(23). Given a number of

input arrays, this program can produce plots according to vari­

ous specified arrangements. A number of smaller programs have

been developed for preparing composite plots of response envel­

opes and time histories. A separate class of small programs

has also been prepared for processing punched data from the

sectional analysis program.

-18-

undertaken, several

answered. For this

out. These analyses

PRELIMINARY STUDIES

Before the parametric studies could be

questions relating to modelling had to be

purpose, preliminary analyses were carried

were made to:

a. explore the possibility of using a lesser number of

lumped masses in the model than the number of floors

in the prototype,

b. examine alternative techniques for modelling the hing­

ing region at the base of the wall and determine the

most appropriate feasible model, and

c. determine the proper integration time step to be used

in the analysis.

Questions (a) and (c) were considered in an effort to reduce

the computer time required for each analysis. Question (b)

assumed importance in this particular study because of the need

to obtain reliable estimates of the expected deformations in

the hinging region and to correlate these with experimental

data.

Exploratory analyses were also carried out to assess the

significance of the so-called P-~ effect on dynamic response.

The basic structure considered in the parametric studies

was also used in the preliminary analyses. This is a hypothe­

tical 20-story building consisting mainly of a series of paral­

lel structural walls, as shown in Fig. 3. The building is 60

ft x 144 ft in plan and rises some 178 ft above the ground.

All story heights are 8 ft-9 in. except the first story, which

is 12 ft.

Number of Lumped Masses in Model

Effect of the number of lumped masses on the accuracy of

the results was investigated using the models shown in Fig. 4.

The 20-mass model represents a full model of the 20-story build­

ing shown in Fig. 3. The reduced models of five and eight

-19-

masses have nodes spaced at closer intervals near the

a more accurate determination of the behavior of the

base for

hinging

region.

Results of dynamic analyses for the 8-mass model compared

very closely to those for the full model. In both cases, the

structure had a fundamental period of 1.4 sec. The 5-mass

model yielded the same maximum top displacement, but somewhat

different moments, shears and plastic rotations in the lower

part of the wall.

Based on these results and considering the desired accuracy

for relative story displacements, it was decided to lump the

masses at alternate floor levels in the upper portion of the

wall. In the lower portion, the masses were lumped at each

floor level. The resulting 12-mass model is shown in Fig. 5.

A comparison of results obtained with this model and the full

20-mass model showed excellent agreement.

Modelling of the Plastic Hinge Region

Proper modelling of the region of potential hinging at and

near the base of the wall is important if a reliable assessment

of the deformation requirements in this critical region is

expected. The model of the plastic hinging region should not

only be realistic but also allow meaningful interpretation of

the dynamic analysis results in terms that relate to measurable

quantities in the experimental investigation.

As previously explained, program DRAIN-2D(12) accounts

for inelastic effects by allowing the formation of concentrated

"point hinges" at the ends of elements when the moments at

these points equal the specified yield moment. The moment­

rotation characteristics of these point hinges can be defined

in terms of a basic bilinear relationship. This relationship

develops into either a stable hysteretic loop or exhibits a

decrease in reloading stiffness with loading cycles subsequent

to yield. Since the latter model represents more closely the

behavior of reinforced concrete members under reversed

inelastic loading, it was used throughout this investigation.

-20-

In modelling an element with bilinear moment-rotation

characteristics using DRAIN-2D, a major problem is the deter­

mination of the properties to be assigned to the hinge. These

properties can be derived by considering the program model and

relating its properties to a real member subjected to the same

set of forces as shown in Fig. 6. The initial hinge stiffness,

Ks ' can then be taken as a very large number so that the two

systems are identical up to the point when yielding occurs. In

the post-elastic range, hinge properties can be derived by im­

posing the condition that total rotation in the model, e 2' be

equal to the total rotation in the real element, e 1. This

yields the following expression for the yield stiffness ratio

of the point hinge, r 2 , (i.e., the ratio of the slope of the

post-yield branch to the slope of the initial branch of the

bilinear moment-rotation curve):

where:

r =22EI-t- (1)

= yield stiffness ratio for the cantileverelement

= rotational stiffness of the point hinge in themodel before yielding

= ratio of the tip moment to the moment at thefixed end

The main difficulty in using the above expression is that

the ratio of the end moments, y, is not known beforehand and,

in fact, varies throughout the analysis. In addition, if the

bending moment in the element is not uniform, the moment­

rotation relationship for the element or a segment of it does

not have the same shape as the sectional moment-curvature rela­

tionship. Thus, the yield stiffness ratio, r l , and the yield

moment, My' corresponding to the moment-rotation relationship

of the element, are unknowns. One way to overcome these pro­

blems is to divide each member into short elements so that the

moments at the two ends are approximately equal and thus y - 1.

-21-

proportional

desired pro-

For this case, the moment-rotation relationship is

to the moment-curvature relationship so that the

perties can be obtained easily.

In this investigation it was found desirable to use the

minimum number of nodal points consistent with an accurate

determination of deformations in the hinging region. This was

done to economize on the cost of each run and thus allow exami­

nation of a wider range of parameter values. Preliminary

studies indicated that if nodal points were established at

every story level near the base of the wall (where hinging is

expected), the ratio of end moments for each segment could be

taken approximately equal to 0.9. For the model with nodal

points as shown in Fig. 7a, (the same as those shown in Fig. 5)

this ratio was used to determine r 1 and M from the moment-- y

curvature relationship and r 2 using Eq. (1). Analyses were

then made using this model and one where the lower region of

the wall was divided into much shorter elements as shown in

Fig. 7b. The curvature is approximately uniform over the

shorter elements.

Results of the dynamic analyses were almost identical for

rotations and forces in the hinging region. The displacements

in the upper stories also compared very well although some

discrepancies occurred in the displacements of the lower

stories. Thus, it was decided that for the parametric studies,

it would be sufficiently accurate to use element lengths equal

to the height of a story near the base of the wall and to assume

a ratio equal to 0.9 for the end moments in a segment.

Integration Time Step

The time step, 6t, to be used in the dynamic analysis is of

primary concern since it affects both the accuracy of the re­

sults and the cost of computer runs. Some preliminary analyses

using simple models indicated that an integration step as long

as 0.02 sec. would result in sufficiently accurate results.

using the final 12-mass model for a structure with funda­

mental period Tl = 1.4 sec., a comparison of results was made

-22-

using values of ~t = 0.02 sec. and ~t = 0.005 sec. The results

showed very good agreement. Plastic deformations were within

one percent and the top displacements within 2.5 percent of each

other. Based on these results, an integration time step ~t =

0.02 sec. was used for structures with initial fundamental

periods of 1.4 sec. or longer.

In using a time step of 0.02 sec., irregularities in the

solution were noticed in two cases -- in walls with significant

discontinuities in stiffness, etc., and in input accelerograms

where large changes in acceleration values occurred in one time

step. For both these cases, the analyses were carried out using

~t = 0.005 sec. For structures with initial fundamental period,

Tl = 0.8 sec., a time step of 0.02 sec. appeared to be too

large, even when extensive yielding occurred. For these cases,

values of ~t = 0.01 and ~t = 0.005 yielded almost identical

results.

The p- ~ Effect

The nonlinear effect of gravity loads on deformations was

examined for a structure with Tl = 1.4 sec., and yield level,

My = 500,000 in.-kips. Although extensive yielding took

place in the structure and the lateral displacements were sig­

nificant, gravity load appeared to have no appreciable effect

on the response. Because of this, the p-~ effect was not con­

sidered in the subsequent analyses.

-23-

PARAMETRIC STUDIES

Once the basic elements of the dynamic analysis model have

been established, the parametric studies could proceed to eval­

uate the relative influence of selected structural and ground

motion parameters on dynamic response. As mentioned, The major

purpose of the parametric studies is to identify the most

significant variables needed in formulation of the design

procedure.

Parameters Considered

Behavior of a building subjected to earthquake motions is

affected by a number of variables related to its dynamic and

structural properties and the characteristics of the ground

motion. In this study those variables expected to have a sig­

nificant effect on the response of the structure, particularly

the force and deformation requirements in individual members as

well as the entire structure are investigated. The variables

considered are:

a. Structure characteristics:

1. fundamental period of vibration, as affected by

stiffness,

2. strength or yield level,

3. stiffness in post-yield range,

4. character of the moment-rotation relationship of

hinging regions,

5. viscous damping,

6. variations in stiffness and strength along the

height,

7. degree of fixity at the base of the structure,

8. height variations (number of stories).

b. Ground motion parameters:

1. intensity,

2. frequency characteristics,

3. duration.

Effects of these parameters on the seismic response of iso­

lated structural walls were investigated by dynamic inelastic

-24-

analyses of suitable mathematical models. The structural models

were obtained by changing values of selected parameters in a

basic reference structure. Properties of the reference struc­

ture and the range of variation in the individual parameters

are discussed in the following sections.

in

in.

20-story

structural

144 ft

8 ft-9

hypothetical

of parallel

is 60 ft x

a

Basic Building Properties

The basic structure considered is

building consisting mainly of a series

walls, as shown in Fig. 3. The building

plan and about 178 feet high. All story heights are

except the first story, which is 12 ft.

For the dynamic analysis, the mass of the structure was

calculated to include the dead weight and 40 percent of the

live load specified for apartment buildings by the Uniform

Building Code(13). This percentage of live load was deemed

reasonable and is consistent with the current specifications

for the design of columns in the lower stories of buildings.

However, in calculating the design lateral forces (UBC Zone 3)

for the purpose of proportioning the wall, only the dead weight

of the building was used, as specified by UBC.

Stiffness of the structural wall in the basic building was

assumed uniform along its height since this provides a better

reference for evaluating the effect of stiffness taper.

The reference structure, here denoted by ISW 1.4, has an

initial fundamental period of 1.4 sec. and a corresponding drift

index (i.e., the ratio of lateral deflection at top to total

height) of approximately 1/950 under the design seismic forces.

Yield moment at the base was assumed equal to 500,000 in.-kips

(56,490 kN·m).*

*The design moment at the base of the wall, on the basis of UBCZone 3 requirements, was calculated as 350,000 in.-kips. Thiscorresponds to a yield moment of approximately 500,000 in.-kipswhen allowance is made for load factors, capacity reductionfactors and the difference between the yield moment and themaximum moment capacity of typical reinforced concrete sections.(1 in.-kip = 0.113 kN·m)

-25-

A constant wall cross section was assumed throughout the

height of the basic structure. However, a reduction in yield

level of sections above the base was included to reflect the

effect of axial load on the moment capacity. This, in effect,

produced a moderate taper in strength. The taper in strength

used in the basic structure represents about the maximum that

can be expected due to axial load only. It was obtained by

examining the interaction diagrams for several types of struc­

tural wall sections with different percentages of longitudinal

steel.

Variation of Structural Parameters

The effects of the different structural and ground motion

parameters on selected response quantities, were determined by

a controlled variation of each parameter in the reference

structure. In particular, shear and rotational ductility demand

at the base of the wall and the interstory distortions were

considered. In most cases, only a single parameter was varied

in the basic structure with the others held constant. Although

this process sometimes resulted in unrealistic combinations of

structural properties, it was considered essential to a proper

evaluation of the effect of each variable on dynamic response.

The range of variation in the values of a given parameter

is important if the results of the parametric study are to have

practical application. In the present study, the values of

each parameter were chosen to represent a reasonable range of

values commonly encountered in practice.

To examine the effect of the fundamental period of vibra­

tion, three other values were assumed in addition to the basic

value of 1.4 sec. These were 0.8, 2.0 and 2.4 sec. The stiff­

ness of the reference structure was modified while keeping

the stiffness distribution and the mass constant to change

the period. As an aid in selecting the appropriate stiffness,

Fig. 8 was prepared. This figure relates the stiffness with

the fundamental period and the drift ratio corresponding to the

-26-

code-specified forces. The period values covered represent a

relatively wide range of structural wall buildings.

To arrive at a reasonable range of values for the other

major structural parameter, the yield level, an examination of

selected structural wall sections was undertaken. Yield

strengths corresponding to selected combinations of rectangular

and flanged wall sections of practical proportions and varying

longitudinal steel percentages (0.5 to 4.0 percent) were deter­

mined for f = 60 ksi (414 X 10 6 N/m2 ) and f' = 4,000 psi (28 X6 2 Y c

10 N/m). Based on the results of these calculations and con-

sidering current design practice, it was decided to use yield

level values ranging from the 500,000 in.-kips (56,490 kN.m) of

the reference structure to 1,500,000 in.-kips (169,570 kN·m).

Intermediate values of 750,000 in.-kips (84,740 kN· m) and

1,000,000 in.-kips (112,980 kN.m) were also considered.

Similar considerations were used in arriving at the ranges

of values for the other structural parameters.

Ground Motion Parameters

As suggested in Ref. 6, a ground motion duration of 10

seconds was used for all analyses except when the effect of

duration was investigated. The effect of intensity of the

ground motion was investigated by normalizing each input in

terms of the 5%-damped spectrum intensity corresponding to the

first 10 seconds of the N-S component of the 1940 El Centro

record, i.e., the "reference intensity", SI ref • Intensities

of 0.75, 1.0 and 1.5 relative to the 1940 El Centro (N-S) were

considered.

With the duration of the ground motion fixed and the inten­

sity normalized, the only other ground motion parameter that

could significantly affect the results is frequency content. A

basis for the broad classification of earthquake accelerograms

according to frequency characteristics was proposed in Ref. 6.

Using this system of classification, a number of records were

chosen early in the study to evaluate the effect of the fre­

quency content of input motions on dynamic structural response.

-27-

Once the effect of this particular parameter on response was

determined, the number of input motions used for investigating

the other parameters could be reduced to one or two. This pro­

cedure was necessary to limit the total number of analyses while

still retaining a reasonable assurance that the calculated re­

sponses would provide a good estimate of structural requirements

under a likely combination of unfavorable conditions.

A complete list of the parameter variations considered is

given in Tables la through Id. The list is divided according

to the fundamental period of the structure. Thus, structures

with a fundamental period of 0.8 sec. are listed under a basic

structure denoted by ISW 0.8, and so on. The third column in

the table shows the variations in the basic value used in the

parametric study.

Dynamic analyses were carried out for each combination of

parameters shown in Table 1. Throughout these analyses it was

assumed that structural elements possess unlimited inelastic

deformation capacity (ductility) since a major objective of the

study is the determination of the deformation requirements cor­

responding to particular values of a parameter.

-28-

DISCUSSION OF RESULTS

Data from the dynamic analyses consist of plots of response

quantities directly relating to the specific objectives of this

study. In selecting the response quantities to be plotted,

primary importance was given to the behavior of the hinging

region at the base of the wall. Generally, the base of the

wall represents the most critical region from the standpoint of

expected deformations and its importance to the behavior of the

wall and other structures which may be attached to the wall.

In addition to quantities characterizing the response of

the hinging region, the horizontal interstory distortions along

the height of the wall have also been recorded. Here, a dis­

tinction is drawn between two measures of interstory distortion.

In isolated structural walls, which exhibit predominantly can­

tilever flexure-type behavior, the interstory "tangential devi­

ation", (i.e., the deviation or horizontal displacement of a

point on the axis of the wall at a given floor level measured

from the tangent to the wall axis at the floor immediately below

it - see Fig. 9), rather than the "interstory displacement",

provides a better measure of the distortion that the wall suf­

fers. In fact, the tangential deviations vary in the same man­

ner as, and are directly reflected in, the bending moments that

are induced by the lateral deflection of the wall.

For open frame structures characterized by a 'shearing type'

deformation (resulting from the flexural action of the individ­

ual columns of the frame), the interstory displacement varies

in about the same manner along the height of the structure as

the tangential deviation and has been used as a convenient index

of the potential damage to both structural and nonstructural

components of this type of building(24). Figure 10 shows the

typical variation with height of these two quantities for a

statically-loaded cantilever wall.

-29-

Selected Results of Analysis

The results of a typical analysis are presented as envelopes

of maximum displacements, forces and ductility requirements

along the height of the structure. In addition, response his­

tory plots of a variety of quantities, including moment-plastic

hinge rotation, moment-nodal hinge rotation and moment versus

shear values have been obtained. To allow convenient comparison

of responses for the parametric study, composite plots of

envelopes and composite response history plots were prepared

from punched cards of each run.

Figures 11 through 21 have been included to give an indica­

tion of the types of results obtained from each analysis through

the plotting options introduced into Program DRAIN-2D(15).

Figure lla, for instance, shows the time history of horizontal

displacement of the top of the reference structure ISW 1.4 (with

fundamental period, Tl = 1.4 sec. and yield level, My =500,000 in.-kips). The structure was subjected to the E-W com­

ponent of the 1940 El Centro record, normalized to yield a

5%-damped spectrum intensity equal to 1.5 times the spectrum

intensity of the N-S component of the same record. Figure lIb

shows the same type plots for the intermediate floor levels of

the same structure. Figures 12a and 12b are time history plots

of the inters tory displacements between the different floor

levels of the same reference structure.

Rotation time histories are of particular interest in the

lower part of the wall. Figure 13 shows the variations with

time of the total rotations from the base of the wall to floor

levels 1 and 2. Figure 14 shows the time history of the plastic

hinge rotation in the first story. Plots of moment and shear

at the base vs. time are shown in Figs. 15 and 16. It is worth

noting the relatively more rapid fluctuation with time of the

shear force when compared to the moment at the base of the wall.

This is an indication of the greater sensitivity of the shear

force to higher modes of response.

Sample plots of two types of moment-rotation curves are

shown in Figs. 17 and 18 for the same reference structure.

-30-

Plots of moment versus total rotation were used to determine

the required rotational ductility at the base of the wall.

The change with time in the cumulative nodal rotation at

the first floor level (split into primary and secondary compo­

nents as illustrated in Fig. 33) is shown in Fig. 19. The nodal

rotation at the first floor level represents the total rotation

in the wall segment between the base and the first floor level

and serves as a convenient measure of the deformation within

this region.

Figure 20 shows the variation with time of the cumulative

area under the base moment-nodal rotation hysteresis loop for

the basic structure ISW 1.4. In calculating the areas for Fig.

20, the base moment was nondimensionalized by dividing by the

corresponding value of the base moment at first yield.

To study the variation of the ratio of moment to shear, as

well as the absolute values of these quantities, moment versus

shear plots were obtained with each analysis. Figure 21 shows

an example of such a plot.

The results of the analyses are presented mainly as

envelopes of maximum values of horizontal story displacements,

interstory displacements, bending moments, horizontal shears,

rotational ductility requirements, and cumulative plastic hinge

rotations over the entire height of the structure.

The rotational ductility requirement for a member as plotted

in these response envelopes is defined as:

II = 8maxr 8 y

(2 a)

where 8 max is

and 8 y is the

moments, this

the maximum rotation at

rotation corresponding to

definition becomes:

the base

yield.

of

In

the

terms

wall

of

II = 1 +r(2 b)

where the yield stiffness ratio, r y ' is the ratio of the slope

of the second, post-yield branch to the slope of the initial or

elastic branch of the primary bilinear moment-rotation curve of

the member.

-31-

To compare response histories for different values of a

particular parameter, histories of normalized forces and defor­

mations were obtained by dividing the value of the particular

response quantity at any time by the corresponding value at

first yield. Figure 25, for example, shows the time variation

of the normalized nodal rotation at the first floor level. The

dotted, horizontal lines through ordinates +1.0 and -1.0 cor­

respond to first yield in each case.

Ground Motion Parameters

Effect of Frequency Characteristics

Tt was considered desirable to investigate the effect of

the frequency characteristics of the input motion on dynamic

response early in the study in order to select the record(s)

for use in analyzing the other parameters. This question is

basic to the problem of determining the critical input motion

in relation to the properties of a given structure. For this

purpose, and to confirm the qualitative observations made in

connection with response spectra of single-degree-of-freedom

systems in Ref. 6, three separate sets of analyses were made.

These are listed in Table 2.

The first set of analyses corresponds to the reference

structure with fundamental period, Tl = 1.4 sec. and consists

of the four accelerograms listed under Set (a) in Table 2. All

the accelerograms were normalized to 1.5 times the 5%-damped

spectrum intensity (ST) of the N-S component of the 1940 El

Centro record;* the normalization factors are listed in Table 2.

The normalized accelerograms are shown in Fig. 22, and the cor­

responding 5%-damped velocity spectra are shown in Fig. 23.

Also shown for comparison is the velocity spectrum for the N-S

component of the 1940 El Centro record. The artificial accel-

*Tn the following discussion, the 5%-damped spectrum intensity(ST) of the first 10 seconds of the N-S component of the 1940El Centro record, for the period range 0.1 sec. to 3.0 sec.,will be denoted by "81 f II (which has a value of 70.15 in. =1781.8 mm). re .

-32-

erogram Sl was designed to have a broad-band spectrum and was

generated using the program SIMQKE developed by Gasparini(25).

Entries in the fourth column of Table 2 indicate the accel­

erogram classification in terms of the general features of its

velocity spectra relative to the initial fundamental period of

the structure, as discussed in Ref. 6. Thus, a "peaking (0)"

classification indicates that the 5%-damped velocity response

spectrum for this accelerogram shows a pronounced peak at or

close to the fundamental period of the structure considered (in

this case, Tl = 1.4 sec.). A "peaking (+)" classification

indicates that the peak in the velocity spectrum occurs at a

period value greater than that of the fundamental period of the

structure.

A "broad-band" classification, as discussed in Ref. 6,

refers to an accelerogram with a 5%-damped velocity spectrum

which remains more or less flat over a region extending from

the fundamental period of the structure to at least one second

greater. A "broad-band ascending" classification is similar to

a broad-band accelerogram, except that the velocity spectrum

exhibits increasing spectral values for periods greater than

the initial fundamental period.

The second and third sets of analyses undertaken to study

the effects of frequency characteristics of the input motion

include two and three accelerograms, respectively. These are

also listed in Table 2. These sets correspond to structures

with fundamental periods of 0.8 sec. and 2.0 sec.

In addition to the above three sets, a set of two analyses

was made using an input motion intensity equal to 0.75 (SI f)re .to illustrate the interaction between the intensity of the input

motion and the yield level of a structure in determining the

critical frequency characteristics of the input motion.

(a) !l = 1.4 sec., My = 500,000* in.-kips

Envelopes of response values for the structure

with period of 1.4 sec. and yield level, My = 500,000

*500,000 in.-kips = 56,490 kN·m.

-33-

500,000 in.-kips, are shown in Fig. 24.* Figures 24a,

24b, and 24c indicate that the E-W component of the

1940 El Centro record, classified as "broad-band

ascending" with respect to frequency characteristics,

produces relatively greater maximum displacements,

interstory displacements and ductility requirements

than the other three input motions considered. How­

ever, the same record produces the lowest value of the

maximum horizontal shear. The artificial accelerogram

Sl produces the largest shear, as shown in Fig. 24d.

Because all structures yielded and the slope of the

second, post-yield branch of the assumed moment­

rotation curve is relatively flat, the moment envelopes

shown in Fig. 24c do not show any significant differ­

ences among the four input motions used.

An idea of the variation with time of the flex­

ural deformation at the base of the wall under each of

the four input motions of Set (a) of Table 2 is given

in Fig. 25. This figure shows the normalized rotations

of the node at floor level "1", which represent the

total rotations occurring in the first story. To plot

the curves in Fig. 25, the absolute values of the

rotations in each case were divided by the correspond­

ing rotation when yielding first occurred. The two

dotted lines on each side of the zero axis (at ordi­

nates +1.0 and -1.0) thus represent the initial yield

level for all cases. The actual location of the

'state point' describing the deformation of the first

story segment relative to its moment-rotation curve at

each instant of time is indicated in Fig. 26, for the

structure subjected to the 1940 El Centro, E-W motion.

It is interesting to note in Fig. 25 that although

the intense motion starts relatively early under the

*In the envelopes for rotational ductility requirements,less than 1.0 represent ratios of the calculated maximumto the yield moment, My'

-34-

valuesmoments

artificial accelerogram Sl (Fig. 22), yielding occurs

first under the 1940 El Centro E-W motion. The rela­

tive magnitude of the rotation at first yielding,

however, is greater under both Sl and the Pacoima Dam

S16E record, a "peaking (0)" accelerogram.

As expected, the 1971 Holiday Orion record,* a

"peaking (+)" accelerogram, produced a much lower re­

sponse during the first few seconds, since the velocity

spectrum for this motion (Fig. 23) peaks at a period

greater than the initial fundamental period (Tl =1.4 sec.) of the structure. As the structure yields

and the effective period increases, however, the re­

sponse under this excitation increases gradually.

It is significant to note in Fig. 25 that as

yielding progresses and the effective period increases,

it is the "broad band ascending" type of accelerogram

(in this case, the 1940 El Centro E-W component) that

excites the structure most severely. Response to the

other types of accelerograms - particularly the peak­

ing accelerograms - tend to diminish.

An indication of the change in fundamental period

of a structure, as the hinging (or yielded) region

progresses from the first story upward, is given by

Fig. 27, for different values of the yield stiffness

ratio, r y = EI 2/EI l or (EI)yield/(EI)elastic' Thefigure is based on the properties of the reference

structure with initial fundamental period, Tl = 1.4

sec. It is pointed out that since the structure goes

through unloading and reloading stages as it oscil­

lates in response to the ground motion (Fig. 17), the

general behavior reflects the effects of the "elastic"

or unloading stiffness, as well as its yield or reload­

ing stiffness. The effect of each stiffness will

*The record obtained at the first floor of the Holiday Inn on8244 Orion Boulevard, Los Angeles, during the San Fernandoearthquake of February 9, 1971.

-35-

each

the

seems

depend on the duration of the response under

stiffness value, as governed by the character of

input motion. When yielding occurs early, .~1~

reasonable to assume that both elastic and yield

stiffnesses play about equal roles in influencing the

"effective period" of the structure. This is particu­

larly true for the type of structure considered here

since the condition at the critical section (i.e., the

base of the wall) determines to a large degree the

response of the structure. In the Takeda model(16)

of the hysteretic loop, the initial portions of the

reloading branches of the moment-rotation loops (Fig.

17) have stiffness values intermediate between the

initial elastic and yield stiffnesses of the primary

curve.

Figure 24f shows the cumulative plastic hinge

rotations, i.e., the sum of the absolute values of the

inelastic hinge rotations over the

period. This parameter reflects

10-second response

the combined effect

of both the number and amplitude of inelastic cycles

and provides another measure of the severity of the

response. The fact that the artificial accelerogram

81 produces a slightly greater cumulative plastic

hinge rotation at the base of the wall than does the

1940 El Centro E-W component, in spite of the lesser

amplitude of the associated maximum rotation (Fig.

24e), indicates that the response to 81 is character­

ized by a relatively greater number of cycles of ine­

lastic oscillation than the response to El Centro E-W.

This is indicated in the more jagged character of the

response history curve corresponding to 81, shown in

Fig. 25. It is a reflection of the considerably

greater number of acceleration pulses over the

10-second duration in 81 than in any of the recorded

accelerograms shown in Fig. 22.

-36-

mode predominating),

requirements at the

of the structure consi-

Tl = 1.4 sec. and a low

of structure considered

of the lower floors are

It can be seen in Fig. 24b that the relative

effect of the parameter considered (in this case, the

frequency characteristics of the input motion) is

similar on both interstory displacements and tangen­

tial deviations, the main difference between the two

quantities being in their distribution along the

height of the structure. For the subsequent cases,

only the horizontal interstory displacement envelopes

are shown. In considering these figures, the signifi­

cance of the interstory displacement relative to the

distortion in an isolated wall, and particularly its

distribution along the height as compared to the cor­

responding tangential deviations, must be borne in

mind.

(b) !l = 0.8 sec., My = 1,500,000* in.-kips

To study the effects of frequency characteristics

for the case of short-period structures with relatively

high yield levels, a "peaking (0)" accelerogram (N-8

component of the 1940 El Centro) and a "broad-band

ascending" type (E-W component of the 1940 El Centro)

were considered.

Figure 28 shows response envelopes of displace­

ment, moments, etc. This figure indicates that the

peaking accelerogram consistently produces a greater

response in the structure than does a broad-band

record. A comparison of Fig. 28e with Fig. 24e, shows

that the ductility requirements are not only signifi­

cantly less for this structure with a high yield

level, but that yielding has not progressed as high up

the structure as in the case

dered under (a), with period

yield level. For the type

here, where the displacements

generally in phase (fundamental

the magnitude of the ductility

*1,500,000 in.-kips = 169,570 kN.m.

-37-

base of the wall is a direct function of the extent to

which yielding has progressed up the height of the

wall.

The greater response of the structure under the

N-S component of the 1940 El Centro (peaking) follows

from the fact that the dominant frequency components

for this motion occur in the vicinity of the period of

the structure, (Fig. 23). In this region, the E-W

component has relatively low-power components. Also,

because of the high yield level of the structure,

yielding was not extensive, particularly under the E-W

component and apparently did not cause the period of

the yielded structure to shift into the range where

the higher-powered components of the E-W motion occur.

On the other hand, Fig. 28e indicates that under the

N-S component of 1940 El Centro, yielding in the

structure extended up to the 4th floor level, as

against the 2nd floor level under the E-W component.

The greater extent of this yielding and the accompany­

ing increase in the effective period of the structure

could easily have put the structure within the next

peaking range of the El Centro N-S component (Fig. 23).

(c) !l = 2.0 sec., My = 500,000* in.-kips

For this structure, the peaking accelerogram used

was the E-W component of the record taken at the first

floor of the Holiday Inn on Orion Boulevard, Los

Angeles, during the 1971 San Fernando earthquake. The

record has a 5%-damped velocity spectrum that actually

peaks at about 1.75 sec. and can thus be classified as

a "peaking (-)" accelerogram relative to the structure.

The other input motion considered is the E-W component

of the 1940 El Centro record ("broad-band ascending").

The response envelopes of Fig. 29 indicate, as in

Set (al with a structure period Tl = 1.4 sec. and

*500,000 in.-kips - 56,490 kN·m.

-38-

My = 500,000* in.-kips, that where yielding is sig­

nificant, the horizontal and interstory displacements,

as well as the bending moments and ductility require­

ments near the base, are greater for the broad band

accelerogram than for the peaking motion. Also, as in

Set (a), the extensive yielding which occurs near the

base results in a reduction of the maximum horizontal

shears. Thus, Fig. 29d, like Fig. 24d, shows the

maximum shears corresponding to the E-W component of

the 1940 El Centro to be less than those for the other

input motions. The greater base shears associated

with these other input motions can be partially attri­

buted to the higher (effective) modes of vibration.

As pointed out earlier, a comparison of Fig. 15 with

Fig. 16 clearly indicates the greater sensitivity of

the horizontal shears, as compared to bending moments

(and displacements), to higher mode response. Figure

23 shows that most of the other input motions consi­

dered have spectral ordinates in the low-period range

that are generally greater than those of the 1940 El

Centro E-W record.

The results of the preceding analyses serve to confirm the

observations made in Ref. 6 relative to the velocity spectrum.

Thus, for structures where extensive yielding occurs, resulting

in a significant increase in the effective period of vibration,

the "broad band ascending" type of accelerogram can be expected

to produce greater deformations than a "peaking" accelerogram

of the same intensity. Where the expected yielding is of

limited extent so that the increase in effective period is

minor, a peaking accelerogram is more likely to produce greater

deformations than a broad band accelerogram of the same inten­

sity. Since the extent of yielding is a function of the earth­

quake intensity, the yield level of the structure, and the fre­

quency characteristics of the input motion, these factors must

be considered in selecting an input motion for a given structure

to obtain a reasonable estimate of the maximum response.

-39-

Interaction Between Intensity and Yield Level

To verify the observation concerning the relationship of

the input motion intensity and the structure yield level, the

reference structure ISW 1.4 (Tl = 1.4 sec., My = 500,000

in.-kips) was subjected to two input motions with an intensity

equal to 0.75 (SI f)' The two motions used were the S16Ere .component of the 1971 Pacoima Dam record and the E-W component

of the 1940 El Centro record. As indicated in Table 2, the

Pacoima Dam record is a "peaking (0)" accelerogram relative to

the initial fundamental period of the structure, while the El

Centro motion is of the "broad band ascending" type.

The resulting envelopes of response, shown in Fig. 30, serve

to further confirm the observation made earlier that when

yielding in the structure is not extensive enough to cause a

significant increase in the effective period of the structure,

the peaking accelerogram is likely to produce the more critical

response. Figure 30e shows that in this case, yielding in the

structure has not extended far above the base when compared to

Set (a) where the input motion was twice as intense (Fig. 24e).

Note that in Set (a) considered earlier, the 1940 El Centro,

E-W record (a "broad-band" accelerogram), with intensity equal

to 1.5 (SI f)' represents the critical motion, while there .Pacoima Dam record (a "peaking" motion) produces a relatively

lower response. By reducing the intensity of the motions by

one-half so that yielding in this structure is significantly

reduced, the Pacoima Dam record becomes the more critical

motion, as Fig. 30 shows.

To summarize, it is pointed out that because the extent of

yielding in a structure is influenced by the yield level of the

structure, My' as well as the intensity of the input motion,

both parameters must be considered when selecting the appropri­

ate input motion, with particular reference to its frequency

characteristics.

In selecting an input motion for use in the analysis of a

structure at a particular site, the probable epicentral distance

-40-

and intervening geology should be considered. These considera­

tions, which affect the frequency content of the ground motion

at the site, may logically rule out the possibility of dominant

components occurring in certain frequency ranges. Because the

high-frequency components in seismic waves tend to attenuate

more rapidly with distance than the low-frequency components,

it is reasonable to expect that beyond certain distances, de­

pending on the geology, most of the high-frequency components

are damped out so that only the low-frequency (long-period)

components need be considered.

Effect of Duration of Earthquake Motion

In studying the effect of duration of the earthquake motion,

the response of the reference structure, with period Tl = 1.4

sec. and M = 500,000 in.-kips, to the first 10 seconds ofy

the E-W component of the 1940 El Centro record was compared

with its response to an accelerogram with a 20-second duration.

Both accelerograms were normalized to yield a spectrum intensity

equal to 1.5 (SI f I·re .The 20-second accelerogram consists of the first 12.48 sec-

onds of the E-W component of the 1940 El Centro record followed

by that portion of the same record between 0.98 sec. and 8.5

sec. The intent in putting together this composite accelerogram

was to subject the structure to essentially the same first 10

seconds of input motion but to extend the period of excitation

using acceleration pulses of about the same intensity as those

occurring in the first 10 seconds. Because the 1940 El Centro,

E-W component has its peak acceleration at about 11.5 seconds,

it was decided to include this peak in the composite record and

add a segment from the more intense portion of the first 10

seconds to make a 20-second record. The inclusion of the peak

acceleration at 11.5 sec. and the addition of the extra 7.5

sec. of fairly intense pulses, however, sufficiently altered

the velocity response spectrum. Because of this change, a nor­

malizing factor smaller than that used for the 10-sec. input

motion was indicated to obtain a spectrum intensity equal to

-41-

1.5 times that of the 1940 El Centro, N-S component. Thus, a

normalizing factor of 1.54 was calculated for the 20-second

composite accelerogram, compared to a factor of 1.88 used for

the 10-second input record. This means that the amplitude of

the pulses in the 10-second input was larger than in the first

10 seconos of the 20-second composite accelerogram by a factor

of 1.88/1.54 or 1.22. A plot of the unnormalized 20-sec. com­

posite accelerogram is shown in Fig. 3la. The corresponding

relative velocity response spectra are given in Fig. 3lb.

In the response envelopes of Fig. 32, the curves corres­

ponding to the 20-sec. composite accelerogram described above

are marked "20 sec.-(al". This figure shows that the displace­

ments, interstory displacements, moments, shears and ductility

requirements are greater for the 10-sec. record than for the

20-sec. composite record having the same spectrum intensity.

In spite of this, the cumulative ductility, i.e., the sum of

the absolute values of the plastic rotations in the hinging

region, is greater for the 20-sec. long record, as shown in

Fig. 32f. This follows from the fact that the structure goes

through a greater number of inelastic oscillations when sub­

jected to longer excitation. The cumulative plastic hinge

rotation plotted in Fig. 32f represents the sum of the "primary"

and the "secondary" plastic rotations illustrated in Fig. 33.

Because it is a measure of the number and extent of the excur­

sions into the inelastic region which the critical segment in a

member undergoes, the cumulative plastic hinge rotation repre­

sents an important index of the severity of deformation asso­

ciated with dynamic response. This is particularly true for

members which tend to deteriorate in strength with repeated

cycles of inelastic deformation, i.e., with relatively short

low-cycle-fatigue lives.

The time history of rotation of the node at the first floor

level of the wall when subjected to the two input motions dis­

cussed above is shown in Fig. 34. The curve marked "10 sec."

actually corresponds to the first 20 seconds of the 1940 El

Centro, E-W record, normalized so that the spectrum intensity

-42-

for the first 10 seconds equals 1.5 times 31 f' As mightre .be expected, the response of the structure during the second 10

secs. of the normalized 1940 El Centro E-W record (marked

"lO-sec." in Fig. 34) decays after 12 seconds. In comparison,

the structure continues to oscillate through several cycles of

relatively large amplitude during the same time interval of the

20-sec. composite accelerogram.

The third curve in Figs. 32 and 34, marked "20 sec.-(b)",

represents the response of the structure to the 20-sec. compo­

site accelerogram when scaled by the same factor used in nor­

malizing the 10-sec. record. Note that the curves marked

"lO-sec." and "20 sec.-(b)" in Fig. 34 coincide over the first

10 seconds.

Both Figs. 32f and 34 indicate that the major effect of

increasing the duration of the large-amplitude pulses in the

input accelerogram is to increase the number of cycles of

large-amplitude deformations which a structure will undergo.

This conclusion assumes that the intensity and frequency charac­

teristics of the additional motion do not differ significantly

from those of the shorter duration input.

Effect of Earthquake Intensity

To examine the effect of earthquake intensity, three sets

of analyses were run corresponding to different combinations of

the fundamental period, T] and the yield level, M. In all- y

cases, the input motion used was the first 10 seconds of the

E-W component of the 1940 El Centro record, normalized to dif­

ferent intensity levels in terms of 31 f're .Table 3 shows the values of the periods and yield levels

assumed for the structure, together with the different inten­

sity levels of input motion used for each set. The response

envelopes for all these cases are shown in Figs. 35, 36 and

37. In all cases, there is a consistent increase in the re­

sponse with increasing intensity, a behavior also observed by

other investigators.

-43-

(see Set (b)

O. 8 sec. pr 0-

Centro record,

level, My =

1. 5 intens i ty

the structure.

Figure 35 shows that the displacements, interstory dis­

placements and ductility requirements increase almost propor­

tionally with increasing intensity. The maximum moments and

shears, however, do not show a proportional increase to reflect

the increase in intensity. Thus, Figs. 35c and 35d show that

an increase in intensity level from 1.0 to 1.5 produces about

the same increase in the maximum moments and shears as an in­

crease from 0.75 to 1.0 in intensity level.

Figure 36e indicates that with a yield

1,000,000* in.-kips, the yielding, even under a

level input motion, does not extend too high up

For this case, the N-S component of the 1940 EI

which has a velocity spectrum that peaks at about

duces a greater response--for the same intensity

of "Effect of Frequency Characteristics").

Structural Parameters

The combinations of significant structural parameters used

in investigating the effect of each parameter on dynamic re­

sponse have been summarized in Table 4. In almost all cases,

the parameter values in each set have been chosen so that only

the parameter of interest is varied while the other variables

remain constant.

The fundamental period, TI , and the yield level, My'

(the basic parameters characterizing the primary force­

deformation curve of the structure), were extensively studied

using different combinations of structural parameters. The

slope of the post-yield branch of the primary bilinear moment­

rotation curve, as defined by the yield stiffness ratio, r,ywas also considered in detail. Also studied was the sensitivity

of the dynamic response to varying degrees of stiffness degrada­

tion in the hinging region of the wall. The effect of the shape

of hysteretic loop was examined by assuming different values

parameters a and S which define the slopes of the unloading and

reloading branches, respectively, of the M-8 loop of potential

*1,000,000 in.-kips = 112,900 kN.m.

-44-

study

the

inelastic hinges (Fig. 3). Other parameters investigated in­

cluded viscous damping, taper in stiffness and strength along

the height of the structure, the fixity condition at the base,

and the number of stories.

Based on the study of the effects of the frequency charac­

teristics of input motions, it was decided to use the first 10

seconds of the E-W component of the 1940 El Centro record, as

input motion for the analyses of the effects of all the struc­

tural parameters on dynamic response. An intensity equal to

1.5 (81 f) was used throughout.re .

Effect of Fundamental Period, TlThe effect of the initial fundamental period of the struc­

ture, Tl , was investigated by using four sets of data as

shown in Table 4, each corresponding to a different yield level,

M . Because of the close interrelationship between these twoy

major structural parameters, it was deemed necessary to

the effects of the structure period under varying values of

yield level.

(al Yield Level, My = 500,000 in.-kips

Corresponding to this yield level, four values of

the fundamental period were used, namely, 0.8, 1.4,

2.0 and 2.4 sees. Even though the stiffness asso­

ciated with a period, Tl , of 0.8 sec. is rather high

for a structural wall section having a yield level of

500,000 in.-kips, this parameter combination was con­

sidered in order to provide some indication of the

behavior of structures which might fall in this range.

Figure 38 shows envelopes of response quantities

for this set. In Fig. 38a and 38b, the maximum hori­

zontal and interstory displacements show a consistent

increase with increasing fundamental period (or de­

creasing stiffness) of the structure. The rotational

ductility requirements, expressed as a ratio of the

maximum rotation to the rotation at first yield, how-

-45-

ever, become greater with decreasing fundamental per­

iod, a trend also observed by Ruiz and penzien(26).

Figure 38e indicates, as do similar plots shown

earlier, that the greater the ductility requirements

at the base, the higher the yielding generally extends

above the base.

Because all structures considered in Fig. 38 have

the same yield level, and have a relatively flat slope

for the post-yield branch of the M-8 curve (r y =0.05), the maximum moments and shears shown in Figs.

38c and 38d do not differ significantly for all four

cases considered. The shorter-period structures show

only slightly greater moments at the base. Figure 38e

also indicates that for structures with relatively low

yield levels, where yielding is significant, ductility

requirements do not decrease significantly with an

increase in period beyond a certain value of the fun­

damental period. Thus, the ductility requirements for

structures with periods of 2.0 and 2.4 sec. are about

the same.

It is worth noting that although Fig. 38e shows

the rotational ductility requirements as increasing

with decreasing period of the structure, the absolute

value of the maximum rotation for the stiff structure

is less than that for the more flexible structure.

The distinction between these two measures of the

deformation requirement is illustrated in Fig. 39, for

the case of structures with My = 500,000 in.-kips.

Figure 38f, for instance, shows the cumulative plastic

hinge rotation, i.e., the sum of the absolute values

of the inelastic hinge rotations, as increasing with

increasing period of the structure. This trend fol­

lows directly from the larger deformations of the more

flexible structures.

Variation with time of the flexural deformation

in the first story for the four structures considered

-46-

are

T l =

mag-

Fig.In

absolute

structure

2.0 sec., by 0.00070 radians. Thus, the

nitude of the rotations for the latter

in this set is shown in Figs. 40a and 40b.

40a, the rotations have been normalized by the respec­

tive yield levels of each structure. For in~tance,

the rotations for the structure with Tl = 0.8 sec.

have been divided by its yield rotation value of

0.00014 radians and those for the structure with

actually 0.00070/0.00014 or 5.0 times those of the

former on the basis of the ordinates shown in this

figure. The variation with time of the actual magni­

tude of the rotations (in radians) for the four cases

considered are shown in Fig. 40b.

(b), (c) and (d) Yield Level, M = 750,000,* 1,000,000y

and 1,500,000 in.-kips, respectively

The same general trends observed in Set (a) above

with respect to displacements, moments and ductility

requirements are also apparent in Figs. 41 through 43

for the higher values of the yield level, My' Thus,

as in Set (a), an increase in period results in an

increase in the horizontal and interstory displace­

ments and a decrease in the moments and the ductility

requirements (expressed as a ratio). Although the

horizontal shears do not change much with changing

period for a given yield level, no clear trend can be

observed insofar as the effect of period variation on

the base shear is concerned. Generally, though, the

shears tend to be slightly higher for shorter-period

structures.

A comparison of Figs.

and 43f shows that while

40f and 41£ with Figs. 42f

for structures with rela-

tively low yield levels (i.e., My = 500,000 and

750,000 in.-kips) the cumulative plastic hinge rota­

tion tends to decrease with decreasing period of the

*750,000 in.-kips - 84,740 kN·m.

-47-

the reverse trend appears to hold as My increases

beyond a certain value. As noted earlier, the ratio

of the maximum rotation to the yield rotation dimin­

ishes with increasing period of the structure, for a

given yield level, My' This means that the relative

extent of yielding diminishes with increasing period

of a structure. Also, as the yield level increases,

the degree of inelastic action generally diminishes,

as might be expected and as indicated by a comparison

of Figs. 38e, 4le, 42e and 43e. Thus, an increase in

both yield level and fundamental period would combine

to reduce the amount of inelastic action in a struc­

ture; and where this combined effect is such as to

make yielding in a structure insignificant, the cumu­

lative plastic rotation becomes less for the long­

period structure than for the short-period structure.

Figure 42e shows that for the structure with My =1,000,000 in.-kips and Tl = 2.0 sec., yielding is

not too significant. Figure 43e indicates that the

structure with My = 1,500,000 in.-kips and Tl =2.4 sec. remains essentially elastic under a base

motion with intensity 51 = 1.5 (51 f).re •

Effect of Yield Level, My

Three sets of yield level values were considered corres­

ponding to fundamental period values of 0.8, 1.4, and 2.0 sec.

The different values included in each set are listed in Table 4.

Envelopes of maximum response values showing the effect of

yield level are given in Figs. 44, 46 and 47 corresponding to

the three values of the fundamental period assumed.

The following general comments apply to all cases shown in

these figures. For the same fundamental period, the horizontal

and interstory displacements decrease sharply as the yield level

increases from 500,000 in.-kips to a value associated with

nominal yielding at the base (a value which tends to decrease

-48-

with increasing value of the fundamental period). This is evi­

dent from a comparison of Figs. 44a and 44b with Figs. 47a and

47b. Above this value, the trend is reversed and an increase

in yield level is accompanied by an increase in horizontal and

inters tory displacements. Maximum moments and shears increase

almost proportionally with the yield level, as shown in

(c) and (d) of Figs. 44, 46, 47. As observed earlier

plots

under

"Effect of Fundamental Period", rotational ductility require­

ments increase significantly as the yield level decreases.

This qualitative trend has also been observed in connection

with the response of single-degree-of-freedom systems (27) .

Figure 44f shows that the cumulative plastic hinge rotations

also increase as the yield level decreases. A normalized time

history plot of the total rotation in the first story is shown

in Fig. 45 for the four structures with fundamental period,

Tl = 1.4 sec.

Fundamental Period and Yield Level Effects - Summary

The interrelationship between the effects of the initial

fundamental period, Tl , and the yield level, My' on the

dynamic response of 20-story isolated structural walls is sum­

marized in Fig. 48. The plots in the figure correspond to only

one input motion, i.e., the E-W component of the 1940 El Centro

record, \'iith the acceleration amplitude adjusted to yield a

5%-damped spectrum intensity equal to 1.5 (SI f). Proper-re •ties of the structures considered are those for the basic

structures listed in Table 1.

Figure 48 shows the effects of both Tl and My on the

maximum horizontal displacement of the top of the structure,

the maximum interstory displacement along the height of the

wall and the maximum moment, (in terms of the yield moment,

My)' the maximum shear, rotational ductility (expressed as a

ratio of the maximum rotation to the corresponding yield rota­

tion) and the cumulative plastic hinge rotation at the base of

the wall. The maximum values of the response parameters are

also listed in Table 5.

-49-

The following significant points, noted earlier in relation

the yield level, My'

per iod (indicating a

structure) results in

fundamentalan increase in the

decrease in the stiffness of the

to the study of the separate effects of the fundamental period

and the yield level, are summarized below for convenience in

considering the curves shown in Fig. 48.

(a) Maximum Top Displacement and Maximum Interstory

Displacement

For a particular value of

an increase in horizontal and interstory displacements.

For a given fundamental period and for relatively

low yield level values, the maximum displacements de­

crease with increasing yield level. Beyond a certain

value of the yield level (which tends to decrease with

increasing value of the fundamental period) associated

with nominal yielding, the above trend is reversed,

i.e., the maximum displacements increase with increas­

ing yield level.

Figure 48a shows a dotted curve representing the

top displacement resulting from the distributed design

base shear, Vb' as specified in the Uniform Building

Code (1976 Edition)--with the factors z, S and I set

equal to 1. O.

(b) Maximum Moment and Rotational Ductility at Base

For a particular yield level value, the ratios

M 1M. Id and e Ie. Id as measures of themax Yle max Ylemaximum moment and rotational ductility at the base,

respectively, generally decrease with increasing fun­

damental period, the decrease being more rapid for

lower yield levels. For a constant fundamental per­

iod, the above ratios consistently increase with de­

creasing yield level.

It should be pointed out that the yield level of

500,000 in.-kips is not too realistic for structural

walls having stiffnesses associated with fundamental

period values less than 1.4 sec.

-50-

(c) Maximum Base Shear

For a particular

base shear decreases

yield

as the

level, My' the maximum

fundamental period in-

creases from 0.8 sec. to a certain value, which value

increases with increasing yield level. Beyond this

value of the fundamental period, the maximum base shear

increases with increasing period of the structure.

(d) Cumulative Plastic Hinge Rotation at Base

Figure 48f indicates, as noted earlier, that for

structures with relatively low yield levels (i.e.,

My = 500,000 and 750,000 in.-kips) in which signifi­

cant yielding occurs, the cumulative plastic hinge

rotation at the base increases with increasing funda­

mental period over the entire period range (0.8 2.4

sec.) considered. For higher values of the yield

level, the cumulative plastic hinge rotation increases

with increasing period up to a point where the com­

bination of high yield level and long period results

in only nominal yielding. Beyond this value of the

fundamental period (which decreases with increasing

yield level) the trend reverses and the cumulative

plastic hinge rotation decreases with increasing fun­

damental period until the point where no yielding

occurs.

Curves corresponding to the design base shear as

specified in the Uniform Building Code (UBC-76),

multiplied by load factors of 1.4 and 2.0 are shown in

Fig. 48d for comparison with the results of the dynamic

analysis.

Effect of Yield Stiffness Ratio, r yThe effect of the slope of the second, post-yield branch of

the primary bilinear moment-rotation curve of the members that

make up the structural wall was investigated. This was accom­

plished by considering a value of the yield stiffness ratio,

r , (i.e., the ratio of the slope of the second, post-yieldy

-51-

branch to the slope of the initial, elastic branch of the M­

curve), equal to 15%, in addition to the value of 5% used for

most cases. As indicated in Table 4, these two values of the

yield stiffness ratio were used with three combinations of the

fundamental period and yield level. In addition, a value of

r y = 0.01 was considered for the case of structures with Tl= 1.4 sec. and My = 500,000 in.-kips.

Response envelopes for the three sets are given in Figs.

49, 50 and 51. These figures show that even with significant

yielding at the base, as indicated by the plots for rotational

ductility, an increase in the value of the yield stiffness ratio

from 5% to 15% generally does not produce any significant effect

of the response. As might be expected, the effect of a change

in the value of the yield stiffness ratio, r y ' is more appar­

ent in structures with relatively low yield levels (My =500,000 in.-kips). There is a substantial (50%) reduction in

the rotational ductility requirement at the base for the long­

period (Tl = 2.0 sec.) structures and a 20% reduction in the

horizontal and interstory displacements for the moderately-long

period (Tl = 1.4 sec.) structures accompanying an increase in

the yield-stiffness ratio from 5% to 15%. The effect of in­

creasing the yield stiffness ratio is less apparent in struc­

tures with relatively high yield levels, as shown by Fig. 50.

A decrease in the yield stiffness ratio from 5% to 1%

results in an increase in the rotational ductility requirement

at the base by a factor of 1.85 (Fig. 4ge). There is a lesser

increase in the cumulative plastic hinge rotation. Except for

these, the effect of r y on the response is relatively minor

for the range of values considered. An increase in the slope

of the post-yield branch of the M-8 curve tends to reduce the

horizontal and interstory displacements as well as the ductil­

ity requirements at the base while increasing the moments and

shears slightly.

-52-

It will be noted that for the Takeda model of the hystere­

tic M-8 loop, the effective stiffness of an element during most

of the response after yielding may be governed not so much by

the primary M-8 curve as by the rules governing the slope of

the reloading stiffness associated with this model. The effect

of r y becomes significant only when the motion of the struc­

ture proceeds in a particular direction long enough for the

primary curve to govern. However, because the post-yield slope

in the bilinear M-8 curve represents the least value of stiff­

ness of the structure, it can be expected to have a greater

effect on the maximum displacement and rotations than the sub­

sequent "reduced" stiffnesses. The effect becomes most pro­

nounced for structures with low yield levels (relative to the

intensity of the input motion). Significant displacements and

ductility requirements can then be expected particularly for

low values of r y . An idea of the percentage of the total

response time, during which the post-yield branch of the pri­

mary curve governs the effective stiffness of the structure,

can be obtained from Fig. 52, which shows the moment-rotation

loop for the base of the structure with Tl = 1.4 sec., My =500,000 in.-kips and r y = 15%.

Effect of Character of M-8 Hysteretic Loop

To investigate the effects of other parameters, the quanti­

ties a and S defining the slopes of the unloading and reloading

branches of the Takeda model hysteretic loop (Fig. 53) were

assigned values of 0.10 and 0.0, respectively. The sensitivity

of the calculated response to variations in the values of these

two parameters was examined by analyzing structures using dif­

ferent values of a and S.

A value of 0.30 was considered for the unloading parameter a

in addition to the basic value of 0.10 (with S = 0), for a

structure with Tl = 1.4 sec. and My = 500,000 in.-kips.

For the reloading parameter S, values of 0.4 and 1.0 were

assumed in addition to the basic value of 0 (with a = 0.10).

-53-

Furthermore, results for a model with a stable bilinear hys­

teretic loop (Fig. 53) were obtained for comparison with the

response of structures with different values of S. The effect

of the reloading parameter S was investigated for two sets of

structures, one set representing low yield levels (My =

500,000 in.-kips, Tl = 1.4 sec.) and the other relatively

higher yield levels (My = 1,000,000 in.-kips, Tl = 0.8

sec.). The various combinations used in this study are listed

also in Table 4.

Figure 54, which shows response envelopes for cases where

the parameter a is allowed values of 0.10 and 0.30 (with S = 0)

indicates practically no difference in the maximum response

values corresponding to the two assumed values of a.

Figures 55 and 56 show the response envelopes for cases

when the reloading parameter S is varied. Both figures indicate

that the effect of this parameter on dynamic response is not

significant.

For structures with low yield level (My = 500,000

in.-kips, Fig. 55), an increase in the slope of the reloading

curve--corresponding to increasing values of s--results in

slight reductions in horizontal and interstory displacements

and cumulative plastic hinge rotations. Where yielding is not

as extensive, as in structures with My = 1,000,000 in.-kips

(compare Fig. 56e with SSe), the effect of variations in S is

even less significant. For this case, the trend with respect

to maximum horizontal displacements is reversed, so that the

structure with the stable bilinear hysteretic loop exhibits

slightly greater displacements than any of the structures with

decreasing-stiffness loops. The cumulative plastic hinge rota­

tions, however, still increase as S decreases (Fig. 56f). Note

that this same trend in the maximum response values was observed

in the study of the effect of the yield level, My (Fig. 44).

In each of the two sets of analyses where the reloading

parameter S was varied, the maximum moments, shears and rota­

tional ductility requirements at the base are practically the

-54-

same in spite of the significant difference in the slopes of

the reloading curves.

The variation with time of the nodal rotations at the first

story level for the structures with Tl = 1.4 sec. and My =500,000 in.-kips are shown in Fig. 57. For all cases, the

figure shows that the maximum response to the particular input

motion used occurs very early, during which time the primary

bilinear curve (identical for all cases) governs; the envelopes

of maximum response values do not differ significantly from

each other. However, after the first major inelastic cycle of

response occurs and the rules assumed for the reloading stiff­

ness in each case take effect, the responses reflect the dif­

ferences in the M-8 hysteresis loops. Thus, after first yield,

the rotations tend to be greater for the lower values of S

(corresponding to reloading branches with flatter slopes or

lesser stiffness).

Moment vs. nodal rotation plots for the structure with a

stable hysteretic loop and that with S= 0.40 are shown in Fig.

58. The corresponding plot for the structure with S = 0 is

shown in Fig. 18.

with

sec.,1.4=structures

Effect of Damping

Viscous damping assumed for the structures in this investi­

gation consists of a linear combination of stiffness-propor­

tional and mass-proportional damping. The damping distribution

among the initial component modes is defined in terms of the

percentage of the critical damping for the first and second

modes, which are assumed to be equal. For most of the analyses,

a damping coefficient of 0.05 was assumed.

To evaluate the effect of damping, a second value of the

damping coefficient, equal to 0.10, was considered. The re­

sponses corresponding to these two values of damping were com­

pared for two combinations of the fundamental period, Tl , and

the yield level, My' as shown in Table 4.

Envelopes of response corresponding to

intermediate period and low yield level (i.e.,

-55-

My = 500,000 in.-kips) are shown in Fig. 59. As observed in

other studies, the general effect of increasing damping in a

structure is to reduce the response. For the structures con­

sidered in this particular set, increasing the damping coeffi­

cient from 0.05 to 0.10 produced only a relatively small (about

12% in the maximum top displacement) reduction in response.

Figure 60 shows the response envelopes of structures with

low period and relatively high yield level, i.e., Tl = 0.8

sec., My = 1,000,000 in.-kips. The effect of increasing the

damping coefficient from 0.05 to 0.10 for this set is likewise

insignificant. When compared with results for the first set of

structures with low yield level, Fig. 60 indicates that an in­

crease in the viscous damping coefficient from 5% to 10% pro­

duces a slightly lower reduction (9%) in the maximum top dis­

placement. However, in terms of deformation requirements, the

increase in damping produces a relatively greater percentage

reduction in the high-yield-level (My = 1,000,000 in.-kips)

structure than the structure where extensive yielding occurs

(My = 500,00 in.-kips): 19% as against 8% for the rotational

ductility requirements and 41% as compared to 13% for the cumu­

lative plastic hinge rotation. This increase in the energy

dissipated through damping as the extent of inelasticity dimin­

ishes, and vice versa, was also noted by Ruiz and penzien(26).

Effect of Stiffness Taper

In examining the effect of other parameters the structures

considered have uniform stiffness throughout their entire

height. To study the effect of a taper in stiffness, the re­

sponse of a structure with the stiffness variation shown in

Fig. 61 was compared to that of a structure with uniform stiff­

ness. A ratio of (EI)base to (EI)top equal to 2.8 was used

(corresponding to AlB = 4.0, Fig. 59), the absolute value of

the stiffness being adjusted to yield a fundamental period of

1.4 sec.

Figure 60, which shows the corresponding response

indicates that for the period and yield level the

-56-

envelopes,

horizontal

and interstory displacements as well as the cumulative

rotations at the base are less for the tapered than

plastic

for the

uniform structure. However, the moments, shears and rotational

ductility requirements at and near the base are greater for the

tapered structure than for the wall with uniform stiffness.

axial

InFig. 61.

of the

Effect of Strength Taper

To evaluate the effect of a taper in the strength (yield

level) of the wall, the following three cases were considered:

(a) A taper ratio (i.e., the ratio of the yield level at

the base to that at the top of the wall) of 2.0, with

equal changes in strength occurring regularly at every

story level. This is the taper used in the reference

structure, which reflects the effects of axial loads

on the moment capacity.

(b) A taper ratio of 3.8, with equal changes in strength

occurring at every fourth story level, in a manner

similar to the stiffness taper shown in

addition, changes reflecting the effect

load occur at every story level.

(c) A taper ratio of 1.0, representing uniform strength

throughout the height of the wall.

in Fig. 63.

requirement in

taper ratio of

taper for the

appears to be

Response envelopes for this set are shown

Except for the increased rotational ductility

the intermediate stories of the structure with a

3.8 as shown in Fig. 63e, the effect of strength

particular period and yield level considered

negligible.

Effect of Fixity Condition at Base

The effect of yielding of the foundation at the base of the

wall was considered by introducing a rotational spring (with

linear M-8 characteristic) at the base of the analytical model.

If the base fixity factor, F, is defined as the ratio of the

moment developed at the base to that which would be developed

-57-

if the base were fully fixed, under the same deformation, the

spring stiffness, Ks ' for the model shown in Fig. 64 is given

by

K = f- F J 3EI ( 3)s \1 Fi') 9,

In the analysis, the (linear) spring representing the rota­

tional restraint of the foundation was modelled by an extension

of the wall below the base having the appropriate flexural

stiffness. This is shown in Fig. 65. The stiffness of the

element used to simulate foundation restraint corresponding to

each degree of base fixity assumed is also shown in the figure

for the two sets considered.

Figure 66 shows a comparison of response envelopes for the

first set, in which the reference structure (T l = 1.4 sec.,

My = 500,000 in.-kips) has a fully fixed base (i.e., F = 1.0),

with cases in which the base fixity factor was assigned values

of 0.75 and 0.50. Note that the initial fundamental period of

1.4 sec. applies only to the fully fixed-base structure. The

calculated fundamental period for the structure with F = 0.75

was 1.43 sec. and for the structure with F = 0.50, 1.52 sec.

Increases in the maximum horizontal and interstory dis­

placements, in almost the same proportion as the increase in

the fundamental period, accompany the decrease in the base

fixity factor, F, from the fully fixed value of 1.0 to 0.75

(20% increase in the maximum top displacement) and then to 0.50

(80% increase). It is interesting to note that for this group

of low-yield-level structures where significant yielding occurs,

relaxation in the base fixity results in increases (though

slight) in maximum moment and shears and in rotational ductility

requirement at the base. In the case of the rotational ductil­

ity requirement, there is a 20% increase accompanying a decrease

in the base fixity factor from 1.0 (fully fixed) to 0.50.

Figure 66f indicates that even though the ductility re­

quirement at the base increases with decreasing base fixity as

-58-

analyses was

sec. and Myfactor equal

shown in Fig. 66e, the cumulative plastic hinge rotation de­

creases with decreasing base fixity. Since the yield rotation

is the same for all three structures in each set and the periods

do not differ significantly from each other, this reversal in

trend for these two measures of deformation may at first appear

contrary to expectation. Figure 67, which shows the variation

with time of the plastic hinge rotation at the bases of the

walls, provides some explanation for this reversal in trend.

The figure shows that although the maximum plastic hinge rota­

tion increases with decreasing base fixity, the number of

oscillations and hence the cumulative plastic hinge rotation

tends to increase as the base fixity increases (and the period

decreases) •

To examine the effect of the base fixity condition on

structures with high yield levels, a second set of

undertaken for a basic structure with Tl = 0.8

= 1,500,000 in.-kips. Values of the base fixity

to 0.75 (Tl = 0.83 sec.) and 0.50 (Tl = 0.88 sec.) were

considered in addition to the fully fixed case.

Figure 68 shows response envelopes for this set. As in Set

(a), Figs. 68a and 68b show the expected increase in maximum

displacements due to a relaxation of the base fixity. However,

for the structures in this set in which yielding was relatively

small (compare Fig. 68e with Fig. 66e), the maximum moments and

shears, as well as the rotational ductility and the cumulative

plastic hinge rotation at the base, tend to decrease with a

reduction in the base fixity factor. A 60% decrease in the

rotational ductility requirement at the base accompanies a re­

duction in the base fixity factor from 1.0 to 0.50.

Effect of Number of Stories

The effect of the number of stories or the height of the

structure on dynamic response was investigated by considering

10-, 30- and 40-story variations of the basic 20-story reference

structure. The stiffness of the structure in each case was

adjusted to yield the same fundamental period (T l

-59-

= 1.4 sec. )

as the basic structure. Two sets of analyses were made, as

listed in Table 4. In the first set, a common value of the

yield level, My' equal to 1,000,000 in.-kips, was assumed for

all four structures. The purpose here was to isolate the effect

of the number of stories on the response. In practice, however,

the strength of walls would normally be expected to increase

with the height of the building. In view of this, a second set

of analyses was made with the yield level in each case adjusted

to result in a ductility ratio of about 4 to 6 at the base of

the wall.

A 12-mass model was used for both the 10- and 40-story

walls, with the masses spaced at every half-story in the lower

two stories of the 10-story wall and at every other floor in

the lower eight stories of the 40-story structure. For the

30-story structure, a 14-mass model was used, with the lumped

masses spaced at every floor in the lower five floors.

Figure 69 shows response envelopes corresponding to the

first set, i.e., with a constant yield level, M = 1,000,000Y

in.-kips, assumed for all four structures. As indicated in

Fig. 6ge, all structures except the 10-story wall yielded at

the bases in varying degrees, the extent of yielding increasing

with increasing height of structure. In the case of the

40-story structure, yielding extended above mid-height.

Except for the 10-story structure which responded elasti­

cally, the maximum displacements and interstory displacements

for the 20-, 30- and 40-story structures were essentially the

same at corresponding floors above the base. For the same

yield level, the maximum bending moments, shears and ductility

requirements (expressed as ratios of maximum to yield rota­

tions) generally increase with increasing height of the struc­

ture. The cumulative plastic hinge rotations, however, tend to

decrease with increasing height of structure. This behavior

follows from the fact that in order to obtain the same funda-

mental period of 1.4 sec., the 40-story wall had to be

tively much stiffer than, say, the 20-story wall. In

case, the sectional stiffness of the 40-story wall had

-60-

rela­

this

to be

15.8 times that of the 20-story wall in order to obtain the same

period, and for the same yield moment had a correspondingly

lesser yield rotation. Thus, for about the same maximum rota­

tion (Fig. 69a), the ductility ratio for the 40-story structure

is greater than that for the 20-story structure. Figure 70

shows a time history plot of the nodal rotations at a point

corresponding to the 2nd floor above the base in each of the

four structures considered under Set (a), as listed in Table 4.

Response envelopes for the second set of analyses--with the

fundamental period, Tl , still equal to 1.4 sec. but with the

yield level, M, adjusted so that the resulting ductilityyratios at the base for all structures fall in the range of 4 to

6--are shown in Fig. 71. The yield level assumed in each case

and the corresponding ductility ratios at the base are listed

below, see also Fig. 7le.

No. of Stories

10

20

30

40

AssumedYield Level, My

(in.-kips)

200,000

750,000

1,500,000

2,500,000

Ductility Ratioat Base

5.4

4.9

6.0

5.9

For the conditions assumed in this set, the maximum dis­

placements at the top of all four structures are about the same,

as shown in Fig. 7la; the slight differences reflect the rela­

tive magnitudes of the base ductility ratio. The equal maximum

deflection at the top for different heights of structure implies

increasing interstory distortions with decreasing height. This

is shown in Fig. 7lb. The increase in the yield level with

height of structure leads to a very regular increase in the

maximum bending moment and shear with the number of stories.

Figure 7lf, however, indicates that the cumulative plastic hinge

rotations at the base tend to increase with decreasing yield

level and structure height, the effect of the former apparently

predominating.

-6).-

SUMMARY

Figures 72 through 83 provide a convenient summary of the

effects of the structural and ground motion parameters consi­

dered on five response quantities. The five response quanti­

ties are the maximum values of the top displacement, interstory

displacement, bending moment, shear force and rotational duc­

tility at the base of the wall. Only in considering the effect

of ground motion duration is the cumulative plastic rotation at

the base shown. It has not been possible to conveniently sum­

marize the effect of the frequency characteristics of input

motions on structural response. However, this parameter has

been adequately covered in the text.

In all of the summary figures (Figs. 72-83), the particular

par.ameter considered is shown as the abscissa. The response

quantities, plotted as ordinates, have all been non dimensional­

ized by using appropriate factors. In these figures, "H" is

the total height of the wall and "h" is the story height.

Intensity and Duration of Ground Motion

Figure 72 shows that the response generally increases with

increasing intensity of the input motion. The magnitude of

r.esponse as well as the increase in response with increasing

intensity is greater for long-period structures with low yield

levels than for stiff, strong (i.e., short-period, high yield

level) structures.

As mentioned earlier and shown in Fig. 73, the effect of

duration of the input motion on response is not too significant,

except on the cumulative plastic rotations.

Structural Parameters

The effect of the fundamental period of the structure on

dynamic inelastic response of isolated walls is summarized in

Fig. 74. This figure indicates the significant increase in

horizontal and interstory displacements with increasing period

(reflecting decreasing stiffness) of a structure. Note that

-62-

there is very little difference in displacements between walls

having different yield levels. The effect of yield level is

more apparent in the maximum moments, shears and rotational

ductility. For these response quantities, the fundamental per­

iod does not have too significant an effect, except for struc­

tures with low yield levels.

The relationship between period and yield level indicated

in Fig. 74 is again apparent in Fig. 75, which shows the effect

of yield level on structural response. Thus, although the

absolute value of the displacements increase with increasing

period, the displacement for any particular fundamental period

changes very little with increasing yield level. On the other

hand, the maximum moment and shear, expressed in terms of the

corresponding yield moment, as well as the rotational ductility,

decrease with increasing yield level.

Figure 76 shows the effect on response of the yield stiff­

ness ratio, r y ' i.e., the ratio of the slope of the second,

post-yield branch of the bilinear primary moment-rotation curve

to the slope of the initial branch. The major effect of this

parameter is on the maximum moment and rotational ductility,

although it is not too significant. As may be expected, an

increase in r y results in an increase in the maximum moment

and a decrease in the rotational ductility demand.

The effect of the parameters a and S defining the slopes of

the unloading and reloading branches of the hysteretic M-8 loop

(Fig. 2b), and viscous damping are shown in Figs. 77 through

79. These figures indicate that for the range of values assumed

for these parameters, their effects on the dynamic response of

isolated walls are negligible.

Figures 80 and 81 show the effect of stepwise reductions in

stiffness and strength along the height of the wall. A taper

in stiffness, when compared to a uniform stiffness throughout

the height of the wall, results in a slight decrease in maximum

displacements and an increase in the maximum moments, shears

and rotational ductility required. The effect of a taper in

strength, for the range of values examined, is negligible.

-63-

The effect of the degree of base fixity, shown in Fig. 82,

is most evident in the maximum displacements, which increase

with decreasing base fixity. The effect of this parameter on

the maximum moments, shears and rotational ductility demand is

not too significant.

The solid curves in Fig. 83 represent the maximum responses

of walls of different height, with their yield levels adjusted

to give a rotational ductility demand at the base of from 4 to

6. For structures so proportioned, the effect of varying height

on the normalized maximum moments and shears (expressed as

ratios to the corresponding My and My/H, respectively) does

not appear to be significant. The maximum displacement, ex­

pressed as a ratio to the corresponding total height, H, tends

to diminish with increasing height of wall. The dashed curves

in Fig. 83 correspond to walls with varying height but the same

yield level at the base. As mentioned, this set does not

represent a realistic condition.

Comparison of Different Parameters

To evaluate the relative importance of the different para­

meters examined in this study, Figs. 84 through 86 were pre­

pared. To compare the effects of the different parameters on

response, it was necessary to assume a practical range of values

for each parameter. The ranges of values assumed for the vari­

ous parameters are shown in the second column of Table 6 and

also in Figs. 84 through 86. These values were based on current

design practice as well as on judgment.

Figure 84 indicates the relative effects of the various

parameters on the maximum displacement at the top of the wall.

Curves corresponding to different values of the yield level,

M , are shown in the figure. The ordinates in the figurey

represent the change in response corresponding to a change in

the value of the particular parameter. The change in response

is expressed as a ratio of the calculated response to the re­

sponse associated with the value of the parameter at the lower

end of its assumed practical range. The change in a parameter's

-54-

value is expressed as a percentage of its assumed practical

range. A zero percent change in the value of a parameter cor­

responds to the lower end of its range. Thus, if the values of

a parameter at the lower and upper ends of its assumed practi­

cal range of values are denoted by Po and Pt , respectively,

and Pi is a particular value of the parameter within the

range,

Value of Parameter Relative to Practical Range =(abscissa in Figs. 84-86)

Similarly, if the calculated response corresponding

denoted by Ro and Ri is the response associated

parameter value Pi' then,

to Po is

with the

Relative Change in Response =(ordinate in Figs. 84-86)

The quantities plotted in Figs. 84 through 86 are listed also

in Table 6.

Figure 84 shows that as far as top displacement is con­

cerned, the most significant structural parameter is the funda­

mental period of vibration, particularly as this is affected by

the stiffness of the structure. The second most significant

structural parameter affecting top displacement is the degree

of base fixity. The most significant ground motion parameter

is the earthquake intensity.

Relative effects of the various parameters on the maximum

base shear is shown in Fig. 85, while Fig. 86 shows the effects

on the rotational ductility demand at the base of the wall.

Both of these figures show that with respect to shear and duc­

tility demand, the most significant structural parameter is the

yield level, My. The fundamental period, TI , exhibits an

important influence for the intermediate range of values. As

in the case of top displacements, earthquake intensity has a

significant influence on both the base shear and the ductility

demand.

-65-

Since the maximum interstory displacement follows the same

trend as the maximum top displacement, the remarks made with

respect to the effects of the different parameters on the latter

also apply to the former. The same relationship holds between

the maximum moment and the rotational ductility demand at the

base.

The above comparison indicates that the two most important

structural variables affecting inelastic response are the fun­

damental period and the yield level. Inelastic response is

also significantly affected by the intensity of the ground

motion.

-66-

CONCLUSIONS

This study of the effects of selected structural and ground

motion parameters is aimed at establishing the relative impor­

tance of these parameters with respect to the inelastic dynamic

response of reinforced concrete structural walls. A total of

60 analyses are included in this study, covering variations in

eleven parameters. The response quantities considered are

those which experience and tests have indicated to be signifi­

cant to the behavior of these walls. By identifying the most

significant parameters affecting behavior, these parametric

studies allow the formulation of the design procedure to be

developed in the subsequent effort(8) based on these few

significant variables.

The major considerations involved in assessing the relative

importance of the various parameters are the magnitude of the

deformation requirements and the accompanying forces. At the

base of the wall, the critical region, the major concerns are

amplitude and number of deformation cycles as well as the

associated moments and shears. Along the height of the struc­

ture, magnitude of the maximum inters tory distortions and the

maximum horizontal displacements are of major interest. Inter­

story distortions,* and in a less direct manner, horizontal

displacements, provide a good index of potential damage in a

building. These response quantities also affect the stability

of the structure.

The following conclusions and general observations can be

made on the basis of the results of this study:

1. The most important structural parameters affecting

inelastic dynamic response of isolated walls are the

fundamental period, Tl , and the yield level, My.

Intensity of the input motion is the principal ground

motion parameter.

*Expressed appropriately as 'interstory tangential deviations'for isolated walls and as 'horizontal interstory displacements'for frame-wall systems.

-67-

2. For the same fundamental period

significant increases in horizontal

result from a decrease in the degree

base of the wall.

and yield level,

displacements can

of fixity at the

mining near-maximum or critical response

design purposes, or in specifying input

use in the analysis of particular types of

3. For the same intensity and duration of the ground mo­

tion, significant increases in response can result

from an input motion having the appropriate frequency

characteristics relative to the period and yield level

of the structure.

Where significant yielding can be expected in a

structure, i.e., yielding that would appreciably alter

the effective period of vibration, an input motion

with a velocity spectrum of the "broad band ascending"

type is likely to produce more severe deformation de­

mands than other types of motion of the same intensity

and duration. For cases where only nominal yielding

is expected, "peaking" accelerograms tend to produce

more severe deformations.

The above considerations are important in deter­

values for

motions for

structures.

4. Shear at the base of an isolated structural wall is

more sensitive to higher mode response and generally

undergoes a greater number of reversals than the mo­

ments or deformations. Because of this, the maximum

base shear can be appreciably lower for input motions

that produce extensive yielding and that are critical

with respect to moments and displacements than for

motions which produce lesser displacements. The cri­

ticality of response with respect to shear will depend

on the relationship of the frequency characteristics

of the input motion to the significant higher effec­

tive mode frequencies of the yielded structure.

-68-

shear forces is of particular

the significant influence that

the deformation capacity of theon

The magnitude of

interest because of

these can have

hinging region.

5. The major effect of the duration of ground motion is

to increase the cumulative plastic rotations in hing­

ing regions. The cumulative plastic rotation is

another useful index of the severity of the deforma­

tion demand in critical regions of a structure and

reflects both the magnitude and the number of cycles

of loading associated with dynamic inelastic response.

6. Deformation requirements increase almost proportionally

with an increase in the intensity of the ground motion.

Determination of the appropriate intensity to use

in a particular case will depend on such factors as

earthquake magnitude, epicentral distance and site

geology, all of which are beyond the scope of this

report. As a matter of fact, the selection of the

proper intensity, duration and frequency characteris­

tics of the input motion for use in the analysis of a

particular case will be determined to a large degree

by factors not considered in this report. The obser­

vations based on this study, however, can aid in

selecting input motion parameters by providing infor­

mation concerning the effect of a particular parameter

on structur.al response.

7. A direct result of incr.easing the stiffness of a

structure and thus decreasing its fundamental period

(assuming essentially the same mass) is the reduction

in the maximum horizontal displacements and the inter­

story distortions along the height of the structure.

The ductility requirement, expressed as a ratio

of the maximum rotation to the rotation at first

-69-

yield, generally increases with decreasing fundamental

period (or increasing stiffness) of a structure. This

trend is mainly a reflection of the smaller absolute

magnitude of the rotations for the stiffer structures.

In structures with relatively low yield levels where

significant yielding occurs, however, the cumulative

plastic rotation in hinging regions tends to decrease

with decreasing period. When the yield level is high

enough so that only insignificant yielding results,

the cumulative plastic hinge rotation tends to decrease

with increasing period.

8. The magnitude of deformation requirements in hinging

regions--in terms of ductility ratios and cumulative

plastic hinge rotations--generally decreases with in­

creasing values of the yield level. Maximum horizontal

displacements and interstory distortions also tend to

diminish with increasing yield levels, as long as

significant yielding occurs. Within a narrow range,

where only nominal yielding occurs, the maximum dis­

placements--particularly in the upper stories of the

structure--tend to increase with increasing yield

level, until a value is reached when no yielding

occurs, i.e., linear elastic response.

Although the yield level in a structure usually

does not vary independently of the fundamental period

(an increase in stiffness is generally accompanied by

an increase in the yield level), the effect of this

major variable is worth noting, because of its inter

action with the initial fundamental period and also

because it affects the dominant or effective period of

a yielding structure. The latter has important bear­

ing on the type of ground motion that is likely to

produce near-maximum or cr.itical response.

-70-

By making possible an assessment of the relative importance

of the different parameters affecting structural response, these

parametric studies allow the subsequent effort, i.e., the deter­

mination of estimates of critical force and deformation require­

ments in hinging regions of structural walls, to concentrate on

a manageable few of the more significant parameters as basis

for the design procedure to be developed.

-71-

ACKNOWLEDGEMENTS

This investigation was carried out in the Engineering

Services Department of the Portland Cement Association, Skokie,

Illinois. Mr. Mark Fintel, then Director of Engineering Ser­

vices, now Director of Advanced Engineering Services, served as

overall Project Director of the combined analytical and experi­

mental program. The help extended by Dr. W. G. Corley, Direc­

tor, Engineering Development Department, in reviewing the draft

of this report is gratefully acknowledged.

The project was sponsored in major part by the National

Science Foundation, RANN Program, through Grant No. ENV74-l4766.

Any opinions, findings and conclusions expressed in this

report are those of the authors and do not necessarily reflect

the views of the National Science Foundation.

-72-

REFERENCES

1. Applied Technology Council, Recommended ComprehensiveSeismic Design Provisions for Buildings, Final ReviewDraft, ATC-3-05, Applied Technology Council, Palo Alto,California, January 7, 1977.

2. Appendix A, Building Code Requirements for ReinforcedConcrete (ACI 318-77), American Concrete Institute, P.O.Box 19150, Redford Station, Detroit, Michigan 48219.

3. The Behavior of Reinforced Concrete Buildings Subjected tothe Chilean Earthquakes of May 1960," Advance EngineeringBulletin No.6, Portland Cement Association, Skokie,Illinois, 1963.

4. Fintel, M., "Quake Lesson from Managua: Revise ConcreteBuilding Design?" Civil Engineering, ASCE, August 1973,pp. 60-63.

5. Anonymous, "Managua, If Rebuilt on SameSurvive Temblor," Engineering News Record,1973, p. 16.

Si te, CouldDecember 6,

6. Derecho, A.T., Fugelso, L.E. and Fintel, M., "StructuralWalls in Earthquake-Resistant Buildings AnalyticalInvestigations, Dynamic ananlysis of Isolated StructuralWalls - INPUT MOTIONS," Final Report to the NationalScience Foundation, RANN, under Grant No. ENV74-14766,Portland Cement Association, January 1978.

7. Derecho, A.T., Ghosh, S.K., Iqbal, M., Freskakis, G.N.,Fugelso, L.E., and Fintel, M., "Structural Walls inEarthquake-Resistant Buildings - Analytical Investigation,Dynamic Analysis of Isolated Structural Walls (Part A:Input Motions and Part B: Parametric Studies)," Report tothe National Science Foundation, RANN, under Grant No.GI-43880, Portland Cement Association, October 1976.

8. Derecho, A.T., Iqbal, M., Ghosh, S.K. and Fintel, M.,"Structural Walls in Earthquake-Resistant BuildingsAnalytical Investigation, Dynamic Analysis of IsolatedStructural Walls DEVELOPMENT OF DESIGN PROCEDUREDESIGN FORCE LEVELS," Final Report to the National ScienceFoundation, RANN, under Grant No. ENV74-14766, PortlandCement Association, March 1978.

9. Oesterle, R.G., Fiorato, A.E., Johal, L.S., Carpenter,J.E., Russell, H.G. and Corley, W.G., "Earthquake­Resistant Structural Walls Tests of Isolated Walls,"Report to the National Science Foundation, Portland CementAssociation, 1976, 44 pp. (Appendix A, 38 pp.; Appendix B,233 pp.).

-73-

10. Derecho, A.T., Iqbal, M., Ghosh, S.K., Fintel, M. andCorley, W.G., "Structural Walls in Earthquake-ResistantBuildings - Analytical Investigation, Dynamic Analysis ofIsolated Structural Walls REPRESENTATIVE LOADINGHISTORY," Final Report to the National Science Foundation,RANN, under Grant No. ENV74-l4766, Portland CementAssociation, March 1978.

11. Derecho, A.T., Freskakis, G.N., Fugelso, L.E., Ghosh, S.K.and Fintel, M., "Structural Walls in Earthquake-ResistantStructures - Analytical Investigation, Progress Report,"Report to the National Science Foundation, RANN, underGrant No. GI-43880, Portland Cement Association, August1975.

12. Ghosh, S.K., Derecho, A.T. and Fintel, M., "PreliminaryDesign Aids for Sections of Slender Structural Walls,"Supplement No.3 to the Progress Report on the PCAAnalytical Investigation, "Structural Walls in Earthquake­Resistant Structures," NSF-RANN Grant No. GI-43880,Portland Cement Association, August 1975.

UniformRoad,

of the

13. International Conference ofBuilding Code, 1973 Edition,Whittier, California, 90601.Code is the 1976 Edition.)

Building Officials,5360 South Workman Mill

(The latest edition

14. Kanaan, A.E. and Powell, G.H., "General Purpose ComputerProgram for Dynamic Analysis of Plane Inelastic Struc­tures," (DRAIN-2D), Report No. EERC 73-6, EarthquakeEngineering Research Center, University of California,Berkeley, April 1973.

15. Ghosh, S.K. and Derecho, A.T., "Supplementary Output Pack­age for DRAIN-2D, General Purpose Computer Program forDynamic Analysis of Plane Inelastic Structures," Supple­ment No. 1 to an Interim Report on the PCA Investigation,"Structural Walls in Earthquake-Resistant Structures,"NSF-RANN Grant No. GI-43880, Portland Cement Association,Skokie, Illinois, August 1975.

16. Takeda, T., Sozen, M.A. and Nielsen, N.N., "ReinforcedConcrete Response to Simulated Earthquakes," Journal ofthe Structural Division, ASCE, Proceedings, Vol. 96, No.ST12, December 1970.

17. Clough, R.W. and Johnston, S.B., "Effect of StiffnessDegradation on Earthquake Ductility Requirements,"Proceedings, Japan Earthquake Engineering Symposium,Tokyo, October 1966.

18. Ghosh, S.K., "A Computer Program for the Analysis ofSlender Structural Wall Sections under Monotonic Loading,"Supplement No. 2 to an Interim Report on an Investigation

-74-

of Structural Walls in EarthquakeNSF-RANN Grant No. GI-43880, PortlandSkokie, Illinois, August 1975.

Resistant StructuresCement Association,

19. Derecho, l".T., "Program DYMFR - IBM 1130Dynamic Analysis of Plane MUltistoryCement Association, 1971 (unpublished).

ProgramFrames,1I

for thePortland

20. Derecho, A.T., "Program DYCAN - IBM 1130 Program for theDynamic Analysis of Isolated Cantilevers," Portland CementAssociation, 1974 (unpublished).

21. Derecho, A.T. and Fugelso, L.E., "Program DYSDF - for theDynamic Analysis of Inelastic Single-Degree-of-FreedomSystems," Portland Cement Association, 1975 (unpublished).

22. Ghosh, S.K., "Program AREAMR forCumulative Areas under Moment-Rotationtive Ductilities," Portland Cement(unpublished) .

the Computation ofCurves and Cumula­Association, 1976

23. Ghosh, S.K., "Program ANYDATA - A General Purpose PlottingRoutine," Portland Cement Association, 1976 (unpublished).

24. Czernecki, R.M., "EarthquakeStructures Publication No.Engineering, MassachusettsJanuary 1973.

Damage to Tall359, DepartmentInstitute of

Buildings,"of Civil

Technology,

25. Gasparini, D., "SIMQKE: A Program for Artificial MotionGeneration," Internal Study Report No.3, Department ofCivil Engineering, Massachusetts Institute of Technology,January 1975.

26. Ruiz, P. and Renzien, J., "Probabilistic Study of theBehavior of Structures During Earthquakes," EarthquakeEngineering Research Center Report No. EERC 69-3,University of California, Berkeley, March 1969.

27. Blume, J.A., Newmark, N.M. and Corning, L.H.,Multistory Reinforced Concrete Buildings for

-75-

Design ofEarthquakeIllinois,

TABLES AND FIGURES

Table 1a Basic Structure Properties and VariationsStructure ISW 0.8

PROPERTIES

STRUCTURAL

Fundamental period

Number of stories

Height

Weight (for masscomputation)

BASIC VALUEOR CHARACTERISTIC

0.8 seconds

20

178.25 ft.

4370 k/wall

VARIATIorlS

Stiffness parameter El 6.346 x 1011 k-in2

Yield level, My 1,000,000 in-k

Yield stfffness ratio, ry 0.05

I500,000 750,000 I1,000,000 1,500,000 in-k

0.15

Character of M-O curve

Damping

Stiffness taper (EI) base(EI) top

(My) baseStrength taper

(My) top

8asic fixity condition

INPUT t~OTION------

Decreasing stiffnessa = .10, P = 0

5% of criti ca1

1.0

2.0

Fully fixed

10% of critical

Intensity SI = 1.5 (Sl ref.)*

Frequency characteristics 1940 El Centro, E··li

1.0 (Sl ref.)

1940 El Centro, N-S1952 Taft, S69EArtificial Ace. SI

Duration 10 seconds

::: 5%-damped spectrum intensi ty - bet\-;een 0.1 and 3.0 secondsccn-espondin9 to the first 10 seconds of the N-S componentof the 1940 El Centro record.

-77-

Table 1b Basic Structure Properties and VariationsStructure ISW 1.4

PROPERTIES

STRUCTURAL

Fundamental period

Number of stories

Height

Weight (for masscomputation)

Stiffness parameter EI

BASIC VALUEOR CHARACTERISTIC

1.4 seconds

20

178.25 ft.

4370 k/wall

2.052 x lOll k_in2*

10.40

VARIATIONS

Yield level, 11y

Yield stiffness ratio, ry

Character of M-8 curve

Damping

. (EI) baseStiffness taper (EI) top

(My) baseStrength taper (My) top

Basic fixity condition

500,000 in-k

0.05

Decreasing stiffnessa= .10, 13= 0

5% of cri ti ca1

1.0

2.0

Fully fixed

750,000 1,000,0001,500,000 in-k, andelastic

0.01, 0.15

a = .30, 13 = 0.4,1.0 and stable loop

10% of critical

2.8

1.0, 3.8

50% and 75% of fullyfixed condition**

* EI = 5.004 x lOll k-in for stiffness taper 2.8

** Period of 1.4 sec. corresponds to fUlly fixed condition.

INPUT MOTION

Intensity 51 = 1.5 (SI ref .)

Frequency characteristics 1940 El Centro, E-W

Duration 10 seconds

-78-

0.75 and 1.0 x (SI ref .)

1971 Holiday Orion, E-W1971 Pacoima Dam, S16E,

Artificial Ace. Sl

20 seconds

Table Ic Basic Structure Properties and VariationsStructure ISW 2.0

PROPERTIES

STRUCTURAL

Fundamental period

Number of stories

Height

Weight (for masscomputation

Stiffness parameter EI

BASIC VALUEOR CHARACTERISTIC

2.0 seconds

20

178.25 ft.

4370 kjV/all

0.990 x lOll k-in2

VARIATImis

Yield level, 11y 500,000 in-k

Yield stiffness ratio, ry 0.05

750,000 1,000,000 in-kand elastic

0.15

Character of M-8 curve

. (EI) baseSt,ffness taper \Ell top

Oecreasing stiffnessu=.10,{l=0

1.0

Strength taper (1'1)y

basetop 2.0

Basic fixity condition

INPUT HOTION

Intensity

Frequency characteristics

Duration

Fully fixed

SI = 1.5 (SI f)re •1940 El Centro, E-W

10 seconds

-79-

1.0 x (Slref.)

1971 Holiday Orion, E-W

Table ld Basic Structure Properties and VariationsStructure ISW 2.4

PROPERTIES

STRUCTURAL

Fundamental period

Number of stories

Height

Weight (for masscomputation)

Stiffness parameter EI

Yield level, My

Yield stiffness ratio, ry

Character of M-e curve

Damping

BASIC VALUEOR CHARACTERISTIC

2.4 seconds

20

178.25 ft.

4370 k/wall

0.684 x loll k_in2

500,000 in-k

0.05

Decreasing stiffnessa = .10, P : 0

5% of critical

VARIATIONS

750,000 1,000,0001,500,000 in-k

1m: of criti ca1

(EI) baseStiffness taper TEl) top 1.0

(My) baseStrength taper (M) top 2.0

YBasic fixity condition

INPUT MOTION

Fully fixed

Intensity SI = 1.5 (SIref.)

Frequency characteristics 1940 E1 Centro, E-W

Durat i on 10 seconds

-80-

Tab

le2

Sum

mar

yo

fIn

pu

tM

otio

nsC

on

sid

ered

inS

tud

yo

fF

req

uen

cyC

hara

cte

rist

ics

c2.

0se

c.(5

00,0

00in

-k)

I <Xl

f-' I

5et

a b

Str

uctu

reP

erio

d,T l

(and

My)

1.4

sec.

(500

,000

in-k

)

0.8

sec.

(l,5

00,0

00in

-k)

Freq

uenc

yC

onte

ntIn

tens

ity

Inpu

t14

0tio

nN

orm

aliz

atio

nC

hara

cter

izat

ion#

Fac

tor*

1971

Paco

ima

Dam

,Pe

akin

g(0

)0.

5951

6Eco

mpo

nent

1971

Hol

iday

Inn

Peak

ing

(+)

3.22

Ori

on,

E-W

com

pone

nt

Art

ific

ial

Acc

eler

ogra

m,

51B

road

band

1.65

1940

ElC

entr

o,B

road

band

,1.

88E-

Wco

mpo

nent

asce

ndin

9

1940

ElC

entr

o,Pe

akin

g(0

)1.

50N

-5co

mpo

nent

1940

ElC

entr

e,B

road

band

,1.

88E-

Wco

mpo

nent

asce

ndin

g

1971

Hol

iday

Inn,

Peak

ing

(-)

3.22

Ori

on,

E-W

com

pone

nt

1940

ElC

entr

o,B

road

band

,1.

88E-

Wco

mpo

nent

asce

ndin

g

Art

ific

ial

Acc

eler

ogra

m,

51B

road

band

1.65

*.~alculated

toyi

eld

a5%

-dam

ped

spec

trum

inte

nsit

y(f

orth

era

nge

0.1

to3.

0se

c.)

equa

lto

1.5

times

the

refe

renc

esp

ectru

min

tens

ity,

51f'

i.e~,

the

5%-d

ampe

dsp

ectr

umin

ten

sity

re•

ofth

eN

-5co

mpo

nent

ofth

e19

40El

Cen

trore

cord

.(5I

ref.

=70

.15

in.)

Inal

lca

ses,

only

a10

-sec

ond

dura

tion

ofea

chac

cele

rogr

amw

asco

nsid

ered

.#

Rel

ativ

eto

the

init

ial

fund

amen

tal

peri

od,

T l,of

the

stru

ctur

eco

nsid

ered

.

Tab

le3

-S

tru

ctu

reP

rop

ert

ies

and

Ear

thq

uak

eIn

ten

sity

Lev

els

Co

nsi

der

ed

Set

Fun

dam

enta

lP

eri

od

,T 1

Yie

ldL

evel

,M

yL

evel

so

fIn

ten

sity

Use

d*

Ia

1.4

sec.

50

0,0

00

in-k

0.7

5,1

.0,

1.5

ro N Ib

0.8

sec.

1,0

00

,00

0in

-k1

.0,

1.5

c2

.0se

c.5

00

,00

0in

-k1

.0,1

.5

*In

term

so

fSI

ref

.

Tab

le4

Sum

mar

yo

fS

tru

ctu

ral

Par

amet

ers

PARN

1ETE

R

Fund

amen

tal

peri

od,

T 1(s

ec.)

VALU

ESOF

PARA

METE

RCO

NSID

ERED

(a)

0.8,

1.4,

2.0,

2.4

(b)

0.8,

1.4,

2.0,

2.4

(c)

0.8,

1.4,

2.0

(d)

0.8,

1.4,

2.4

REMA

RKS

My'

500,

000

in-k

My'

750,

000

in-k

My•

1,00

0,00

0in

-k

My=

1,50

0,00

0in

-k

(ry

"0.

05)

I(a

)50

0,00

0,1,

000,

000,

1,50

0,00

00

0 wY

ield

lev

el,

M(b

)50

0,00

0,75

0,00

0,1,

000,

000

I(i

n-k)

y1,

500,

000

(Ela

stic

)

(c)

500,

000,

750,

000,

1,00

0,00

0

T 1•

0.8

sec.

T 1•

1.4

sec.

T 1•

2.0

sec.

Yie

ldS

tiff

nes

sra

tio

,r y

(a)

0.0

5,0

.15

(b)

0.01

,0.

05,

0.15

(c)

0.0

5,0

.15

(T1

•0.

8se

c.

My•

1,00

0,00

0.in

-k

(T1

=1.

4se

c.

My=

500,

000

in-k

(T1

'2.

0se

c.

My•

500,

000

in-k

Tab

le4

(co

nt'

d.)

Sum

mar

yo

fS

tru

ctu

ral

Par

amet

ers

I co ..,. I

PARA

MET

ER

Para

met

ers

char

acte

rizi

ngsh

ape

ofM

-Shy

ster

etic

loop

:Ta

keda

mod

elpa

ram

eter

sa

and

B

Vis

cous

dam

ping

(per

cent

ofcr

itic

al)

Sti

ffne

ssta

per

EI(b

ijse)

ITlt

Oii

J

Str

engt

hta

per

My(b

ase)

My

(top

)

VALU

ESO~

PARA

MET

ERCO

NSID

ERED

(a)

a"

0.10

.0.

30

(b)

p"

O.0.

4.1.

0&

stab

lelo

op

(c)

.p"

0,0.

4,1.

0&

stab

lelo

op

(a)

5,10

(b)

5,10

(a)

1.0.

2.8

(a)

1.0,

2.0,

3.8

REMA

RKS

{

T1"

1.4

sec.

.My

"50

0.00

0in

-k

fl"0

{

T 1"

1.4

sec.

My"

500.

000

in-k

a"0

.10

{

T1"

0.8

sec.

My"

1,00

0,00

0in

-k

a;;

0.1

0

(T 1

"1.

4se

c.

My"

500,

000

in-k

[T

1"

0.8

sec.

My"

1,00

0,00

0in

-k

(T1

"1.

4se

c.M

"500

.000

in-k

y.

(Tl

"1.

4se

c.

My(b

ase)

"50

0,00

0in

-k

Tab

le4

(can

t'd

.)Su

mm

ary

of

Str

uctu

ral

Par

amet

ers

I(X

)

U1 I

PARAI~ETER

Bas

eF

ixit

yco

ndit

ion

(per

cent

offu

llfi

xit

y)

Num

ber

ofst

orie

s

VALU

ESOF

PARA

METE

RCO

NSID

ERED

(a)

50,

75,

100

(b)

50,

75,

100

(a)

10,

20,

30,

40

(b)

10,

20,

30,

40

REMA

RKS

(

T1•

1.4

sec.

*

My=

500,

000

in-k

rT1

•0

.8se

c.*

lMy

•1,

500,

000

in-k

rT1

=1.

4se

c.

lMy

=1,

500,

000

in-k

{

T1=

1.4

sec.

Mad

just

edto

yiel

dd~

ctil

ity

rati

os,,-

4-

6

*Ref

ers

toth

efu

lly-

fixe

dco

ndit

ion.

Table 5 Effect of Fundamental Period and Yield Levelon Dynamic Response of 20-Story Isolated

Structura 1 Hall s

Earthquake Input: 1940 El Centro, E-W Component (IO sec, SI = 1. 5 51 &)reI.

Fundamental Period, T1.~(s~e~c~.1 ___Yield Level, My(in-kipc,) 0.8

Initial

1.4 2.0 2.4

Max. Top Oisplacement (in.)

.500,000

750,000

1,000,000

1,500,000

Elastic

8.3

4.7

4.0

4.5

5.2

15.0

10.9

9.2

11.0

11.0

20.5

17.0

20.0

23.1

23.2

25.1

23.7

22.4

24.5 (E)*

24.5

Max. Interstory DisDlacesent (in.)

500,000

750,000

1,000,000

1,500,000

Elastic

0.47

0.28

0.24

0.29

0.35

0.86

0.65

0.58

0.75

0.76

1.24

1.12

1.25

1.57

1. 57

1. 56

1. 51

1. 51

1. 76 (E)

1. 76

Max. Horizontal Shear at Base (kips)

500,000

750,000

1,000,000

1,500,000

Elastic

1030

1190

1330

1610

1700

830

950

1190

1300

1420

1050

1070

1120

1150

1240

1060

1179

1240

1300 (El

1300

*Elastic, i.e., structure did not yield.

-86-

Table 5 (cont'd.) Effect of Fundamental Periodand Yield Level on Dynamic

Response of 20-StoryIsolated Structural Walls

Earthquake Input: 19·10 E1 Centro, E-W Component (Io sec., SI = 1.5 SIref.)

Yield level, 1,1 Initial Fundamental Peri ad, T1 (sec.)(in-k) Y 0.8 1.4 2.0 2.4

Moment at 8ase (in-k)

500,000 813,000 711,000 651,000 648,000

750,000 996,000 941,000 862,000 903,000

1,000,000 1,214,000 1,149,000 1, 126,000 1,107,000

1,500,000 1,701,000 1,533,000 1,609,000 1,519,000 (E)

Elastic 2,419,000 1,677,000 1,732,000 1,519,000

Rotational Ductility Ratio

500,000 13.6 8.1 6.2 5.7

750,000 6.3 4.9 2.9 3.92

1,000,000 4.1 2.9 2.5 2.1

1,500,000 2.6 1.1 1.4 1.0

Cum. Plastic Hinge Rotation at 8ase (radians)

500,000 1. 06xlO-2 1.50x10-2 1. 76xlO-2 2.31xI0,2

750,000 .92xl0-2 1.01xlO-2 1. 13xlO,2 1. 69xl0·2

1,000,000 0.73xlD-2 0.89XIG-~ .66xl0·2 1. 02xl0-2

1,500,000 .60xl0,2 0.04xlO-2 .09xlO-2 O.

-87-

Table 6 - Relative Changes in Response Quantities Due toChanges in Selected Parameters

Parameter Values Normalized ChangeParameter Pract i Cd 1 Range in Response

Considered %of Range Top dls- Base Rotational Remarli:splacement Shear Ouctil i ty

0.8 0 0 0 0

1.4 38 0.79 0.14 0.32 M .Y

2.0 75 1.46 0.01 I 0.48 500,000 in-k

2.4 100 2.01 0.02 0.52

0.8 0 0 0 0

1.4 38 1.29 0.18 0.22 M .Fundamenta 1

y

period~ 0.8 - 2.4 sec. 2.0 75 3.58 0.10 0.54 750,000 in-kT1 (sec.)

2.4 100 5.00 0 0.37

0.8 0 0 0 0

1.4 33 1.28 O.l! 0.3 M .Y

2.0 75 3.94 0.16 0.4 1.000,000 in-k.

2.4 100 4.55 0.07 0.5

0.8 I 0 I 0 0 0

1.4 0.38 1.45 0.19 0.58 M =y

2.0 75 4.13 0.28 0.461.500,000 1n-1o:.

2.4 100 4.45 0.19 0.63

0.75 0 I 0 a a1.00 33 0.15 0.16 0.34 T1 = 0.8 sec.

.75 - 1.5 1.50 100 O.l! 0.32 0.59(106 in - k)

0.75 0 0 0.16 0

1.00 33 0.16 0.03 I 0.41 TI =: 1.4 sec.1.50 100 0.01 0.33 0.77

Yield moment 0.5 0 0 0 0H

Y 0.5-1.5 (106 in-k) 0.75 25 I 0.17 0.32 0.53 TI = 2.0 sec.1.0 50 0.03 0.47 0.60

1.5 100 0.13 0.63 0.77

0.5 0 a 0 0

0.5-1.0 (106 in-k) 0.75 50 0.05 0.25 0.31 TI

= 2.4 sec.1.00 100 O.l! 0.41 0.63

-

-88-

Table 6 (cont'd.) Relative Changes in Response QuantitiesDue to Changes in Selected Parameters

0.01 a a a aYield

stiffness 0.01-(l..20 0.05 21 0.06 0.04 0.44ra tic. ry

0.15 74 0.24 0.08 0.65

0.02 a 0 0 a

Damping 0.02-0.15 of the 0.05 23 0.07 0.04 0.04 Ti = 1.4 sec..critical

0.10 52 0.17 O.l! 0.11 My =1.0 0 0 a 0 500,000 in-I:.

Stiffness taper 1.0-2.82.8 100 0.25 0.22 0.52

1.00 a a 0 aStrength taper 1.0-3.75 2.05 28 0.03 0.03 0.05

3.75 100 0.04 a 0.07

75 a a a 0 TI ,. 1.4 sec.

100 100 0.18 0.03 0.05 M = 500,000 in-I<.Base fixity 75% - lOO~

75 a a a a Tt ,. 0.3 sec.

100 100 0.45 o.oe 0.31 My ,. 1,500.000 in-I<.

Unloading 0.1 0 0 0 0 Iparameter, a. 0.1-0.3

100 I 0.02 a 0.01 T1 1.40.3 = sec.

0 a a a a t,' .YReloading

0.0-1.0 0.4 40 0.02 a 0.10 500,000 i n-kparameter. S

1.0 I 100 0.12 a a

Ground motion 10 a 0 a 0 Tt :: 1.4 sec.10 - 20 sec.

duration 20 laC 0.21 0.07 0.22 M = 500,000 in-kv

0.75 0 a a 0 11 '" 1.4 sec.1.00 33 0.58 0.09 0.7 M = 500 ,000 in-k

Ground motion yintens ity 0.75-1.5 (5I)ref 1.50 100 2.25 0.40 1.11

1.00 33 0.21 0.22 0.41 11 :c 0.8 sec.

1.50 100 0.63 0.57 1.24 My ,. 1,000,000 in-x

-89-

Fig. 1 Moment - Rotation Relationship

--

M ---'-~

e

Fig. 2a Stable Hysteretic Loop

R

M

/

l/ ....­l--

--

e

Fig. 2b Decreasing Stiffness Hysteretic Loop

-90-

(6 @ 7.3m 43.9m)

I• Eb

t=-:=-= ---- =----~---_. -' r<J..==== ---- .= - -=---=-- -'--- ---- 0 (1)<!)~

\,

Structural Wall AJ6 @ 24'-0"= 1441-0"

=

PLAN

8" -l(203 mmrr ~""""'====~?"'l

=r<J EI'-,d<!) l!1

<!)

"" E=(J'l!"-:,

-(1) (\J

@ @(J'l~

SECTION A-A

Fig. 3 Twenty Story Building with IsolatedStructural Walls

-91-

--l· ---~

Fioor Level

Bose

~:I--- --16 - - --

14

12 --- --­

10

a6

4 - ---

2- -

.- -- -- -"

20 - Mosses a-Mosses 5-Mosses

Fig. 4 Preliminary Models for 20 Story Building

Floor Level 20;18

16%14

12'108

64.3.Base ;L

Fig. 5 Final Model - 12 Masses

-92-

~

~

......c EIQ) KsE0 I~ beam

Rotation e(a) Model In computer program

(b) Bilinear element

......cQ)

Eo~

--"""'"1-' r1 EI

EI1

Rotation e

Fig. 6 computer Model of Bilinear Element(After Yield)

-93-

2

3

5

---Base

---4

Floor level---6

ypicalElement

r

T

~ff/ff~ 'l'/'

TypicalElement

~--l---- --

(0)

12 Element Model

(b)

25 Element Model

Fig. 7 Models for Lower Portion of Structural Wall

-94-

r~1 I /

EI

II I

l~I I w1

--I.

_6~----~

N.

-

.S I "'" ~ T4.

0o x w

r.-

--

--

----

:;l2.

0I

I"---l

---=.

._--~

__

/jI

..-;X

II

-1

-.-

.1!

([)

aI

II

I!

!I

I,_

1.0

1.41.

82.

22.

60

80

016

002

40

03

20

0F

unda

men

tal

Per

iod

(sec

)D

rift

Rat

io1

/8

----

(I)

0 0.6

N:6

.0c: I "'"~ 1

4 .0

x w 0,2.

0II

I '"c: ­..-

I ""U1 I

Fig

.8

Fu

nd

am

en

tal

Peri

od

-S

tiff

ness

-D

rift

Rela

tio

nsh

ip

Her. IntersteryDisplacement

Interstory TonqentiolDeviation

-t"""~----- Floor Level i + I

1-------- Floor Level i + I

Fig. 9 Illustrating Tangential Deviation

.06Relative

Displacement

oH.'------l

HorizontalDisplacement

0'-----

2.0 I'.89'1

2.0

Tangential

16 16DeViation

12. 12.

Q; Q;> >~

.....J

~ 8 l5 800 0u: u: lnterstory

Displacement

4 4

Fig. 10 Tengential Deviations versusInterstory Displacement

-96-

15 (a) Displacement of Top of Structure

10.08.04.0 6.0TIME (seconds)

2.0o-t5

-to

-;;;".<:: 10 T

1" = 1.4 sec.§

M = 500,000 in-kto- yZ 1940 El Centro, E-WW 5~

SI = 1. 5 (SI f)W() re •

:JCl. 0~cC0:I:

-5

2

10.08.0

T 1 = 1.4 sec.

M = 500,000 in-ky

1940 El Centro, E-W

SI = 1.5 (SI f)re •

(b) Horizontal Nodal Displacements

4.0 6.0TIME (seconds)

2.0o-5

-4

-;;;".<::

~ 0to-Zw::Ew -Iu«..JCl.en Floor leveli5

-2cC0:I:

-3

Fig. 11 Time Histories of Horizontal Displacements - Bldg. ISW 1.4

-97-

2

"' T1 = 1.4 sec.'"..c: Floor levels 6-8u

M 500,000 in-kc: =<::.4-6 y

t-- 1940 E1 Centro, E-WZ 2-4w SI = 1. 5 (SI f):: re.WU<t..Ja.'"i5ci0:cw 0>

~w0::

(a) Relative Horizontal Displacements Between Nodes

-I0 2.0 4.0 6.0 8.0 10.0

TIME (seconds)

-J,--f'-- --

Floor levels 14-16

16-1818-20

(b) Relative Horizontal DisplacementsBetween Nodes

T 1 = 1.4 sec.

M = 500,000 in-ky

1940 E1 Centro, E-W

SI = 1. 5 (SI f)re •

2

o+_/_~-

t-­ZW

C5U<ta:'"i5cio:cw>~..JWa::

-Io 2.0 4.0 6.0

TIME (seconds)8,0 10.0

Fig. 12 Time Histories of Relative Horizontal Displacement-- Bldg. ISW 1.4

-98-

4 Floor level 2

1 T1 = 1.4 sec.

M = 500,000 in-kY

3 1940 El Centro. E-W~

'? SI = 1. 5 (SI f)2 re •><

"' 2c:

"'5.gZ0F~0 0 ~Cl:

-I

-2o 2.0 4.0

TIME6.0

(seconds)8.0 10.0

Fig. 13 Time Histories of Nodal Rotations - Bldg. ISW 1.4

..,2 2><

"'c:

"'5.,g.zo!;i5 .la::w~zJ:I­2W

fli 0+----,::l(,,)

T 1 = 1.4 sec•

M ;;: 500~ 000 in-ky

1940 E1 Centro. E-W

SI = 1. 5 (SI f)re •

-Io 2.0 4.0 6.0

TIME (seconds)8.0 10.0

Fig. 14 Time History of Plastic Hinge Rotation at Base ­Bldg. ISH 1. 4

-99-

8T

11.4= sec.

M = 500.000 in-~~ y., 6Q 1940 El Centro. E-W

" 51 = 1. 5 (51 f)'" re •.,

4.c".5,'".9- 2.>:~

I­Z~ 0 -f-"~----'\­o::;:

(!). 2z-CizwOJ -4

-6o 2.0 4.0 6.0

TIME (second.)8.0 10.0

Fig. 15 Time History of Bending Moment at Base - Bldg. ISW 1.4

10

NQ

" 5'"c.ElJJ'-'Ct:0lJ..

Ct:<tW:r: 0(JJ

-5

T 1 = 1.4 sec.

M = 500,000 in-ky

1940 El Centro, E-W

51 = 1.5 (SI f)re •

f\

"N_V. - - . - .-'V\) ~ ~

~~

- J I,

I ! I,

Io 2.0 4.0 6.0

TIME (seconds)8.0 10.0

Fig. 16 Time History of Horizontal Shear at Base - Bldg. ISW 1.4

-100-

T 1 = 1.4 .ec.

M = 500,000 in-ky .1940 E1 Centro, E-W

51 = 1. 5 (51 f)re.

8~.,Q 6>Cen.9--'",

"c'='...:2UJ 2:00::t(!)

0z5zUJtD -2

-4

-6-1.0 o 1.0

CURRENT HINGE ROTATION

2.0

(radians 1I JO-3)

3.0 4.0

·80 T1 = 1.4 sec.

6 M = 500,000 in-k60 y

1940 El Centro, E-W

4 51 = 1. 5 (SI f)40 re.

2 20

Fig. 17 Moment-Plastic Hinge Rotation Relationship - Bldg. ISW 1.4

~ 10 TkN·m

.., 100Q 8xenCo

~~

-1.0 o 1.0

NODAL ROTATION (radians x 10-3 )

2.0 3.0

Fig. 18 Moment-Nodal Rotation Relationship - Bldg. ISW 1.4

-101-

(!) TOTALT

1= 1,4 sec. t. PRIMARY-(\J 2.0 ¢ SECONDARY'0

My = 500,000 in-k><.,<::0'60 1.5~~

Z0i=~0a: /.0lJJ>!;;t-J::J::;: 0.5a

16.012.08.0TIME (seconds)

4.0

0.0 Q-===",---«O_--+---~__+---_~__-+-__~_-4

0.0

Fig. 19 Cumulative Nodal Rotations versus Time for Node atFirst Story Level - Bldg. ISW 1.4

120

N 100Q><

'"~, 80<::=W>a:::J

60uCl)

I::;:

a: 40waz::J<tw 20~

T1

= 1.4 sec.

My = 500,000 in-k

(!) TOTAL6 PRIMARY¢ SECONDARY

16.012.08.0TIME (seconds)

4.0O'---~-+-~'-----'---+---~---+---~--4

o

Fig. 20 Cumulative Rotational Energy versus Time for Nodeat First Floor Level - Structure ISW 1.4

-102-

Mmnent vs. Shear at Base

x XXX

XX X X

X X X X XX X X

10.08.0

x

xx x >It

x

x Xx I

x xx

xx x

xx

xx

xx

x

6.0MOMENT(kip-inches x 105)

xx

x

xx

x

~

x x xx

x xx

x>\(

xx

x

4.0

BENDING

x

x~ x

x x

xx

xx

x xx

x x

,x

x

xx x x

x

x

x

xx

2.0

x x x'"xx x x

x x x Xx xXx x

Xx x x 'l<x

x

x

x

x

x

x

x

x

xXX X X X X

X XX X X

20

80

o0.0

won:e 40

n:::l5I(f)

Q 60><<f>a.6

Fig. 21 Simultaneous Moment and Shear Values at Baseat Each Time Interval During Response

Bldg. ISW 1. 4

-103-

-0: %0 00

8II OJ 2.5 Ng"~ ~~ u

w 1940 El Centro, E-W~ ~

0 0 "~ x 1. 88~~

"" ... ~~ ·OJw~

~ ·20 .'0

-0 20 008II ZS,Ng" OJ

II ~

~u

1940 El Centro, N-Sw0 0 •"

~~ x 1. 5n~

~ "~ -10 ... !~~ ·20 .'0

20 00

-0g 10 "!:l" NO 1971 Pacoima Dam, S16E~ o -~ 0

~~ u x 0.594w

~

~ ·'0 ... "~~ ~

~ "d·:0 .."~

~.>0 .?>

·0 20 008i;l

"' Z5 "/Q"~~ ~

~ 0~ 1971 Holiday Orion, E-W0 •" 3.22~ ~ x•

~ "ffi ·OJ ... !~v~ -20 .'0

o

~ -'0

!.20t~ .·~:t_~-t---+---+-+---+--+-t--- - I I I I I

2 :. IS 7 fa

T(~ SECOf'CS

Artificial Accelerogram, SIx 1. 66

Fig. 22 Ten-Second Duration Normalized Accelerograms

-104-

80

70

1940

E1

Cen

tro

.E

-W"

1.8

8------

19

40

E1

Co

ntr

o,

N.S

"1

.50

-'-

-'-

19

11

Paco

ima

Dam

,S

16

E"

0.5

94

_.._

...

1911

Ho

lid

ayO

rlan

,E

-W"

3.2

2---

1952

Taf

t.S

69

E"

3.1

1a•••••••••••••••

Art

ific

ial

Acc

eler

og

ram

,51

x1

.66

20

00

~'0 <ll

rJ) .....

1000

E E

50

0

1500

'-

........

.......

.

S'lo

Dam

ping/\ i\.'"

"'.;

\A

,'"

1\

/-',.//\

\'"

'.,

,\/

0'·

'\/'..

.....

/..,1

\rA

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,.

/'.'

.\.

/~

/\

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j.J\"

\,

/\/\

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...\.

,,/

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..J"..

....

...

•J.

...J..

Jl)/

;'\.....

.{',"

/........

n'

..I

,.

/',.

/...

",.1\

r"";'W

\·\;

,\/~.._

.""'..,/

/~'"

I'.I

:!.

"'.

..'/V

......,

~,.....

~~_

",'.

.."

,.y,

.'~,

J""

1 M'~

./'

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J'.....

".

.I

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--'<'

:"...

.._._.~

I.".

/','

\I

..,'.

...._

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.

f/"IV'

...'

\,.1

........

.!,../

.......

...;ff

Y\I

"-"

-"_

00

-fir

\./i~1:., if

I

W f" :I ,:,:

60

20 10

>­ I- g40

-l

W > ~3

0:2

:

.~ U <ll

rJ) ';::

50

:.::.

I I-'

o U1 I

o o1.

02.

0P

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(se

con

ds)

3.0

4.0

Fig

.23

Rela

tiv

eV

elo

cit

yR

esp

on

seS

pectr

afo

rF

irst

Ten

-Seco

nd

so

fN

orm

ali

zed

Inp

ut

Mo

tio

ns

Dam

.

Orla

n.

EIC

entro

,E

-W

Pac

oim

a

SI6

E

Hol

iday

E-W

30

1j=

1.4

sec.

My

=5

00

,00

0in

-k

Bas

edon

Rel

ativ

eD

ispl

acem

ents

Bas

edon

Tan

gent

ial

Dev

iatio

ns

20

" IlI

IIi

SI

III

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1

i--i

lI

1971

~111

I•

I

I",

I1

94

0,,I,~

I),

I_1

971

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10

I /.

II'

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,/

f/'

If,'-

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/I

II

I

Max

.In

ters

tory

Dis

plac

emen

t(in

.)

20 515

., ~ ..J

to,., ~ o - en

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94

0E

IC

on

tra

,E

-W

T I=

1.4

sec.

My·

50

0,0

00

in-k

5

10

02

00

30

04

00

(mm

)o

r!

II

II

II

I

o5

.010

.015

.0

Max

.H

oriz

onta

lD

ispl

acem

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(in.)

20

I91

1971

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rla

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15t

1971

Pa

coim

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am,

Sl6

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a..

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,., ~ 0 -lf)

(a)

(b)

Fig

.24

Eff

ect

of

Fre

qu

en

cy

Ch

ara

cte

risti

cs

of

Gro

un

dM

oti

on

60

(kN

,Icr,

10.0

'---

-<15

.0

(kip

sxId

)

T 1•

1.4

sec.

My

•5

00

,00

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.-k

20

--'-

+­

50

Max

.H

ori

zon

tal

She

ar

oo

15

(kN

.m'1

03)

>I

I

60.0

00

.0

(in

-Id

psx

104

)

20

20

40

II

II

20.0

40

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Mox

.B

endi

ngM

omen

t

oo

1520

Q)

51>

Q)

Q)

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J10

Q)

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10

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Pac

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aO

am,

SI6

E--

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71H

oli

day

Orl

an,

E-W

~1

94

0EI

Cen

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(f)

(f)

T1

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4se

c.19

71P

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am

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50

0,0

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,E

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~19

71H

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day

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on

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I

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(d)

Fig

.24

(co

nt'

d.)

Eff

ect

of

Fre

qu

en

cy

Ch

ara

cte

risti

cs

of

Gro

un

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oti

on

20

20

1i

T I~

1.4

sec.

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fAse

c.I

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My

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00

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IO)

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SI

I"7~

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1.0

2.0

3.0

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atio

nal

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tili

ty

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OJ

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1\/

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71H

oli

day

Orl

an,

E-W

00

'-'-

I0

0-

-~~I

(/)

--

1971

Pac

oim

aD

am,

SI6

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__

1971

PC

lcol

ma

DO

M,

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19

40

EIC

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(e)

(f)

Fig

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(co

nt'

d.)

Eff

ect

of

Fre

qu

en

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Ch

ara

cte

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cs

of

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NO

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Flo

or

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1971

Ho

lid

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1971

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I8

t""

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4

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II

II

II

II

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t

0.0

102.

03

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05.

0T

IME

(sec

onds

)

6.0

70

8.0

9.0

10

0

Fig

.2

5N

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ali

zed

Ro

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on

sin

Fir

st

Sto

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us

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II

II

II

II

I

0.0

Fig

.26

No

dal

Ro

tati

on

His

tory

for

Str

uctu

reS

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jecte

dto

19

40

El

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po

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6.0 ,..-----,--,--,,----nr--rr-.--,------r---,

5.0

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2.0

I' 5.7 x 107;n.4

E . /. 2I' 3600 kips 10.

A' 1500 in2

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W2o '194kips

1'1, .... 19

1.0 L-_--L-'--_-'L-_---"__'-'-_-'--'--_--'-__-'

o 10 20 30 40 50 60 70Height of hinge(feetl

Fig. 27 Fundamental Period versus Height of Yield Hinge,20-Mass Cantilever

-111-

2.5

5.0

7.5

10(m

m)

o'

'I'I

II

II

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mm

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.28

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of

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.28

(co

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d.)

Eff

ect

of

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qu

en

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Ch

ara

cte

risti

cs

of

Gro

un

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1.5

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1940

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(can

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ffect

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Fig

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Eff

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d.)

Eff

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of

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(mm

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qu

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.)In

tera

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of

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of

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en

cy

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ara

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Fig. 31a Twenty-Second Composite Accelerogram

80

Datnping ;;; 0%20/05%

60

-0..~

:S>-... 40U0.JW>

20

4.03.01.0

o~----+----+------+--------<0.0

Fig. 31b Velocity Response Spectra for 20-SecondComposite Accelerogram

-121-

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2E

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of

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of

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of

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T2

0

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Eff

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of

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of

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8

Accum. primary rotations = LO+, LO­

Accum. secondary rotations = Lb+, 2:b-

Fig. 33 Components of Cumulative Plastic Rotationsfor Takeda Stiffness Degrading Model

-125-

20

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(bl

./

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tal

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Eff

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5+

"J;=

0.8

sec.

\\

5T

1=

0.8

sec.

My

=1

,00

0,0

00

In.-k

My

=1

,00

0,0

00

in.-k

01

50

100

"'"

(kN

.mtI

O O"

20

40

(kN

1102

)I

I!

I0

05.

010

.015

.00

5.0

10.0

15.0

Max

.B

endi

ngM

omen

t(I

n-ki

psI

105

)M

ax.

Hor

izon

tal

She

ar(k

ips

11

02)

15+

\"

15

20

.2

0

'iiiQ

l>

>j

Q)

I10

...J

10\-

'W

»»

\-'

~~

I0

0i7i

-(f)

(c)

(d)

Fig

.36

(co

nt'

d.)

Eff

ect

of

Inte

nsit

yo

fG

rou

nd

Mo

tio

n

20

20

/6

1;•

0.8

sec.

My

•1,

000,

000

in~k

~=

0.8

sec.

My

=1

,00

0,0

00

in.-k

15

~i"I

\\'":> (l

)

...J

10

'"... 0 -C/)

8.0

6.0

-3(

rod.

-10

)

SI

•1

.0

•1.

5

2.0

4.0

Cum

.Pi

astic

Rot

otio

ns

(f)

5 01~1

II

o6

.02.

04

.0

Rot

otlo

nol

Duc

tility

(e)

o!

I--

-....

=-=:

-.,.

Io5

Fig

.3

6(c

on

t'd

.)E

ffect

of

Inte

nsit

yo

fG

rou

nd

Mo

tio

n

20

TI

/2

0I

T 1=

2.0

sec.

j/

:1/

My

=50

0,O

OO

in"k

"I

-15

Qi >

'"cP

>--

l10

'"I

--l

10f-

'

>,

W

....W

>,

0....

-I

0

(f)

-(f)T 1

=2.

0se

c.

1515

+/

/M

y=

50

0,0

00

in~k

200

40

060

0(m

m)

01

102

03

0(m

m)1

o!

,I

II

II

II

!I

I

010

.020

.030

.00

Q5

1.0

1.5

Max

.H

orlz

onto

lD

ispl

acem

ent

(In.

)M

ax.

Inte

rsta

ryD

ispl

acem

ent

(In.

l

(al

(b)

Fig

.37

Eff

ect

of

Inte

nsit

yo

fG

rou

nd

Mo

tio

n

20

~2

0

I\"

'"T 1

=2

.0se

c.

10-I

\.""

-M

y=

50

0,0

00

in~k

10

60

(kN

,102 )

,I 15.0

40

20

5.0

10.0

Max

.H

oriz

onta

lS

hear

(kip

s<1

02)

oo

(kN

'm,1

03)

~

eo.o

40

20 2M

4M

6M

Max

.B

endi

ngM

omen

t(I

n-ki

ps<

104

)

'"Q

;>

>'"

'"I

-J10

'-J

10t-

'W

'"'"

'"'-

'-I

aa

--

(f)

(f)

1j=

2.0

sec.

0+M

y=

50

0,0

00

In~k

\\

5

(c)

(d)

Fig

.3

7(co

nt'

d.)

Effect

of

In

ten

sit

yo

fG

rou

nd

Mo

tio

n

20

20

~=

2.0

sec.

My

=5

00

,00

0in

,k15

T I=

2.0

sec.

My

=5

00

,00

0in~k

15

iii

Qi

>>

Ia>

a>..

J10

..J

10/-

'W (J

l>

0>

.I

~~

00

iii-CIl

5 oI~

====

-=7=

tI

o2

.04

.06

.0

Rot

atio

nal

Duc

tility

5·

51•

1.0

•1.

5

01~1

Io

1.0

2.0

300

Cum

.P

last

icR

otat

ions

(ra

d.x

102

)

(e)

If)

Fig

.3

7(c

on

t'd

.)E

ffect

of

Inte

nsit

yo

fG

rou

nd

Mo

tio

n

20I

/2

0

//

/I

<.I

!I<.I

<.l~

..'"

'"~

'<t

<Xl

~N

0'"

IS+

c/

..../

",:/

n~

/1

5+

II ...-

Qj

.>>

>'"

j10

I...

J10

I-'

W>

.>

.0

"\~

~

I0

0

-Vi

(J)

My

=5

00

,00

0in

,j{ip

s

5+

/1

//

5

(b)

20

04

00

60

0(m

rri)

o'

II

II

II

010

.020

.030

.0

Max

.H

oriz

onta

lD

ispl

acem

ent

{in.>

(al

10 II

0.5

Max

.In

ters

tory

1.0

Dis

plac

emen

t(i

n.)

1.5

Fig

.38

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

1

20

20

15.0

60

(kN

'10

2)

20

5.0

10.0

Max

.H

oriz

onta

lS

hear

(kip

s.10

2)

My

=5

00

.00

0in

~ki

ps

oI

,I

I.1

111

[I

o

15

My

=5

00

,00

0In

.-kip

s

(kN

'm,

103

)0

1I

I\\\

'\!

I'

Io

5.0

10.0

I~.O

Max

.B

endi

ngM

omen

t(i

n-k

ips

x10

5)

I~

Qi > Q)

10..J

Q)

"t\

,.,>

'-

Q)

0

WT

I=

0.8

.ec.

..J

-CIlI

J2

0••

c.

f--'

,.,

"0

.

~2'

.4••

c.

w'-

-T,

•2

.4

-..J

0 -2

.0,e

e.

IC

Il

1.4

'''c.

1M

//

/"0

.8.ee

.~+

(c)

(d)

Fig

.38

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

20

20

I~

My

'50

0,0

00

In"k

ips

I~

My.

50

0,0

00

In.-k

ips

Q)

0;

>>

Q)

Q)

I.J

10.J

10f-

'W

>.

>.

co~

~

Ii?

0 -(f

)(f

)

~T

1=

0.8

sec.

•1.

4

,t:\-~;;:

,o

1.0

2.0

3.0

Cum

.Pl

astic

Rot

atio

ns(r

ad.'

101

I~O

10.0

~.o

Rot

atio

nal

Duc

tility

0'

~-....

...I

........

Io

~

(e)

(f)

Fig

.38

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

Momentat

Base stiff structure, lj = 0.8 sec.

------ 2~~";'F==:::::~-:=.:-:.:-:.:~-----T-----

Yield Rotation Max. ROfatione y (red.l erne. Ired.)

I .00014 .00135

2 .00070 .00448

flexible structure, T, = 2.0 sec.

Nod 01 Rotat ion

Ist Floor Level

MEASURES OF DEFORMATION

Ductility Ratio vs. Absolute Rotations

Iernex .M, = = 9.6 but note that:

ely

2 Iernex . = 3.3 ernex .2

M2ernex.

6.4= =e2

y

Fig. 39 Rotational Ductility Ratio versus Maxim~~ Absolute Rotationas Measures of Deformation

-139-

15 10

IT

"0

.8s

ec

,1

"1

.4se

c.

"2

.0

5+

"2

.4se

c.

zl~

IQ~

Itr~

0I-

'I- 0

-I."

.0:

:0

0-'

IW >=

-5 10~

"5

00

,00

0in~k

NO

DA

LR

OT

AT

ION

SIst

Flo

or

Lev

el

9.0

8.0

7.0

5.0

6.0

TIM

E(s

econ

ds)

4.0

3.0

2.0

1.0I

II

II

II

II

I-l 10

.00.

0

Fig

.4

0a

No

rmali

zed

Ro

tati

on

sin

Fir

st

Sto

ryv

ers

us

Tim

efo

rD

iffe

ren

tV

alu

es

of

Fu

nd

am

en

tal

Peri

od

,T

l

6,

.T

1=

2.4

sec.

=2

.0se

c.

=1

.4se

c.

=0

.8se

c.

.--

4

to '02

>< if> co .<e "I

0

I-'

~

r~

~z

0I-

'0

If= <r r- 0 0

:

-2 -4

MY

=5

00

,00

0iu

"k

NO

DA

LR

OT

AT

ION

S1

stF

loo

rL

ev

el

"_~

I__

.._

_-

----'

1.02.

04

.05.

06.

0T

IME

(sec

onds

)7.

08.

09

.010

.04"

Fig

.4G

bA

ctu

al

Valu

es

of

Ro

tati

on

inF

irst

Sto

ryv

ers

us

Tim

efo

rD

iffe

ren

tV

alu

es

of

Fu

nd

am

en

tal

Peri

od

,T

l

My

:7

50

,00

0in

,.k

2.0

IOC

.1

.4 ••c.

o.e

••c.

<lJ > <lJ

.-J

1020 o

T1

10

>,~ o -CIl

My

:7

50

,00

0in

,.k

,; Ii ~

51520

"iii > <lJ

10I

-JI-

' "'">

,tv

~

I0 -CIl

Max

.H

oriz

onta

lD

ispl

acem

ent

(In.!

(a)

(mm

)

1.0

:;0

20

10

0.0

1.0

Max

.In

ters

tory

Dis

plac

emen

t(in

.)

(b)

o!

II

II

!I

'Io

(mm

l--

--; 30.0

60

0

20.0

40

02

00

JI 10.0

01

-'-

-­

o

Fig

.4

1E

ffect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

1

15.0

60

(kN

'102

)!

I

My

=7

50

,00

0in

:-k

20 l5

.010

.02

Max

.H

oriz

onta

lS

hear

(kip

sx

10)

T.0

2.4

soc.

•2

.0se

c.

•1.

4se

c.

•o.

eso

c.

o o

151520

150

(kN

'm'1

03)

,I 115.0

o10

.0

(in-k

ips

x105

)

My

=7

50

,00

0in

~k

Mom

ent

50

5.0

Max

.B

endi

nll

o o

51520

lii

"j\

'">

>j

'"I

....I

10f-

-'o!

'>>

.>

.w

~~

I0

0

--

(fl

(fl

(c)

(d)

Fig

.4

1(c

an

t'd

.)E

ffect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

20

20

My

~7

50

,00

0in

.-kM

y•

75

0,0

00

In:-k

1515 J~£L

II

o1.

02.

03

.0-2

Cum

.Pl

astic

Rot

atio

ns(r

od.x

/O)

6.0

2.0

4.0

Rot

otio

nol

Duc

tility

ol~,

Q; >

Q)

Q)

>

-l

10Q

)

I-l

10>

.f-

'

...".

>.

0".

...-

I0

(/)

- en

.~rli

~0.8

:sec

.2

.4sa

c.

1.4

S6

0.

H\I

\/

•2

.0.e

c.2

.0ae

c.~

1.4

sec.

2.4..

•sc

.•

0.8

sec

.

(e)

(f)

Fig

.41

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

I f-'

",

U1 I

Q) > Q)

-l >, ... o .... l/

)

20

1u "~ ~ 0 "

10+

,..-

10 5M

y•

',OO

O,O

OO

ln.-k

20

u "~15+

~-Iu " ~ 'I; -

,..-

IQ

) > Q)

-l

10>

, ... 0 ..- l/)

5

My

=1

,00

0,0

00

in.-k

20

04

00

60

0(m

m)

oI

,I

'I

!I

10.0

20.0

30

.0

Max

.H

orIz

onta

lD

ispl

acem

ent

(In.)

(a)

102

03

0(m

m)

oI

!I

'I

JI

0.5

1.0

1.6

Max

.In

ters

tory

Dis

pla

cem

en

t(i

n)

(b)

Fig

.4

2E

ffect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,1'

1

My

=1

,00

0,0

00

in.-k

oI

l!,o

I4

pI

'-.L

-1(k

NxIO

~)6

.010

.016

.0

Max

.H

ori

zon

tol

Sh

ea

r(k

ips

x,d)

6

I~

l!0

(kN

'mxI

03)

100

10.0

16.0

Mom

ent

(In-

kips

x10

5)

60

6.0

Max

.B

endi

ng

My

=1

,00

0,0

00

Inck

aI

!I

'I

""

....

.!I

616l!0

Q)

ro>

>ro

Q)

-l

10I

-l

10>

.f-

'

...""

>.

0'"

...~

I0

Cf)

~ Cf)

Ie)

(d)

Fig

.42

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

20

20

My

•1

,00

0,0

00

In~k

My'

1,0

00

,00

0in

.-k

I~

15

'"'"

>>

'"Q

)

I-l

10-l

10f-

' ..,.>

.>

.-.

J...

...0

0I

~.,..

(J)

(J)

,-T

1•

2.0

sec.

,0

.6

..c.

..

1.4

10

0.

5 oflJ

jI

I

0.5

1.0

I.~

-2C

um.

Pla

stic

Ro

tati

on

s(

rod

.'10

)

6.0

2.0

4.0

Rot

atio

nal

Du

ctili

ty

5 o'

I~

==---

-.-I

o

(e)

(f)

Fig

.42

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

l

My

=1

,50

0,0

00

inck

5

.2

0

My

=1

,50

0,0

00

in~k

520

cJ ~ ~

15+

~I

_,,:/

.tj

15~

'V.

*....,

a:;Q

)>

>Q

)Q

)..

J10

..J

10I

>0

/-'

>0

""'-

'-0

00

0-

-1

Ul

Ul

(a)

Max

.H

oriz

onta

lD

ispl

acem

ent

(in,)

20

04

00

20

40

60

(mm

l0

1,

I!

I!

01.0

2.0

3.0

Max

.In

ters

tory

Dis

plac

emen

t(i

n.)

(b)

Fig

.4

3E

ffect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

, L

My

=1

,50

0,0

00

In,..

k

oI

'4,~

I~

e,o

I(k

N'1

,02l

o10

.020

.030

.0

Max

.H

oriz

onta

lS

hear

(kip

sx

102

)

1020

My

•.1

.50

0,0

00

In~k

oI

I~I

\\\

2p,

(kN

":U

I041

o10

.02

0.0

30

.05

Max

.B

endi

ngM

omen

t(I

n-k

ips'

10)

1020

a:; > Q)

10...

J

"j\

>.

•2

.4••

c.

...T

I

Q)

0

"

>

...1.

4••

c.

Q)

(/)

I

=

I...

Jf-

-'

~..

----

'o,

ese

c.

"">

.\D

...

o~

0I

-(/)T,

=o.

eao

c.\

\\/

=1.

4se

c.0+

•2.

4S6

C.

(c)

(d)

Fig

.43

(co

nt'

d.)

Eff

ect

of

Fu

nd

am

en

tal

Peri

od

of

Vib

rati

on

,T

1

20

.W

6.0

4.0

(ro

d.

x16~

My

D1

,50

0,0

00

In~k

2.0

Cum

.P

last

icR

otat

ions

15

'"> <lJ

10-l ;>,

.... 0 .... en

5j;-

TI=

2.4

86

C.

•1.

4se

c.

=0.

8se

c.

My

=1,

500,

000

In;o

k

01~~

Io

I~

~~

Rot

atio

nal

Duc

tility

0+\

--\-

\---

----

-1j

=2

.4se

c.

--------

•1.

4••

e.

•O

.BIt

c.

15

Qj > <lJ

I-l

10I-

'Lf

1;>

,0

I. 0

I-en

(e)

(f)

Fig

.43

(co

nt'

d.)

Eff

ect

of

Pu

nd

am

en

tal

Peri

od

of

vib

rati

on

,T

1

I f-'

U1 I-' I

a:; fi; --J >.~ .2 (f)

10 5

My

•5

00

,00

0In

-k

T1=

1.4

sec.

a:; j 1:- o -(f)

10 5

T,=

1.4

sec.

Max

.H

oriz

onta

lD

ispl

acem

ent

(in.)

Max

.In

ters

tory

Dis

plac

emen

t(In

.)

102

03

0(m

m)

oI

II

'I

'I

o0

.51.

0U

l

20

010

03

00

(mm

)o

'!

I!

II

Io

5.0

10.0

15.0

(a)

(b)

Fig

.4

4E

ffect

of

Yie

ldL

ev

el,

My

20

I1'\0

I~11(kN"02)

o!

1I·

1o

5.0

10.0

15.0

2M

ax.

Hor

izon

tal

She

ar(k

ips'

10)

20

10.0

20.0

30.0

5M

ax.

Ben

ding

Mom

ent

(In-

kips

xIO

)

20

T2

0,~,

T1

=1.

4se

c.I

T,

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(f)

Fig

.49

(can

t'd

.)E

ffect

of

Slo

pe

of

po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

20

TH

20

IT

1'

0.8

sec.

I~-J

.0\

'f

My

'1,

000,

000

in~k

I~

'"Q

)

>>

<l)

<l)

I....

J1

0....

J10

f-' '"

;.,

;.,

-..]

~~

I0

0-

-(J

)(J

)

T1

'0

.8se

c.

5+II

My

'1

,00

0,0

00

in.-

k5

r:

y 5%

(mm

) ---i 3.0

60

2.0

40

1.0

20

O!

'I

II

I

o15

0(m

m)

..>

...-

.; 6.0

100

-'-

t­

4.0

50

-'+ 2.0

oo

Max

.H

ori

zon

tal

Dis

pla

cem

en

t(in

.)

(a)

Max

.In

ters

tory

Dis

pla

cem

en

t(i

n.)

(b)

Fig

.5

0E

ffect

of

Slo

pe

of

Po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

202

0

Max

.B

en

din

gM

om

en

t

(c)

5.0

Max

.H

ori

zon

tal

Sr,

ea

r

15.0

kN'1

02

)--l

10.0

2(k

ips

x10

)

40

(d)

20

T1

=0

.8se

c.

My

~',0

00

,00

0Ci o

o

1015

>.

'­ o<l> > <l>

...J

Ul-

f y=

15%

r--5

%

3kN

'm,I

O)

~15

.0

(in

-kip

sx

105

)

10.0

50 4

­5

.0

T,=

0.8

sec.

My

=1

,00

0,0

00

in7k

5 oo

15·

., > <l> ...J

10I

>.

I-'

~

'"0

00

-I

Ul

Fig

.5

0(C

an

t'd

.)E

ffect

of

Slo

pe

of

po

st-

Yie

ldB

rCln

cho

f!1

om

en

t-R

ota

U,o

nC

urv

e

20~

20

Tr

=0.

8se

c.T,

=0

.8se

c.

My'

1,00

0,00

0in

.-kM

y=

1,00

0,00

0in

.-k

15

+\

15

'"'"

".

".

'"'"

I..

J10

..J

10I-

'>

.'"

>.~

~

u:>

00

I~

~

(/)

(/)

=5

('/0

55 0'

I'>

===.

10

00

2.0

4.0

6.0

2.0

4.0

6.0

8.0

Ro

tati

on

al

Du

cti

lity

-3C

um.

Pla

stic

Ro

tati

on

(ro

d.x

10)

(e)

(f)

Fig

.50

(co

nt'

d.)

Eff

ect

of

Slo

pe

of

Po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

T1

=2

.0se

c.

My

=5

00

,00

0in

.-k15

<-

20

1520

Q)

In'

>>

Q)

Q)

I-.

J1

0-

.J10

>-'

>.

>.

-..J

~~

00

01

~~

(J)

(J)

TI

=2

.0se

c.

5+

IIM

y=

50

0,0

00

in,.

k5

20

04

00

600

102

03

0(m

m)

oI

,I

II

lI

o0.

51.

01.

5

Max

.H

oriz

onta

lD

ispl

acem

ent

(in.)

(a)

Max

.In

ters

tory

Dis

plo

cem

on

t(in

.)

(b)

Fig

.5

1E

ffect

of

Slo

pe

of

Po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

20

'2

0

60

(kN

,(02

)I

I10

.015

.02

(kip

sx

10)

40

(d)

5.0

Max

.H

ori

zon

tal

Sh

oa

r

20

T,~

2.0

soc.

My

~5

00

,00

0In

"k

5 00

'"> <1)

...J

10>

.~ o -U)

Max

.B

endi

ngM

oman

t

(c)

oI

2,0I

4,0

I6,0

I"'"

a~N'

~'(0

5)o

20.0

40

.060

.06

0.0

(in

-kip

sx

104

)

15

+~

15

<ll > <ll

--I

10I !-'

>.

-J~

/-'

0

I-

IT,

=2

.0se

c.U

)

My

=5

00

,00

0in

"k

5

Fig

.51

(Co

nt'

d.)

Eff

ect

of

Slo

pe

of

Po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

20

T2

0

T,

c2

.0se

c.T,

=2

.0se

c.

My

=5

00

,00

0in

.-kM

y=

50

0,0

00

in~k

15

+\\

15

0)

0)

>>

0)

0)

..J

10..

J10

I f-'

'"'"

--J

~~

IV0

0-

-I

(j)

(j)

5

=t~%

:.5

%

5

r y=

15%

=5%

0'

I

'"-=

-,..

I0

02.

04

.06.

00

1.0

2.0

3.0

Ro

tati

on

alD

ucti

Iity

Cum

.P

last

icR

ota

tio

ns

(rad

.•,0

2)

(e)

(f)

Fig

.5

1(c

on

t'd

.)E

ffect

of

Slo

pe

of

Po

st-

Yie

ldB

ran

ch

of

Mo

men

t-R

ota

tio

nC

urv

e

I f-'

-J

W I

10

~

'" .9- "'" I5

.£ '"0 = I- Z w :;;:

0 :;;:

C)

0z 0 z w U

J

-5

T1

=1.

40

sec.

M=

50

0,0

00

in-k

yr

y=

15%

-4-2

o2

46

CU

RR

EN

TH

ING

ER

OTA

TIO

N(r

adia

nsx

10-4

)8

Fig

.5

2M

om

en

t-R

ota

tio

nL

oo

pfo

rF

irst

Sto

ryH

ing

e

I·ap

• IM

III KoI

I

88rec

a8p

8p

(a) unloading stiffness

8

j:•

Ku /.: KoItI .

/ :

I·­···l~

M

~ .,.....-•••••••,

•.I / I ./ f

: / I / : 8+: I 1,/: rec. I / :fl Ku ...a......~••

•j 1/..•.••L--- -' stable loop

8 rec

" .

(b) reloading stiffness

Fig, 53 Takeda Model Parameters a and 6

-174-

2.0

2.0

2.0

103

0(m

m)

01

rI

!I

II

o0.

51.

01.

5

Max

.In

ters

tory

Dis

plac

emen

t(in

,)

15.0

(mm

)3

0

10.0

2.0

5:0

10

Max

.H

oriz

onta

lD

ispl

acem

ent

(in,)

;/I~

II

15t

tI

.n/Y

I

15

+

a; > <l.l

10..

J

/>

.

"I~

'"

0

>

(jj

T=

1.4

sec.

'"...J

jJ

I

I.

k

I-'

>.

5tM

=5

00

,00

0In

~

--J

~

=1.

4se

c.

y

U1

0I

(jj

1j My

=5

00

,00

0in

~k/

5+

(a)

(b)

Fig

.54

Eff

ect

of

Pla

sti

cH

ing

eM

-0H

yste

reti

cL

oo

pU

nlo

ad

ing

Para

mete

rsa

~T,

=1.

4se

c.

My

=5

00

,00

0in

.-k

~20

"2

0

I'"

1j=

1.4

sec.

I~~

'"M

Y=

50

0,0

00

in.-

kI

III~

~!"I

\.(j

j ,. Q)

...J

10

-.J

,.,

,.,~

~

0)

00

I-

Cii(f

)

204

0o o

20

-'-+

­5.

0

Max

.H

oriz

onla

lS

hear

40

10.0

(kip

s'

102

)

60

(kN

,102 )

!r 15

.0

(c)

(d)

Fig

,54

(co

nt'

d.)

Eff

ect

of

Pla

sti

cH

ing

eM

-8H

yste

reti

cL

Oop

Un

load

ing

Para

mete

rsa

1.5

0.5

Cum

.P

last

icR

atat

ions

01:

Ir

ro

0'

I.....

II

o~

1M1

M

Rot

atio

nal

Duc

tility

20l

20

I

15+

\1;

~1.

4se

c.

~=

1.4

sec.

My

~5

00

,00

0in~k

IM

y=

50

0,0

00

in"k

15

Q; > '"

I..

J10

>-'

03

">

.

> '"

"~

..J

0I

iii

10

-a

=0

.30

>.

~ 0 -0

.10

(j)

I'"

..x

lC=a

=0

.30

5

.,/

=0.

10

Ie)

If)

Fig

.5

4Ic

on

t'd

.)E

ffect

of

Pla

sti

cH

ing

eM

-eH

yste

reti

cL

oo

pU

nlo

ad

ing

Para

mete

rsa

2°T

',

n2

0

BIl

inea

rS

lab

Ie

f3=

1.0

VJB

ilin

ear

Sla

ble

=0

.4f3

=1.

0

IH=.o~

•0

.4

=0

Max

.In

ters

tory

Dis

plac

emen

t(in

,)

102

03

0(m

m)

oI

II

"I

Io

MID

L5

30

10.0

Dis

pla

cem

en

t(in

.)

5.0

Max

.H

ori

zon

tal

or

,I

II

I(m

m) I 15

.0

Q} > Q)

...J

Q)

I10

>

!-'

>-

Q)

--.J

~

...J

10

co0

I-

>-

m

~ 0 -T,

=1.

4se

c.m

//jT,

=1.

4se

c.

H/f

fM

y=

50

0,0

00

In.-k

My

=5

00

,00

0In

,.k

5

(a)

(b)

Fig

.5

5E

ffect

of

Pla

sti

cH

ing

eM

-8H

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rfJ

60

(kN

x102

),

I10

.015

.0

(kip

s,1

02l

20

....I.

...+­

5.0

Max

.H

ori

zon

tal

Sh

sar

oo

Max

.B

en

din

gM

om

en

t

oI

2,0,

4,0

16,

0I

~N.mx,o~l

o20

.04

0.0

60

.08

0.0

(in

-kip

s'1

04)

20

..

T,=

1.4

sec.

20

1J

"'-M

y=

50

0,0

00

in.-k

"\T

1=

1.4

sec.

My

=5

00

,00

0ia

.-k

15

OJ > OJ

OJ

...J

"](3

.0

> '"

0.4

...J

10>

.

=~

1

0

1.0

f-'

>.

Sil

inea

r

-j.

~

(f)

--.J

0S

tab

leI.

J:)

-I

en

(3•

1.0

55

+

"""=

0,4

•0

(cl

(d)

Fig

.5

5(c

on

t'd

.1

Effect

of

Pla

sti

cH

ing

eM

-8H

yste

reti

cL

oo

pR

elo

ad

i.n

gP

ara

mete

rf3

20

20

T

T,=

1.4

sec.

IT)

"1.

4se

c.M

y=

50

0,0

00

in.-k

M=

50

0,0

00

in.-k

y,

15.).

\15

1.5

0.5

1.0

Cum

.P

lost

icR

ota

tio

ns

(rad

."1

02)

J~I

II

o0

'I

=:"

,.I

Io

M1

M1

M

Ro

tatio

na

lD

ucti

lity

~~

>>

~~

I....

I10

....I

10I-

'>

.>

.ro

~~

o0

0I

Z;;r-

Bil

inoa

rS

tob

ieen

(3=

1.0

Sf

\~

/~=

0.4

5.J

\\/

/(3

=1

.0

o1\

=0

.4

=0

(e)

(f)

Fig

.5

5(c

on

t'd

.)E

ffect

of

Pla

sti

cH

ing

eM

-eH

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rJ3

6T,=

0.8

sec.

My

=1,

000,

000

in,k

42

~

,., ~ o iii

I~O(mm)

--'--

-i 6.0

'"> '"..J10

100

...L

.-f-­

4.0

T,=

0.8

sec.

My

=1

,00

0,0

00

in.-k

~o 4­

2.0

~ oo

20

T,

,2

0

f3=

0,

0.4

,a

I.0

""""II

If3

•0

=0

,48l

linoo

rSt

able

"'//

J=

/,0

IHIV

Bil

inea

rSt

able

a; :0- '"

,..

J10

f-'

())

,..f-

'~

,.2 en

Max

.H

oriz

onta

lD

ispl

acem

ent

(in,)

(a)

(b)

Fig

.5

6E

ffect

of

Pla

sti

cH

ing

eM

-SH

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rt5

20

"2

0

T,=

0.8

sec.

M=

1,00

0,00

0in

,ky

15

+'\.

,~

a.;0:

;>

><

l)<

l)

...J

10...

J10

, f-'

>,

ro>

,~

~

N.2

0,

(f)

iii

Bill

near

'.S

lab

le

1.0

=0

a0

.4

40

20

~

"Ii=

0.8

sec.

M=

1,00

0,00

0in

,..k

yB

Ilin

ear

Sla

ble

'00

50

f3=

0,

0.4

,a

1.0

4(k

N·m

KIO

·)o

1,

I!

I"

tI

o5.

010

.015

.0

Max

.B

endi

ngM

omen

t(i

n-k

ips

x10

6)

5

(c)

(d)

Fig

.56

(co

nt'

d.)

Eff

ect

of

Pla

sti

cH

ing

eM

-eH

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rJ3

•

20

20

T

Tl

=0

.8se

c.

T,=

0.8

sec.

M=

1,0

00

,00

0in

.-kI

M=

1,00

0,00

0in

.-k

y

yIS

+15

+\\

Cum

.P

last

icR

otat

ions

J':=

=-4

;;:1

\)

Io

2.0

4.0

6.0

B.O

-3(r

od

.'/0

)

01I~

Io

2.0

4.0

6.0

Rot

atio

nal

Duc

tility

0303

a;> O

JI

..J

10..

J10

I-'

OJ

,.,,.,

w~

~

I0

0iii

iii{3

=o

a1.

0

skr-

;Bl1

Ineo

rS

tab

le

5+\\

/"

=0

.4r-

(3=

1.0

=0

.4B

Iline

arS

tabl

e11

11/

/\

=0

(e)

(f)

Fig

.56

(co

nt'

d.)

Eff

ect

of

Pla

sti

cH

ing

eM

-eH

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rf3

51T

1=

1.4

sec.

M=

50

0,0

00

inrk

Iy

4

NO

DA

LR

OT

AT

raN

S1

st

Flo

or

Lev

el

-I

':'3

~I

i21

f3=

0

~=

0.,4

z 0

=1

.0

f=I

~f-

'

Dil

inca

rst

ab

le

co0

~

0::

I

I

\\

~~

i\~

0..

'~, \

-20.

01.

02.

03.

04.

05.

0TI

ME

(sec

onds

)6.

07.

08

.09

.010

.0

Fig

.57

Ro

tati

on

inF

irst

Sto

ryv

ers

us

Tim

efo

rD

iffe

ren

tV

alu

es

of

the

Pla

sti

cH

ing

eM

-$H

yste

reti

cL

oo

pR

elo

ad

ing

Para

mete

rS

Mornent at Basevs.

Nodal Rotation at Fir st Floor Level

10 !~ : j~ OJ.¥ ,I

~ :ILJJ::;;:

~ 0 +-1------:-f-AI----r'71"'---f;f;j!---A7----:;:-T--::-,l.----~ozwn:: -2wco::;;:w

::;: - 4 I (a) ~ = 0.4. a = 0

-6 ----'==t----'------+---<-----l-2.0 -lO 0.0 1.0

NODAL ROTATION (radians x 10-3)2.0 3.0

2.01.0

/L__-J~------J-.J-:---r~(b) Bilinear Stable Loop

Moment at Basevs.

Nodal Rotation at First Floor Level

2

4

o

10

: 1

-4

-2

-6 j-8 -----~------~--

-3.0 -2.0 -1.0 0.0N,)DAL ROTATION (radians x 10-3 )

It)

Q><</)<l).c<.>

.f:Ia.

:>2~

I-Zw

'.. ;;;;0;;;;0zwn::wco::;:w::;:

Fig. 58 Momellt at base versus Nodal Rotation at First Story Levelfor Different M-8 Curve Characteristics

-185-

"T;=

1.4

sec.

My

=5

00

,00

0in

.-k

102

03

0(m

m)

oI

tI

It'

Io

0.5

1.0

1.5

Mox

.In

lers

lory

Dis

plac

emen

l(in

.)

20

20

010

0

20

*/~

o'

}.

,-

'"I~

+cP

I~

",,<>

""-

'!S

'~~

{J<i

0 U 0>

0;

0;

oS 0-

>>

E'"

'"0

...J

10...

J10

QI >-'

>-0

0>-

~~

'"0

0I

-iii

(f)

TI

=1.

4se

c.

My

=5

00

,00

0in~k

~+

//

5

(al

(hI

Fig

.59

Eff

ect

of

Dam

pin

g

20

20

60

(kN

X102

)I

, 15.0

10.0

(kip

sx

102)

40

5.0

Max

.H

oriz

onta

lS

hear

oo

I2

04

06

0~N.m'103l

o'

I'I

!I

Io

20.0

40

.06

0.0

80

.0

Max

.B

endi

ngM

omen

t(i

n-

kips

x10

4)

1S

t~~(

15I

II

0:; >

T I

Ij

10

\=

1.4

sec.

r-'

00

0:;

My

=5

00

,00

0in

.-k

--J

>.

>"

I5

'"

(jj

...J

10

>.

~ 0 (jj

5T I

=1.

4se

c.

My

=5

00

,00

0in

,.k

\.\.

5

(c)

(d)

Fig

.5

9(c

on

t'd

.)E

ffect

of

Dam

pin

g

20

12

0

1.4

sec.

1.4

sec.

T,=

T,=

My

=5

00

,00

0In

.:-k

My

=5

00

,00

0in

.-k

15

+\

15

I~

oI

tI

oM

WI~

Cum

.P

last

icR

otat

ions

(rod

.x10

2)

oJI.

....

....

....

....

...

II

o5.

010

.015

.0

Rot

atio

nal

Duc

tility

OJ

OJ

>>

'"'"

1-l

'0-l

10,.... 0

0>

.>

.0

0~

51

0 iiiiii

Dam

pIng

Coe

f.=

10%

5~

jDO

mP

lnq

Coo

t.=

IOG

/ o

5+\"-

/=

5%=

5%

(e)

(f)

Fig

.59

(can

t'd

.)E

ffect

of

Dam

pin

g

Dam

pIng

Coe

t.:::

0%

U=

10

%1520

=10

%..I

Dam

pln9

Cae

f.

=5

b /o/~I

1520

(ij ij;

10-l

I:>

,

(ij

"I~ 0

>

iii

Q)

=0.

8se

c.

-l

1j

I

=0

.8se

c.

II

=1

,00

0,0

00

in~k

f-"

:>,

T 1

d

co~

=1

,00

0,0

00

in"k

My

'"0

Iiii

My

5+II

50

100

150(

mm

)0

14

812

(mm

)0

'q

,I

'1

II

,I

,I

02.

04

.08.

00

2.0

40

6.0

Max

.H

oriz

onta

lD

ispl

acem

ent

(in,)

-IM

ax.

Inte

rsto

ryD

ispl

acem

ent

On.

x10

)

(a)

(b)

Fig

.6

0E

ffect

of

Dam

pin

g

,

40

=

20

Dam

ping

Coe

f.=

5

20

(kN

.mxI0

4)

o0\b.

o\\

!20

.0~b.o

Mox

.B

endi

ngM

omen

t(in

-kip

sx

105

)

5

20

,2

0

T 1

,~~

=0.

8se

c.

My

=1,

000,

000

in"k

I~

T 1=

0.8

sec.

15+

\

My

=1

,00

0,0

00

in.-k

15

Q)

"IID

;>>

Q)

Q)

-l

-l

10I f-'

>.

Dam

ping

Cae

f.=

5%>

.<

D~

~

00

=10

°/"

0I

-\

-(/

)(/

)

(c)

(d)

Fig

.6

0(c

an

t'd

.)E

ffect

af

Dam

pin

g

20

T2

0

1j•

0.8

sec.

T,•

0.8

sec.

My

=1

,00

0,0

00

Inck

My

=1

,00

0,0

00

Inck

I~+

~10

01~

Ia

2.0

4.0

G.O

Rot

atio

nal

Duc

tility

I f-'

\0 f-' I

'"> j >.

L o (i)

10 o''\

Dam

ping

Coo

f.•

0%

~•

10

%

a:; j >.

L o (i)

10 o

Dam

ping

Coo

f.•

10

%

II~%

I'-'

II

o.~

=-r-

I'"

Ia

2.0

4.0

G.O

B.O

Cum

.P

last

icR

ot.

(in.•

103

)

(e)

(f)

Fig

.6

0(C

on

t'd

.)E

ffect

of

Dam

pin

g

EI (top)

r----:~

B \Floor level 20

16

12

Fig. 61 Stiffness Taper

-192-

I i-' '" w I

20 15

'"> a.> -.J

10

,., .... a -(f)5

~=

1.4

sec.

My

=5

00

,00

0in

:k

20 15

11I

E ~ 0 '"c '"Q

; > a.> -.J

10

'".... a (i)

5T I

=1.

4se

c.

M=

50

0,0

00

in"k

y

100

20

03

00

(m"'

)0

110

203

0(m

m)

a',

II

I,

I,

II

II

I

a5,

010

.015

.0.

00.

51.

01.

5

Max

.H

oriz

onta

lD

ispl

acem

ent

(In.)

Max

.In

ters

tory

Dis

plac

emen

t(i

ll)

(a)

(b)

Fig

.62

Eff

ect

of

Sti

ffn

ess

Tap

er

20

.,2

0

I"

T,=

1.4

sec.

IT 1

=1.

4se

c.

IOf

\M

Y"

50

0,0

00

in"k

\M

y"

50

0,0

00

in.-k

I~

(l)

"j\

Q;

>>

(l)

(l)

1...

J...

J10

/-'

'0>-

,.,"'"

~....

I.Q

0

enen

~ o o10

0

~~

10~

Max

.B

endi

ngM

omen

t(i

n-k

ips'

10~)

150

(kN

·rilx

I03

)I

I 15.0

a o oM

Max

.H

oriz

onta

lS

hear

10.0

(kip

s.10

2)

60

(kN

xI02

)!

I 1M

(c)

(d)

Fig

.6

2(c

an

t'd

.)E

ffect

of

Sti

ffn

ess

Tap

er

20

T2

0

T I=

1.4se

c.T I

=I.

4se

c.

My

=5

00

,00

0in

.kM

y•

50

0,0

00

In.-k

15+I

15

Q)

"Jl

0;

>>

Q)

Q)

--l

--l

10I f-' '"'

,.,,.,

U1~

~

.E0

I-

(/)

(/)

01'"

:0

GI

o0

.51.

51.

5

Cum

.P

last

icR

otat

ions

(ro

d.'

102

)

Tap

ered

0'

I..

....

...

I:>

.-I

oM

U15~

RO

latio

nal

Duc

tility

5

(e)

(f)

Fig

.62

(co

nt'

d.)

Eff

ect

of

Sti

ffn

ess

Tap

er

20T

/20

T1

=1.

4se

c.(M

y)b

a••

=2.

0(M

y)ba

sec

50

0,0

00

in.-k

(My)

tap

=3.

e

IH//

15U

n/f

arm

1020

30(m

m)

01

1I

.I

•0

0.5

1.0

1.5

Max

.In

ters

tory

Dis

pla

cem

en

t(i

n.)

(b)

(a)

Max

.H

ori

zon

tal

Dis

pla

cem

en

t(I

n.)

o!

1l?0

I29

01

3QO

(mm

)I

o5.

010

.015

.0

'" > '"(M

y)b

a••

..J

10=

2.0

,.,'"

~

>

lMy)t

op

0

JJ

Q)

•3.

8-

..J

10

Cfl

JI

,.,

1.4

sec.

f-'

'"U

nif

orm

T 1=

""0

=5

00

,00

0in~k

m-

1C

fl

(My)

base

5

Fig

.63

Eff

ect

of

Str

en

gth

Tap

er

Un

ifo

rm

60

(kN

.102

)I

I10

.016

.0

(kip

s'

102

)

40

20

6.0

Mox

.H

oriz

onta

lS

he

ar

o o

20

(Myl

bO'.

(My)

top

•2

.0

")DF

""""·•

3.8

- Q) :>- Q)

...J

10,., ... 0 -

I\\

\T,

•1.

4se

c.CI

l

..(M

ylba

se•

50

0,0

00

in~k

6

.3

kN'm

,IO

l"-'

---l 60

.0

60

60.0

(in

-kip

s.1

04,

•3.

e

•2.

0(M

ylb

o,.

(MY

)IO

P

20

40

!I

'I

20.0

40

.0

Max

.B

endi

ngM

omen

t

T,•

1.4

sec.

(Mfb

ase

'50

0,0

00

in.-k

1620 6 0

0

Q) :>­ Q)

...J

'0,., ... o -CIl

I I-' '" "I

(c)

(d)

Fig

.6

3(can

t'd

.)E

ffect

of

Str

en

gth

Tap

er

20

20

Tj

=1.

4se

c.

(~)base

•5

00

,00

0in

.-k

Tj

=1.

4se

c.

(My)

base

•5

00

,00

0In

.-k·

1515

Q) > Q)

1U

....J

10

> 1U

>.

....J

10

...1

(Myl

ba••

0

f-'

3.8

->

.

•(/

)

(My

lba.

.

<!)

...~

,I

3.6

00

0

(My}

,ap

•

,-

(My

l,op

en

•2

.05

•2.

05+

I\.

'-../

Unl

farm

/U

nifo

rm

0'

,.....

..........

.........

I,

05.

010

.015

.01.0

2.0

3.0

Ro

tatio

na

lD

uct

ility

-2C

um.

Pla

stic

Ro

tati

on

s(r

ad.

xlO

)

(e)

(f)

Fig

.6

3(c

an

t'd

.)

Eff

ect

af

Str

en

gth

Tap

er

P_~..,

spring. Ks

F( pfl

//

////

EI

FP---...,

2)1- -\

//

//II

EI

Fu lIy fixed Partially fixed

Fig. 64 Definition of Base Fixity Factor, F

T1r 1.4 sec

My • 500,000 in.- k

12(Ellwoll

•.205 x 10

-.,.,Co!:::

elementsimulotingfoundationrestraint

{Ells

TI

• 0.8 sec.

My. 1,500,000 in.-k

'1/1 I I I

Bese(EI)s k_in2Fixity (%) TI (sec.)

ICO co ! 1.40

75 .31 xlO" 1,,3

50 .104 x 1011 1.52

BaseFixity (%.) {EllS T, (sec.l

100 co 0.80

75 .9Ex 1011 0.83 I50 .32 x lO" 0.88 I

Fig. 65 Models Used in Study of Effect of Base Fixity Condition

-199-

20

T

/2

0

'"''",,'....#

1j=

1.4

sec.

·75

"/0

fixed--

My

=5

00

,00

0in.

-kba

se-

~O%

fl••

dI

I

fI

..fo

rfi

lll1

fin

dca

seI~

+bo

lO7

7I~

(jj

(jj

> .5>

I10

.5N

10

0~

0

>0

I0

...i7i

0 iiiT j

=1.

4se

c.-

fully

fl••

db

an

75

%flx

4d

I>+

/I

/M

y=

500,

000

in~k

~b

an

•fo

rfU

llyfl

••d

ca•• 75

0(in

m)

01

102

03

0.

(mm

).

II

II

30.0

0o.~

1.0

1.5

Max

.In

ters

tory

Dis

plac

emen

t(i

n)

(b)

50

02

50

(a)

oIf

'I'

I'

Io

10.0

20.0

Max

.H

oriz

onta

l.D

ispl

acem

ent

(in.)

Fig

.66

Eff

ect

of

Base

Fix

ity

Co

nd

itio

n

fUlly

fixed

bo,.

75%

fixed

bose

~o%

flxod

base

5 oI

2,~llJ

4.0,

(kN'IO~1

o5

,010

.015

.02

Mox

.H

oriz

onta

lS

hear

(kip

sxl

O)

7fiQ /o

fIxe

dba

s"~

fully

fl:~

edb

ass

-"\

~o,.o

fixed

bose

~

I2

04

06

0~N'm'I~)

o'

II

II

II

o2

00

40.0

60

.08

0.0

Mox

.B

endi

ngM

omen

tO

n-ki

psx

104

)

:i

20

20

..

=1.

4se

c,·

1.4

se

c,·

T 1

I,\

T,=

M=

50

0,0

00

in.-k

My

=5

00

,00

0in

cky

,\

I..

for

fUlly

fl.ed

CO

lO

15+

III

..fo

rfU

llyfi

ud

calO

l:>f

,,""-

0:;

Qj

>>

<lJ

<lJ

I..

J10

..J

10IV 0

>.

>.

I-'

~~

I0

0

--

(/)

(/)

(c)

(d)

Fig

.6

6(c

on

t'd

.)E

ffect

of

Base

Fix

ity

Co

nd

itio

n

20

T2

0

T 1=

1.4

sec.

..T

1=

1.4

sec.

..

My

=5

00

PO

Oin

7kM

y=

50

0,0

00

in.-k

10+

I..

.fo

rfu

llyfix

edca

••10

•fo

rfu

llyfl

x.d

ca••

1.5

75

%fi

xed

bos

o

fUlly

fixe

dbo

••

1.0

-2(r

od.

xlO

l

,.---

50

%fix

edb

ose

0.5

Cum

.P

last

icR

otat

ions

5r--

fully

fixed

base

75%

fixed

base

50

%flx

adba

..

o 01

I>.

:>..

........

1I

o5.

010

010

.0

Rot

atio

nal

Duc

tility

OJ

ojl

Q;

:>:>

OJ

OJ

I-l

-l

10tv 0

>.

>.

tv~

~

I0

0-

-U

lU

l

(e)

(f)

Fig

.66

(co

nt'

d.)

Eff

ect

of

Base

Fix

ity

Co

nd

itio

n

"' 'Q '" V> co " :.0 "~ w UJ « co f-- « z 0 f-- ~ 0

Ia::

'"W

a'-

9w

Zr

I <.> t1 <r ..J

(L

T1

=1.

01se

c.

4t

M;

50

0,0

00

in,k

y

3 2 a

~

PL

AS

TIC

HIN

GE

RO

TA

TIO

NS

AT

BA

SE

50

%fi

xed

base

75

%fi

xed

base

full

yfi

xed

base

'j...

'"

r ., rr f

-I0

.01,

02.

03.

04.

05.

06,

0T

IME

(sec

onds

)7.

08.

09.

010

.0

Fig

.67

Pla

sti

cH

ing

eR

ota

tio

ns

at

Base

vers

us

Tim

efo

rD

iffe

ren

tD

eg

rees

of

Base

Fix

ity

/4--~0%

fl,.

db

a,.

1:5%

./flx

.dba

..

fully

fr..

dba

..

:51020

T 1=

0.8

sec.

.,

My

=1,

500,

000

in,.k

15'

•fa

rfu

llyfl.

.dca

••

,., ~ a (j)<ii > .5

TI

=0

.8se

c.·

My

=1

,50

0,0

00

inck

..fo

r11.

111)'

fixed

case

i'-5

0%

,fix

edbQ

88

~!C

,7

5%

fixed

base

IIL

l!fu

llyfi

xod

bas

e

5I~

20

Q) > Q)

....J

10I N

,.,0

~

"'0

I(j)

100

20

03

00

48

12

Mox

.H

oriz

onto

lD

ispl

acem

ent

(in,)

(a)

(b)

Fig

.68

Eff

ect

of

Bas

eF

ixit

yC

on

dit

ion

01

10"&

2.0

(kN

·m.1

04)

01

40

WB

O(k

Nx

102

)•

II

'I

'II

I!

I

010

.02

0.0

30

.00

100

20

.0'3

0.0

Max

.B

endi

ngM

omen

t(I

n-ki

ps'1

05

)M

ax.

Hor

izon

tal

She

ar(k

ips

x10

2)

(c)

(d)

0:;

0:;

>>

<l)

<l)

..1

-J10

-J10

tv 0>

.>

.U

1L

.L

.

10

\0

enen

50

%fi

xed

bas

e50

%fi

xed

bOGe

5+

'\\\

575

%fi

xed

base

75

Q /ofix

edb

ase

lu.l

lyfl

x.d

bo••

lull

yli

xed

bo••

Fig

.68

(co

nt'

d.)

Eff

ect

of

Base

Fix

ity

Co

nd

itio

n

6.0

4.0

-3(r

ad.x

10l

.I50

%tix

edba

se

,,--

75%

fixed

basi

fully

fixe

dbo

••

2.0

Cum

.P

last

icR

otat

ions

5

3.0

2.0

Duc

tility

r-

50

0 / Dflx

odba

se

75%

fixed

base

fUlly

fixe

dba

..

1.0

Rot

atio

nal

5 01~

Io

20~

20

~=

0.8

sec.

i'T,

=0

.8se

c.•

My

=1

,50

0,0

00

in.-k

M=

I,5

00

,00

0in

:ky

..fo

rfu

llyfi

xed

CO

le15

•fo

rfu

lly

fixed

co..

'"

"j\

0;

>>

'"'"

...J

...J

10I N

>.

>.

0~

~

'"0

0

Iiii

iii

(e)

(f

l

Fig

.68

(co

nt'

d.)

Eff

ect

of

Base

Fix

ity

Co

nd

itio

n

(in

.)

30

20

0.5

1.0

Inte

rsto

ryD

isp

lace

me

nt

(b)

Max

.

40

II

T1

=1.

4se

c.

M~

1,0

00

,00

0In

.-k

y

30

tI

40

Sto

rl..

-3

0S

torl••

Q) > Q)

--J

40

>.~ 0 -Cf)

wi/I~

20S

tori

e.

10S

torl

••

75

0(m

m)

--l.

....

-j 30

.0

TI

~1.

4se

c.

My

=1,

000,

000

in:-k

50

0..

..l­ 20

.0

10S

torl..

20

Sto

rl••

250

.Lt­

10.0

Max

.H

ori

zon

tal

Dis

pla

cem

en

tO

n.)

(a)

30

Sto

rl••

40

Sto

rl••

00

1040

.30

Q) > Q)

I--

JN

20

0>

.-J

~

I0 - Cf)

Fig

.69

Eff

ect

of

Num

ber

of

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ries

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40

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4se

c.T,

=1.

4se

c.

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000

inck

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tal

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ea

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rie.

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.69

(co

nt'

d.)

Eff

ect

of

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mb

ero

fS

tori

es

40

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0

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=1.

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c.

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00

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stic

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atia

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9(c

on

t'd

.)

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ect

of

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mb

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l

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ds)

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.7

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al

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tati

on

sat

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on

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rD

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re

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Fig

.7

1E

ffect

of

Num

ber

of

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ries

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40

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=1.

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c.

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Fig

.7

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on

t'd

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ffect

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54

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ary

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y=5

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c..M

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ness

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0a

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