tJ)F/tU+-780085 PB 283 705 - NISTtJ)F/tU+-780085 PB 283 705 A Report on Research Sponsored by...
Transcript of tJ)F/tU+-780085 PB 283 705 - NISTtJ)F/tU+-780085 PB 283 705 A Report on Research Sponsored by...
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
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(Formerly NTIS-35)Department of Commerce
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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-)
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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?
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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
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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.
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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
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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
~
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o
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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
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o,
N.S
"1
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19
11
Paco
ima
Dam
,S
16
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0.5
94
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...
1911
Ho
lid
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1952
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69
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Art
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ial
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og
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1000
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50
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ping/\ i\.'"
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esp
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afo
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Ten
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Dam
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Pac
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30
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Bas
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Rel
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ispl
acem
ents
Bas
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Tan
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Dev
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20
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(b)
Fig
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Eff
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of
Fre
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en
cy
Ch
ara
cte
risti
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of
Gro
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dM
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on
60
(kN
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(kip
sxId
)
T 1•
1.4
sec.
My
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zon
tal
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ar
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15
(kN
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03)
>I
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00
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20
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.24
(co
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d.)
Eff
ect
of
Fre
qu
en
cy
Ch
ara
cte
risti
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of
Gro
un
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20
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1971
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(co
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d.)
Eff
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of
Fre
qu
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Ch
ara
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risti
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of
Gro
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S1st
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1971
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(sec
onds
)
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70
8.0
9.0
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Fig
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5N
orm
ali
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Ro
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on
sin
Fir
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us
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.26
No
dal
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His
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for
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40
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A' 1500 in2
~'o .833
W, '220 kips
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
o1.
02.
03
.04
.0-I
Mox
.In
ters
tory
Dls
ploc
emon
t(in
.•IO
l
50
100
150(
mm
)'I
II
II
2.0
4.0
6.0
Mox
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oriz
onta
lD
ispl
acem
ent
(in.)
oo
20
T/
/2
0
1940
EI
Cln
tro
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-ST 1
=0
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c.
1940
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Cln
tro.
E-W
My.
1,50
0,00
0in
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J15
(I)
(I)
:-:-
(I)
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lot
19
40
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Cln
lro.
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r-'
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0I
1940
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Cen
lro.
N-S
--
CIl
TI
=0
.8se
c.C
Il
5+/
/M
y'
1.50
0,00
0In
.-k5
(al
(b)
Fig
.28
Eff
ect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
oI
4,~I/1
l~(k
N',1
02l
o10
,02
0,0
30
,02
Max
.H
oriz
onta
lS
he
ar
(kip
sx1
0)
102
0(k
N·m
.104
j0
'I
I,).,
II
~
o10
,02
0,0
3.0
Mox
.B
endi
ngM
om
en
t(I
n-k
ips
x10
5)
20
1'
20
TI
=0
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c,
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0,8
sec,
My
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000
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in,k
III.
L\\
15
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tro,
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0
-I
\--\---1
94
0EI
Con
lro,
E-W
-(/
)(/
)
IIIi
(c)
(d)
Fig
.28
(co
nt'
d.)
Eff
ect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
1.5
1.0
-2(r
od.x
10)
1940
EIC
entr
.,N
-S
~1940
EIC
on
tr.,
E-W
0.5
Cum
.Pl
astic
Rot
atio
n
5 00
4.0
~.o
5•. 01,~,
Io
1.0
2.0
Rot
atio
nal
Duc
tility
20
T
T 1=
0.8
sec.
My
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000
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1940
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=0
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en
>
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15
00
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40EI
Cen
lr.,
Q)
.J10
Y,
,00
0In
:-k
E-W
>. ... 0 -en
(e)
(f)
Fig
.28
(can
t'd
.lE
ffect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
25
05
00
750(
mm
)0
110
203
0(m
m)
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010
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30
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0.5
1.0
/.0
Max
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oriz
onta
lD
ispl
acem
ent
(in.)
Max
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ters
tory
01sp
loce
men
t(in
.)
(a)
(b)
20
20T
//
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entr
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y=5
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/T
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2.0
sec.
My'
50
0,0
00
In~k
Fig
.29
Eff
ect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
20~
20
T,=
2.0
sec.
My
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sec.
15t
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endi
ngM
omen
t(in
-kip
sx
104
)
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40
20 5
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Max
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oriz
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hear
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ntr
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1971
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iday
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ldoy
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I f-'
f-' '" I
(c)
(d)
Fig
.2
9(c
on
t'd
.)E
ffect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
20
20
Rot
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nal
Duc
tility
01'-
h',~
o2.
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06.
0
SI
19
40
EIC
entr
o,E
-IV
1971
Hol
iday
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an,
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T,=
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sec.
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1971
Hol
iday
Orl
an.
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SI
5r19
40EI
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lra.
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(f)
Fig
.29
(co
nt'
d.)
Eff
ect
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
of
Gro
un
dM
oti
on
(mm
)12
64
60
40
20
T2
0
1971
Poco
lma
,.]19
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ma
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1940
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sec.
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00
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sec.
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ref.
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y=
50
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00
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51=
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1re
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2.0
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Max
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ters
tory
Di6
plac
emen
tlin
.x10
1 )
oI
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(b)
Fig
.3
0In
tera
cti
ng
Eff
ects
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
and
Inte
nsit
yo
fG
rou
nd
Mo
tio
n
60
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150
50
20
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lS
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(kip
sx
101
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c.
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00
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115+
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lref
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sec.
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516£
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l6E
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Fig
.3
0(c
on
t'd
.)In
tera
cti
ng
Eff
ects
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
and
Inte
nsit
yo
fG
rou
nd
Mo
tio
n
20~
20
TI
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4se
c.T
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4se
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00
,00
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00
,00
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16
+\
SI
a0
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(51
ref.
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ef.
)
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00
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1940
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1940
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1971
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8.0
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last
icR
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(ra
d.•
10"'1
(e)
(f)
Fig
.3
0(c
on
t'd
.)In
tera
cti
ng
Eff
ects
of
Fre
qu
en
cy
Ch
ara
cte
risti
cs
and
Inte
nsit
yo
fG
rou
nd
Mo
tio
n
25
20
15
o
oJQ
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o..~E 5.szo!;;a::wG:l -5uu<t
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-15
0.0 4.0 8.0 12.0
TIME (seconds)
16.0 20.0
Fig. 31a Twenty-Second Composite Accelerogram
80
Datnping ;;; 0%20/05%
60
-0..~
:S>-... 40U0.JW>
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4.03.01.0
o~----+----+------+--------<0.0
Fig. 31b Velocity Response Spectra for 20-SecondComposite Accelerogram
-121-
20 10
Q; >
IC
ll
f-'
-l
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I~ £ (j)
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(mm
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ters
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plac
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0
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20.0
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0
Max
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oriz
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lD
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acem
ent
(a)
(b)
Fig
.3
2E
ffect
of
Du
rati
on
of
Gro
un
dM
oti
on
20
20
T\
T1
•1.
4se
c.•
1.4
sec.
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ax.
Ben
ding
Mom
ent
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kip
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~ Q)
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10
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(d)
Fig
.32
(co
nt'
d.)
Eff
ect
of
Du
rati
on
of
Gro
un
dM
oti
on
20
T2
0
T,
=1.
4se
c.T
1=
1.4
sec.
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00
,00
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My
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nal
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ty
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)
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so·c
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(b)
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Fig
.32
(co
nt'
d.)
Eff
ect
of
Du
rati
on
of
Gro
un
dM
oti
on
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
-sec.-
(bl
./
20
-sec.-
tal
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......
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.,,~
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~ 20
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Fig
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4R
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nin
Fir
st
Sto
ryv
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us
Tim
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ren
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l
20
0
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plac
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t
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m)
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(in.)
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(b)
Fig
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5E
ffect
of
Inte
nsit
yo
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rou
nd
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tio
n
T 1=
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sec.
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l5 oo
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20
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(d)
Fig
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(co
nt'
d.)
Eff
ect
of
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yo
fG
rou
nd
Mo
tio
n
202
0
16
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sec.
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1.4
sec.
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00
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rad.
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Fig
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5(c
on
t'd
.)E
ffect
of
Inte
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yo
fG
rou
nd
Mo
tio
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//
20I
T I=
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5.0
Max
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acem
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(in.l
(a)
(b)
Fig
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6E
ffect
of
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nsit
yo
fG
rou
nd
Mo
tio
n
5+
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sec.
\\
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00
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50
100
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40
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10.0
15.0
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n-ki
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)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,
=1.
4se
c.
15
t\\
\~
15
0;
a;>
>Q
>Q
>
I-l
10M
=5
00
,00
0In
,k-l
"]\~~
f-'
Y~t'o
\\~
In»
=7
50
,00
0»
N~
~o
"0
;,;.
I0
=1
,00
0,0
00
0"
0q,
~d'-
"(j
j-
=1
,50
0,0
00
IJl
00
0 0E
la.t
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Fig
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(can
t'd
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ffect
af
Yie
ldL
ev
el,
My
20
20
T,=
1.4
sec.
1;=
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sec.
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rad.
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5
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last
icR
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Fig
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(co
nt'
d.)
Eff
ect
of
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el,
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00
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sec.
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IO)
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00
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(co
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d.)
Eff
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of
Yie
ldL
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el,
My
202
0
~=
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sec.
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last
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Fig
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Eff
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Yie
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(can
t'd
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ffect
af
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el,
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20
20
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2.0
sec.
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(co
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d.)
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281
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tera
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Eff
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of
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l,
an
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se
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(Co
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d.)
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Eff
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of
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nd
am
en
tal
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od
,T
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esp
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8(C
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t'd
.)In
tera
cti
ve
Eff
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of
Fu
nd
am
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tal
Peri
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l,an
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Fig
.49
Eff
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of
Slo
pe
of
Po
st-
Yie
ldB
ran
ch
of
Mo
men
t-R
ota
tio
nC
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tal
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ar(k
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x10
2)
(kN
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(d)
Fig
.4
9(c
on
t'd
.lE
ffect
of
Slo
pe
of
po
st-
Yie
ldB
ran
ch
of
Mo
men
t-R
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tio
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15H
15
3.0
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last
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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
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~~
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
Sto
ries
40~
40
\T,
=1.
4se
c.T,
=1.
4se
c.
My
=1,
000,
000
inck
My
=1,
000,
000
In,.k
30
10.0
:20.
03
0.0
Mox
.H
orl
zon
tal
Sh
ea
r(k
ips
x10
:2)
(d)
oI
'Yt
Ilap
(kN
'IO:2
)o
II
I
4I
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