6 Analytical Studies
6. 1 Overview
Analytical studies were conducted as part of this project
to serve two broad objectives: (1) to assess the effects of
damage from a prior earthquake on the response of
single degree of freedom oscillators to a subsequent,
hypothetical performance level earthquake, and (2) to
evaluate the utility of simple, design oriented methods
for estimating the response of damaged structures.
Previous analytical studies were also reviewed.
To assess the effects of prior damage on response to a
performance level earthquake, damage to a large
number of single degree of freedom (SDOF) oscillators
was simulated. The initially “damaged” oscillators were
then subjected to an assortment of ground motions. The
response of the damaged oscillators was compared with
that of their undamaged counterparts to identify how the
damage affected the response.
The oscillators ranged in initial period from 0. 1 to 2. 0
seconds, and the strength values were specified such
that the oscillators achieved displacement ductility
values of 1, 2, 4, and 8 for each of the ground motions
when using a bilinear force displacement model. The
effects of damage were computed for these oscillators
using several Takeda based force displacement models.
Damage was parameterized independently in terms of
ductility demand and strength reduction.
Ground motions were selected to represent a broad
range of frequency characteristics in each of the
following categories: Short duration (SD) records were
selected from earthquakes with magnitudes less than
about 7, while long duration (LD) records were
generally selected from stronger earthquakes. A third
category, forward directivity (FD), consists of ground
motions recorded near the fault rupture surface for
which a strong velocity pulse may be observed very
early in the S wave portion of the record. Six motions
were selected for each category, representing different
frequency characteristics, source mechanisms, and
earthquakes occurring in locations around the world
over the last half century.
The utility of simple, design oriented methods for
estimating response was evaluated for the damaged and
undamaged SDOF oscillators. The displacement
coefficient method is presented in FEMA 273 (FEMA,
1997a) and the capacity spectrum and secant stiffness
methods is presented in ATC 40 (ATC, 1996).
Estimates of peak displacement response were
determined according to these methods and compared
with computed values obtained in the dynamic analyses
for the damaged and undamaged structures. In addition,
the ratio of the peak displacement estimates of damaged
and undamaged structures was compared with the ratio
obtained from the displacements computed in the
nonlinear dynamic analyses.
This chapter summarizes related findings by previous
investigators in Section 6. 2. The dynamic analysis
framework is described in detail in Section 6. 3, and
results of the nonlinear dynamic analyses are presented
in Section 6. 4. The design oriented nonlinear static
procedures are described in Section 6. 5, and the results
of these analyses are compared with the results
computed in the dynamic analyses in Section 6. 6.
Conclusions and implications of the work are presented
in Section 6. 7.
6. 2 Summary of Previous
I Findings
Previous studies have addressed several issues related to
this project. Relevant analytical and experimental
findings are reviewed in this section.
6. 2. 1 Hysteresis Models
Studies of response to recorded ground motions have
used many force displacement models that incorporate
various rules for modeling hysteretic response. By far,
the most common of these are the bilinear and stiffness
degrading models, which repeatedly attain the strengths
given by the monotonic or envelope force displacement
relation. The response of oscillators modeled using
bilinear or stiffness degrading models is discussed
below.
6. 2. 1. 1 Bilinear and Stiffness Degrading
Models
Many studies (for example, Iwan, 1977; Newmark and
Riddell, 1979; Riddell, 1980; Humar, 1980; Fajfar and
Fischinger, 1984; Shimazaki and Sozen, 1984; and
Minami and Osawa, 1988) have examined the effect of
the hysteresis model on the response of SDOF
structures. These studies considered elastic perfectly
plastic, bilinear (with positive post yield stiffness), and
stiffness degrading models such as the Takeda model
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and the Q model, as well as some lesser known models.
For the nonlinear models used in these studies, the post
yield stiffness of the primary curve ranged between 0
and 10% of the initial stiffness. It is generally found that
for long period structures with positive post yield
stiffness, peak displacement response tends to be
independent of the hysteresis model, and it is
approximately equal to the peak displacement of linear
elastic oscillators having the same initial stiffness. For
shorter period structures, however, peak displacement
response tends to exceed the response of linear elastic
oscillators having the same initial stiffness. The
difference in displacement response is exacerbated in
lower strength oscillators. Fajfar and Fischinger (1984),
found that for shorter period oscillators, the peak
displacements of elastic perfectly plastic models tend
to exceed those of degrading stiffness models (the Q
model), and these peak displacements tend to exceed
those of the bilinear model. Riddell (1980), reported
that the response of stiffness degrading systems tends to
“go below the peaks and above the troughs” of the
spectra obtained for elastoplastic systems.
The dynamic response of reinforced concrete structures
tested on laboratory shake tables has been compared
with the response computed using different hysteretic
models. The Takeda model was shown to give good
agreement with measured response characteristics
(Takeda et al., 1970). In a subsequent study, the Takeda
model was shown to match closely the recorded
response; acceptable results were obtained with the
less complicated Q Hyst model (Saiidi, 1980). Time
histories computed by these models were far more
accurate than those obtained with the bilinear model.
Studies of a seven story reinforced concrete moment
resisting frame building damaged in the 1994
Northridge earthquake yield similar conclusions.
Moehle et al. (1997) reported that the response
computed for plane frame representations of the
structure most nearly matched the recorded response
when the frame members were modeled using stiffness
degrading models and strength and stiffness degrading
force displacement relationships; dynamic analysis
results obtained using bilinear force displacement
relationships were not sufficiently accurate.
Iwan (1973) examined the effect of pinching and
yielding on the response of SDOF oscillators to four
records. It was found that the maximum displacement
response of oscillators having an initial period equal to
one second was very nearly equal to that computed for
bilinear systems having the same initial stiffness and
yield strength. For one second oscillators having
different system parameters and subjected to different
earthquake records, the ratio of mean degrading system
peak displacement response to bilinear system response
was 1. 06, with standard deviation of 0. 14. Iwan noted
that for periods appreciably less than one second, the
response of degrading systems was significantly greater
than that for the corresponding bilinear system, but
these effects were not quantified.
Iwan (1977) reported on the effects of a reduction in
stiffness caused by cracking. Modeling the uncracked
stiffness caused a reduction in peak displacement
response for shorter period oscillators with
displacement ductility values less than four, when
compared with the response of systems having initial
stiffness equal to the yield point secant stiffness.
Humar (1980) compared the displacement ductility
demand calculated for the bilinear and Takeda models
for SDOF and multi degree of freedom (MDOF)
systems. For the shorter period SDOF oscillators, the
displacement ductility demands exceeded the strength
reduction factor, particularly for the Takeda model.
Five and ten story frames were designed with girder
strengths set equal to 25% of the demands computed in
an elastic analysis, and column strengths were set
higher than the values computed in an elastic analysis.
The Takeda model, which included stiffness
degradation, generally led to larger interstory drifts and
girder ductility demands than were computed with the
bilinear model.
The studies described above considered hysteretic
models for which the slope of the post yield portion of
the primary curve was greater than or equal to zero.
Where negative post yield slopes are present, peak
displacement response is heightened (Mahin, 1980).
The change in peak displacement response tends to be
significantly larger for decreases in the post yield slope
below zero than for similar increases above zero. Even
post yield stiffness values equal to negative 1 % of the
yield stiffness were sufficient to cause collapse. These
effects were found to be more pronounced in shorter
period systems and in relatively weak systems.
Rahnama and Krawinkler (1995) reported findings for
SDOF structures subjected to 15 records obtained on
rock sites. They found that higher lateral strength is
required, relative to elastic demands to obtain target
displacement ductility demands, for oscillators with
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negative post yield stiffness. The decrease in the 6. 2.
1. 2 Strength Degrading Models
strength reduction factor is relatively independent of
The response of structures for which the attainable
vibration period and is more dramatic with increases in
target displacement ductility demand. These effects
strength is reduced with repeated cyclic loading is
discussed below.
depend on the hysteresis model; the effect of negative
post yield stiffness on the strength reduction factor is
much smaller for stiffness degrading systems than for Parducci and Mezzi (1984) used elasto plastic force displacement models to examine the effects of strength
bilinear systems. They note that stiffness degrading degradation. Yield strength was modeled as decreasing
systems behave similarly to bilinear systems for linearly with cumulative plastic deformation. Using
positive post yield stiffness, and they are clearly accelerograms recorded in Italian earthquakes, The
superior to systems with negative values of post yield authors found that strength degradation causes an
stiffness.
increase in displacement ductility demand for the
stronger, shorter period oscillators. For weaker
Palazzo and DeLuca (1984) found that the strength oscillators, strength degradation amplifies ductility
required to avoid collapse of SDOF oscillators demand over a broader range of periods. The more rapid
subjected to the Irpinia earthquake increased as the degradation of strength, the greater the increase in
post yield stiffness of the oscillator became ductility demand. An analogy can be made with the
increasingly negative. Xie and Zhang (1988) compared
the response of stiffness degrading models (having zero
findings of Shimazaki and Sozen (1984): when strength
post yield stiffness) with the response of models having degradation occurs, the increase in ductility demand can
a negative post yield stiffness. The SDOF oscillators be kept small for shorter period structures if sufficient
were subjected to 40 synthetic records having duration
strength is provided.
varying from 6 to 30 seconds. It appears that Xie and
Zhang found that for shorter period structures, negative Nakamura and Tanida (1988) examined the effect of
strength degradation and slip on the response of SDOF
post yield stiffness models were more likely to result in
oscillators to white noise and to the 1940 NS El Centro
collapse than were the stiffness degrading models for motion. Figure 6 1 plots the force displacement
all durations considered.
response curves obtained in this study for various
combinations of hysteresis parameters for oscillators
with a 0. 2 sec period. The parameter D controls the
r +1OZ D C . 0 D C 0. 2 Q D C 0. 4 D C 0X. 6
r 0 DC 0. 0ram D C . O2 D C 0. 6 D C 0. 0 D C 0. 2 D C O. 4 D C0. 6
0 t +D C 0. 4 + r 0 0 X r 10% 0 2J
D 0. 6 l0. 0 SeC, 0. 75 (E D. C. tr C. O. 6 DUO DSC 0. 0 IDC UX2 DEC ot. r C 0. 6 D 0. 6 Tt. 0. 2 see C 0. 375 (El Centro NS. D 0. T. 0. 2se, C 0. 375 (El Ceotro NlS) a e. S. a 02 C E o
Effect of Load Degradation on Response under El Centro Wave.
Figure 6 1 Effect of Hysteretic Properties on Response to 1940 NS El Centro Record (from Nakamura, 1988)
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amount of slip, C controls the degraded loading
stiffness, and as and a, control the unloading stiffness
for the slip and degrading components of the model. It
is clear that peak displacement response tends to
increase as slip becomes more prominent, as post yield
stiffness decreases or even becomes negative, and as
loading stiffness decreases.
Rahnama and Krawinkler (1995) modeled strength
degradation for SDOF systems as a function of
dissipated hysteretic energy. Strength degradation may
greatly affect the response of SDOF systems, and the
response is sensitive to the choice of parameters by
which the strength degradation is modeled. Results of
such studies need to be tied to realistic degradation
relationships to understand the practical significance of
computed results.
6. 2. 2 Effect of Ground Motion Duration
As described previously, Xie and Zhang (1988)
subjected a number of SDOF oscillators to 40 synthetic
ground motions, which lasted from 6 to 30 seconds. For
stiffness degrading and negative post yield stiffness
models, the number of collapses increased, as ground
motion duration increased. The incidence of collapse
tended to be higher for shorter period structures than
longer period structures. Shorter duration ground
motions that were just sufficient to trigger the collapse
of short period structures did not trigger the collapse of
any longer period structures.
Mahin (1980) reported on the evolution of ductility
demand with time for SDOF oscillators subjected to
five synthetic records, each having a 60 second
duration. Peak evolutionary ductility demands were
plotted at 10 second intervals for bilinear oscillators;
ductility demand was found to increase asymptotically
toward the peak values obtained at 60 seconds. This
implies that increases in the duration of ground motion
may cause relatively smaller increases in ductility
demand.
Sewell (1992) studied the effect of ground motion
duration on elastic demand, constant ductility strength
reduction factors, and inelastic response intensity, using
a set of 262 ground motion records. He found that the
spectral acceleration of elastic and inelastic systems is
not correlated with duration, and that strength reduction
factors can be estimated using elastic response
ordinates. These findings suggest that the effect of
duration on inelastic response is contained within
representations of elastic response quantities.
6. 2. 3 Residual Displacement
Kawashima et al. (1994) studied the response of bilinear
systems with periods between 0. 1 and 3 seconds that
were subjected to Japanese ground motion records.
According to this study, residual displacement values
are strongly dependent on the post yield stiffness of the
bilinear system; that is, systems with larger post yield
stiffness tend to have significantly smaller residual
displacements, and systems with zero or negative post
yield stiffness tend to have residual displacements that
approach the peak response displacement. They also
found that the magnitude of residual displacement,
normalized by peak displacement, tends to be
independent of displacement ductility demand, based
on displacement ductility demands of two, four, and six.
The results also indicated that the magnitude of residual
displacement is not strongly dependent on the
characteristic period of the ground motion, the
magnitude of the earthquake, or the distance from the
epicenter.
In shake table tests of reinforced concrete wall and
frame wall structures, Araki et al. (1990) reported that
residual drifts for all tests were less than 0. 2% of
structure height. These tests included wall structures
exhibiting displacement ductility demands up to about
12 and frame wall structures exhibiting displacement
ductility demands up to about 14. The small residual
drifts in this study were attributed to the presence of
restoring forces (acting on the mass of the structure),
which are generated as the wall lengthens when
displaced laterally. Typical response analyses do not
model these restoring forces. These results appear to be
applicable to systems dominated by flexural response.
However, larger residual displacements have been
observed in postearthquake reconnaissance.
6. 2. 4 Repeated Loading
In the shake table tests, Araki et al. (1990) also
subjected reinforced concrete wall and frame wall
structures to single and repeated motions. It appears that
a synthetic ground motion was used. It was found that
the low rise structures subjected to repeated shake table
tests displaced to approximately twice as much as they
did in a single test. For the mid rise and high rise
structures, repeated testing caused peak displacements
that were approximately 0 to 10% larger than those
obtained in single tests.
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Wolschlag (1993) tested three story reinforced concrete
walls on a shake table. In one test series, an undamaged
structure was subjected to repeated ground motions of
the same intensity. In the repeat tests, the peak
displacement response at each floor of the damaged
specimen hardly differed from the response measured
for the initially undamaged structure.
Cecen (1979) tested two identical ten story, three bay,
reinforced concrete frame models on a shake table. The
two models were subjected to sequences of base
motions of differing intensity, followed by a final test
using identical base motions. When the structures were
subjected to the repeated base motion, the peak
displacement response at each story was only slightly
affected by the previous shaking of the same intensity.
When the two structures were subjected to the same
final motion, peak displacement response over the
height of the two structures was only slightly affected
by the different prior sequences. Floor acceleration
response, however, was prone to more variation.
Mahin (1980) investigated the analytical response of
SDOF oscillators to repeated ground motions. He
reported minor to moderate increases in displacement
ductility demand across all periods, and weaker
structures were prone to the largest increases. For
bilinear models with negative post yield stiffness,
increased duration or repeated ground motions tended
to cause significant increases in displacement ductility
demand (Mahin and Boroschek, 1991).
6. 3 Dynamic Analysis
Framework
6. 3. 1 Overview
This section describes the dynamic analyses determining
the effects of damage from prior earthquakes on
the response to a subsequent performance level
earthquake. In particular, this section describes the
ground motion and hysteresis models, the properties of
the undamaged oscillators, and the assumptions and
constructions used to establish the initially damaged
oscillators. Results of the dynamic analyses are
presented in Section 6. 4.
6. 3. 2 Dynamic Analysis Approach
The aim of dynamic analysis was to quantify the effects
of a damaging earthquake on the response of a SDOF
oscillator to a subsequent, hypothetical, performance
event earthquake. Two obvious approaches may be
taken: the first simulates the damaging earthquake, and
the second simulates the damage caused by the
damaging earthquake.
To simulate the damaging earthquake, oscillators can be
subjected to an acceleration record that is composed of
an initial, damaging ground motion record, a quiescent
period, and a final ground motion record specified as
the performance level event. This approach appears to
simulate reality well, but it is difficult to determine a
priori how to specify the intensity of the damaging
ground motion. One rationale would be to impose
damaging earthquakes that cause specified degrees of
ductility demand. This would result in oscillators
having experienced prior ductility demand and residual
displacement at the start of the performance level
ground motion.
In the second approach, taken in this study, the force
displacement curve of the oscillator is modified
prescriptively to simulate prior ductility demand, and
these analytically “damaged” oscillators are subjected
to only the performance level ground motion. To
identify the effects of damage (through changes in
stiffness and strength of the oscillator force
displacement response), the possibility of significant
residual displacements resulting from the damaging
earthquake was neglected. Thus, the damaging
earthquake is considered to have imposed prior ductility
demands (PDD), possibly in conjunction with strength
reduction or strength degradation, on an initially
undamaged oscillator. Initial stiffness, initial unloading
stiffness, and strength of the oscillators at the start of the
performance level ground motion may be affected.
Response of the initially damaged structure is
compared with the response of the undamaged structure
under the performance level motion. This approach
presumes that an engineer will be able to assess changes
in lateral stiffness and strength of a real structure based
on the nature of damage observed after the damaging
earthquake.
While a number of indices may be used to compare
response intensity, peak displacement response is
preferred here because of its relative simplicity, its
immediate physical significance, and its use as the basic
parameter in the nonlinear static procedures (described
in Section 6. 5). The utility of the nonlinear static
procedures is assessed vis à vis their ability to estimate
accurately the peak displacement response.
It should be recognized that predicting the capacity of
wall and infill elements may be difficult and prone to
uncertainty, whether indexed by displacement, energy,
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or other measures. When various modes of response
may contribute significantly to an element s behavior,
existing models may not reliably identify which mode
will dominate. Uncertainty in the dominant mode
necessarily leads to uncertainty in estimates of the
various capacity measures.
6. 3. 3 Ground Motions
Several issues were considered when identifying ground
motion records to be used in the analyses. First, the
relative strength of the oscillators and the duration of
ground motion are thought to be significant because
these parameters control the prominence of inelastic
response. Second, it is known that ground motions rich
in frequencies just below the initial frequency of the
structure tend to exacerbate damage, because the period
of the structure lengthens as yielding progresses. Third,
information is needed on the characteristics of structural
response to near field motions having forward
directivity effects.
The analyses were intended to identify possible effects
of duration and forward directivity on the response of
damaged structures. Therefore, three categories of
ground motions were established: short duration (SD),
long duration (LD), and Forward Directivity (FD). The
characteristics of several hundred ground motions were
considered in detail in order to select the records used in
each category. Ground motions within a category were
selected to represent a broad range of frequency
content. in addition, it was desired to use some records
that were familiar to the research community, and to use
some records obtained from the Loma Prieta,
Northridge, and Kobe earthquakes. Within these
constraints, records were selected from a diverse
worldwide set of earthquakes in order to avoid
systematic biases that might otherwise occur. Six time
series were used in each category to provide a statistical
base on which to interpret response trends and
variability. Table 6 1 identifies the ground motions that
compose each category, sorted by characteristic period.
Record duration was judged qualitatively in order to
sort the records into the short duration and long duration
categories. The categorization is intended to
discriminate broadly between records for which the
duration of inelastic response is short or long. Because
the duration of inelastic response depends
fundamentally on the oscillator period, the relative
strength, and the force displacement model, a suitable
scalar index of record duration is not available.
The physical rupture process tends to correlate ground
motion duration and earthquake magnitude. It can be
observed that earthquakes with magnitudes less than 7
tended to produce records that were categorized as
short duration motions, while those with magnitudes
greater than 7 tended to be categorized as long duration
motions.
Ground motions recorded near a rupturing fault may
contain relatively large velocity pulses if the fault
rupture progresses toward the recording station.
Motions selected for the forward directivity category
were identified by others as containing near field pulses
(Somerville et al., 1997). Recorded components aligned
most nearly with the direction perpendicular to the fault
trace were selected for this category.
The records shown in Table 6 1 are known to come
from damaging earthquakes. The peak ground
acceleration values shown in Table 6 1 are in units of
the acceleration of gravity. The actual value of peak
ground acceleration does not bear directly on the results
of this study, because oscillator strength is determined
relative to the peak ground acceleration in order to
obtain specified displacement ductility demands.
Identifiers in Table 6 1 are formulated using two
characters to represent the earthquake, followed by two
digits representing the year, followed by four characters
representing the recording station, followed by three
digits representing the compass bearing of the ground
motion component. Thus, IV40ELCN. 180 identifies the
South North component recorded at El Centro in the
1940 Imperial Valley earthquake. Various magnitude
measures are reported in the literature and repeated here
for reference: ML represents the traditional local or
Richter magnitude, Mw represents moment magnitude,
and Ms represents the surface wave magnitude.
Detailed plots of the ground motions listed in Table 6 1
are presented in Figures 6 2 through 6 19. The plots
present ground motion acceleration, velocity, and
displacement time series data, as well as spectral
response quantities. In all cases, ground acceleration
data were used in the response computations, assuming
zero initial velocity and displacement. For most records,
the ground velocity and displacement data presented in
the figures were prepared by others. For the four records
identified with an asterisk ( ) in Table 6 1, informal
integration procedures were used to obtain the ground
velocity and displacement values shown.
(Text continued on page 120)
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Table6 1 Recorded Ground Motions Used in the Analyses
Identifier Earthquake Mag. Station T Component PGA Epic. Char.
Date (g) Dist. Period
l m (k) (sec)
Short Duration (SD)
WN87MWLN. 090 Whittier Narrows ML= 6. 1 Mount Wilson 90 0. 175 18 0. 20
1Oct 87 Caltech Seismic Station
BB92CIVC. 360 Big Bear Ms=6. 6 Civic Center Grounds 360 0. 544 12 0. 40
28 Jun 92
SP88GUKA. 360 Spitak Ms=6. 9 Gukasyan, Armenia 360 0. 207 57 0. 55
7 Dec 88
LP89CORR. 090 Loma Prieta Ms=7. 1 Corralitos 90 0. 478 8 0. 85
17 Oct 89 Eureka Canyon Rd.
NR94CENT. 360 Northridge MW=6. 7 Century City 360 0. 221 19 1. 00
17 Jan 94
IV79ARY7. 140 Imperial Valley ML=6. 6 Array #7 14 140 0. 333 27 1. 20
15 Oct79
Long Duration (LD)
CH85LLEO. 010 Central Chile Ms=7. 8 Llolleo Basement of 1 010 0. 711 60 0. 30
3 Mar 85 Story Building:
CH85VALP. 070 Central Chile Ms=7. 8 Valparaiso University of 070 0. 176 26 0. 55
3 Mar 85 Santa Maria
IV40ELCN. 180 Imperial Valley ML=6. 3 El Centro 180 0. 348 12 0. 65
18 May 40 Irrigation District
TB78TABS. 344 Tabas M=7. 4 Tabas 344 0. 937 <3 0. 80
16 Sep 78
LN92JOSH. 360 Landers M=7. 5 Joshua Tree 360 0. 274 15 1. 30
28 Jun 92
MX85SCTI. 270 Michoacan MS=8. 1 SCTI Secretary of Communication 270 0. 171 376 2. 00
19 Sep 85 and Transportation
Forward Directivity (FD):
LN92LUCN. 250 Landers M=7. 5 Lucerne 250 0. 733 42 0. 20
28 Jun 92
6
IV79BRWY. 315 Imperial Valley ML=6. Brawley Municipal Airport 315 0. 221 43 0. 35
15 Oct 79
LP89SARA. 360 Loma Prieta Ms=7. 1 Saratoga 360 0. 504 28 0. 40
17 Oct 89 Aloha Avenue
NR94NWHL. 360 Northridge MW=637 Newhall 360 0. 589 19 0. 80
17 Jan 94 LA County Fire Station
NR94SYLH. 090 Northridge MW=6. 7 Sylmar County Hospital 090 0. 604 15 0. 90
17 Jan 94 Parking Lot
KO95TTRI. 360 Hyogo Ken Nambu ML= 7. 2 Takatori kisu 360 0. 617 11 1. 40
17 Jan 95
Indicates that informal integration procedures were used to calculate the velocity and displacement histories
shown in Figures 6 2 through 6 19.
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WN87MWLN. 090
0 10 2 0 30 40 so 60
10 20304 50 50
0 10 20 30 40
0 10 2 0 3 0 40 50 60
Time (sec) Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2
1200 I I I2 Damping 2% Damping
5 0 1000. 10% Damping 20% Damping.
40 800 30 60 0 2 0 400 10 200 0 0 L . 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (eec) Period, (eec)
Figure 6 2 Characteristics of the WN87MWLN. 090 (Mount Wilson) Ground Motion
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SP88GUJKA 360 2 00
I 1 2
150 Ground. Acceleration (cm see
100
50
I 0
5 0
100
15 0
2 00
250L I I II I
0 10 2 0 3 0 40 50 60
15 1 0 Ground Velocity (cm sec)
5 0 5 1 0
1 5 20
O 10 20 30 4 0 50 60
3 Ground Displacement (cm)
2 1
0
1
2
3
0 10 2 0 3 0 40 5 0 60
Time (sec)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2
9 0 900
2% Damping 2% Damping
80. 5% Damping 800 z 5% Damping… 10%
Damping 10% Damping
7 0 20% Dam ing 700 20% Damping
6 0 600
50 500 j.
40 4 0 0.
I 3 0 0 3
20 200.
10 100.
0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 I. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 4 Characteristics of the SP88GUKA. 360 (Spitak) Ground Motion
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H85VALP. 070
CH85VALP. 070
200
150
100
50
50
100
150
200
0 10 20 30 40 5060
15
10
5
0
5
10
15
10 2030 40 50 60
4. Displacement (cm) | Ground
10 203040 50 60
Time (see)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2)
120. 1000 II
2% Damping
900. 5% Damping 100
10% Damping 800
20% Damping.
7 0 0. 80. 600 8 4 1 60
500 400 A
40 2 0 0. 200
20 100 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 9 Characteristics of the CH85VALP. 070 (Valparaiso University) Ground Motion
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TB78TABA. 344
1000
400
600
600
0 10 20 30 40 50 60
150
Ground Velocity (cm sec)
100
50
0
50
100
150 10 20 30 40 5
I0 10 20 30 40 50 646
80
60.
40
60.
80.
100
0 10 20 30 40 50 60
Time (sec)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2)
5000 I 1 1 1
450 I I T I I
2% Damping 500 2%Damping
400: 5%
Dampin 4500 5% Damping 10%
Damping 10% Damping 350 p 20%
Damping 4000. 20% Damping.
300 1. 1 3500
3000.
250
2500 I.
200
2000.,
150. 1500, y. i
100
1000.
500.
L I
0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 11 Characteristics of the TB78TABS. 344 (Tabas) Ground Motion
FEMA 307 Technical Resources
Chapter 6:. Analytical Studies
LN92LUCN. 250
800
600 (cm (sec4
era&t1on.
400 200 0
200 I I. I. I
400
600
800I
D 10 20 30 40 50 61
Ground
40
20
20
40
60
80. 1 111,
100.
120.
140
1I0 20 30 40 50 6
40
20nd Displacement
20.
40.
60.
80
100
120
140
160
0 10 20 30 40 50 60
Time (sec) Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2
180 4000
2% Dpi I 2% Damping
160. 5% Damping … 2% Damping
10% Damping: 3500 10% Damping
140 fl
3000
20% Damping.
120:.
I 2500
100
2000
80. I IT
1500
:
60 I. 1000
40
500
:.:.
20
I I
0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 14 Characteristics of the LN92LUCN. 250 (Lucerne) Ground Motion
FEMA 307
Technical Resources
114 Technical Resources FEMA 307
Chapter 6: Analytical Studies
LP89SARA. 360
300
200. Ground Acceler. & Ion. (CM. SeC2.
100
0
10 0
200
300
400
500
I) 10 20 30 40 50 63
30
Velocity
20. J. Ground. (cm/sec)
10
0 r 11
1 0
20
30
40
50
0 10 20 30 40 50 6 (
20
Ground Displacement (cm)
15
10
5
0
5
10
0 102030405060
Time (sec)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm seC2
200 1 1 I. 2000 I 1I I
2% D ping Damping
5%Dampin
180 5% ng 1800 2% 10%
Damping 160
20% g. 1600. 20% Damping. 140
L. 1400.
120 12000
100. 7.
80.
20
0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure6 16 Characteristics of the LP89SARA. 360 (Saratoga) Ground Motion
FEMA 307
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Technical Resources FEMA 307
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NR94NWHL. 360
600
Ground Acceleration (cm sec
400
200
0
200
400
600
0 10 20 30 40 50 6 D
60
40. Ground. Velocity. I. cm sec
20
I 0
20
40
60
80
100
0 10 20 30 40 50 60
40
Ground Disp ac t (cm)
I
30
,.
20
10
0.
10
20,.
30
0 10 20 3040 50 60 Time (sec)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm seC2) 450 3000
2% Damping 400 5% Damping 10% Damping
2500 350
Po 20Da>V, 5w mpi.
300 t 2000 r
250, 2w
1500
200
150 1000
I I 1 100
I. I I. 500
I. 50
O, 0 L
0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 17 Characteristics of the NR94NWHL. 360 (Newhall) Ground Motion
FEMA 307 Technical Resources
Chapter 6: Analytical Studies
NR94SYLM. 090
600
500 lrat o
400
300
200
100
100
200
300.
400
0 10 20 30 40 50 60
60
1
20.
0
20
40
60
80
20 30 40 50 60
15
0 10 0 >Ground Displacement (m
1
10 5
1 I.
0
I I I. I
5
10
15
20
0 102030405060
Time (sec)
Equivalent Velocity (cm sec) Pseudo Acceleration (cm sec2)
250 2000 1 I I
2% Damping
1800. 5% Damping
10% Damping 200
1600. 20% Damping 1400
150. 1200 +
1000.I
j.
9.
100. 800 800
600
50 400.
200
00. 0
0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
Period, (sec) Period, (sec)
Figure 6 18 Characteristics of the. 090 (Sylmar Hospital) Ground Motion
Technical Resources FEMA 307
Chapter 6: Analytical Studies
The characteristic period, Tg, of each ground motion
was established assuming equivalent velocity spectra
and pseudo acceleration spectra for linear elastic oscillators
having 5% damping. The equivalent velocity, Vm,
is related to input energy, Em, and ground acceleration
and response parameters by the following expression:
I mV = Em =m cXidt (6 1)
2 m
where m= mass of the single degree of freedom
oscillator, xg = the ground acceleration, and k = the
relative velocity of the oscillator mass (Shimazaki and
Sozen, 1984). The spectra present peak values
calculated over the duration of the record.
The characteristic periods were determined according to
engineering judgment to correspond approximately to
the first (lowest period) peak of the equivalent velocity
spectrum, and, at the same time, the period at which the
transition occurs between the constant acceleration and
constant velocity portions of a smooth design spectrum
fitted to the 5% damped spectrum (Shimazaki and
Sozen, 1984; Qi and Moehle, 1991; and Lepage, 1997).
Characteristic periods were established prior to the
dynamic analyses.
Other criteria are available to establish characteristic
periods. For example, properties of the site,
characterized by variation of shear wave velocity with
depth, may be used to establish Tg. Alternatively, the
characteristic period may be defined as the lowest
period for which the equal displacement rule applies,
and thus becomes a convenient reference point to
differentiate between short and long period systems.
6. 3. 4. Force Displacement Models
The choice of force displacement model influences the
response time history and associated peak response
quantities. Ideally, the force displacement model should
represent behavior typical of wall buildings, including
strength degradation and stiffness degradation.
Actual response depends on the details of structural
configuration and component response, which in turn,
depend on the material properties, dimensions, and
strength of the components, as well as the load
environment and the evolving dynamic load history
(which can influence the type and onset of failure). The
objective of the dynamic analyses is to identify basic
trends in how prior damage affects system response in
future earthquakes. Fulfilling this objective does not
require the level of modeling precision that would be
needed to understand the detailed response of a
particular structure or component. For this reason, we
selected relatively simple models that represent a range
of behaviors that might be expected in wall buildings.
Three broad types of system response can be
distinguished:
Type A: Stiffness degrading systems with positive
post yield stiffness (Figure 6 20a).
Type B: Stiffness degrading systems with negative
post yield stiffness (Figure 6 20b).
Type C: Pinched systems exhibiting strength and
stiffness degradation Figure 6 20c).
(a) Stiffness Degrading (b) Stiffness Degrading (c) Stiffness and Strength
(positive post yield stiffness) (negative post yield stiffness) Degrading (with pinching)
Figure 6 20 Force Displacement Hysteretic Models
Technical Resources FEMA 307
Chapter 6: Analytical Studies
Type A behavior typically represents wall systems
dominated by flexural response. Type B behavior is
more typical of wall systems that exhibit some
degradation. In response with increasing displacement;
degradation may be due to relatively brittle response
modes. Type C behavior is more typical of wall systems
that suffer degradation of strength and stiffness,
including those walls in which brittle modes of response
may predominate.
Type A behavior was represented in the analyses using
the Takeda model (Takeda et al., 1970) with post yield
stiffness selected to be 5% of the secant stiffness at the
yield point (Figure 6 21a). Previous experience
(Section 6. 2. 1) indicates that this model represents
stiffness degradation in reinforced concrete members
exceptionally well. In addition, it is widely known by
researchers, and it uses displacement ductility to
parameterize stiffness degradation. The Takeda model
features a trilinear primary curve that is composed of
uncracked, cracked, and yielding portions. After
yielding, the unloading stiffness is reduced in
proportion to the square root of the peak displacement
ductility. Additional rules are used to control other
aspects of this hysteretic model. This model is
subsequently referred to as “Takeda5 “.
Type B behavior was represented in the analyses using
the Takeda model with post yield stiffness selected to
be 10% of the yield point secant stiffness
(Figure 6 21b). This model is subsequently referred to
as “TakedalO”.
F 4 k 0. 05k F
Type C behavior was represented in the analyses by a
modified version of the Takeda model (Figure 6 21c).
The behavior is the same as for Type A, except for
modifications to account for pinching and cyclic
strength degradation. The pinching point is defined
independently in the first and third quadrants
(Figure 6 22). The pinching point displacement is set
equal to 30% of the current maximum displacement in
the quadrant. The pinching point force level is set equal
to 10% of the current maximum force level in the
quadrant. Cyclic strength degradation incorporated in
this model is described in Section 6. 3. 6. This model is
subsequently referred to as “TakPinch”.
Collectively, the Takeda5, TakedalO, and TakPinch
models are referred to as degrading models in the body
of this section. For these models, dynamic analyses
were used to identify the effects of prior damage on
response to future earthquakes. The analyses covered a
number of relative strength values, initial periods of
vibration, damage intensities, and performance level
earthquakes. For all dynamic analyses, damping was set
equal to 5% of critical damping, based on the period of
vibration that corresponds to the yield point secant
stiffness.
In addition, a bilinear model (Figure 6 23) was selected
to establish the strength of the degrading oscillators,
which were set equal to the strength required to achieve
bilinear displacement ductility demands of 1 (elastic), 2,
4, and 8 for each reference period and for each of the 18
ground motions. The bilinear model does not exhibit
stiffness or strength degradation. Besides establishing
Ak Yield Point F 4k 0. 10k A (a) Takeda Model (+5%) (b) Takeda Model ( 10%) (c) Takeda Pinching Model (Takeda5 ) (TakedalO) (TakPinch)
Figure 6 21 Degrading Models Used in the Analyses
FEMA 307 Technical Resources
Chapter 6: Analytical Studies k Yield Point A 0. 05k
Figure 6 22 Bilinear Model Used to Determine Strengths of Degrading Models
Maximum prior displacement cycle I 0. 3A., g
Figure 6 23 Specification of the Pinching Point for the Takeda Pinching Model
the strength of the oscillators, this model serves two
additional purposes. First, results obtained in this study
with the bilinear model can be compared with those
obtained by other researchers to affirm previous
findings and, at the same time, to develop confidence in
the methods and techniques used in this study. Second,
the bilinear model provides a convenient point of
departure from which the effects of stiffness and
strength degradation can be compared.
6. 3. 5 Undamaged Oscillator Parameters
To identify effects of damage on response, it is first
necessary to establish the response of initially
undamaged oscillators to the same ground motions. The
response of the undamaged oscillators is determined
using the degrading models of Figure 6 21 for the
performance level ground motions.
The yield strength of all degrading models is set equal
to the strength required to achieve displacement
ductility demands (DDD) of 1 (elastic), 2, 4, and 8
using the bilinear model. This is done at each period
and for each ground motion. For any period and ground
motion considered, the yield strength of the initially
undamaged models is the same, but only the bilinear
model achieves the target displacement ductility
demand. Where the same target displacement ductility
demand can be achieved for various strength values, the
largest strength value is used, as implemented in the
computer program PCNSPEC (Boroschek, 1991).
The initial stiffness of the models is established to
achieve initial (reference) vibration periods of 0. 1, 0. 2,
0. 3, 0. 4, 0. 5, 0. 6, 0. 8, 1. 0, 1. 2, 1. 5, and 2. 0 seconds.
These periods are determined using the yield point
secant stiffness for all the models considered.
For the undamaged Takeda models, the cracking
strength is set equal to 50% of the yield strength, and
the uncracked stiffness is set equal to twice the yield
point secant stiffness (Figure 6 24).
Technical Resources FEMA 307
Chapter 6: Analytical Studies
Yield Point
F 0. 5FY
Cracking Point
A Aax A y
Specification of the Uncracked Stiffness, Cracking Strength, and Unloading Stiffness for the Takeda
Figure 6 24 Models 6. 3. 6 Damaged Oscillator Parameters
Damage is considered by assuming that the force
displacement curves of the oscillators are altered as a
result of previous inelastic response. Reduction in
stiffness caused by the damaging earthquake is
parameterized by prior ductility demand. Strength
degradation is parameterized by the reduced strength
ratio.
Each of the initially undamaged degrading oscillators is
considered to have experienced prior ductility demand
(PDD) equal to 1, 2, 4, or 8 as a result of the damaging
earthquake. The construction of an initially damaged
oscillator force displacement curve is illustrated for a
value of PDD greater than zero in Figure 6 25. The
prior ductility demand also regulates the unloading
stiffness of the Takeda model until larger displacement
ductility demands develop.
The analytical study considered damaging earthquakes
of smaller intensity than the performance level
earthquake. Consequently, the PDD values considered
must be less than or equal to the design displacement
ductility (DDD). Thus, an oscillator with strength
established to achieve a displacement ductility of 4 is
analyzed only for prior displacement ductility demands
of 1, 2, and 4. The undamaged Takeda oscillators
sometimes had ductility demands for the performance
level earthquake that were lower than their design
values (DDD). Again, because the damaging earthquake
is considered to be less intense than the performance
level event, oscillators having PDD in excess of the
undamaged oscillator response were not considered,
further.
The Takeda models of the undamaged oscillators
represent cracking behavior by considering the
uncracked stiffness and the cracking strength. The
effects of cracking in a previous earthquake were
assessed by comparing the peak displacement response
Undamaged Damaged
Figure 6 25 Construction of Initial Force Displacement Response for Prior Ductility Demand > 0 and Reduced
Strength Ratio = 1 123
FEMA 307 Technical Resources Technical Resources 123
Chapter 6: Analytical Studies
Initially Undamaged
I Initially Damaged
Figure 6 26 Construction of Initial Force Displacement Response for PDD> 0 and RSR< 1 for Takeda5 and TakedalO Models
of initially uncracked oscillators to the response of should more closely approximate the response of oscillators that are initially cracked; that is, Takeda the ideal model. oscillators having a PDD of one. When larger PDD
values are considered, the reductions in initial loading 2. TakPinch Oscillators: Rather than begin with a
and unloading stiffness are determined in accordance reduced strength, a form of cyclic strength degradation
was explicitly modeled for the Takeda Pinching
with the Takeda model.
oscillators. A trilinear primary curve was established
It is not obvious what degree of strength degradation is (Figure 6 27), identical to the envelope curve consistent with the PDDs, nor just how the degradation used in the Takeda5 model. The curve exhibits of strength should be modeled to represent real cracking, a yield strength determined from the
structures. We used two approaches to gauge the extent response of the bilinear models, and a post yield
to which strength degradation might affect the response: stiffness equal to 5% of the yield point secant stiffness. A secondary curve is established, having the
1. Takeda5 and TakedalO Oscillators: The initial same yield displacement and post yield stiffness as
strength of the damaged models was reduced to try the primary curve, but having yield strength equal
to capture the gross effects of strength degradation to the reduced strength ratio (RSR) times the prion response. The initial response of the damaged mary yield strength. For displacements less than the
oscillator was determined using the construction of current maximum displacement in the quadrant, a
Figure 6 26. The resulting curve may represent a reduced strength point is defined at the maximum
backbone curve that is constructed to approximate displacement at 0. 54 (l RSR) Fy above the secondary
the response of a strength degrading oscillator. For curve strength, where n is the number of cycles
example, a structure for which repeated cycling approaching the current maximum displacement.
causes a 20% degradation in strength relative to the The oscillator may continue beyond this displace
primary curve may be modeled as having an initial ment, and once it loads along the primary curve, n
strength equal to 80% of the undegraded strength. is reset to one, to cause the next cycle to exhibit
If the backbone curve is established using the strength degradation. The term (1 RSR) Fyis simply
expected degraded strength asymptotes, then the strength difference between the primary and
modeled structure tends to have smaller initial secondary curves, and the function O. 5 represents
stiffness and larger displacement response relative an asymptotic approach toward the secondary curve
to the ideal degrading structure. Consequently, the with each cycle. In each cycle, the strength is
modeled response is expected to give an upper reduced by half the distance remaining between the
bound to the displacement response expected from current curve and the secondary curve. Pinching
the ideal model. If, instead, the backbone curve is and strength degradation are modeled independently selected
to represent an average degraded response, in the first and third quadrants.
using typical degraded strength values rather than
the lower asymptotic values, the computed response
Technical Resources FEMA 307
Chapter 6: Analytical Studies
(RSR) Fy YiC
Figure 6 27 Strength Degradation for Takeda Pinchir
For the TakPinch models, strength degradation is
modeled with and without PDD. When PDD is
present, the oscillator begins with n equal to one.
This represents a single previous cycle to the PDD
displacement, and corresponds to initial loading
towards a reduced strength point halfway between
the primary and secondary curves at the PDD
displacement (Figure 6 28).
For the other degrading models, strength reduction is
considered possible only for PDDs greater than zero.
The parameter RSR is used to describe strength.
degradation in the context of the Takeda Pinching
models and strength reduction in the context of the
other degrading models. For this study, values of RSR
were arbitrarily set at 100%, 80%, and 60%.
Oscillators were referenced by their initial, undamaged
vibration periods, determined using the yield point
secant stiffness, regardless of strength loss and PDDs.
Note that changes in strength further affect the initial
stiffness of the damaged oscillators.
While the values of the parameters used to model Type
A, B, and C behaviors, as well as the hysteresis rules
themselves, were chosen somewhat arbitrarily, they
were believed to be sufficiently representative to allow
meaningful conclusions to be made regarding the
effects of prior damage on response characteristics of
various wall structures. Values of RSR and PDD were
selected to identify trends in response characteristics,
not to represent specific structures.
0. 5 (l RSR) Fy Prior maximum displacement cycle
Current cycle A. A ig Model 6. 3. 7 Summary of Dynamic Analysis
Parameters
Nonlinear dynamic analyses were conducted for SDOF
systems using various force displacement models,
various initial strength values, and for various degrees
of damage. The analyses were repeated for the 18
selected ground motion records. The analysis
procedures are summarized below.
1. Initially undamaged oscillators were established at
eleven initial periods of vibration, equal to 0. 1, 0. 2,
0. 3, 0. 4, 0. 5, 0. 6, 0. 8, 1. 0, 1. 2, 1. 5, and 2. 0 seconds.
At these periods, the strength necessary to obtain
design displacement ductilities (DDDs) of 1 (elastic),
2, 4, and 8 were obtained using the bilinear
model for each earthquake. This procedure establishes
44 oscillators for each of 18 ground motions.
2. The responses of the oscillators designed in step 1
were computed using the three degrading models
(Takeda5, TakedalO, and TakPinch). The yield
strength of the degrading oscillators in this step is
identical to that determined in the previous step for
the bilinear model. The period of vibration of the
degrading oscillators, when based on the yield
point secant stiffness, matches that determined in
the previous step for the bilinear model.
3. Damage is accounted for by assuming that the
force displacement curves of the oscillators are
altered as a result of previous inelastic response.
The extent of prior damage is parameterized by
PDD. For some cases, the strength of the oscillators
is reduced as well. Each of the initially undamaged,
degrading oscillators was considered to have experienced
a PDD equal to 1, 2, 4, or 8, but not in 2
excess of the ductility demand for which the oscillators
Technical Resources
FEMA 307 Technical Resources
Chapter 6: Analytical Studies
Primary Curve
Pinching Point
Reduced Strength Point
5 0c5 (1 RSR) Fy
Initial Damaged Response
Ay 0. 3 (PDD) A (PDD) Ay A
Figure 6 28 Construction of Initial Force Displacement
Response for PDD> 0 and RSR< 1 for Takeda Pinching Model
was designed. The effects of cracking on 6. 4 Results Of Dynamic
response were determined by considering a PDD of Analyses
one. Where larger PDDs are considered, reductions
in the initial loading and unloading stiffness were
determined in accordance with the Takeda model. 6. 4. 1 Overview and Nomenclature
4. Strength degradation was modeled explicitly in the This section describes results obtained from the
TakPinch model. In the Takeda5 and TakedalO dynamic analyses. Section 6. 4. 2 characterizes the
models, strength degradation was approximated by ground motions in terms of strength and displacement
reducing the initial strength of the damaged demand characteristics for bilinear oscillators, in order
Takeda5 and TakedalO models. RSRs equal to to establish that the ground motions and procedures
100%, 80%, and 60% were considered. Although used give results consistent with previous studies.
the strength reduction considered in the Takeda 5 Section 6. 4. 3 discusses the response of the Takeda
and TakedalO models does not model the evolution models in some detail, for selected values of
of strength loss, it suggests an upper bound for the parameters. Section 6. 4. 4 presents summary response
effect of strength degradation on response characteristics statistics for the Takeda models for a broader range of
parameter values.
6. 3. 8 Implementation of Analyses Several identifiers are used in the plots, as follows:
Over 22, 000 inelastic SDOF analyses were conducted Records:
using a variety of software programs. The strength of
the oscillators was determined using constant ductility SD= Short duration ground motions.
iterations for the bilinear oscillators using the program LD= Long duration ground motions.
PCNSPEC (Boroschek, 1991), a modified version of FD= Forward directivity ground motions.
NONSPEC (Mahin and Lin, 1983). Response of the
Takeda models was computed using a program DDD: Design Displacement Ductility. Strength
developed by Otani (1981). This program was modified was determined to achieve the specified
at the University of Illinois to include the effects of DDD response for bilinear oscillators having PDD, pinching, and strength degradation and to identify post yield stiffness equal to 5% of the collapse states for models with negative post yield initial stiffness. Values range from 1 to 8. stiffness.
PDD: Prior Ductility Demand. This represents a
modification of loading and unloading
stiffness, to simulate damage caused by
126 Technical Resources FEMA 307
Chapter 6: Analytical Studies
previous earthquakes. Values range from 1
to 8, but not in excess of DDD.
RSR: Reduced Strength Ratio. This represents a
reduction or degradation of strength and
associated changes in stiffness. Values
ranges from 100% to 60%, as detailed in
Figures 6 26, 6 27, and 6 28.
Displacements:
dd = Peak displacement response of
undamaged oscillator
d d=Peak displacement response of damaged
oscillator
de = Peak displacement response of elastic
oscillator having stiffness equal to the
yield point secant stiffness of the corresponding
Takeda oscillator
Space constraints limit the number of included figures.
Selected results for oscillators designed for a
displacement ductility of 8 are presented below. Elastic
response characteristics are presented as part of the
ground motion plots in Figures 6 2 to 6 19
6. 4. 2 Response of Bilinear Models
Figures 6 29 to 6 31 present the response of bilinear
models to the SD, LD, and FD ground motions,
respectively. The ratio of peak displacement of the
inelastic model to the peak displacement response of an
elastic oscillator having the same initial period,
ddlde, is presented in the upper plot of each figure. The
lower plot presents the ratio of elastic strength demand
to the yield strength provided in order to attain the
specified DDD, which in this case equals 8.
When the strength reduction factor, R, has a value of 8,
the inelastic design strength is 1 8 of the elastic
strength. For DDD = 8, an R = 8 means that the reduced
inelastic design strength and the resulting oscillator
ductility are equal. If R is greater than 8, say 12, for
DDD = 8, then the reduced inelastic design strength of
the structure can be 1 12 of the expected elastic strength
to achieve an oscillator ductility of 8. That is, for any R,
the structure can be designed for 1IR times the elastic
needed strength to achieve a ductility of DDD.
Response to each ground motion is indicated by the
plotted symbols, which are ordered by increasing
characteristic period, Tg. It was found that the
displacement and strength data are better organized
when plotted against the ratio T Tg instead of the
reference period, T The plots present data only for
T Tg < 4 in order to reveal sufficient detail in the range
T Tg < 1. The trends shown in Figures 6 29 through 6 31
resemble those reported by other researchers, for
example, Shimazaki and Sozen (1984), Miranda (1991),
and Nassar and Krawinkler (1991). However, it can be
observed that the longer period structures subjected to
ground motions with forward directivity effects show a
peak displacement response in the range of
approximately 0. 5 to 2 times the elastic structure
response, somewhat in excess of values typical of the
other classes of ground motion. Additionally, strength
reduction factors, R, tend to be somewhat lower for the
FD motions, representing the need to supply a greater
proportion of the elastic strength demand in order to
maintain prespecified DDDs.
6. 4. 3 Response of Takeda Models
The Takedamodels were provided with lateral strength
equal to that determined to achieve specified DDDs of
1, 2, 4, and 8 for the corresponding bilinear models,
based on the yield point secant stiffness.
Prior damage was parameterized by prior ductility
demand (PDD), possibly in conjunction with strength
reduction or strength degradation, which is
parameterized by RSR. PDD greater than zero (damage
present) and RSR less than one (strength reduced or
degrading) both cause the initial period of the oscillator
to increase. When previous damage has caused
displacements in excess of the yield displacement
(PDD>1), even small displacements cause energy
dissipation through hysteretic response. No further
attention is given to those oscillators for which the
imposed PDD exceeds the response of the undamaged
oscillator, and these data points are not represented on
subsequent plots.
6. 4. 3. 1 Response of the Takeda5 Model
It is of interest to observe how structures proportioned
based on the bilinear model respond if their force
displacement response is represented more accurately
by a Takeda model. This interest is based in part on the
widespread use of the bilinear model in developing
current displacement based design approaches.
Figures 6 32 through 6 34 present the response of
Takeda5 models in which the oscillator strength was set
to achieve a bilinear displacement ductility demand of
8. The upper plot of each figure shows the ratio of peak
displacement response to the peak response of an elastic
(Text continued on page 134)
Technical Resources
Technical Resources
FEMA 307
Chapter 6: Analytical Studies
Records=SD; DDD=8; PDD==1; RSR=; Model=Bilinear
O WN87MWLN. 090
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Period Ratio, Tl7g
IF Records=SD; DDD=8; PDD=O; RSR=1; Model=Bilinear
30
o WN87MWLN. 090 I
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0 | 0
15 c v OV 0O e VA
0 cm 10
C AA 0 V 0l A
0s 5 00 0. V
5 V 0 0 123 4
Period Ratio, T Tg
Figure 6 29 Response of Bilinear Oscillators to Short Duration Records (DDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
Technical Resources FEMA 307
Chapter 6: Analytical Studies
I Records=LD; DDD=8; PDD=0; RSR=1; Model=Bilinear
7 0 CH85LLEO. 010
6 0 CH85VALP. 070
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5
LN92JOSH. 360
0) A A
MX85SCT1. 270
Z 4
A A
fi A A
1:3
3 A 0 A
J.
0)
22
V & 0 o
04 0
0 00A 0 IO 0
OI
I
1 2 3 zI
Period Ratio, TIT,
Records=LD; DDD=8; PDD=0; RSR=1; Model=Bilinear
30
C CH85LLEO. 010 A
O CH85VALP. 070
cc 25
A IV40ELCN. 180 0
v TB78TABS. 344
0 20 LN92JOSH. 360.
UC:
MX85SCT1. 270
0
Q 15
CC
10 V 00v Ea
ci)
5 AD V 0 V. 0 v
JJ A AA A V
0
0 23 4
Period Ratio, TITj
Figure 6 30 Response of Bilinear Oscillators to Long Duration Records (DDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307 Technical Resources 129
Chapter 6: Analytical Studies
Records=FD; DDD=8; PDD=O; RSR=1; Model=Bilinear
7
OCLN92LUCN. 250
6. O IV 79B RW Y. 315
A LP89SARA. 360
V NR94NWHL. 360
5
NR94SYLH. 090
KO95TTRI. 360
IM4
8d
A At 3 ob
2 v v A A o w
0 A 0
v 0 0
1 VA V V AV
0 0 1 2 3 4
Period Ratio, T T,
Records=FD; DDD=8; PDD=O; RSR=1; Model=Bilinear
3C I o LN92LUCN. 250
O IV79BRWY. 315
25P a LP89SARA. 360
v NR94NWHL. 360
o O NR94SYLH. 090. i
c 20I
0 KO95TTRI. 360
0 C:
3 150
I c 10 A
2 tOA
0 co
5
v A 0 l
0 I2 3 4
Period Ratio, TIT,
Figure 6 31 Response of Bilinear Oscillators to Forward Directive Records (DDD= 8) DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
Technical Resources FEMA 307
Chapter 6: Analytical Studies
Records LD; DDD=8; PDD=O; RSR=1; Model=Takeda5
I IU
o CH85LLEO. 010
O CH85VALP. 070
8 A IV40ELCN. 180
V TB78TABS. 344
O LN92JOSH. 360
6. S.
MX85SCT1. 270.
8P 4. O.
Q.
CC A
O 2 vo v v v
0 l 2 3 4
Period Ratio, T T
9
Records=LD; DDD=8; PDD=O; RSR=1; Models=Takeda5 & Bilinear
6. 0. CH85VALP. 070
Qv IV40ELCN. 180
v 5.: v TB78TABS. 344
LN92JOSH. 360
MX85SCT1. 270
0 13 0 X E3 A
Period Ratio, TIT,
VA AA V A VAIV0ECN18
0 SReodDDD= z 3 1) esTaea Blna
Long Duratlon; 8; andD RSR=i l
02 0 0 3 4
132ALto Ahc A F 3 o 1 2 3 4
Period Ratio, TIT, Figure 6 33 Displacement Response of Takeda Models Compared with Elastic Response ahd Bilinear Response for
Long Duration Records (DDD= 8 and RSR= 1)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio 132 Technical Resources FEMA307
Chapter 6: Analytical Studies
Records=FD; DDD=8; PDD=O; RSR=1; Model=Takeda5 10
O3 LN92LUCN. 250
O IV79BRWY. 315
A LP89SARA. 360
8
V NR94NWHL. 360
o NR94SYLH. 090
to V 0; A) w
49 0A 4 b0o. t.
V V 0 0 A 3 0 0
2 % V A
7 0 A
0 O 1 2 3 4
Period Ratio, T Tg
Records=FD; DDD=8; PDD=O; RSR=1; Models=Takeda5 & Bilinear
73 LN92LUCN. 250
6 0 IV79BRWY. 315
A LP89SARA. 360
7 NR94NWHL. 360
A g X: v
NR94SYLH. 090
KO95TTRI. 360
A 0 0 A AAA 1
1 0 V v V Is S
VOv. V 2 l 2 3 4
Period Ratio, TIT,
Figure 6 34 Displacement Response of Takeda Models Compared with Elastic Response and Bilinear Response for
Forward Directive Records (DDD= 8 and RSR= 1)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
07 Technical Resources 13
FEMA 307 Technical Resources 133
Chapter 6: Analytical Studies analog, dd de. The upper plots of Figures 6 32 through 6 34 are analogous to those presented in Figures 6 29
through 6 3 1.
The lower plots of Figures 6 32 through 6 34 show the
ratio of the Takeda5 and bilinear ultimate displacements,
dd Takeda5 Bilinear It is clear that peak displacements
of the Takeda model may be several times
larger or smaller than those obtained with the corresponding
bilinear model.
The effect of damage on the Takeda5 model is shown in
Figures 6 35 through 6 40, for Takeda5 oscillators that
were initially designed for a bilinear DDD of 8. The
upper plot of each figure shows the response without
strength reduction (RSR = 1); the lower plot shows
response for RSR = 0. 6.
Figures 6 35 through 6 37 show the effect of cracking
on response. The displacement response, dd, of
Takeda5 oscillators subjected to a PDD of one is
compared with the response of the corresponding
undamaged Takeda5 oscillators, dd. Where no strength
degradation occurs (RSR = 1), cracking rarely causes an
increase in displacement demand; for the vast majority
of oscillators, cracking is observed to cause a slight
decrease in the peak displacement response. Reductions
in strength typically cause a noticeable increase in
displacement response, particularly for low T Tg.
Figures 6 38 through 6 40 show the effect of a PDD of
8 on peak displacement, d d, relative to the response of
the corresponding undamaged oscillators. Prior damage
is observed to cause modest changes in displacement
response where the strength is maintained (RSR = 1);
displacements may increase or decrease. Where
displacements increase, they rarely increase more than
about 10% above the displacement of the undamaged
oscillator for the short duration and long duration
motions. For the forward directivity motions, they
rarely increase more than about 30% above the
displacement of the undamaged oscillator. The largest
displacements tend to occur more frequently for T 0. Force displacement plots for
the first 10 sec of response of each oscillator are
provided in the lower part of the figure, using the same
PDD legend. It can be observed that even though the
undamaged oscillators initially have greater stiffness,
their displacement response tends to converge upon the
response of the initially damaged oscillators within a
few seconds. The displacement response of the
damaged oscillators tends to be in phase with that of the
initially undamaged oscillators, and maximum values
tend to be similar to and to occur at approximately the
same time as the undamaged oscillator peaks. Thus, it
appears that prior ductility demands have only a small
effect on oscillator response characteristics and do not
cause a fundamentally different response to develop.
6. 4. 3. 2 Response of the TakPinch Model
Figures 6 46 to 6 48 plot the ratio, d dldd, of damaged
and undamaged displacement response for the TakPinch
models having DDD = 8 and PDD = 8, for RSR = 1 and
0. 6. Figure 6 49 plots the displacement time history of
TakPinch oscillators subjected to the NS component of
the 1940 El Centro record, and Figure 6 50 plots results
for oscillators having cyclic strength degradation given
by RSR = 0. 6. These oscillators have a reference period
of one second, DDD = 8, and various PDDs.
By comparison with the analogous figures for the
Takeda5 model (Figures 6 38 to 6 40 and 6 43), it can
be observed that: (1) for RSR = 1 (no strength
degradation), the effect of PDD on displacement
response is typically small for the Takeda5 and
TakPinch oscillators, and (2) the effect of cyclic
strength degradation, as implemented here, is also
relatively small. Thus, the observation that prior
(Text continued on page 151)
Technical Resources FEIVA 307
Chapter 6: Analytical Studies
I I Records=SD; DDD=8; PDD=1; RSR=1; Model=Takeda5
Excludescases where pnor damage (PDD) exceeds undamaged
3 WN87MWLN. 090
O BB92CIVC. 360
A SP88GUKA. 360
1. 5 C. 1. I. 11 1. 1. 1. 1. I I ll “I ll. I I l:1 v LP89CORR. 090
O NR94CENT. 360
IV79ARY7. 140
0t
1 A A. A t A 0 AI V a A, A I. I. A
M A, Q: b
0. 5
C
0 l 2 3 4
Period Ratio, T T,
5
Records=SD; DDD=8; PDD=1; RSR=0. 6; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds undamaged response
O WN87MWLN. 090
O BB92CIVC. 360
4 A SP88GUKA. 360
V LP89CORR. 090
| NR94CENT. 360
3 IV79ARY7. 140
0 2 o. s.
e 0 i
0
0
0 I 2 3 4
Period Ratio, T Tg
Figure 6 35 Effect of Cracking Without and With Strength Reduction on Displacement Response of Takeda5 Models,
for Short Duration Records (DDD= 8 and PDD= 1)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307 Technical Resources
Chapter 6: Analytical Studies
Records=LD; DDD=8; PDD=1; RSR=1; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds undamaged response:
O CH85LLEO. 010
O CH85VALP. 070
A IV40ELCN. 180
1. 5. V TB78TABS. 344
O LN92JOSH. 360
A MX85SCT1. 270
1 r 6. iE 0 5>fiSo. A v
b 1 i oo
s o 0 0 1 2 3 4
Period Ratio, TIT,
Records=LD; DDD=8; PDD=1; RSR=0. 6; Model=Takeda5
Excludes cases where prior damage (PDP) exceeds undamaged response
CH85LLEO. 010 A 0 CH85VALP. 070
A IV40ELCN. 180
4. A. V TB78TABS. 344
LN92JOSH. 360
3. A. MX85SCT1. 270 V
VA 0 0 1 2 3 4
Period Ratio, T IT
Figure 6 36 Effect of Cracking Without and With Strength Reduction on Displacement Response of Takeda5 Models,
for Long Duration Records (DDD= 8 and PDD= 1)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
136 Technical Resources FEMA 307
136 Technical Resources FEMA 307
Chapter 6: Analytical Studies
Records=FD; DDD=8; PDD=1; RSR=1; Model=Takeda5
2 Excludes cases where prior damage (PDD) exceeds undamaged response
3 LN92LUCN. 250
O IV79BRWY. 315
A LP89SARA. 360
1. 5 V NR94NWHL. 36
NR94SYLH. 090
KO95TTRI. 360
A. 1 o D O a 0 A
0. 5 I 0 0 1 2. 3 4I
Period Ratio, TI T
Records=FD; DDD=8; PDD=1, RSR=0. 6; Model=Takeda5
5 Excludes cases where prior damage (PDD) exceeds undamaged responses
Ol LN92LUCN. 250
O IV79BRWY. 315
4 A LP89SARA. 360
V NR94NWHL. 360
NR94SYLH. 090
KO95TTRI. 360
II. 3 2 A D
VA 0 00
vOV Ev 0 v.
V4 V. O 0. So. X OA 4 A; V sQ;
0 I 0 1 2 34
Period Ratio, T T
Figure 6 37 Effect of Cracking Without and With Strength Reduction on Displacement Response of Takeda5 Models,
for Forward Directive Records (DDD= 8 and PDD= 1)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307 Technical Resources 137
Chapter 6: Analytical Studies
Records=SD; DDD=8; PDD=8; RSR=1; Model=Takeda5
2 Excludes: cases response:
Where prior damage (PDD) exceeds undamaged
C WN87MWLN.090
O BB92CIVC.360
A SP8BGUKA.360
1.5 I. I I I I 1 I l
V LP89CORR.090
0 NR94CENT.360
IV79ARY7.140
V E, I I I. V I W s a 0 0 0 I I
V V 0.5 1 I I S.I
0 1 2 3 z I
Period Ratio, TIT,
Records=SD; DDD=8; PDD=8; RSR=0.6; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds
O WN87MWLN.090
O BB92CIVC.360
4 A SP88GUKA.360
V LP89CORR.090
NR94CENT.360
I. A IV79ARY7.140
bt 2 A, A A A Va A o 0
I VD A 0 i 0 1 2 34 Period Ratio, T T,
Figure 6 38 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of Takeda5 Models, for Short Duration Records (DDD= 8 and PDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio Technical Resources FEMA 307
Chapter6: Analytical Studies
Records=LD; DDD=8; PDD=8; RSR=1; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds undamaged response.
o CH85LLEO.010
O CH85VALP.070
A IV40ELCN.180
1.5 v TB78TABS. 344
0 LN92JOSH. 360
A v MX85SCT1. 270
t 1 V 0tA 0e 00 I 9 o I
0.5 0 0 l2 34 I
Period Ratio, TIT,
Records=LD; DDD=8; PDD=8; RSR=0.6; Model=Takeda5
5 Excludes cases where prior damage (PDD) exceeds undamaged response
o CH85LLEO.010
O CH85VALP.070
4 A IV40ELCN.180
v TB78TABS.344
LN92JOSH.360
MX85SCT1.270
3 0. 01 A 2 v 0
v 0i5, v 0 vY A 0 0. v. 03. 0.
1 A, PAAkA 0 0 0 1 234
Period Ratio, T Tg
Figure 6 39 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of Takeda5 Models, for Long Duration Records (DDD= 8 and PDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307 Technical Resources
Chapter 6: Analytical Studies
Records=FD; DDD=8; PDD=8; RSR=1; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds undamaged response.
o LN92LUCN.250
O IV79BRWY.315
A LP89SARA.360
1.5 V NR94NWHL.360
A a v r NR94SYLH.090
0 0 K095TTRI.360
1va0. 1 A 0 0 A
0.5 0.5 0
0 1 2 3 4
Period Ratio, TIT,
Records=FD; DDD=8; PDD=8; RSR=0.6; Model=Takeda5
Excludes cases where prior damage (PDD) exceeds undamaged response
El LN92LUCN.250
O IV79BRWY.315
4 A LP89SARA.360
V NR94NWHL.360
NR94SYLH.090
A 095TTR 360
3 9.3.6.0
2 0 OVO E
0 0 0 1 2 3 4
Period Ratio, TIT,
Figure 6 40 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of Takeda5 Models, for Forward Directive Records (DDD= 8 and PDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307
Technical Resources
Technical Resources FEMA 307
Chapter 6: Analytical Studies
IV40ELCNDDD 8 T 1.5 sec. TakedaS
Displacement (cm)
15 pd 10, g1
S 0 n < d8
5 10, 1
15 20 0 5 10 15 20 25 30 35 40C
15 p. pddO10 1ps;
d + 5
0 5 10 15 j I J I I I j
L I I 2 0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Time (sec)
Force (N) Force (N)
40 r, r 40
30;
30. 40
20
10
0
10
01 1.
4
20 j r
20 30
40 T i I I I I
20 15 10 5 0 10 15 20 15 10 5 0 5 10 15
Displacement (cm) Displacement (cm
Force (N) Force (N)
40 40
L 30, 1 iS.
30
20 20
10 10
L, 0
0
X¢. 5.
10 10
20. 20
30
30
4A
An 20 15 10 5 0 5 10 15 20 15 10 5 0 5 10 15
Displacement (cm) Displacement (cm)
Figure 6 44 Effect of Damage on Response to El Centro (IV40ELCN. 180) for Takeda5, T=1.5 sec (DDD= 8)
DOD = Design Displacement Ductility
FEMA 307
Technical Resources
144 Technical Resources FEMA 307
Chapter 6: Analytical Studies
IV40913CNDDD 8 T 2.0 sea, Takeda5
Displacement (cm)
15
10
5
I R I I I pdd8
0
5
10
15
0 5 10 15 20 25 30 35 41D
15 I
pddO
10 i T pddlpdd4
p d.
2 j.
0 5
paS
5, y.
10. I
15
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Time (sec)
Force (N) Force (N)
25 25
20 I 20
15 15
10
10.I. 10.
5 S1,
x.
15 4; i0
10
15
I. 205. X f.
20
25
0
15 10 5 O 5 10 15 15 10 5 0 5 10 15
Displacement (cm) Displacement (cm)
Force (N) Force (N)
25
20.
15 iI
; 15 X
f
20.
20
5 X
10 15 10
15.
20 15; 10 5.
25
25
15 10 5 0 5 10 15 0 5 10 15
Displacement (cm) Displacement (cm)
Figure 6 45 Effect of Damage on Response to El Centro (IV40ELCN. 180) for Takeda5, T=2.0 sec (DDD= 8)
DDD = Design Displacement Ductility
FEMA 307 Technical Resources
Chapter 6: Analytical Studies
Records=SD; DDD=8; PDD=8; RSRM1; Model=TakPinch 5
Excludes cases where prior damage (PDD) exceeds undamaged
response
3 WN87MWLN.090
O BB92CIVC.360
A SP88GUKA.360
4 v LP89CORR.090
O NR94CENT.360
3 IV79ARY7.140
2
0
I I AC A
V
A 0O A 13 A AI &
0
0 1 2 3 4I
Period Ratlo, T IT
Records=SD; DDD=8; PDD=8; RSR=0.6; Model=TakPinch
Excludes cases where prior damage (PDD) exceeds undamaged
response. WN87MWLN.090
O BB92CIVC.360
A SP88GUKA.360
4 V LP89CORR.090 4
I O NR94CENT.360
IV79ARY7.140
3 0 2 0
: A 0 A
I 0 1 2 34
Period Ratio, T T
Figure 6 46 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of TakPinch Models, for Short Duration Records (DDD= 8 and PDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
FEMA 307 Technical Resources
146 Technical Resources FEMA 307
Chapter 6: Analytical Studies
5Records=LD; DDD=8; PDD=8; RSR=1; Model=TakPinch
Excludes cases where prior damage (PDD) exceeds undamaged response
0 CH85LLEO.010
0 CH85VALP.070
A IV40ELCN.180
v TB78TABS.344
LN92JOSH.360 b E
MX85SCT1.270
vo o 1 a 0t 0sOo A
o+0 0a; i0
0 l2 3 4
Period Ratio, T T Records=LD; DDD=8; PDD=8; RSR=0.6; Model=TakPinch
Excludes cases where prior damage (PDD) exceeds undamaged response 11C8LE.1 OCH85VALP.4 A IV
40ELCN.180
v TB78TABS.344
O LN92JOSH.360
@; MXESSCT1.270
AO V v e++v v o V, V.
ow§4 : 003
0C8VL.70 El
0 1 2 3 4
Period Ratio, TIT,
Figure 6 47 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of TakPinch Models, for Long Duration Records (DDD= 8 and PDD= 8) DDD
= EDesign
Displacement Ductility; PDD= Prior Ductility Demands RM Reduced Strength Ratio
FEMA 307 Technical Resources 147
Chapter 6: Analytical Studies
Records=FD; DDD=8; PDD=8; RSR=1; Model=TakPinch
Excludes cases where prior damage (PDD) exceeds undamaged response OLN92WCN.250
o IV79BRWY.315
A LP89SARA.360
4 V NR94NWHL.360
: NR94SYLH.090
K095TTRI.360
3
2
V 0
I Y pO0 O
V cW A 00; A
b
0
0
o 2 3 4
0
Period Ratio, T T,
Records=FD; DDD=8; PDD=8; RSR=0.6; Model=TakPinch
Excludes cases where prior damage (PDD) exceeds undamaged response O LN92LUCN250
O IV79BRWY.315
4 A LP89SARA.360
V NR94NWHL.360
NR94SYLH.090
3 A KO95TTRI.360 @
3 06 A¢o AV O0
0 I I
0 1 2 3 4
Period Ratio, TITg
Figure 6 48 Effect of Large Prior Ductility Demand Without and With Strength Reduction on Displacement
Response of TakPinch, for Forward Directive Records (DDD= 8 and PDD= 8)
DDD = Design Displacement Ductility; PDD = Prior Ductility Demand; RSR = Reduced Strength Ratio
14 ehia
Technical Resources FEMA 307
Chapter 6: Analytical Studies
IV40ELCNDDD=8 T l.0 sec, Takeda Pinching RSR 1
Displacement (cm)
15
A 10
S
:t g.
0
5
10
15
0 5 10 15 20 25 30 35 4 (1
15 2
10
5
0 i i i ip
d
5
I 10
1 r
0.0 2.0 4.0 6.0 8.0 10.0
Time (sec)
Force (N) Force (N)
80. 80
60. 60.
40 40;
20.
1. 0
O20, 20
40 40.
60 60 I.
80 i i i i. 80
15 10 5 0 5 10 15 15 10 5 0 5 10 15
Displacement (cm) Displacement (cm)
Force (N) Force (N)
80 80 I I I I
60 40.40
z. 0
+.
20
20
0,
20,
20O i.
40 40 I.
60,
60
80
1.5 10 5 0 5 10 15 15 10 5 0 5 10 15
Displacement (cm) I Displacement (cm)
Figure 6 49 Effect of Damage on Response of TakPinch Model to El Centro (IV40ELCN. 180) for
T=1.0sec and RSR= I (DDD=8)
DDD = Design Displacement Ductility
FEMVA 307 Technical Resources 149
Chapter 6: Analytical Studies
IV40ELCNDDD 8T 1.O sec, Takeda Pinching RSR 0.6
Displacement (cm) 15 10 5 0 10d
5 10
0 5 10 15 25. 30 35 4D
F i t" 20, v 15 10 5
0, I i 5 10 15
0.0 2.0 4.0 6.0 8.0 10.0
Time (sec)
Forrce (N) Force (N)
80 80 1
60. 60 40
40, 7C; 4
20 20 0
20 20
40 40
60.
60.
80 1 80 i i II
5 10 5 0 5 10 15 15 10 5 0 5 10 15
Displacement (cm) Displacement (cm)
Force (N) Force (N)
80 80 I
60 60
40 i 40
20
20 0
0 1
20 20.r
40 40
60
80 80
I15
10 5 0 5 10 15 1.5 10 5 0 5 10 15
Displacement (cm) Displacement (cm)
Figure 6 50 Effect of Damage on Response of TakPinch Model to El Centro (IV40ELCN. 180) for
T=1.0 sec and RSR = 0.6 (DDD= 8)
DDD = Design Displacement Ductility
Technical Resources FEMVA 307