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Accounting for Ground-Motion Spectral Shape
Characteristics in Structural Collapse Assessment
through an Adjustment for Epsilon
Curt B. Haselton, M.ASCE
1
; Jack W. Baker, M.ASCE
2
; Abbie B. Liel, M.ASCE
3
; and
Gregory G. Deierlein, F.ASCE
4
Abstract: One of the challenges of assessing structural collapse performance is the appropriate selection of ground motions for use in the
nonlinear dynamic collapse simulation. The ground motions should represent characteristics of extreme ground motions that exceed the
ground-motion intensities considered in the original building design. For modern buildings in the western United States, ground motions
that cause collapse are expected to be rare high-intensity motions associated with a large magnitude earthquake. Recent research has shown
that rare high-intensity ground motions have a peaked spectral shape that should be considered in ground-moti on selection and scaling. One
method to account for this spectral shape effect is through the selection of a set of ground motions that is specific to the building’s fundamental
period and the site hazard characteristics. This selection presents a significant challenge when assessing the collapse capacity of a large
number of buildings or for developing systematic procedures because it implies the need to assemble specific ground-motion sets for each
building. This paper proposes an alternative method, whereby a general set of far-field ground motions is used for collapse simulation, and the
resulting collapse capacity is adjusted to account for the spectral shape effects that are not reflected in the ground-motion selection. The
simplified method is compared with the more direct record selection strategy, and results of the two approaches show good agreement. DOI:
10.1061/(ASCE)ST.1943-541X.0000103. © 2011 American Society of Civil Engineers.
CE Database subject headings: Ground motion; Structural failures; Assessment; Spectral analysis.
Author keyword s: Ground motions; Spectral shape; Epsilon; Collapse assessment; Performance assessment; ATC-63; FEMA P695.
Introduction and Goals of Study
One of the challenges in assessing structural collapse capacity by
nonlinear dynamic analysis is the selection and scaling of ground
motions for use in the analysis. Baker and Cornell (2005) have
shown that spectral shape, in addition to ground-motion intensity,
is a key characteristic of ground motions affecting structural
response. In particular, for a given ground-motion hazard level
(e.g., a 2% chance of exceedence in 50 years), the shape of the
uniform hazard spectrum (UHS) can be quite different from the
shape of the mean or expected response spectrum of a real ground
motion having an equally high spectral amplitude at a single period
(Baker 2005; Baker and Cornell 2006). Spectral shape character-
istics are especially important for structural collap se assessments
because at high amplitudes these differences are most signifi-
cant. Therefore, when assessing the probability of collapse under
high-amplitude motions, the choice of ground motions significantly
affects the collapse assessment.
To illustrate the distinctive spectral shape of rare ground
motions, Fig. 1 shows the acceleration spectrum of a Loma Prieta
ground motion. [The motion shown in Fig. 1 is from the Saratoga
station and is owned by the California Department of Mines and
Geology and included in the Pacific Earthquake Engineering
Research Center (PEER) Next Generation Attenuation (NGA) da-
tabase (PEER 2008). For this illustration, this spectrum was scaled
by a factor of þ1:4. This scaling is for illustration purposes only,
and epsilons should be computed by using unscaled spectra.] The
Loma Prieta spectrum has a rare spectral intensity at 1.0 s of 0.9 g,
which has only a 2% chance of exceedance in 50 years. The figure
also shows the mean expected spectrum predicted by the Boore
et al. (1997) attenuation prediction that is consistent with the event
magnitude, distance, and site characteristics associated with this
ground motion. Fig. 1 shows that this extreme ground motion
has a much different shape than the mean predicted spectrum.
In particular, the spectrum for this record has a “peak” from approx-
imately 0.6 to 1.8 s and lower intensities relative to the predicted
spectrum at other periods. The intensity at 1.0 s, exceeded with a
2% likelihood in 50 years, is in the peaked region of the spectrum,
and at this period the observed Sað1sÞ¼0:9 g is much higher than
the mean expected Sað1sÞ¼0:3 g; at other periods away from the
peak, spectral values are closer to the mean expected Sa. This
peaked shaped arises because ground motions that have an
above-average intensity do not necessarily have equally large inten-
sities at other periods.
At a 1.0 s period, the spectral value of the Loma Prieta record is
1.9 standard deviations above the predicted mean spectral value
from the attenuation relationship so this record is said to have
1
Dept. of Civil Engineering, California State Univ. Chico, Chico, CA
95929.
2
Dept. of Civil and Environmental Engineering, Stanford Univ.,
Stanford, CA 94305.
3
Dept. of Civil, Environmental, and Architectural Engineering, Univ. of
Colorado, Boulder, CO 80309.
4
Dept. of Civil and Environmental Engineering, Stanford Univ.,
Stanford, CA 94305.
Note. This manuscript was submitted on October 13, 2008; approved on
August 31, 2009; published online on October 2, 2009. Discussion period
open until August 1, 2011; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Structural Engineer-
ing, Vol. 137, No. 3, March 1, 2011. ©ASCE, ISSN 0733-9445/2011/3-
332–344/$25.00.
332 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011
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“ε ¼ 1:9 at 1.0 s.” ε (i.e., epsilon) is defined as the number of
logarithmic standard deviations between the observed spectral
value and the mean Sa prediction from a ground-motion prediction
or “attenuation” model. Similarly, this record has ε ¼ 1:1 at 1.8 s.
Thus, the parameter ε is a function of the ground-motion record, the
ground-motion prediction model to which it is compared, and the
period of interest.
Just as ε is a function of the period, the relationship between ε
and the spectral shape depends upon the period considered. For
example, a motion with εð1sÞ¼2:0 would tend to have a peak
near a period of 1 s, and a motion with εð2sÞ¼2:0 would tend
to have a peak near a period of 2 s. Because ground motions are
inherently random, this relationship between ε and the spec tral
shape (shown in Figs. 1 and 2) is not necessarily evident for indi-
vidual ground motions, but is evident and statistically defensible
when examining average trends in large data sets of recorded
ground motions (Baker and Jayaram 2008).
The “peaked” spectral shape of rare ground motions observed in
Fig. 1 is general to non-near-field sites in coastal California. In par-
ticular, such sites typ ically exhibit values of ε between 1 and 2 for
the motions with a 2% in 50 years intensity levels. These posit ive ε
arise from the fact that the return period of the ground motion (i.e.,
2,475 years for a 2% in 50 years motion) is much longer than the
return period of the earthquake that causes the ground motion (i.e.,
typical earthquake return periods that govern the high seismic haz-
ard are 150–500 years in California). Accordingly, record selection
for structural analyses at such sites should reflect the expectation of
ε ¼ 1–2 for 2% in 50 years motions.
This paper focuses on the consideration of the spectral shape
through the parameter ε for the purposes of collapse assessment
through nonlinear dynamic analysis. A prediction of structural col-
lapse requires a set of ground motions in which the amplitude of
each ground motion in the set is scaled to an increasing intensity
until it causes collapse. The collapse capacity of an individual
ground-motion record is denoted by the corresponding intensity
on the basis of the spectral acceleration at the first-mode period
of the building S
a;col
ðT
1
Þ. The structure’s collapse capacity is then
defined by the mean and dispersion of the collapse capacities of the
individual records. [Strictly speaking, the “mean” used throughout
this paper is defined as the geometric mean (i.e., the exponential of
the mean of the logarithms). This mean is equal to the median
of a lognormal distribution so it is also sometimes referred to as
the “median.”] The proposed approach for selecting and sealing
records and characterizing spectral shape through the ε parameter
is predicated on defining the ground-motion intensities by using
SaðT
1
Þ.
As described subsequently, previous research has shown that the
consideration of this peaked spectral shape significantly increases
the computed collapse capacit y of a structure relative to the results
obtained by using motions without a peaked spectral shape.
For cases in which these rare motions (i.e., those with ε values
approaching 2.0) govern the performance assessment, such as when
assessing the collapse risk of modern buildings in the seismic
regions of California, properly acc ounting for the expected þε
is critical.
The most direct approach to account for spectral shape in struc-
tural analysis is to select ground motions that have εðT
1
Þ values that
match the target εð T
1
Þ obtained from a hazard analysis for the
intensity level of interest, measured at the fundamental period of
the structure. An alternative approach is to select and scale ground
motions by an intensity measure other than SaðT
1
Þ, which accounts
for a spectral shape in either an implicit or an explicit manner.
Possible intensity measures include inelastic spectral displacement
(Tothong 2007) or Sa values averaged over a period range (Baker
and Cornell 2006). However, because the SaðT
1
Þ intensity measure
is widely used to describe the seismic hazard, the goal of this study
is to develop an alternative approach to define and characterize the
ground motions for analysis.
The proposed approach is intended to (1) permi t the use of a
general ground-m otion set for structural analysis selected in-
dependently of ε values, and (2) then correct the collapse capacity
estimates to account for the spectral shape. The correction adjust-
ment is calculated by using εðT
1
Þ, which is computed for a given site
and hazard level through the disaggregation of the seismic hazard
for the site. Development of this proposed approach was motivated
by related studies (FEMA 2008; Haselton and Deierlein 2007, chap-
ters 6–7) that involved assessing the collapse safety of a large set of
buildings with differing fundamental periods. Because of the large
number of buildings and a desire to generalize the site characteris-
tics as seismic design categories (SDC), selecting unique ground-
motion sets for each of the buildings was not feasible.
This paper first discusses how the spectral shape and ε
are related and then illustrates how the spectral shape affects the
calculated structural collapse capacity. Next considered are the
representative spectral shapes and the ε values expected for various
sites and hazard levels. A regression method is propos ed to account
for the effects of the spectral shape on collapse by applying a cor-
rection factor to the mean collapse capacities obtained by using a
generic ground-motion record set. The regression method is then
applied to 111 buildings to develop a simplified method to adjust
the collapse capacity through an ε correction factor.
Previous Research on the Epsilon Parame ter and
Spectral Shape Effects on Collapse Assessment
How Spectral Shape Relates to the Epsilon
Values of Ground Motions
Fig. 1 shows the spectral shape of a single Loma Prieta ground-
motion record that is consistent with a 2% in 50 years intensity
level at 1.0 s and has εð1sÞ¼1:9. This figure suggests that a
positive ε value tends to be related to a peak in the acceleration
spectrum around the period of inte rest. Recent studies have verified
the statistical robustness of this relationship between a positive ε
and a peaked spectral shape by using multiple ground motions.
To illustrate, Fig. 2 compares the mean spectral shape of three
0 0.5 1 1.5 2
0
0.5
1
1.5
2
Sa
component
[g]
Period [s]
Spectral Peak
ε = +1.9 at 1.0s
ε = +1.1 at 1.8s
Observed Loma Prieta
Spectrum with 2% in 50
yr Sa(1s)
Mean + 2σ
Mean BJF
Mean - 2σ
Fig. 1. Comparison of an observed spectrum from a Loma Prieta
motion with spectra predicted by Boore et al. (1997); after Haselton
and Baker (2006)
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ground-motion sets containing 78 motions, 20 motions, and 20 mo-
tions, respectively: (1) a set selected without regard to ε [i.e., gen-
eral far-field set, desc ribed in FEMA (2008)], (2) a set selected to
have εð1sÞ¼þ2, and (3) a set selected to have εð2sÞ¼þ2. The
general far-field set is approximately epsilon-neutral. To facilitate
comparison, the record sets were scaled such that the mean Sað1sÞ
for Set 2 and Sað2sÞ for Set 3 were matched to the respective val-
ues of Set 1. Fig. 2 shows that the spectral shapes are distinctly
different when the records are selected with or without regard
to ε. When the records have positive ε values at a specified period,
their spectra tend to have a peak at that period. This shape is much
different than a standard uniform hazard spectral shape. Baker and
Cornell (2005) developed a statistically rigorous method to predict
this expected spectral shape, which is termed the conditional mean
spectrum (CMS) because it is conditioned on an Sa value at a speci-
fied period.
How Spectral Shape (Epsilon) Affects Collapse
Capacity
Selecting ground motions with peaked spectral shapes typical of
rare ground motions, as represented by positive εðT
1
Þ values,
has been shown to significantly increase collapse capacity pre-
dictions for which capacity is defined SaðT
1
Þ. Conceptually, the
difference in collapse capacity can be explained by comparing
the spectral shapes of the epsilon-neutral set to the two positive
epsilon sets anchored at 1.0 and 2.0 s. These sets are shown as
long-dashed, solid, and short-dashed lines, respectively, in Fig. 2.
For example, if a building period is 1.0 s and the ground-motion
records are scaled to a common value of Sað1sÞ, the spectral values
of the positive epsilon set (represented by a solid line in Fig. 2) are
smaller than those of the epsilon-neutral set (represented by the
long-dashed line) for SaðT > 1sÞ. The spectral values at longer
periods are significant because the effective period will elongate
as the structure becomes damaged. Similarly, the smaller spectral
values for shorter periods (i.e., T < 1 s) for the positive epsilon set
(represented by a solid line in Fig. 2) are significant because they
will impact the contribution of higher modes with T < T
1
.
Four studies have documented the effect of epsilon on nonlinear
collapse simulations. Bak er and Cornell (2005) studied the effects
of various ground-motion properties on the collapse capacity of a
7-story nonductile reinforced concrete (RC) frame building with a
fundamental period T
1
of 0.8 s. They found that the mean collapse
capacity increased by a facto r of 1.7 when an εð0:8sÞ¼2:0
ground-motion set was used in place of a set selected without re-
gard to epsilon which had the mean εð0:8sÞ¼0:2. Goulet et al.
(2007) studied the collapse safety of a modern 4-story RC frame
building with a period of T
1
¼ 1:0 s and compared the collapse
capacities for a ground-motion set with a mean εð1:0sÞ¼1:4
to another set that had a mean εð1:0sÞ¼0:4. The set with
εð1:0sÞ¼1:4 resulted in a mean collapse capacity that was
1.3–1.7 times larger than that of the εð1:0sÞ¼0:4 set in which
the range was associated with variations in building design and
modeling attributes. Haselton and Baker (2006) used a ductile,
but degrading, single-degree-of-freedom oscillator with a period
of T
1
¼ 1:0 s to demonstrate that a ε ð1:0sÞ¼2:0 ground-motion
set resulted in a 1.8 times larger mean collapse capacity compared
to using a ground-motion set selected without regard to ε which had
the mean εð 1 :0sÞ¼0:2. Likewise, Zareian (2006) investigated the
effects that ε had on the collapse capacities of generic frame and
wall structures. For a selected 8-story frame and 8-story wall build-
ing, he showed that a change from εðT
1
Þ¼0:0toεðT
1
Þ¼1:5 re-
sulted in a factor of 1.5–1.6 increase in mean collapse capacity.
The ε parameter has also been considered for the prediction of a
response from near-fault ground motions but was found to not fully
quantify the impact of forward-directivity velocity pulses on struc-
tural response (Baker and Cornell 2008). The app roach proposed in
this paper should not be applied to near-fault motions with large
forward-directivity velocity pulses.
What Epsilon Values to Expect for a Specific Site
and Hazard Level
Illustration of Concept by Using a Characteristic Event
To illustrate the relationship between expected ε, site, and hazard
level, consider an idealized site in which the ground-motion hazard
is dominated by a single characteristic event:
• Characteristic event return period = 200 years
• Characteristic event magnitude = 7.2
• Closest distance to fault = 11.0 km
• Site soil conditions—V
s
30
¼ 360 m=s
• Building fundamental period of interest = 1.0 s
Fig. 3 shows the predicted mean spectrum and spectra for
mean 1 and 2 standard deviations (i.e., 1ε and 2 ε), given
occurrence of the characteristic event. The mean predicted ground
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
Period [s]
Sa [g]
Mean General Far-Field Set
Mean Positive
ε
Set (for T
1
= 1.0s)
Mean Positive
ε
Set (for T
1
= 2.0s)
Fig. 2. Comparison of spectral shapes of ground-motion sets selected
with and without considering ε after Haselton and Baker (2006)
0 0.5 1 1.5 2
0
0.5
1
1.5
2
Sa [g]
Period [s]
Mean + 2σ
Mean BJF
Mean - 2σ
2% in 50 yrs. (2,475 yr)
10% in 50 yrs. (475 yr)
50% in 5 yrs. (7.2 yr)
Fig. 3. Boore et al. (1997) ground-motion predictions for the
characteristic event, predicted lognormal distribution at T ¼ 1:0s,
and spectral accelerations for the 2% in 50 years and other hazard levels
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motion is Sað1sÞ¼0:40 g when using the Boore et al. (1997)
attenuation model. This figure also includes a superimposed log-
normal distribution of Sað1sÞ representing the predicted distribu-
tion of Sað1sÞ values, with a logarithmic standard deviation of
0.57, expected from an event with this magnitude and distance.
The Sað1sÞ values associated with less frequent ground motions
(i.e., 2% in 50 years) are associated with the upper tail of the dis-
tribution of Sað1sÞ for this event.
In general, when the return period of the characteristic earth-
quake (e.g., 200 years) is much shorter than the return period of
the ground motion of interest (e.g., 2,475 years), then the ground
motion of interest will have a positive ε. This statement is easily
illustrated for an idealized site. When a single characteristic event
dominates the ground-motion hazard, the mean ret urn period (RP)
of the ground motion Sa ≥ x is related to the character istic event as
follows:
1
RP
Sa≥x
¼
1
RP
Characterstic Event
½PðSa ≥ xjCharacteristic EventÞ
ð1Þ
The return period for a 2% in 50 years motion, computed
by using the standard Poissonian occurrence assumption, is
PðSa > x in time tÞ¼1 expðt=RP
Sa>x
Þ, where t ¼ 50 years
and PðSa > x in time tÞ¼0:02. This results in a return period,
RP
Sa>Sa2=50
, of 2,475 years. The return period of the characteristic
event is 200 years. From Eq. (1 ) then, ð1=2;475 yearsÞ¼
ð1=200 yearsÞð0:081Þ. Only 8% of motions that come from
the characteristic earthquake are at least as large as the 2% in
50 years motion. An 8% probability of exceedance corresponds
to 1.43 standard deviations above the mean value, or εð1sÞ¼
1:43. A change in site soil conditions would affect the predicted
spectral accelerations because of the change in the attenuation pre-
diction, but it would not change the ε value because the ratio of the
return periods of the ground motion of interest and the return period
of the earthquake would be unchanged. The situation is more com-
plicated for realistic sites with more earthquake sources, but in gen-
eral the ε value associated with a design Sa level does not change
significantly when the site conditions are varied.
The expected ε value depends strongly on the return period of
the ground motion of interest. Fig. 3 shows that a 10% in 50 years
motion (i.e., a return period of 475 years) is associated with
Sað1sÞ¼0:46 g and ε ð1sÞ¼0:3. For a much more frequent
50% in 5 years motion (i.e., a return period of 7.2 years), Sað1sÞ¼
0:15 g and εð1sÞ¼1:7. For cases in which rare motions drive
the performance assessment, such as with the collapse assessment
of modern buildings, it is likely that the ground motion will fall into
the “positive ε” category.
Eq. (1) also shows that the expected ε value depends on the
return period of the characteristic event. In coastal California, earth-
quake return periods of 200 years are common, but in the eastern
United States, large earthquake return periods are longer. These
longer return periods in the eastern United States will cause the
expected ε values for extreme (i.e., rare) ground motions to be
smaller.
Expected Epsilon Values from the United States
Geological Survey
Unlike the idealized site considered in the preceding section, most
locations have several causal earthquake sources that contribute
significantly to the ground-motion hazard, as well as having more
complex distributions of magnitude. For the general case, expected
ε values must be computed by disaggregating the results of the
seismic hazard analysis.
The USGS conducted seismic hazard analyses across the United
States and used disaggregation to determine the mean ε and
ε
0
val-
ues for various periods and hazard levels of interest (Harmsen et al.
2002; Harmsen 2001). Fig. 4 shows the
ε
0
for a 2% in 50 years
Sað1sÞ intensity in the western United States fo r Site Class B
(i.e., rock sites). Values of
ε
0
ð1sÞ¼0:50–1:25 are typical in most
of the western United States, except for the high seismic coastal
regions of California, for which the typical values are
ε
0
ð1sÞ¼
1:0–1:75 with peak values as high as 2.0. In the eastern United
States, typical values of
ε
0
ð1sÞ are 0.75–1.0, with some values
reaching up to 1.25, as shown in Fig. 5(a). Expected
ε
0
ð1sÞ values
fall below 0.75 for the New Madrid Fault Zone, for portions of the
eastern coast, for most of Florida, for southern Texas, and in areas
in the northwest portion of the map. The effect of period is illus-
trated by comparing Fig. 5(a) for
ε
0
ð1sÞ to Fig. 5(b) for
ε
0
ð0:2sÞ,
which shows that typical
ε
0
ð0:2sÞ are slightly lower and more var-
iable than
ε
0
ð1sÞ.
To further quantify the expected
ε
0
values in various regions of
the United States, the numeric data used to create the described
maps were examined. The data consists of expected
ε
0
values
for periods of 0.2 and 1.0 s at the centroid of each zip code. Table 1
summarizes the subsets of these data for Seismic Design Catego-
ries B, C, and D, as defined by the International Building Code
(2003). The table provides the average
ε
0
values and the spectral
accelerations for four ground-motion hazard levels: 10, 2, 1, and
0.5% in 50 years for each SDC. The number of zip codes in each
SDC, whic h is a general measure of building inventories, is also
listed.
Because the fault characteristics on the western coast of the
United States vary from those in other parts of the country (i.e.,
the recurrence inter vals of the seismic events is shorter), Table 1
also shows the data for SDC D sites in California and in selected
California cities. On average, the
ε
0
values are consistently higher
in California when compared with other geographic locations of
SDC D sites, and the
ε
0
values for many of the highly populated
California cities are often even higher than the California average.
For example, the
ε
0
ð1sÞ value for the 2% in 50 years hazard in San
Francisco is 1.5, whereas the average value for SDC D sites is 0.99.
Values in Oakland, San Jose, and Riverside are even higher, rang-
ing between 1.65 and 1.95.
Fig. 4. Predicted
ε
0
values from disaggregation of ground-motion
hazard, for the western United States in which the values are for a
1.0 s period and a 2% in 50 years motion after Harmsen et al. (2002)
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Figs. 4 and 5, and Table 1 illustrate the expected
ε
0
values for
Site Clas s B (i.e., rock sites). These values should be generally
applicable to other site conditions provided that the variability
of the ground motions is similar to that of Site Class B. For Site
Classes in which the variability in the ground motions differs from
that of Site Class B (e.g., soft soil under very high levels of
shaking), additional study is required to determine how the
expected
ε
0
values may vary from those for Site Class B.
Target Epsilon Values
The expected or target
ε
0
value for use in a building response as-
sessment depends on the site and hazard level of interest. Thus, the
target ε should be determined on the basis of the hazar d level that
corresponds to the building performance level considered. For
example, when computing the probability of collapse under a
ground motion with a 2% frequency of exceedence in 50 years,
P½CjSa ¼ Sa
2=50
, the appropriate target hazard level is the 2%
in 50 years intensity. When computing the mean annual frequency
of collapse λ
col
, the appropriate target hazard level is more difficult
to determine. Ideally, one would increment the target
ε
0
value for
the various levels of Sa when integrating over the hazard curve.
Alternatively, as an approximate approach, one could use the target
hazard level that most significantly influenc es λ
col
, which will be a
function of both the site and the collapse capa city of the structure.
Haselton and Deierlein (2007, chapter 5) looked at this question
for two example 4-story RC frame buildings at a site in Los
Angeles, and for those buildings and site, the ground-motion inte n-
sity level at 60% of the mean collapse capacity was the most
dominant contributor to the calculation of λ
col
. In their example,
this corresponded to motions that have roughly 1.5 times the
Fig. 5. Mean predicted
ε
0
values from disaggregation of ground-motion hazard for the eastern United States in which the values are for (a) 1.0 s; and
(b) 0.2 s periods and a 2% in 50 years motion after Harmsen et al. (2002)
Table 1. Mean Predicted
ε
0
Values for Periods of 0.2 and 1.0 s, Sorted by Seismic Design Category, with Additional Detail Given for California Sites and
Selected California Cities
Average ε values Average Sa values
ε
0
(0.2 s) ε
0
(1.0 s) Sa(0.2 s) [g] Sa(1.0 s) [g]
Seismic design
category ε
10=50
ε
2=50
ε
1=50
ε
0:5=50
ε
10=50
ε
2=50
ε
1=50
ε
0:5=50
Sa
10=50
Sa
2=50
Sa
1=50
Sa
0:5=50
Sa
10=50
Sa
2=50
Sa
1=50
Sa
0:5=50
Number
Zip Code
data points
SDC B 0.14 0.42 0.49 0.55 0.31 0.80 0.94 1.04 0.06 0.18 0.26 0.39 0.02 0.06 0.08 0.11 20,142
SDC C 0.11 0.51 0.63 0.75 0.23 0.74 0.88 1.00 0.11 0.31 0.46 0.66 0.04 0.10 0.14 0.19 7,456
SDC D 0.25 0.88 1.09 1.27 0.33 0.99 1.21 1.39 0.50 1.05 1.35 1.68 0.18 0.38 0.49 0.62 6,461
SDC D, CA 0.67 1.12 1.30 1.46 0.89 1.35 1.52 1.67 0.81 1.42 1.73 2.07 0.31 0.55 0.68 0.81 2,273
San Francisco,
SDC D
0.88 1.57 1.79 1.95 0.75 1.50 1.75 1.94 1.13 1.78 2.07 2.37 0.52 0.89 1.07 1.25 16
Oakland, SDC D 0.75 1.50 1.75 2.00 0.95 1.65 1.89 2.13 1.56 2.60 3.07 3.55 0.60 1.01 1.21 1.41 10
Berkeley, SDC D 0.67 1.41 1.66 1.91 0.90 1.58 1.82 2.04 1.55 2.62 3.11 3.65 0.59 1.01 1.22 1.43 3
San Jose, SDC D 1.11 1.67 1.84 1.94 0.97 1.64 1.86 2.06 1.23 1.92 2.24 2.59 0.47 0.79 0.94 1.10 29
Los Angeles,
SDC D
0.66 1.17 1.39 1.62 0.90 1.33 1.50 1.70 1.12 1.99 2.43 2.92 0.39 0.69 0.85 1.02 58
Riverside, SDC D 1.35 1.77 1.87 1.88 1.41 1.95 2.12 2.22 1.17 1.74 2.02 2.32 0.47 0.72 0.83 0.94 8
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spectral acceleration of a 2% in 50 years ground motion and
corresponding characteristic ε values typically larger than two.
Approaches to Account for Epsilon in Collapse
Assessment
Two alternative methods of accounting for ε are illustrated by ap-
plication to the collapse assessment of an 8-story RC frame model.
The design and model were developed by Haselton and Deierlein
(2007, ID 1011 in chapter 6) and consists of a three-bay special
moment resisting perimeter frame (SMF) with 6.1 m (20 ft) bay
widths, a tributary seismic mass floor area of 669 m
2
(7,200 sq ft),
and a fundam ental period T
1
of 1.71 s. Haselton and Deierlein
(2007) provided more details regarding the nonlinear structural
modeling and the methodology used for predicting collapse.
The site used is in northern Los Angeles, is typical of the non-
near-field regions of coastal California (Goulet et al. 2007), and
has National Earthquake Hazard Reduction Program (NEHRP)
Category D soil. The model’s primary purpose was to compute the
conditional collapse probability for a 2% in 50 years ground motion,
which is Sað1 :71 sÞ¼0:57 g. Hazard disaggregation provides a
target epsilon of ε ¼ 1:7 for this level of ground motion.
Method 1: Ground-Motion Set Selected with the Target
Epsilon
Select ground motions with ε values that are consistent with those
expected for the site and hazard level of interest. When selecting
records, we used the εðT
1
Þ values computed with the Abrahamson
and Silva (1997) ground-motion prediction equation. We
selected a positive ε ground-motion set to include 20 ground
motions having a mean εðT
1
Þ¼1:7 where T
1
¼ 1:71 s and each
individual record has εðT
1
Þ > 1:25. We imposed additional selec-
tion criteria including both the minimum earthquake magnitude and
the site class. Haselton and Deierlein (2007, chapter 3) documented
the motions included in this ground-motion set and provided the
complete list of selection criteria.
Fig. 6 shows the resulting collapse capacity distribution
predicted by subjecting the 8-story RC SMF to the 20 ground
motions of the positive ε set. The collapse capacity for a single
ground-motion record is defined as the minim um S
a
ðT
1
Þ value
that causes the building to become dynamically unstable, as
evidenced by excessive drifts. This figure shows both the individual
collapse capacities of the 20 records and a fitted lognormal
distribution. The mean collapse capacity is S
a;col
ðT
1
Þ¼1:15 g,
and the standard deviation of the logarithm of collapse capacities
(denoted σ
LNðSa;colÞ
) is 0.28. This dispersion, termed record-to-
record variability, is associated with variation in ground-motion
properties other than SaðT
1
Þ. For the 2% in 50 years SaðT
1
Þ¼
0:57 g, the conditional probability of collapse is quite low and
equal to 0.5%.
Method 2: General Ground-Motion Set with
Adjustments for Epsilon
Motivation and Overview of Method 2
Method 1 may not be feasible or practical in all situations, as it
requires the selection of a specific ground-motion set for a specified
period T
1
at a specified site with a target ε. For example, related
work in the Applied Technology Council 63 Project (FEMA 2008)
involved a collapse assessment of approximately 100 buildings,
with differing fundamental periods, for generic seismic design
categories. In such a study, the selection of a specific ground-
motion set for each building is not practical; nor is it desirable
because the goal is to generalize the collapse assessment results
across seismic design categories.
Method 2 uses a general ground-motion set, selected without
regard to ε values, and then corrects the calculated structural
response distribution to account for the
ε
0
expected for the specific
site and hazard level. This method can be applied to all types of
structural responses (e.g., interstory drifts and plastic rotations),
but this study focuses on the prediction of collapse capacity.
Method 2 is outlined as follows:
1. Select a general far-field ground-motion set without regard to
the ε values of the motions. These are termed the general set.
The general set should have a large number of motions to pro-
vide a statistically significant sample and to ensure that the re-
gression analysis in Step 3 is accurate.
2. Calculate the collapse capacity by nonlinear dynamic ana-
lyses, by using the incre mental dynamic analysis method
(Vamvatsikos and Cornell 2002) to scale records and organize
the results in a cumulative distribution that is characterized
by the mean and record-to-record dispersion of the collapse
capacity.
3. Perform a linear regression analysis between the collapse
capacity of each record LN½S
a;col
ðT
1
Þ and the εðT
1
Þ of the
record. This analysis establishes the relationship between
the mean LN½S
a;col
ðT
1
Þ and the ε ðT
1
Þ value.
4. Adjust the collapse capacity distribution, by using the regres-
sion relationship, to be consistent with the target εðT
1
Þ for the
site and hazard level of interest.
General Far-Field Ground-Motion Set and Comparison
with Positive ε Set
The general set used in this study consisted of 78 strong far-field
motions that were selected without consideration of their ε values.
Haselton and Deierlein (2007, chapter 3) documented these mo-
tions and provided the complete list of selection criteria. A subset
of 44 of these ground motions was also used in the “Applied Tech-
nology Council-63 (ATC-63) Project” (FEMA 2008) as part of a
procedure to validate seismic provisions for structural design. The
expanded set of 78 records was used to achieve more accurate re-
gression trends between the collapse capacity and the ε values, but
fewer may suffice. Fig. 7 compares the mean response spectra of
the general set with the positive ε ground-motion record set de-
scribed in Method 1. For comparison, both sets have been scaled
0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
Sa(T=1.71s) [g]
P[collapse]
Empirical CDF
Lognormal CDF
Fig. 6. Predicted collapse capacity distribution for the example 8-story
reinforced concrete frame, computed by using the positive ε ground-
motion set
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so that each ground motion has the same SaðT
1
Þ¼0:57 g at
T ¼ 1:71 s. The peaked shape of the positive ε set, relative to
the general set, is evident.
Application of Method 2 to Assess the Collapse
of an 8-Story RC SMF Building
When subjected to the general set, the 8-story RC SMF building
(T
1
¼ 1:71 s) has a mean collapse capacity μ
Sa;colðT
1
Þ
of 0.72 g
and a dispersion in capacity of σ
LNðSa;colÞ
¼ 0:45. The 2% in
50 years intensity for this site is Sað1:71 sÞ¼0:57 g, so the prob-
ability of collapse for this level of motion is 29%. Recall that the
probability of collapse under the 2% in 50 years motion when
analyzed by using the positive ε set was only 0.5%. The collapse
capacity prediction from the general set still needs to be adjusted to
be consistent with the target εðT
1
Þ for Method 2.
The collapse capacity LN½S
a;col
ðT
1
Þversus the correspon-
ding εðT
1
Þ values for each record are shown in Fig. 8. Also shown
is a linear regression (Chatterjee et al. 2000) between LN½S
a;col
ðT
1
Þ
and εðT
1
Þ, which follows an approach previously proposed
by Zareian (2006). The relationship between the mean of
LN½S
a;col
ðT
1
Þ and εðT
1
Þ can be described as
μ
0
LN½Sa;colðT
1
Þ
¼ β
0
þ β
1
· εðT
1
Þð2Þ
where β
0
¼0:356 and β
1
¼ 0:311 in this example. β
1
represents
the slope between ε and the collapse capacity such that larger val-
ues of β
1
indicate a greater significance of ε in the prediction of the
collapse capacity.
To adjust the mean collapse capacity for the target εðT
1
Þ¼1:7,
Eq. (2) can be evaluated for the target
ε
0
ðT
1
Þ, resulting in the
following adjusted mean of LN½S
a;col
ðT
1
Þ:
μ
0
LN½Sa;colð1:71 sÞ
¼ β
0
þ β
1
· ½
ε
0
ðT
1
Þ
¼0:356 þ 0:311 · ½1:7¼0:173 ð3Þ
The adjusted mean collapse capacity can now be computed by
taking the exponential of Eq. (3)
Mean
0
Sa;colðT
1
Þ
¼ expðμ
0
LN½Sa;colðT
1
Þ
Þ¼expð0:173Þ¼1:19 g ð4Þ
The calculation for the ratio of the adjusted to the original mean
collapse capacity is
Ratio ¼
expðμ
0
LN½Sa;colðT
1
Þ
Þ
expðμ
LN½Sa;colðT
1
Þ
Þ
¼
1:19 g
0:72 g
¼ 1:65 ð5Þ
where μ
LN½Sa;colðT
1
Þ
is computed directly from the collapse simula-
tion results by using the general set of ground motions and
μ
0
LN½Sa;colðT
1
Þ
is the value adjusted by the regression analysis for
the target
ε
0
ðT
1
Þ value. The calculated increase in the mean col-
lapse capacity from 0.72–1.19 g (a ratio of 1.65) has a significant
impact on the collapse performanc e assessment.
The dis persion in the collapse capacity computed directly from
the records is σ
LN½Sa;colðT1Þ
¼ 0:45, but this capacity is also reduced
by the adjustment to the target
ε
0
ðT
1
Þ. The reduced conditional
standard deviation can be computed as follows (Benjamin and
Cornell 1970, Eq. 2.4.82):
σ
0
LN½Sa;colðT
1
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fσ
LN½Sa;colðT
1
Þ;reg
g
2
þðβ
1
Þ
2
ðσ
ε
Þ
2
q
ð6Þ
where σ
LN½Sa;colðT1Þ;reg
¼ 0:36 is computed from the residuals of the
regression analysis shown in Fig. 8, and σ
ε
is the standard deviation
of the εðT
1
Þ values from disaggregation for a site and hazard
level. For the example site used in this study, σ
ε
is estimated to
be 0.35 for the 2% in 50 years intensity of ground motion.
Calculations for the reduced standard deviation in Eq. (7) show that
the original record-to-record dispersion in collapse capacity (i.e.,
σ
LN½Sa;colðT1Þ;reg
) is more dominant than the effects of the dispersion
in the expected ε value (i.e.,
ffiffiffiffiffiffiffiffiffiffi
β
2
1
σ
2
ε
p
)
σ
0
LN½Sa;colðT
1
Þ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð0:36Þ
2
þð0:31Þ
2
ð0:35Þ
2
q
¼ 0:38 ð7Þ
The reduced dispersion is 15% lower than the dispersion in the col-
lapse capacity computed directly from the records, which was
σ
LN½Sa;colðT1Þ
¼ 0:45. Relative to the increase in the mean collapse
capacity described in the preceding, this decrease in dispersion
from 0.45–0.38 has only a moderate impac t on collapse perfor-
mance assessment, which is most apparent near the tails of the
collapse capacity distribution.
-2 -1 0 1 2
-1
-0.5
0
0.5
LN[Sa
col
(T
1
=1.71s)]
ε
AS
(T
1
=1.71s)
Observation
Regression
5/95% CIs on Mean
Fig. 8. Relationship between the collapse capacity quantified as spec-
tral acceleration and ε for each ground-motion record computed by
using Abrahamson and Silva (1997), including linear regression relat-
ing LN½S
a;col
ðT
1
Þ to εðT
1
Þ
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
Period [seconds]
Sa [g]
General Far-Field Set
Positive
ε
Set (for T
1
= 1.71s)
Fig. 7. Comparison of mean spectra for the general set and positive ε
set of ground motion
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Comparison of the Two Methods
Fig. 9 overlays the predicted collapse capacity distributions
obtained from Methods 1 and 2 for the 8-story RC frame. The fig-
ure also includes the collapse predictions of Method 2 before the
adjustment for ε. Fig. 10 is similar to Fig. 8, but for comparison,
Fig. 10 includes the data for the positive ε set of ground motions.
Together, Figs. 9 and 10, and Table 2 show that the two methods
produce nearly identical results, with the predictions of the mean
collapse capacity differing by only 4%. The dispersion in the col-
lapse capacity [σ
LNðSa;colÞ
] differs from 0.28 for Method 1 to 0.38
for Method 2. From the writers’ past experience, it is not expected
that such a large observed difference occurs. The large difference in
dispersion could be attributable to the smaller num ber of ground
motions in the positive ε set for Method 1. The probabilities of
collapse associated with the 2% in 50 years motion are similar
(0.5 and 2.4%), and when the collapse CDF is integrated with
the site hazard curve for the example site, the mean annual rates
of collapse λ
col
differ only by a factor of 2, as shown in Fig. 9.
These differences are negligible when compared to a factor of
23 in the overprediction of λ
col
that results from not accounting
for the proper ε. In addition, data from Haselton and Deierlein
(2007, chapter 6) show that even minor differences in the structural
design can cause the λ
col
prediction to change by a factor of 1.5–
2.2, which is similar to the difference in results from the two meth-
ods compared here.
Simplified Method to Account for Effects of Epsilon
Motivation and Overview
The preceding section showed that we can obtain roughly the same
collapse capacity predictions by either (a) selecting records with
appropriate ε values (Method 1) or (b) using general ground
motions and then applying a correction factor to account for the
appropriate ε (Method 2). Method 2 is useful because it can acc ount
for the target ε without needing to select a unique ground-motion
set for each building period and site. However, as described in the
preceding section, Method 2 requires a significant effort to compute
εðT
1
Þ values for each ground-motion record and then to perform a
regression analysis to relate S
a;col
ðT
1
Þ to εðT
1
Þ. To provide a more
practical method for adjusting the collapse capacity, a simplified
version of Method 2 can be used to determine the appropriate
adjustment factors for the collapse capacity distribution without
requiring the computation of the εðT
1
Þ values for each record
and then performing a regression analysis. The simplified method
uses an empirical equation to estimate β
1
from Eq. (2) and an
approximate value of σ
LN½Sa;colðT1Þ
to correct the collapse capacity
distribution.
Building Case Studies to Develop the Simplified
Method for ε Adjustment
The complete Method 2 was applied to three sets of RC frame
buildings, a total of 111 buildings, to develop a simplified adjust-
ment approach. They included:
• Sixty-five modern RC SMF buildings ranging in height from
1–20 stories. Thirty of these buildings were code-conforming
buildings that were representative of current design [ASCE
7-05 (ASCE 2005) and ACI 318-05 [American Concrete
Institute (ACI) 2005]] in high seismic regions of California
-2 -1 0 1 2
-1
-0.5
0
0.5
LN[Sa
col
(T
1
=1.71s)]
ε
AS
(T
1
=1.71s)
Observation (General Set)
Regression (General Set)
5/95% CI (General Set)
Observation (Positive
ε
Set)
Fig. 10. Relationship between the spectral acceleration and ε from
Fig. 8 including the collapse results predicted when directly by using
the positive ε set of ground motions
0 0.5 1 1.5 2 2.5
0
0.2
0.4
0.6
0.8
1
Sa(T=1.71s) [g]
P[collapse]
No
ε
adjustment (empirical)
No
ε
adjustment (lognormal)
After
ε
adjustment (lognormal)
Selection for
ε
(empirical)
Selection for
ε
(lognormal)
Fig. 9. Comparison of collapse capacity distributions predicted by
using the two methods; Method 2 results are shown before and after
the adjustment to the target
ε
0
ðT
1
Þ
Table 2. Comparison of Collapse Risks for the Example 8-Story RC SMF Building, Predicted by Using the Two Proposed Methods and without Treatment
of ε
Method Mean Sa;col (1.71 s) σ
LN
(Sa;col) P½CjSa
2=50
λ
col
[10
4
]
Method 1 1.15 0.28 0.005 0.28
Method 2 1.20 0.38 0.024 0.50
Predictions without ε adjustment 0.72 0.45 0.29 6.3
Ratio: Method 2 to Method 1 1.0 1.2 5 2
Ratio: Without adjustment to Method 1 0.63 1.6 58 23
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(Haselton and Deierlein 2007, chapter 6). The other 35 RC SMF
buildings (4- and 12-story) were designed to meet revised
structural design requirements, including variations to design
strength requirements, interstory drifts, and strong column-
weak beam ratio (Haselton and Deierlein 2007, chapter 7).
• Twenty code-conforming ordinary moment frame (OMF) build-
ings ranging from 2–12 stories, which were representative of
buildings in the eastern United States. These designs were de-
veloped as part of the “Applied Technology Council-63 Proj ect ”
(FEMA 2008).
• Twenty-six nonductile RC frame buildings, which were repre-
sentative of existing 1967-era buildings, ranging from 2–12
stories in hig h seismic regions of California (Liel 2008).
The collapse analysis was conducted for each building and the
regression analysis was applied to LN½S
a;col
ðT
1
Þ versus εðT
1
Þ to
determine the factor β
1
as defined in Eq. (2). A selected subset
of these values is shown in Table 3. The mean β
1
value for the
65 RC SMF buildings was β
1
¼ 0:28. This value is exceptionally
stable with a coefficient of variation value of only 0.14 over the
wide variety of buildings of varying heights and design. The sta-
bility of the β
1
values indicates that the influence of ε (i.e., spectral
shape) on the collapse capacity is fairly consistent among buildings
with similar levels of inelastic deformation capacity. The mean
value for the 20 RC OMF buildings was β
1
¼ 0:19, which is
40% lower than the more ductile SMF buildings. The mean value
for the 1967-era buildings was β
1
¼ 0:18, which is quite similar the
RC OMF frames. The lower β
1
values indicate that ε has less of an
influence on the collapse capacities for the RC OMF and the 1967-
era RC frame buildings, both of which have less inelastic deforma-
tion capacity as compared to the RC SMF buildings. Building de-
formation capacities, as quantified by the ultimate roof drift ratio,
are also reported in Table 3. Note that RDR
ult
is the roof drift ratio
at 20% strength loss, as predicted by using static pushover analysis
(e.g., RDR
ult
¼ 0:047 for the pushover shown in Fig. 13). Table 3
shows that buildings with a larger deformation capacity RDR
ult
have higher values of β
1
.
Developing Components of the Simplified Method
Prediction of β
1
The significance of ε, as reflected in the β
1
parameter, is larger for
buildings with a higher deformation capacity because ductile build-
ings soften, and their effective period increases before collapse,
which makes the spectral shape, specifically spectral values at
T > T
1
, more important to the structural response. The trend be-
tween β
1
and RDR
ult
is illustrated in Fig. 11(a) for four sets of RC
SMF buildings, each set with the same height. These data show a
trend for deformation capacities up to RDR
ult
¼ 0:04, and suggest
that deformation capacity in excess of this (i.e., RDR
ult
> 0:04)
does not influe nce β
1
.
β
1
also tends to be larger fo r taller buildings because of the
significance of higher mode effects on the dynamic response of
tall buildings, thereby making the spectral shape for periods less
Table 3. Results for a Subset of the 111 Buildings Showing the Relationship between Building Deformation Capacity RDR
ult
and β
1
, a Measure of the
Significance of εðT
1
Þ in Collapse Capacity Predictions in which β
1
Is Obtained from Regression Analysis
Design information RC SMF buildings 1967 RC frame buildings RC OMF buildings
Number of stories Framing system RDR
ult
β
1
σ
LN;reg
=σ
LN
RDR
ult
β
1
σ
LN;reg
=σ
LN
RDR
ult
β
1
σ
LN;reg
=σ
LN
2 Perimeter 0.067 0.26 0.82 0.035 0.22 0.86 0.024 0.28 0.95
Space 0.085 0.26 0.81 0.019 0.16 0.91 0.019 0.09 0.97
4 Perimeter 0.038 0.27 0.83 0.013 0.18 0.90 0.016 0.24 0.92
Space 0.047 0.26 0.83 0.016 0.20 0.88 0.011 0.27 0.97
8 Perimeter 0.023 0.31 0.81 0.007 0.16 0.97 0.009 0.12 0.82
Space 0.028 0.32 0.79 0.011 0.18 0.95 0.014 0.19 0.95
12 Perimeter 0.026 0.29 0.84 0.005 0.10 0.97 0.009 0.17 0.97
Space 0.022 0.25 0.86 0.010 0.16 0.95 — 0.16 —
Mean of subset: 0.033 0.27 0.82 0.012 0.17 0.93 0.014 0.18 0.95
Mean of full set: — 0.28 ——0.18 ——0.19 —
0 0.02 0.04 0.06 0.08
0
0.1
0.2
0.3
0.4
Ultimate Roof Drift (RDR
ult
)
β
1
4-story perimeter frame
4-story space frame
12-story perimeter frame
12-story space frame
0 5 10 15
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Number of Stories
β
1
RDR
ult
= 0.018
RDR
ult
= 0.020
RDR
ult
= 0.032
RDR
ult
= 0.032
RDR
ult
= 0.038
RDR
ult
= 0.045
(a) (b)
Fig. 11. Relationship between (a) β
1
and building deformation capacity RDR
ult
; and (b) β
1
and number of stories
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than T
1
an important consideration. To investigate the impact of
building height, separate from deformation capacity, Fig. 11(b)
compares the β
1
values of six pairs of 4- and 12-story RC SMF
buildings that have the same RDR
ult
values. These data show a clear
trend between β
1
and building height, for five of the six sets of
buildings considered.
Standard linear regression analysis was used to calculate
LNðβ
1
Þ as a function of RDR
ult
and building height, on the basis
of the data from all 111 buildings to create the predictive equation
for β
1
(Chatterjee et al. 2000). We then applied judgmental correc-
tions to better replicate the trends with deformation capacity and
building height (see Fig. 11). These corrections were required be-
cause of the limited number of data points available to reflect the
separate trends of height and building deformation capacity. The
functional form of Eq. (8) captures the nearly linear effects of
height and the nonlinear effects of RDR
ult
for buildings with a
lower deformation capacity. The resulti ng equation for β
1
becomes
^
β
1
¼ð0:4ÞðN þ 5Þ
0:35
ðRDR
ult
Þ
0:38
ð8Þ
where N = number of stories limited to N ≤ 20 on the
basis of available data and RDR
ult
= roof drift ratio at 20%
base shear strength loss from the static pushover analysis
[RDR
ult
¼ minðRDR
ult
; 0:04Þ] and the observation from Fig. 11
(a) that the trend saturates at a value of 0.04. Note that the appli-
cation of static pushover analysis to taller buildings is limited be-
cause of the important impact of higher modes, but it is utilized here
to approximate the building deformation capacity.
The effects of height and deformation capacity tend to counter-
act one another, which is why β
1
is fairly consistent for the set of 30
code-conforming RC frame buildings varying from 1 to 20 stories.
In Fig. 12, the ratio of observed β
1
to the predicted β
1
from Eq. (8)
is plotted against the building deformation capacity and the number
of stories, which shows that Eq. (8) provides reasonab le predictions
for most of the 111 buildings used in this study. However, β
1
is sig-
nificantly underpredicted (i.e., conservative) for three of the 1-story
buildings, but is accurate for the fourth 1-story building. It would be
useful to extend this study to include a larger number of short period
buildings to further validate the proposed relationship.
Prediction of σ
0
LN½Sa;colðT1Þ
The data in Table 3 show that accounting for ε reduces the
dispersion in collapse capacity. This reduction in dispersion is
reduced by about 10–15% for ductile RC SMF buildings and
5% for nonductile buildings. For simplicit y, it is proposed to ignore
this effect and to compute the dispersion directly from the general
set, that is, to assume that
σ
0
LN½Sa;colðT
1
Þ
≈ σ
LN½Sa;colðT
1
Þ
ð9Þ
Proposed Simplifi ed Method
This section summarizes the proposed simplified method for
adjusting the collapse capacity to reflect an appropriate spectral
shape with an illustration for a 4-story RC SMF space frame.
1. Build a structural model that is robust and able to simulate
structural collapse. Calculate the building period and perform
a static pushover analysis with a reasonable load pattern to
determine the roof drift ratio RDR
ult
at 20% of lateral
strength loss. For this example a 4-story RC SMF building,
T
1
¼ 0:94 s. Calculations for the static pushover analysis were
conducted by using the lateral load pattern recommended by
ASCE 7-05 (ASCE 2005) resulting in the pushover curve
shown in Fig. 13 where RDR
ult
¼ 0:047.
2. Perform nonlinear dynamic analyses to predict the collapse ca-
pacity by using the FEMA P695 (FEMA 2008) far-field set of
44 records. [Alternatively, one could use the larger general set
of 78 records. However, our analyses have shown that the two
sets result in nearly identical mean and dispersion of collapse
capacity. The reason for using the larger set in this paper was to
better predict the regression line between LN½S
a;col
ðT
1
Þ and
εðT
1
Þ; this additional information is not required in the sim-
plified method.] Compute the natural logarithm of the collapse
capacity for each record, and then compute the mean and
the standard deviation of these values for all records (i.e.,
μ
LN½Sa;colðT1Þ
and σ
LN½Sa;colðT1Þ
). For the example 4-story RC
SMF building, the results of the nonlinear dynamic collapse
analyses are shown as follows:
0 0.05 0.1
0
0.5
1
1.5
2
[
β
1
Obs.] / [
β
1
Pred.]
Ultimate Roof Drift (RDR
ult
)
0 5 10 15 20
0
0.5
1
1.5
2
[
β
1
Obs.] / [
β
1
Pred.]
Number of Stories
(a) (b)
Fig. 12. Ratio of observed/predicted β
1
, plotted against (a) building deformation capacity RDR
ult
; and (b) number of stories
0 0.02 0.04 0.06
0
50
100
150
200
250
Base Shear (kips)
Roof Drift Ratio
RDR
ult
= 0.047
Fig. 13. Static pushover curve for an example 4-story RC SMF
building (ID 1008)
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μ
LN½Sa;colðT1Þ
¼ μ
LN½Sa;colð0:94 sÞ
¼ 0:601 ð10Þ
σ
LN½Sa;colðT1Þ
¼ σ
LN½Sa;colð0:94 sÞ
¼ 0:40 ð11Þ
The mean collapse capacity can be computed from the loga-
rithmic mean as follows:
Mean
½Sa;colð0:94 sÞ
¼ expfμ
LN½Sa;colð0:94 sÞ
g¼1:82 g ð12Þ
3. Estimate β
1
by using Eq. (8). For the 4-story RC SMF
example
^
β
1
¼ð0:4ÞðN þ 5Þ
0:35
ðRDR
ult
Þ
0:38
ð13Þ
RDR
ult
¼ 0:04 ð14Þ
^
β
1
¼ð0:4Þð4 þ 5Þ
0:35
ð0:04Þ
0:38
¼ 0:254 ð15Þ
4. Determine the target mean ε value εðT
1
Þ
;target
for the site and
hazard level of interest. For the example 4-story RC SMF, we
assumed that the target is ½εðT
1
Þ
;target
¼1:9, which is similar
to an expected ε value of a 2% in 50 years ground-motion level
in Riverside, California (see Table 1).
5. Adjust for the difference between the target ε value and the ε
values of the ground motions used in the collapse simulation.
To do this, the mean ε value from the general set of records
εðT
1
Þ
;records
is required. The mean ε values for the general
set of records is shown in Fig. 14. From this figure, one
can read the value of
εðT
1
Þ
;records
. For the example building
T
1
¼ 0:94 s, and because the collapse simulation is calculated
by using the 78 general record set,
εðT
1
Þ
;records
¼ 0:17. Any set
of ground motions could be used provided that
εðT
1
Þ
;records
is known.
6. Compute the adjusted mean collapse capacity. This adjusted
capacity accounts for the difference between the mean ε of
the general set of records
εðT
1
Þ
;records
and the target ε values
that come from disaggregation
ε
0
ðT
1
Þ. The following
equations illustrate this calculation for the example 4-story
RC SMF
μ
0
LN½Sa;colðT1Þ
¼ μ
LN½Sa;colðT1Þ
þ
^
β
1
½
ε
0
ðT
1
Þ
εðT
1
Þ
;records
ð16Þ
μ
0
LN½Sa;colð0:94 sÞ
¼ 0:601 þ 0:254ð1:9 0:17Þ
¼ 1:040 ð17Þ
Mean
0
Sa;colð0:94 sÞ
¼ expfμ
0
LN½Sa;colð0:94 sÞ
g¼expð1:040Þ
¼ 2:83 g ð18Þ
The ratio of the adjusted to unadjusted mean collapse capacity
can also be computed by using Eqs. (12) and (18), as follows:
Ratio ¼
Mean
0
Sa;colðT
1
Þ
Mean
½Sa;colðT
1
Þ
¼
Mean
0
Sa;colð0:94 sÞ
Mean
½Sa;colð0:94 sÞ
¼
2:83 g
1:82 g
¼ 1:55 ð19Þ
7. Compute the dispersion in the collapse capacity by using
Eq. (10). In this step, we propose to simply use the value com-
puted directly from the nonlinear dynamic analyses, where
σ
0
LN½Sa;colðT
1
Þ
≈ σ
LN½Sa;colðT
1
Þ
¼ 0:40 ð20Þ
Comparison of the Simplified Method and Method 2
For comparison, applying Method 2 to this same building by using
data from Haselton and Deierlein (2007, chapter 3) would result in
very similar results to the Simplified Method. The full regression
analysis results yield β
1
¼ 0:257, which agrees very well with the
simplified value of
^
β
1
¼ 0:254. The corresponding mean collapse
capacity from Method 2 is 2.63 g as compared to the simplified
value of 2.83 g. This difference of about 8% is reasonable for most
applications, particularly in contrast to the alternative of neglecting
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Period (T
1
) [s]
Mean of
ε
(T
1
)
,records
General Far-Field Set (78 records)
Reduced ATC-63 Far-Field Set (44 records)
Fig. 14. Mean ε values for the full and reduced versions of general set of ground motions
εðT
1
Þ
;records
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the spectral shape effects. The calculated disper sion from Method 2
is σ
0
LN½Sa;colðT
1
Þ
¼ 0:35, which is about 10% lower than the slightly
conservative value of 0.40 used in the simplified method. The con-
ditional probability of collapse for the 2% in 50 years ground-
motion level Sað0:94 sÞ¼0:87 g is effectively zero in both cases,
0.2% and 0.1%, respectively.
Summary, Limitations, and Future Work
The consideration of spectral shape is critical in the selection and
scaling of ground motions for use in a collapse assessment by non-
linear dynamic analysis. This paper presents that the spectral shape
characteristics can be included in a collapse assessment through
consideration of the parameter ε, which is a measure of how
ground-motion acceleration spectra vary from the mean predictions
provided by ground-motion attenuation relationships. For an exam-
ple 8-story RC SMF building, accounting for the ε adjustment in-
creased the mean collapse capacit y by a factor of 1.6, decreased the
conditional probability of collapse for the 2% in 50 years ground
motion, P½CjSa
2=50
, from 29% to 0.5%, and decreased the mean
annual frequency of collapse by a factor of 23.
The most direct approach to account for the ε-effect in collapse
assessment is to select ground motions whose εðT
1
Þ values match
those of the building site, the collapse intensity Sa, and the struc-
tural period of interest. However, this approach is often impractical
and sometimes infeasible when assessing the collapse performance
of buildings with varying vibration periods at multiple sites and
under varying ground-motion intensities. An alternative simplified
approach is proposed that applies an adjustment to the collapse
capacity by using the target εðT
1
Þ, which eliminates the necessity
of considering εðT
1
Þ in the selection of the ground-motion records.
Two variants of the ε-adjustment method are proposed, one of
which is a simplified version of the other.
To develop and validate the proposed ε-adjustment method, the
collapse capacities of three sets of RC frame buildings were inves-
tigated including (a) 65 modern RC ductile special moment frames,
(b) 26 nonductile 1967-era RC frames, and (c) 20 RC limited-
ductility ordinary moment frames. These 111 buildings range in
height from 1–20 stories with fundamental vibration periods rang-
ing from 0.4–4.4 s, with most periods less than 3.0 s. We simulated
the collapse capacity of each building for 78 ground-motion re-
cords, and then used regression analysis to find the relationsh ip
between the collapse intensity S
a;col
ðT
1
Þ and the corresponding
εðT
1
Þ for each building and ground motion. The resulting collapse
capacities calcula ted through this regression technique, called
Method 2 in this paper, are shown to agree well with the results ob-
tained by using a ground-motion set selected to have the target ε.
A simplified version of Method 2 was developed, in which a
semiempirical equation [Eq. (8)] was used to calculate the εðT
1
Þ
collapse adjustment factor in lieu of conducting regression analy-
ses. Generalized regression analyses conducted by using data from
the collapse capacities of the 111 case study buildings were used to
develop this equation. The resulting semiempirical equation
[Eq. (8)] reflects variations in building height and deformation
capacity, the latter of whic h is determined by using a pushover
analysis. The proposed simplified method allows the analyst to
use a general ground-motion set, selected wi thout regard to ε,to
calculate an unadjusted building collapse capacity by using nonlin-
ear dynamic analysis, and then to correct this capacity by using an
adjustment factor to reflect the expected εðT
1
Þ for the building site
and collapse hazard intensity, S
a;col
ðT
1
Þ. The general set of far-field
strong ground motions from the FEMA P695 (FEMA 2008) are
suggested for applying this simplified procedure.
Whereas Method 2 is general in its applicability, the simplified
method should be utilized only for structures and ground motions
similar to those to which it was developed and calibrated. The
development was limited to moment frame buildings, ranging in
height from 1 to 20 stories and ranging in periods from 0.4–
3.0 s. The ground motions and target ε values used in the study
are generally represent ative of Site Classes B, C, and D, with a
focus on ε values in the range of ε ¼ 0toþ2:0. The simplified
method should not be used for other site classes, particularly soft
soil sites, or for sites with target ε values outside of the noted range
without appropriate ground-motion selection and recalibration of
the adjustment factor for these conditions.
An implicit assumption of the proposed techniques is that the
spectral acceleration at the fundamental period of the building,
SaðT
1
Þ, is used to scale the ground motions and quantify the col-
lapse intensity. This assumption is fundamental to the definition of
the ε adjustment factor. For tall or irregular buildings, there may be
multiple dominant periods of response, the effects of which warrant
further study. For example, if three periods dominate the structural
response of a tall building, perhaps the collapse assessment could
be completed once for each of the three periods, and the controlling
case could be used.
This work is currently adapted for use in the “ATC-63 Project”
(FEMA 2008) to provide codified guidelines and procedures for the
collapse capacity prediction of buildings. The goal of the ATC
Project is to use the codified collapse prediction procedures to
determine the appropriate prescriptive design requirements (e.g.,
the R factor) for newly proposed structural systems.
This research could also be extended to look more closely at
impacts of spectral shape ε on the collapse behavior of short period
buildings. Additionally, this method was developed with the pri-
mary goal of identifying a generalized collapse assessment to
evaluate the relative safety among groups of buildings located
on comparable sites. Further work would be useful to extend this
method for a case-specific collapse analysis of specific buildings at
particular sites. This extension may involve the selection of records
to match the target spectral shape directly (Baker and Cornell
2006), including factors such as site class, which may significantly
alter the shape.
Acknowledgments
This research was supported primarily by the Earthquake Engineer-
ing Research Centers Program of the National Science Foundation
under award number EEC-9701568 through the Pacific Earthquake
Engineering Research Center (PEER). The research findings were
also supported by related studies conducted for the “ATC-63 Pro-
ject,” which is supported by the Federal Emergency Management
Agency. Any opinions, findings, and conclusions or recommenda-
tions expressed in this material are those of the writers and do not
necessarily reflect those of the National Science Foundation or the
Federal Emergency Management Agency.
The writers also acknowledge the contributions of Nico Luco,
Stephen Harmsen, and Arthur Frankel of the USGS, who provided
the mean
ε
0
data used in this research; the suggestions and advice
of Dr. Charlie Kircher and other members of the “ATC-63 Project;”
and the assistance of Jason Chou and Brian Dean in conducting the
structural collapse analyses used in this study.
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