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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

334 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011

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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)

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011 / 335

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

336 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011

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

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011 / 337

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

338 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011

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

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011 / 339

(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

340 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011

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)

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011 / 341

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

342 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / MARCH 2011

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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