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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2007; 36:12351254

Published online 6 March 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.680

Probabilistic estimation of maximum inelastic displacementdemands for performance-based design

Jorge Ruiz-Garca1,, and Eduardo Miranda2

1Facultad de Ingenier a Civil, Universidad Michoacana de San Nicol as de Hidalgo, Edificio C, Planta Baja,

Cd. Universitaria, Morelia 58040, M exico2 Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020, U.S.A.

SUMMARY

A probabilistic approach to estimate maximum inelastic displacement demands of single-degree-of-freedom(SDOF) systems is presented. By making use of the probability of exceedance of maximum inelasticdisplacement demands for given maximum elastic spectral displacement and the mean annual frequencyof exceedance of elastic spectral ordinates, a simplified procedure is proposed to estimate mean annualfrequencies of exceedance of maximum inelastic displacement demands. Simplifying assumptions arethoroughly examined and discussed. Using readily available elastic seismic hazard curves the procedurecan be used to compute maximum inelastic displacement seismic hazard curves and uniform hazard spectraof maximum inelastic displacement demands. The resulting maximum inelastic displacement demandspectra provide a more rational way of establishing seismic demands for new and existing structures whenperformance-based approaches are used. The proposed procedure is illustrated for elastoplastic SDOFsystems having known-lateral strength located in a region of high seismicity in California. Copyright q2007 John Wiley & Sons, Ltd.

Received 14 December 2005; Revised 12 December 2006; Accepted 9 January 2007

KEY WORDS: performance-based design; inelastic single-degree-of-freedom systems; inelastic displace-ment ratios; uniform hazard spectra; seismic hazard

1. INTRODUCTION

It is well known that one of the largest sources of uncertainty in estimating the effects of earthquakes

on man-made structures lies on the estimation of the characteristics of future earthquake ground

motions that can occur at a specified site. A probabilistic seismic hazard analysis (PSHA) represents

a rational and quantitative procedure to estimate the hazard of earthquake ground motions at

Correspondence to: Jorge Ruiz-Garca, Facultad de Ingeniera Civil, Universidad Michoacana de San Nicolas deHidalgo, Edificio C, Planta Baja, Cd. Universitaria, 58040 Morelia, Mexico.

E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

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1236 J. RUIZ-GARCIA AND E. MIRANDA

a site [1 4]. Using the geometry and location with respect to the site of all possible seismic

sources, the probability distribution of earthquake magnitudes at each source and attenuation

relationships, a conventional PSHA permits the estimations of the mean annual frequency (MAF)

of exceedance of a certain peak ground motion parameter (e.g. peak ground acceleration, etc.)

or a linear elastic response spectral ordinate (e.g. pseudo-acceleration, Sa) by integration over allpossible sources, earthquake magnitudes and distances. However, conventional PSHA only provides

probabilistic estimates of demands on linear elastic systems while most structures are likely to

undergo significant inelastic deformations in the event of strong, or even moderate, earthquake

ground motions. Therefore, it is of utmost importance to develop rational methods to estimate

lateral displacement demands on inelastic systems.

Although most structures do not behave like single-degree-of-freedom (SDOF) systems, various

studies have shown that inelastic SDOF systems may provide the basis for estimating global

deformation demands of buildings [511]. Based on this observation, non-linear static procedures

have been introduced in recent seismic recommendations for design of new structures as well as

for assessment and rehabilitation of existing structures [1217] in which roof displacements are

estimated from maximum displacements of SDOF inelastic systems. In these design provisions

the estimation of maximum inelastic SDOF displacements is done through simplified procedures

by either applying modification factors on maximum SDOF elastic displacement demands or by

considering equivalent SDOF systems with elongated fundamental period and increased damping

ratio [18]. However, the inherent uncertainty introduced by approximating maximum inelastic

displacements from maximum elastic displacements is neglected.

The objective of this work is to propose a simplified approach to estimate site-specific MAF of

exceedance hazard curves of maximum inelastic displacement demand of SDOF systems by using

readily available information of the elastic seismic hazard curve at a specific site. In particular, this

study presents a procedure to obtain maximum inelastic displacement seismic hazard curves and

uniform hazard spectra of peak inelastic displacement demand to be used during the performance-

based design process of new structures or during the seismic evaluation and rehabilitation phase of

existing structures. This investigation makes use of statistical results of inelastic displacement ratioscomputed from the dynamic response of inelastic SDOF systems having a wide range of periods

of vibration and lateral strength when subjected to a relatively large suite of ground motions. The

proposed approach explicitly takes into account the epistemic uncertainty in estimating elastic

displacement demands as well as the epistemic uncertainty that exists in estimating inelastic

displacement demands from maximum elastic displacement demands. The procedure is illustrated

by obtaining probabilistic estimates of maximum lateral deformation demands of inelastic SDOF

systems having elastoplastic hysteretic behaviour when subjected to ground motions recorded on

rock or firm soil conditions. However, the procedure is quite general and can be used for other

hysteretic behaviours and other site conditions.

2. FORMULATION OF SIMPLIFIED APPROACH TO ESTIMATE (i )

The mean annual rate by which the maximum inelastic displacement demand, i , exceeds a certain

lateral displacement, i , can be estimated as follows:

(i )=

0

P[i>i |Sd = sd; T,Cy] d(sd) (1)

Copyright q 2007 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2007; 36:12351254

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PROBABILISTIC ESTIMATION OF MAXIMUM INELASTIC DISPLACEMENT DEMANDS 1237

where (sd) is the site-specific MAF of exceedance of the elastic spectral displacement, Sd,

evaluated at sd, P[i>i |Sd = sd; T,Cy] is the probability ofi exceeding i , conditioned on the

fundamental period of the system, T, the yield strength coefficient, Cy (defined as the lateral yielding

strength of the system, Fy, normalized by its weight), as well as on the maximum spectral elastic

displacement demand, Sd. The mean annual rate (or mean frequency) of exceedance computedfor a wide range of displacement (or pseudo-acceleration) spectral ordinates is what is known as

seismic hazard curve which is the main output of a conventional PSHA [3, 4].

In order to numerically integrate Equation (1) it is convenient to re-write it as follows:

(i )=

0

P[i>i |Sd = sd; T,Cy]

d(sd)dSd dsd (2)

where the term inside the absolute sign is the derivative of the elastic seismic hazard curve with

respect to the spectral displacement and evaluated at an intensity sd. Seismic hazard curves for

rock sites with average shear wave velocities between 760 and 1500 m/s (i.e. for sites that can

be classified as site class B according to [13]) are readily available for any geographical location

in the United States from the United States Geological Survey [19]. For other site conditions,a site-specific seismic hazard analysis can be performed or alternatively frequency-dependent

modification factors can be used to take into account site effects [2022]. Therefore, the additional

information required in estimating MAF of inelastic displacements using Equation (2) is on the

estimation of P[i>i |Sd = sd; T,Cy] which represents the likelihood that the maximum inelastic

displacement will be larger than i given that the systems properties and the elastic displacement

(or pseudo-acceleration) spectral ordinate are known.

Ruiz-Garca and Miranda [23] recently conducted a statistical study that permits the estimation

of maximum inelastic displacement demand as i = CR Sd, where CR is the constant relative

strength inelastic displacement ratio given by

CR =i

Sd(3)

In Equation (3), maximum inelastic displacement demands are computed for SDOF systems

with constant yield strength coefficient, Cy, and ground motions are scaled to the same elastic

displacement demand or, alternatively, for SDOF systems with constant yield strength relative to

the strength required to maintain the system elastic (i.e. constant relative strength) using unscaled

earthquake records. In both approaches the relative lateral strength is measured by the lateral

strength ratio, R, which is defined as follows:

R =Sa

Cy g(4)

where Sa is the spectral pseudo-acceleration ordinate and g is the acceleration due to gravity. In

the proposed procedure, it is assumed that the expected value ofi can be obtained as the product

of CR times Sd, which implies that CR and Sd are independent random variables and that there isa lack of correlation between them. Furthermore, in computing the conditional probability term in

Equation (2), it is assumed that it is equivalent to compute P[CR>cR|Sd = sd; T,Cy], where cR is

defined as: cR = i/sd. Therefore, Equation (2) can be expressed as follows:

(i )=

0

P[CR>cR|Sd = sd; T,Cy]

d(sd)dSd dsd (5)


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It should be noted that although here R is referred to as lateral strength ratio, in reality it is

simply a measure of the ground motion intensity of the ground motion relative to the yielding

capacity of the system. This relative ground motion intensity can be expressed in terms of forces, as

done in Equation (4), or by assuming that the spectral pseudo-acceleration ordinate is equal to the

elastic spectral displacement ordinate times the squared frequency of the system. Thus, it can alsobe expressed as the ratio of the elastic spectral displacement to the systems yield displacement,

y, as follows:

R =Sd

y(6)

Therefore, Equation (5) can be used to estimate the MAF of exceedance of maximum inelastic

displacements for SDOF systems with a certain yield strength coefficient Cy or with a certain yield

displacement, y.

A key assumption in the proposed procedure is that i just depends on both the structural

properties of the system and on the level of ground motion intensity measured by Sd, but it

is not significantly affected by other ground motion parameters such as earthquake magnitude

nor distance to the source. More formally this means that the probability of i exceeding i is

assumed conditionally independent of earthquake magnitude and distance to the source which are

the main parameters affecting linear displacement spectral ordinates. This assumption allows us to

decouple the probabilistic estimation of the seismic hazard at a given site for elastic systems (i.e.

right-hand side in the integrand in Equations (1), (2) and (5)) from the probability of exceedance

of i conditioned on the elastic displacement. The two components are then combined using

the total probability theorem [24]. Therefore, Equation (5) can be viewed as an extension of

conventional PSHA. It should be mentioned that this simplifying assumption has been extensively

used in the past (e.g. [25, 26]). In particular, this simplification has been used to formulate the

probabilistic basis of the SAC/FEMA guidelines for steel moment-resisting frame buildings [27]

and extensively used in research developed at the Pacific Earthquake Engineering Research (PEER)

Center [2830]. In addition, recent statistical studies [23, 31, 32] provided foundation on theaforementioned assumption. For example, Ruiz-Garca and Miranda [23] studied the effect of

earthquake magnitude and distance to the source on CR using a ground motion database containing

216 records gathered from 12 Californian historical earthquakes with seismic wave magnitude

ranging from 5.8 to 7.7 and distance to the horizontal projection of the rupture (the so-called

Joyner and Boore distance) ranging from 1.0 to 117.6 km. Among their conclusions, they observed

that distance to the rupture had a negligible effect on CR ordinates while for systems with periods

of vibration longer than about 1 s earthquake magnitude had also a negligible effect on CR; but

that for periods smaller than 1.0 s some effects of magnitude were observed and that these effects

increased with increasing lateral strength ratio (i.e. as the system becomes weaker with respect

to the intensity of the ground motion). More recently, Chopra and Chintanapakdee [31] as well

as Medina and Krawinkler [32] also concluded that neither earthquake magnitude nor distance

to the source had a significant effect on median values of CR, with exception of systems withperiods of vibration shorter than 0.3 s [32]. They employed the same ground motion database in

their statistical studies which included records from events having earthquake moment magnitude

from 5.8 to 7.0 and distance to the source between 13 and 60 km. Because of the small effect

on magnitude previously observed by the authors [23] for periods smaller than 1.0 s, we have

opted to use an approximate sign in Equations (1), (2) and (5) instead of an equal sign. Since the

importance of this simplifying assumption, it is further discussed in Sections 5.1 and 5.2.


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It should be noted that for small mean annual rate of exceedance (smaller than 0.01), the

annual probability of exceedance ofi is numerically very close to the MAF of exceedance ofi .

Therefore, Equation (2) provides a very good approximation to the annual probability of exceeding

maximum inelastic displacement demand.

3. STATISTICAL PARAMETERS OF CR

To evaluate the conditional probability in the integrand of Equation (5), statistical information

of CR is necessary. The authors have performed comprehensive statistical studies of inelastic

displacement ratios computed by using earthquake ground motions recorded on different site

conditions (e.g. firm and soft soil conditions) as well as considering near-fault ground motions

[23, 3335]. These studies have provided quantitative information about the central tendency and

dispersion of CR for a wide range of periods of vibration, lateral strength ratios and hysteretic

behaviours. A relatively large number of earthquake ground motions were used in those studies

in order to assess not only the central tendency of CR but also its variability and probabilitydistribution for a given period of vibration and lateral strength ratio. A brief summary of the main

statistical results (i.e. central tendency and dispersion) previously obtained by the authors [35] on

inelastic displacement ratios computed from the dynamic response of elastoplastic SDOF systems

subjected to 240 acceleration time histories compiled from recording stations placed on rock or

firm soil sites from 12 Californian historical earthquakes with moment magnitude from 6.0 to 7.7

and distance to the horizontal projection of the rupture ranging from 9.0 to 117.6 km is presented

in the next section. It should be noted that in order to provide robust estimates of central tendency

and dispersion, outliers were removed from the statistical results according to the recommendations

provided by ANSI standards [36].

3.1. Central tendency of CR

Several measures of central tendency can be obtained from statistical studies of CR such as the

sample mean, the counted median and the geometric mean (i.e. the mean of the natural logarithm

of the data) [24]. For instance, Figure 1 shows the geometric mean of CR corresponding to 240

earthquake acceleration time histories. From the figure, it can be seen that the CR factors are

characterized by being larger than one (i.e. maximum inelastic displacements are larger than

maximum elastic displacements) in the short-period spectral region and relatively close to one

(i.e. maximum inelastic displacements are approximately equal to maximum elastic displacement)

for medium- and long-period systems. For periods smaller than 1.0 s inelastic displacement ratios

are strongly dependent on the period of vibration and on the lateral strength ratio. In general, in

this spectral region maximum inelastic displacements become much larger than maximum elastic

displacements as the lateral strength ratio increases (i.e. as the lateral strength decreases with

respect to the lateral strength required to maintain the system elastic) and as the period of vibrationdecreases. Furthermore, inelastic displacement ratios tend toward infinity as the period of vibration

goes to zero, which means that existing structures with very short periods may undergo very large

inelastic displacement demands relative to their elastic counterparts unless structures in this spectral

region are designed to remain elastic or nearly elastic. It should be noted that in this period region

the use of the equal displacement rule (i.e. maximum inelastic displacement demand is equal to

the maximum elastic displacement demand) could result in significant underestimations of the


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SITE CLASSES AB, C, D

geometric mean of 240 ground motions

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

E[CR]

R = 6.0

R = 5.0

R = 4.0

R = 3.0R = 2.0

R = 1.5

Figure 1. Geometric mean of inelastic displacement ratios for 240 ground motions recorded inNEHRP site classes AB, C and D.

maximum inelastic displacement demand. It is also important to mention that the limiting period

dividing spectral regions where the equal displacement rule is applicable from those where this rule

is not only inapplicable but also unconservative (produces an underestimation of the maximum

inelastic displacement demand) depends primarily on the lateral strength ratio. In general, this

limiting period increases as the lateral strength ratio increases. For further sample mean and

counted median results for CR the reader is referred to [35].

3.2. Variability of CR

While the geometric mean of inelastic displacement ratios is very important, as they represent

a measure of central tendency, it is equally important to know its dispersion, also known as

record-to-record variability, about the central tendency of CR. An effective way to quantify the

dispersion is through the standard deviation of the natural logarithm of CR, ln CR , This parameter

is particularly effective in characterizing the variability of a random variable if it is lognormally

distributed. Figure 2 shows ln CR corresponding to all ground motions recorded in site classes AB,

C and D. It can be seen that dispersion increases as the level of lateral strength ratio increases,

but it tends to saturate. Dispersion is particularly high for systems with short period of vibration

(e.g. T

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SITE CLASSES AB, C, D

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s ]

lnCR

R = 6.0

R = 5.0

R = 4.0

R = 3.0R = 2.0

R = 1.5

Figure 2. Logarithmic standard deviation of inelastic displacement ratios for 240 ground motions recordedin NEHRP site classes AB, C and D.

having very long periods should be expected. However, most Civil Engineering structures will

have periods well below the periods where this reduction in dispersion occurs.

4. SIMPLIFIED EQUATIONS TO ESTIMATE CENTRAL TENDENCY

AND DISPERSION OF CR

In this study, simplified non-linear equations are proposed to estimate the central tendency and

the dispersion of inelastic displacement ratios as a function of the period of vibration, T, and

the lateral strength ratio, R. It should be noted that the proposed equations can be used to fitany measure of central tendency or dispersion through non-linear regression analysis. Then, the

following equation was proposed to estimate the central tendency of CR:

CR =

1, R 1

1 +

1

1 T2

(R 1), R>1

(7)

where 1 and 2 are parameters whose estimates are obtained through non-linear regression analysis.

In addition, the following simplified non-linear equation is proposed to estimate the dispersion

of CR:

CR =

0, R 11

1+

1

2 (T + 0.1)

, R>1

(8a)

where

= 3 [1 exp(4(R 1))] (8b)


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and 1, 2, 3 and 4 are parameters that can be similarly obtained through non-linear regression

analysis. It should be noted that the functional form of Equations (8a) and (8b) captures the

observed trend ofln CR as illustrated in Figure 2. For instance, the parameter accounts for the

saturation ofln CR as the level of lateral strength ratio increases in the medium- and long-period

regions. Equations (7) and (8) correspond to a surface in the CRRT and the CR RT space,

respectively. In this investigation, non-linear least-square regression analysis that minimizes the

difference between the measured values and the estimated response using Equations (7) and (8)

was conducted using the LevenbergMarquardt method [37]. Thus, parameter estimates 1 and

2 and their 95% confidence intervals obtained from three measures of central tendency (i.e.

sample mean, counted median and geometric mean) of CR are reported in Table I. In addition,

the following parameter estimates to be used in Equations (8a) and (8b) using ln CR data were

found: 1 = 5.876, 2 = 11.749, 3 = 1.957 and 4 = 0.739. The fitted equations to estimate central

tendency, corresponding to geometric mean of CR, and lnCR are shown in Figures 3 and 4. It

should be mentioned that the functional form of Equation (7) has been recently suggested to

Table I. Parameter estimates and 95% confidence intervals for Equation (7)corresponding to three different measures of central tendency computedfrom 240 ground motions.

Central tendency 1 2 c.i. (1) c.i. (2)

Sample mean 35.79 2.12 32.40, 38.19 2.07, 2.17Counted median 79.12 1.98 65.60, 91.68 1.89, 2.07Geometric mean 49.03 1.87 44.73, 53.33 1.82, 1.92

1.0

2.0

3.0

4.0

5.06.0

0.10.5

0.91.3

1.7

2.2

3.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

CR

RPERIOD[s]

Figure 3. Central tendency of inelastic displacement ratios for elastoplastic systems computed withEquation (7) and fitted parameters for geometric mean given in Table I.


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1.0

2.0

3.0

4.0

5.0

6.0

0.10.5

0.9

1.3

1.7

2.2

3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

RPERIOD[s]

Figure 4. Dispersion (logarithmic standard deviation) of inelastic displacementratios computed with Equation (8).

improve coefficient C1 in FEMA 356 document [16], but with different parameter estimates 1

and 2.

5. EVALUATION OF SIMPLIFIED ASSUMPTIONS

As mentioned previously, three main simplifying assumptions have been made in the formulationof Equation (5). The first simplified assumption states that CR is conditionally independent of

earthquake magnitude, M, and distance to the rupture, D, which implies that the conditional

expectation of CR on a given earthquake magnitude event at a specified distance to the rupture,

E[CR|M= m, D = d], would be approximated by the unconditional expected value of CR, E[CR].

The second simplified assumption is that there is a lack of correlation between CR and Sd (i.e. CRand Sd are statistically uncorrelated) and they can be assumed as independent random variables.

Additionally, the proposed procedure assumes that the empirical cumulative probability distribution

of CR can be adequately approximated by a parametric cumulative distribution function (CDF).

Thus, it is necessary to verify the validity of these simplifying assumptions and to point out possible

limitations of the proposed simplified probabilistic approach. For that purpose, this section examines

the simplifying assumptions by making use of statistical results of CR computed for elastoplastic

systems subjected to ground motions recorded on rock or firm site conditions and reported in [35].

5.1. Effect of earthquake magnitude on CR

To evaluate the influence of earthquake magnitude on CR, the statistical significance of adding

the earthquake magnitude as an additional explanatory variable on CR (i.e. response variable) was

obtained by plotting fitted residuals computed by using the fundamental period of vibration and


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Figure 5. Inelastic displacement ratio fitted residuals versus earthquake magnitude for

T = 0.6 s and six levels of relative lateral strength: (a) R = 1.5; (b) R = 2.0; (c) R = 3.0;(d) R = 4.0; (e) R = 5.0; and (f) R = 6.0.

the lateral strength ratio as baseline predictor variables versus earthquake magnitude (i.e. prospec-

tive predictor) and then performing conventional linear regression analysis on the prospective

explanatory variable [38]. Fitted residuals were computed as ei = CR,i CR, where CR,i is the

computed inelastic displacement ratio for the i th ground motion and CR is the estimation of CRobtained from Equation (7). For convenience, residuals were obtained in a logarithmic base. Thus,

hypothesis testing was performed using the slope of the fitted line and considering rejecting the

null hypothesis for a p-value smaller or equal than 5% significance level. In general, it was found

that the earthquake magnitude might be statistically significant on CR ratios for systems in the

short spectral period region (T

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-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

CR,

MW

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

Figure 6. Correlation between CR and earthquake magnitude (Mw) from 240 ground motionsrecorded in rock and firm sites.

Figure 7. Inelastic displacement ratio fitted residuals versus distance to the rupture forT = 0.6 s and six levels of relative lateral strength: (a) R = 1.5; (b) R = 2.0; (c) R = 3.0;

(d) R = 4.0; (e) R = 5.0; and (f) R = 6.0.

general it is smaller and can be considered negligible for a wide range of systems. For example,

the (log) residuals of the inelastic displacement ratio versus D are shown in Figure 7. In addition,

the Pearsons correlation coefficient between CR and D as a function of the period of vibration

and lateral strength ratio is shown in Figure 8. It can be seen that, in general, the inelastic

displacement ratio is positively correlated with D for all periods and that the level of correlation

increases slightly as the lateral strength ratio increases. This feature could be explained since Sd


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-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

R = 6.0

R = 5.0

R = 4.0

R = 3.0

R = 2.0

R = 1.5

CR,

D

Figure 8. Correlation between CR and distance to the rupture, D (km), from 240 ground motionsrecorded in rock and firm sites.

decreases as distance to the rupture increases and, for a given distance to the rupture, peak inelastic

displacement demands increases for weak systems than those of strong systems. It can also be

observed that the correlation between CR and D is, in general, smaller than the correlation between

CR and earthquake magnitude and that the level of correlation is very close to zero for periods

larger than 0.7 s. However, it should be noted that the aforementioned observations only apply to

the range of distance to the rupture, between 9.0 and 117.7 km, considered in the ground motion

database.

5.3. Statistical correlation between CR and Sd

In the formulation of Equation (5) it is assumed that there is negligible statistical correlation

between CR and Sd and, thus, they can be treated as independent random variables. Thus, to

compute the Pearsons correlation coefficient between CR and Sd, CR,Sd , it is considered that the

acceleration time histories contained in the ground motion ensemble used in this investigation are

representatives of the seismic hazard environment (i.e. earthquake magnitude range, distance to

the rupture range, and soil conditions) considered in computing (sd). Therefore, CR,Sd computed

from all 240 ground motions recorded in firm sites is shown in Figure 9. From the figure, it can

be seen that CR,Sd is not constant and, furthermore, CR and Sd are negatively correlated (i.e. CR

decreases as Sd increases) over the whole range of periods. In addition, it should be noted that thelevel of correlation is not significantly influenced by the level of lateral strength ratio for periods

of vibration shorter than 1.0 s. However, small influence of the lateral strength ratio on CR,Sd is

observed for periods of vibration longer than 1.0 s since CR,Sd increases as the lateral strength

ratio increases. In general, for periods longer than about 0.5 s the correlation between CR and Sd is

smaller than 25%, which means that CR and Sd are weakly correlated and they might be assumed

as independent random variables.


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-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

R = 6.0

R = 5.0

R = 4.0

R = 3.0R = 2.0

R = 1.5

CR,

Sd

Figure 9. Correlation between CR and Sd computed from 240 ground motionsrecorded in rock and firm sites.

5.4. Probability distribution of CR

An explicit consideration of the uncertainty involved in the estimation of inelastic demands for

structures subjected to earthquake ground shaking through a probabilistic framework requires

the characterization of the conditional probability of exceeding a given seismic demand level

of interest. For example, Miranda [39] studied the empirical (sample) probability distribution of

inelastic strength demands obtained from statistical results of non-linear SDOF systems undergoing

constant ductility demands when subjected to a set of 124 earthquake ground motions. The author

found that parametric probability distributions such as Lognormal, Gamma, Gumbel type I andWeibull were adequate to represent the empirical cumulative probability distribution of inelastic

strength demands.

In this investigation, the empirical cumulative probability distribution of CR was obtained

considering results of CR, for a given period of vibration and lateral strength ratio, from all 240

earthquake ground motions as a random sample drawn from the population and by assuming each

CR value as an independent outcome. Next, all 240 CR observations were sorted in ascending

order and each observation, i , was assigned a probability equal to i/(n + 1) as plotting positions

of the ordered statistics, where n corresponds to 240 observations. From the resulting plots,

it was observed that the empirical cumulative distribution of CR follows a skewed distribution

(i.e. asymmetric) with longer tails moving toward upper values. Since the lognormal CDF has

been extensively employed to characterize the cumulative distribution of seismic demands, this

parametric CDF was evaluated in this study to determine if they can characterize the empiricalcumulative distribution of CR. To verify whether the candidate CDF is adequate, the well-known

KolmogorovSmirnov (KS) goodness-of-fit test [24] was used in this investigation.

In general, it was found that the lognormal probability distribution provided an adequate fit

to the empirical distribution. For example, a comparison of the empirical distribution of CRwith respect to lognormal (using counted median and logarithmic standard deviation of CR as

statistical parameters) corresponding to a short-period strong system (T = 0.5s and R = 2.0) and a


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long-period weak system (T = 2.0 s and R = 6.0) is shown in Figure 10. Therefore, it was decided

to use the lognormal probability distribution to characterize the empirical cumulative distribution

of CR, which can be evaluated as follows [24]:

P[CR>cR|T,Cy] = 1 ln(cR)

ln CRln CR

(9)

where is the standard normal CDF, ln CR is the mean of the natural logarithm of CR and ln CRis the standard deviation of the natural logarithm of CR.

5.5. Evaluation of proposed functional models to estimate conditional probability of CR

Since our objective is to compute the continuous maximum inelastic displacement hazard curve for a

wide range of periods of vibration and levels of relative lateral strength, it is of interest to verify if the

proposed simplified equations introduced in Section 4 to estimate central tendency and dispersion

of CR can also provide a good fit when assuming both a lognormal distribution (Equation (9)) but

not using it with sample statistical measures but rather with approximate statistical measures (i.e.

central tendency and dispersion) computed with Equations (7) and (8). For example, Figure 11(a)

T=0.5 s, R = 2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

C(a) (b)

P[C

|T,

R]

data

Lognormal fit

K-S test, 10%

significance level

T=2.0s, R = 6.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

C

P[C

|T,

R]

data

Lognormal fit

K-S test, 10%

significance level

Figure 10. Evaluation of lognormal fit to the conditional probability distribution of CR:(a) T = 0.5 s, R = 2.0 and (b) T = 2.0 s, R = 6.0.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0c

P[C

|T=0.5s,

R

=

4]

Data, T=0.5s

Lognormal with parameters from proposedequationsK-S test, 90% confidence

0.0

0.2

0.4

0.6

0.8

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0c(a) (b)

P[C

|T=-.5s,

R

=

4]

Data, T=0.5s

Lognormal with parameters from sample dataK-Stest, 90% confidence

Figure 11. Fitting of the empirical distribution of CR for a short-period system assuming lognormalCDF: (a) using statistical parameters from sample data and (b) using statistical parameters from

proposed Equations (7) and (8).


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R = 2

0.0

(a) (b)

(c)

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

CR

Data

p=90%

p=70%

p=50%

p=30%

p=10%

R = 4

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

CR

Data

p=90%

p=70%

p=50%

p=30%

p=10%

R = 6

0.0

1.0

2.0

3.0

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

PERIOD [s]

CR

Data

p=90%

p=70%

p=50%

p=30%

p=10%

Figure 12. Comparison of counted percentiles and percentiles of CR assuming lognormal CDF usingstatistical parameters estimated from Equations (7) and (8) for: (a) R = 2; (b) R = 4; and (c) R = 6.

shows the empirical probability distribution of CR as well as the lognormal fit computed with

sample parameters for a short-period system (T = 0.5 s) and R = 4.0. It can be observed thatthe lognormal CDF adequately follows the empirical distribution for this period of vibration and

level of lateral strength ratio. In addition, a comparison of the same empirical distribution of CRand its corresponding lognormal fit computed with statistical parameters estimated by proposed

Equations (7) and (8) and using parameter estimates obtained from counted median and the sample

logarithmic standard deviation is illustrated in Figure 11(b). From the figure, it can be observed

that the use of proposed equations to estimate statistical parameters of the lognormal distribution

also leads to a good agreement with respect to the empirical distribution of CR. The graphic

representation of the KS test corresponding to a 90% confidence level is also shown in both

figures. It should be mentioned that the adequacy of using Equations (7)(9) to approximate the

empirical cumulative distribution of CR was also verified for other periods of vibration and lateral

strength ratios. Therefore, it is believed that the functional forms of Equations (7) and (8), with

adequate parameter estimates, provide a good way to estimate the central tendency and dispersionparameters of the conditional probability distribution of CR.

One way of considering the dispersion ofCR consists on computing inelastic displacement ratios

corresponding to different percentiles. For example, CR values associated with different counted

percentiles computed from statistical results have been reported by the authors [23, 35]. In this

investigation, percentiles of CR were computed by assuming a lognormal distribution with central

tendency and dispersion parameters estimated from Equations (7) and (8). A comparison between


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CR spectra corresponding to five different percentiles computed from the statistics of CR and those

computed from the lognormal assumption (continuous line) are shown in Figure 12. From the

figure, it can be seen that a very good estimation is obtained by assuming that CR is lognormally

distributed.

6. EVALUATION OF THE PROPOSED SIMPLIFIED PROBABILISTIC APPROACH

6.1. Maximum inelastic displacement seismic hazard curves

The Stanford Campus located in Northern California, which is surrounded by several active faults

including the San Andreas and Hayward faults, was chosen to illustrate the proposed methodology.

Thus, the elastic seismic hazard curve for pseudo-acceleration spectral ordinates, (sa), correspond-

ing to five periods of vibration (T = 0.2, 0.3, 0.5, 1.0 and 2.0 s) was obtained from Frankel and

Leyendecker [19]. However, the cited reference only provides discrete values of (sa), while inte-

gration of Equation (5) requires a continuous seismic hazard curve and, thus, a continuous function

is desirable. Then, a continuous function of the spectral pseudo-acceleration seismic hazard curvewas obtained by fitting a fourth-order polynomial model of the following form:

ln (sa)= 0 + 1 ln sa + 2 (ln sa)2 + 3 (ln sa)

3 + 4 (ln sa)4 (10)

Conventional linear regression analysis was employed to obtain the set of parameter estimates

that provide the best fit of each seismic hazard curve corresponding to each period of vibration.

Next, the MAF of exceeding a certain elastic displacement demand, (sd), was also obtained for

each of the aforementioned periods of vibration. Therefore, a total of 25 seismic hazard curves

of maximum inelastic displacement demand, (i ), corresponding to five periods of vibration

(T = 0.2, 0.3, 0.5, 1.0 and 2.0 s) and five yielding strength coefficients (Cy = 0.1, 0.2, 0.4, 0.6 and

0.8) were computed through numerical integration of Equation (5). As a remainder, maximum

inelastic seismic hazard curves allow to estimate the MAF of exceeding a threshold displacement

demand which also represent approximately the annual probability of exceeding a certain dis-placement demand. For illustration purposes, two maximum inelastic displacement demand hazard

curves and its linear elastic spectral displacement seismic hazard curves counterpart correspond-

ing to two periods of vibration (T = 0.5 and 1.0 s) are shown in Figure 13. For a short-period

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.1 1.0 10.0 100.0

Maximum inelastic displacement, [cm]

USGS, T=1.0s

Cy=0.8

Cy=0.6

Cy=0.4

Cy=0.2Cy=0.1

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.1 1.0

(a) (b)

10.0 100.0

Maximum inelastic displacement, [cm]

(i)

USGS, T=0.5s

Cy=0.8

Cy=0.6

Cy=0.4

Cy=0.2Cy=0.1

(i)

Figure 13. Maximum inelastic displacement demand seismic hazard curves computed usingthe proposed approach corresponding to five levels of lateral strength for two periods of

vibration: (a) T = 0.5 s and (b) T = 1.0 s.


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(i)

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.1 1 10 100 1000

Maximum Inelastic Displacement,(a) (b) [cm]

Cy = 0.1

T = 0.2s

T = 0.3s

T = 0.5s

T = 1.0s

T = 2.0s

(i)

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.1 1 10 100 1000

Maximum Inelastic Displacement, [cm]

Cy = 0.4

T = 0.2s

T = 0.3s

T = 0.5s

T = 1.0s

T = 2.0s

Figure 14. Maximum inelastic displacement seismic hazard curves computed using the proposed approachfor five different periods of vibration and two levels of lateral strength: (a) Cy = 0.1 and (b) Cy = 0.4.

weak system (Cy = 0.1, T = 0.5 s) the MAF of exceeding 10 cm (4 in) is 2.88 times the MAFof exceeding the same displacement of a strong system (Cy = 0.8) with the same period of

vibration. Both systems are expected to behave non-linearly for this seismic displacement demand

level. Another example of maximum inelastic displacement demand hazard curves, as a function

of period of vibration, for two levels of Cy developed in this study are shown in Figure 14. As

expected, it can be observed that (i ) depends on both the lateral strength and the period of

vibration of the systems. For instance, for a long-period (T = 2.0 s) weak (Cy = 0.1) system the

MAF of exceeding 10 cm (4 in) is about 2.64, 7.28, 15.89 and 28 times the MAF of exceeding

the same inelastic displacement demand of a system with the same strength but with periods of

vibration of 1.0, 0.5, 0.3 and 0.2 s, respectively. On the other hand, a stronger system (Cy = 0.4)

with a period of vibration of 2.0 s would experience an inelastic displacement demand of 10 cm

with a exceedance probability of about 2.36, 3.71, 5.42 and 6.57 times than that of a system with

identical strength but with periods of vibration of 1.0, 0.5, 0.3 and 0.2 s, respectively. Furthermaximum inelastic displacement demand hazard curves developed for the same site can be found

in Reference [35].

6.2. Uniform hazard spectra of maximum inelastic displacement demand

Maximum inelastic displacement demand hazard curves can be used to build maximum inelastic

displacement uniform hazard spectra (MID-UHS). MID-UHS represents the maximum inelastic

displacement demand ordinates corresponding to the same probability of exceedance, as a function

of period of vibration and the lateral strength of the system. For example, MID-UHS spectra

corresponding to five yield strength coefficients and for 10 and 2% exceedance probability in

50 years (e.g. return periods of 475 and 2475, respectively) are shown in Figure 15. As can

be expected, weaker systems (i.e. with low yield strength coefficient) are more susceptible toexperience larger maximum inelastic displacement demands than stronger systems (i.e. with high

yield strength coefficient). For example, for a 10% chance of being exceeded in 50 years, a system

with T = 0.5 s and Cy = 0.1 will experience maximum inelastic displacement demand 2.22 times

larger than that experienced by the same system but with Cy = 0.8. However, for a 2% probability

of exceeding in 50 years, the same system with Cy = 0.1 would experience maximum inelastic

displacement demands twice as high as a system with the same period but with Cy = 0.8.


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10% in 50 yrs.

0

10

20

30

40

50

60

70

80

90

0.0 0.5 1.0 1.5 2.0 2.5

PERIOD(a) (b)[s]

i[cm]

Cy = 0.1

Cy = 0.2

Cy = 0.4

Cy = 0.6Cy = 0.8

Elastic

2% in 50 yrs.

0

10

20

30

40

50

60

70

80

90

0.0 0.5 1.0 1.5 2.0 2.5

PERIOD [s]

i[cm]

Cy = 0.1

Cy = 0.2

Cy = 0.4

Cy = 0.6

Cy = 0.8

Elastic

Figure 15. Maximum inelastic displacement uniform hazard spectra computed using the proposed approachcorresponding to two return periods: (a) 10% in 50 years and (b) 2% in 50 years.

Finally, it should be pointed out that MID-UHS are very useful for the seismic assessment ofexisting structures. In addition, this type of spectra can be used to establish performance limit-states

based on maximum inelastic displacement demands.

7. CONCLUSIONS

A simplified probabilistic approach to estimate maximum inelastic displacement demands of SDOF

systems was introduced in this paper. The proposed approach permits the computation of maximum

inelastic displacement demand hazard curves, (i ), which provide MAF of exceeding various

levels of inelastic displacement demands by using readily available information on the elastic

seismic hazard at a specific site. The proposed approach can also be used to compute uniformhazard spectra of maximum inelastic displacement demands corresponding to different return

periods.

The proposed simplified approach makes use of constant-strength inelastic displacement ratios,

CR, that permit the estimation of maximum inelastic displacement demand, i , from maximum

elastic displacement demand, Sd. From this study, it was found that the empirical probability

distribution of CR can be adequately approximated by using a parametric lognormal distribution

in order to compute the conditional probability of CR exceeding a certain level of cR given the

fundamental period and the relative lateral strength of the SDOF system. It was shown that by

using simple equations for computing the parameters of the conditional probability distribution of

CR, seismic hazard curves and uniform hazard spectra for maximum inelastic displacements can

be easily computed using already available elastic seismic hazard curves.

The influence of earthquake magnitude and distance to the rupture on CR was carefully examined.It was found that earthquake magnitude has a negligible effect on CR ordinates for periods of

vibration longer than about 0.5 s and with lateral strength ratios smaller than 4. However, some

statistically significant dependence ofCR on magnitude was observed for weakshort-period systems

(i.e. weak systems relative to the ground motion intensity). Similar observations were found for

the dependence of CR on distance to the rupture, but in general the effects were observed to be

smaller. This means that the proposed approach leads to very good results for periods longer than


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0.5 s but could lead to unconservative results for weakshort-period structures if subjected to large

magnitude earthquakes. For these systems it is not possible to separate the conditional probability

of exceeding the inelastic displacement given the elastic displacement from the estimation of the

elastic seismic hazard at the site. More accurate estimation of maximum inelastic displacement

demands of weak short-period systems may require the development of attenuation relationshipsof inelastic displacement ordinates, such as that very recently proposed by Tothong and Cornell

[40], to perform rigorous PSHA.

For illustration purposes, the proposed approach was used to obtain probabilistic estimates of

inelastic displacement demands of elastoplastic SDOF systems located on a firm soil site in a region

of high-seismicity in Northern California. However, it is believed that the suggested approach is

quite general and it can be used for other hysteretic behaviours and for soil conditions that differ

from those considered in this study (e.g. soft soil sites).

ACKNOWLEDGEMENTS

The first author acknowledges financial support from the Consejo Nacional de Ciencia y Tecnologa(CONACYT) in Mexico to pursue his doctoral studies at Stanford University under supervision of thesecond author. The authors are grateful to two anonymous reviewers that provided useful comments thathelped to improve the final version of this paper.

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