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Engineering Structures 32 (2010) 11–20

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Probabilistic estimation of residual drift demands for seismic assessment ofmulti-story framed buildingsJorge Ruiz-García a,∗, Eduardo Miranda ba Facultad de Ingeniería Civil, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C, Planta Baja, Cd. Universitaria, 58040 Morelia, Mexicob Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020, USA

a r t i c l e i n f o

Article history:Received 26 August 2008Received in revised form16 June 2009Accepted 13 August 2009Available online 17 September 2009

Keywords:Residual drift demandsSeismic assessmentInelastic intensity measureResidual drift hazard curve

a b s t r a c t

This paper presents the implementation of a probabilistic approach to estimate residual drift demands(e.g. residual roof, residual drift at specific stories, and maximum residual drift over all stories) duringthe seismic performance-based assessment of existing multi-story buildings. The approach combinesresidual drift demand fragility curves obtained froman inelastic intensitymeasure, incorporates explicitlythe aleatory uncertainty (i.e. record-to-record variability) inherent in the estimation of residual driftdemands at the end of the seismic excitation, with maximum inelastic displacement seismic hazardcurves to obtain site-building-specific residual drift demand hazard curves which express the meanannual frequency of exceeding residual drift demands. Recognizing the evolution of central tendencyand dispersion of residual drift demands with changes in the ground motion intensity, this proceduremakes use of functional models that capture that variation. It is shown that the relationship betweentransient (maximum) and residual (permanent) drift demands depends on the mean annual frequency ofexceedance and the building’s number of stories for a similar lateral load resisting system.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In modern performance-based seismic assessment methodolo-gies for the evaluation of existing structures, it is necessary toadequately estimate relevant seismic demands related to build-ing performance under different seismic hazard levels. For exam-ple, it is widely accepted that while the amplitude of maximum(transient) interstory drift ratio (IDRmax) is correlated with dif-ferent damage states sustained for structural elements as well assome drift-sensitive non-structural elements, peak floor acceler-ation (PFA) is related to failure of some classes of non-structuralcomponents. Furthermore, for advanced performance-based seis-mic assessment procedures it is particularly important to estimatethe likelihood that relevant seismic demands of a particularman-made structure would exceed pre-defined performance limitstates under different levels of ground motion intensity. Thisassessment approach requires a probabilistic framework wherethe estimation of seismic demands of interest and its inherentuncertainty (e.g. record-to-record variability) should be explicitlyincorporated.Inspired by Probabilistic Seismic Hazard Analysis (PSHA) [1–3],

which allows the estimations of the mean annual frequency of

∗ Corresponding author. Tel.: +1 52 443 3223500x4339; fax: +1 52 443 3221002.E-mail address: jruizgar@stanfordalumni.org (J. Ruiz-García).

exceedance (MAF) of a certain peak ground motion parameter(e.g. peak ground acceleration, etc.) or a linear elastic responsespectral ordinate (e.g. pseudo-acceleration, Sa) taking into accountthe hazard of earthquake ground motions at a given site,extensions have been proposed for computing the MAF ofnonlinear structures exceeding, for example, performance limitstates [4–6], damage measures [7], maximum drift demands(e.g. [8,9]), as well as, in a more general context, decision variablesfor performance-based seismic assessment. In particular, theprobabilistic approach proposed by Cornell and his co-workers(e.g. [5–8]) named Probabilistic Seismic Demand Analysis (PSDA),has been the basis for extensive research currently developed atthe Pacific Earthquake Engineering Research (PEER) Center [10,11]and the basis of the SAC/FEMA guidelines for seismic evaluation ofsteel moment-resisting frame buildings in the United States [12].It should be noted that PSDA employs simplifying assumptionsthat lead to a closed-form solution of the integral for computingthe MAF of nonlinear response parameters [6,13]; in particular,it assumes that: (1) the central tendency of the engineeringdemand parameter of interest (e.g. peak inter-story drift) withchanges in the intensitymeasure (e.g. pseudo-acceleration spectralordinate at the structure’s fundamental period of vibration) islinear in the log–log space, and (2) dispersion (i.e. record-to-recordvariability) of the engineering demand parameter is constant withchanges in the ground motion intensity. Very recently, alternativeapproaches have been proposed, aimed at improving PSDA and,

0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.08.010

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12 J. Ruiz-García, E. Miranda / Engineering Structures 32 (2010) 11–20

as a consequence, calculations of MAF of peak drift and peak flooracceleration demand [9] and the MAF of structural collapse [14].It should be noted that performance-based seismic assessment

procedures for existing structures emphasize the estimation ofpeak (transient) lateral drift demands [15]. However, earthquakefield reconnaissance has evidenced that permanent (residual) lat-eral displacement demands after earthquake excitation (e.g. resid-ual roof drift ratio ormaximum residual inter-story drift ratio) alsoplay an important role in defining the seismic performance of astructure and can have important consequences. For instance, sev-eral dozen damaged reinforced concrete (RC) buildings in MexicoCity had to be demolished after the 1985 Michoacan earthquakebecause of the technical difficulties with straightening and repair-ing buildings with large permanent drifts [16]. Years later, Okadaet al. [17] reported that several low-rise reinforced concrete (RC)buildings suffered light structural damage but experienced rela-tively large residual deformations as a consequence of the 1995Hyogo-Ken Nambu earthquake, even though they had sufficientdeformation capacity. These field observations imply that the to-tal expected economic losses computed from peak drift demandsand peak floor acceleration demands could be smaller than thosecomputed, taking into account permanent drift demands due tothe necessity of demolishing structures having excessive perma-nent deformations although they did not experience severe struc-tural damage. Furthermore, residual displacement demands havebeen identified as one of the most important response parametersin the evaluation of the residual capacity of building and bridgedamaged structures to sustain aftershocks [18,19]. Therefore, sev-eral researchers have highlighted that the estimation of residualdrift demands should also play an important role during the de-sign of new buildings (e.g. [20,23]) and the evaluation of the seis-mic structural performance of existing buildings (e.g. [21,22,25]).Motivated by earthquake field reconnaissance observations, re-

searchers have performed analytical investigations aimed at gain-ing further understanding on the parameters that influence theamplitude and height-wise distribution of residual drift demandsin existing multi-story buildings (e.g. [20–23]). They have re-ported that the residual drift demand amplitude and distributionover the height depends on the component hysteretic behavior[20–23], building frame mechanism [20–22], structural over-strength [21,22] as well as the ground motion intensity [20–22].Particularly, Ruiz-Garcia and Miranda [21,22] have noted that theevaluation of residual drift demands in regular moment-resistingframe models involves large levels of uncertainty (i.e. record-to-record variability) in its estimation and, moreover, this un-certainty is larger than that associated with the estimation ofmaximum (transient) drift demands. Thus, the evaluation of resid-ual deformation demands into seismic performance-based assess-ment methodologies requires a probabilistic approach where therecord-to-record variability is explicitly incorporated. In this con-text, very little research has been done to obtain probabilisticdescriptions of residual drift demands [21,24].Themain purpose of this paper is to discuss the implementation

of a probabilistic approach for obtaining estimates of residualdrift demands to be used during the seismic performance-basedassessment of existing multi-story frame buildings. Since previousanalytical studies have shown that the estimation of residualdrift demands involves large levels of record-to-record variabilityand, furthermore, it varies with changes in the ground motionintensity, this study followed the approach suggested by Aslaniand Miranda [9]. Thus, the following specific goals were stated:(a) to investigate a suitable intensity measure, (b) to select aparametric probability distribution in order to characterize theempirical probability distribution of residual drift demands, and(c) to characterize the variation of central tendency and dispersion(i.e. record-to-record variability) of residual drift demands withchanges in the groundmotion intensity. The next sections describethe implementation of the probabilistic approach for estimatingresidual drift demands.

2. Probabilistic estimation of residual drift demands

2.1. Probabilistic framework

In the currently well-known seismic performance-based as-sessment approach suggested at the Pacific Earthquake Engineer-ing Center [10,11] in the United States, the mean annual frequencyof exceedance of a engineering demand parameter (EDP) of inter-est exceeding a certain level edp can be computed as follows:

ν(EDP > edp) ∼=∫ ∞0

P (EDP > edp | IM = im)

×

∣∣∣∣dν(IM)dIM

∣∣∣∣ dIM (1)

where IM denotes the ground motion intensity measure, P(EDP >edp | IM = im) is the conditional probability that a EDP exceeds acertain level of edp given that the IM is evaluated at the groundmotion intensity measure level im. In addition, ν(IM)refers tothe mean annual frequency of exceedance of the IM, which alsorepresents the seismic hazard at a specific site. In this context,while the first term in the right-hand side of Eq. (1) can be obtainedfromprobabilistic estimates of the EDP of interest, the second termin Eq. (1) represents the slope in the seismic hazard curve, whichcan be computed from conventional Probabilistic Seismic HazardAnalysis (PSHA), evaluated at the groundmotion intensity level im.Therefore, Eq. (1) is suitable for computing the mean annual

frequency of exceeding a given residual drift demand, ∆r , for anexisting building (i.e. with known lateral strength) as follows:

ν(∆r > δ) =

∫ ∞0

P(∆r > δ | IM = im; T1, Cy

)×

∣∣∣∣dν(IM)dIM

∣∣∣∣ dIM (2)

In the above expression, P(∆r > δ | IM = im; T1, Cy) isthe probability of ∆r exceeding a defined residual deformationdemand conditioned on the fundamental period of vibration of theexisting building, T1, the yielding strength coefficient, Cy, and theground motion intensity measure evaluated at level im, which iscommonly known as the fragility curve.In order to apply Eq. (2) for obtaining probabilistic descriptions

of residual drift demands, two key steps should be addressed:(a) to select an appropriate intensity measure, and (b) to obtainbuilding–specific residual drift demand fragility curves based onthe selected ground motion intensity measure. The followingsection discusses these key steps.

2.2. Generic framemodels for probabilistic estimation of residual driftdemands

For the purpose of obtaining information regarding the selec-tion of an intensity measure and for obtaining building-specificresidual drift demand fragility curves, the statistical results onresidual drift demands obtained from rigorous nonlinear dynamictime history analyses of four one-bay generic frame building mod-els having three different number of stories (N = 3, 9, and 18), aswell as a flexible and rigid version of a 9-story frame model wereconsidered in this research. The building’s fundamental period ofvibration was obtained from empirical period formulas suggestedby Chopra and Goel [26], which corresponds to mean-minus andmean-plus-one-standard-deviation of fundamental periods of vi-brations measured in instrumented steel moment-resisting framebuildings. All generic frames were designed according to currentseismic provisions for structures located in a region of high seis-micity in California [15]. Special attention was given to provide a

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a b

Fig. 1. Central tendency of elastic spectra obtained from the ground motion set employed in this study: (a) normalized spectral acceleration, (b) spectral displacement.

Table 1Fundamental period of vibration, T1 , roof yield displacement, δy,roof , and normalizedmodal participation factor, Γ1φ1 obtained for each generic frame considered in thisstudy.

N T1 (s) δy,roof (cm) Γ1φ1

3 0.50 5.1 1.239-R 1.19 16.3 1.359-F 1.90 26.4 1.3518 2.00 30.5 1.37

realistic lateral height-wise stiffness distribution, similar to thatfound in existing multi-story frame buildings, which controls thefundamental modal shape of each framemodel. Thus, each genericbuilding was modeled as a two-dimensional centerline frame us-ing the computer software RUAUMOKO [27]. A lumped-plasticityapproach was assumed for taking into account inelastic deforma-tions in frame elements (i.e. inelastic deformationwas restricted toplastic hinges at both ends of beam and column elements). A non-degrading elastic-perfectly plastic moment-curvature hystereticrelationship, which is typical hysteretic behavior representativeof steel components that do not exhibit early local buckling, wasassumed at each plastic hinge. The flexural yielding moment ca-pacity in the elements was determined from story shear forceswith the lateral static force distribution obtained from current seis-mic provisions in the United States [15]. While the main structuralproperties of the analyzed buildings are given in Table 1, a de-tailed description of the design process andmodeling assumptionsof the generic framedmodels is available in [21]. Later, each build-ing frame model was subjected to a suite of 40 earthquake groundmotions, referred to as LMSR-N, recorded during 5 historical earth-quakes in California. All earthquake ground motions included inthe LMSR-N set were recorded on stiff soil or soft rock correspond-ing to soil type D according to FEMA recommendations (e.g. [15]).This set comprises motions recorded in earthquake events withmomentmagnitude ranging from6.5 to 7.0 andwith source-to-sitedistances ranging from 13 km to 40 km, which it can be consideredrepresentative of a typical moderate and large magnitude-smalldistance seismic environment in California. It should be noted thatnone of the records included in the ground motion set exhibitedpulse-type near-fault features. Fig. 1 shows central tendency ofthe spectral acceleration and displacement spectra obtained fromthe LMSR-N ground motion set. Blue dots in both spectra indicatethe elastic ordinates corresponding to each generic frame modelconsidered in this study. Information on the main ground motionfeatures of each earthquake ground motion can be found in [28].

2.3. Selection of intensity measure for probabilistic estimation ofresidual drift demands

An important component in Eq. (2) is the selection of anappropriate parameter to characterize the intensity of the ground

motion, which is also known as intensity measure (IM). Severalstudies have used the spectral pseudo-acceleration, Sa, or thespectral displacement, Sd, of a linear elastic 5% damped single-degree-of-freedom (SDOF) system having a fundamental period ofvibration of the structure, T1, as groundmotion intensity measures(i.e., Sa(T1) or Sd(T1)) since seismic hazard curves developed byseismologists are commonly expressed in terms of Sa (e.g. [6]).However, several alternative IM’s have been recently proposedin the literature (e.g. [29,30]) aimed at providing smaller record-to-record variability in the evaluation of the SRP parameter ofinterest as well as negligible dependence between the SRP andground motion parameters (i.e., earthquake magnitude, distanceto the source, etc) than Sa(T1). In particular, some researchers havenoted that ground motions scaled to reach the same maximuminelastic displacement demand of an equivalent elastoplastic SDOFsystem having the same initial lateral stiffness (i.e., fundamentalperiod of vibration, T1), ∆i(T1), leads to smaller record-to-recordvariability than other proposed IMs [29,30]. In spite of its improvedefficiency, ∆i(T1) has not been employed as IM for computingthe MAF of exceedance of decision variables due to the lack ofa site-specific maximum inelastic displacement hazard curves,ν(∆i). This critical step can be overcome if a ground motionprediction equation (GMPE) for maximum inelastic displacementdemand is available for conducting a PSHA. Recently, Tothongand Cornell [31] developed a GMPE for maximum inelasticdisplacement demand of bilinear SDOF systems having a post-yield stiffness equal to 5% of initial stiffness, which can beincorporated into PSHA to compute ν(∆i). As an alternative, Ruiz-Garcia and Miranda [21,32] developed a simplified procedure forobtaining site-specific maximum inelastic displacement demandhazard curves for constant-relative strength SDOF systems havingelastoplastic behavior. Therefore, Sd(T1) and ∆i(T1) where chosenas candidates IM to be used in Eq. (2).A comparison of the variation of median residual roof drift

demand, θr,roof , as well as counted 16th and 84th percentile bands,with changes in the ground motion intensity computed from theresponse of a 18-story one-bay generic frame model (T1 = 2.0 s)using both Sd(T1) and ∆i(T1) as IMs is shown in Fig. 2. Fromthe figure, it can be seen that the record-to-record variabilityusing both IM’s is not constant and it tends to increase as theintensity of the ground motion increases. In particular, the use ofSd(T1) as intensitymeasure leads to large levels of record-to-recordvariability. However, the use of∆i(T1) as IM leads to smaller levelsof record-to-record variability as compared to Sd(T1). Moreover,the number of outliers (i.e. very large values compared to therest of the sample) is considerably reduced when using ∆i(T1).In addition, a similar comparison for the variation of maximumresidual drift demand, RIDRmax, for the same building model isillustrated in Fig. 3. Similar results were obtained for the 3-storyand 9-story generic frame models [21,22].

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14 J. Ruiz-García, E. Miranda / Engineering Structures 32 (2010) 11–20

a b

Fig. 2. Variation of θr,roof with changes in IM: (a) Sd(T1); and (b)∆i(T1).

a b

Fig. 3. Variation of RIDRmax with changes in IM: (a) Sd(T1); and (b)∆i(T1).

From the aforementioned observations, it is believed that theuncertainty in the estimation of residual drift demands is reducedwhen ∆i(T1) is employed as IM. Then, it is appealing to expressEq. (2) in the following form:

ν(∆r > δ) =

∫ ∞0

P(∆r > δ | ∆i(T1) = δi; T1, Cy

)×

∣∣∣∣dν(∆i(T1))d∆i(T1)

∣∣∣∣ d∆i(T1) (3)

In the above alternative expression ν(∆i(T1)) is the site-specificmaximum inelastic displacement demand hazard curve, as afunction of the specific fundamental period of vibration of thesystem and the yield strength coefficient of the structure, whichshould be available.

2.4. Conditional probability distribution of residual drift demands

The next step in the development of Eq. (3) consists of obtaininga probabilistic description of the distribution of residual drift de-mands conditioned in the selected ground motion intensity mea-sure. For instance, the empirical probability distribution of RIDRmaxcorresponding to a 9-story flexible frame model (T1 = 1.90 s) isshown in Fig. 4. The empirical cumulative probability distributionof RIDRmaxwas obtained by considering drift values as independentoutcomes. Sample data was then sorted in ascending order andplotted with a probability equal to i/(n+1), where i is the positionof the drift ratio and n is the size of the sample. It can be seen that inboth cases the empirical distribution is not symmetricwith respectto the 50th percentile (i.e. sample median) and they have longertails moving towards the upper values. Thus, skewed parametricprobability distributions such as lognormal, Weibull or Rayleigh

could be adequate to characterize the empirical cumulative prob-ability distribution. For example, Fig. 4 also shows the fittedprobability distribution using lognormal, Weibull, and Rayleighprobability distribution functions. Similar plots were obtained forother generic frame models at different ground motion inten-sities. The well-known Kolmogorov–Smirnov (K–S) goodness-of-fit test [33] was used in this investigation to verify whether thecandidate probability distributions are adequate to characterizethe empirical cumulative probability distribution of residual driftdemands.Even though both the lognormal and Weibull probability

distributions satisfy the K − S test, the lognormal probabilitydistribution was chosen to describe the probability distributionof residual drift demands since it has the convenience overother skewed probability distributions (e.g. Weibull or Gumbeldistribution) that can be fully defined from two parameters whichexplicitly represent the central tendency and the dispersion, orspread, of the sample distribution. In this study, it was found thatthe sample geometric mean and the standard deviation of thenatural logarithmof the data as parameters of central tendency anddispersion of residual drift demands provided better fitting withrespect to the sample distribution as follows:

P (∆r > δr) = 1− Φ(ln(δr)− µln δr

σln δr

)(4)

For illustration purposes, Fig. 5(a) and (b) show a comparison ofthe fitted lognormal probability distribution ofRIDRmax for both the9-R and 9-F generic framemodels employing the sample geometricmean and the counted median as a measure of central whilethe logarithmic standard deviation is employed as a measure ofdispersion. The graphic representation of the K − S test (at a 10%significance level) is also shown in both Figures.

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J. Ruiz-García, E. Miranda / Engineering Structures 32 (2010) 11–20 15

Fig. 4. Comparison of parametric and empirical probability distribution obtainedfor a 9-story generic frame model.

2.5. Statistical parameters of residual drift demands as a function ofthe ground motion intensity

Ruiz-Garcia and Miranda [21,22] have shown that the samplestatistical measures (i.e. central tendency and dispersion) ofresidual drift demands changewith variation of the groundmotionintensity. For instance, the variation of median RIDR and σln RIDRfor 5 different story levels of the 9-story generic stiff frame modelwith changes in the groundmotion intensity is illustrated in Fig. 6.It can be seen that median RIDR computed in the seventh andninth story level grows nonlinearly as the groundmotion intensityincreases, whereas median RIDR of the first and third story growsalmost linearly at a much faster rate than the aforementionedstories with changes in the ground motion intensity. The lattertrend is a reflex of the residual drift concentration in the bottomstories as the ground motion intensity increases and, in general,both trends reflect the type of frame mechanism. On the otherhand, dispersion seems to increase or decrease depending on the

level of ground motion intensity and location along the height.For example, dispersion tends to decrease for ∆i(T1) between 20and 30 cm, but it tends to increase for ∆i(T1) smaller than 20 cm(Fig. 6(b)).Therefore, the variation of sample statistical measures with

changes in the ground motion intensity should be reflected in theparameters employed to estimate the building-specific conditionalprobability of exceeding a given residual drift demand thresholdgiven in Eq. (4). For that purpose, the following functionalmodels are proposed for describing the variation of the centraltendency of residual drift demands with changes in the intensitymeasure [5–11,34]:

µ̃ = α1α∆i(T1)2 (∆i(T1))α3 (5)

µ̃ = a(∆i(T1))b (6)

while coefficients α1, α2, α3 in Eq. (5) can be obtained fromnonlinear regression analysis, coefficients a and b in Eq. (6)are obtained from conventional linear regression analysis in thelog–log domain that, indeed, implies a linear relationship betweenµ̃ and IM . It should be noted that Eq. (6) has been extensivelyemployed for performance-based seismic assessment of buildingsand bridges (e.g. [5–8,10,11,34]).The fitted variation of the geometric mean of RIDRmax with

changes in the ground motion intensity using Eqs. (5) and (6)obtained for the 3-story model (α1 = 0.12, α2 = 0.68, α3 = 2.91;a = 0.20, b = 1.37) and the 18-story model (α1 = 0.37,α2 = 0.79, α3 = 2.01; a = 0.46, b = 1.14) is shown in Fig. 7.It can be seen that while the functional form of Eq. (5) capturesreasonably well the variation of median RIDRmax with changes inthe intensity of the ground motion, the use of Eq. (6) might leadto underestimations or overestimations, depending on the levelof ground motion intensity, to predict median RIDRmax for bothbuilding models.

a b1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.0 1.0 2.0 3.0 4.0

RIDRmax [%]

Δi (T1 ) = 48 cm

9-Story (T1 = 1.185 s)

Δi (T1 ) = 78 cm

9-Story (T1 = 1.902 s)

5.0 6.0 7.0 8.0 9.0 0.0 1.0 2.0 3.0 4.0

RIDRmax [%]

5.0 6.0 7.0 8.0 9.0

P(RIDRmax | Δi)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

P(RIDRmax | Δi)

Fig. 5. Fitting of the parametric lognormal CDF of RIDRmax for two 9-story generic frame models: (a) T1 = 1.185 s; and (b) T1 = 1.902 s.

RIDR [%]

9-Story (T1 = 1.185 s) 9-Story (T1 = 1.185 s)

0 10 20 30 40 50 60 70 80

a b2.5

2.0

1.5

1.0

0.5

0.0

Δi (T1 ) [cm]

σln RIDR

0 10 20 30 40 50 60 70 80

2.5

2.0

1.5

1.0

0.5

0.0

Δi (T1 ) [cm]

Fig. 6. Variation of central tendency and dispersion of RIDR computed for five stories of the GF-9R building model: (a) Median RIDR; and (b) dispersion of RIDR (σln RIDR).

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a b

Fig. 7. Evaluation of Eqs. (5) and (6) to estimate RIDRmax for two generic frame models: (a) 3-story (T1 = 0.5 s); and (b) 18-story (T1 = 2.0 s).

a b

Fig. 8. Evaluation of the functional form of Eqs. (5) and (6) to estimate dispersion of RIDRmax: (a) GF-3R (T1 = 0.5 s); and (b) GF-18R (T1 = 2.0 s).

a b

Fig. 9. Evaluation of lognormal fitting CDF of RIDRmax using Eqs. (2) and (7) for two 9-story generic frame models: (a) T1 = 1.185 s; and (b) T1 = 1.902 s.

In addition of evaluating changes of central tendency thefeasibility of using the functional forms given by Eqs. (5) and (6)to characterize the variation of dispersion as a function of groundmotion intensity was also investigated. For example, Fig. 8 showsthe fitted variation of σln RIDRmax for the same 3-story (α1 = 1.05,α2 = 1.15, α3 = −0.94; a = 1.29, b = −0.23) and 18-storygeneric frame models (α1 = 0.82, α2 = 1.04, α3 = −0.56;a = 0.87, b = −0.49). It can be seen that both the power andthree-parameter functional forms provides a reasonable fit of thevariation of σln RIDRmax with changes on ∆i(T1). Therefore, it alsoproposed to evaluate the evolution of dispersion of residual driftdemands using the same functional formgiven in Eq. (5) as follows:

σ̃ ln δ r = γ1γ∆i(T1)2 (∆i(T1))γ3 . (7)

It is important tomention that in order to employ the functionalform given in Eqs. (5)–(7), at least three different levels of groundmotion intensity should be used to obtain the parameter estimates.It is also recommended that two of these levels of ground motion

intensity correspond to approximately the limits of the range ofinterest [9].For illustration purposes, the probability of exceeding RIDRmax,

assuming a lognormal distribution with probability parameterscomputed with Eqs. (5) and (7) and those obtained from sampledata, for the 9-story stiff and flexible generic frame models,corresponding to a predefined intensity measure (∆i(T1) = 48 cmfor GF-9R and∆i(T1) = 78 cm for GF-9F) are shown in Fig. 9. It canbe seen that the use of Eqs. (5) and (7) leads to adequate probabilityparameter estimates and, thus, to a reasonable representation ofthe probability of exceeding RIDRmax (Fig. 9(a) and (b)).

3. Illustration of residual drift demand hazard curves

The proposed approach allows us to obtain different residualdrift demand hazard curves that would be useful for decisionmak-ing process during the seismic assessment of existing buildings.For example, building-specific hazard curves of maximum residual

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Fig. 10. Maximum inelastic displacement demand hazard curve corresponding toT = 0.5 s and five levels of yield strength coefficient (after Ref. [32]).

roof drift, ν(θr,roof ), maximum residual inter-story drift ratio overall stories, ν(RIDRmax), or residual inter-story drift ratio at selectedstory levels, ν(RIDRi) can be obtained, which represent the meanannual frequency (MAF) of exceeding a given level of residual driftdemand. Such residual drift demand hazard curves can be directlycomparedwithmaximumdrift demand hazard curve counterpartsdeveloped in parallel.To illustrate the proposed probabilistic approach, residual drift

demand hazard curves for the short-period (3-story, T1 = 0.5 s)and the long-period (18-story, T1 = 2.0 s) generic frame modelswere computed by performing numerical integration of Eq. (3).It should be noted that the proposed probabilistic approach re-quires that a site-fundamental period-yield strength-specific max-imum inelastic displacement hazard curve, ν(∆i), is availablesince ∆i(T1) was employed as ground motion intensity measure.These ν(∆i) curves can be obtained from probabilistic seismic haz-ard analysis employing ground motion attenuation relationshipssuch as that developed by Tothong and Cornell [31], or the re-cently proposed simplified approach suggested by Ruiz-Garcia andMiranda [21,32]. For instance, Fig. 10 shows the elastic displace-ment hazard curve provided by the United States Geological Sur-vey (USGS) for the Stanford Campus site [35] as well as maximuminelastic displacement hazard curves computed byRuiz-Garcia andMiranda [21,32] for an elastoplastic SDOF systems with funda-mental period of 0.5 s and five different levels of yield lateralstrength.

3.1. Residual drift demand hazard curves

The yield strength coefficient, Cy, of the 3-story (T1 = 0.5 s)and 18-story (T1 = 2.0 s) generic framemodels was obtained froma nonlinear static (i.e. pushover) analysis being approximately 0.8and 0.2, respectively. Therefore, ν(∆i) hazard curves developed byRuiz-Garcia and Miranda [21,32] corresponding to the combina-tion of T1 and Cy were used for computing residual roof drift de-mand, ν(θr,roof ), and maximum residual interstory drift demand,ν(RIDRmax), hazard curves for both frame models. For illustrationpurposes, both residual drift hazard curves for the short-periodgeneric framemodel are shown in Fig. 11. The residual drift hazardcurves shown in these figure allow a probabilistic assessment ofresidual drift demands, inwhich both theuncertainty in the groundmotion hazard at a given site and the uncertainty (i.e. record-to-record variability) in the seismic response of a specific building areexplicitly taken into account.

Fig. 11. Residual drift hazard curves for short-period 3-story generic framemodel.

Recently proposed performance-based assessment procedures,such as those developed at the Pacific Earthquake Engineer-ing Center [10,11], assume that dispersion of relevant engi-neering demand parameters remains constant with changes inthe ground motion intensity. Then, it is interesting to exam-ine the influence of assuming that dispersion in the estima-tion of residual drift demands does not evolve with changesin intensity of the ground motion. For example, Fig. 12(a)and (b) show ν(θr,roof ) and ν(RIDRmax) hazard curves com-puted for the 18-story generic frame models, respectively, as-suming constant dispersion varying with the ground motionintensity (i.e. computedwith Eq. (7)) and two levels of constant dis-persion with changes in the ground motion intensity. It should bementioned that previous statistical results obtained by the authorshave shown that the record-to-record variability of residual driftdemands is very high, but it is larger for residual roof drift demandsthan for maximum residual interstory drift demands [21,22].From the figures, it can be seen that assuming constant dispersionmight lead to underestimations or overestimation in theMAF curvewith respect to the MAF curve assuming non-constant dispersiondepending on the level of residual drift demands. Therefore, it is be-lieved that the variation of both central tendency and dispersion ofresidual drift demands with changes in the intensity of the groundmotion should be addressedwhile computing residual drift hazardcurves (Fig. 12).

3.2. Comparison of residual and maximum drift demand hazardcurves

Following a similar procedure to the development of resid-ual drift demand hazard curves, roof drift demand, ν(θroof ), andmaximum interstory drift demand, ν(IDRmax), hazard curves werealso developed in parallel for the same 3-story short-period and18-story long-period building models. Thus, a direct comparisonof both residual (permanent) and maximum (transient) drift de-mand hazard curves can be done during the performance-basedseismic assessment of existing structures. For example, Fig. 13(a)shows a comparison between ν(θr,roof ) and ν(θroof ) computed forthe 3-story short-period frame model while a similar comparisonis shown in Fig. 13(b) corresponding to the 18-story long-periodbuilding model. It can be seen that the level of θr,roof as well as thedifference between θroof and θr,roof for a given MAF of exceedanceis different for each generic frame model. Similarly, a comparisonof maximum and residual interstory drift demand hazard curvescomputed for the same building models are shown in Fig. 14(a)and (b). It can also be observed that the difference between RIDRmaxand IDRmax depends on the MAF of exceedance and the building’speriod of vibration. This observation implies that transient and

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a b

Fig. 12. Comparison of residual drift hazard curves considering variable and constant variation of dispersion with changes in ground motion intensity for the 18-storygeneric frame model: (a) ν(θr,roof ); and (b) ν(RIDRmax).

a b

Fig. 13. Comparison of θroof and θr,roof hazard curves obtained from two generic building models: (a) 3-story; and (b) 18-story.

a b

Fig. 14. Comparison of IDRmax and RIDRmax hazard curves obtained from two generic building models: (a) 3-story; and (b) 18-story.

permanent drift limits associated to performance levels, such asthose recommended in FEMA 356 [15], should also depend on thefundamental period of vibration besides the lateral-load resistingstructural system.

3.3. Issues for computing residual drift demand hazard curves

Before concluding this section, the authorswould like to discussthe following issues that might have an influence on computingresidual drift demand hazard curves:

(1) Modeling of generic frames. Recent studies have highlightedthe influence of several modeling issues at computing per-manent deformations of reinforced concrete elements. Yazganand Dazio [36] noted that member modeling using a lumped

plasticity approach led to larger permanent displacementswhen using a fiber-element modeling. In addition, they notedthat the implementation of hysteretic rules for simulatingstiffness-degrading member load-deformation features in dif-ferent structural analysis software packages might lead to dif-ferent estimations of permanent displacements. Thus, cautionshould be taken when developing residual drift demand haz-ard curves for multi-story buildings that include degradingfeatures in the component hysteretic behavior.

(2) Uncertainty in component capacities. Residual drift demandfragilities presented in this study only included the aleatoryuncertainty due to the record-to-record variability in theestimation of residual drift demands as a source of uncertainty.However, during the evaluation of existing buildings theactual member capacities might not be known due to the

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uncertainty in the mechanical properties of the materials thatmay have been deteriorated, actual construction detailing, etc.The epistemic uncertainty due to capacity of the structuralmembers could be included while computing residual driftdemands. Recent studies have noted that this source ofuncertainty could be important if the structure is driven tothe collapse threshold (e.g. [37]). To overcome this issue, non-destructive tests could be carrying out to obtain informationfor estimating actual member capacities.

4. Summary and conclusions

Previous analytical studies have shown that the uncertaintyin the estimation of residual drift demands of multi-story framessubjected to earthquake excitation is very large, even larger thanthat of maximum drift demands. Thus, this uncertainty should beexplicitly taken into account for the performance-based seismicassessment of existing buildings.A rational probabilistic procedure that allows the incorporation

of the inherent aleatory uncertainty due to the record-to-recordvariability in the estimation of residual drift demands (i.e. roofresidual drift demand, maximum residual drift demand at all sto-ries, and residual drift demands at specific stories) of multi-storyframes was employed in this study for computing residual driftdemand hazard curves. It was shown that an inelastic intensitymeasure based on the peak inelastic displacement demand of anequivalent elastoplastic single-degree-of-system having the samefundamental period of vibration of the analyzed building allowsdecreasing the record-to-record variability during the estimationof residual drift demands of multi-story frame models under dif-ferent levels of ground motion intensity. In addition, it was con-cluded that the empirical probability distribution of residual driftdemands can be characterized with a two-parameter lognormalprobability distribution in order to compute the conditional ex-ceedance probability of residual drift demands. Recognizing thatboth the central tendency and the dispersion of residual driftdemands vary with changes in the ground motion intensity, athree-parameter functional model was found adequate in orderto estimate the statistical parameters of the lognormal probabilitydistribution.The probabilistic procedure outlined in this paper was used

to compute residual drift demand hazard curves for 3- and 18-story generic frame models having non-degrading elastoplasticcomponent hysteretic behavior. For comparison purposes, tran-sient (maximum) drift hazard curves for the same tested buildingmodels were developed in parallel since comparing both building-specific transient and residual drift hazard curves provides a betterway of assessing the seismic performance of existing structures.It was shown that the level of residual drift demand dependson the mean annual frequency of exceedance and the building’sfundamental period of vibration, which means that residual driftlimit-states associated to different structural performance levels,as those provided in the FEMA 356 [15] recommendations in theUnited States for performance-based seismic assessment of exist-ing structures, should depend on both the lateral load resistingsystem and the fundamental period of vibration.

Acknowledgements

The first author would like to express his gratitude to theConsejo Nacional de Ciencia y Tecnología (CONACYT) and theUniversidad Michoacana de San Nicolas de Hidalgo in Mexico for thesupport provided in developing the research reported in this paper.

References

[1] Esteva L. Criteria for construction spectra for seismic design. In: Proc. 3rd panAm. symp. structures. 1967 [in Spanish].

[2] Esteva L. Basis for the formulation of decision in seismic design. Report182. Mexico city (Mexico): Institute of Engineering, National AutonomousUniversity of Mexico; 1968 [in Spanish].

[3] Cornell CA. Engineering seismic risk analysis. Bull Seismol Soc Amer 1968;58(5):1583–606.

[4] Esteva L, Ruiz SE. Seismic failure rate ofmultistory frames. J Struct Engrg 1989;115(2):268–84.

[5] Bazzurro P, Cornell CA. Seismic hazard analysis of nonlinear structures I:Methodology. J Struct Engrg 1994;120(11):3320–44.

[6] Cornell CA. Calculating building seismic performance reliability: A basis formulti-level design norms. In: Proceedings of the eleventh world conf. onearthquake engineering, Paper No. 2122. 1996.

[7] Shome N, Cornell CA, Bazzurro P, Carvallo JE. Earthquakes, records, andnonlinear response. Earthq Spectra 1998;14(3):469–500.

[8] Luco N, Cornell CA. Effects of connection fractures on SMRF seismic driftdemands. J Struct Engrg 2000;126(1):127–36.

[9] Aslani H, Miranda E. Probabilistic-based seismic response analysis. Eng Struct2005;27:1151–63.

[10] Cornell CA, Krawinkler H. Progress and challenges in seismic performanceassessment. PEER News 2000;3(2).

[11] Deierlein GG. Overview of a comprehensive framework for performanceearthquake assessment. Report PEER 2004/05. Pacific Earthquake EngineeringCenter; 2004. p. 15–26.

[12] Cornell CA, Jalayer F, Hamburger RO, Foutch DA. The probabilistic basis forthe 2000 SAC/FEMA steel moment frame guidelines. J Struct Eng (ASCE) 2002;128(4):526–33.

[13] MacGuire RK. Seismic hazard and risk analysis. Earthquake EngineeringResearch Institute, MN-10; 2004.

[14] Bradley BA, Dhakal RP. Error estimation of closed-form solution for annual rateof structural collapse. Earthq Eng Struct Dyn 2008;37(15):1721–37.

[15] Federal Emergency Management Agency (FEMA). Prestandard and commen-tary for the seismic rehabilitation of buildings. Report FEMA 356.Washington,DC, 2000.

[16] Rosenblueth E, Meli R. The 1985 Mexico earthquake: Causes and effects inMexico City. Concrete Internat (ACI) 1986;8(5):23–34.

[17] Okada T, Kabeyasawa T, Nakano T, Maeda M, Nakamura T. Improvement ofseismic performance of reinforced concrete school buildings in Japan—Part 1Damage survey and performance evaluation after the 1995Hyogo-KenNambuearthquake. In: Proceedings of the twelfth world conference on earthquakeengineering, Paper 2421. 2000.

[18] Luco N, Bazzurro P, Cornell CA. Dynamic versus static computation ofthe residual capacity of a mainshock-damaged building to withstand anaftershock. In: Proceedings of the thirteenth world conference on earthquakeengineering, Paper No. 2405. 2004.

[19] Mackie K, Stojadinovic B. Residual displacement and post-earthquake capacityof highway bridges. In: Proceedings of the thirteenth world conference onearthquake engineering, Paper No. 1550. 2004.

[20] Pampanin S, Christopoulos C, Priestley MJN. Performance-based seismicresponse of framed structures including residual deformations. Part II:Multiple-degree-of-freedom systems. J Earthq Eng 2003;1(7):119–47.

[21] Ruiz-García J, Miranda E. Performance-based assessment of existing struc-tures accounting for residual displacements. Report no. 153. Stanford(CA): The John A. Blume Earthquake Engrg Ctr, Stanford Univ.; 2005.http://blume.stanford.edu/Blume/TechnicalReports.htm.

[22] Ruiz-Garcia J,Miranda E. Evaluation of residual drift demands in regularmulti-story frame buildings for performance-based assessment. Earthq Eng StructDyn 2006;35(13):1609–29.

[23] Pettinga D, Christopoulos C, Pampanin S, PriestleyMJN. Effectiveness of simpleapproaches inmitigating residual deformations in buildings. Earthq Eng StructDyn 2006;36(12):1763–83.

[24] Uma SM, Pampanin S, Christopoulos C. A Probabilistic framework forperformance-based seismic assessment of structures considering residualdeformations. In: 1st. European conference on earthquake engineering andseismology 2006. Paper No. 731.

[25] Yazgan U, Dazio A. Utilization of residual displacements in the post-earthquake assessment. In: 14th. world conference on earthquake engineer-ing. 2008. Paper 05-01-0445.

[26] Chopra AK, Goel RK. Building period formulas for estimating seismicdisplacements. Earthq Spectra 2000;16(2):533–6.

[27] Carr AJ. RUAUMOKO-inelastic dynamic analysis program. Christchurch, NewZealand: Dept. of Civil Engineering, University of Canterbury; 2008.

[28] Medina R, Krawinkler H. Seismic demands for nondeteriorating framestructures and their dependence on ground Motions. Report PEER 2003/13.Pacific Earthquake Engineering Center; October, 2003.

[29] Miranda E, Aslani H, Taghavi S. Assessment of seismic performance in terms ofeconomic losses. Report PEER 2004/05. Pacific Earthquake Engineering Center.p. 149–60.

[30] Luco N, Cornell CA. Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions. Earthq Spectra 2007;23(2):357–92.

[31] Tothong P, Cornell CA. An empirical ground motion attenuation relation forinelastic spectral displacement. Bull Seismol Soc Amer 2006;96(6):2146–64.

Author's personal copy

20 J. Ruiz-García, E. Miranda / Engineering Structures 32 (2010) 11–20

[32] Ruiz-Garcia J, Miranda E. Probabilistic estimation of maximum inelasticdisplacement demands for performance-based design. Earthq Eng Struct Dyn2007;36(9):1235–54.

[33] Benjamin JR, Cornell CA. Probability, statistics, and decision for civil engineers.New York: McGraw-Hill Book Company; 1970.

[34] Mackie K, Stojadinovic B. Improving seismic demand models through refinedintensity measures. In: Proceedings of the thirteenth world conference onearthquake engineering, Paper No. 1556. 2004.

[35] Frankel AD, Leyendecker EV. Seismic hazard curves and uniform hazardresponse spectra for theUnited States. CD-ROMV3.01, USGS, Denver, CO, 2001.

[36] Yazgan U, Dazio A. Comparison of different finite-element modelingapproaches in terms of estimating the residual displacements of RC structures.In: Proceedings of the eightUSnational conference on earthquake engineering,Paper No. 309. 2006.

[37] Dolsek M. Incremental dynamic analysis with consideration of modelinguncertainties. Earthq Eng Struct Dyn 2008;38(6):805–25.