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Probabilistic Capacity Models and Fragility Estimatesfor Reinforced Concrete Columns based on Experimental

ObservationsPaolo Gardoni1; Armen Der Kiureghian, M.ASCE2; and Khalid M. Mosalam, M.ASCE3

Abstract: A methodology to construct probabilistic capacity models of structural components is developed. Bayesian updatingto assess the unknown model parameters based on observational data. The approach properly accounts for both aleatory anuncertainties. The methodology is used to construct univariate and bivariate probabilistic models for deformation and shear capcircular reinforced concrete columns subjected to cyclic loads based on a large body of existing experimental observations. Tbilistic capacity models are used to estimate the fragility of structural components. Point and interval estimates of the fragformulated that implicitly or explicitly reflect the influence of epistemic uncertainties. As an example, the fragilities of a typicalcolumn in terms of maximum deformation and shear demands are estimated.

CE Database keywords: Probability; Concrete columns; Cyclic loads; Deformation; Concrete, reinforced; Capacity; Models.

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Introduction

Predictive capacity models in current structural engineering ptice are typically deterministic and on the conservative siThese models typically were developed using simplified mechics rules and conservatively fitting to available experimental dAs a result, they do not explicitly account for the uncertaininherent in the model and they provide biased estimates ofcapacity. While these deterministic models have been successused to design safe structures, the needs of modern strucengineering practice, and especially the advent ofperformance-based design concept, require predictive capmodels that are unbiased and explicitly account for all the pvailing uncertainties.

In this paper, we present a Bayesian framework for the deopment of multrivariate probabilistic capacity models for strutural components that properly account for all the prevailingcertainties, including model errors arising from an inaccurmodel form or missing variables, measurement errors, and sttical uncertainty. With the aim of facilitating their use in practicrather than developing new capacity models, we develop cortion terms to existing deterministic capacity models in commuse that properly account for the inherent bias and uncertainthese models. Methods for assessing the model parameters obasis of observed experimental data are described. Through

1Graduate Student, Dept. of Civil & Environmental EngineerinUniv. of California, Berkeley, CA 94720.

2Taisei Professor of Civil Engineering, Dept. of Civil & Environmental Engineering, Univ. of California, Berkeley, CA 94720.

3Assistant Professor, Dept. of Civil & Environmental EngineerinUniv. of California, Berkeley, CA 94720.

Note. Assocaite Editor: George Deodatis. Discussion open uMarch 1, 2003. Separate discussions must be submitted for indivipapers. To extend the closing date by one month, a written requestbe filed with the ASCE Managing Editor. The manuscript for this pawas submitted for review and possible publication on February 26, 2approved on January 4, 2002. This paper is part of theJournal of Engi-neering Mechanics, Vol. 128, No. 10, October 1, 2002. ©ASCE, ISS0733-9399/2002/10-1024–1038/$8.001$.50 per page.

1024 / JOURNAL OF ENGINEERING MECHANICS / OCTOBER 2002

J. Eng. Mech. 2002.

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use of a set of ‘‘explanatory’’ functions, we are able to identterms that correct the bias in an existing model and providesight into the underlying behavioral phenomena. Althoughmethodology described in this paper is aimed at developprobabilistic capacity models, the approach is general and caapplied to the assessment of models in many engineering mecics problems. As an application, this methodology is used tovelop probabilistic shear and deformation capacity modelsreinforced concrete~RC! columns under cyclic loading.

Fragility is defined as the conditional probability of failure oa structural member or system for a given set of demand vables. The probabilistic capacity models are used in a formulato estimate the fragility of structural components with specattention given to the treatment of aleatory and epistemic untainties. Univariate fragility curves and bivariate fragility cotours are estimated for a representative bridge column underclic loading.

Capacity Models

In the context of this paper, a ‘‘model’’ is a mathematical expresion relating one or more quantities of interest, e.g., the capacof a structural component, to a set of measurable variablex5(x1 ,x2 , . . . ), e.g., material property constants, member dimesions, and imposed boundary conditions. The main purpose omodel is to provide a means for predicting the quantities of inest for given deterministic or random values of the variablesx.The model is said to be univariate when only one quantity is topredicted and multrivariate when several quantities are to bedicted. We begin our discussion with the univariate form of tmodel and then generalize to the multrivariate case.

A univariate capacity model has the general form

C5C~x,Q! (1)

where Q denotes a set of parameters introduced to ‘‘fit’’ thmodel to observed data andC is the capacity quantity of interesThe functionC(x,Q) can have a general form involving algebraexpressions, integrals, or differentials. Ideally, it should be

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rived from first principles, e.g., the rules of mechanics. Ratthan developing new models, in this paper we adopt commoused deterministic models, to which we add correction terms.believe this approach will facilitate the use of the resulting probilistic models in practice. With this in mind, we adopt the geeral univariate model form

C~x,Q!5 c~x!1g~x,u!1s« (2)

whereQ5(u,s), u5(u1 ,u2 , . . . ), denotes the set of unknowmodel parameters, c(x)5selected deterministic modeg(x,u)5correction term for the bias inherent in the determinismodel that is expressed as a function of the variablesx and pa-rametersu, «5random variable with zero mean and unit vaance, ands represents the standard deviation of the model erNote that for givenx,u ands, we have Var@C(x,Q)#5s2 as thevariance of the model.

In formulating the model, we employ a suitable transformatof the quantity of interest to justify the following assumptions:~1!the model variances2 is independent ofx ~homoskedasticity assumption!, and ~2! « has the normal distribution~normality as-sumption!. Box and Cox~1964! suggest a parameterized famiof transformations for this purpose. However, in practice,model formulation itself often suggests the most suitable transmation. Diagnostic plots of the data or the residuals against mpredictions or individual regressors can be used to verify the sability of an assumed transformation~Rao and Toutenburg, 1997!.

As defined earlier, the functiong(x,u) corrects the bias in thedeterministic modelc(x). Since the deterministic model usualinvolves approximations, the true form ofg(x,u) is unknown. Inorder to explore the sources of bias in the deterministic modelselect a suitable set ofp ‘‘explanatory’’ basis functionshi(x), i51, . . . ,p, and express the bias correction term in the form

g~x,u!5(i 51

p

u ihi~x! (3)

By examining the posterior statistics of the unknown parameu i , we are able to identify those explanatory functions thatsignificant in describing the bias in the deterministic model. Nthat, while the bias correction term is linear in the parametersu i ,it is not necessarily linear in the basic variablesx.

A structural component may have several capacity measwith respect to the demands placed on it. For example, a rforced concrete column has different capacities relative to faiin shear, bending, reinforcing bar slip or excessive deformatFor the analysis of such a component, we formulateq-dimensional multrivariate capacity model in the form

Ck~x,uk ,S!5 ck~x!1gk~x,uk!1sk«k ,

k51, . . . ,q (4)

where

gk~x,uk!5(i 51

pk

ukihki~x!, k51, . . . ,q (5)

With the exception of the new termS, all entries in the aboveexpressions have definitions analogous to those of the univamodel. S denotes the covariance matrix of the variablessk«k ,k51, . . . ,q. The set of unknown parameters of the model in E~4! is Q5(u,S), where u5(u1 , . . . ,uq) and uk

5(uk1 , . . . ,ukpk). Considering symmetry,S includesq unknown

J. Eng. Mech. 2002.

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variancessk2, k51, . . . ,q, and q(q21)/2 unknown correlation

coefficientsrkl , k51, . . . ,q21, l 5k11, . . . ,q.

Uncertainties in Model Assessment and Prediction

In assessing a model, or in using a model for prediction purpoone has to deal with two broad types of uncertainties: aleauncertainties~also known as inherent variability or randomnes!and epistemic uncertainties. The former are those that are inhein nature; they cannot be influenced by the observer or the maof the observation. Referring to the model formulations in tpreceding section, this kind of uncertainty is present in the vablesx and partly in the error terms«k . The epistemic uncertainties are those that arise from our lack of knowledge, our delibate choice to simplify matters, from errors in measuriobservations, and from the finite size of observation samples.kind of uncertainty is present in the model parametersQ andpartly in the error terms«k . The fundamental difference betweethe two types of uncertainties is that, whereas aleatory uncerties are irreducible, epistemic uncertainties are reducible, e.guse of improved models, more accurate measurements and cotion of additional samples. Below, we describe the specific tyof uncertainties that arise in assessing capacity models. Forplicity in the notation, we use the formulation of a univariamodel.

Model Inexactness

This type of uncertainty arises when approximations are induced in the derivation of the deterministic modelc(x). It hastwo essential components: error in the form of the model, e.glinear expression is used when the actual relation is nonlinear,missing variables, i.e.,x contains only a subset of the variablethat influence the quantity of interest. In Eq.~2!, the termg(x,u)provides a correction to the form of the deterministic modwhereas the error terms« represents the influence of the missinvariables as well as that of the remaining error due to the inexmodel form. Since the missing variables are inherently randthat component of« that represents the influence of the missivariables has aleatory uncertainty, whereas the component resenting inexact model form has epistemic uncertainty. In practit is difficult to distinguish between the two uncertainty compnents of«. However, after correction of the model form with thtermg(x,u), one can usually assume that most of the uncertainherent in« is of aleatory nature. The coefficients representsthe standard deviation of the model error arising from modelexactness.

Measurement Error

As we will see shortly, the parameters of the model are asseby use of a sample of observationsCi of the dependent variablefor observed valuesxi , i 51, . . . ,n, of the independent variablesThese observed values, however, could be inexact due to errothe measurement devices or procedures. To model these ewe let Ci5Ci1eCi andxi5 xi1exi be the true values for thei thobservation, whereCi andxi are the measured values andeCi andexi are the respective measurement errors. The statistics omeasurement errors are usually obtained through calibratiomeasurement devices and procedures. The mean values oferrors represent biases in the measurements~systematic error!,whereas their variances represent the uncertainties inherent i

JOURNAL OF ENGINEERING MECHANICS / OCTOBER 2002 / 1025

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measurements. In most cases the random variableseCi andexi canbe assumed to be statistically independent and normally disuted. The uncertainty arising from measurement errorsepistemic in nature, since improving the measurement deviceprocedures can reduce it.

Statistical Uncertainty

The accuracy of estimation of the model parametersQ dependson the observation sample sizen. The smaller the sample size, thlarger the uncertainty in the estimated values of the parameThis uncertainty can be measured in terms of the estimatedances of the parameter. Statistical uncertainty is epistemic inture, as it can be reduced by further collection of data.

Bayesian Parameter Estimation

The philosophical framework for the approach developed inpaper is based on the Bayesian notion of probability. The evengoal of developing probabilistic capacity models and fragility etimates is seen in the context of making decisions with regarthe performance-based design of new structures or the retrofirehabilitation of existing structures. In this context, it is essenfor the approach to be capable of incorporating all types of avable information, including mathematical models of structuralhavior, laboratory test data, field observations, and subjectivegineering judgment. It is equally important that the approaaccount for all the relevant uncertainties, including those thataleatory in nature and those that are epistemic. The Bayeframework employed in this paper is ideally suited for this ppose.

In the Bayesian approach, the parametersQ are estimated byuse of the well-known updating rule~Box and Tiao 1992!

f ~Q!5kL~Q!p~Q! (6)

wheref (Q)5posterior distribution representing our updated stof knowledge aboutQ; L(Q)5likelihood function representingthe objective information onQ contained in a set of observationp(Q)5prior distribution reflecting our state of knowledge aboQ prior to obtaining the observations; andk5@*L(Q)p(Q)dQ#215normalizing factor. The likelihood is afunction that is proportional to the conditional probability of maing the observations for a given value ofQ. Specific formulationsof this function are described in the next section. The prior dtribution may incorporate any information aboutQ that is basedon our past experience or engineering judgment. When no sinformation is available, one should use a prior distribution thas minimal influence on the posterior distribution, so that infences are unaffected by information external to the observatiFor the set of parametersQ5(u,S), it is generally assumed thau and S are approximately independent so thatp(Q)'p(u)p(S). Using Jeffrey’s rule~1961!, Box and Tiao~1992!have shown that the noninformative prior for the parametersu islocally uniform such thatp(Q)>p(S). Furthermore, they haveshown that the noninformative prior for the elementsS i j of S hasthe form

p~S!}uSu2(q11)/2 (7)

where u•u denotes the determinant. When the unknown paraeters are the standard deviationss i and the correlation coeffi-cients r i j , the noninformative prior takes the form~Gardoni2002!


J. Eng. Mech. 2002.

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p~S!}uRu2(q11)/2)i 51

q1

s i(8)

whereR5@r i j #5correlation matrix. For a univariate model, thabove simplifies to

p~s!}1

s(9)

We note that the posterior distribution is centered at a pothat represents a compromise between the prior informationthe data, and the compromise is increasingly controlled bydata as the sample size increases. For large or even modesized samples, a fairly drastic modification of the prior distribtion may only lead to a minor modification of the posterior dtribution ~Box and Tiao 1992!.

Once the posterior distribution ofQ is determined, one cancompute its mean vector and covariance matrix. We denote tasMQ andSQQ , respectively. Computation of these statistics,well as the normalizing constantk is not a simple matter, as irequires multifold integration over the Bayesian kernL(Q)p(Q). An algorithm for computing these statistics is dscribed in Gardoni~2002!.

Likelihood Functions

As mentioned earlier, the likelihood is a function that is proptional to the conditional probability of the observations for givvalues of the model parameters. Formulation of the likelihofunction depends on the type and form of the available informtion. Here, we start by considering the univariate model wexact measurements. The effect of measurement error is thecorporated in an approximate manner. The formulation is nextended to multrivariate models.

In observing the state of a structural component in a laboratest or in the field with respect to a specific mode of failure, oof three possible outcomes may be realized:~1! the demand ismeasured at the instant of failure, in which case the measdemand represents the component capacity;~2! the componentdoes not fail, in which case the measured demand represelower bound to the component capacity; and~3! the componenthas failed under a lower demand than measured, in which casmeasured demand represents an upper bound to the compcapacity. These observations are categorized as three typedata, as described below.

Failure Datum

Observed value of the capacityCi , for a givenxi , measured atthe instant when the component fails. Using Eq.~2! we haveCi

5 c(xi)1g(xi ,u)1s« i or s« i5r i(u), where r i(u)5Ci2 c(xi)2g(xi ,u) and« i denotes the outcome of the model error termthe i th observation.

Lower Bound Datum

Observed lower boundCi to the capacity for a givenxi , when thecomponent does not fail. In this case we haveCi, c(xi)1g(xi ,u)1s« i or s« i.r i(u).

Upper Bound Datum

Observed upper boundCi to the capacity for a givenxi , when thecomponent is known to have failed at a lower demand levelthis case we haveCi. c(xi)1g(xi ,u)1s« i or s« i,r i(u).

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Lower and upper bound data are often referred to as censdata.

With exact measurements, and under the assumption of stically independent observations, the likelihood function for tunivariate model has the general form

L~u,s!} )failure data

P@s« i5r i~u!#

3 )lower bound data

P@s« i.r i~u!#

3 )upper bound data

P@s« i,r i~u!# (10)

Since« has the standard normal distribution, we can write


H 1

swF r i~u!

s G J3 )

lower bound dataFF2

r i~u!

s G3 )

upper bound dataFF r i~u!

s G (11)

wherew(•) and F(•), respectively, denote the standard normprobability density and cumulative distribution functions.

Now consider the case where measurement errors are preDenoteCi and xi as the measured values in thei th observationandeCi andexi as the corresponding measurement errors. Withloss of generality, we assume the measurements have beenrected for any systematic error, so that the means ofeCi andexi

are zero. Letsi2 andSi denote the variance ofeCi and the cova-

riance matrix ofexi , respectively. As it should be evident, wallow dependence between the measurement errors for diffevariables at each observation; however, we assume independbetween the measurement errors at different observationsalso assume that the error terms are normally distributed. Fofailure data we haveCi1eCi5 c( xi1exi) 1g( xi1exi ,u)1s« i ,for the lower bound data we haveCi1eCi, c( xi1exi)1g( xi

1exi ,u)1s« i and for the upper bound data we haveCi1eCi

. c( xi1exi)1g( xi1exi ,u)1s« i . Defining

r i~u,exi !5Ci2 c~ xi1exi !2g~ xi1exi ,u! (12)

the conditions for the three types of data can be written ass« i

2eCi5r i(u,exi), s« i2eCi.r i(u,exi), and s« i2eCi,r i(u,exi),respectively. Unfortunatelyr i(u,exi) in general is a nonlineafunction of the random variablesexi , which makes the computation of the likelihood function enormously more difficult. Tovercome this difficulty, under the assumption that the errorsexi

are small in relation to the measurementsxi , we use a first-orderapproximation to expressr i(u,exi) as a linear function ofexi .Using a Maclaurin series expansion aroundexi50, we have

r i~u,exi !>Ci2 c~ xi !2g~ xi ,u!2@¹xic~ xi !1¹xi

g~ xi ,u!#exi

5 r i~u!1¹xir i~u!exi (13)

where¹x denotes the gradient row vector with respect tox andr i(u)5Ci2 c( xi)2g( xi ,u). The conditions for the three types odata can now be written ass« i2eCi2¹xi

r i(u)exi5 r i(u), s« i

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2eCi2¹xir i(u)exi. r i(u), and s« i2eCi2¹xi

r i(u)exi, r i(u), re-spectively. The left-hand sides of these expressions are a norandom variable with zero mean and variances2(u,s)5s21si

2

1¹xir i(u)Si¹xi

r i(u)T. Hence, in presence of measurement errothe likelihood function takes the form


H 1

s~u,s!wF r i~u!

s~u,s!G J

3 )lower bound data

FF2r i~u!

s~u,s!G

3 )upper bound data

FF r i~u!

s~u,s!G (14)

We now consider the multrivariate model in Eq.~4! underexact measurements. For thei th observation, any of theq capac-ity measures can be either directly observed, observed fbelow ~lower bound data!, or observed from above~upper bounddata!. However, these observations in general are dependencause of the correlation between the model error terms«k . Fig. 1illustrates a conceptual representation of the various ways thadata for a bivariate capacity model may appear. The curved lindicate the limit states for the two failure modes and the arwith varying intensities of shading indicate regions of failure anonfailure with respect to each mode. The dots indicate hypothcal data points. It can be seen that the data points can be i2

59 different categories~i.e., lower bound–lower bound datalower bound–failure data, lower bound–upper bound data, e!.More generally, the data points for aq-variate model can be of amost 3q different types.

For the i th observation of thekth capacity model, definer ki(uk)5Cki2 ck(xi)2gk(xi ,uk), where Cki is the measuredvalue of thekth capacity or its lower or upper bound. Also, let«ki

be the outcome of the error term for thekth capacity model in thei th observation. Noting that any of theq capacity terms can bemeasured as a failure datum, lower bound datum, or upper bodatum, the likelihood function takes the form

L~Q!} )observationi

P$ù failure

datak

@sk«ki5r ki~uk!#

ùlower bounddatak

@sk«ki.r ki~uk!# ùupper bounddatak

@sk«ki,r ki~uk!#%

(15)

Fig. 1. Representation of data types


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Table 1. Probability Terms for Bivariate Capacity Model with Lower Bound and Failure Data

Capacity model 2

Capacity Model 1

Failure datum Lower bound

Failure Datum 1

s1u2wFr1i~u!2m1u2

s1u2G 1

s2wF r 2i~u!

s2G FF2

r 1i~u!2m1u2

s1u2G 1

s2wF r 2i~u!

s2G

Lower BoundFF2

r 2i~u!2m2u1

s2u1G 1

s1wF r 1i~u!

s1G E

r 2i

`

FF2r 1i~u!2m1uz

s1u2G 1

s2wS z

s2Ddz

Note: mku l5rkl(sk /s l)r li , sku l5skA12rkl2 , k,l 51,2, andm1uz5r12(s1 /s2)z.

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whereQ5(u1 , . . . ,uq ,S). The events in the above expressiongeneral are dependent because of the correlation between«ki fordifferent indicesk. The probability term for each observation cabe computed using the multinormal probability density andmulative distribution functions. As an example, Table 1 listsexpressions for these terms for a bivariate model with only lowbound and failure data. Naturally, the required computationalfort for evaluating the likelihood function grows with increasindimension of the model.

In the presence of measurement error, the likelihood funcfor the multrivariate model remains similar to that in Eq.~15!with the term sk«ki for the i th observation of thekth modelreplaced by

sk«ki2eCki2¹xir ki~uk!exi (16)

and r ki(uk) replaced byr ki(uk), which is equivalent tor i(u)for the kth model. The term Eq.~16! is a zero mean normarandom variable with variance sk

2(uk ,sk)5sk21ski

2

1¹xir ki(uk)Si¹xi

r ki(uk)T. Furthermore, the terms for thekth

and l th models have the covariancerklsks l5rklsks l

1¹xir ki(uk)Si¹xi

r l i (ul)T, whererkl denotes the correlation coe

ficient. For the bivariate model described in Table 1, the formlation remains the same withsk , rkl , and r ki(uk) replaced bysk , rkl and r ki(uk), respectively.

Model Selection

For the sake of simplicity of notation, the discussion in this stion is focused on the univariate model. However, the concediscussed are equally applicable to a multrivariate model.

The probabilistic model in Eqs.~2! and~3! requires the selection of the deterministic modelc(x) and a set of explanatoryfunctionshi(x), i 51, . . . ,p. For prediction purposes, the seletion process should aim at a model that is unbiased, accurateeasily implementable in practice. Furthermore, from a statiststandpoint, it is desirable that the correction termg(x,u) have aparsimonious parameterization~i.e., have as few parametersu i aspossible! in order to avoid loss of precision of the estimates aof the model due to inclusion of unimportant predictors andavoid overfit of the data.

The model form in Eq.~2! is unbiased by formulation. Furthermore, a good measure of its accuracy is represented bstandard deviations. Specifically, among a set of parsimonioucandidate models, the one that has the smallests can be consid-ered to be the most accurate. Therefore, an estimate of the pa


J. Eng. Mech. 2002.

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eter s, e.g., its posterior mean, can be used to select the maccurate model among several viable candidates.

The explanatory functionshi(x) should be selected so as t

enhance the predictive capacity of the deterministic modelc(x).It is appropriate to select terms that are thought to be missin

c(x). Ideally, rules of mechanics should be used in formulatthe explanatory functions. However, in many cases relianceintuition is necessary. It is also desirable thathi(x) have the same

dimension asc(x) so thatu i are dimensionless. It is best to stathe model assessment process with a comprehensive candform of g(x,u) and then simplify it by deleting unimportant termor combining terms that are closely correlated. A stepwise dtion process may proceed as follows:

1. Compute the posterior statistics of the model parameteu5(u1 , . . . ,up) ands.

2. Identify the termhi(x) whose coefficientu i has the largestposterior coefficient of variation. The termhi(x) is the leastinformative among all the explanatory functions, so one mselect to drop it fromg(x,u).

3. If u ihi(x) is retained, determine the largest absolute vacorrelation coefficienturu iu j

u5maxkÞiuruiuku betweenu i and

the remaining parametersuk , kÞ i . A value of uru iu ju close

to 1, say 0.7<uru iu ju, is an indication that the information

contents inhi(x) andhj(x) are closely related and that thestwo explanatory functions can be combined. On the othand, a value ofuru iu j

u small in relation to 1, sayuru iu ju

<0.5, is an indication that the information content inhi(x)is not closely related to that in the remaining terms. If 0<uru iu j

u, one can choose to replaceu i by

ui5mui1ruiuj

sui

suj

~uj2muj! (17)

wheremu iandsu i

5posterior mean and standard deviation

u i , respectively. The above expression provides the bestear predictor ofu i as a function ofu j ~Stone 1996!. Thisreduces one parameter ing(x,u).

4. Assess the reduced model of step 2 or 3 by estimatingparameters. If the posterior mean ofs has not increased byan unacceptable amount, accept the reduced model anturn to step 2 or 3 for possible further reduction of thmodel. Otherwise, the reduction is not desirable andmodel form before the reduction is as parsimonious as psible.

There is considerable room for judgment in the above procedThis is a part of the art of model building.

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Assessment of Structural Component Fragility

For a structural component, fragility is defined as the conditioprobability of attaining or exceeding prescribed limit states fogiven set of boundary variables. Following the conventionaltation in structural reliability theory~Ditlevsen and Madsen1996!, let gk(x,Q) be a mathematical model describing thekthlimit state of interest for the structural component, where as inprevious sectionsx denotes a vector of measurable variables aQ denotes a vector of model parameters. This function is defisuch that the event$gk(x,Q)<0% denotes the attainment or exceedance of thekth limit state by the structural component. Usally x can be partitioned in the formx5(r ,s), wherer is a vectorof material and geometrical variables, ands is a vector of demandvariables such as boundary forces or deformations.

Using the capacity models described earlier, the limit stfunctions for a structural component can be formulated as

gk~r ,s,Q!5Ck~r ,s,Q!2Dk~r ,s! k51, . . . ,q (18)

whereDk(r ,s) denotes the demand for thekth failure mode. Forexample, for failure in shear of a reinforced concrete columCk(r ,s,Q)5 maximum shear force that the column can sustawhereasDk(r ,s)5maximum applied shear force. Note that boquantities can be functions of other demand variables, e.g.,plied bending moment or axial force. Therefore, the functioCk(r ,s,Q) and Dk(r ,s) generally could include bothr and s asarguments.

The fragility of the structural component is stated as

F~s,Q!5PFøk $gk~r ,s,Q!<0%us,QG (19)

whereP@Aus# denotes the conditional probability of eventA forthe given values of variabless. The uncertainty in the event fogiven s arises from the inherent randomness in the capacity vablesr , the inexact nature of the limit state modelgk(r ,s,Q) ~orits submodels!, and the uncertainty inherent in the model paraeters Q. We have expressed the fragility as a function of tparameters so as to highlight the fact that an estimate offragility depends on how we treat the model parameters. Varestimates of fragility can be developed depending on howtreat the parameter uncertainties. These are described in thelowing subsections.

Point Estimates of Fragility

A point estimate of the fragility is obtained by ignoring the ucertainty in the model parameters and using a point estimateQ inplace ofQ. Most commonly the posterior meanMQ or the maxi-mum likelihood estimate~MLE! QMLE is used. We denote thcorresponding point estimate of the fragility as

F~s!5F~s,Q! (20)

The uncertainty in this estimation arises from the intrinsic vaability in r and from the random model correction terms«k ,which is essentially aleatory in nature. In the special case wvariablesr are deterministically known,F(s) can be computed interms of the multinormal probability distribution of«k . Moregenerally, a multifold integral involving the joint distribution ofrand «k over the failure domain must be computed. Methodsnumerical computation of such probability terms are well devoped in the field of structural reliability~Ditlevsen and Madsen1996!.

J. Eng. Mech. 2002.

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Predictive Estimate of Fragility

The point estimate of fragility in Eq.~20! does not incorporate theepistemic uncertainties inherent in the model parametersQ. Toincorporate these uncertainties, we must considerQ as randomvariables. The predictive estimate of fragilityF(s) is the expectedvalue ofF(s,Q) over the posterior distribution ofQ, i.e.,

F~s!5E F~s,Q! f ~Q!dQ (21)

This estimate of fragility incorporates the epistemic uncertainin an average sense. However, it does not distinguish betweedifferent natures of the aleatory and epistemic uncertainties.

Bounds on Fragility

Uncertainty inherent in the fragility estimate due to the epistemuncertainties is reflected in the probability distribution ofF(s,Q)relative to the parametersQ. Exact evaluation of this distributionunfortunately requires nested reliability calculations~Der Ki-ureghian 1989!. Approximate confidence bounds can be obtainby first-order analysis in the manner described below.

The reliability index corresponding to the conditional fragiliin Eq. ~19! is defined as

b~s,Q!5F21@12F~s,Q!# (22)

whereF21(•) denotes the inverse of the standard normal cumlative probability. In generalb(s,Q) is less strongly nonlinear inQ than F(s,Q) is. Using a first-order Taylor series expansioaround the mean pointMQ , the variance ofb(s,Q) is approxi-mately given by

sb2~s!'¹Qb~s!SQQ¹Qb~s!T (23)

where¹Qb(s)5gradient row vector ofb(s,Q) at the mean pointand SQQ denotes the posterior covariance matrix. The gradivector¹Qb(s) is easily computed by first-order reliability analysis ~see Ditlevsen and Madsen 1996!. Bounds on the reliabilityindex can now be expressed in terms of a specified numbestandard deviations away from the mean. For example,b(s)6sb(s), whereb(s)5F21@12F(s)# denotes the mean61 SDbounds of the reliability index. Transforming these back intoprobability space, one obtains

$F@2b~s!2sb~s!#, F@2b~s!1sb~s!#% (24)

as ‘‘1 SD’’ bounds of the fragility estimate. These bounds aproximately correspond to 15% and 85% probability levels.

Applications

Large uncertainty is inherent in predicting the capacity of Rstructural components under repeated cyclic loading~Park andAng 1985!. At the same time, a large body of valuable expemental data is available that has not been fully utilized. Thfacts have motivated us to employ the methodology presentethe previous sections to develop deformation and shear capmodels for RC circular columns under cyclic loading. This spcific class of structural component is selected because of tpredominant use for bridge structures in many seismically acregions of the world.

As described in ‘‘Capacity Models’’, the probabilistic modecan be built upon existing deterministic models. The deformatcapacity model used in this study is based on the notion of


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Table 2. Ranges of Variables from Database

Variable Symbol Range

Compressive strength of concrete~MPa! f c8 18.9– 42.2Yield stress of longitudinal reinforcement~MPa! f y 207– 607Ultimate strength of longitudinal reinforcement~MPa! f su 396–758

Yield stress of transverse reinforcement~MPa! f yh 207– 607

Longitudinal reinforcement ratio~%! r l 0.53–5.50Volumetric transverse reinforcement ratio~%! rs 0.17–3.00Slenderness ratio H/Dg 1.09– 10.00Ratio of gross to core diameters Dg /Dc 1.05–1.31Axial load ratio 4P/pDg

2f c8 0.00–0.87

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composing the total displacement of the RC column into its bacomponents. Specifically, the column displacement is considto be composed of elastic and inelastic components, with the etic component itself consisting of contributions from the flexuand shear deformations and from the longitudinal reinforcingslip. For the shear capacity model, owing to the complex naturthe underlying load transfer mechanisms, a unique consemodel does not exist. Here, we consider two alternative determistic models used in practice and assess objective measurtheir relative qualities. The more accurate shear capacity modsubsequently used together with the deformation capacity mto formulate a bivariate deformation–shear capacity model.nally, the probabilistic capacity models are used to compunivariate fragility curves and bivariate fragility contours forrepresentative bridge column.

Experimental Data

The behavior of RC columns under the effect of repeated cyloading has been the focus of experimental research by a lnumber of investigators for many years. A large collectionthese experimental data is organized at the World Wide Webhttp://maximus.ce.washington.edu/;peera1/, where referencesthe original publications for each tested column are listed. Attime of this writing, the database contained the results of cylateral load tests on 134 circular or octagonal columns, 188 rangular columns, 11 retrofitted columns, and 4 spliced columfrom 74 different experimental studies conducted by 115 invegators. During these experiments, all columns were subjecteconstant axial loads. For the purpose of this study, out of thetested columns with circular or octagonal cross sections, we onally considered the first 117 that were available at the timethis database, columns 38, 69, 75, and 82 did not include sreinforcement, column 106 had some missing data, and colu108 and 111 were subjected to tensile axial load. These coluwere excluded in the present study. Furthermore, column 53selected as a sample column for the subsequent fragility anaand, hence, was also excluded. Following an initial analysis,data for columns 50, 51, and 52 were identified as outlierscause the reported test data appeared to be inconsistent withsonable predictions. These columns were also excluded fromther consideration. Thus, the analysis reported in this studbased on the data from the remaining 106 columns.

Reported in the database are the material properties and getry of each test column. The ranges of the important variablesthe considered columns are listed in Table 2. In this tableHrepresents the equivalent cantilever length~clear column height!andDg andDc are the gross and core column diameters, resp


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tively. For octagonal cross sections, the largest circle that canincluded in the cross section is used. The database reportsapplied constant axial loadP, the cyclic lateral load–deformationrelationships, and the mode of failure~shear, flexure, or combinedshear flexure! for all the tested columns. Since these are all labratory experiments, the measurement errors were judged tosmall in relation to the uncertainties in the models and wereglected.

Deformation Capacity Model

Following common practice~Park and Paulay 1975; Lynn et a1996!, we define the deformation capacity of a column asdisplacementD corresponding to a drop in the lateral force restance equal to 20% of its peak value. Fig. 2~a! illustrates thisdefinition for a cyclically loaded structural component.

The lateral load–deformation relationships from the expemental database were examined to determineD for each of the106 tested columns. Three categories of observations were itified. First are columns for which the lateral force resistancereached and followed by strength degradation up to the thresdrop of 20%. Fig. 2~b! represents a representative case. This tof observation is identified as ‘‘failure’’ datum. Second are cumns whose lateral force resistance is not reached becauspremature load reversal. Fig. 2~c! illustrates such a case, wherstiffness deterioration occurs without reaching the lateral foresistance. The measured displacement in this case~i.e., that cor-

Fig. 2. Deformation capacity definition and data types

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responding to 80% of the peak lateral load! is obviously a lowerbound to the deformation capacityD. This type of observation isidentified as a ‘‘lower bound’’ datum. Third are columns whodeformation capacity is not reached because of limitation inapplied maximum displacement. Fig. 2~d! illustrates such a caseObviously, the measured maximum displacement providelower bound to the deformation capacityD. This kind of obser-vation is also a ‘‘lower bound’’ datum.

To develop the deformation capacity model, we employdrift ratio capacityd5D/H. This is a dimensionless quantityconvenient for model formulation. Letd(x) be an existing deter-ministic model for predicting d, where x5( f c8 , f y , . . . ,H,Dg , . . . ,P) is the set of constituent materiageometry and load variables. Considering the non-negative naof the deformation capacity, the logarithmic variance-stabiliztransformation is selected among other possible transformatto formulate a homoskedastic model. Thus, we adopt the mform

ln@d~x,Q!#5 ln@ d~x!#1gd~x,u!1s« (25)

where Q5(u,s)5set of unknown model parameters angd(x,u), s, and«, are as described earlier. Because of the eployed logarithmic transformation, one can show thats is ap-proximately equal to the coefficient of variation~COV! of thedrift ratio. The following two subsections describe the formution used ford(x) andgd(x,u).

Deterministic Model

It is common practice to decompose the ultimate displacemcapacity of a column into two components: the elastic componDy due to the onset of yield, and the inelastic componentDp dueto the plastic flow, as illustrated in Fig. 3 for a single RC columbridge bent. Accordingly,

d~x!51

H~Dy1Dp! (26)

For a RC column responding as a cantilever~i.e., with a singlecurvature! with fixed base, the yield displacement is compriseda flexural componentD f based on a linear curvature distributioalong the full column height~Fig. 3!, a shear componentDsh dueto shear distortion, and a slip componentDsl due to the localrotation at the base caused by slipping of the longitudinalreinforcement. These three components are illustrated in FigThus, we have

Fig. 3. Decomposition of lateral displacement of single-columbridge bent

J. Eng. Mech. 2002.

e

sl

tt

.

Dy5D f1Dsh1Dsl (27)

Given the curvaturefy at yield, the flexural component of thdisplacement is given by

D f513 fyl eff

2 (28)

wherel eff5H1YP5effective length of the column, in which YPdenotes the depth of the yield penetration into the column b~Fig. 3!. The latter term accounts for the additional rotation of tcritical section resulting from strain penetration of the longitunal reinforcement into the column footing. According to Priestlet al. ~1996!, YP is estimated as YP50.022f ydb, wheredb5diameter of the longitudinal reinforcement having yield stref y , which must be expressed in units of MPa.

The shear deformation is obtained from the well-known epression

Dsh5VyH

GAve(29)

where Vy5shear force at yield,G5shear modulus of concreteand Ave5effective shear area. The latter is computed asAve

5kIksAg, whereAg5gross cross sectional area,ks50.95shapefactor for a circular cross section, and the factorkI reflects theincreased shear deformation in a flexurally cracked RC coluDue to lack of specific research data, it is usually assumed thareduction in shear stiffness is proportional to the reductionflexural stiffness ~Priestley et al. 1996! such that kI5I e /I g ,where I e5effective moment of inertia determined from thmoment–curvature relationship of the column cross sectiondemonstrated in Fig. 5, andI g5gross moment of inertia of thecross section.

The contribution to the yield deformation due to slippagethe longitudinal reinforcing bars is related to the local rotation

Fig. 4. Components of yield displacementDy for reinforced concretecolumn

Fig. 5. Generic moment–curvature diagram


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the base of the columnasl ~Fig. 4!. We adopt the assumptions bPujol et al.~1999!, whereby the bond stress at yield is uniformdistributed and is given bym51.08Af c8 when MPa units are usedand the column rotates about the neutral axis of the flexurcritical section when slip takes place. These assumptions leaasl5(fyf ydb)/(8m). Accordingly,

Dsl5fyf ydbH

8.64Af c8(30)

The contributions from postelastic flexural behavior, diagotension cracking, and yield penetration are manifested in thecalled plastic hinge rotationap , as shown in Fig. 3. An equivalent rectangular plastic curvature is commonly assumed~Priestleyet al. 1996!. Accordingly, the plastic deformationDp in Eq. ~26!is obtained from

Dp5apH5fpl pH (31)

where l p50.08H1YP>0.044f ydb5equivalent plastic hingelength~Priestley et al. 1996!, in which f y in the lower bound limitmust be expressed in units of MPa, andfp5fu2fy5plasticcurvature, wherefu denotes the ultimate curvature.

A generic moment–curvature relationship with elastiperfectly plastic idealization is illustrated in Fig. 5 showing tdefinitions of fy and fu . In this figure,M y5moment that in-duces the first yielding in the column longitudinal reinforcemefy85corresponding curvature, andMI5ideal ~theoretical! mo-ment capacity corresponding to the idealized yield curvaturefy .In experiments on RC columns conducted by Priestley and P~1987! and Watson and Park~1994!, spalling of the concretecover, which normally precedes the yielding of the longitudinreinforcement, was found to take place when«c>0.005, where«c

is the longitudinal compressive strain of the extreme concfiber. In the present studyMI and the correspondingfy are con-servatively determined by setting«c50.005. For each tested RCcolumn, we compute all the moments and the correspondingvatures defined above using fiber–element section ana~Thewalt and Stojadinovic 1994!.

From the elastic–perfectly plastic idealization in Fig. 5,fy isdetermined from the linear extrapolationEIe5M y /fy85MI /fy .On the other hand, the ultimate curvaturefu corresponds to«c

5«cu, where«cu accounts for the confining effects of the tranverse reinforcement. This is conducted using the energy balargument of Mander et al.~1988! leading to the conservative estimate

«cu50.00411.4rsf yh«su

f cc8(32)

wherers54Ah /DcS5volumetric ratio of the confining steel, inwhich Ah5cross-sectional area of the transverse reinforcemandS5longitudinal spacing of the hoops or spirals,«su5strain atthe maximum tensile stress of the transverse steel, which is cmonly taken as 0.12~Priestley et al. 1996!, and f cc8 5compressivestrength of the confined concrete, defined according to Manet al. ~1988! as

f cc8 5 f c8S 2.254A117.94f l8

f c822

f l8

f c821.254D (33)

where f l85Kef yhrs /25effective lateral confining stress, in whicKe50.955confinement effectiveness coefficient for circular setions.


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Model Correction

The termgd(x,u) on the right-hand side of Eq.~25! is intended tocorrect the bias inherent in the deterministic model ln@d(x)#. Weselect the linear form in Eq.~3! for this function, whereu5(u1 , . . . ,up) is a vector of unknown parameters anh1(x), . . . ,hp(x) are selected ‘‘explanatory’’ functions. To capture a potential bias in the model that is independent of the vablesx, we selecth1(x)51. To detect any possible under- ooverestimation of the individual contributions defined in Eqs.~26!and~27! to the total deformation, we select the next four explantory functions ash2(x)5D f /H, h3(x)5Dp /H, h4(x)5Dsh/H,andh5(x)5Dsl /H. Additional explanatory functions are selecteto capture the possible dependencies of the bias in ln@d(x)# ondifferent factors characterizing the behavior of the column.selecth6(x)5Dg /H to account for the possible effect of the apect~slenderness! ratio. To capture the possible effect of the idalized elastic–perfectly plastic shear forceVI5MI /H, we intro-duce h7(x)54VI /(pDg

2f t8), where f t850.5Af c8 in MPa units isthe tensile strength of concrete. To account for the possible inences of the confining transverse reinforcement and the corewe selecth8(x)5rs( f yh / f c8)(Dc /Dg). To explore the effect ofthe longitudinal reinforcement, we chooseh9(x)5r l f y / f c8 . Fi-nally, to capture the effects of the material properties we emph10(x)5 f y / f c8 andh11(x)5«cu. Note that these explanatory functions are all dimensionless. As a result, the parametersu are alsodimensionless. While one could select additional explanatfunctions or different forms of these functions, we believe tselected ones are sufficiently broad to capture all the factorsmay significantly influence the deformation capacity of RC cumns.

Parameter Estimation

Having defined the deterministic modeld(x) and the correctiontermgd(x,u), we are now ready to assess the probabilistic moin Eq. ~25!, i.e., estimate its parametersQ5(u1 , . . . ,u11,s) bythe Bayesian updating formula described in ‘‘Bayesian ParamEstimation’’. Having no prior information on these parametewe select the noninformative prior in Eq.~9!. We note that, giventhe large amount of observed data, any reasonable choice oprior has practically no influence on the posterior estimates ofparameters.

In ‘‘Model Selection’’ section, a stepwise deletion proceduwas described for reducing the number of terms ingd(x,u) toachieve a compromise between model simplicity~few correctionterms! and model accuracy~small s!. Sinces is approximatelyequal to the COV of the predicted drift ratio, the accuracy ofmodel is not expected to improve by including a term that haCOV much greater thans.

Fig. 6 summarizes the stepwise term deletion process fordeformation model. For each step, the figure shows the posteCOV of the model parametersu i ~solid dots! and the posteriormean of the model standard deviations ~open square!. At step 1with the complete 12-parameter model, the posterior mean ofs is0.306 and the parameter with the largest COV~50.71! is u9 . Tosimplify the model, we drop the termu9h9(x). This is indicatedby a cross symbol in Fig. 6. In step 2, we assess the redu11-parameter model. The posterior mean ofs now is 0.314,which indicates no appreciable deterioration of the model, andparameter with the highest COV now isu2 . We remove the termu2h2(x) and continue the same procedure. After eight steps,

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find the largest COV~for parameteru7! to be close in magnitudeto s. This is an indication that further reduction may deteriorthe quality of the model. Stopping at this step, we are left withterms u1h1(x), u7h7(x), u8h8(x), and u11h11(x). At this stagethe mean ofs is 0.379.

Analysis with the above reduced model reveals thatu8 andu11

are strongly correlated~r520.85!. Considering the definitions oh8(x) andh11(x) and Eq.~32!, a strong correlation between theparameters is not surprising. As a further simplification, usingposterior estimates of the four-parameter model,u8 , is expressedby its linear regression onu11 @see Eq.~17!# as u8526.03521.034u11. Thus, the reduced correction term takes the form

gd~x,u!5u11u7

4VI

pDg2f t8

1~26.03521.034u11!rsf yhDc

f c8Dg

1u11«cu (34)

with only three unknown parameters.Table 3 lists the posterior statistics of the parametersQ

5(u1 ,u7 ,u11,s) of the reduced model. The following observtions are noteworthy:~1! The positive mean ofu1 indicates that,independent of the variablesx, the deterministic modeld(x)tends to underestimate the deformation capacity of the colu~2! The positive estimate ofu7 indicates that the deterministimodel tends to underestimate the effect of the idealized sforce VI ~corresponding toMI!. This is expected in view of theconservative assumption regarding the concrete strain in d

Fig. 6. Stepwise deletion process for deformation capacity mowhere a superposed cross~3! indicates term to be removed

J. Eng. Mech. 2002.

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mining MI andfy , as described in ‘‘Deterministic Model’’. Thenegative estimate ofu11 indicates that the deterministic modetends to underestimate the contribution of transverse reinfoment and overestimate the contribution of the ultimate concstrain.

Fig. 7 shows a comparison between the measured anddicted values of the drift ratio capacities for the test columbased on the deterministic~left chart! and the probabilistic~rightchart! models. For the probabilistic model, median predictio(«50) are shown. The failure data are shown as solid dotsthe censored data are shown as open triangles. For a pemodel, the failure data should line up along the 1:1 dashedand the censored data should lie above it. The deterministic mon the left is strongly biased on the conservative side since mof the failure and many of the censored data lie below theline. The probabilistic model on the right clearly corrects thbias. The dotted lines in the right figure delimit the region with1 SD of the median. We note that a majority of the failure dpoints fall within 1 SD limits and that most of the censored daare above the 1:1 line. While the conservatism inherent indeterministic deformation capacity model might be approprifor a traditional design approach, for a performance-based demethodology unbiased estimates of the capacity are essentialconstructed probabilistic model is unbiased and properly accofor all the prevailing uncertainties.

Shear Capacity Models

In this section we construct and compare two probabilistic shcapacity models for RC circular columns. For this purpose,classify the maximum lateral load measured in each experimas a ‘‘failure’’ datum if the tested column failed in shear, and alower-bound ‘‘censored’’ datum if the tested column failed

Table 3. Posterior Statistics of Parameters in Deformation Mode

Parameter MeanStandarddeviation

Correlation Coefficient

u1 u7 u11 s

u1 0.531 0.119 1u7 0.701 0.204 20.37 1u11 248.4 13.6 20.59 20.38 1s 0.383 0.050 20.04 0.14 0.20 1

Fig. 7. Comparison between measured and predicted drift ratio capacities based on deterministic~left! and probabilistic~right! models


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flexure or in a combined flexural–shear failure mode. In thetabase used for this analysis, 57 out of the 106 tested columnin the latter data category.

To develop a dimensionless model for the shear force capaV, we consider the normalized quantityn5V/(Agf t8), whereAg

is the gross cross-sectional area andf t850.5Af c8 is the tensilestrength of concrete in MPa units. Owing to the non-neganature of the shear capacity, we choose the logarithmic variastabilizing transformation to make the model homoskedasThus we adopt the model form

ln@n~x,Q!#5 ln@ n~x!#1gn~x,u!1s« (35)

where Q5(u,s)5set of unknown model parametern(x)5existing deterministic model for predictingn, andgn(x,u),s , and« are as defined earlier. Due to the logarithmic transfmation used,s is approximately equal to the coefficient of varition of n. The following subsections describe the formulationsn(x) andgn(x,u).

Deterministic Models

Two predictive models for the shear capacity of RC columnsconsidered. The first model was proposed by the ASCE–AJoint Task Committee 426~1973! and is widely used in practiceThe second model, proposed by Moehle et al.~1999, 2000!, is arefinement of the Federal Emergency Management Agency~FEMA 1997! model.

The ASCE–ACI model is based on the well-known approaof considering the shear capacity as the sum of a contribufrom the concreteVc and a contribution from the transverse steVs , i.e.,

V5Vc1Vs (36)

whereV denotes the deterministic prediction of the shear capa~often denoted nominal capacityVn!. According to Priestly et al.~1996!, in circular bridge columns the contribution from the cocrete is governed by the shear force required to initiate flexushear cracking and can be expressed as

Vc5nbAe1Md

a(37)

wherenb5(0.067110r t)Af c8<0.2Af c8 with units of MPa is the‘‘basic’’ shear strength of concrete, in whicr t50.5r l5longitudinal tension reinforcement ratio,Ae5effectiveshear area taken as 0.8Ag for circular sections~Priestly et al.1996!, Md5PIg /(Agyt)5PDg/85decompression moment witthe axial loadP and yt5Dg/2, and a5M /V5shear span ex-pressed as the ratio of the moment to shear at the critical se~for a cantilever columna5H!. The contribution from the transverse steel in Eq.~36! is based on the well-known truss model ais given by

Vs5An f yhDe

S(38)

whereAn52Ah5total area in a layer of the transverse reinforcment in the direction of the shear force,De5effective depth com-monly taken as 0.8Dg for circular cross sections, andS5spacingof transverse reinforcement. Substituting Eqs.~37! and~38! in Eq.~36!, we have

V50.8nbAg10.125PDg

H11.6

Ahf yhDg

S(39)


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The second deterministic model, proposed by Moehle e~1999, 2000!, is a refinement of the FEMA 273~1997! model.This model accounts for the reduction in the shear strength duthe effects of flexural stress and redistribution of internal forcescracking develops. Here the concrete contributionVc is obtainedby setting the principal tensile stress in the column equal tof t8 .Considering the stress transformation in Fig. 8~a!, interaction be-tween flexural and shear stresses, and strength degradation wthe plastic hinge, the final form of this model is

Vc5kS f t8

a/DeA11

P

f t8AgD Ae (40)

where the aspect ratioa/De is limited to the range 1.7–3.9~Moe-hle et al. 2000! and k is a factor included to account for thstrength degradation within the plastic hinge region as a funcof the displacement ductilitymD5D/Dy5d/dy , as defined in Fig.8~b!.

We consider two alternatives for the drift ratio to be usedthe expression formD . The first is simplyd5d(x), the determin-istic deformation model described in ‘‘Deterministic Model’’. Thsecond is the median of the probabilistic deformation capamodel,d5d(x)exp@gd(x,MQ)#, where the parameters are fixedthe posterior mean estimates,MQ . In the two papers by Moehleet al. ~1999, 2000!, different approaches are proposed to intrduce the effect of strength degradation onVs . In Moehle et al.~1999!, a reduction factor of 0.5 is applied toVs , whereas inMoehle et al.~2000! Vs is reduced by the factork. In this paper,we do not introduce any reduction factor forVs in the determin-istic model. However, we assess possible modification ofVs dueto strength degradation through a properly selected explanafunction in the model correction term, as discussed in the folloing subsection.

Model Correction

The model correction termgn(x,u) in Eq. ~35! is intended tocapture the inherent bias in ln@n(x)#. We select the linear form inEq. ~3!. To capture a potential constant bias in ln@n(x)#, we selecth1(x)51. To account for a possible correction in the contributiof the longitudinal steel we selecth2(x)5r l , and to account forany correction in the effect of the axial load we selecth3(x)5PDg /(Agf t8H). Furthermore, to account for any modificatioin the contribution from the transverse steel, we selecth4(x)5An f yhDg /(Agf t8S). Note that the explanatory functions aagain dimensionless, making the parametersu i also dimension-less. While additional or different explanatory functions couldselected, we believe that the above functions capture the msignificant factors that may influence the shear capacity ofcolumns.

Fig. 8. Shear failure model by Moehle et al.~1999, 2000!

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Table 4. Reduced Model Correction Terms and Posterior Means and Standard Deviations ofs Selected for Shear Models

Deterministic shear capacity model gn(x;u) Mean ofs Standard deviation ofs

ASCE–ACI 426 model u11u2h2 0.189 0.019

Deformation-dependent model withd5d(x) used incomputingk

u2h21u4h4 0.179 0.019

Deformation-dependent model withd5d(x)exp@gd(x;MQ)#used in computingk

u2h21u4h4 0.153 0.013

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Parameter Estimation

We now estimate the parametersQ5(u1 , . . . ,u4 ,s) for theshear capacity models using Bayesian inference. Owing tolack of prior information, the noninformative priorp(Q)}s21 isselected. The stepwise deletion procedure described in‘‘Model Selection’’ section is used to detect superfluous explatory functions, which are then dropped to simplify each modThe reduced form ofgn(x,u) is different for the two deterministicshear capacity models. Furthermore, the parameter estimatepend on whetherd5d(x) or d5d(x)exp@gd(x,MQ)# is used tocompute the factork. Table 4 lists the expression ofgn(x,u) andthe posterior mean and standard deviation ofs for each model.

As mentioned earlier, a measure of the predictive accuraceach model is the posterior mean ofs. The last two columns ofTable 4 show that the deformation-dependent shear capamodel withd5d(x)exp@gd(x;MQ)# used in computing the factok has the smallest error standard deviation and, therefore, ismost accurate model. In the remainder of this paper, we preresults only for this model.

Fig. 9 summarizes the stepwise deletion process for thelected shear capacity model. For each step, the figure showCOV of the model parametersu i ~solid dots! and the mean of themodel standard deviations ~open square!. At step 1 with thecomplete five-parameter model, the COV ofu3 is 0.340 and themean ofs is 0.136. To simplify the model, we drop the teru3h3(x). In step 2 for the reduced four-parameter model,mean ofs is 0.144, which indicates no appreciable deterioratof the model. The parameter with the highest COV~50.348! nowis u1 . We remove the termu1h1(x) and continue the same procedure. At step 3 we find the largest COV~for parameteru4! to beof the same order of magnitude as the mean ofs, which indicatesthat further simplification is not justified. Thus, the reduced mocorrection term is

gn~x,u!5u2r l1u4An f yhDg /Agf t8S (41)

Fig. 9. Stepwise deletion process for shear capacity model, whsuperposed cross~3! indicates term to be removed

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Table 5 lists the posterior statistics of the parametersu2 , u4 , ands for the reduced model.

The following noteworthy observations can be made frompreceding results:~1! the fact that the explanatory functionh1

51 is not informative indicates that there is no constant biasthe deterministic model.~2! The presence ofh2(x)5r l with apositive coefficient in Eq.~41! is an indication that the contribution of the longitudinal reinforcement to the shear capacityunderestimated in the deterministic model. This could be duethe fact that the deformation-dependent model was calibrausing rectangular column data~Moehle et al. 1999!, for which thecontribution of longitudinal reinforcement is known to be leimportant than that for circular columns. Interestingly, for tASCE–ACI 426 model, which includes a term representingcontribution ofr l , the posterior mean ofu2 is 4.43, which is farsmaller than the estimate 23.1 for the selected model.~3! The factthat the explanatory functionh3(x)5PDg /(Agf t8H) is not infor-mative is an indication that the effect of the axial force is accrately accounted for in the deterministic model through the traformation of stresses by the Mohr’s circle@Fig. 8~a!#. ~4! Thepresence of the explanatory functionh4(x)5An f yhDg /(Agf t8S)with a negative coefficient in Eq.~41! represents the effect ostrength degradation in the contribution from the transverse sthat is needed for a more accurate prediction.

Fig. 10 shows a comparison between the measured anddicted values of the normalized shear capacities for the testumns based on the deterministic~left chart! and probabilistic~right chart! shear capacity models. Same definitions as in Figapply. It is seen that the deterministic model on the left is stronbiased on the conservative side. The probabilistic model onright clearly corrects this bias. The dotted lines in this figudelimit the region within 1 SD of the model.

Bivariate Deformation–Shear Capacity Model

In this section we construct a bivariate deformation–shear caity model that accounts for the correlation between the two moerrors. The subscriptsd and n are used to indicate quantitierelated to the deformation and shear capacities, respectivUsing Eq.~25! and~35!, the bivariate capacity model is written a

ln@d~x,ud ,sd ,r!#5 ln@ d~x!#1gd~x,ud!1sd«d (42a)

Table 5. Posterior Statistics of Parameters in Selected Shear Mo

Parameter Mean Standard deviation

Correlation Coefficient

u2 u4 s

u2 23.1 1.2 1u4 20.614 0.120 20.87 1s 0.153 0.013 20.13 0.06 1


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Fig. 10. Comparison between measured and predicted shear capacities based on deterministic~left! and probabilistic~right! models

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ln@n~x,un ,sn ,r!#5 ln@ n~x!#1gn~x,un!1sn«n (42b)

where all terms are as defined earlier andr5correlation coeffi-cient between the model errors«d and«n . The unknown modelparametersQ5(ud ,sd ,un ,sn ,r) are estimated by Bayesian inference. Having already determined the informative explanafunctions for each model, the reduced model correction termEqs.~34! and ~41! are used. Owing to the lack of prior information, we select the noninformative prior p(Q)}(12r2)23/2/(sd sn).

Table 6 lists the posterior statistics of the parametersQ. Asexpected, the estimates ofud , un , sd , and sn are nearly thesame as the corresponding estimates for the individual modThe negative sign of the posterior mean of the correlation coecient shows that the deformation and shear capacities are ntively correlated. This indicates that, relative to their median vues, a column with high deformation capacity is likely to havelow shear capacity, and vice versa.

Univariate and Bivariate Fragility Estimates

The probabilistic deformation and shear capacity models deoped in the preceding sections can be used to assess the frafunction for any circular column with specified geometry, marial properties, and applied compression force. ‘‘AssessmenStructural Component Fragility’’ describes the computatioframework for this purpose. An example column with geomeand material properties that are representative of currently cstructed RC highway bridge columns in California is conside


J. Eng. Mech. 2002.

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a-

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f

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~Naito 2000!. The column has the longitudinal reinforcement rar l51.99%, gross diameterDg51,520 mm, and ratio of gross tocore diametersDg /Dc51.07, clear heightH59,140 mm, andvolumetric transverse reinforcement ratiors50.65% with yieldstress of transverse reinforcementf yh5493 MPa. To account formaterial variability, we assume the compressive strength of ccretef c8 is lognormally distributed with mean 35.8 MPa and 10COV, and the yield stress of longitudinal reinforcementf y is log-normal with mean 475 MPa and 5% COV. To account for vaability in the axial load, we assumeP is normally distributed withmean 4,450 kN~corresponding to 7% of the axial capacity bason the gross cross-sectional area! and 25% COV. Finally, to ac-count for variability in construction, we assume the effective mment of inertiaI e is lognormal with mean 2.12631011 mm4 and10% COV.

As defined in ‘‘Assessment of Structural Component Fragity’’, fragility is the conditional probability of failure given one omore measures of demands5(s1 ,s2 , . . . ) . Thepredictive fra-

gility estimateF(s) is the expected fragility estimate with respeto the distribution of the model parametersQ. This estimate ac-counts for the effect of epistemic uncertainties~uncertainty in themodel parameters! in an average sense. An explicit account of tvariability in the fragility estimate due to epistemic uncertaintiis provided by confidence bounds at specified probability levIn the following, we present these estimates for the exampleumn.

Figs. 11 and 12, respectively, show the univariate fragicurves with respect to drift ratio demandd and normalized sheademandn for the example column. The solid lines represent

.166

Table 6. Posterior Statistics of Parameters in Bivariate Deformation–Shear Model

ud,1 ud,7 ud,11 sd un,2 un,4 sn r

Mean 0.512 0.828 -50.8 0.383 21.6 -0.584 0.189 20.535Standard deviation 0.054 0.198 11.5 0.048 1.4 0.180 0.019 0

Correlation coefficientsud,7 20.38ud,11 20.60 20.51sd 20.41 0.36 0.07un,2 0.02 20.10 0.07 20.06un,4 0.12 0.00 20.11 20.01 20.84sn 0.06 0.05 20.10 0.06 0.02 0.03r 0.14 20.29 0.12 20.26 0.12 0.01 20.15

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predictive estimatesF(d) andF(n) and the dashed lines indicatthe 15 and 85% confidence bounds. The dispersion indicatethe slope of the solid curve represents the effect of aleatorycertainties~those present inf c8 , f y , P, I e , «d , and«n! and thedispersion indicated by the confidence bounds represents thfluence of the epistemic uncertainties~those present in the modeparametersQ!.

The bivariate deformation–shear fragility estimates for theample column are obtained using the bivariate capacity modeveloped in the ‘‘Bivariate Deformation–Shear CapacModel’’ section. The fragility in this case is defined as failurethe column, in either deformation or shear mode, for a given pof deformation and shear demands. Fig. 13 shows the contourof the predictive fragility surfaceF(d,n) in terms of the drift ratiodemandd and normalized shear demandn. Each contour in thisplot connects pairs of values of the demandsd andn that give riseto a given level of fragility in the range 0.1–0.9. Significant iteraction between the two failure modes, particularly at highmand levels, is observed.

Conclusions

A comprehensive Bayesian framework for constructing univarand multrivariate predictive capacity models based on experimtal observations is formulated. The models are unbiased and

Fig. 11. Fragility estimate for deformation failure of exampcircular reinforced concrete column

Fig. 12. Fragility estimate for shear failure of example circulreinforced concrete column

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plicitly account for all the prevailing uncertainties. With the aiof facilitating their use in practice, the probabilistic models aconstructed by developing correction terms to existing determistic models in common use. Through a model selection procthat makes use of a set of ‘‘explanatory’’ functions, terms theffectively correct the bias in the existing model forms are idetified and insight into the underling behavioral phenomenagained.

Although the methodology presented in this paper is aimeddeveloping probabilistic capacity models for structural compnents, the approach is quite general and can be applied toassessment of models in many engineering fields. As an apption, univariate and bivariate probabilistic models for the defmation and shear capacities of RC circular columns subjectecyclic loading are developed using existing experimental data

The probabilistic capacity models are used in a formulationassess the fragility of structural components, with due consiation given to the different natures of aleatory and epistemiccertainties. Point estimates of the fragility based on posteriortimates and predictive analyses, as well as confidence intervafragility that reflect the influence of epistemic uncertainties,presented. This methodology is used to estimate univariate fraity curves and bivariate fragility contours for an example Rbridge column. The effect of the cyclic nature of the loadingimplicitly accounted for through the experimental data used.

Acknowledgments

This work was supported primarily by the Earthquake Engineing Research Centers Program of the National Science Foution under Award No. EEC-9701568 to the Pacific EarthquaEngineering Research~PEER! Center at the University of Cali-fornia, Berkeley. Opinions and findings presented are those ofwriters and do not necessarily reflect the views of the sponsoPEER.

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Fig. 13. Contour plot of predictive deformation-shear fragilitsurface of example circular reinforced concrete column


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