Engineering Structures 74 (2014) 13–22

Contents lists available at ScienceDirect

Engineering Structures

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

Probabilistic capacity models and fragility estimates for RC columnsretrofitted with FRP composites

http://dx.doi.org/10.1016/j.engstruct.2014.05.0050141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: tabande2@illinois.edu (A. Tabandeh), gardoni@illinois.edu

(P. Gardoni).

Armin Tabandeh ⇑, Paolo GardoniDepartment of Civil and Environmental Engineering, University of Illinois, 3118 Newmark Civil Engineering Laboratory, 205 N. Mathews Ave., Urbana, IL 61801, United States

a r t i c l e i n f o

Article history:Received 15 May 2013Revised 8 May 2014Accepted 8 May 2014Available online 28 May 2014

Keywords:FRP retrofitBridge columnProbabilityMarkov Chain Monte Carlo (MCMC)Bayesian approachFragility

a b s t r a c t

This paper proposes a probabilistic formulation to assess the effectiveness of the fiber reinforced polymer(FRP) retrofit schemes in enhancing the structural performance of reinforced concrete (RC) bridgecolumns. Two probabilistic models are proposed to predict the deformation capacities of retrofittedcolumns. One deformation model corresponds to the flexural failure and the other considers the bondfailure in the lap-splice region. A Markov Chain Monte Carlo (MCMC) simulation method is used to esti-mate unknown model parameters in the context of a Bayesian updating approach. The probabilisticcapacity models are used to estimate the fragility curves of three example columns. In this paper, fragilityis defined as the conditional probability of failure for given deformation demand. The results compare thecolumn fragilities before and after the application of the retrofit measure. The results from the examplecolumns indicate that the use of FRP composites considerably reduced the fragility for the bond failuremode and is also beneficial but with a moderate impact when considering the flexural failure.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction experiencing a significant lateral deformation. Providing additional

Around the world a number of reinforced concrete (RC) struc-tures have been built with little or no consideration to seismicloading in part because the region was believed to be of low seis-micity or due to inadequate design specifications [1]. However,more recent events have shown that regions that were tradition-ally considered to be of low seismicity might actually experiencelarge earthquakes. For example, the May 2012 Emilia Romagnaearthquakes, struck the Northern Italy, with two main shocks,Mw � 5:8 and Mw � 5:9, and a sequence of aftershocks in a regionthat was traditionally believed to be of low seismicity [2]. Likewise,the September 2010, Mw � 7:1, and the February 2011, Mw � 6:2,earthquakes near Christchurch, New Zealand, occurred on previ-ously unknown faults in a region historically considered to be oflow seismicity [3]. Consequences of past events have raised con-cerns about the vulnerability of existing structures and the needto undertake seismic retrofit programs.

Deficiencies associated with vulnerable structures includewidely spaced transverse reinforcements, inadequate confinementof the core concrete, and short lap splices. In such structures, a pre-mature failure may occur when the column loses its structuralintegrity before reaching its lateral load carrying capacity and

concrete confinement (e.g., retrofitting with fiber reinforced poly-mer (FRP) composites or steel jackets) increases the ductilitycapacity and can avert or delay the formation of undesirable brittlefailure modes. Retrofit with FRP composites has become one of thepreferred retrofitting schemes because of (1) their favorable prop-erties (such as high strength/weight ratio and corrosion resistance[4–6]); (2) the fact that they increase the ductility and strength,and improve the concrete-reinforcement bond behavior withoutchanging the structural stiffness [5–10]; (3) their ease of installa-tion [5,6,10,11]; and (4) their lower expected cost in comparisonto the traditional retrofitting strategies [6]. However, a cost-benefitanalysis, that justifies the use of FRP composites as the retrofitmeasure of existing RC bridges, requires quantification of thereduction in failure probability of the retrofitted bridges. Estimat-ing the reduction in the probability of failure requires a probabilis-tic formulation that accounts for the relevant uncertainties in thematerial properties, geometry, and structural behavior. The proba-bilistic assessment of un-retrofitted RC bridges has already beenthe subject of numerous studies (e.g., [12–20]). In addition, a fewauthors have also studied the reliability of flexural members (i.e.,beams) retrofitted with FRP composites (e.g., [21–24].) However,the current approaches have four main limitations: (1) the model-ing and estimation are structure-specific. Consequently, the resultof the assessment for a particular structure cannot be used toassess the performance of another structure; (2) the modeling iscarried out at the system level (i.e., the full structure.) Therefore,

Table 1Ranges of variables from database.

Variable Symbol Range

Compressive strength of concrete (MPa) f 0c 16.7–44.8Yield stress of longitudinal reinforcement (MPa) fy 300–481Yield stress of transverse reinforcement (MPa) fyh 210–492FRP tensile strength (MPa) fFRP 425–4433Lap-splice length (mm) ls 305–760Longitudinal reinforcement ratio (%) ql 1.23–5.79FRP volumetric ratio (%) qf 0.03–10.49Slenderness ratio H=Dg 1.5–6Axial load ratio 4P= pD2

g f 0c� �

0.05–0.34

14 A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22

the estimates of vulnerability cannot take advantage of experimen-tal test data that are normally available for the structural compo-nents (i.e., individual members); (3) the available approachesgenerally assume an arbitrary function (often a normal or lognor-mal cumulative distribution function) to express the vulnerabilityas a function of some intensity measures of the loading and simplyestimate the function parameters. As a result the parameters haveno direct physical interpretation; (4) these approaches do notproperly account for all the uncertainties that are involved.

This paper proposes two novel deformation capacity models forFRP retrofitted RC columns that account for the relevant uncertain-ties. One deformation model corresponds to the flexural failure andthe other considers the bond failure in the lap-splice region. Basedon experimental data [5,6,25,26], the flexural failure and the con-crete-reinforcement bond failure are found to be the two typicalfailure modes of RC columns retrofitted with FRP composites undercyclic loading. The proposed probabilistic capacity models aredeveloped by combining information from existing mechanisticmodels with information about the performance of FRP retrofittedcolumns from laboratory test data. A Bayesian approach is used toestimate unknown model parameters using the test data. One ofthe distinctive features of the Bayesian approach, used in thispaper, is the ability to update existing models with new informa-tion when new data become available. The Bayesian updating isimplemented using an efficient Markov Chain Monte Carlo(MCMC) simulation method [27]. The proposed models identifythe most influential parameters affecting the capacity of retrofittedcolumns. As an application of the developed capacity models, theconditional failure probability (fragility) of three example columns,before and after retrofit, are developed for given values of thedeformation demand, expressed in terms of drift ratio. The fragilitycurves objectively quantify the differences in the reliability andshow the effectiveness of FRP composites as a retrofitting strategy.As a result, the developed fragility curves provide valuable infor-mation that can facilitate the decision-making process for the seis-mic retrofit of bridge columns.

Following this introduction, the next section discusses the exper-imental data used for the Bayesian updating. Next, we present theproposed probabilistic capacity models, including a discussionabout the general formulation, variance stabilizing transformations,model calibration and model selection. Finally, the paper presentsthe assessment of the structural fragility of RC columns retrofittedwith FRP composites and, as an illustration, the fragilities of threeexample columns, before and after retrofit, are compared.

2. Data for constructing probabilistic capacity models

For developing the proposed capacity models, we use data fromexperimental tests on FRP retrofitted columns available in the lit-erature [5,6,25,26]. The data include the hysteretic responses of41 FRP retrofitted columns with circular cross sections, out ofwhich 29 columns failed in flexure (8 columns with a lap-spliceat the column base and 21 with no lap-splice), 11 columns experi-enced bond failure in the lap-splice region, and 1 test on a columnwith no lap-splice was stopped before failure due to the limitationof the test rig displacement. For each column, the data also includethe applied axial load, the column geometry and material proper-ties. Table 1 gives the range of the relevant parameters, where His the equivalent cantilever length (clear column height), Dg isthe gross column diameter and P is the applied constant axial load.

3. Formulation of probabilistic capacity models

To assess the ability of a structure to withstand future hazards,we first need to estimate the structure’s capacity. Gardoni et al.

[12] proposed a general formulation for developing probabilisticand unbiased capacity models. In order to facilitate its applicationin practice and to account for the current available knowledge, themodels are developed starting from accepted deterministic expres-sions. Correction terms are then introduced to correct for theinherent bias and a model error term is added to capture theremaining variability in the residuals due to, for example, inaccu-racy of the model form, missing variables, and statistical uncertain-ties. The general formulation of the univariate probabilisticcapacity model is as follows:

Ck x;Hkð Þ ¼ ck xð Þ þ ck x; hkð Þ þ rkek k ¼ f ; b ð1Þ

In this expression, Ck x;Hkð Þ is the capacity measure of interest;the index k denotes the failure mode of interest, (i.e., k ¼ f for theflexural failure and k ¼ b for the bond failure); x is a set of measur-able variables including material properties, member dimensions,and imposed boundary conditions; Hk ¼ hk;rkð Þ is the set ofunknown model parameters that need to be estimated; ck xð Þ isan existing deterministic capacity model; ck x; hkð Þ is the correctionterm for the bias inherent in the deterministic model; and rkek isthe model error, in which rk is the standard deviation of the modelerror which is assumed to be independent of x (homoskedasticityassumption) and ek is a standard normal random variable (normal-ity assumption). The normality and homoskedasticity assumptionsand the assumption of an additive form of Eq. (1) typically can besatisfied using an appropriate variance stabilizing transformation.For this purpose, [28] suggested a parameterized family of trans-formations where a positive variable dk (i.e., the drift ratio capac-ity) transforms to Ck ¼ dk

k. This family of transformations iscontrolled by the parameter k. To satisfy the homoskedasticity,normality and additive assumptions, we use a logarithmic trans-formation to produce the variables Ck x;Hkð Þ and ck xð Þ. The suitabil-ity of the logarithmic transformation (i.e, k ¼ 0) is also verified bymeans of diagnostic plots (e.g., [29]).

According to Gardoni et al. [12], the following functional form,linear in hk, is typically sufficiently flexible for writing ck x; hkð Þ:

ck x; hkð Þ ¼Xp

i¼1

hk;ihk;i xð Þ ð2Þ

where hk;i xð Þ ’s are a set of explanatory functions obtained using anappropriate Box–Cox transformation of basis functions of x; gk;i xð Þ(i.e., hk;i ¼ gkk;i

k;i xð Þ in which Kk ¼ kk;1; . . . ; kk;p

� �is a vector of

unknown exponents), and hk;i ’s are the components of the vector hk.Further, we construct a bivariate model based on the univariate

models in Eq. (1) that also accounts for the correlation coefficient,q, between ef and eb as

Cf x; hf ;rf ;q� �

¼ cf xð Þ þ cf x; hf� �

þ rf ef

Cb x; hb;rb;qð Þ ¼ cb xð Þ þ cb x; hbð Þ þ rbeb

(ð3Þ

where H ¼ Hf ;Hb;q� �

and K ¼ Kf ;Kb� �

are the vectors of allunknown parameters.

Table 2Probability terms for bivariate capacity model (adapted from [12]).

Capacity model l Capacity model k

Failure datum Lower bound

Failure datum 1rkjl

u rk;q hkð Þ�lkjlrkjl

h i1rl

u rl;q hlð Þrl

h iU � rk;q hkð Þ�lkjl

rkjl

h i1rl

u rl;q hlð Þrl

h iLower bound U � rl;q hlð Þ�lljk

rljk

h i1rk

u rk;q hkð Þrk

h i R1rl;q

U � rk;q hkð Þ�lkj1rkjl

h i1rl

u 1rl

h id1

Note: lkjl ¼ q rk=rlð Þrl;q; rkjl ¼ rk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� q2

p; k; l ¼ f ; b; and lkj1 ¼ q rk=rlð Þ1.

A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22 15

3.1. Calibration of Box–Cox transformation, Bayesian updating, andmodel selection

We can find Kk and Hk simultaneously with a nonlinear regres-sion, treating Kk and Hk in the same way. In this case Kk and Hk arere-estimated at each step in the model selection process (discussedlater in this section). However, providing accurate estimation of Kk

and Hk simultaneously can be computationally challenging. Fur-thermore, the increased number of unknown parameters (consid-ering Kk in addition to Hk) might deteriorate the accuracy of theestimations (e.g., [30]). To circumvent such issues, we propose tofind Kk first, to satisfy the normality assumption, and then use apoint estimate, at the closest rounded value in the 95% confidenceinterval, when re-estimating Hk in the model selection process.This approach is faster than, and the estimates are comparableto, those obtained using the nonlinear regression.

To estimate Kk, we use the multivariate generalization of theBox–Cox method to transform all gk;i xð Þ ’s simultaneously to jointlynormal functions hk;i xð Þ ’s. Following Velilla [31], we use the max-imum likelihood criterion in this paper to make inferences aboutthe parameters Kk. The likelihood function in this method can bewritten, following Weisberg [32], as

L Kk;Mk;Vkð Þ /Yn

q¼1

1

2p Vkj jð Þ1=2

� exp �12

hk xq� �

�Mk

� �T V�1k hk xq

� ��Mk

� �� ð4Þ

where Mk and Vk are the mean vector and covariance matrix of the

random functions hk xq� �

¼ gkk;1k;1 xq� �

; . . . ;gkk;p

k;p xq� �h iT

. Estimates of Mk

and Vk can be obtained by maximizing L Kk;Mk;Vkð Þ and have thefollowing forms [32]:

cMk Kkð Þ ¼ 1n

Xn

q¼1

hk xq� �

bVk Kkð Þ ¼ 1n

Xn

q¼1

hk xq� �

�cMk Kkð Þh i

hk xq� �

�cMk Kkð Þh iT

� 8>>>><>>>>: ð5Þ

where cMk Kkð Þ and bVk Kkð Þ are the sample mean and the samplecovariance matrix, respectively. Substituting these estimates intothe Eq. (4) and rearranging terms, we find the log-likelihood as

ln L Kkð Þ½ � / � n2

ln 2pð Þ � n2

ln bVk Kkð Þ � �

� n2

ð6Þ

This equation can be maximized to obtain the estimates of Kk byminimizing the determinant of bVk Kkð Þ over values of Kk.

Once we found the Kk, we can transform the explanatory func-tions and move forward to estimate Hk. In the Bayesian approach,the unknown model parameters, Hk, are estimated using the fol-lowing updating rule [30]:

f Hkð Þ ¼ nL Hkð Þp Hkð Þ ð7Þ

where f Hkð Þ is the posterior distribution that reflects the updatedstate of information about Hk; L Hkð Þ is the likelihood function thatcaptures the information from the data; p Hkð Þ is the prior distribu-tion that represents the information available before collecting thedata; and n ¼

RL Hkð Þp Hkð ÞdHk

� ��1 is a normalizing factor. TheBayesian inference relies on the feasibility of computing the poster-ior statistics. If L Hkð Þp Hkð Þ not being proportional to a familiar prob-ability density function (PDF), computing n can be challenging inthe case of multidimensional problems. In this paper, we use anadaptive delayed rejection MCMC simulation method, also knownas the DRAM method [27], to obtain the posterior statistics of theunknown model parameters.

In writing L Hkð Þ, data need to be divided into equality datawhen the capacity of interest is observed and lower bound data

(or censored data) when lower values are observed and not theactual capacity of interest. For instance, when a column cannotbe pushed up to the failure point, due to the limitation of the strokeof the actuator, what we observe in the test is a lower bound capac-ity of the column and not the actual capacity. Additional detailabout the treatment of censored data can be found in [12]. Forthe flexural model, there are 29 equality data and 12 lower bounddata. Similarly, for the bond model, there are 11 equality data and8 lower bound data. Under the assumption of statistically indepen-dent observations, L Hkð Þ can be written, following Gardoni et al.[12], as

L Hkð Þ /Y

failure datum q

1rk

urk;q hkð Þ

rk

� ��

�Y

lower bound datum q

U � rk;q hkð Þrk

� �ð8Þ

where, rk;q hkð Þ ¼ Ck;q � ck xq� �

� ck xq; hk

� �is the prediction’s residual

for the observation q. In the case of bivariate capacity model, thelikelihood function, L Hkð Þ, for each observation takes the formshown in Table 2 (adapted from Gardoni et al. [12]). When thereis no prior information about the distribution of unknown modelparameters, we use a noninformative prior distributionp Hkð Þ / 1=rk for univariate models and p Hð Þ / 1= qrf rb

� �for the

bivariate model [12].Inclusion of unimportant terms in ck x; hkð Þ might lead to a loss

of precision in the estimates, larger values of rk, and over-fittingthe data. A parsimonious form of ck x; hkð Þ (with as few explanatoryfunctions as possible) can be developed using a stepwise deletionprocess. Following Gardoni et al. [12], we begin with the completemodel (i.e., all p candidate explanatory functions) and successivelyeliminate one function hk;i xð Þ at a time on the basis of the posteriorstatistics (i.e., the posterior coefficient of variation, COV) of the cor-responding coefficient hk;i. After each elimination, the remainingexplanatory functions are refitted to the data and hk;i ’s are re-esti-mated. Since rk is approximately equal to the COV of the predictedcapacity, the accuracy of the model is not expected to improve byincluding a term that has a COV much larger than rk. However,when the largest COV for a parameter becomes close in magnitudeto rk, eliminating an extra term can lead to an unacceptableincrease in rk. What constitutes an unacceptable increase in rk issomewhat subjective and depends on the desired model accuracy,the level of uncertainty in the other variables in the model (e.g., thevariability in the model parameters, hk), and the desired level ofparsimony. Furthermore, we can check for the pairwise linear cor-relations between the remaining parameters. If two parameters, hk;i

and hk;j, are strongly correlated (i.e., qhk;i ;hk;j

P 0:7), we can linearlycombine them as follows:

hk;i ¼ lhk;iþ qhk;i ;hk;j

rhk;i

rhk;j

hk;j � lhk;j

� �ð9Þ

where, lhk;iand rhk;i

are the posterior mean and standard deviationof hk;i, respectively.

The interpretation of the individual signs and values of multipleregression coefficients is problematic. Specifically, the coefficients

Table 3Box–Cox transformation of explanatory functions in the deformation capacity modelfor the flexural failure.

Initial explanatory function Estimatedkf ;i

Transformed explanatoryfunction

gf ;2 xð Þ ¼ Dfl=H 1:00 hf ;2 xð Þ ¼ Dfl=H

gf ;3 xð Þ ¼ Dp=H 0:44hf ;3 xð Þ ¼ Dp=H

� �0:44

gf ;4 xð Þ ¼ Dsh=H 0:50hf ;4 xð Þ ¼ Dsh=H

� �0:5

gf ;5 xð Þ ¼ Dsl=H 0:25hf ;5 xð Þ ¼ Dsl=H

� �0:25

gf ;6 xð Þ ¼ Dg=H �1:45 hf ;6 xð Þ ¼ Dg=H� ��1:45

gf ;7 xð Þ ¼ ql fy=f 0c� �

1:00 hf ;7 xð Þ ¼ ql fy=f 0c� �

gf ;8 xð Þ ¼ f 0c=fFRP 0:00 hf ;8 xð Þ ¼ ln f 0c=fFRP� �

gf ;9 xð Þ ¼ 2tFRP=Dg� �

fFRP=f 0c� �

1:00 hf ;9 xð Þ ¼ 2tFRP=Dg� �

fFRP=f 0c� �

gf ;10 xð Þ ¼ ecu 0:59 gf ;10 xð Þ ¼ ecuð Þ0:59

16 A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22

should not be simply interpreted as a direct relationship betweenthe response and corresponding explanatory function, particularlyin the case of high positive or negative correlations between coef-ficients [33]. Therefore, the physical interpretation of the signs andvalues of each coefficient needs to be further investigated.

3.2. Deformation capacity for flexural failure

Flexural failure of FRP retrofitted columns is associated with theloss of confinement in the plastic hinge region (e.g., [7]). In thismechanism, concrete crushing occurs along with the failure of FRPcomposites and the rupture or buckling of the longitudinal rein-forcement. Flexural failures typically occur with some residual dis-placement ductility. Thus, a flexural failure is more desirable thanother more brittle failure modes (e.g., [7]). Using Eq. (1), we can writea probabilistic deformation capacity model for the flexural failure as

ln df x;Hf� �� �

¼ ln df xð Þh i

þ cf x; hf� �

þ rf ef ð10Þ

where df x;Hf

� �¼ Df =H is the drift ratio capacity for the flexural

failure; Df is the corresponding deformation capacity;df xð Þ ¼ Df =H is an existing deterministic model for predicting df ;and Df is the associated deterministic deformation capacity.

3.2.1. Deterministic modelFor a bridge pier responding as a cantilever and subject to hor-

izontal forces, the lateral deformation capacity can be estimated as

df xð Þ ¼ 1H

Dy þ Dp þ Dsl

� �ð11Þ

where Dy is the elastic component due to the onset of yield; Dp isthe inelastic component due to the plastic flow; and Dsl is the con-tribution of the fixed-end rotation due to the longitudinal reinforce-ment slippage from the anchorage zone when the end sectionreaches the ultimate curvature.

The term Dy can be written as

Dy ¼ Dfl þ Dsh ð12Þ

where Dfl is the flexural component based on a linear curvature dis-tribution along the equivalent column height; and Dsh is the contri-bution of the shear deformation to the yield displacement. Giventhe yield curvature, /y, the flexural component of the yield displace-ment can be written as

Dfl ¼13

/yl2e ð13Þ

where le ¼ H þ YP is the effective column length, in which YP is thelength of yield penetration estimated as YP ¼ 0:022f ydb [34]. Theshear component, Dsh, for columns with FRP confinement isobtained from Biskinis and Fardis [35] empirical expression, validfor circular columns.

Dsh ¼ 0:0027H 1�min 1;2

15HDg

� �� �ð14Þ

The second term in Eq. (11), Dp, can be computed based on its relationto the sectional curvature. The plastic curvature, /p, is defined as/p ¼ /u � /y; where /u is the ultimate curvature. It is assumed thatthe plastic curvature occurs along the plastic hinge length lp esti-mated as lp ¼ 0:07H þ 8:16db for columns with FRP confinement[11], where db is the diameter of the reinforcement. Based upon a sec-tion analysis, Binici [10] proposed the following closed-form expres-sion for the ultimate curvature of FRP confined circular sections:

/u ¼0:0034 4:44�eg2 þ 0:65þ 3:84�e I þ 0:59ð Þ½ �gþ 0:56I þ 0:33

� �Dg 0:44nþ 0:32I þ 0:04gþ 0:02ð Þ

ð15Þ

where �e ¼ eFRP=0:002ð Þ0:45 in which eFRP is the rupture strain of theFRP composite; g ¼ 2EFRPeFRPtFRPð Þ= Dgf 0c

� �in which EFRP and tFRP are

the tensile modulus and thickness of the FRP composite, respec-tively; I ¼ ql fy=f 0c

� �; and n ¼ 4P= pD2

g f 0c� �

. Given the ultimate curva-ture, from Eq. (15), the plastic deformation, Dp, can be written as

Dp ¼ /plp H � 0:5lp� �

¼ /u � /y

� �lp H � 0:5lp� �

ð16Þ

Finally, Dsl for FRP confined members at the ultimate curvature canbe computed according to Biskinis and Fardis [35] as

Dsl ¼/udbfyH

10ffiffiffiffif 0c

p ð17Þ

3.2.2. Model correctionAs initial candidate explanatory functions, we select gf ;1 xð Þ ¼ 1

to capture the potential bias in the model, independent from x;gf ;2 xð Þ ¼ Dfl=H;gf ;3 xð Þ ¼ Dp=H, gf ;4 xð Þ ¼ Dsh=H, and gf ;5 xð Þ ¼ Dsl=Hto correct a possible bias in each of the deterministic terms;gf ;6 xð Þ ¼ Dg=H to account for the possible dependence of the biason the slenderness of the column; gf ;7 xð Þ ¼ ql fy=f 0c

� �and

gf ;8 xð Þ ¼ f 0c=fFRP to capture the possible effects of longitudinal rein-forcement and the strength of the composite material;gf ;9 xð Þ ¼ 2tFRP=Dg

� �fFRP=f 0c� �

to account for the possible influenceof the confinement; and gf ;10 xð Þ ¼ ecu to capture the possibledependence of the bias on the concrete properties, where ecu isthe ultimate strain of the FRP confined concrete. The explanatoryfunctions hf ;i xð Þ’s are then obtained as described earlier in thepaper. To compare the results obtained using the proposedapproach and a simultaneous nonlinear regression, we also per-formed the time-consuming simultaneous estimation of Kf andHf . The results confirmed that the values of Kf do not change sig-nificantly over the deletion process. Table 3 summarizes the trans-formed functions, hf ;i xð Þ’s.

3.2.3. Parameter estimation and model selectionThe unknown parameters, Hf , are estimated using Eq. (7). Choe

et al. [18] developed a probabilistic model for the flexural capacityof un-retrofitted columns. Since the explanatory functions consid-ered in this paper are different, in order to account for the effects ofthe FRP composites, we consider a noninformative prior for hf .However, because the model in Choe et al. [18] and the proposedcapacity model are predicting the same response variable, thequality of the prediction in Choe et al. [18], captured by the modelstandard deviation, can provide some indications of the expectedaccuracy of the proposed model before the model is actually cali-brated, using the experimental data. In fact, the model in Choeet al. [18] is a special case of the model proposed in this paper

A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22 17

where the thickness of the FRP composite is set to zero (i.e.,tFRP ¼ 0). Therefore, we use the posterior estimate for the standarddeviation in [18] as a prior in the Bayesian updating. We also notethat because there is a limited number of data with FRP confine-ment, if we use a noninformative prior, the standard deviation ofthe model error will be artificially small. Using an informativeprior, obtained from a larger dataset, helps obtain an estimate ofthe model standard deviation that is more realistic and likely closerto the one that we would have if more data were available andcould be used for the model calibration. In the complete modelwith all candidate explanatory functions, hf ;2 has the largest COV(=2.07); hence, to simplify the model, we eliminate the termhf ;2hf ;2 xð Þ. Next, we assess the reduced model. Eventually after ninesteps, the model selection process identifies hf ;1 xð Þ and hf ;9 xð Þ asthe explanatory functions needed in an accurate but parsimoniousmodel. Fig. 1 summarizes this stepwise deletion process. For eachstep, the figure shows the COV of the model parameters (soliddots) and the posterior mean of the model standard deviation(open square). In addition, analysis of the posterior statistics ofthe reduced model reveals that hf ;1 and hf ;9 are correlated(q1;9 ¼ �0:80). As a further simplification, these two terms are lin-early combined, using Eq. (9), giving the following expression forcf x; hf

� �:

Fig. 1. Stepwise deletion process of the deformation capacity for the flexuralfailure.

Table 4Posterior statistics of parameters in the deformation capacity model for the flexuralfailure.

Parameters Mean Standard deviation Correlation coefficient

hf ;1 rf

hf ;1 1.072 0.106 1rf 0.337 0.036 �0.057 1

Fig. 2. Comparison between the measured and predicted drift ratio capacities based

cf x; hf

� �¼ hf ;1 þ �0:109� 2:406hf ;1

� � 2tFRP

Dg

fFRP

f 0c

� �ð18Þ

Table 4 lists the posterior statistics of the parametersHf ¼ hf ;1;rf

� �. The following observation is noteworthy: The posi-

tive mean of hf ;1 indicates that, independent from the variables x,the deterministic model, df xð Þ, tends to underestimate the defor-mation capacity of a column with FRP confinement. In addition,the mean of hf ;9 is �2:69, which indicates that df xð Þ tends to over-estimate the contribution of FRP composites on the confinement.Fig. 2 shows a comparison between the measured and predictedvalues of the drift ratio capacities for the test columns based onthe deterministic (left plot) and probabilistic (right plot) models.For a perfect model, the failure data would line up along the 1:1solid line and the censored data would lie above it. Visual inspec-tion of the data shows the bias in df xð Þ and that the proposed prob-abilistic model effectively corrects this bias. For the probabilisticmodel, the failure data points are evenly distributed within theone standard deviation limits and almost all of the censored dataare above the 1:1 line.

For the sake of comparison, we have also provided the estimatesof the unknown parameters with a nonlinear regression (i.e., esti-mating kf ;9 simultaneously with hf ;1 and hf ;9) in Table 5. In addition,Fig. 3 compares the predicted capacities based on the proposedapproach and the nonlinear regression. We can see that eventhough the individual estimates of the unknown parameters aredifferent (see Tables 4 and 5), the values of the predicted capacitiesare in close agreement. We should also note that due to the strongcorrelation between kf ;9 and hf ;1 and hf ;9, obtained with the nonlin-ear regression, comparing individual estimates between the twomodels is problematic. The high correlation coefficients mask theactual impact of individual terms on the capacity and do not letthe values speak for themselves.

3.3. Deformation capacity for bond failure

The combined action of the reinforcing steel and the neighbor-ing concrete requires a load transfer mechanism that is referred toas a bond and is typically idealized as a continuous stress field thatdevelops in the vicinity of the steel-concrete interface [36]. The

on deterministic (left) and probabilistic (right) models for the flexural failure.

Table 5Posterior statistics of parameters in the deformation capacity model for the flexuralfailure, using nonlinear regression.

Parameter Mean Standard deviation Correlation coefficient

hf ;1 hf ;9 kf ;9 rf

hf ;1 1:607 0:290 1hf ;9 �2:821 0:329 �0:732 1kf ;9 0:581 0:141 �0:813 0:271 1rf 0:320 0:039 �0:155 0:113 �0:023 1

Fig. 3. Comparison between the predicted drift ratio capacities based on nonlinearregression and proposed model for the flexural failure.

Table 6Box–Cox transformation of explanatory functions in the deformation capacity modelfor the bond failure.

Initial explanatory function Estimatedkb;i

Transformed explanatoryfunction

gb;2 xð Þ ¼ Dfl=H 1:00 hb;2 xð Þ ¼ Dfl=H

gb;3 xð Þ ¼ Dp=H 1:00 hb;3 xð Þ ¼ Dp=H

gb;4 xð Þ ¼ Dsh=H 1:00 hb;4 xð Þ ¼ Dsh=H

gb;5 xð Þ ¼ Dsl=H 0:00 hb;5 xð Þ ¼ ln Dsl=H� �

gb;6 xð Þ ¼ 2tFRP=Dg� �

fFRP=f 0c� �

1:00 hb;9 xð Þ ¼ 2tFRP=Dg� �

fFRP=f 0c� �

gb;7 xð Þ ¼ db=ls 1:00 hb;7 xð Þ ¼ db=lsgb;8 xð Þ ¼ ecu 0:00 hb;8 xð Þ ¼ ln ecuð Þ

18 A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22

transfer of the interface shear stresses along the reinforcing steelcan cause the formation and propagation of splitting cracks inthe concrete. When a bond failure occurs, the lap spliced reinforce-ment cannot develop the full tensile capacity. Subsequently, arapid bond degradation and hysteretic pinching occur, which leadto the loss of the lateral load-carrying capacity [37].

The RC columns subject to a cyclic loading, experience a severelocalized bond demand in the lap-splice region [37]. Therefore, it isessential to assess the deformation capacity of columns accountingfor a possible bond failure. This section constructs a deformationmodel that accounts for the bond failure. Using Eq. (1), we canwrite a probabilistic deformation capacity model for the bond fail-ure as follows:

ln db x;Hbð Þ½ � ¼ ln db xð Þh i

þ cb x; hbð Þ þ rbeb ð19Þ

where db x;Hbð Þ ¼ Db=H, Db; db xð Þ ¼ Db=H, Db;Hb ¼ hb;rbð Þ,cb x; hbð Þ;rb, and eb have similar definitions as the terms in Eq.(10) but are associated with a bond failure.

3.3.1. Deterministic modelThe deterministic model, db xð Þ, has the same definition as df xð Þ,

given in Eq. (11), where the definition of Dy also remains the sameand is computed using Eqs. (12)–(14). On the other hand, the def-initions of Dp and Dsl are modified to account for the bond failure.In particular, Dp is computed using Eq. (16) and Dsl is computedusing Eq. (17), both with a new value of /u.

To compute /u, accounting for the bond failure, we first need tofind the strain es in the longitudinal reinforcement that is lap splicedin the plastic hinge region. To compute es, we write the equilibriumequation for the extreme spliced reinforcement in tension withlength ls. The equilibrium equation is between the force associatedwith the bond stress, ubond, that is ubondpdbls, and the axial force inthe reinforcement, that is fspd2

b=4; where, fs is the induced stress inthe reinforcement. Following Harajli [38], the three components ofubond are: (1) the average bond stress due to the unconfined concrete,uun ¼ 0:1þ 0:25cover=db þ 4:15db=lsð Þ

ffiffiffiffif 0c

p; (2) the contribution of

transverse reinforcements, utr ¼ Atrfyh

ffiffiffiffif 0c

p= 41:6stnsdbð Þ � 0:25

ffiffiffiffif 0c

p;

in which, Atr is the area; st is the spacing of the transverse reinforce-ment; ns is number of spliced reinforcement bars in tension taken ashalf of the total spliced reinforcement bars; and (3) the additionalcontribution from FRP confinement, uFRP ¼ 2tFRPEFRP

ffiffiffiffif 0c

p= 8000nsdbð Þ

� 0:4ffiffiffiffif 0c

p. Using the equilibrium equation and the upper limit spec-

ified by the design code [39], fs can then be written as

fs ¼lsdb

EFRPtFRP

1000nsdbþ Atrfyh

10:4stnsdbþ cover

dbþ 0:4

� �þ 16:6

� � ffiffiffiffif 0c

q� 1:25f y ð20Þ

After computing fs, the corresponding strain, es, can be obtainedfrom the steel stress–strain constitutive relation.

To find /u, we first use the section analysis, as described inBinici [10], to find the depth of the compression zone, c, and thenrelate /u to es, using rules of mechanics, as

/u ¼es

w� cð21Þ

where w ¼ Dg � cover. Given es and c, the ultimate curvature can bewritten as

/u ¼es

w� Dg 1:7 �0:38þ 0:44wð Þ ð22Þ

where, w ¼ nþ I þ 0:4gþ 0:21ð Þ= 0:32I þ 0:37gþ 0:19ð Þ is the anglebetween the cross-sectional radii that delimit the compressionzone, in which n; g, and I are the same as defined in Eq. (15).

3.3.2. Model correctionAs initial candidate explanatory functions, we select gb;1 xð Þ ¼ 1

to capture a potential bias in the model that is independent of x;gb;2 xð Þ ¼ Dfl=H; gb;3 xð Þ ¼ Dp=H, gb;4 xð Þ ¼ Dsh=H, and gb;5 xð Þ ¼ Dsl=Hto examine the potential bias associated with the individual com-ponents of db xð Þ;gb;6 xð Þ ¼ 2tFRP=Dg

� �fFRP=f 0c� �

to account for the pos-sible influences on the bias of the FRP confinement; gb;7 xð Þ ¼ db=ls

to explore the contribution on the bias from the splice length; andgb;8 xð Þ ¼ ecu to correct for the effects on the bias of the materialproperties. Before moving on to the model selection, the trans-formed explanatory functions, hb;i xð Þ, are obtained as describedearlier in the paper. Table 6 summarizes the transformed functionshb;i xð Þ’s.

3.3.3. Parameter estimation and model selectionGiven that no prior information is available about Hb, we con-

sider a noninformative prior distribution in Eq. (7). In this sectionwe shorten the list of hb;i xð Þ’s to obtain a parsimonious model.Fig. 4 shows the posterior COV of hb;i’s (as dots) and mean of rb

(as an open square) at each step in the deletion process. The dele-tion process is stopped at the 7th step, keeping hb;1hb;1 xð Þ andhb;2hb;2 xð Þ in the model. The examination of the correlation coeffi-cient indicates a high dependence (q1;2 ¼ �0:98) between hb;1

and hb;2. Therefore, we combine them, using Eq. (9), and cb is leftwith only one model parameter.

cb x; hbð Þ ¼ hb;1 þ 12:83� 107:14hb;1� � Dfl

H

!ð23Þ

Table 7 lists the posterior statistics of the parametersHb ¼ hb;1;rb

� �. The positive mean of hb;1 suggests, independent

from the variables x, the deterministic model, db xð Þ, tends tounderestimate the deformation capacity. Furthermore, the nega-tive mean of hb;2 (¼ �58:95) indicates that db xð Þ tends to overesti-mate the contribution of the flexural deformation. There is a

Fig. 4. Stepwise deletion process of the deformation capacity for the bond failure.

Table 8Posterior statistics of parameters in the deformation capacity model for the bondfailure, using nonlinear regression.

Parameter Mean Standard deviation Correlation coefficient

hb;1 hb;2 kb;2 rb

hb;1 �0:47 0:238 1hb;2 0:002 0:001 �0:822 1kb;2 �1:18 0:548 �0:096 0:579 1rb 0:14 0:039 0:153 �0:101 0:044 1

Fig. 6. Comparison between the predicted drift ratio capacities based on nonlinearregression and proposed model for the bond failure.

Table 7Posterior statistics of parameters in the deformation capacity model for the bondfailure.

Parameters Mean Standard deviation Correlation coefficient

hb;1 rb

hb;1 0:67 0:212 1rb 0:14 0:035 �0:003 1

A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22 19

possible physical interpretation of this latter observation. The yieldcurvature, /y, in RC columns are often ill-defined because of crack-ing of concrete and sequential yielding of reinforcement bars [40].In the literature, there are various definitions for the yield curva-ture and each works well under particular circumstances. Thereis no unique definition that works well everywhere. While addi-tional work and in particular an experimental investigation wouldbe needed to better understand this issue, we believe that the sim-ple model used in this paper, following [10], overestimates /y for acolumn that fails in bond. This bias is corrected in the proposedmodel. However, as noted earlier, the interpretation of the individ-ual signs and values of the multiple regression coefficients is prob-lematic. Therefore, the above interpretation would need to befurther investigated. Fig. 5 shows a comparison between the mea-sured and predicted values of drift ratio capacities for the test col-umns based on the deterministic and probabilistic models.

Table 8 provides the estimates of the unknown parameters witha nonlinear regression (i.e., estimating kb;2 simultaneously with hb;1

and hb;2) and Fig. 6 compares the predicted capacities based on theproposed approach and the nonlinear regression. The predictedcapacities with the two models are in close agreement. As dis-cussed before, due to the strong correlation between kb;2 and hb;1

and hb;2 in the nonlinear regression method, we should avoid com-paring individual estimates of the unknown parameters in the twomodels.

Fig. 5. Comparison between measured and predicted drift ratio capacities based

3.4. Bivariate flexure-bond deformation capacity model

All 41 data points can be used for the flexural model but out ofthem only 19 can be used for the bond model. Therefore, we usethe 22 data points that cannot be used for the bond model todevelop an informative prior distribution for Hf , a noninformativeprior is used for Hb and the remaining 19 data points are used todevelop the bivariate likelihood function according to Table 2.The bivariate model uses the reduced model forms in Eqs. (18)and (23) and K ¼ Kf ;Kb

� �are the same as those in the univariate

models. Table 9 lists the posterior statistics of the parameters H.As expected, the estimates of Hf , and Hb are nearly identical tothe corresponding estimates for the univariate models. In addition,the bivariate model provides an estimate of the correlation coeffi-cient, q , between ef and eb.

4. Fragility estimates of the FRP-retrofitted columns

The developed probabilistic capacity models can be used to com-pute the fragility of FRP-retrofitted columns. Fragility functions

on deterministic (left) and probabilistic (right) models for the bond failure.

Table 9Posterior statistics of the bivariate deformation capacity model.

hf ;1 rf hb;1 rb q

Mean 1.078 0.331 0.656 0.146 0.036Std. dev. 0.105 0.038 0.201 0.035 0.554

Correlation coefficienthf ;1 1rf �0:26 1hb;1 �0:02 0:02 1rb 0:05 0:02 �0:03 1q �0:00 0:00 �0:13 0:16 1

20 A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22

express the conditional probability of meeting or exceeding a pre-scribed limit state for a given value of the demand measure. Evalu-ating the fragility of columns before and after retrofit providesvaluable insights on quantifying the effectiveness of the retrofitmeasure in improving the performance of the retrofitted columnsand mitigating their vulnerabilities.

Following Ditlevsen and Madsen [41], we define a limit statefunction gk x;Hkð Þ such that the event gk x;Hkð Þ � 0f g indicatesthe failure in the kth failure mode. By partitioning x as x ¼ r; sð Þ,

Fig. 7. Fragility estimate for (a) example Column 1, (b) example C

where r is a vector of material and geometrical variables and s isa vector of demand variables such as boundary forces or deforma-tions, the limit state function can be written as

gk r; s;Hkð Þ ¼ Ck r; s;Hkð Þ � Dk r; sð Þ ð24Þ

where Ck is computed using Eqs. (10) and (19) and Dk denotes thegiven demand for the kth failure mode. The column fragility canthen be written as

F s;Hð Þ ¼ P[

k

gk r; s;Hkð Þ � 0js;Hkf g" #

ð25Þ

where P AjB½ � denotes the conditional probability of the event A forthe given event B. As discussed in Gardoni et al. [12], the predictive

fragility estimate, eF sð Þ, that incorporates the uncertainties in H, canbe computed as the expected value of F s;Hð Þ over the posterior dis-

tribution of H as eF sð Þ ¼R

F s;Hð Þf Hð ÞdH. Further, to show the effectof the epistemic uncertainty in H, we compute the confidencebounds of F s;Hð Þ. Following Gardoni et al. [12], the confidence

bounds can be written as U �~b sð Þ � rb sð Þh i

;U �~b sð Þ þ rb sð Þh in o

olumn 2, and (c) example Column 3, before and after retrofit.

Table 10Summary of design random variables for the example columns.

Variables Mean Distribution COV (%)

Column 1 Column 2 Column 3

f 0c 34.5 MPa 20 MPa 35.8 MPa Lognormal 10fy 303.2 MPa 425.2 MPa 475 MPa Lognormal 5fyh 303.2 MPa 426.2 MPa 493 MPa Lognormal 5ql 0.0252 0.0188 0.0199 – –Dg 610 mm 760 mm 1520 mm – –H 3658 mm 3250 mm 9140 – –P 1780 kN 1361 kN 4450 kN Normal 25lp 437 mm 405 mm 1105 mm Lognormal 20ls 381 mm 760 mm – – –tFRP 5.1 mm 0.83 mm 2.0 mm – –eFRP 0.0105 0.015 0.0105 Lognormal 15EFRP 124 GPa 231 GPa 124 GPa Lognormal 15

Fig. 8. Difference between fragilities before and after retrofit for the three examplecolumns when either of flexural or bond failure occur.

A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22 21

where ~b sð Þ ¼ U�1 1� eF sð Þh i

is the generalized reliability index cor-

responding to eF sð Þ, and r2b sð Þ � rHb sð ÞRHHrHb sð ÞT is the variance

of the reliability index, b s;Hð Þ, obtained using a first-order Taylorexpansion around the posterior mean of model parameters, MH.In the last expression, rHb sð Þ is the gradient row vector of b s;Hð Þat the mean point MH; and RHH denotes the posterior covariancematrix. The bounds approximately correspond to 0.15 and 0.85quantiles.

Fig. 7 shows the predictive fragility curves of three examplecolumns with respect to the drift ratio demand (i.e., Dk ¼ ln dð Þfor both k ¼ f and k ¼ b) before and after they are retrofitted.The fragilities for the as-built columns are developed usingEqs. (10) and (19) with tFRP ¼ 0. Columns 1 and 2 (modeled after[7,42], respectively) represent non-seismically designed columnswith a lap-splice at the column base. Column 3 (modeled after[43]) represents a seismically designed column with no lap-splice.Table 10 shows a summary of the geometry and the materialproperties of the example columns and, when appropriate, theirstatistical information to account for their inherent randomness.

Fig. 7(a) shows the fragilities associated with the first examplecolumn. The left plot shows the fragility for each individual modeof failure and the right plot shows the fragility considering eitherthe flexural or bond mode of failure developed using the bivariatemodel. The thin lines correspond to the as-built condition and thethick lines correspond to the retrofitted condition. The right plotalso shows the confidence band created by the confidence boundsat different values of d. The insets in the right plot are zoom-inwindows of fragilities. They are showing fragilities associated withsmaller drift ratio demands that are not so clear in the full-sizewindow and are of more interests in structural engineeringapplications.

The left plot shows that for the as-built column the bond modeof failure dominates. In addition, the retrofit measure significantlyreduces the fragility for each mode of failure and, as a result, alsothe fragility that considers either mode of failure is also reduced(as shown also in the right plot). The retrofit is more effective inincreasing the mean value of the bond capacity than of the flexuralcapacity (i.e., the 0.5 probability of failure occurs at a lower driftratio demand for the flexural failure mode than the bond failuremode). However, because of the smaller rb the bond still controlsat d > 0:07. These observations are consistent with the experimen-tal outcomes reported by Seible et al. [7]. Specifically, both the as-built and the retrofitted columns experienced a bond failure and itwas observed that while the FRP composite improved the overallperformance of the column and reduced the vulnerability, thisimprovement was not sufficient to change the failure mode.

Similarly, Fig. 7(b) shows the results for Column 2. The resultsare similar to those already presented for Column 1. However,

the flexural and bond fragilities for the retrofitted column arenow closer. This is consistent with the experimental observationsreported by Chang et al. [42] who reported that while the as-builtcolumn failed in bond the retrofitted column now failed in flexure.Finally, Fig. 7(c) shows the fragility for Column 3. Since this columndoes not have a lap-splice, only the flexural fragilities are provided.

To show the quantitative improvement in the fragility, wheneither modes of failure occur, Fig. 8 provides the differencebetween fragilities before and after retrofit for the three examplecolumns. Comparing the reduction in the fragility of seismicallydesigned column (i.e., Column 3) with non-seismically designedcolumns (i.e., Columns 1 and 2), it can be seen that FRP retrofitmeasure is more effective in improving the performance ofnon-seismically designed columns. Though the thickness of FRPcomposites in the three example columns differs, they are alldetermined following a consistent design approach. Furthermore,sensitivity analysis, which is not provided in the paper, shows thatfurther increase of the FRP composite thickness, beyond thedesigned values, does not change the fragilities significantly.

5. Conclusion

Two probabilistic models are proposed to predict the deforma-tion capacity of FRP retrofitted RC columns. One capacity modelpredicts the deformation capacity corresponding to a flexural fail-ure and the other model predicts the deformation capacity forbond failure. To develop the probabilistic models, simple correc-tion terms are added to deterministic models to compensate fortheir conservative bias. The correction terms are constructed start-ing from candidate explanatory functions developed using theBox–Cox transformation. A stepwise deletion process is then usedto eliminate the least informative terms and simplify the models,while maintaining an acceptable level of accuracy. As an applica-tion of the probabilistic models, fragility curves are developed forthree example columns. Two columns are representative of non-seismic design and have a lap-splice at the column base. One col-umn is seismically designed and does not have a lap-splice. Thefragilities quantify the effect of FRP retrofit measure in reducingthe probability of failure of RC columns. For the considered levelsof FRP thickness, the results indicate that (1) the FRP retrofittingmeasure considerably reduces the fragility for the bond failure;and (2) the FRP confinement is also beneficial, but with a moremoderate impact, when considering the flexural failure. The fragil-ity analysis also evaluated the efficiency of the retrofitting schemeconsidering either mode of failure. Again, the results clearly quan-tify the reduction in the probability of failure due to the FRP retro-fitting measure.

22 A. Tabandeh, P. Gardoni / Engineering Structures 74 (2014) 13–22

References

[1] Padgett JE. Seismic vulnerability assessment of retrofitted bridges usingprobabilistic methods. PhD thesis. Georgia Institute of Technology, Atlanta,GA; 2007.

[2] Burrato P, Vannoli P, Fracassi U, Basili R, Valensise G. Is blind faulting trulyinvisible? Tectonic-controlled drainage evolution in the epicentral area of theMay 2012, Emilia-Romagna earthquake sequence (Northern Italy). AnnGeophys 2012;55(4):525–31.

[3] Beavan J, Fielding E, Motagh M, Samsonov S, Donnelly N. Fault location and slipdistribution of the 22 February 2011 Mw 6.2 Christchurch, New Zealand,earthquake from geodetic data. Seismol Res Lett 2011;82(6):789–99.

[4] Ellingwood BR. Toward load and resistance factor design for fiber-reinforcedpolymer composite structures. J Struct Eng 2003;129(4):449–58.

[5] Haroun MA, Elsanadedy HM. Behavior of cyclically loaded squat reinforcedconcrete bridge columns upgraded with advanced composite-material jackets.J Bridge Eng 2005;10(6):741–8.

[6] Haroun MA, Elsanadedy HM. Fiber-reinforced plastic jackets for ductilityenhancement of reinforced concrete bridge columns with poor lap-splicedetailing. J Bridge Eng 2005;10(6):749–57.

[7] Seible F, Priestley MJN, Hegemier GA, Innamorato D. Seismic retrofit of RCcolumns with continuous carbon fiber jackets. J Compos Constr1997;1(2):52–62.

[8] Xiao Y, Ma R. Seismic retrofit of RC circular columns using prefabricatedcomposite jacketing. J Struct Eng 1997;123(10):1357–64.

[9] Harries KA, Ricles JR, Pessiki S, Sause R. Seismic retrofit of lap splices innonductile square columns using carbon fiber-reinforced jackets. ACI Struct J2006;103(6):874–84.

[10] Binici B. Design of FRPs in circular bridge column retrofits for ductilityenhancement. Eng Struct 2007;30(3):766–76.

[11] Binici B, Mosalam KM. Analysis of reinforced concrete columns retrofitted withfiber reinforced polymer lamina. Compos Part B: Eng 2007;38(2):265–76.

[12] Gardoni P, Der Kiureghian A, Mosalam KM. Probabilistic capacity models andfragility estimates for reinforced concrete columns based on experimentalobservations. J Eng Mech 2002;128(10):1024–38.

[13] Ramanathan K, DesRoches R, Padgett JE. Analytical fragility curves formultispan continuous steel girder bridges in moderate seismic zones.Transport Res Rec: J Trans Res Board 2010;2202(1):173–82.

[14] Mackie KR, Lu J, Elgamal A. Performance-based earthquake assessment ofbridge systems including ground-foundation interaction. Soil Dyn Earthq Eng2012;42:184–96.

[15] Tondini N, Stojadinovic B. Probabilistic seismic demand model for curvedreinforced concrete bridges. Bull Earthq Eng 2012;10(5):1455–79.

[16] Lee T-H, Mosalam KM. Probabilistic fiber element modeling of reinforcedconcrete structures. Comput Struct 2004;82(27):2285–99.

[17] Gardoni P, Mosalam KM, Der Kiureghian A. Probabilistic seismic demandmodels and fragility estimates for RC bridges. J Earthq Eng 2003;7(1):79–106.

[18] Choe DE, Gardoni P, Rosowsky D. Closed-form fragility estimates, parametersensitivity, and Bayesian updating for RC columns. J Eng Mech2007;133(7):833–43.

[19] Zhong J, Gardoni P, Rosowsky D. Closed-form seismic fragility estimates,sensitivity analysis and importance measures for reinforced concrete columnsin two-column bents. Struct Infrastruct Eng 2012;8(7):669–85.

[20] Huang Q, Gardoni P, Hurlebaus S. Probabilistic seismic demand models andfragility estimates for reinforced concrete highway bridges with one single-column bent. J Eng Mech 2010;136(11):1340–53.

[21] Okeil AM, El-Tawil S, Shahawy M. Flexural reliability of reinforced concretebridge girders strengthened with carbon fiber-reinforced polymer laminates. JBridge Eng 2002;7(5):290–9.

[22] Atadero RA, Karbhari VM. Probabilistic based design for FRP strengthening ofreinforced concrete. Special Publication SP-230-42; 2005. p. 723–42.

[23] Wang N, Ellingwood BR, Zureick AH. Reliability-based evaluation of flexuralmembers strengthened with externally bonded fiber-reinforced polymercomposites. J Struct Eng 2010;136(9):1151–60.

[24] Billah AHMM, Alam MS, Bhuiyan MAR. Fragility analysis of retrofitted multi-column bridge bent subjected to near fault and far field ground motion. JBridge Eng 2012;18(10):992–1004.

[25] Chung YS, Park CK, Meyer C. Residual seismic performance of reinforcedconcrete bridge piers after moderate earthquakes. ACI Struct J 2008;105(1):87.

[26] Fahmy MFM, Wu Z, Wu G. Seismic performance assessment of damage-controlled FRP-retrofitted RC bridge columns using residual deformations. JCompos Constr 2009;13(6):498–513.

[27] Laine M. Adaptive MCMC methods with applications in environmental andgeophysical models. PhD thesis. Lappeenranta University of Technology,Lappeenranta, Finland; 2008.

[28] Box GEP, Cox DR. An analysis of transformations. J Roy Stat Soc. Ser B(Methodol) 1964;26:211–52.

[29] Rao CR, Toutenburg H. Linear models, least squares and alternatives. Springer;1997.

[30] Box GEP, Tiao GC. Bayesian inference in statistical analysis, vol. 40. NY: JohnWiley & Sons; 2011.

[31] Velilla S. A note on the multivariate Box–Cox transformation to normality. StatProbab Lett 1993;17(4):259–63.

[32] Weisberg S. Applied linear regression, vol. 528. NY: John Wiley & Sons; 2005.[33] Ryan TP. Modern engineering statistics. New York: Wiley-Interscience; 2007.[34] Priestley MJN, Seible F, Calvi GM. Seismic design and retrofit of

bridges. NY: John Wiley & Sons; 1996.[35] Biskinis D, Fardis MN. Effect of lap splices on flexural resistance and cyclic

deformation capacity of RC members. Beton Stahlbeton 2007;102(1):51–9.[36] Lowes LN, Moehle JP. Evaluation of retrofit of beam-column T-joints in older

reinforced concrete bridge structures. ACI Struct J 1999;96(4):519–32.[37] Cho J-Y, Pincheira JA. Inelastic analysis of reinforced concrete columns with

short lap splices subjected to reversed cyclic loads. ACI Struct J2006;103(2):280–90.

[38] Harajli MH. Bond strengthening of lap spliced reinforcement using externalFRP jackets: an effective technique for seismic retrofit of rectangular orcircular RC columns. Constr Build Mater 2009;23(3):1265–78.

[39] A. Committee, Building code requirements for structural concrete andcommentary (ACI 318M-05): an ACI standard. American Concrete Institute;2005.

[40] Elnashai AS, Di Sarno L. Fundamentals of earthquake engineering. UK: JohnWiley & Sons; 2008.

[41] Ditlevsen O, Madsen HO. Structural reliability methods, vol. 178. NY: JohnWiley & Sons; 1996.

[42] Chang KC, Chung LL, Lee BJ, Tsai KC, Hwang JS, Hwang SJ. Seismic retrofit studyof RC bridge columns. In: International training program for seismic design ofbuilding structures. Taipei, Taiwan.

[43] Naito CJ. Experimental and computational evaluation of reinforced concretebridge beam-column connections for seismic performance. PhD thesis.University of California, Berkeley; 2000.