Seismic fragility estimates for reinforced concrete bridges subject to corrosion

Structural Safety 31 (2009) 275–283

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Seismic fragility estimates for reinforced concrete bridges subject to corrosion Do-Eun Choe a, Paolo Gardoni a,*, David Rosowsky a, Terje Haukaas b a b

Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, United States Department of Civil Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC, Canada V6T 1Z4

a r t i c l e

i n f o

Article history: Received 24 April 2007 Received in revised form 3 July 2008 Accepted 14 October 2008 Available online 28 November 2008 Keywords: Reinforced concrete columns Corrosion Probabilistic demand models Shear capacity Drift capacity Sensitivity analysis

a b s t r a c t The paper develops novel probabilistic models for the seismic demand of reinforced concrete bridges subject to corrosion. The models are developed by extending currently available probabilistic models for pristine bridges with a probabilistic model for time-dependent chloride-induced corrosion. In particular, the models are developed for deformation and shear force demands. The demand models are combined with existing capacity models to obtain seismic fragility estimates of bridges during their service life. The estimates are applicable to bridges with different combinations of chloride exposure condition, environmental oxygen availability, water-to-cement ratios, and curing conditions. Model uncertainties in the demand, capacity and corrosion models are accounted for, in addition to the uncertainties in the environmental conditions, material properties, and structural geometry. As an application, the fragility of a single-bent bridge typical of current California practice is presented to demonstrate the developed methodology. Sensitivity and importance analyses are conducted to identify the parameters that contribute most to the reliability of the bridge and the random variables that have the largest effect on the variance of the limit state functions and thus are most important sources of uncertainty. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Corrosion of the reinforcement in reinforced concrete (RC) structures is a matter of increasing concern. Corrosion is a longterm process that effectively weakens structural elements and increases their vulnerability to extreme loads. The concerns include serviceability and safety limit-states, as well as economic costs due to maintenance and repair. In this paper, particular attention is devoted to bridges subject to a seismic hazard. Due to the uncertainties in the corrosion process, the structural properties, and the demands on the structures due to an impending earthquake, it has been difficult to predict the seismic fragility of deteriorating RC bridges. The objectives and contributions of this paper are to (1) develop probabilistic demand models for RC bridges subject to earthquake ground motion that include the time-dependent effects of corrosion, (2) estimate the ensuing seismic fragility of RC bridges, (3) identify the parameters, i.e., structural properties, environmental factors, and model parameters, that have the highest influence on the seismic fragility estimates, and (4) identify the most important random variables that have the largest effect on the variance of the limit state functions. Several models have been developed to quantify and account for corrosion in the design, construction, and maintenance of RC structures. In particular, Tuutti [1] and Liu and Weyers [2] devel* Corresponding author. Tel.: +1 979 845 4340; fax: +1 979 845 6554. E-mail address: [email protected] (P. Gardoni). 0167-4730/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2008.10.001

oped deterministic models for the corrosion of reinforcing steel. Thoft-Christensen et al. [3,4] and DuraCrete [5] developed probabilistic models for the corrosion process. Stewart and Rosowsky [6], Vu and Stewart [7], Enright and Frangopol [8,9] developed probabilistic models for the corrosion of bridge slabs, beams and girders. Probabilistic demand models for the relevant modes of failures (deformation and shear) are needed to develop the seismic fragility estimates of bridges. To develop estimates that are applicable at any time during the bridges service life, this paper develops probabilistic models for the deformation and shear demand of RC columns that incorporate probabilistic models for the corrosion process. Gardoni et al. [10] developed probabilistic demand models for shear and deformation by employing deterministic demand models used in practice as a starting point. Additional terms were included to explicitly describe the inherent systematic and random errors in the demand predictions. In this paper, probabilistic demand models of deteriorating RC bridge systems are developed by incorporating in the models by Gardoni et al. [10] a probabilistic model for chloride-induced corrosion [5] and a time-dependent corrosion rate function [7]. The deterioration models used to develop the demand models incorporate uncertainties both in the structural properties and the material deterioration processes. This is significant because of the well-known presence of considerable uncertainty in these constituents. Fragility estimates are obtained by assessing the conditional probability that the deformation or shear demand will exceed

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the corresponding capacity for a given value of the spectral acceleration Sa, which is used here as a measure of intensity of the ground motions. The capacity is assessed using the probabilistic capacity models developed by Choe et al. [11], which already incorporates effects of corrosion. As an application, a bridge design that is typical of current California practice is used to demonstrate the time-variant fragility assessment methodology. The fragility estimates consider different combinations of chloride exposure condition, environmental oxygen availability, water-to-cement ratios, and curing conditions. Model uncertainties in both the capacity and the corrosion models are considered, in addition to the uncertainties in the environmental conditions, material properties, and structural geometry. It is emphasized that the models developed in this paper are applicable to both existing and new RC bridges. Importantly, they may be used for the prediction of the service-life of existing and new structures as well as general-purpose life-cycle cost analysis for RC structures. There are seven sections in this paper. The first section describes a corrosion initiation model used to assess the deterioration of RC bridge columns over time. The second section discusses how the loss of cross-sectional area is computed for a given corrosion initiation time, Tcorr. The third section develops novel probabilistic demand models for the seismic demand in deformation and shear for RC bridge columns subject to corrosion. The fourth section reviews previously developed deformation and shear capacity models. The fifth section provides a general formulation of the seismic fragility estimations for RC bridges subject to corrosion by writing appropriate limit state functions that use both the newly developed demand models and the currently available capacity models. As an application, the sixth section estimates the seismic fragility of an example bridge typical of current California practice. Finally, in the seventh section, sensitivity and importance analyses are conducted. Novelties include the probabilistic modeling of the corrosion initiation time, the development of new probabilistic demand models, the formulation and estimation for a specific example of the seismic fragility of RC bridges during their service life, and the assessment of sensitivity and importance measures that provide physical insight into the effect of corrosion on RC bridge systems. 2. Corrosion initiation model This study uses the probabilistic model of chloride-induced corrosion presented by DuraCrete [5] and the time-variant corrosion rate function by Vu and Stewart [7] to predict the corrosion status of RC members. The corrosion model is extended to estimate the probability distribution of Tcorr. The model includes uncertainties in the structural parameters, environmental conditions, and model parameters. The original model for the corrosion initiation time reads [4]

"

T corr

1   2 #1n 2 dc C cr 1 ¼ XI  erf 1 ; Cs 4ke kt kc D0 ðt 0 Þn

ð1Þ

where dc is the reinforcement cover depth, ke is an environmental factor, kt represents the influence of test methods to determine the empirical diffusion coefficient D0, kc is a parameter that accounts for the influence of curing, t0 is the reference period for D0, n is the age factor, XI is a model uncertainty coefficient to account for the idealization implied by Fick’s second law, Cs = Acs(w/b) + ecs is the chloride concentration on the surface, w/b is the water-tobinder ratio, Acs and ecs are model parameters, Ccr is the critical chloride concentration, and erf() is the error function. DuraCrete [5] also provides the probability distributions for the parameters in Eq. (1) (provided in Appendix 1 for completeness). Eq. (1) is used in this pa-

per to compute the cumulative distribution function (CDF) and the probability density function (PDF) of the corrosion initiation time:

CDF : FT corr ðt corr Þ ¼ P½T corr 6 tcorr 

ð2Þ

and

PDF : f T corr ðt corr Þ ¼

@FT corr ðtcorr Þ : @tcorr

ð3Þ

In the next section, the deteriorated member properties (e.g., the corroded reinforcement area) are determined by the timedependent corrosion rate function developed by Vu and Stewart [7], given a realization of the corrosion initiation time. In particular, for time instances less than Tcorr, the cross-section is assumed to be the same as the pristine cross-section. For time instances greater than Tcorr the reinforcement bars are assumed to have a reduced cross-sectional area, in accordance with Vu and Stewart [7]. The reduced reinforcement area is used in the probabilistic models as described later in this paper. 3. Change in diameter of reinforcing steel over time The time-dependent corrosion rate developed by Vu and Stewart [7] is used to compute the loss of steel cross-section area over time. The corrosion current density at time t is expressed as

icorr ðtÞ ¼ 0:85icorr;0 ðt  T corr Þ0:29 ;

t P T corr ;

ð4Þ

where icorr,0 denotes the corrosion current density at the initiation of corrosion propagation; namely

icorr;0 ¼

37:5ð1  w=cÞ1:64 dc

lA=cm2 ;

ð5Þ

where w/c represents the variable water-to-cement ratio and dc is cover depth, which is the distance from the surface of steel bar to the surface of concrete structure. Note that according to Eq. (4), the corrosion rate diminishes with time because corrosion products formed around the bar impede the diffusion of iron ions. Following Choe et al. [11], after the corrosion process initiates, the diameter of the reinforcement is assumed to decrease over time. In summary, the diameter of the reinforcement at a general time t for given corrosion initiation time Tcorr can be computed as

8 > < db 0 1:64 db ðtjT corr Þ ¼ db 0  1:0508ð1w=cÞ ðt  T corr Þ0:71 dc > : 0

t 6 T corr ; T corr < t 6 T f ; t > Tf ;

9 > = > ;

ð6Þ where db0 is the diameter of the reinforcement at time t = 0, and Tf is the time when db(t|Tcorr), in theory, reaches zero, that is T f ¼ T corr þ dbi fd=½1:0508ð1  w=cÞ1:64 g1=0:71 . A similar equation can also be used to compute the diameter of the spiral reinforcement dsp(t|Tcorr) as a function of the diameter of the reinforcement at time t = 0, dsp0 . 4. Probabilistic demand models for corroding RC bridges This section first reviews currently available probabilistic demand models for pristine bridges. Then novel probabilistic models are proposed for corroding RC bridges. The proposed models build on the available models for pristine bridges and incorporate both the model to estimate the diameter of the reinforcement at a general time t for given corrosion initiation time Tcorr given in Eq. (6) and the probability distribution of Tcorr given in Eq. (1). 4.1. Probabilistic demand models for pristine bridges Gardoni et al. [10] developed probabilistic models for the seismic deformation and shear demands of pristine RC bridges

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for a given Sa. The demand models are constructed using deterministic demand models already available in practice and correction terms designed to capture the inherent systematic (bias) and random errors. The models incorporate nonlinear static push-over analysis followed by a nonlinear response spectrum analysis [12,13]. For a bridge system with s single-column bents, the deformation (d) and shear (v) demand models for given Sa are written as

^ ðx0 jSa Þ þ hDd1 þ hDd2 d ^Dd ðx0 jSa Þ þ rDd e ; Did ðx0 ; HD jSa Þ ¼ d id id i ¼ 1; . . . ; s;

ð7Þ

^iv ðx0 jSa Þ þ hDv 1 þ hDv 2 d ^iv ðx0 jSa Þ Div ðx0 ; HD jSa Þ ¼ d ^ ðx0 jSa Þ þ rDv e ; i ¼ 1; . . . ; s: þ hDv 3 d iv

id

Probabilistic demand models of corroding RC bridges are developed by integrating the previously described probabilistic model for chloride-induced corrosion with the probabilistic demand models for pristine conditions described above. The probabilistic demand models for a corroding RC column at time t are written as 1

Dik ðt; x0 ; HD jSa ; T corr ÞfT corr ðt corr ÞdT corr ;

ð9Þ

0

where Dik ðt; x0 ; HD jSa ; T corr Þ denotes the kth conditional demand (either shear or deformation) for given Sa and Tcorr, and fT corr ðtcorr Þ is the PDF of Tcorr defined in Eq. (3). Assuming that the demands are unaffected by corrosion for t 6 T corr , Eq. (10) can be rewritten as

Dik ðt; x0 ; HD jSa Þ ¼ Dik ðx0 ; HD jSa Þ½1  F T corr ðtÞ Z t þ Dik ðt; x0 ; HD jSa ; T corr ÞfT corr ðtcorr ÞdT corr

ð10Þ

0

where Dik(x0, HD|Sa) is the demand for the pristine structure, and Dik ðt; x0 ; HD jSa ; T corr Þ can be written using Eqs. (7) and (8) and the simplifications of these equations provided by Gardoni et al. [13] as

Did ðt; x0 ; HD jSa ; T corr Þ ¼ 0:61 þ 3:90hDd2 ^ ðt; x0 jSa ; T corr Þ þ ð1 þ hDd2 Þd id

þ rDd eid ;

i ¼ 1; . . . ; s;

i ¼ 1; . . . ; s:

ð12Þ

The terms in Eqs. (12) and (13) have analogous definitions to these in Eqs. (7) and (8) except that now they are computed at time t, given Tcorr. In particular, the pristine diameter of the reinforcing steel d b 0 is replaced by db(t|Tcorr) compute using Eq. (6). Furthermore, the set of unknown model parameters HD is assumed not to vary with time. 5. Probabilistic capacity models for corroding RC columns

4.2. Probabilistic demand model for corroding bridges

Dik ðt; x0 ; HD jSa Þ ¼

iv

^id ðt; x0 jSa ; T corr Þ þ rDv eiv ; þ hDv 3 d

ð8Þ

In Eqs. (7) and (8) the logarithmic transformation is used to stabilize the variance of the model, that is Did ðx0 ; HD jSa Þ ¼ ln½dðx0 ; HD Þ and Div ðx0 ; HD jSa Þ ¼ ln½v ðx0 ; HD Þ are the natural logarithms of the predicted drift and shear demands for the ith column of a bridge system, d ¼ D=H is the drift demand, in which D is the displacement demand and H is the clear column height, v ¼ V=ðAg ft0 Þ is the normalized shear demand, in which V is the shear demand, Ag is the gross cross-sectional area, and ft0 is the tensile strength of concrete. The vector x0 represents a set of basic variables (e.g., material properties, member dimensions, and imposed boundary conditions) at the time of construction. The set of unknown parameters HD ¼ ðHDd ; HDv ; RÞ is introduced to fit the models to observed ^id ðx0 Þ and data, where hDd = (hDd1, hDd2) and hDv = (hDv1, hDv2, hDv3), d ^ ðx0 Þ are the natural logarithms of the deterministic demand estid iv mates for ith column bent of the bridge system, rDdeid and rDveiv are the model errors where, eid and eiv are a random variable with zero mean and unit variance, rDd and rDv represent the standard deviation of the model error and the covariance matrix R contains both rDd and rDv, and the correlation coefficient between eid and eiv. Further details and statistics for the unknown parameters are provided in Gardoni et al. [10].

Z

^ ðt; x0 jSa ; T corr Þ þ hDv 1 Div ðt; x0 ; HD jSa ; T corr Þ ¼ d iv ^ ðt; x0 jSa ; T corr Þ þ hDv 2 d

ð11Þ

To estimate the fragility of a bridge, the values from the developed demand models are compared with the values of the corresponding capacities. Choe et al. [11] developed probabilistic capacity models for corroding RC columns. These models will be used here to develop fragility estimates. For completeness, the models are briefly reviewed in this section. 5.1. Probabilistic capacity models for pristine columns Gardoni et al. [14] and Choe et al. [15] developed probabilistic models for the deformation and shear capacity of pristine RC columns. The general form of the probabilistic capacity models is

C k ¼ C k ðx0 ; HC Þ;

ð13Þ

where, Ck is the capacity of interest, and HC is a set of unknown model parameters introduced to fit the capacity models to observed data. 5.2. Probabilistic demand models for corroding columns Choe et al. [11] extended the probabilistic capacity models for pristine RC columns to include the effects of corrosion. As described in the previous section for the demand model, for a given value of Tcorr, the deformation and shear capacity models are written as [11]

C d ðt; x0 ; HC jT corr Þ ¼ ^cd ðt; x0 jT corr Þ þ cCd ðt; x0 ; HCd jT corr Þ þ rCd eCd ; ð14Þ C v ðt; x0 ; HC jT corr Þ ¼ ^cv ðt; x0 jT corr Þ þ cCv ðt; x0 ; HC v jT corr Þ þ rCv eC v ; ð15Þ where, as for the demand models, the logarithmic transformation is used to stabilize the variance of the model, that is C d ðt; x0 ; HC jT corr Þ ¼ ln½dðt; x0 ; HC jT corr Þ and C v ðt; x0 ; HC jT corr Þ ¼ ln½v ðt; x0 ; HC jT corr Þ, d ¼ D=H is now the drift capacity, in which D is the displacement capacity, v ¼ V=ðAg ft0 Þ is the normalized shear capacity, in which V is the shear capacity, HC ¼ ðHCd ; HC v ; RÞ is a set of unknown model parameters introduced to fit the models to observed data, ^cd ðt; x0 jT corr Þ and ^cv ðt; x0 jT corr Þ denote the selected deterministic capacity models, which are expressed as the natural logarithm of the deterministic deformation and shear capacities, i.e., ln½^ dðt; x0 jT corr Þ and ln½v^ ðt; x0 jT corr Þ, respectively. The deterministic model for deformation capacity, ^ dðt; x0 jT corr Þ, includes the elastic component at the onset of yield as well as the inelastic component due to the plastic flow for a single corroded RC column. The elastic component of the drift considers (a) a flexural component based on a linear curvature distribution along the full column height, (b) a shear component of deformation due to shear distortion, and (c) a slip component; that is, the deformation due to the local rotation at the base caused by slipping of the longitudinal bar reinforcement. The quantities cCd ðt; x0 ; HCd jT corr Þ and cC v ðt; x0 ; HC v jT corr Þ represent

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correction terms introduced to capture the bias inherent in the deterministic models. Additional details on the capacity modeling are available in Choe et al. [11]. As in Eq. (10) for the proposed demand model, a generic probabilistic capacity model for a corroding RC column at time t is written as a function of time, t, as follows:

C k ðt; x0 ; Hk Þ ¼

Z

L

Ds

Kabt.

A

Kabt.

H

D

1

C k ðt; x0 ; Hk jT corr ÞfT corr ðt corr ÞdT corr :

A`

ð16Þ

A − A`

0

Assuming that the conditional capacity C k ðt; x0 ; Hk jT corr Þ during t 6 T corr is the same as the pristine capacity Ck(x0, Hk). Hence, Eq. (17) is rewritten as

C k ðt; x0 ; HÞ ¼ C k ðx0 ; Hk Þ½1  F T corr ðtÞ Z t þ C k ðt; x0 ; Hk jT corr ÞfT corr ðtcorr ÞdT corr :

Ksoil

ð17Þ

Fig. 1. Example single-bent over pass.

0

6. Fragility estimates Following Gardoni et al. [10], the conditional probability of failure for the pristine structure is written in terms of a given spectral acceleration Sa as

  F 0 ðSa ; HÞ ¼ P [ [fg ik ðx0 ; H; Sa Þ 6 0gjSa ; i

ð18Þ

k

where H = (HC, HD) and

g ik ðx0 ; H; Sa Þ ¼ C ik ðx0 ; HC Þ  Dik ðx0 ; HD jSa Þ;

k ¼ d;

v i ¼ 1; . . . ; s:

ð19Þ

For the purpose of the present study, the fragility of deteriorating RC bridge systems is expressed as

  Fðt; Sa ; HÞ ¼ P [ [fg ik ðt; x0 ; H; Sa Þ 6 0gjt; Sa ; i

k

ð20Þ Table 1 Variables for the pristine single-bent bridge.

where

g ik ðt; x0 ; Hk ; Sa Þ ¼ C ik ðt; x0 ; HC Þ  Dik ðt; x0 ; HD jSa Þ; k ¼ d;

v i ¼ 1; . . . ; s:

g ik ðt; Sa ; x0 ; HÞ ¼ g 0ik ðx0 ; H; Sa Þ½1  F T corr  Z t þ ½C ik ðt; x0 ; HC jT corr Þ 0

 Dik ðt; x0 ; HD jSa ; T corr ÞfT corr ðt corr ÞdT corr :

ð22Þ

The epistemic uncertainty in the models parameters H is incorporated in the fragility estimate by constructing predictive fragility ~ Sa Þ following Gardoni et al. [14] as estimates Fðt;

Z

Fðt; Sa ; HÞfH ðHÞdH:

Description

Parameter

Value/ distribution

Span length (right and left) Span-to-column height ratio Column-to-superstructure dimension ratio Concrete nominal strength Reinforcement nominal yield strength Initial longitudinal reinforcement ratio of column Initial transverse reinforcement ratio of column Soil stiffness based on NEHRP groups (FEMA-273, 1997) Additional bridge dead load (as a ratio of the dead weight)

L (mm) L/H D/Ds fc0 (MPa) fy (MPa)

Ksoil

18,300 2.4 0.75 LN(27.6, 2.76) LN(448.2, 22.4) 2.0% 0.7% B

r

N(0.1, 0.025)

ð21Þ

Using Eqs. (11) and (18), we can obtain the following relation between the limit state functions of the deteriorated and the pristine bridges:

~ Sa Þ ¼ Fðt;

bridge shown in Fig. 1 is a single-bent highway overpass with geometry and material properties that are representative of currently constructed highway bridges in California. The bridge was designed by Mackie and Stojadinovic [18] according to Caltrans’ Bridge Design Specification and Seismic Design Criteria [19]. Table 1 lists the design parameters of interest for the considered bridge. To estimate the corrosion initiation and propagation we assume that the bridge is constructed in a tidal zone with water-to-cement ratio of the concrete material equal to 0.5 with 1 day curing time, and that the zone is subjected to many humid-dry cycles Table 2. Fig. 2 shows the capacity degradation and the increase in the demand (computed for 2.0 g) of the example RC bridge system as a function of time. The figure shows the mean estimates along with the confidence bounds computed as ±1 standard deviation of the model error, rDk. The variations in the capacity and the demand

ð23Þ

The predicted fragility is then the expected value of F(t, Sa, H) over the distribution of H. In this paper, the analyses are carried out using OpenSees [16]. This is a comprehensive, open-source, object-oriented finite element software that also has reliability and response sensitivity capabilities [17]. In addition, OpenSees is extended in this study with the implementation of the probabilistic models for corrosion initiation, corrosion rate, and loss of reinforcement area. 7. Application In this section the developed methodology is applied to assess the seismic fragility of an example RC bridge. The selected RC

ql0 qs0

Table 2 Variables in the diffusion model used to estimate the corrosion initiation of the example bridge. Distribution

Mean

Normal 38.1 (mm) dc = Cover dept of concrete column Gamma 0.924 ke = Environmental correction factor Beta 2.4 kc = Curing time correction factor Normal 0.832 kt = Correction factor for tests Normal 473 D0 = Reference diffusion coefficient at (mm2/yr) t0 = 28 days n = Aging factor Beta 0.362 Normal 0.90 Ccr = Critical chloride content (mass-% of binder) Ccs = Chloride surface concentration (linear function of Acs and ecs,% binder) Normal 7.758 Acs = Parameter used in Ccs ecs = Parameter used in Ccs Normal 0

St. dev. 11.4 (mm) 0.155 0.7 0.024 43.2 (1012 m2/s) 0.245 0.15 by weight of 1.36 1.105

279

Deformation Demand and Capacity

5 0.14

t = 100 (year)

t = 0 (year)

D.-E. Choe et al. / Structural Safety 31 (2009) 275–283

0.12 0.10

2.5 2

0.06

1.5 1

0.04 20

40

60

80

0.5 0

100

0.7

0.6

0.6

3

0.7

0.7

0.6

3.5

Sa

0.7

0.7

4

0.08

0

0.7

4.5

0.6

0.6

0.6

0.5

0.5

0.5

0.5 0.4 0.4 0.3

0.5 0.4 0.4 0.3

0.5 0.4 0.4 0.3

0.3

0.2

0.2

0.3

0.2

0.2

0.1

20

40

0.3 0.2

0.2 0.1 60

80

01

100

Time t (Year)

Time (Year)

(a) Deformation failure mode

(a) Deformation (dashed), and deformation

and shear (solid) failure mode 5

Shear Demand and Capacity

0 .1 5

4.5 0.6

4

0 .1

3.5

0.5

t = 100 (year)

t = 0 (year)

0.7

0.4 0.3 0.2

3

3

Sa

2.5 2

0 .1 1

1.5

0.09

1 0.5 0

0.1 0

20

40

60

80

0.0 7 0.06

0.0 5 20

40

60

80

100

Time t (Year)

100

Time (Year)

(b) Shear failure mode

(b) Shear failure mode Fig. 4. Contour plot of the predictive fragility estimates. Fig. 2. Capacity degradation (solid line) and demand increment for Sa = 2.0 g (dash line) for the example RC bridge due to corrosion.

0.8

S a = 5 .0

0.7

S a = 4.0

0.6

S a = 3.0

0.5

S a = 2.0

0.4

0.16

S a = 5 .0

0.14

P(Failure|Sa )

P(Failure|Sa )

in the deformation failure mode are shown in the top plot. The variations in the capacity and the demand in the shear failure mode are shown in the bottom plot. In a heuristic manner, the plots in Fig. 2 illustrate that the overlapping area between the capacity and the demand distributions increase, which implies that the fragility increases over time due to corrosion. To account for uncertainties in the material properties we assume the following probability distributions: the compressive strength of concrete, fc0 , has the lognormal distribution with mean 27.6 MPa and 10% coefficient of variation, and the yield stress of the longitudinal reinforcement, fy, has the lognormal distribution with mean 448.2 MPa and 5% coefficient of variation. To consider the variability in the axial load for the single-bent overpass we assume the additional bridge dead load has the normal distribution with mean equal to 10% of the dead weight and a 25% coefficient of variation. The parameter values that enter into the probabilistic

models for the selected environmental and material conditions are assumed in accordance with the values provided by DuraCrete [5] (Appendix 1). ~ Sa Þ for the example single-bent bridge over Fig. 3 shows Fðt; time for given Sa for the deformation failure mode (dashed lines), the shear failure mode (dash-dotted lines), and the combined drift and shear failure mode (solid lines). Interestingly, we observe that the deformation failure model dominates the fragility. This is consistent with the design approach used by Caltrans [19]. As expected, we also observe that the fragility increases with time due to the corrosion of the reinforcement. ~ Sa Þ as a function of time t Fig. 4 shows the contour plots of Fðt; and spectral acceleration Sa. The contour lines connect pairs of values of time t and Sa that are associated with the same level of ~ Sa Þ. The top plot in Fig. 4 shows the fragility contours for the Fðt; deformation failure mode (dashed lines) and for the combined drift and shear failure mode (solid lines). The bottom plot in Fig. 4 shows the fragility contours for the shear failure mode. It can be

0.12

S a = 4 .0

S a = 3.0

S a = 2.0

0.10 0.08

S a = 1.0

0.06

0.3

S a = 1.0

S a = 0.5

0.04

0.2 0

20

40

60

80

100

Time (Year)

0.1 0.0 0

20

40

60

80

100

Time (Year) Fig. 3. Fragility estimates given Sa for deformation (dashed), shear (dash-dotted), and deformation and shear (solid) failure mode at intervals 1.2 g of Sa.

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D.-E. Choe et al. / Structural Safety 31 (2009) 275–283

~ Sa Þ seen that, due to the effects of corrosion, the same values of Fðt; can be obtained over time for smaller values of Sa. Finally, Fig. 5 compares the fragilities of the pristine and the deteriorated bridge using the developed probabilistic demand models. In the figure, the pristine bridge is compared with the bridge 100 years after construction. Again, changes in the fragilities as a function of time reflect the capacity degradation and demand increment for the different failure modes. 8. Sensitivity and importance measures It is argued that the use of sophisticated prediction models should generally be accompanied by the assessment of sensitivity and importance measures to assess the confidence in the results and to gain physical insights. Furthermore, sensitivity and importance measures can be employed in the decision making process for maintenance scheduling, as well as in a life-cycle cost analysis to determine where resources should be allocated to minimize the fragility of the structures in a network. The following distinction is made between sensitivity and importance measures. Sensitivity

measures are with respect to model parameters. They are essentially the derivative of the reliability index b or the corresponding probability with response to the model parameters. As such, sensitivity measures have different units and in general cannot be compared. Importance measures provide a ranking of the importance of random variables based on their effect on the variance of the limit state function. The sensitivity of the reliability index b is expressed as rðx0c ;Hf Þ b, which is the vector of derivatives of b with respect to the parameters ðx0c ; Hf Þ. For this formulation, the vector x0 is partitioned as x0 ¼ ðx0c ; x0p Þ, where x0c is the vector of deterministic parameters in the limit-state function and x0p is the vector of random variables. Also Hf is a set of distribution parameters, e.g., means, standard deviations, and correlation coefficients. In first-order reliability analysis (FORM), rðx0c ;Hf Þ b is obtained according to the reliability sensitivity analysis developed by Hohenbichler and Rackwitz [20] and Bjerager and Krenk [21]. In this study the reliability index is a function of time: b = b(t). In this case, the sensitivity is also a function of time and is expressed as rðx0c ;Hf Þ bðtÞ. The derivatives of the FORM reliability approximation of the failure probability, F~1 , is then obtained by the chain rule of the differentiation as

rðx0c ;Hf Þ F~1 ðtÞ ¼ u½bðtÞrðx0c ;Hf Þ b;

ð24Þ

0.8

t = 100 years

0.7

P(Failure|Sa)

0.6

t = 0 year

0.5 0.4

t = 100 years t = 0 year

0.3 0.2 0.1 0.0 0

1

2

3

4

5

Sa Fig. 5. Fragility estimates for the example bridge for deformation (dashed), shear (dash-dotted), and deformation and shear (solid) failure modes.

0.020

where /() is the standard normal PDF. Although the components of the gradient rðx0c ;Hf Þ bðtÞ (or rðx0c ;Hf Þ F~1 ðtÞ) have different units and thus cannot be employed for ranking of the parameters, it is valuable to plot the sensitivity of the fragility to changes in various parameters over time. Fig. 6 shows the sensitivity over time, reflecting the effects of corrosion. In the first years, the sensitivity with respect to the corrosion parameters (the parameters related to the corrosion initiation model) is shown to increase rapidly, since these parameters regulate the beginning of the corrosion propagation phase. Once the corrosion phase has begun, their sensitivities diminish. Moreover, the corrosion initiation is observed to have a more pronounced effect on the sensitivity measures for the deformation failure mode than for the shear failure mode.

E(XI)

0.015

0.04

E(n)

0.010

∇ ( x0 c ,Θ f )β(t ) 0.005

E(Ccr)

0.000

E(kC)

-0.005 -0.010

E(n) E(XI)

0.02

dC

∇ ( x0 c ,Θ f )β(t )

0.00 -0.02 -0.04

E(kt)

-0.06

0

10

20

E(kt)

30 -0.08

Time (year)

0

20

0.010

60

80

100

80

100

0.08

E(n)

0.07

E(Ccr)

E(n)

0.06

0.005

0.05

E(XI )

∇ ( x0 c ,Θ f )β(t )

∇ ( x0 c ,Θ f )β(t ) 0.04

E(D0)

0.000

0.03

E(kc) -0.005

40

Time (year)

(a) Deformation failure mode

0.02

E(kt) 0

E(Ccr)

0.01

10

20

Time (year)

30

0.00 -0.01

E(kt) 0

20

40

60

Time (year)

(b) Shear failure mode Fig. 6. Sensitivity measures of the means of diffusion variables for deformation (top) and shear (bottom) failure modes for Sa = 2.0 g.

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D.-E. Choe et al. / Structural Safety 31 (2009) 275–283

Fig. 7 illustrates the sensitivity with respect to the parameters related to the structural model. The parameters include the mean of several random variables, as well as several deterministic structural parameters. Different variations over-time are observed for

the deformation and shear failure modes. For the deformation failure mode, the sensitivity to changes in dsp0 increases over time and the sensitivity to changes in db0 is approximately constant with a soft peak at the transition from the diffusion phase to the corrosion phase. In contrast, for the shear failure mode, the reliability index becomes less and less sensitive to changes in db0 as the corrosion develops. The trends observed in Fig. 7 are consistent with the form of the capacity model used in this study. Specifically, the effect of the longitudinal reinforcement on the shear capacity is included as a correction term in the capacity models developed by Gardoni et al. [14]. In that study, the significance of this term was assessed using 106 experimental observations through a Bayesian framework. In the present study, the effect of the longitudinal reinforcement on the capacity is confirmed by its sensitivity for the deteriorated structure. Furthermore, a negative value of the sensitivity with respect to the spacing of the transverse reinforcement, S, is observed. As expected, this implies that the fragility decreases when S increases. Each random variable has a different contribution to the variability of the limit state function. Important random variables have a larger effect on the variability of the limit state function than less important ones. Following Der Kiureghian and Ke [22] and Haukaas and Der Kiureghian [23], a measure of importance c can be defined as

0.7 0.6

dsp0

0.5 0.4

∇( x0 c ,Θ f )β(t ) 0.3 0.2

db0

0.1

E(fy)

0.0

E(f 'c)

S

-0.1 0

20

40

60

80

100

Time (year)

(a) Deformation failure mode 0.7 0.6

db0

dsp0

0.5

cT ¼

0.4

∇( x0 c ,Θ f )β(t ) 0.3

E(f 'c)

0.0

E(fy)

S

-0.1 0

20

40

60

80

ð25Þ

where z is the vector of the random variables, z = (xp, H) and Ju*,z* is the Jacobian through which the probability is transformed from the original space z into the standard normal space u, with respect to the coordinates of the most likely failure realization, z* , SD0 is the standard deviation matrix of the equivalent normal variables z0 which are defined by the linearized inverse transformation z0 = z* + Jz*,u*(u  u*) at the most likely failure realization. The matrix SD0 consists of the elements that are the square root of the corresponding diagonal elements of the covariance matrix R0 ¼ Jz ;u JTz ;u of the variables z0 .

0.2 0.1

aT Ju ;z SD0 ; kaT Ju ;z SD0 k

100

Time (year)

(b) Shear failure mode Fig. 7. Sensitivity measures of the means of structural variables for deformation (top) and shear (bottom) failure mode for Sa = 2.0 g.

0.0010

0.03

γ

0.02

Acs

ke

kt

0.0000

γ

D0

0.01

Ccr

-0.0005 XI

n -0.0010

dC

εcs

kc

0.0005

0

kc

0.00 10

20

XI

30

Time (year)

n

-0.01 0

20

40

60

80

100

80

100

Time (year)

(a) Deformation failure mode 0.0010 0.0005

γ

εcs

kc

0.025

Acs

0.020

ke

0.015

D0

XI --0.0005

dC

0.010

kt

0.0000

γ

Ccr

0.005

kc

0.000 -0.005

n

-0.010

-0.0010 0

5

10

15

20

Time (year)

25

30

-0.015

n

-0.020 0

(b) Shear failure mode

20

40

60

Time (year)

Fig. 8. Importance measures for the diffusion variables for deformation (top) and shear (bottom) failure modes for Sa = 2.0 g.

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D.-E. Choe et al. / Structural Safety 31 (2009) 275–283

0.05

dC

0.00

γ

f 'c fy

-0.05 -0.10 -0.15

γ

θCδ7

θDδ2 θCδ1

θCδ11

0

10

20

30

Time (year)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5

εDδ

θDδ1

dC

θCδ11 εCδ 0

20

40

60

80

100

80

100

Time (year)

(a) Deformation failure mode 1.0

0.05 σDv 0.00 σ Cv

f 'c

dC

εDv

0.8 0.6

γ -0.05

θCv2

γ

θCv4

-0.10

θDv1

0.2

θDv2

-0.15

0.4

0.0

0

20

40

60

80

100

Time (year)

-0.2

εCv 0

20

40

60

Time (year)

(b) Shear failure mode Fig. 9. Importance measures of the structural variables for deformation (top) and shear (bottom) failure modes for Sa = 2.0 g.

Fig. 8 shows components of the importance vector c for the random variables of the diffusion model as a function of time. It is observed that the parameters related to the deformation failure mode are highly significant for the corrosion initiation process, while those for the shear failure mode are less important. It is concluded that for the bridge under consideration the shear failure mode is less affected by corrosion compared with the deformation failure mode. Fig. 9 presents the values of the importance vector associated to the random variables that represent the structural model. For the deformation failure mode the uncertainties in eDd and eCd are seen to be the most important. This observation is in agreement with the results in Choe et al. [15,11]. It is furthermore observed that the uncertainties in fc0 and fy are more important for the corrosion process than those in H. The bottom plot in Fig. 9 is associated with the shear failure mode, in which it is observed that eDv and eCv represent the most important sources of uncertainty. It is also observed that the importance of hDv3, which represents the effect of deformation in the shear failure mode, increases as the corrosion propagates. This implies that the two failure modes are correlated and that this relationship is affected by the corrosion process.

Condition

Distribution

Mean[mm2/yr]

St. dev.[10-12 m2/s]

w/c=0.4 w/c=0.45 w/c=0.5

Normal Normal Normal

220.9 315.6 473

25.4 32.5 43.2

9. Conclusions

n: Aging factor

This paper develops probabilistic models for seismic demand in the deformation and shear of corroding RC bridge systems. It is shown that the demands on the bridge increase as the corrosion process unfolds. The demand models are applicable to bridges with different combinations of chloride exposure condition, environmental oxygen availability, water-to-cement ratios, and curing conditions. The demand models are used to develop predictive seismic fragility estimates for corroding RC bridges. The fragility estimates account for both the capacity degradation and the demand increase due to corrosion. Model uncertainties in the demand, capacity

and corrosion models are accounted for, in addition to the uncertainties in the environmental conditions, material properties, and structural geometry. The models are applicable to existing and new columns that are subject to current or future deterioration. As an application, the presented models and methodology are applied to an example bridge structure. Fragility estimates are obtained at different time during the service life of the bridges along with sensitivity and importance measures. It is observed that the presence of corrosion alters the sensitivity and importance measures over time. The developed fragility estimates and the sensitivity and importance measures can be employed in service-life and life-cycle cost analyses. Appendix 1 Do: Reference diffusion coefficient at t0 = 28 days

Condition

Distribution

Mean

St. dev.

A

B

All

Beta

0.362

0.245

0

0.98

ke: Environmental correction factor Condition

Distribution

Mean

St. dev.

Submerged Tidal Splash Atmospheric

Gamma Gamma Gamma Gamma

0.325 0.924 0.265 0.676

0.223 0.155 0.045 0.114

D.-E. Choe et al. / Structural Safety 31 (2009) 275–283

kc: Curing time correction factor Condition

Distribution

Mean

St. dev.

A

B

curing curing curing curing

Beta Beta Deterministic Beta

2.4 1.5 1.0 0.8

0.7 0.3

1.0 1.0

4.0 4.0

0.1

0.4

1.0

1day 3day 7day 28day

kt: Correction factor for tests Condition

Distribution

Mean

St. dev.

All

Normal

0.832

0.024

XI: Modeling uncertainty Condition

Distribution

Mean

St. dev.

All

Lognormal

1

0.05

Acs and ecs: Parameters in Cs Condition

Submerged Tidal Splash Atmospheric

Distribution

Normal Normal Normal Normal

ecs

Acs Mean

St. dev.

Mean

St. dev.

10.348 7.758 7.758 2.565

0.714 1.36 1.36 0.356

0 0 0 0

0.58 1.105 1.105 0.405

Ccr: Critical chloride content (mass % of binder)

Constantly saturated

Constantly humid or many humid-dry cycles

w/c ratio

Distribution

Mean

St. dev.

0.30 0.40 0.50 0.30 0.40 0.50

Normal Normal Normal Normal Normal Normal

2.30 2.10 1.60 0.50 0.80 0.90

0.20 0.20 0.20 0.10 0.10 0.15

References [1] Tuutti K. Corrosion of steel in concrete Swedish. Cement and Concrete Research Institute Stockholm; 1982.

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[2] Liu Y, Weyers RE. Modeling the time-to-corrosion cracking in chloride contaminated reinforced concrete structures. ACI Mater J 1998; 95(6):675–81. [3] Thoft-Christensen P, Jensen FM, Middleton CR, Blackmore A. In: Frangopol DM, Corotis RB, Rackwitz R, editors. Assessment of the reliability of concrete slab bridges reliability and optimization of structural systems. Pergamon; 1997. [4] Thoft-Christensen P. Corrosion and cracking of reinforced concrete. In: Lifecycle performance of deteriorating structures, assessment, design and management, ASCE; 2003. [5] DuraCrete. Statistical quantification of the variables in the limit state functions. The European Union Brite EuRam 3 contract BRPR-CT95-0132 Project BE95-1347 Report no BE95-1347/R7; May 2000. [6] Stewart MG, Rosowsky DV. Structural safety and serviceability of concrete bridges subject to corrosion. J Infrastruct Syst ASCE 1998;4(4): 146–55. [7] Vu KAT, Stewart MG. Structural reliability of concrete bridges including improved chloride-induced corrosion models. Struct Safety 2000;22(4): 313–33. [8] Enright MP, Frangopol DM. Probabilistic analysis of resistance degradation of reinforced concrete bridge beams under corrosion. Eng Struct 1998;20(11): 960–71. [9] Enright MP, Frangopol DM. Service-life prediction of deteriorating concrete bridges. J Struct Eng ASCE 1998;124(3):309–17. [10] Gardoni P, Mosalam KM, Der Kiureghian A. Probabilistic seismic demand models and fragility estimates for RC bridges. J Earthquake Eng 2003;7(Special Issue 1):79–106. [11] Choe D, Gardoni P, Rosowsky D, Haukaas T. Probabilistic capacity models and seismic fragility estimates for RC columns subject to corrosion. Reliab Eng Syst Safety 2007;93:383–93. [12] Chopra AK, Goel RK. Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures: SDF systems report no PEER1999/02 Pacific Earthquake Engineering Research Center University of California Berkeley; 1999. [13] Fajfar P. A nonlinear analysis method for performance-based seismic design. Earthquake Spect 2000;16(3):573–92. [14] Gardoni P, Der Kiureghian A, Mosalam KM. Probabilistic capacity models and fragility estimates for RC columns based on experimental observations. J Eng Mech ASCE 2002;128(10):1024–38. [15] Choe D, Gardoni P, Rosowsky D. Closed-form fragility estimates parameter sensitivity and bayesian updating for RC columns. J Eng Mech ASCE 2007; 133(7):833–43. [16] McKenna F, Fenves GL. An object-oriented software design for parallel structural analysis. In: Proceedings of the 2000 structures congress and exposition advanced technology in structural engineering, Philadelphia, Pennsylvania; 2000. [17] Ditlevsen O, Madsen HO. Structural reliability methods. New York: Wiley; 1996. [18] Mackie K, Stojadinovic B. Probabilistic seismic demand model for California highway bridges. J Bridge Eng ASCE 2001;6(6):468–81. [19] Caltrans. Seismic Design Criteria. Ed. 1.1. California Department of Transportation; 1999. [20] Hohenbichler M, Rackwitz R. First-order concepts in system reliability. Struct Safety 1983;1(3):177–88. [21] Bjerager P, Krenk S. Parameter sensitivity in first order reliability theory. J Eng Mech ASCE 1989;115(7):1577–82. [22] Der Kiureghian A, Ke J-B. Finite-element based reliability analysis of frame structures. In: Proceedings of ICOSSAR‘85, 4th international conference on structural safety and reliability, Kobe, Japan; 1995. p. 395–404. [23] Haukaas T, Der Kiureghian A. Parameter sensitivity and importance measures in nonlinear finite element reliability analysis. J Eng Mech ASCE 2005;131(10): 1013–26.