Corrosion-induced cracking fragility of RC bridge with improved concrete carbonation and steel reinforcement corrosion models

Engineering Structures 208 (2020) 110313

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Corrosion-induced cracking fragility of RC bridge with improved concrete carbonation and steel reinforcement corrosion models

T

⁎

Bo Suna,b, Ru-cheng Xiaob, , Wei-dong Ruana, Peng-bo Wangc a

Department of Civil Engineering, Zhejiang University of Technology, D515 Structural Engineering Building, 288 Liuhe Road, Hangzhou, Zhejiang 310000, China Department of Bridge Engineering, Tongji University, 619 Bridge Building, 1239 Siping Road, Shanghai 200092, China c Beijing Urban Construction Design & Development Group Co., Limited, No. 5, Fuchengmen Beidajie, Xicheng District, Beijing 100034, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Reinforced concrete bridge Concrete carbonation Steel reinforcement corrosion Probabilistic model Cracking fragility

Corrosion-induced cracking is one important limit state in the durability performance analysis of the reinforcement concrete (RC) structures. A comprehensive probabilistic approach is established for the corrosioninduced cracking fragility analysis of the RC bridge in the urban atmospheric condition with improved concrete carbonation and steel reinforcement corrosion models. The improved deterioration models have a probabilistic formation by adding correction and error terms to the existing deterministic models. The deterministic parts are selected from reviews and discussions on existing models. The correction terms are sets of explanatory functions representing the influencing factors related to the potential bias in the deterministic models. The statistical distributions of unknown model parameters are calibrated and the optimum collections of the explanatory functions are selected through the Bayesian rule based on long-term data from onsite tests or natural exposure experiments. The probabilistic analysis approach for the corrosion-induced cracking fragility are generated based on the improved deterioration models. Fragility curves, parameter sensitivities and random variable importance are achieved for the example RC bridge. The results show that increases on the concrete strength, cover depth and steel bar diameter, or decrease on the CO2 density, are efficient countermeasures to improve the durability performance of the RC bridge against corrosion-induced cover cracking and the uncertainty of the problem mainly comes from the concrete carbonation model.

1. Introduction Steel reinforcement corrosion is a long-lasting threat to RC bridges during life time and includes two stages: initiation stage and propagation stage [1]. For the first stage, concrete carbonation is considered in this paper since it is a commonly occurred phenomenon in the urban area (atmospheric condition) with the fast growth of CO2 concentration due to the industrial prosperity [2]. Concrete carbonation is a chemical process that weakens the protection provided by the concrete to the reinforcement and initializes the corrosion of the reinforcement. For the second stage, corrosion propagation and accumulation of the rust products lead to cracking of the concrete cover. Moreover, corrosion-induced cracking is one important limit state in the durability performance analysis of the RC bridge. The deterioration process from concrete carbonation, corrosion initiation, rust accumulation to cover cracking contains uncertainties from environmental, material and structural properties. While there are abundant researches in the deterministic sense [3–5] (experimentally or theoretically), the

⁎

probabilistic studies on the carbonation-related durability performance of RC bridge in the urban atmospheric condition are in limited numbers. Most researchers focus on chloride-related corrosion in the marine environment when doing probabilistic analysis of the RC bridge [6–8]. However, the amount of RC bridges in the urban cities is much larger than that in the marine conditions and carbonation is a parallel phenomenon to the chloride penetration even in the marine area. Thus a comprehensive probabilistic evaluation on the carbonation-related durability performance for the RC bridges with respect to the corrosioninduced cracking limit state should attract more attentions. Some researchers set the depassivation of the steel reinforcement as the limit state and study the durability performance of RC structures in the first stage. Kwon and Na [2] obtained the carbonation durability failure through Monte Carlo simulation based on concrete carbonation velocities measured from filed investigations. Na et al. [9] proposed a stochastic approach for predicting service life of RC structure subjected to carbonation with the spatial variation of material, geometric properties of RC structures, and environmental factors. Teplý et al. [10]

Corresponding author. E-mail addresses: [email protected] (B. Sun), [email protected] (R.-c. Xiao), [email protected] (W.-d. Ruan), [email protected] (P.-b. Wang).

https://doi.org/10.1016/j.engstruct.2020.110313 Received 11 August 2019; Received in revised form 29 December 2019; Accepted 29 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

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Nomenclature

z∗, u∗

Roman case letters

Greek case letters

Cl cCO2 DCO2 dc db0 d1 Ec Eef fcuk fct fc28 Icorr kp mCO2 RH s T t ti tp Wrust Wcrit w c X x xd xr z

Cl− content CO2 density around concrete surface CO2 diffusion coefficient in the concrete concrete cover depth uncorroded steel bar diameter free pore band thickness at steel/concrete interface concrete original elastic modulus concrete effective elastic modulus concrete characteristic compression strength concrete tensile strength concrete 28-day compression strength corrosion rate rust accumulation coefficient amount of CO2 absorbed by unit concrete relative humidity given boundary condition temperature service time corrosion initiation time corrosion propagation time rust product amount critical rust product amount water to cement ratio concrete carbonation depth design parameter deterministic design parameter random design parameter random variable

σ ε αrust ρrust φcr νc β λ Θ, θ θg θz

design point

standard variation standard normal random variable rust type coefficient mass density of the rust product creep coefficient of concrete Poisson’s ratio of concrete reliability index importance measure model parameter deterministic design parameter distribution parameter

Function case letters

F (∙) fragility ∼ predictive fragility estimate F (∙) f (∙) posterior distribution g (∙) limit state function explanatory function hi (∙) Jacobian matrix J(∙) likelihood function L (∙) prior distribution p (∙) X (∙), Ico rr (∙) improved probabilistic model  (∙), Icô rr (∙) deterministic model X σε model error correction term γ (∙) gradient vector ∇ (∙)

are generated based on the improved models of concrete carbonation and steel reinforcement corrosion. Fragility curves with both the predictive estimates and confidence bounds are obtained for the example bridge based on the proposed probabilistic approach and improved deterioration models. Parameter sensitivity and random variable importance are also analyzed to provide guidance on the design and model optimization for the durability performance of the RC bridge. The rest of the paper is organized as follows. Section 2 and Section 3 review existing models and generate the improved models for the concrete carbonation and steel reinforcement corrosion. Section 4 discusses the fragility analysis approach based on the improved deterioration models; fragility curves, parameter sensitivities and random variable importance are obtained for the example bridge to illustrate this process. Section 5 draws some general conclusions.

studied the durability design and provide comments on the assessment of RC structures in relation to existing codes. Some researchers move a further step to include the steel reinforcement corrosion stage in the durability probability analysis. Teplý [11] studied the effect of and compromise between the service life and the time-varying reliability level of deteriorating RC structures. Niu [12] established efficient corrosion models and studied the corrosion initiation and cracking probabilities of RC structures. Marques and Costa [13] conducted both safety factor and probabilistic approaches to estimate the service life periods of RC structures with carbonation performance properties determined from experimental projects. However, those valuable attempts have limitations: (1) The concrete carbonation and steel reinforcement corrosion models are simply chosen from the existing models without detailed discussions and remain deterministic. (2) The uncertainties accounted in those works mainly focus on aleatory aspect on material and structural properties. (3) The epistemic uncertainties arise from the modeling and analysis process (assumptions and simplifications) are commonly ignored, which are actually important part in the probabilistic analysis. In this paper we develop improved deterioration models for the concrete carbonation and steel reinforcement corrosion by adding correction terms to the existing deterministic models. The deterministic parts are selected based on reviews/discussions of the existing models and the correction terms are sets of explanatory functions representing the influencing factors widely concerned in the current research to maximize the scientificity and adoptability of the improved models. The statistical distributions of the unknown model parameters are calibrated and the optimum collections of the explanatory functions are selected through the Bayesian rule based on long-term data from the onsite tests or natural exposure experiments. A probabilistic analysis approach for the corrosion-induced cracking fragility of the RC bridge

2. Concrete carbonation 2.1. Theoretical model The carbon dioxide (CO2) in the atmosphere will penetrate into the concrete cover and react with the carbonatable substance in the concrete [14]. The carbonation depth X is used to measure the carbonation process. The widely used theoretical model for X is derived by adopting the assumption that the diffusion process obeys the Fick’s first law [1]

X=

2DCO2 cCO2 mCO2

t (1)

where X = distance from the carbonation frontier to the concrete surface at service time t , DCO2= diffusion coefficient of CO2 in the concrete and cCO2= CO2 density around the concrete surface. mCO2 is the amount of CO2 absorbed by unit volume of concrete and a theoretical solution is 2

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2.3. Probabilistic model formation

given by Papadakis [15] according to the mass-balance condition of the carbonatable substance

X=

2DCO2 cCO2 [Ca (OH )2 ]0 + 3[CSH ]0 + 3[C3 S ]0 + 2[C2 S ]0

t

Following the general probabilistic model development process proposed by Sun et al. [22,23] and Gardoni et al. [24,25], we define the probabilistic model formula for the concrete carbonation depth by adding correction and error term to the existing deterministic model

(2)

where [Ca (OH )2]0 , [CSH ]0 , [C3 S ]0 and [C2 S ]0 are the initial densities of the carbonatable substance in the concrete.

 (x) + γ (x, θ) + σε X (x, Θ) = X X

(3)

where X (x, Θ)= improved probabilistic model for the carbonation depth, x= design parameters accounting for the aleatory uncertainties in the carbonation analysis, Θ = (θ, σ )= unknown model parameters  (x)= existing determithat need detailed calibration and selection, X nistic model, γX (x, θ)= correction term and σε is the error term which accounts for the epistemic uncertainties together with Θ. Three assumptions are applied to the above probabilistic model formation:

2.2. Empirical models The exact values of the model parameters DCO2 , [Ca (OH )2]0 , [CSH ]0 , [C3 S ]0 and [C2 S ]0 in the theoretical models are usually hard to achieve. Thus engineers have to turn to efficient empirical models in the engineering practice. The existing empirical models can be divided into three types with respect to the key factors used to predict the carbonation depth: (1) test-based model, (2) w/c-based model and (3) strength-based model. Some of the existing empirical models for the concrete carbonation are listed in Table 1 for better understanding of the discussions below. Table 1 shows that the test-based models require specific characteristics achieved from the laboratory test to conduct the carbonation prediction, e.g., the carbonation resistance in the FIB model [16]/CEB model [17]. The dependency of the laboratory test restricts the application of the test-based models in the design practice and the high cost for the repeatability makes them unsuitable for a probabilistic design scheme. The other two types of models choose the commonly used characteristics of the concrete (water to cement ratio or compression strength) to construct the prediction models, together with other possible influencing factors including the material composition, curing condition and environmental condition. However, more discussions are needed on those model parameters: some of them are not convenient to achieve in the design practice; some of them may have minor influence on the prediction results of the carbonation depth. Moreover, all the existing models are in a deterministic sense, which is inconsistent with the probabilistic requirement of the modern design. The discussions above show that improvements are required on the existing models for the prediction of the concrete carbonation depth. The improved model should be probabilistic and the model parameters should be well calibrated and selected to reflect the most sensitive characteristics influencing the concrete carbonation process.

(1) Homoscedasticity assumption – the model variance σ 2 (a quantitative measure of the model error) is independent of the design  (x) that is related to specific parameter x . All the potential bias in X design properties are considered and corrected by the correction term γX (x, θ) . (2) Normality assumption – the model prediction error of the improved probabilistic model has a normal distribution (ε ~N (0, 1) ). The prediction results for a given set of design value x are unbiased (uniformly distributed around the median prediction).  (x) , correction (3) Additivity assumption – the deterministic model X term γX (x, θ) and error term σε are in an additive form. The linear relationship of the model components is convenient for the formulation of the parameter calibration and selection process. Proper transformation of model input x and output X is required in order to satisfy the three assumptions. In this paper we adopt the natural logarithm as the variance-stabilizing transformation and Eq. (3) is rewritten as

 (x)] + γ (x, θ) + σε ln[X (x, Θ)] = ln[X X

(4)

The proposed probabilistic model for the concrete carbonation depth is unbiased and accounts for all types of uncertainties.

Table 1 Existing empirical models for the concrete carbonation. Authors

Type

Model

Description

FIB (2010) [16]

test-based

X=

−1 2k e k c RNAC,0 cCO2 W (t ) t

CEB 238 (1997) [17]

test-based

X=

k 0 k1 k2

Teply (2010) [10]

w/c-based

X=

2cCO2

k e= environmental function, k c= execution transfer parameter, W (t )= weather function and RNAC,0 is defined as the carbonation resistance obtained from the laboratory test k 0= testing method factor, k1= relative humidity factor, k2= curing factor and RC65 is the tested carbonation resistance in CEB 238 c= cement content, w= water content, P= concrete admixture content, k= concrete admixture factor, ρc = mass density of cement and f (RH )= relative humidity factor

t

RC 65

1.09DCO2 cCO2 10−6 0.218(c + kP )

t 3

⎛

DCO2 = 6.1 ×

⎞

w − 0.267(c + kP ) 10−6 ⎜ ⎟ c + kP +w ⎟ ⎜ 1000 ρc ⎝ ⎠

× f (RH )2.2 Japan model (1963) [18]

w/c-based

Zhu (1992) [19]

w/c-based

CECS (2007) [20]

strengthbased

Smolczyk (1962) [21]

strengthbased

w c > 0.6 X = rc ra rs

w c − 0.25 0.3(1.15 + 3w c )

w c ⩽ 0.6

4.6w c − 1.76 7.2

X=

X = rc ra rs

(

w γ1 γ2 γ3 12.1 c

)

− 3.2

t

w c= water cement ratio, rc= cement type factor, ra= aggregate type factor and rs= concrete admixture factor

t

t

X = 3K CO2 K kl K kt K ks KF

γ1= cement type factor, γ2= fly ash factor and γ3= environment factor see Appendix A.

58 × T1 4RH1.5 (1 − RH ) ⎛ − 0.76⎞ t ⎝ fcuk ⎠

X = 250 ⎛ ⎝ ⎜

1 Fc

−

1 Fg

⎞ t ⎠

Fc= concrete compression strength and Fg= critical un-carbonized strength of concrete

⎟

3

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environmental condition, we select h4 (x) = cCO2 c0 to capture the possible influence of CO2 density, h5 (x) = T T0 to reflect the potential influence of temperature and h6 (x) = RH RH0 to represent the influence of relative humidity, where T0 = 13oC and RH0 = 0.7 are the reference annual temperature and relative humidity respectively. The data {Xi , x i} used to calibrate the model parameter Θ for the probabilistic carbonation depth model are all collected from the onsite carbonation tests on the real bridges in service [29–35]. The onsite data reflect the deteriorative properties of the concrete in the exact service conditions. The test results of the concrete carbonation depth at different service times are collected with the corresponding concrete properties and environmental conditions. The data ranges collected are listed in Table 2.

Table 2 Data ranges collected for the concrete carbonation depth. Variable

Symbol

Range

Service time (year) Characteristic strength (MPa)

t fcuk cCO2 T RH

2–50 15–55

CO2 density (%) Annual temperature (oC) Annual relative humidity

0.03–0.0432 1.2–21.4 0.5–0.81

2.4. Deterministic model

 (x) is adopted to account for the existing The deterministic model X knowledge and experience on the concrete carbonation process. Thus  (x) should be selected or constructed based on the existing models X discussed in Sections 2.1 and 2.2 to reflect the physical rules in the carbonation process and maximize the acceptance of the improved probabilistic model. The theoretical and empirical models clearly show a linear relationship between the carbonation depth and the square root of time, i.e.,

 (x) = K (x) t X

2.6. Model parameter calibration and selection The unknown model parameters Θ = (θ, σ ) are defined as set of random variables to capture the epistemic uncertainties. With the se (x) , {hi (x)} and the onsite carbonation data {Xi , x i} , the unlected X known statistical properties of Θ are estimated through the Bayesian approach [36]

(5)

f (Θ) = κL (Θ) p (Θ)

where K (x) is the carbonation coefficient function describing the influences of different factors on the concrete carbonation, e.g., K (x) = 2DCO2 cCO2 mCO2 in Eq. (1). The CO2 density cCO2 is easy to quantify in the engineering practice and is an important influencing factor that widely been used in the theoretical and empirical models; thus we adopt it in the deterministic model. For other factors, considering that under the natural logarithm transformation, they could be transferred into the correction term γX (x, θ) and well calibrated/selected during the model parameter calibration and selection process. The probabilistic carbonation depth model in Eq. (4) is rewritten as

ln[X (x, Θ)] = ln

cCO2 t + γX (x, θ) + σε c0 ⏟ X (x)

f (Θ) where is the posterior distribution of Θ and κ = [ ∫ L (Θ) p (Θ) dΘ]−1 is a normalizing factor. f (Θ) incorporates two aspects of available information: (1) the prior knowledge on Θ before the consideration of the onsite test data, which is represented by the prior distribution p (Θ) , and (2) the objective information from the onsite carbonation tests, which is represented by the likelihood function L (Θ) . Since the measured data set {Xi , x i} (the ith observation) is the exact carbonation depth Xi with respect to the ith set of design value x i , L (Θ) has the following form L (Θ) ∝

2.5. Correction term and test data The correction term γX (x, θ) is adopted to capture the carbonation  (x) . A coefficient function K (x) and correct the potential bias in X linear form for γX (x, θ) is adopted in this paper k

∑ θi hi (x) i=1

∏ P [σεi = Xi − Xi (xi) − γX (xi, θ)]

(9)

i (x i) − γ (x i, θ) is the prediction bias of the deterministic where Xi − X X  (x) for the ith observation. More details on the Bayesian model model X parameter calibration process can be found in Sun et al. [22,23] and Gardoni et al. [24,25]. The posterior distribution f (Θ) is computed by the Markov Chain Monte Carlo (MCMC) simulation [37] based on the Bayesian approach described above. For p (Θ) , a non-informative prior is selected in order to minimize the influence of p (Θ) since our prior knowledge on it is unavailable [36]. A stepwise model selection process is applied to select the most sensitive explanatory functions {hi (x)} (the optimum combination) step by step for the improved probabilistic model:

(6)

where c0 = 0.03% is the standard CO2 density in the air.

γX (x, θ) =

(8)

(7)

(1) For the first step, calibrate the posterior statistics of the model parameter Θ for the whole bunch of explanatory functions {hi (x)} ; (2) In the second step (first deletion step), delete the most insensitive explanatory function hi (x) (least informative) with the largest coefficient of variation for θi and recalibrate the model parameter for the rest explanatory functions; (3) Repeat the deletion process step by step. The optimum combination for the explanatory functions arises when there is an unacceptable increase on the model error (indicated by the posterior mean of σ ) in the next step; (4) The selected model keeps a balance between model accuracy and model conciseness.

where Σ[θi hi (x)], i = 1, ...,k , can be treated as a transformation of K (x) and {hi (x)} is a set of “explanatory” functions capturing specific factors that may influence the prediction accuracy of the concrete carbonation depth. The explanatory function hi (x) should be selected to reflect the possible influencing factors indicated by the existing models in Table 1 and based on the available test data. Firstly, h1 (x) = 1 is selected to represent the constant part in K (x) and capture the potential bias that is independent of x . Secondly, some literatures [13,26] pointed out that the relationship between the carbonation depth X and service time t may not strictly following the square root order. We select h2 (x) = ln t to evaluate this hypothesis. Other explanatory functions account for the two main aspects of factors influencing the concrete carbonation: concrete quality and environmental condition. For the concrete quality, since the characteristic compression strength is a comprehensive index reflecting the concrete quality and is more convenient to measure/record in the engineering practice compared with the material composition, we select h3 (x) = fcuk fC30 , where fC30 = 30 Mpa is the characteristic compression strength for concrete grade C30 [27,28]. For the

The stepwise calibration and selection process for X (x, Θ) is shown in Fig. 1. The mean value for σ (representing the model accuracy) and coefficient of variation (COV) for θi (representing the sensitivity of θi hi (x) ) obtained are shown in Fig. 1 for each step. For Step 1, the mean value of σ is 0.370 and the largest COV = 2.07 appears for θ6, showing that h6 (x) is the most insensitive explanatory function related to the 4

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beside the 1:1 line and two dashed lines are drawn to shown the ± 1 SD of the median predictions. The improved model of concrete carbonation depth is unbiased and more accurate. Moreover, the formation of the improved model is probabilistic in nature and considers all types of uncertainties. 3. Steel reinforcement corrosion 3.1. Corrosion initiation The high alkalinity of newly built concrete provide a protective passive layer (concrete cover) to the steel reinforcement against environmental corrosion. The carbonation process decreases the alkalinity of the concrete and thus destroys the protective effect of the concrete cover [11]. The steel corrosion happens when the carbonation frontier reaches the surface of the steel reinforcement, i.e.,

X (xX |t = ti , ΘX ) = dc

where ti is the corrosion initiation time and dc is the distance from the surface of the concrete to the steel reinforcement. Based on the improved model for X (xX , ΘX ) in Eq. (11), ti is obtained as

Fig. 1. Stepwise model selection process for X (x, Θ) where a superposed cross (×) indicates term to be removed.

ln ti (x ti, Θti) =

+ σX εX

) (13)

0.5 + θX 2

The corrosion of the steel reinforcement will propagate after the steel is depassivated (t > ti ). During this phase, the corrosion depth is used to describe the propagation process [38,39], i.e.,

(10)

p (t ) =

∫t

t

i

0.0116Icorr dt

(14)

where t − ti = tp is the corrosion propagation time and Icorr is the corrosion rate measured as a current density of the corroded steel bar. Icorr is an important quantity in describing the corrosion propagation of the steel reinforcement. The existing models for the corrosion rate of the steel reinforcement are listed in Table 4. The existing models can be divided into two main types: constant models and time-variant models. The various influencing factors considered in the existing models mainly reflect the surrounding conditions of the steel reinforcement including the concrete condition and environmental condition. However, those existing models remain deterministic in nature and the influencing factors are not discussed and selected in a systematical and comprehensive manner. Improvements should be applied on the corrosion rate model following the general process in Section 2.

2.7. Model summary The final form of the improved model for the concrete carbonation depth is

f cCO2 ⎤ t + θX 1 + θX 2 ln t + θX 3 cuk + σX εX c0 ⎥ f30 ⎦

f30

3.2. Corrosion propagation

The posterior statistics (mean, standard deviation and correlation coefficient) of model parameters Θ = (θ1, θ2 , θ3 , σ ) for the selected γX (x, θ) are listed in Table 3. The selection of θ1 indicates that there is a constant value in the carbonation coefficient K (x) that is independent of any possible influencing factors. The selection of θ2 proves that the carbonation depth X dose not strictly follow a linear relationship with respect to the square root of service time t . The corrected exponent value of t is 0.5 + θ2 based on the onsite data in this paper. The survival of θ3 shows that the compression strength is an important property influencing the carbonation performance of the concrete and capturing the potential bias in the deterministic model.

ln[X (xX , ΘX )] = ln ⎡ ⎢ ⎣

fcuk

where ti (x ti, Θti) is the probabilistic model for the corrosion initiation time of the steel reinforcement, in which x ti = (cCO2, fcuk , dc ) and Θti = ΘX .

fcuk f30

(

ln(dc c0 cCO2 ) − θX 1 + θX 3

prediction bias. Thus θ6 h6 (x) is removed from γX (x, θ) and the recalculated mean value of σ remains 0.370, i.e., the same model accuracy is obtained with a more parsimonious model form (fewer number of model parameters). Repeating this stepwise deletion process until at Step 4, the mean value of σ (0.366) is very close to the largest COV of θ2 (0.414) and a further improvement on γX (x, θ) (Step 5) will bring an increase on the model error. So the optimum γX (x, θ) is obtained at Step 4

γX (x, θ) = θ1 + θ2 ln t + θ3

(12)

3.3. Improved corrosion propagation model

(11)

We write the improved model for the corrosion rate of the steel reinforcement in a probabilistic form with the natural logarithm

where xX = (t , cCO2, fcuk ) and ΘX = (θX 1, θX 2 , θX 3 , σX ) . A comparison between the improved/probabilistic model and the existing empirical/deterministic model (CECS model [20]) based on the predicted and measured value of concrete carbonation depth is shown in Fig. 2. For an ideal model, the solid dots should perfectly stand along the 1:1 line. The prediction results of the CECS model in Fig. 2a clearly show large bias: most solid dots line upon the 1:1 line. The CECS model tends to overestimate the carbonation depth due to a conservative consideration, which is actually not suitable in a probabilistic design scheme. The prediction results of the improved model in Fig. 2b is obtained with ε = 0 (median prediction). The improved model clearly corrects the bias of the empirical model: the solid dots line more evenly

Table 3 Posterior statistics of model parameters for X (x, Θ) Parameter

θ1 θ2 θ3 σ

5

Mean

2.30 –0.217 –1.41 0.366

Standard deviation

0.405 0.090 0.189 0.038

Correlation coefficient θ2 θ1 θ3

σ

1.0 –0.93 –0.86 0.02

1.0

1.0 0.64 –0.02

1.0 –0.01

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Fig. 2. Comparison between (a) CECS model and (b) improved model based on measured and predicted concrete carbonation depth.

transformation

Icorr ,0 =

̂ (x)] + γI (x, θ) + σε ln[Icorr (x, Θ)] = ln[Icorr

(15)

(1 − w c )−1.64 dc

Moreover, the influencing factors f1 , f2 , ...,fk can be transferred into γI (x, θ) under the logarithm transformation. Thus Eq. (15) can be rewritten as

where the indexes are defined earlier in Section 2. The review of existing models suggests a multiply structure for the ̂ (x) deterministic part Icorr

ln[Icorr (x, Θ)] = ln

̂ (x) = f1 f2 . ..fk Icorr ,0 Icorr

(17)

(16)

where Icorr,0 is the initial corrosion rate and f1 , f2 , ...,fk are k influencing factors. For Icorr,0 , since the water to cement ratio w c is an important property influencing the porous structure of the concrete, which has decisive effect on the diffusion of O2, the equation suggested by Guo and Trejo [38]/Vu and Stewart [44] is adopted

(1 − w c )−1.64 + γI (x, θ) + σε dc ̂ ⏟(x) Icorr

(18)

For the correction term γI (x, θ) , h1 (x) = 1 is selected to capture the potential bias that is independent of x and other explanatory functions are from the possible influencing factors f1 , f2 , ...,fk suggested by the existing models and available data. We select h2 (x) = ln tp to capture the potential influence of the corrosion propagation time, h3 (x) = Cl ClTh and h4 (x) = (Cl ClTh)2 to represent the influencing

Table 4 Existing empirical models for the corrosion rate Authors

Type

Andrade&Alonso (2004) [40] Ahmad&Bhattacharjee (2000) [41]

constant constant

Guo&Trejo (2014) [38]

time-variant

Model

Description

< 0.1/0.1 ~ 0.5/0.5 ~ 1.0/ > 1.0 uA/cm

Icorr = 37.726 + 6.12 × 2.231CA2 B + 2.722B 2C2 A = (c − 300) 50, B = (w c − 0.65) 0.075 C = (%CaCl2 − 2.5) 1.25 6

Icorr = [e−6000(mc − 0.75) ] ⎛ ⎝ ×⎡e ⎢ ⎣

2283

−T

time-variant

CECS (2007) [20]

time-variant

Li (2004) [43]

time-variant

Vu&Stewart (2000) [44]

time-variant

Liu&Weyers (1998) [45]

time-variant

c )−1.64 ⎤ dc ⎦

) sin[2π (t − a )]

p s high low ×⎡ + 7.6⎤ 8.6(t p − as ) ⎣ ⎦ ln Icorr = 8.617 + 0.618 ln C (dc , t ) − 3034 T − 0.000105R c R c = exp[8.03 − 0.549 ln(1 + 1.69Ct )]

ln Icorr = 8.617 + 0.618 ln C (dc , t ) −3034 (T + 273) − 5 × 10−3R c + ln mcl

C (dc , t )= Cl− content at the surface of the steel reinforcement and Ct = Cl− content of the concrete cover see Appendix A. NA

Icorr = 0.3683 ln tp + 1.1305

α = 0.85, β=–0.3 and A= 37.8/27 for RH = 0.75/0.8

Icorr = Icorr ,0 α (t − ti ) β Icorr ,0 =

For negligible/low/moderate/high corrosion zone respectively %CaCl2 is the percentage of CaCl2 in unit weight of cement

Cl= Cl− content, mc= concrete humidity, ClTh= Cl− threshold for the corrosion initiation, Thigh= average high temperature, Tlow= average low temperature and as= adjusting factor of seasoning effect

Cl + ClTh ⎞ 2ClTh ⎠

1 −1 (1 − w ( 284.15 T )⎤⎡

⎥ ⎦⎣

(T

Zhou et al. (2010) [42]

2

A (1 − w c )−1.64 dc

NA

ln 1.08Icorr = 7.89 + 0.7771 ln 1.69Cl −3006 T − 0.000116R c + 2.24tp−0.215

Yalcyn&Ergun (1996) [46]

time-variant

NA

Icorr (t ) = Icorr ,0 exp(−1.1 × 10−3tp) Icorr ,0 = 0.53

6

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where db0 is the diameter of the uncorroded steel bar and αrust is the rust type coefficient which is equal to the molecular mass ratio between the steel (Fe) and rust (Fe(OH)2 or Fe(OH)3). Based on Eq. (20)–(22), the prediction model for the rust product expansion Wrust (xwr , Θwr ) is

effect of chloride content in concrete, h5 (x) = T T0 to reflect the possible influence of environmental temperature and h6 (x) = RH RH0 to capture the potential bias related to relative humidity. It worth mention that since the ohmic resistance of the concrete R c is hard to measure and can be reflected by water to cement ratio and chloride/moisture content [38], it is not included in {hi (x)} in this paper. The data {Icorr , x i} used to estimate the statistical distributions of the model parameters for the corrosion rate are from the long-term tests conducted by Liu and Weyers [45]. The 5-year natural exposure experiment on sets of RC slab samples obtained the corrosion current densities of the steel reinforcement under different influencing factors and the data ranges collected are listed in Table 5. The stepwise calibration and selection process for Icorr (x, Θ) is ̂ (x) , {hi (x)} and collected data shown in Fig. 3, based on the selected Icorr {Icorr , x i} . The optimum combination of {hi (x)} is obtained at Step 2 (with σ = 0.281) and a further deletion of θ4 h4 (x) will cause an unacceptable raise in σ . The selected γI (x, θ) is shown in Eq. (19) and the posterior statistics of model parameters Θ = (θ1, θ2 , θ3 , θ4 , θ5 , σ ) are listed in Table 6. The most sensitive influencing factors are identified and the following observations are noteworthy: (1) The selection of θ1 ̂ (x) , which is in indicates that a constant bias exists in the selected Icorr accordance with the parameter A in Vu&Stewart model [44]. (2) The survival of θ2 proves that a time-variant model is more suitable for the corrosion rate. (3) The selection of both θ3 and θ4 shows that the prê (x) has a nonlinear relationship with the chloride diction bias in Icorr content in the concrete. (4) The survival of θ5 indicates that the environmental temperature is an important/sensitive factor influencing the prediction results of the corrosion rate of the steel reinforcement.

Wrust (xwr , Θwr ) =

Cl Cl ⎞ T + θ4 ⎛ + θ5 ClTh T0 ⎝ ClTh ⎠ ⎜

Cl Th

F (s, Θ) = P [g (x, Θ) ⩽ 0 |s, Θ]

Cl 2 ClTh

( )

g (x, Θ) = Wcrit (xwc) − Wrust (xwr , Θwr )

d f

(20)

2

)

(26)

4.2. Example structure The example structure used to conduct the corrosion-induced cracking fragility analysis is the typical simple-supported RC bridge in Shanghai, China. The concrete grade of the RC beam is C30 and the distance from the concrete surface to the middle of the steel reinforcement is 50 mm. The grade for the steel reinforcement is HRB335 and the steel bar diameter is 18 mm. The design parameters x can be divided into random design parameters x r and deterministic design parameters x d , i.e., x = (x r , x d) . The selection and statistical distributions (shown in Table 7) of the random design parameters x r = (fc 28 , dc , db0, d1) are based on the Unified Standard for Reliability Design of Highway Engineering Structures (GB/T 50283-1999) [49] and the existing literatures [38,48]. Other design parameters are all set to be deterministic and their values are listed in Table 8. The

3.5. Rust product expansion The corrosion product (rust) of the steel reinforcement will accumulate around the corroded steel bar. The rust products have more volume than the original steel, resulting a volumetric expansion effect and finally lead to the cracking of the concrete [47]. The accumulation speed of the rust has an inverse relationship to the amount of the rust product Wrust due to the protective effect of the growing rust to the ionic diffusion [48], i.e.,

Table 5 Data ranges collected for the corrosion rate

(21)

where kp= rust accumulation coefficient is a function of the corrosion rate

kp = 0.098πdb0 Icorr αrust

2

where ρrust = mass density of the rust product, d1= thickness of the free pore band at the steel/concrete interface, fct = 0.5 fc28 = tensile strength of the concrete, in which fc28 is the 28-day compression strength, Eef = Ec (1 + φcr )= effective elastic modulus of the concrete, in which Ec , φcr are the original elastic modulus and creep coefficient of the concrete respectively, a = (db0 + 2d1) 2 , b = a + dc , ρst = mass density of the steel reinforcement and νc= Poisson’s ratio of the concrete.

where xI = (t , w c, Cl, T , x ti) and ΘI = (θI 1, θI 2 , θI 3 , θI 4 , θI 5 , σI , Θti) . A comparison between the improved model and the existing CECS model [20] is shown in Fig. 4. Fig. 4a shows that the prediction results of the CECS model tends to underestimate the corrosion rate; most of the solid dots are under the 1:1 line. This is unacceptable in the lifecycle design practice for the RC structures. The proposed model in this paper (Fig. 4b) clearly corrects this potential bias and properly reflects the uncertainties in the corrosion propagation stage of the steel reinforcement.

kp dWrust = dt Wrust

(

b +a πρrust db0 ⎡d1 + Ec ct 2 2 + νc ⎤ b −a ef ⎣ ⎦ Wcrit (xwc) = 1 − αrust ρrust ρst

T

0

(25)

where x = xwc ∪ xwr and Θ = Θwr . Wcrit (xwc) is the critical amount of the rust product for cover cracking and the model suggest by Liu [48] is adopted in this paper

(19)

+ θI 5 T + σI εI

(24)

where F (s, Θ)= conditional probability of cover cracking, s= boundary values and g (x, Θ) is the limit state function defined as the amount of the rust product reaches a certain threshold

⎟

+ θI 4

(23)

ti

For the RC bridge structure, the corrosion-induced cracking fragility is defined as the conditional probability of attaining the cracking limit state caused by the rust product expansion for a given set of boundary values

+ θI 1 + θI 2 ln[t − ti (x ti, Θti)]

+θI 3 Cl

ti

4.1. Limit state function

The final form of the improved model for the corrosion rate is (1 − w c )−1.64 ⎤ dc ⎦

i

4. Cracking fragility analysis

3.4. Model summary

ln[Icorr (xI , ΘI )] = ln ⎡ ⎣

t

∫t (x ,Θ ) [0.098πdb0 Icorr (xI , ΘI ) αrust ] dt

where xwr = (db0, xI ) and Θwr = ΘI .

2

γI (x, θ) = θ1 + θ2 ln tp + θ3

2

(22) 7

Variable

Symbol

Range

Propagation time (year)

tp

0–5

Water to cement ratio Cl− content (weight ratio to cement) (%) Annual temperature (F) Annual relative humidity Concrete cover depth (mm)

w c Cl T RH dc

0.41–0.45 0.082–1.46 28.3–103.2 0.76–0.98 50.8–76.2

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Table 7 Random design parameters Name

Symbol

Distribution

Mean

COV

Concrete 28-day compression strength (Mpa) Concrete cover depth (mm) Uncorroded steel bar diameter (mm) Free pore band thickness (mm)

fc28

Normal

39.1

0.173

dc db0 d1

Normal Normal Normal

41.7 18 0.0125

0.050 0.050 0.050

Table 8 Deterministic design parameters Name

Fig. 3. Stepwise model selection process for Icorr (x, Θ) where a superposed cross (×) indicates term to be removed. Table 6 Posterior statistics of model parameters for Icorr (x, Θ) Parameter

θ1 θ2 θ3 θ4 θ5 σ

Mean

0.844 –0.515 1.329 –0.177 1.121 0.281

Standard deviation

0.075 0.024 0.068 0.017 0.052 0.013

Symbol

Value

Characteristic strength (MPa)

fcuk

30

Water to cement ratio Rust mass density (mg/mm3) Steel mass density (mg/mm3) Concrete elastic modulus (MPa) Poisson’s ratio Creep coefficient Annual temperature (°C) CO2 density (%) Rust type coefficient

w c ρrust ρst Ec νc φcr T cCO2 αrust

0.4 3.6 7.85 30,000 0.2 2.0 15.7 0.0507 0.57

4.3. Fragility estimate and bound

Correlation coefficient

θ1

θ2

θ3

θ4

θ5

σ

1.00 –0.10 –0.42 0.38 –0.80 0.06

1.00 0.03 0.00 –0.20 0.06

1.00 –0.98 –0.05 –0.03

1.00 0.05 0.03

1.00 –0.06

1.00

The consideration of epistemic uncertainties represented by Θ in the improved models leads to a random nature in F (s, Θ) , i.e., the conditional probability F (s, Θ) for the given boundary condition s is a random quantity with respect to Θ instead of a deterministic value. The ∼ predictive fragility estimate F (s) [22,24] is used to represent the average cracking probability level by calculating the expected value of F (s, Θ) with respect to the posterior distribution of Θ

∼ F (s) = environmental parameters are taken from the typical outdoor condition in Shanghai, China and the parameters related to the corrosion-induced cracking analysis are referred to the existing literatures [38,47,48,50,51]. It worth mention that since this paper mainly focus on carbonation-induced corrosion in the urban atmospheric condition, the chloride ion content in the concrete is ideally assumed to be zero (Cl = 0%) and not further discussed in the following paper.

∫ F (s, Θ) f (Θ) dΘ

(27)

and the ± 1 SD confidence bounds (approximately corresponding to 15% and 85% probability levels) are used to represent the uncertainty levels caused by the epistemic uncertainties

∼ ∼ {Φ[−β (s) − σβ (s)], Φ[−β (s) + σβ (s)]}

(28)

∼ ∼ where β (s) = Φ−1 [1 − F (s)]= reliability index, Φ[∙]= cumulative

Fig. 4. Comparison between (a) CECS model and (b) improved model based on measured and predicted corrosion rate. 8

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where α = −∇G (u∗) ∥∇G (u∗)∥= unit vector of ∇G (u∗) , ∇G (u∗)= gradient vector of the limit state function G (u) (in the standard normal space) at the design point u∗, Ju, θz (z∗)= Jacobian matrix of the random variables u (in the standard normal space) to their distribution parameters θz at z∗ and ∇θg g (z∗)= gradient vector of the limit state function g (z) (in the original design space) with respect to the deterministic design parameters θg at z∗. Thus parameter sensitivity ∇θ F (z∗) can be obtained by applying firs-order derivation on F (z∗) = Φ[−β (z∗)]

density function of the standard normal distribution and σβ (s) is the ∼ standard deviation of β (s) . A detail calculation of σβ (s) from the first order reliability analysis process [52] can be referred to Gardoni et al. [24]. The fragility estimates and bounds of the corrosion-induced cracking for the example structure are obtained through the Finite Element Reliability Using MATLAB (FERUM package) [53] based on the improved models and general process described above. Time effect and harmful substance in the environment are important generalized “loading effects” for the durability analysis of the RC bridge. Thus the service time t and CO2 density cCO2 are selected for the boundary value s . Fig. 5 shows the fragility curves for the corrosion-induced cracking of the example RC bridge with respect to the service time t . The solid line ∼ is the curve for the predictive estimates F (t ) and the two dashed lines are the 15% and 85% confidence bounds. Fig. 5a shows the fragility curves in the normal service life of 0 to 100 years for the RC bridge and an expanded time scope up to 500 years is shown in Fig. 5b. The predictive estimates and bounds with respect to the CO2 density cCO2 are shown in Fig. 6 at the service time t = 100 (year). Fig. 7 shows the contour plot of the fragility surface F (t , cCO2) with respect to both t and cCO2 . Every point in each contour (solid line) connects pair of values on t and cCO2 that gives rise to a specific level of predictive fragility. Obvious interaction between the two “load inputs” is observed, especially at low boundary value levels.

∇θ F (z∗) = −φ (β ) ∇θ β (z∗)

where φ (∙) is the standard normal probability density function. The sensitivity curves ∇Θ β (t ) of the corrosion-induced cracking for the example bridge with respect to the model parameters Θ are shown in Fig. 8. The fragility analysis results in Fig. 5 show that the cracking probability under 10 year is close to zero. Thus the service period from 0 to 10 year is not included in the sensitivity analysis for computation efficiency. Fig. 8 shows that during the service life, the sensitivity (absolute value discussed here and in the following) for θX 2 increases sharply, the sensitivities for θI2 and σI have a decreasing trend. For σX , the sensitivity increases in the early age and slowly recovers and decreases in the service period. Fig. 9 shows the sensitivity curves ∇x β (t ) with respect to the design parameters x . The analysis results suggest that increases on the concrete strength fcuk , cover depth dc and steel bar diameter db0 , or decrease on the CO2 density cCO2 , are efficient countermeasures to improve the durability performance of the RC bridge against corrosion-induced cover cracking during the service period. For the evolution laws, the sensitivities for dc and db0 decrease slowly during life time, and the changes for fcuk and cCO2 are relatively moderate.

4.4. Parameter sensitivity Sensitivity analysis shows the changing trend of fragility with respect to the parameters and derives the most sensitive parameters that are important for the design optimization and management decision in the engineering practice [54]. Parameter sensitivity is defined as the gradient vector ∇θ F (z∗) of the fragility F (s) with respect to the parameters θ at the design point z∗. The parameters θ for the probabilistic model formation in this paper include the deterministic design parameters θg = (x d, s) and the distribution parameters θz for the random variables z = (x r , Θ, ε ) , i.e., θ = (θg , θz) . The gradient vector of the reliability index ∇θ β (z∗) can be obtained through the first order reliability process [55] with respect to different types of the parameters

∇θz β (z∗) = αT Ju, θz (z∗) ∇θg β (z∗) = −

4.5. Random variable importance The uncertainty of the corrosion-induced cover cracking problem comes from the accumulation and propagation of the uncertainties presented in the random variables z = (x r , Θ, ε ) in the limit state function g (x, Θ). However, the effects of different random variables may be different. Some of them may be important; they account for most part of the uncertainties of the problem. Others may be less important; the uncertainties presented in them can be ignored to improve the computation efficiency of the probability analysis. An importance measure λ is defined to evaluate this phenomenon [56]

(29)

1 ∇θ g (z∗) ∥∇G (u∗) ∥ g

(31)

(30)

Fig. 5. Fragility curves for the corrosion-induced cracking of example bridge with respect to service time. 9

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Fig. 6. Fragility curves for the corrosion-induced cracking of example bridge with respect to CO2 density.

λT =

αT Ju∗, z∗SD′ ∥αT Ju∗, z∗SD′∥

(32)

where Ju∗, z∗ is the Jacobian matrix of u∗ to z∗ (probability transformation from the original space to the standard normal space) and SD′ is the standard deviation of the equivalent normal variables z′, in which z′ = z∗ + Jz∗, u∗ (u − u∗)= linearized inverse transformation at z∗. In the random variable importance analysis for the example bridge, the random design parameters x r = (fc 28 , dc , db0, d1) are assumed to be independent. For the random model parameters ΘX and ΘI , their crosscorrelation statistics are listed in Tables 3 and 6; εX and εI are independent standard normal random variables in the error terms. Fig. 10 shows the importance curves of the random variables z for the corrosion-induced cracking fragility in the service period of the example RC bridge. The results indicates that the uncertainty of the cover cracking problem mainly comes from the concrete carbonation model X (xX , ΘX ) : model parameters θX1, θX 2 , θX 3 and error term σX εX . It worth mention that except for θX 2 , the changing trends for those important random model parameters are relatively moderate during the life time of the RC bridge. The cover depth dc and the model error εI for the steel corrosion rate model Icorr (xI , ΘI ) also account for part of the uncertainty sources. For the other random variables, their importance and contributions on the uncertainty level of the engineering problem are

Fig. 7. Contour plot of the fragility surface F (t , cCO2) .

Fig. 8. Sensitivity curves for the model parameters Θ . 10

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Fig. 9. Sensitivity curves for the design parameters x .

Fig. 10. Importance curves for the random variables z . 11

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against corrosion-induced cover cracking; (3) The uncertainty of the cover cracking problem mainly comes from the concrete carbonation model. The cover depth and the model error for the steel corrosion rate model also account for part of the uncertainty sources. For the other random variables, their importance and contributions on the uncertainty level of the engineering problem are relatively small.

relatively small. 5. Conclusion This paper proposes a comprehensive probabilistic approach for the corrosion-induced cracking fragility analysis of the RC bridge in the urban area with improved concrete carbonation and steel reinforcement corrosion models. Reviews and discussions are conducted on the existing deterioration models. Improved deterioration models have a probabilistic formation with deterministic part, correction term and error term. The explanatory functions in the correction term come from the influencing factors suggested by the existing models and the most sensitive ones (optimum combination) are selected by a stepwise model calibration and selection process. Comparisons and discussions are conducted between the improved deterioration models and the existing models based on the prediction results and measured long-term data. Results show that the improved models are unbiased, more accurate and properly account for all types of uncertainties, satisfying the probabilistic requirements of the modern design. Fragility curves, parameter sensitivities and random variable importance are achieved for the example RC bridge based on the proposed approach and improved concrete carbonation and steel reinforcement corrosion models. The following findings in the analysis results are noteworthy:

CRediT authorship contribution statement Sun Bo: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft. Xiao Ru-cheng: Resources, Data curation, Writing - review & editing. Ruan Wei-dong: Investigation, Validation. Wang Peng-bo: Visualization. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors wish to thank all the members of the Long-span Bridge Center at the Tongji University, for helpful discussions and suggestions. The research work was supported in part by National Natural Science Foundation of China (Grant 51908503 and 51909236), Natural Science Foundation of Zhejiang Province (Grant LQ19E090009) and Innovative Experiment Project of ZJUT (Grant SYXM1723). Opinions and findings presented are those of the authors and do not necessarily reflect the views of the sponsors.

(1) Fragility analysis results with respect to service time and CO2 density indicate significant interaction between the two “load inputs”, especially at low boundary value levels; (2) Increases on the concrete strength, cover depth and steel bar diameter, or decrease on the CO2 density, are efficient countermeasures to improve the durability performance of the RC bridge Appendix A. Corrosion models in CECS (2007) A.1. Concrete carbonation depth

The concrete carbonation depth X at service time t is calculated by the following equation: (A.1)

X=k t where k is the carbonation coefficient considering a set of influencing factors:

58 k = 3K CO2 Kkl Kkt Kks KF T1 4RH1.5 (1 − RH ) ⎜⎛ − 0.76⎞⎟ ⎠ ⎝ fcuk

(A.2)

where

K CO2 – CO2 density factor, K CO2 = cCO2 0.03 ; Kkl – location factor, Kkl = 1.4 for corner of the component and 1.0 for other area; Kkt – curing factor, Kkt = 1.2 ; Kks – stress factor, Kks = 1.0 for compression condition and 1.1 for tension condition; KF – fly ash factor, KF = 1.0 + 13.34F 3.3 , F is the fly ash content (weight ratio). A.2. Corrosion rate The corrosion rate Icorr at service time t for a chloride penetration environment is calculated by the following equation:

ln Icorr = 8.617 + 0.618 ln C (dc , t ) − 3034 (T + 273) − 5 × 10−3R c + ln mcl

(A.3)

d C (dc , t ) = C0 + (Cs − C0 ) ⎡1 − erf ⎛ c ⎞ ⎤ ⎢ 2 ⎝ Dt ⎠ ⎥ ⎣ ⎦

(A.4)

R c = kR (1.8 − Ct ) + 10(RH − 1)2 + 4

(A.5)

⎜

⎟

where

C (dc , t ) – Cl− content at the surface of the steel reinforcement; C0 – original Cl− content in the concrete; Cs – Cl− content at the concrete surface; 12

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D – Cl− diffusion coefficient; R c – ohmic resistance of the concrete; kR – factor reflecting concrete quality, kR = 11.1 when w c = 0.3~0.4 or for C40 ~ C50 concrete, kR = 5.6 when w c = 0. 5~0.6 or for C20 ~ C30 concrete; Ct – Cl− content in the concrete cover (mean value); mcl – local environment factor, mcl = 4.0~4.5 for outdoor environment in the marine atmospheric zone and 2.0 ~ 2.5 for indoor environment. Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engstruct.2020.110313.

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