Probabilistic durability assessment of concrete structures in marine environments: Reliability and sensitivity analysis

A probabilistic framework for durability assessment of concrete structures in marine environments was proposed in terms of reliability and sensitivity analysis, which takes into account the uncertainties under the environmental, material, structural and executional conditions. A time-dependent probabilistic model of chloride ingress was established first to consider the variations in various governing parameters, such as the chloride concentration, chloride diffusion coefficient, and age factor. Then the Nataf transformation was adopted to transform the non-normal random variables from the original physical space into the independent standard Normal space. After that the durability limit state function and its gradient vector with respect to the original physical parameters were derived analytically, based on which the first-order reliability method was adopted to analyze the time-dependent reliability and parametric sensitivity of concrete structures in marine environments. The accuracy of the proposed method was verified by comparing with the second-order reliability method and the Monte Carlo simulation. Finally, the influences of environmental conditions, material properties, structural parameters and execution conditions on the time-dependent reliability of concrete structures in marine environments were also investigated. The proposed probabilistic framework can be implemented in the decision-making algorithm for the maintenance and repair of deteriorating concrete structures in marine environments.


Introduction
Chloride induced reinforcement corrosion has been recognized as one of the most significant deterioration mechanisms of concrete structures in marine environments (Oslakovic et al., 2010;Teply and Vorechovska, 2012;Ryan and O'Connor, 2013). Reinforcement corrosion often induces cracking of concrete cover, reduction of cross-section of reinforcing bar, loss of bond between reinforcement and concrete, etc. As a result, the serviceability and load-carrying capacity of concrete structures in marine environments could be reduced (Engelund et al., 1999;Saassouh and Lounis, 2012;Zhao and Jin, 2006).
Reinforcement corrosion induced deterioration of concrete structures can be divided into two stages-the initiation and the propagation periods (Yu et al., 2014;Ji et al., 2014Ji et al., , 2015. The former refers to the depassivation of steel reinforcement induced by the ingress and the accumulation of chloride ions, while the latter starts from corrosion initiation to critical steel loss or concrete cracking/spalling. Compared with the first stage, the propagation period is relatively short. Therefore, the chloride diffusion process has been frequently used to indicate durability and service life of reinforced concrete structures (Nogueira and Leonel, 2013). The penetration of chloride ions into concrete is a complex process which depends on a large number of parameters including properties of concrete (e.g., composition, porosity and microstructure), degree of concrete pore saturation, depth of concrete cover, and exposure conditions. Owing to the uncertainties that inherently exist in the environments and construction practices, a considerable level of variability has been observed in the above parameters (Saassouh and Lounis, 2012;Papakonstantinou and Shinozuka, 2013;Shafei et al., 2013). Ait-Mokhtar et al. (2013) performed a comprehensive statistical analysis of chloride diffusion coefficient of concrete specimens, which indicated that the chloride diffusion coefficient shows a significant variation with a coefficient of variation (COV) up to 0.254. Goltermann (2004) determined the variations of surface chloride concentration and chloride diffusion coefficient of concrete samples taken from a range of columns on Danish highway road bridges constructed between 1968 and 1990. It was found that the COVs of the above parameters are up to 40%, which prevents reliable deterministic assessment and predictions of durability of concrete structures.
By considering these uncertainties, the durability of concrete structures can be more realistically represented by probabilistic approaches (Bentz, 2003). Pan et al. (2014) discussed the influence of aggregate properties including grade, shape, content and distribution on the chloride diffusion coefficient by a simulation of random aggregate structure. Kirkpatrick et al. (2002) determined the time to first repair and subsequent rehabilitation of concrete bridge decks exposed to chloride deicer salts based on the statistical data collected from 10 bridge decks built in Virginia. It was found that the time to corrosion initiation is highly sensitive to the chloride diffusion coefficient, but not as much to the depth of concrete cover. Jin et al. (2008) presented a path probability model to predict the probability distribution of steel reinforcement based on the Monte Carlo simulation (MCS) technique. Ryan and O'Connor (2013) discussed the time to chloride induced corrosion for different self-compacting concretes by adopting the results of natural chloride migration testing as statistical input parameters in the probabilistic deterioration model. Shafei et al. (2013) proposed a stochastic computational framework to investigate the chloride-induced corrosion of reinforced concrete structures based on three-dimensional finite-element models. However, it involves complex and time-consuming multi-field coupling analysis, such as chloride ion, heat, humidity and carbon dioxide. Recently, the MCS and the first-order reliability method (FORM) have been adopted to evaluate the probabilistic durability of concrete structures. Val and Trapper (2008) evaluated the probabilistic initiation time of chloride-induced corrosion based on the MCS. Bastidas-Arteaga et al. (2011) introduced the chloride penetration model into a stochastic framework based on the MCS with Latin hypercube sampling. Lu et al. (2011) assessed the probabilistic lifetime of reinforced concrete structures with steel corrosion and cover cracking based on the MCS. It showed that the time to corrosion initiation follows a Lognormal distribution, while that the time from corrosion initiation to cover cracking as well as the time for the crack to develop from a hairline crack to a limit crack width can be described by the Weibull distributions. Tikalsky et al. (2005) investig-ated the influence of variation in diffusion coefficient and depth of concrete cover on the chloride penetration of concrete bridge decks, based on the MCS and the Fick's second law of diffusion. Williamson et al. (2009) investigated the influence of concrete and steel types on the probabilistic corrosion service life of bridge decks. Furthermore, Saassouh and Lounis (2012) presented two simplified semi-analytical probabilistic models to investigate the uncertainty of key parameters for chloride induced corrosion. Nogueira and Leonel (2013) developed a coupled model based on mechanical behavior simulation and reliability algorithms to allow probabilistic analysis of reinforced concrete structures.
It should be noted that the above models mainly focus on the uncertainty of critical chloride concentration, surface chloride concentration, chloride diffusion coefficient or concrete cover depth, but mostly ignore the uncertainty of initial chloride concentration, time-varying nature of chloride diffusion coefficient, especially the influences of environmental conditions and material properties. However, for marine concrete structures immersed into the sea water, there are four typical environment zones with different surface chloride concentrations and availabilities of oxygen and water, such as the submerged, splash, tidal, and atmospheric zones. Despite extensive research, the influences of environmental conditions, material parameters and execution conditions on the time-dependent reliability of concrete structures in marine environments still remain unclear due to various factors and uncertainties (Moodi et al., 2014).
The objectives of this paper are twofold: (1) to present a probabilistic framework for durability assessment of concrete structures in marine environments in terms of reliability and sensitivity analysis, which takes into account the uncertainties under environmental conditions, material properties, structural parameters and execution conditions; (2) to investigate the influences of environmental conditions, material properties, structural parameters and execution conditions on the time-dependent reliability of concrete structures in marine environments through a comprehensive analysis. The proposed probabilistic framework provides a more realistic evaluation of durability and can be implemented in a variety of decision-making algorithms required for the maintenance and repair of deteriorating concrete structures in marine environments.

Time-dependent probabilistic model of chloride ingress
The chloride transport mechanism in concrete is a complex phenomenon and may occur in several forms, such as ionic diffusion, capillary absorption, electrical migration and permeation due to hydraulic pressure heads (Saassouh and Lounis, 2012). However, the diffusion process is often assumed to be the dominant mode of chloride transportation. The most commonly used mathematical expression to describe chloride ingress into concrete is based on the Fick's second law of diffusion (Bastidas-Arteaga et al., 2011), which describes the time-dependent distribution of chloride ions within the concrete as: where x is the distance from the exposed surface (mm); t is the exposure time (a); C(x, t) is the chloride concentration (%, with mass percentage of binder throughout this study) at the distance x and time t; erf(·) is the error function; C 0 is the initial chloride concentration (%) within concrete, which is a Lognormal random variable; C s is the surface chloride concentration (%) related to the material and environmental conditions, which is a Lognormal random variable and can be predicted by (DuraCrete, 1998) C s = A (w=b) ; (2) where w/b is water-to-cement ratio; A is the regression coefficient related to environmental conditions and binder types.
It should be noted that the Fick's second law of diffusion is used merely as a simple mathematical tool for the analysis of chloride profiles. Under this perspective, the value of D a (t) in Eq. (1) is normally called the apparent diffusion coefficient, since it quantifies the influence of various transport mechanisms on chloride penetration (Bermudez and Alaejos, 2010;Papakonstantinou and Shinozuka, 2013). As taking material, execution, test method, and curing time into account, D a (t) can be defined by (DuraCrete, 1998) where k e is the environmental factor influenced by the environmental condition and binder type, which is a Gamma random variable; k t is the test method factor, which relates the rapid laboratory compliance to a real material compliance under defined executional and environmental conditions; k t is a Normal random variable; k c is the execution factor, which is a Beta random variable; t 0 is the curing period (a), which is usually adopted as 28 d = 0.0767 a; D 0 is the chloride coefficient (mm 2 /a) at the reference time t 0 and under the standard environmental, curing and test conditions, which is a Normal random variable; n is the age factor influenced by binder type and environmental condition, which is a Beta random variable. The statistical characteristics of the above random variables can be found in Dur-aCrete (2000). According to Eqs. (1) and (3), the time-dependent probabilistic model for chloride ingress within concrete can be described as: For durability analysis of concrete structures, the chloride concentration at steel surface is more desirable. In this case, the distance x is replaced by the cover depth d c .

Durability limit state of marine concrete structures
If the durability limit state of concrete structures in marine environments is defined as the time when the chloride concentration at the steel surface reaches a threshold value, the durability limit state function G(X, t) can be expressed as: (5) where X is a random vector composed of nine random variables, such as the initial chloride concentration C 0 , surface chloride concentration C s , critical chloride concentration C cr , apparent chloride diffusion coefficient D 0 , concrete cover depth d c , and the age factor n. According to Eq. (5), the time-dependent probability of failure for concrete structures in marine environments concerning the durability limit state can be expressed as: where d c is the depth of concrete cover (mm), which is a Lognormal random variable; C(d c , t) is the chloride concentration (%) at the steel surface and time t; C cr is the critical chloride concentration, which is a Normal random variable with mean and standard deviation of 0.48% and 0.15%, respectively (DuraCrete, 2000); f C (s) and (r) are the probability density functions for random variables C(d c , t) and C cr , respectively.
According to Eq. (6), the time-dependent reliability index of concrete structures in marine environments concerning the durability limit state can be expressed as: where β(t) is the reliability index at time t; Φ -1 [·] is the inverse standard Normal cumulative distribution function. It is difficult to determine the probability density function f C (s) of chloride concentration at the steel surface analytically, since it is affected by lots of influential factors. As a result, the reliability index cannot be computed by Eqs. (6) and (7) straightway. In this study, the first-order reliability method (FORM) is adopted to compute the time-dependent reliability index of concrete structures in marine environments.

Reliability analysis based on the FORM
The FORM addresses the reliability problem defined in Eqs. (6) and (7) by means of two key operations: one is to find the design point in the transformed uncorrelated standard Normal space; the other is to approximate the limit state surface at this point and make use of the properties of the standard Normal space to obtain the probability estimate. The reliability index β is obtained by linearizing the performance function at the point on the limit state surface nearest the origin, which can be described as the following constrained optimization problem: where Z * is the design point vector in the uncorrelated standard Normal space, which can be determined by where the superscripts k and k+1 denote the iterative steps; is the gradient vector of the limit state surface in the standard Normal space, which is expressed as: As shown in Eq. (8), an essential ingredient in the FORM for reliability analysis is to find the so-called design point. The search for this point is performed in an uncorrelated standard Normal space. However, the durability limit state function defined in Eq. (5) for concrete structures in marine environments involves various non-normal variables, such as Lognormal random variables (e.g., C 0 and C s ), Gamma random variable (e.g., k e ) and Beta random variables (e.g., k c and n). Hence, it is necessary to transform those non-normal random variables from the original physical space into the independent standard Normal space.

Transformation of correlated non-normal variables
According to the isoprobabilistic transformation, the relationship between an arbitrary distribution random variable X i in the original physical space and a standard Normal random variable Y i is (Der Kiureghian et al., 2006) where Φ(·) and Φ -1 (·) are the standard Normal cumulative distribution function (CDF) and the inverse standard Normal CDF, respectively; and are the marginal cumulative distribution function of X i and its corresponding inverse function, respectively.
According to Eq. (12), the correlated arbitrary distribution random vector in the original physical space can be transformed into a random vector in the correlated standard Normal space. If Y is a n-dimensional standard Normal random vector with the joint probability density function having zero means, unit standard deviations and a correlation matrix ρ y , the approximated joint probability density function of the correlated random vector X is (Der Kiureghian et where is the joint probability density function of the random vector X. According to the chain rule of differentiation, the joint probability density function can be expressed as: where φ(·) is the probability density function of the standard Normal random variable; is the marginal probability density function of X i .
After obtaining the correlation matrix ρ y (Der Kiureghian and Liu, 1986), the relationship between the correlated standard Normal random vector Y and the independent standard Normal random vector where λ and A are the eigenvalue and eigenvector of the matrix ρ y obtained by the eigenvalue decomposition. The inverse transformation from vector Z to vector X is According to Eqs. (15) and (16), the random variables can be transformed between the original physical space and the independent standard Normal space.

Gradient vector of durability limit state function
As shown in Eq. (10), the gradient vector of the limit state surface is an essential ingredient in the FORM for reliability analysis, which can be expressed as: where is the gradient vector of the durability limit state function G(X) with respect to the original physical vector , which can be obtained by the direct differential method based on Eq. (5) as follows: Furthermore, J X, Z in Eq. (17) is the Jacobian matrix of the original physical vector X with respect to the independent standard Normal random vector Z, which can be expressed as: 4.4 Sensitivity analysis of reliability index Sensitivity analysis is useful for many purposes, including for identification of important sources of uncertainty and for optimal design. In order to reflect the importance of the random variables in the original physical vector X to the reliability index of concrete structures in marine environments concerning the durability limit state, the following sensitivity coefficient vector can be defined where D is the diagonal matrix whose diagonal elements are the standard deviations of the vector X; J Z, X is the Jacobian matrix of the independent standard Normal random vector Z with respect to the original physical vector X, which can be expressed as: where α is the negative normalized gradient row vector defined by The sensitivity coefficient vector γ is an important measure of various governing parameters in the original physical space with respect to the reliability index. The parameter x i is a resistance variable if γ i is positive, while x i is a load effect variable when γ i is negative. The reliability index of concrete structures in marine environments can be improved by adjusting the governing parameters appropriately. According to the Nataf transformation and the FORM, the flow chart of iterative procedures for reliability and sensitivity analysis of concrete structures in marine environments concerning the durability limit state is shown in Fig. 1. Here, ε 1 and ε 2 are predefined tolerance, which are usually adopted as 0.0001.

Validation of accuracy
An ordinary Portland cement (OPC) concrete structure is located in the marine splash zone, whose water-to-binder ratio w/b=0.4, and initial curing period t 0 =28 d=0.07671 a.
crete structures increases with the increasing concrete cover depth. For example, the reliability index of marine concrete structure at time 100 a increases from 0.3696 to 2.1042 when increases from 40 mm to 70 mm. As expected, the reliability index of marine concrete structures decreases with the increment of service life. Taking =50 mm as an example, the reliability index of marine concrete structures decreases from 2.6450 to 1.6210 and 1.0549, respectively, when the service life increases from 20 a to 60 a and 100 a.
It should be noted that the FORM approximates the limit state surface as a tangential hyperplane at the design point and its accuracy is primarily dependent on the nonlinearity of the limit state surface around the design point. This is because the dominant contribution to the failure probability comes from the neighborhood of the design point, where the probability density in the standard Normal space achieves its maximum value. On the other hand, the SORM approximates the limit state surface as a hyperboloid, which can reflect the nonlinearity around the design point up to the second order. For nonlinear problems, the SORM usually gives a better approximation than the FORM. However, the SORM is more complex and more time-consuming to implement than the FORM. As shown in Table 2, the results obtained by the FORM agree with those obtained by the SORM and the MCS for the reliability analysis on concrete structures in marine environments, with the relative error almost within 5%. Hence, the FORM can be used for the reliability analysis of marine concrete structures concerning the durability limit state, and the following analysis are all based on the FORM without special descriptions.

Probability characteristics of chloride concentration
Taking the marine concrete structure with =220.75 mm 2 /a and =60 mm as an example, the correlation between the chloride concentration at the steel surface and various basic random variables can be determined based on the results of the MCS, as shown in Fig. 2. It is clear that the correlation coefficients between C(60 mm, 100 a) and C 0 , C s , D 0 , k e , k c or k t are positive, while those between C(60 mm, 100 a) and n or d c are negative. Meanwhile, the absolute value of the correlation coefficient between C(60 mm, 100 a) and n is the largest (i.e., 0.7523), and that between C(60 mm, 100 a) and k t is the smallest (i.e., 0.0514).
In order to determine the probability distribution type of the chloride concentration at the steel surface, the Kolmogorov-Smirnov (K-S) test is used to decide if it comes from a population with a specific distribution. An attractive feature of the K-S test is that the distribution of the K-S test statistic itself does not depend on the underlying cumulative distribution function being tested. In this study, the K-S test is defined by H 0 : the chloride concentration follows a specified distribution and H a : the chloride concentration does not follow the specified distribution. The K-S test statistics is defined as: (x) and (x) are the theoretical and empirical cumulative distributions of the distribution being tested. The hypothesis H 0 is rejected if the test statistics D is larger than the critical value D n (n, α). Here, D n (n, α) is the critical value of the K-S test with significance level α and sample size n.
The tested results of seven types of probability distributions, including Normal, Lognormal, Weibull, Gamma, Type I Largest Value, Exponential, and Rayleigh distributions are summarized in Table 3. The critical values of the K-S test for different significance levels and sample sizes are listed in Table 4. According to Tables 3 and 4, the chloride concentration at the steel surface can be approximately described as Lognormal, Weibull, or Gamma distribution, but it does not obey Normal, Type I Largest Value, Exponential, or Rayleigh distribution. The probability density functions of the chloride concentration at the steel surface with Lognormal, Weibull, and Gamma distribution are shown in Fig. 3. It shows that the mean values of different distributions are comparable, but the Gamma distribution exhibits larger variability.

Parametric sensitivity analysis of reliability index
According to Eq. (23), the sensitivity coefficient vector of reliability index for marine concrete structure with re-µ D 0 µ d c spect to the random variables in the original physical space can be obtained. Taking the marine concrete structure with =220.75 mm 2 /a and =60 mm as an example, the sensitivity coefficients of reliability index with respect to the mean value and standard deviation of random variables are shown in Fig. 4. It is obvious that the mean values of k e , C cr , C 0 and k t as well as the standard deviations of C cr and k e have significant effects on the reliability index of concrete structures in marine environments.

Influence of structural and material parameters
The influences of the apparent chloride diffusion coefficient (material parameter) and the depth of concrete cover (structural parameter) on the reliability of marine concrete structure are discussed first. The reliability indices of marine concrete structure during 20 a and 100 a are shown in Fig. 5, when increases from 126.14 mm 2 /a (4.0×10 -12 m 2 /s) to 315.36 mm 2 /a (10.0×10 -12 m 2 /s) and increases from 40 mm to 70 mm. As shown in Fig. 5, the mean value of the apparent chloride diffusion coefficient has a signific- Fig. 2. Correlation between the chloride concentration at the steel surface and random variables.
ant effect on the reliability index of concrete structures in marine environments. Taking =50 mm as an example, the reliability index of marine concrete structure at the service time of 100 a decreases from 1.9274 to 0.5063, when increases from 126.14 mm 2 /a to 315.36 mm 2 /a. The reliability index of marine concrete structure decreases nonlinearly with the incensement of service time. Particularly, the reliability index decreases rapidly after 20 a when the apparent chloride diffusion coefficient is large (e.g., 315.36 mm 2 /a), while it does not vary obviously when the apparent chloride diffusion coefficient is small (e.g., 126.14 mm 2 /a) even the service time is as long as 80 a, especially in the situation when the depth of concrete cover is large (e.g., 70 mm). This is due to the fact that the chloride concentration at the steel surface is very small when the apparent chloride diffusion coefficient is small and the depth of concrete cover is large.
As discussed above, the apparent chloride diffusion coefficient and the depth of concrete cover are two important parameters for the durability of concrete structures in marine environments. It is assumed that the target reliability index is 1.5 for the durability of marine concrete structure. The reliability index of marine concrete structure at service time 100 a is larger than the target reliability index, when =126.14 mm 2 /a and =50 mm. However, the reliability index of marine concrete structure at service time 50 a is larger than 1.5, but that at service time 100 a is smal- ler than 1.5, when =220.75 mm 2 /a and =50 mm. More seriously, the reliability index of marine concrete structure with =220.75 mm 2 /a and =50 mm is smaller than 1.5 even at service time 50 a. On the other hand, the reliability indices of marine concrete structures with =126.14 mm 2 /a, 220.75 mm 2 /a and 315.36 mm 2 /a are 2.6535, 2.1042 and 1.5469, respectively, when =70 mm and service time is 100 a, which indicates that the durability of concrete structures in marine environments can be guaranteed when the depth of concrete cover is large enough. According to the above analysis, it is clear that the durability of concrete structures in marine environments can be guaranteed by selecting the appropriate combination of the apparent chloride diffusion coefficient and the depth of concrete cover.
The influence of water-to-cement ratio w/b (material parameter) on the reliability index of concrete structures in marine environments is investigated here. The OPC concrete structure with w/b=0.4 and t 0 =28 d=0.07671 a is located in the splash zone. The statistical characteristics of random variables related to w/b are listed in Table 5, while the other random variables are listed in Table 1.
When the marine concrete structures with different combinations of the apparent chloride diffusion coefficient and the depth of concrete cover, such as Case 1 ( =220.75 mm 2 /a and =50 mm), Case 2 ( =126.14 mm 2 /a and

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YU Bo et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 63-73 =50 mm) and Case 3 ( =220.75 mm 2 /a and =60 mm), the influence of w/b on the reliability index of concrete structures in marine environments is shown in Fig. 6. It is clear that the water-to-cement ratio has a significant effect on the reliability index of concrete structures in marine environments. The reliability index decreases with the increase of service time and water-to-cement ratio (w/b).

Influence of environmental conditions
For marine concrete structures immersed in sea water, there are four typical environment zones with different surface chloride concentrations and availabilities of oxygen and water, such as submerged, splash, tidal, and atmospheric zones. Since the durability of marine concrete structures in atmospheric zone is usually less serious than those in the other three zones, only the influences of environmental conditions, including submerged, tidal, and splash zones on the reliability index of concrete structures in marine environments are investigated here. The OPC concrete structure with w/b=0.4 and t 0 =28 d=0.07671 a is considered here. The statistical characteristics of random variables related to environmental conditions are listed in Table 6, while the other random variables are listed in Table 1.
The reliability indices of marine concrete structures under different environmental conditions are shown in Fig. 7, when the concrete structures in marine environments with different combinations of apparent chloride diffusion coefficient and depth of concrete cover including Case 1 ( = 220.75 mm 2 /a and =50 mm), Case 2 ( =126.14 mm 2 /a and =50 mm) and Case 3 ( =220.75 mm 2 /a and =60 mm). As shown in Fig. 7, the reliability index of marine concrete structures under tidal zone is smaller than those under submerged and splash zones. This is due to the fact that the surface of concrete in the tidal zones is exposed to wetting and drying cycles which provide the optimal conditions (chlorides, water and oxygen) for corrosion to initiate. Furthermore, it also shows that the reliability index of concrete structures in marine environments decreases with the increase of service time and apparent chloride diffusion coefficient, and increases with the increase of the depth of concrete cover.

Influence of execution conditions
The influences of the execution condition (i.e., initial curing period t 0 ) on the reliability index of concrete structures in marine environments are investigated in this section. The OPC concrete structure located in the splash zone is of w/b=0.4, while the initial curing period t 0 adopts 1 day, 3 days and 28 days, respectively. The statistical characteristics of execution factor are listed in Table 7, while the other random variables are listed in Table 1.
The influence of the initial curing period t 0 on the reliability index of concrete structures in marine environments is shown in Fig. 8, when the marine concrete structures with different combinations of apparent chloride diffusion coefficients and depths of concrete cover, including Case 1 ( = .75 mm 2 /a and =40 mm) and Case 2 ( =315.36 mm 2 /a and =50 mm). It is clear that the reliability index of concrete structures in marine environments increases with the increment of initial curing period, especially within the first 40 years. However, the influence of initial curing period on the reliability index of marine structures decreases with service time. For example, the initial curing period has an unapparent effect on the reliability index after 80 years. The reliability indices of marine concrete structures with = 220.75 mm 2 /a and =40 mm as well as =315.36 mm 2 /a and =50 mm are considerable, which also implies that the durability of concrete structures in marine environments can be guaranteed by selecting the appropriate combination of apparent chloride diffusion coefficient and the depth of concrete cover.

Conclusions
A probabilistic framework for durability assessment of concrete structures in marine environments was proposed by combining the first-order reliability method (FORM) with the time-dependent probabilistic model of chloride ingress. The influences of environmental conditions, material properties, structural parameters and execution conditions on the time-dependent reliability of concrete structures in marine environments were investigated through a comprehensive analysis. According to the analysis results, the following conclusions can be drawn.
(1) For reliability analysis of concrete structures in marine environments concerning the durability limit state, the results obtained by the FORM agree with those obtained by the SORM and the MCS. Meanwhile, the FORM is more efficient than the SORM and the MCS. Hence, the FORM is recommended for the reliability analysis of concrete structures in marine environments.
(2) The reliability index of the concrete structures in marine environments increases with the decrease in waterto-cement ratio and apparent chloride diffusion coefficient, or the increase of the initial curing period and the depth of concrete cover. The durability of concrete structures in marine environments can be guaranteed by selecting an optimum combination among the apparent chloride diffusion coefficient, water-to-cement ratio, depth of concrete cover and initial curing period.
(3) Results of sensitivity analysis show that the mean Beta 0.79 0.08 # 0.00 1.00 20.14 5.35 Note: Beta distribution is denoted as Beta (a, b, q, r); Data marked with "*" are computed with the coefficient of variation equal to 0.3; Data marked with "#" are computed with the coefficient of variation equal to 0.1. values of environmental factor, critical chloride concentration and initial chloride concentration, as well as the standard deviations of critical chloride concentration and environmental factor have significant effects on the reliability of concrete structures in marine environments.
(4) The chloride concentration at the steel surface can be approximately described as the Lognormal, Weibull, or Gamma distribution, but it does not obey the Normal, Type I Largest Value, Exponential, or Rayleigh distribution.
(5) The reliability index of concrete structures in marine environments of tidal zone is smaller than those in splash and submerged zones, due to the fact that it provides the optimal conditions (chlorides, water and oxygen) for corrosion initiation and propagation.