Bayesian Assessment of the E ﬀ ects of Cyclic Loads on the Chloride Ingress Process into Reinforced Concrete

: Chloride-induced corrosion and load induced concrete cracking a ﬀ ect the serviceability and safety of reinforced concrete (RC) structures. Once these phenomena occur simultaneously, the prediction of RC structures’ lifetimes becomes a major challenge. The objective of this paper is to propose a methodology to evaluate the e ﬀ ect of loading and cracking on the mechanism of chloride ion penetration in concrete. The proposed methodology will be based on Bayesian networks. Bayesian networks are useful to update the lifetime assessment based on experimental data as well as to characterize the uncertainties of the input parameters of a chlorination model including a chloride di ﬀ usion acceleration factor. The proposed methodology is illustrated with experimental data coming from tests on RC beams subjected to static and cyclic loading before being in contact with a solution containing chloride ions. The characterized parameters are then used to evaluate the e ﬀ ect of these two loading conditions (static and cyclic) on the corrosion initiation time and the corrosion initiation probability. The results obtained indicate that the proposed methodology is capable of integrating loading and chlorination test data for the determination of the probabilistic parameters of a model in a comprehensive way.


Introduction
Reinforced concrete (RC) is a resistant and durable material that is widely used in the construction of different types of structure. It is one of the most used materials in the world because of the availability of its components and its ease of construction. However, structures placed in coastal and offshore areas are exposed to chloride-induced corrosion. The penetration of chloride ions is one of the major causes of deterioration of RC structures [1,2]. It causes a local reduction in the reinforcement and an accumulation of corrosion products at the interface between concrete and steel, which lead to tensile stresses that initiate cracks. The different effects of chloride-induced corrosion produce a significant reduction in the service life and structural safety, as well as an increase of maintenance costs [3][4][5]. Reinforced concrete structures, generally designed for a lifetime of 50 to 100 years, could begin to deteriorate after 20 to 30 years when are in contact with chloride ions [6][7][8]. In addition, structures in service are subject to mechanical stresses such as cyclic loads, which cause cracks in the reinforced been used to estimate the effects of mechanical loading on the chloride ingress mechanism by using experimental data. The experimental data presented in the paper comes from a previous research study on the combined effects of chlorination and cracking detailed in [4].
The paper is organized as follows. The first part of the document summarizes the chloride diffusion models in sound and cracked concrete (Section 2). Section 3 gives a general description of the experimental tests by presenting the equipment, method and inspection data that will be used for identification purposes. In Section 4, we detail the proposed Bayesian network that will be used to characterize the model parameters. Finally, Section 5 deals with the results of the identification of input variables for different loading cases and their effects the probability of corrosion initiation.

Chloride Ingress for Uncracked Concrete
The diffusion of chloride ions in saturated concretes is described by the second Fick's law [26] with the assumption that concrete is a homogeneous and isotropic material [31]: where C is the free chloride concentration, t is time, D is the diffusion coefficient of the chloride ions, and x is the depth of the concrete in the diffusion direction. With the following initial conditions: (1) the concentration is zero at the beginning of the exposure t 0 , and (2) the chloride surface concentration is constant; the concentration of free chloride ions C(x, t) at depth x and at time t for a semi-infinite medium can be expressed by an analytical solution of Fick law using the error function: where C s is the chloride concentration at the surface, and erf(·) is the error function. This solution makes possible to calculate chloride profiles at given times and depths as well as the corrosion initiation time t ini . t ini is the time at which the chloride concentration at the surface of the reinforcements reaches a threshold value C th . This threshold value represents the concentration of chlorides for which the passive layer of the steel is destroyed and the corrosion reaction starts. The corrosion initiation time is, therefore, estimated for the chloride concentration equal to the threshold value C th and x equal to the cover depth of the concrete c, according to the following equation: Collepardi's model (Equation (2)) is only valid for saturated concrete exposed to a constant concentration of chloride on the surface. In reality, for a heterogeneous material, such as concrete, these conditions are rarely observed and the concentration of chlorides on the surface varies over time. Although this model does not take into account several important parameters such as the aging state of concrete and environmental factors (temperature, humidity) [32,33], it will be used to identify the variables required because the beams of the tests are subjected to a constant concentration of chloride on the surface of the concrete. In addition, this solution (Equation (2)) remains complex enough to illustrate the proposed methodology. More elaborated models could be considered but they require additional experimental data to determine supplementary model parameters.

Chloride Ingress for Cracked Concrete
In recent years, much research has been devoted to the development of models for predicting the cracking effects on chloride ingress and corrosion phenomenon (corrosion initiation, propagation, Appl. Sci. 2020, 10, 2040 4 of 19 corrosion rate, etc.). Otieno et al. [34] have developed empirical models for predicting the rate of corrosion induced by chloride ions in cracked RC structures. These models incorporated the influence of several geometries of concrete cracks and cover depth (with the crack width/concrete cover ratio [35,36]), as well as the quality of concrete on the corrosion speed. This study was useful to provide recommendations about the best combination of the aforementioned parameters to meet the desired durability performance of the structure. Kurumatani et al. [37] have proposed a numerical method that allowed simulating the chloride ingress into concrete by an unstable diffusion analysis, taking into account damage associated with 3D internal cracks. The crack propagation analysis with a damage model based on fracture mechanics was also taken into account to reproduce the 3D geometry of the internal cracks. Several authors have also developed models of corrosion or chloride transport into cracked concrete (saturated or unsaturated) at the mesoscale, to consider the heterogeneity of concrete in more than two phases [38][39][40].
Several experiments as well as field studies have shown that cracking leads to rapid penetration of chloride ions and early degradation of structures [41][42][43]. For example, Gowripalan et al. [35] conducted chloride diffusing tests on cracked RC beams, and concluded that the crack width/concrete cover ratio may be a deterministic parameter for assessing the durability of cracked reinforced concrete. His research has shown that there is a hyperbolic relationship between this ratio and the chloride threshold value in cracked concrete. The introduction of cracking parameters in chlorination models is therefore useful for estimating the time of corrosion initiation taking into account the combined effects of chloride ingress and concrete cracking. To model the diffusion of chlorides in cracked concrete, we propose to use the simplified model described by Equation (4) in which we add an acceleration factor α which multiplies the diffusion coefficient D: The acceleration factor is higher than 1 for cracked conditions to consider that major chloride diffusion is expected in a cracked concrete. Since there are significant uncertainties related to this parameter depending on the concrete properties, fabrication of the structural component, loading history, etc. this parameter should be modeled as a random variable to be identified from experimental data.

General Description
The tests were carried out on 12 RC beams (300 × 120 × 1500 mm, Figure 1), designed according to Chinese standards GB/T 50081-2016 [44]. The different stages of the experimental tests are presented in the form of a flowchart in Figure 2 (for more details see [4]). Stage I includes the fabrication and curing (28 days) of the reinforced concrete specimens. RC specimens have made from commercial concrete with a target strength of 40 MPa at 28 days and a water/cement ratio 0.44 chosen according requirements of [45] for the concrete elements in a chloride-containing environment. The composition and additional details of concrete are provided in Table 1. of several geometries of concrete cracks and cover depth (with the crack width/concrete cover ratio [35,36]), as well as the quality of concrete on the corrosion speed. This study was useful to provide recommendations about the best combination of the aforementioned parameters to meet the desired durability performance of the structure. Kurumatani et al. [37] have proposed a numerical method that allowed simulating the chloride ingress into concrete by an unstable diffusion analysis, taking into account damage associated with 3D internal cracks. The crack propagation analysis with a damage model based on fracture mechanics was also taken into account to reproduce the 3D geometry of the internal cracks. Several authors have also developed models of corrosion or chloride transport into cracked concrete (saturated or unsaturated) at the mesoscale, to consider the heterogeneity of concrete in more than two phases [38][39][40]. Several experiments as well as field studies have shown that cracking leads to rapid penetration of chloride ions and early degradation of structures [41][42][43]. For example, Gowripalan et al. [35] conducted chloride diffusing tests on cracked RC beams, and concluded that the crack width/concrete cover ratio may be a deterministic parameter for assessing the durability of cracked reinforced concrete. His research has shown that there is a hyperbolic relationship between this ratio and the chloride threshold value in cracked concrete. The introduction of cracking parameters in chlorination models is therefore useful for estimating the time of corrosion initiation taking into account the combined effects of chloride ingress and concrete cracking. To model the diffusion of chlorides in cracked concrete, we propose to use the simplified model described by Equation (4) in which we add an acceleration factor α which multiplies the diffusion coefficient D: The acceleration factor is higher than 1 for cracked conditions to consider that major chloride diffusion is expected in a cracked concrete. Since there are significant uncertainties related to this parameter depending on the concrete properties, fabrication of the structural component, loading history, etc. this parameter should be modeled as a random variable to be identified from experimental data.

General Description
The tests were carried out on 12 RC beams (300 × 120 × 1500 mm, Figure 1), designed according to Chinese standards GB/T 50081-2016 [44]. The different stages of the experimental tests are presented in the form of a flowchart in Figure 2 (for more details see [4]). Stage I includes the fabrication and curing (28 days) of the reinforced concrete specimens. RC specimens have made from commercial concrete with a target strength of 40 MPa at 28 days and a water/cement ratio 0.44 chosen according requirements of [45] for the concrete elements in a chloride-containing environment. The composition and additional details of concrete are provided in Table 1.   In stage II (Figure 2), the specimens were subjected to no loading (4 beams), static loads (single load of 18 kN, 4 beams) and cyclic loads (500,000 loading cycles 5.4k N to 18 kN, 4 beams) using a three-point loading test. The control and record of static and cyclic load magnitudes were carried out by a load cell [46].  In stage II (Figure 2), the specimens were subjected to no loading (4 beams), static loads (single load of 18 kN, 4 beams) and cyclic loads (500,000 loading cycles 5.4k N to 18 kN, 4 beams) using a three-point loading test. The control and record of static and cyclic load magnitudes were carried out by a load cell [46].
After the mechanical tests, the beams were exposed to chloride ions by wetting and immerging the beams in a 3.5%~5% NaCl solution (Stage III in Figure 2). The beams were first covered with a sponge wetted with 3.5%~5% NaCl solution and dried. Afterwards, they were completely immersed in a pool filled with this same solution and dried. A total of four wetting/drying cycles and three immersion/drying cycles were performed during 388 days [47].
Once the chloride exposures were completed, four cylinders (100 mm diameter × 120 mm height) were drilled in the center zone of each beam (Stage IV in Figure 2). In Stage V, the cylinders were cut in 10 mm slices. These slices were oven dried and ground to powder to make a complete blend [4]. The resulting powders were used to measure the total content of concrete chlorides at four depths (5 mm, 15 mm, 25 mm and 35 mm). Three categories of tests were studied according to the loading conditions: uncharged tests, static and cyclic loading. In total, 12 chloride profiles were measured for each category.

Test Results
The results of the tests presented in this section will be used to illustrate the developed methodology for the identification purposes. After the loading tests, the maximum width of the crack varied between 0.15 and 0.25 mm for the cyclic load case, and between 0.04 and 0.06 mm for static load case. The results of the chloride measurements in the concrete as a function of the depth for each load case are detailed in Appendix A (Tables A1-A3).
Based on these experimental data, Wang et al. [4] estimated the input parameters of a simple chloride diffusion model (Collepardi's model [26,31]). These parameters are the concentration of chloride ions at the surface C s and the chloride diffusion coefficient D. The results of the study given in the following section presents the models of chloride ion diffusion in the sound (uncracked) and cracked concrete considered in this work. Table 2 showed that the loadings increase the values identified for the mean and the standard deviation of these parameters compared to the unloaded state. Therefore, we propose in the following section a procedure based on Bayesian networks that could be used towards this aim.

Basics of Bayesian Networks
A Bayesian network is the graphical representation of the influence of one event, one fact, or one variable on another. It is a directed acyclic graph (DAG) composed of parent and child nodes (existing real events) modeled as random variables, and oriented arrows that represent the causal relationship between the nodes. To each child node is associated a conditional probability with respect to its parent node and the set of nodes is defined on the probabilized space (Ω, X) such that: where Ω is the samples space, X is the set of random variables {X 1 , . . . , X N }, and pa(X i ) is the set of parents of the X i nodes (for more information see [48]). Figure 3 illustrates a simple three-node Bayesian network corresponding to three random variables Y, X 1 , and X 2 where X 1 and X 2 are the child nodes of the parent node Y. The joint probability of the events (Y, X 1 , X 2 ) of this network is given as the product of conditional probabilities: where P(X i |Y ) is the conditional probability of X i knowing Y.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 19 where P(X |Y) is the conditional probability of X knowing Y. Bayesian networks are useful to update probabilities in the network by integrating new information of observed variables called evidences. In our study, the methodology will allow us to integrate information from experimental trials to characterize random variables.

Proposed Bayesian Network Configuration for Random Variable Characterization
The proposed configuration of the Bayesian network will depend on the loading conditions of the tests. In this case, we have two configurations depending on whether the beam is loaded or unloaded ( Figure 4). The input parameters to be determined C , D and α are modeled as parent nodes of the Bayesian networks according to the loading case. The child nodes represent the chloride profiles at the four measurement depths of C(x , t)) and are the same for all loading conditions: x = 5 mm, 15 mm, 25 mm and 35 mm) in both cases. All inspections are undertaken at the same time after exposure t = 388 days. For tests without load, we observe in Figure 4 that we have six nodes including two parent nodes (C and D) and four child nodes (C(x , t),C(x , t), C(x , t) and C(x , t)). For tests with load (static or cyclic), the parameter α is added as a parent node giving seven nodes for this network (Figure 4). We consider that all the nodes are discrete and independent. All nodes are divided into a number of states in a range whose boundaries were chosen large enough to cover most possible values of the different nodes. As the observed data are limited, the number of states has been adjusted to get more accurate results. The details of the discretization and the prior information of the different nodes are summarized in Table 3; Table 4 for unloaded and loaded conditions, respectively. The a priori Bayesian networks are useful to update probabilities in the network by integrating new information of observed variables called evidences. In our study, the methodology will allow us to integrate information from experimental trials to characterize random variables.

Proposed Bayesian Network Configuration for Random Variable Characterization
The proposed configuration of the Bayesian network will depend on the loading conditions of the tests. In this case, we have two configurations depending on whether the beam is loaded or unloaded ( Figure 4). The input parameters to be determined C s , D and α are modeled as parent nodes of the Bayesian networks according to the loading case. The child nodes represent the chloride profiles at the four measurement depths of C(x i , t)) and are the same for all loading conditions: x i = 5 mm, 15 mm, 25 mm and 35 mm) in both cases. All inspections are undertaken at the same time after exposure t = 388 days. For tests without load, we observe in Figure 4 that we have six nodes including two parent nodes (C s and D) and four child nodes (C(x 1 , t), C(x 2 , t), C(x 3 , t) and C(x 4 , t)). For tests with load (static or cyclic), the parameter α is added as a parent node giving seven nodes for this network (Figure 4).
where P(X |Y) is the conditional probability of X knowing Y. Bayesian networks are useful to update probabilities in the network by integrating new information of observed variables called evidences. In our study, the methodology will allow us to integrate information from experimental trials to characterize random variables.

Proposed Bayesian Network Configuration for Random Variable Characterization
The proposed configuration of the Bayesian network will depend on the loading conditions of the tests. In this case, we have two configurations depending on whether the beam is loaded or unloaded ( Figure 4). The input parameters to be determined C , D and α are modeled as parent nodes of the Bayesian networks according to the loading case. The child nodes represent the chloride profiles at the four measurement depths of C(x , t)) and are the same for all loading conditions: x = 5 mm, 15 mm, 25 mm and 35 mm) in both cases. All inspections are undertaken at the same time after exposure t = 388 days. For tests without load, we observe in Figure 4 that we have six nodes including two parent nodes (C and D) and four child nodes (C(x , t),C(x , t), C(x , t) and C(x , t)). For tests with load (static or cyclic), the parameter α is added as a parent node giving seven nodes for this network (Figure 4). We consider that all the nodes are discrete and independent. All nodes are divided into a number of states in a range whose boundaries were chosen large enough to cover most possible values of the different nodes. As the observed data are limited, the number of states has been adjusted to get more accurate results. The details of the discretization and the prior information of the different nodes are summarized in Table 3; Table 4 for unloaded and loaded conditions, respectively. The a priori We consider that all the nodes are discrete and independent. All nodes are divided into a number of states in a range whose boundaries were chosen large enough to cover most possible values of the different nodes. As the observed data are limited, the number of states has been adjusted to get more accurate results. The details of the discretization and the prior information of the different nodes are summarized in Table 3; Table 4 for unloaded and loaded conditions, respectively. The a priori distributions of parent nodes were modeled as uniform distributions for the Bayesian network under unloaded conditions and the acceleration factor α for the Bayesian network under loaded conditions. With the uniform distribution, there is any assumption on the a priori shape of these parameters in order to obtain not skewed results after the characterization process. The a posteriori outputs (histograms) of C s and D for the Bayesian network under unloaded conditions are used as a priori distributions for these parameters in the Bayesian network under loaded conditions (Table 4). This procedure is justified in Section 4.2.2.

Proposed Characterization Methodology
The proposed methodology for characterizing the different parameters (C s , D and α) is illustrated in Figure 5. The Bayesian networks shown in Figure 4 allow us to integrate the observed information on the child nodes (chloride measurements at different depths) in order to update the a posteriori probabilities of their parent nodes. The proposed methodology is composed by two steps, the first one aims at determining C s and D from unloading test data. It is supposed that the parameters identified during this step are representative of the exposure conditions (i.e., C s ) and chloride diffusivity (i.e., D) for uncracked concrete. The factor α is identified in Step 2 using the evidence of tests with loading (static or cyclic). The a posteriori histograms of C s and D in Step 1 are used as prior information of Step 2. This procedure allows us to focus the characterization process on the acceleration factor α. At the end of each step, a posteriori histograms of the variables are used to characterize the distribution, mean and standard deviation of each parent node.

Results and Discussion
The objective of this section is to apply the proposed methodology to evaluate and quantify the effect of loading on the chloride ingress mechanism. In Section 5.1, we present and discuss the results of the characterization of the input parameters of the chloride ingress model. The identified values are then used to calculate the time and probability of corrosion initiation in Section 5.2.

Parameter Characterization
The output histograms of C , D, and α are also updated after each step of the procedure ( Figures  6, 7 and 8, respectively). The changes in Figure 6 and 7 after updating are due to the fact that in Step 2 we have added additional information on these two parameters for the loaded tests. These a posteriori histograms will be used to characterize the mean and standard deviation of the model parameters by considering loading conditions.

Results and Discussion
The objective of this section is to apply the proposed methodology to evaluate and quantify the effect of loading on the chloride ingress mechanism. In Section 5.1, we present and discuss the results of the characterization of the input parameters of the chloride ingress model. The identified values are then used to calculate the time and probability of corrosion initiation in Section 5.2.

Parameter Characterization
The output histograms of C s , D, and α are also updated after each step of the procedure (Figures 6-8, respectively). The changes in Figures 6 and 7 after updating are due to the fact that in Step 2 we have added additional information on these two parameters for the loaded tests. These a posteriori histograms will be used to characterize the mean and standard deviation of the model parameters by considering loading conditions. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 19 Figure 5. Flowchart of the proposed methodology.

Results and Discussion
The objective of this section is to apply the proposed methodology to evaluate and quantify the effect of loading on the chloride ingress mechanism. In Section 5.1, we present and discuss the results of the characterization of the input parameters of the chloride ingress model. The identified values are then used to calculate the time and probability of corrosion initiation in Section 5.2.

Parameter Characterization
The output histograms of C , D, and α are also updated after each step of the procedure (Figures 6, 7 and 8, respectively). The changes in Figure 6 and 7 after updating are due to the fact that in Step 2 we have added additional information on these two parameters for the loaded tests. These a posteriori histograms will be used to characterize the mean and standard deviation of the model parameters by considering loading conditions.   The mean and standard deviation of C , D, and α obtained from the Bayesian networks are summarized in Table 5 for each case (without loading, static, and cyclic loading). For parameters C and D, the results of step 1 (without load) are very close to the values given in Table 2. It is also noted that the mean and standard deviation of C vary little from one loading case to another, which makes sense because all beams have been exposed to the same constant concentration of Cl -. The parameters of D (µ and σ), are close for all cases meaning that the proposed Bayesian methodology has well separated the effects of cracking from the diffusion phenomenon which depends on the formulation and fabrication (pouring, curing, etc.) of the concrete beams. The Bayesian model allows characterization of a factor that increases the chloride diffusivity into concrete due to the presence of load-induced cracks and has a higher value for the cyclic loading. Indeed, the mean values of this factor (α) obtained from the Bayesian network are 1.73 and 1.77, respectively, for static and cyclic loading (Table 5). In both loading cases, the application of the two-step methodology makes it possible to further optimize the characterization of the parameters C and D by integrating the results of the tests with and without loading. Table 5. Mean and standard deviation of a posteriori values.
Step One Step 2: Static Loading   The mean and standard deviation of C , D, and α obtained from the Bayesian networks are summarized in Table 5 for each case (without loading, static, and cyclic loading). For parameters C and D, the results of step 1 (without load) are very close to the values given in Table 2. It is also noted that the mean and standard deviation of C vary little from one loading case to another, which makes sense because all beams have been exposed to the same constant concentration of Cl -. The parameters of D (µ and σ), are close for all cases meaning that the proposed Bayesian methodology has well separated the effects of cracking from the diffusion phenomenon which depends on the formulation and fabrication (pouring, curing, etc.) of the concrete beams. The Bayesian model allows characterization of a factor that increases the chloride diffusivity into concrete due to the presence of load-induced cracks and has a higher value for the cyclic loading. Indeed, the mean values of this factor (α) obtained from the Bayesian network are 1.73 and 1.77, respectively, for static and cyclic loading (Table 5). In both loading cases, the application of the two-step methodology makes it possible to further optimize the characterization of the parameters C and D by integrating the results of the tests with and without loading. Table 5. Mean and standard deviation of a posteriori values.
Step One Step 2: Static Loading  The mean and standard deviation of C s , D, and α obtained from the Bayesian networks are summarized in Table 5 for each case (without loading, static, and cyclic loading). For parameters C s and D, the results of step 1 (without load) are very close to the values given in Table 2. It is also noted that the mean and standard deviation of C s vary little from one loading case to another, which makes sense because all beams have been exposed to the same constant concentration of Cl − . The parameters of D (µ and σ), are close for all cases meaning that the proposed Bayesian methodology has well separated the effects of cracking from the diffusion phenomenon which depends on the formulation and fabrication (pouring, curing, etc.) of the concrete beams. The Bayesian model allows characterization of a factor that increases the chloride diffusivity into concrete due to the presence of load-induced cracks and has a higher value for the cyclic loading. Indeed, the mean values of this factor (α) obtained from the Bayesian network are 1.73 and 1.77, respectively, for static and cyclic loading (Table 5). In both loading cases, the application of the two-step methodology makes it possible to further optimize the characterization of the parameters C s and D by integrating the results of the tests with and without loading. Table 5. Mean and standard deviation of a posteriori values.
Step One Step 2: Static Loading  (2)) with the values of Table 2, the proposed Bayesian approach with Equation (4) and the identified values (Table 5), and the experimental measurement values (Figure 9) for all loading cases. We note in Figure 9 that the 10% and 90% percentiles of the two models remain very close for all loading cases, and all points of the data fall within the 10% and 90% percentile ranges of both approaches. This indicates that the model used is complex enough to represent this experiment. Moreover, both percentiles are closer to the data for the larger depths (x = 25 or 35 mm). This is due to the fact that the model does not take into account the effects of the convection zone for depths near the surface of the concrete. This result indicates that the addition of a new random variable (α) did not significantly modify the percentiles in comparison to the Collepardi's model. This means that the proposed approach was able to distinguish between material and concrete cracking related effects and uncertainties in a comprehensive way. Figures 10-12, compare the a posteriori histograms of the child nodes C(x, t) obtained from the proposed Bayesian approach with Equation (4) and the identified values (Table 5), and those obtained from Monte Carlo simulations with Equation (2). We can see that for all the loading cases, the probability densities follow the same distributions. There is a slight difference between the densities of the BN and those of the Monte Carlo simulations, which changes with the depth and depending on the loading case. For all cases, this difference is small for the depths of x = 5 mm and 15 mm; as the depth increases, the difference between the densities more visible. In the case of static loading and for all depths, we notice that the difference decreases compared to the unloaded case; which shows that the values of the parameters C s and D are optimized after the integration of the loaded test data. However, for cyclic loading, it increases slightly compared to the other two cases; which may be due to the relative uncertainties to the complex process of opening and closing of cracks during loading cycles. Therefore, we can conclude that the methodology allows these parameters to be updated taking into account uncertainties associated with the model and measurement methods.  (2)) with the values of Table 2, the proposed Bayesian approach with Equation (4) and the identified values (Table  5), and the experimental measurement values (Figure 9) for all loading cases. We note in Figure 9 that the 10% and 90% percentiles of the two models remain very close for all loading cases, and all points of the data fall within the 10% and 90% percentile ranges of both approaches. This indicates that the model used is complex enough to represent this experiment. Moreover, both percentiles are closer to the data for the larger depths (x = 25 or 35 mm). This is due to the fact that the model does not take into account the effects of the convection zone for depths near the surface of the concrete. This result indicates that the addition of a new random variable (α) did not significantly modify the percentiles in comparison to the Collepardi's model. This means that the proposed approach was able to distinguish between material and concrete cracking related effects and uncertainties in a comprehensive way. Figures 10, 11 and 12, compare the a posteriori histograms of the child nodes C(x, t) obtained from the proposed Bayesian approach with Equation (4) and the identified values (Table 5), and those obtained from Monte Carlo simulations with Equation (2). We can see that for all the loading cases, the probability densities follow the same distributions. There is a slight difference between the densities of the BN and those of the Monte Carlo simulations, which changes with the depth and depending on the loading case. For all cases, this difference is small for the depths of x = 5 mm and 15 mm; as the depth increases, the difference between the densities more visible. In the case of static loading and for all depths, we notice that the difference decreases compared to the unloaded case; which shows that the values of the parameters C and D are optimized after the integration of the loaded test data. However, for cyclic loading, it increases slightly compared to the other two cases; which may be due to the relative uncertainties to the complex process of opening and closing of cracks during loading cycles. Therefore, we can conclude that the methodology allows these parameters to be updated taking into account uncertainties associated with the model and measurement methods.     (2) and Equation (4) for the unloaded case. Figure 10. Distribution of chloride profiles with Equation (2) and Equation (4) for the unloaded case.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19 Figure 11. Distribution of chloride profiles with Equation (2) and Equation (4) percentile for the static load case.

Reliability Analysis
The phenomenon of initiation to corrosion corresponds to the depassivation of steels by chloride ions. The limit state function defining corrosion initiation is written as follows: where C (X, t) represents the total chloride concentration at the concrete cover depth c, at time t. This limit state function is used to estimate the probability of corrosion initiation.

Time of Corrosion Initiation
The corrosion initiation time t is estimated once the chloride concentration at the cover concrete depth is equal to the threshold value C (see Equation (3)). In the case of cracked concrete, the acceleration factor α is considered in the assessment of t as follows: Monte Carlo simulations are used to evaluate the corrosion initiation time by considering the parameters C , D, α and C as random variables. C , α and D follow log-normal distributions

Reliability Analysis
The phenomenon of initiation to corrosion corresponds to the depassivation of steels by chloride ions. The limit state function defining corrosion initiation is written as follows: where C (X, t) represents the total chloride concentration at the concrete cover depth c, at time t. This limit state function is used to estimate the probability of corrosion initiation.

Time of Corrosion Initiation
The corrosion initiation time t is estimated once the chloride concentration at the cover concrete depth is equal to the threshold value C (see Equation (3)). In the case of cracked concrete, the acceleration factor α is considered in the assessment of t as follows: Monte Carlo simulations are used to evaluate the corrosion initiation time by considering the parameters C , D, α and C as random variables. C , α and D follow log-normal distributions

Reliability Analysis
The phenomenon of initiation to corrosion corresponds to the depassivation of steels by chloride ions. The limit state function defining corrosion initiation is written as follows: where C tc (X, t) represents the total chloride concentration at the concrete cover depth c, at time t. This limit state function is used to estimate the probability of corrosion initiation.

Time of Corrosion Initiation
The corrosion initiation time t ini is estimated once the chloride concentration at the cover concrete depth is equal to the threshold value C th (see Equation (3)). In the case of cracked concrete, the acceleration factor α is considered in the assessment of t ini as follows: Monte Carlo simulations are used to evaluate the corrosion initiation time by considering the parameters C s , D, α and C th as random variables. C s , α and D follow log-normal distributions [32,49,50] with the mean and standard deviation given in Table 5; and C th follows a uniform distribution with mean 0.4% and a coefficient of variation (COV) = 0.19% [51]. The value of the concrete cover considered is c = 40 mm and corresponds to the value considered for the design of the beams [4]. Probability density values of t ini , obtained from Monte Carlo simulations for the different loading cases are given in Figure 13. It is noted that the static and cyclic loadings decrease the mean of the corrosion initiation time respectively by 1.1 and 1.31 years compared to unloaded tests. Standard deviations of corrosion initiation time are also decreased as a function of loading. These values are close to the experimental results which indicate that the mean corrosion initiation time for static and cyclic loading decreases by 1 and 1.5 years, respectively, compared to the unloaded case. Table 6 shows the mean and standard deviation values obtained with the parameters characterized for the three loading cases. [32,49,50] with the mean and standard deviation given in Table 5; and C follows a uniform distribution with mean 0.4% and a coefficient of variation (COV) = 0.19% [51]. The value of the concrete cover considered is c = 40 mm and corresponds to the value considered for the design of the beams [4]. Probability density values of t , obtained from Monte Carlo simulations for the different loading cases are given in Figure 13. It is noted that the static and cyclic loadings decrease the mean of the corrosion initiation time respectively by 1.1 and 1.31 years compared to unloaded tests. Standard deviations of corrosion initiation time are also decreased as a function of loading. These values are close to the experimental results which indicate that the mean corrosion initiation time for static and cyclic loading decreases by 1 and 1.5 years, respectively, compared to the unloaded case. Table 6 shows the mean and standard deviation values obtained with the parameters characterized for the three loading cases.   Figure 14 shows the corrosion initiation probability curves for the different loading cases. As expected, the probability of initiation to corrosion (PIC) increases with time and loading. This is due to the accumulation of chloride ions in the area near the reinforcements during the exposure time. Cracks in the concrete facilitate the access of oxygen and water, necessary for the oxidation reactions and the formation of rust [52]. For example, for the experimental exposure conditions, the times to reach PIC = 0.6, are 996, 575 and 505 days for, respectively, the tests without loading, with static loading and with cyclic loading. Indeed, the exposure conditions of tests are extreme because the values of the surface chloride concentration are very high [16]. The lifetime is significantly reduced for the loaded tests. These results are in agreement with experimental observations which indicate that, for certain rebars, corrosion signs have been observed at the end of the tests (t = 388 days).   Figure 14 shows the corrosion initiation probability curves for the different loading cases. As expected, the probability of initiation to corrosion (PIC) increases with time and loading. This is due to the accumulation of chloride ions in the area near the reinforcements during the exposure time. Cracks in the concrete facilitate the access of oxygen and water, necessary for the oxidation reactions and the formation of rust [52]. For example, for the experimental exposure conditions, the times to reach PIC = 0.6, are 996, 575 and 505 days for, respectively, the tests without loading, with static loading and with cyclic loading. Indeed, the exposure conditions of tests are extreme because the values of the surface chloride concentration are very high [16]. The lifetime is significantly reduced for the loaded tests. These results are in agreement with experimental observations which indicate that, for certain rebars, corrosion signs have been observed at the end of the tests (t = 388 days). Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 19 Figure 14. Probability of initiation corrosion.

Conclusions and Perspectives
This study proposed a methodology for the probabilistic characterization of the input parameters of a simple chlorination model including an acceleration factor for the diffusion coefficient of chloride in concrete. On the basis of the results obtained, the following conclusions are drawn: 1. The chloride content at different depths increases when the beams are loaded and for larger loading intensity. 2. The methodology, based on the Bayesian network approach, allows integrating data from experimental trials to determine the parameters of a model. It was also useful to separate the cracking effects from the diffusion of chloride ions mechanism through an acceleration factor. 3. The characterized means of the parameters C and D, close to the experimental values, show the usefulness of the Bayesian approach for this type of study. 4. The acceleration factor increases with the intensity of the load and is higher for the cyclic load, which resulted in larger width cracks on the beams. 5. Static and cyclic loads reduced the corrosion initiation time by 1.1 and 1.31 years, respectively, compared to the unloaded case.
In addition, one limitation of this study is the use of a simple chlorination model (Collepardi model) that does not take into account several parameters such as concrete aging and environmental conditions. Further work should consider chlorination models more representative of the chloride diffusion process. Another aspect to improve the methodology is to consider mechanics-based cracking models and to combine it with chlorination models. With these improvements, the acceleration factor could take into account crack characteristics (width, length, density, etc.), crack initiation and propagation mechanisms, and loading in a comprehensive way.

Conclusions and Perspectives
This study proposed a methodology for the probabilistic characterization of the input parameters of a simple chlorination model including an acceleration factor for the diffusion coefficient of chloride in concrete. On the basis of the results obtained, the following conclusions are drawn: 1.
The chloride content at different depths increases when the beams are loaded and for larger loading intensity.

2.
The methodology, based on the Bayesian network approach, allows integrating data from experimental trials to determine the parameters of a model. It was also useful to separate the cracking effects from the diffusion of chloride ions mechanism through an acceleration factor. 3.
The characterized means of the parameters C s and D, close to the experimental values, show the usefulness of the Bayesian approach for this type of study.

4.
The acceleration factor increases with the intensity of the load and is higher for the cyclic load, which resulted in larger width cracks on the beams.

5.
Static and cyclic loads reduced the corrosion initiation time by 1.1 and 1.31 years, respectively, compared to the unloaded case.
In addition, one limitation of this study is the use of a simple chlorination model (Collepardi model) that does not take into account several parameters such as concrete aging and environmental conditions. Further work should consider chlorination models more representative of the chloride diffusion process. Another aspect to improve the methodology is to consider mechanics-based cracking models and to combine it with chlorination models. With these improvements, the acceleration factor could take into account crack characteristics (width, length, density, etc.), crack initiation and propagation mechanisms, and loading in a comprehensive way. Funding: This research was funded by the Regional Council of 'Pays de la Loire' within the framework of the BUENO 2018-2021 research program (Durable Concrete for Offshore Wind Turbines).

Conflicts of Interest:
The authors declare no conflict of interest.