## 3 Chloride Ingress Model
The apparent chloride diffusion coefficient can be expressed as:  
\[D_{app}(t)=k_{e}\cdot D_{RCM,0}\left(\frac{t_{0}}{t}\right)^{\alpha} \tag{4}\]  
where \(k_{e}\) is the environmental variable [-]; \(D_{RCM,0}\) is the chloride migration coefficient at the reference point of time [m\({}^{2}\)/s]; \(t_{0}\) is the reference point of time [y]; and \(\alpha\) is the ageing exponent [-]. The environmental variable, \(k_{e}\), takes the form:  
\[k_{e}=\exp\left[b_{e}\left(\frac{1}{T_{ref}}-\frac{1}{T_{real}}\right)\right] \tag{5}\]  
where \(b_{e}\) is the temperature coefficient [K]; \(T_{ref}\) is the reference temperature [K]; and \(T_{real}\) is the temperature of the structural element or the ambient air [K]. To detect and categorize the values of the chloride ingress parameters in existing concrete bridges, an adjustment is required. Such an adjustment is necessary since in codes and norms, no correlation of chloride ingress parameters to formerly used exposure classes, concrete classes, cement types, etc. is given. Unfortunately, a detailed description of the correlation between chloride transport characteristics and parameters resulting from former and new codes is beyond the scope of this paper. Thus, the parameter identification and the process of obtaining prior probabilities of depassivation, \(P_{dp}\), were in this paper based on a general case. For quantification and a detailed description of parameters of the employed model, see [1; 2; 3; 10; 11; 39; 40; 41]. Furthermore, for a detailed insight into how the chosen parameters influence the outcome of the used model, see [42]. As shown in Equation (2), the limit state of chloride-caused depassivation is assumed to be reached when the chloride content, \(C(c,t)\), at the reinforcement depth surpasses the critical chloride content, \(C_{crit}\). Both of these variables are random, and hence, form a random limit state function, \(g(c,t)\). The probability of failure, \(p(t)\), is regarded as the probability of the limit state function, \(g(c,t)\), being lower than zero, as shown in Equation (6):  
\[p_{f}(t)=p_{dp}=p[g<0] \tag{6}\]  
The reliability (safety) index, \(\beta\), was introduced to demonstrate how often the standard deviation, \(\dot{s}_{g}(t)\), of the limit state function, \(g(c,t)\), may be placed between zero and the mean value, \(m_{g}(t)\), as shown in Equation (7):  
\[\beta(t)\approx\frac{m_{g}(t)}{s_{g}(t)} \tag{7}\]  
The reliability index, \(\beta(t)\), can be used for determining the initial (prior) probability of depassivation, \(P_{dp}\), the value of which has not yet been affected by the additional information obtained in tests and measurements. Assuming that the limit state, \(g(c,t)\), is normally distributed, the probabilities of depassivation can be estimated through the reliability index, \(\beta(t)\), as follows:  
\[P_{dp}=p_{f}(t)\approx\Phi(u=-\beta) \tag{8}\]where \(\Phi(u)\) is the normal distribution function, which gives the probability that a standard normal variate assumes a value in the interval \([0,u]\). Although the service life of a structure is the sum of the initiation and propagation period, by the limit state given in Equation (3), only the initiation period is taken into account, deeming the propagation period to be comparatively very short. The evolution of different forms of deterioration, once corrosion has been initiated, does not always occur at the same rate. Hence, the consequences of reaching the limit state, cost of repair, and probability of the occurrence of damage are not always the same. For the exposure classes XD1 and XD2, according the European Standard EN 206-1 [43] (i.e., for predominantly dry or wet conditions), the consequences and probability of corrosion occurrence are not expected to be significant, because the corrosion rate is limited either due to the lack of moisture or the lack of oxygen. However, in exposure class XD3, higher consequences and probabilities of the occurrence of corrosion are expected because it is likely that sufficient moisture and oxygen are present.