## 3 Chloride Ingress Model  
The chloride ingress model used in the paper is based on [PERSON]'s second law of diffusion, believing that the transport of chlorides in concrete is mainly controlled by diffusion [34]. [PERSON] et al. [35] were the first to apply [PERSON]'s solution [36] for [PERSON]'s second law, which resulted in a prediction model based on the error function. Although it is generally accepted today, the model still exhibits certain shortcomings, which are described in [37; 38]. A full probabilistic design approach for modelling chloride-induced depassivation is based on the limit state given in Equation (2). In the approach, the actual chloride content at the depth of the reinforcing steel is compared to the critical chloride content: \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable:** & **Description:** \\ \hline \(P_{Lap}\) & Initial probability of depassivation; \\ \(P(D_{\text{max}}\mid D_{\text{dep}})\) & Probability of damage (racking) given that depassivation occurred; \\ \(P(D_{\text{dep}}\mid T_{\text{max}})\) & Probability of depassivation given that the measurements indicated no depassivation (inaccuracy of the test); \\ \(P(D_{\text{dep}}\mid T_{\text{max}})\) & Probability of no depassivation given that the measurements indicated no depassivation (accuracy of the test); \\ \(P(D_{\text{dep}}\mid T_{\text{max}})\) & Probability of depassivation given that the measurements indicated depassivation (accuracy of the test); \\ \(P(D_{\text{dep}}\mid T_{\text{max}})\) & Probability of no depassivation given that the measurements indicated depassivation (inaccuracy of the test); \\ \(P_{\text{ap}}\) & Probability of successful repair; \\ \(C_{\text{max}}\) & Cost of measurement (test); \\ \(C_{\text{rp},p}\) & Cost of proactive repair; \\ \(C_{\text{rp},p}\) & Cost of reactive repair; \\ \(C_{\text{rp},r}\) & Cost of reactive repair; \\ \(U_{i}\) & Total cost. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables and probabilities used for the analysis. \begin{table}
\begin{tabular}{c c} \hline \hline
**Outcome:** & **Formulation:** \\ \hline Outcome 1 & \(E_{1}=(1-P_{Lap})*U_{1}\) \\ Outcome 2 & \(E_{2}=P_{\text{dep}}*(1-P(D_{\text{max}}\mid D_{\text{dep}}))*U_{2}\) \\ Outcome 3 & \(E_{3}=P_{\text{dep}}*P(D_{\text{max}}\mid D_{\text{dep}})*U_{3}\) \\ Outcome 4 & \(E_{4}=(1-P_{\text{dep}})*P(D_{\text{no-dep}}\mid T_{\text{max}})*U_{4}\) \\ Outcome 5 & \(E_{5}=P_{\text{dep}}*P(D_{\text{dep}}\mid T_{\text{max}})*(1-P(D_{\text{max}} \mid D_{\text{dep}}))*U_{5}\) \\ Outcome 6 & \(E_{6}=P_{\text{dep}}*P(D_{\text{dep}}\mid T_{\text{max}})*P(D_{\text{max}} \mid D_{\text{dep}})*U_{6}\) \\ Outcome 7 & \(E_{7}=(1-P_{\text{dep}})*P(D_{\text{no-dep}}\mid T_{\text{max}})*1*U_{7}\) \\ Outcome 8 & \(E_{8}=P_{\text{dep}}*P(D_{\text{dep}}\mid T_{\text{max}})*P_{\text{rp}}*U_{8}\) \\ Outcome 9 & \(E_{9}=P_{\text{dep}}*P(D_{\text{dep}}\mid T_{\text{max}})*(1-P_{\text{rp}})*U_{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical formulation of the outcomes in the decision tree formulated in Figure 1.  
\[g(c,t)=C_{crit}-C(c,t) \tag{2}\]  
The content of chlorides at a corresponding depth \(x\) is given by Equations (3)-(5):  
\[C_{x}(x,t)=C_{0}+(C_{S,\Delta x}-C_{0})\cdot\left[1-erf\frac{x-\Delta x}{2\cdot \sqrt{D_{app}(t)\cdot t}}\right], \tag{3}\]  
where \(C_{crit}\) is the critical chloride content [wt.-%/c]; \(C_{0}\) is the initial chloride content [wt.-%/c]; \(C_{S,\Delta x}\) is the chloride content at depth \(\Delta x\) [wt.-%/c]; \(x\) is the depth with a corresponding content of chlorides (\(x=c\)) [mm]; \(\Delta x\) is the depth of the convection zone [mm]; \(D_{app}(t)\) is the apparent chloride diffusion coefficient [m\({}^{2}\)/s]; and \(t\) is the time [s].