## 5 Case Study
### Carbonation Propagation
Figure 6: Time of wetness _ToW_ for the weather station Gansendorf, expressed in the percentage of number of rainy days. Carbonation rate \(k_{NAC}\) was determined based on Table 2, considering that according to the design documents, the concrete class B160 was used in the wings. A w/c ratio of 0.75 and the cement type CEM I were assumed, in the belief that this cement type was most common at the time of construction. Such presumed parameters resulted in normally distributed carbonation rate \(k_{NAC}\) with a mean value of 8.0 and a standard deviation of 1.1. It is not known which exact cement strength class was used, nor what the ambient temperature was during construction. However, since the resistivity of the concrete was proven to be low by the performed test, a short curing time is assumed, i.e., \(t_{c}\) of 2 days. The value of carbonation concentration of ambient air C\({}_{a}\) was taken to be the maximum proposed for rural areas, thus being normally distributed with the mean value of 0.042% and a standard deviation of 0.005. The overall parameters for the carbonation-induced depassivation model gathered for the wings of the Weikendorf Bridge are presented in Table 14. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & & **Unit** & **Distribution Type** & **Mean Value \(m\)** & **Standard Deviation \(s\)** \\ \hline Relative & \(RH_{l}\) & \% & Constant & 65 & - \\ humidity—laboratory & \(f_{e}\) & - & Constant & 5 & - \\ Exponent & \(g_{e}\) & - & Constant & 2.5 & - \\ Regression exponent & \(b_{w}\) & - & Lognormal & 0.446 & 0.163 \\ Exponent & \(b_{c}\) & - & Normal & \(-\)0.567 & 0.024 \\ Reference time & \(t_{0}\) & years & Constant & 0.0767 & - \\ CO\({}_{2}\) & \(C_{l}\) & vol.\% & Constant & 0.04 & - \\ concentration—laboratory & \(C_{a}\) & mm & Normal & 30 & 6 \\ Concrete cover & \(c\) & mm & Normal & 72.4 & 2.2 \\ Relative humidity of & \(RH_{a}\) & \% & Normal & 0.145 & - \\ ambient air & \(T_{0}W\) & \% & Constant & 0.10 & - \\ Time of wetness & \(T_{0}W\) & \% & Constant & 0.145 & - \\ Probability of & \(p_{dr}\) & \% & Constant & 0.10 & - \\ driving rain & \(k_{NAC}\) & mm/years\({}^{0.5}\) & Normal & 8 & 1.1 \\ Carbonation rate & \(k_{c}\) & days & Constant & 2 & - \\ C\({}_{2}\) concentration of & & \(C_{a}\) & vol.\% & Normal & 0.042 & 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Parameters of carbonation-induced depassivation for the wings of the Weikendorf Bridge. Figure 7: Average relative humidity \(RH_{a}\) for weather stations in the state of Lower Austria in the period 1981–2010, measured at 7:00 a.m. and 2:00 p.m. [58]. Carbonation depth at the age 46 years \(x_{c}\)(46) calculated using the parameters in Table 14 resulted in a mean value of 40.80 mm and a standard deviation of 13.87 mm. Such calculated results closely match the highest measured value of carbonation in point W4, as presented in Table 12. Moreover, from the parameters shown in Table 14, using Equation (17), it is possible to calculate the duration of time of initiation \(t_{ini}\). For the probabilistic analysis of time of initiation \(t_{ini}\), as well as for other probabilistic analysis in the article, the probabilistic engine FReET (Feasible Reliability Engineering Tool) was used [61]. The determined statistical parameters of the duration of time of initiation \(t_{ini}\) are presented in Table 15 as lognormally distributed. As shown in Figure 1, the time of initiation \(t_{ini}\) represents the sojourn time of condition state one \(T_{1}\) of the semi-Markov process.