## 5 Case Study
### Condition Prediction  
Once the input parameters are determined, it is possible to perform the analysis using the semi-Markov process model. To do so, the sums of the sojourn times \(T_{1\to i}\). have to be numerically determined. These sums for the wings of the Weikendorf Bridge are presented in Table 19.  
The PDFs and SFs (in Equations (7)-(11)) of the sojourn time sums \(T_{1\to i}\) were used to calculate single-time step transition probabilities \(p_{ij}^{t,t+1}\). For instance, the probability of transition from the state 3 to 4 in the period between the 38 th and 39 th year is given in Equation (26):  
\[p_{3,4}^{38,39}\ =\ \frac{f_{1\to 3}(38)}{S_{1\to 3}(38)-S_{1\to 2}(38)}\ =\ 0.1470. \tag{26}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Robustness Class (ROC)** & **Phase** & **Mean Value \(m\)** & **Standard Deviation \(s\)** & **Location Parameter \(\upmu\)** & **Scale Parameter \(\upsigma\)** \\ \hline ROC 3 & \(t_{prop\_all}\) & 50.82 & 22.89 & 3.84 & 0.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 18: Duration of propagation until the event of collapse \(t_{prop\_coll}\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(T_{1\to i}\) & \(\mathbf{\Sigma\;t}\) & **Mean Value \(m\)** & **Stdard Deviation \(s\)** & **Location** & **Scale** \\  & & & **Deviation \(s\)** & **Parameter \(\upmu\)** & **Parameter \(\upsigma\)** \\ \hline \(T_{1\to 2}\) & \(t_{ini}+t_{prop\_era}\) & 19.54 & 125 & 2.85 & 0.49 \\ \(T_{1\to 3}\) & \(t_{ini}+t_{prop\_spta}\) & 27.99 & 197 & 3.26 & 0.38 \\ \(T_{1\to 4}\) & \(t_{ini}+t_{prop\_coll}\) & 61.90 & 24.91 & 4.05 & 0.39 \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Sums of sojourn times \(T_{1\to i}\). of the process of carbonation-induced corrosion in the wings of the Weikendorf Bridge. Once all the single-time step probabilities are known, it is possible to populate the transition matrices \(P^{t,t+1}\), as shown for the probability matrix \(P^{38,39}\) in Equation (27):  
\[p^{38,39}\ =\ \left[ \begin{array}{cccc}p_{1,1}^{38,39}&p_{1,2}^{38,39}&0&0&0 \\ 0&p_{2,2}^{38,39}&p_{2,3}^{38,39}&0&0\\ 0&0&p_{3,3}^{38,39}&p_{3,4}^{38,39}&0\\ 0&0&0&p_{4,4}^{38,39}&p_{4,5}^{38,39}\\ 0&0&0&0&1\end{array} \right]\ =\ \left[ \begin{array}{cccc}0.918&0.081&0&0&0 \\ 0&0.674&0.326&0&0\\ 0&0.853&0.147&0\\ 0&0&0&0.981&0.019\\ 0&0&0&0&1\end{array} \right]. \tag{27}\]  
The probability that the process will assume one of the states \(i\) in the subsequent time period \(k\) can be calculated using Equation (3). For instance, when predicting the condition of the wings at the age \(t=38\) years for the subsequent period \(k\) of 20 years, the \(n\)-dimensional vector expressed in Equation (3) takes the form:  
\[p(58)\ =\ p(38)\cdot P^{38,39}\cdot P^{39,40}\ldots P^{57,58}\ =\ [0\ 0\ 0.034\ 0.626\ 0. 340]. \tag{28}\]  
The obtained values in Equation (28) show that the probabilities that the wings will be in each one of five possible condition states in the year 2025 are equal to \(p_{1}(58)=p_{2}(58)=0\%\), \(p_{3}(58)=3.4\%\), \(p_{4}(58)=62.6\%\), and \(p_{5}(58)=34.0\%\). After all the transition matrices \(P^{t,t+1}\) are known, it is possible to analyze the condition of the wings for all subsequent periods \(k\), as depicted in Figure 9a. The shaded part of Figure 9a presents the elapsed time, i.e., the time before the inspection in 2005 when the CS3 was first assigned. If the same structure is regarded as newly built, then the state condition vector takes the form \(p(0)=(1\ 0\ 0\ 0\ 0)\). Applying such a state condition vector, the conditions for the subsequent periods \(k\) were obtained, as depicted in Figure 9b. One can easily notice that if the structure is observed as new, the most probable condition state at the time \(t=38\) years is CS4, in contrast to CS3, which was assigned based on visual inspection at the age of 38 years.