## 4 Numerical Studies
### Calculation of the Time-Varying Reliability of Concrete Carbonation
#### 4.4.2 Time-Dependent Reliability Index and the Failure Probability of the Aqueduct  
Table 8 lists the results of the large reliability index (larger than 3) of the proposed method and the direct fourth-moment method at time \(t=15\)a, as well as the MCS results for comparison. It is seen that the REs of the proposed method and the fourth-moment method are 2.07% and 14.62%, respectively. It is demonstrated that the proposed method \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Times/a** & **Mean** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Skewness** & **Kurtosis** \\ \hline
10 & 17.5526 & 7.0438 & 0.9526 & 4.6624 (\(1\times 10^{8}\) times) \\
15 & 16.4546 & 7.0424 & 0.9480 & 4.6540 (\(2\times 10^{7}\) times) \\
20 & 15.5577 & 7.0645 & 0.9479 & 4.6495 (\(1\times 10^{7}\) times) \\
25 & 14.7855 & 7.0706 & 0.9449 & 4.6446 (\(1\times 10^{7}\) times) \\
30 & 14.1056 & 7.0666 & 0.9361 & 4.6124 (\(2\times 10^{6}\) times) \\
35 & 13.5002 & 7.0790 & 0.9416 & 4.6398 (\(2\times 10^{6}\) times) \\
40 & 12.9538 & 7.0867 & 0.9370 & 4.6287 (\(2\times 10^{6}\) times) \\
45 & 12.4530 & 7.0917 & 0.9287 & 4.5826 (\(2\times 10^{6}\) times) \\
50 & 11.9942 & 7.0996 & 0.9347 & 4.6354 (\(2\times 10^{6}\) times) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The first four moments estimated with the MCS. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Times/a** & **Mean** & \begin{tabular}{c} **Relative** \\ **Error /\%** \\ \end{tabular}  & \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular}  & \begin{tabular}{c} **Relative** \\ **Error /\%** \\ \end{tabular}  & \begin{tabular}{c} **Skewness** \\ **Error /\%** \\ \end{tabular}  & \begin{tabular}{c} **Relative** \\ **Error /\%** \\ \end{tabular}  & \ \begin{tabular}{c} **Kurtosis** \\ **Error /\%** \\ \end{tabular}  & \begin{tabular}{c} **Relative** \\ **Error /\%** \\ \end{tabular}  \\ \hline
10 & 17.5537 & 0.0063 & 7.0277 & 0.2286 & 0.9579 & 0.5564 & 4.6814 & 0.4075 \\
15 & 16.4591 & 0.0273 & 7.0507 & 0.1179 & 0.9507 & 0.2848 & 4.6958 & 0.8982 \\
20 & 15.5610  
has better accuracy than the direct fourth-moment method when the large reliability index needs to be estimated. Also, the calculation times of the proposed method are 19, while the times of the fourth-moment method are 21, and those of the MCS method are 2e7 (COV of Pf is 0.0385). This further demonstrates the efficiency of the proposed method. Moreover, Figures 6 and 7 plot the curves of the failure probabilities and the time-dependent reliability indices with respect to the different times, respectively. It can be observed that the results of the proposed method are in good agreement with those of the MCS. As shown in Figure 7, when calculating the 50-year service life of the aqueduct, the reliability index curve obtained using the fourth-moment method significantly deviates from that of the MCS, with a maximum relative error of 17.18%. In contrast, the reliability index curve calculated with the proposed method closely matches the MCS results, with a maximum relative error of only 3.32%. This demonstrates the rationality and accuracy of the proposed method for calculating high reliability indices. Since the proposed method only requires 19 calculations, whereas the MCS requires millions of calculations to achieve high precision, it is evident that the proposed method can reduce the computation time while maintaining accuracy. Figure 8 shows the REs of the proposed method and the fourth-moment method at different times. From the figure, it can be seen that throughout the entire investigation range, the RE of the proposed method is much lower than that of the fourth-moment method, especially when the time is small, as the corresponding large reliability index is estimated. Based on the reliability index curves obtained using the method proposed in this paper, it can be observed that after 30 years, the reliability index of the Shazibei aqueduct falls below 3, which is lower than the target reliability index set during the structural design phase.