## 4 Numerical Example
### Bridge System Reliability Estimation of the Single Failure Criterion
#### 4.2.3 Failure Criterion III  
Failure criterion IIIinvolves a series-parallel system, which assumes that failure of any three adjacent slab segments will cause the bridge system to fail. According to the copula theorem, the copula parameters of the selected copula functions were evaluated, as shown in Table 11. The AIC and BIC values were calculated according to Equations (4) and (5), the results of which are shown in Table 12. Similarly, according to the principle of minimum AIC and BIC values, the Gaussian copula was the best-fitting copula for the three performance functions, as seen by comparing the AIC and BIC values. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(Z_{i}Z_{j}\) & **Joint Failure Probability \(P_{f}\)** & \(Z_{i}Z_{j}\) & **Joint Failure Probability \(P_{f}\)** \\ \hline \(Z_{1}Z_{2}\) & \(3.989\times 10^{-8}\) & \(Z_{5}Z_{6}\) & \(4.459\times 10^{-6}\) \\ \(Z_{2}Z_{3}\) & \(1.418\times 10^{-7}\) & \(Z_{6}Z_{7}\) & \(6.060\times 10^{-7}\) \\ \(Z_{3}Z_{4}\) & \(6.075\times 10^{-7}\) & \(Z_{7}Z_{8}\) & \(1.408\times 10^{-7}\) \\ \(Z_{4}Z_{5}\) & \(4.459\times 10^{-6}\) & \(Z_{6}Z_{9}\) & \(3.985\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The joint failure probability of each pair of failure modes based on a Gaussian copula. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(Z_{i}Z_{j}\)} & \multicolumn{4}{c}{**Copula Parameters \(\theta\)**} \\ \cline{2-6}  & **Gaussian** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline \(Z_{1}Z_{2}\) & 0.9790 & 8.194 & 6.759 & 27.58 \\ \(Z_{2}Z_{3}\) & 0.9786 & 7.674 & 6.821 & 27.46 \\ \(Z_{3}Z_{4}\) & 0.9785 & 7.732 & 6.759 & 27.28 \\ \(Z_{4}Z_{5}\) & 0.9776 & 7.861 & 6.546 & 26.26 \\ \(Z_{5}Z_{6}\) & 0.9785 & 7.920 & 6.778 & 27.33 \\ \(Z_{6}Z_{7}\) & 0.9779 & 7.432 & 6.780 & 27.00 \\ \(Z_{7}Z_{8}\) & 0.9778 & 7.690 & 6.746 & 27.19 \\ \(Z_{8}Z_{9}\) & 0.9789 & 7.750 & 6.778 & 27.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The copula parameters for the candidate copulas. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(Z_{i}Z_{j}\)} & \multicolumn{4}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} \\ \cline{2-6}  & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** \\ \hline \(Z_{1}Z_{2}\) & \(-3172.17\) & \(-3167.264\) & \(-2638.153\) & \(-2633.245\) & \(-2987.18\) & \(-2982.272\) & \(-2894.185\) & \(-2889.277\) \\ \(Z_{2}Z_{3}\) & \(-3148.243\) & \(-3143.335\) & \(-2532.662\) & \(-2527.755\) & \(-3005.002\) & \(-3000.094\) & \(-2884.315\) & \(-2879.408\) \\ \(Z_{3}Z_{4}\) & \(-3146.562\) & \(-3141.654\) & \(-2543.441\) & \(-2538.533\) & \(-2992.757\) & \(-2987.849\) & \(-2873.682\) & \(-2868.774\) \\ \(Z_{4}Z_{5}\) & \(-3103.555\) & \(-3098.648\) & \(-2574.042\) & \(-2569.134\) & \(-2932.939\) & \(-2928.031\) & \(-2815.509\) & \(-2810.602\) \\ \(Z_{5}Z_{6}\) & \(-3145.400\) & \(-3140.492\) & \(-2584.306\) & \(-2579.398\) & \(-2985.515\) & \(-2980.608\) & \(-2872.403\) & \(-2867.496\) \\ \(Z_{6}Z_{7}\) & \(-3116.444\) & \(-3111.536\) & \(-2482.662\) & \(-2477.754\) & \(-2993.222\) & \(-2988.314\) & \(-2854.561\) & \(-2849.653\) \\ \(Z_{7}Z_{8}\) & \(-3112.185\) & \(-3107.277\) & \(-2526.177\) & \(-2521.269\) & \(-2985.223\) & \(-2980.315\) & \(-2858.875\) & \(-2853.967\) \\ \(Z_{8}Z_{9}\) & \(-3161.455\) & \(-3156.548\) & \(-2549.842\) & \(-2544.934\) & \(-3001.664\) & \(-2996.757\) & \(-2898.705\) & \(-2893.797\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The akaike information criterion (AIC) and bayesian information criterion (BIC) values for the candidate copulas. Then, based on the Gaussian copula function, the joint failure probability of each group of failure modes was obtained according to Table 4 and Equation (7), as shown in Table 13. According to the ternary Gaussian copula and Equation (11), the system failure probability \(P_{f_{\rm III}}\) of the case bridge under failure criterion III was \(4.356\times 10^{-6}\).