## 4 Numerical Example
### Bridge System Reliability Estimation of the Single Failure Criterion
#### 4.2.2 Failure Criterion II  
Failure criterion II involves a series-parallel system, which assumes that failure of any two 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 8. The AIC and BIC values were calculated according to Equations (4) and (5), the results of which are shown in Table 9. The best-fitting copula function was selected based on the principle of minimum AIC and BIC values. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(Z_{i}Z_{j}\)} & \multicolumn{3}{c}{**[PERSON]**} & \multicolumn{3}{c}{**[PERSON]**} & \multicolumn{3}{c}{**[PERSON]**} & \multicolumn{3}{c}{**[PERSON]**} \\ \cline{2-9}  & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** \\ \hline \(Z_{1}Z_{2}\) & \(-3172.172\) & \(-3167.264\) & \(-2638.153\) & \(-2633.245\) & \(-2987.18\) & \(-2982.272\) & \(-2894.185\) & \(-2889.277\) \\ \(Z_{1}Z_{3}\) & \(-3149.584\) & \(-3144.676\) & \(-2601.126\) & \(-2596.218\) & \(-2987.19\) & \(-2982.282\) & \(-2902.385\) & \(-2897.477\) \\ \(Z_{1}Z_{4}\) & \(-3169.802\) & \(-3164.894\) & \(-2580.177\) & \(-2575.269\) & \(-2998.766\) & \(-2993.858\) & \(-2917.606\) & \(-2912.698\) \\ \(Z_{1}Z_{5}\) & \(-3153.864\) & \(-3148.956\) & \(-2543.111\) & \(-2538.203\) & \(-2996.096\) & \(-2991.189\) & \(-2867.761\) & \(-2862.853\) \\ \(Z_{2}Z_{3}\) & \(-3148.243\) & \(-3143.335\) & \(-2532.662\) & \(-2527.755\) & \(-3005.002\) & \(-3000.094\) & \(-2884.315\) & \(-2879.408\) \\ \(Z_{2}Z_{4}\) & \(-3109.616\) & \(-3104.708\) & \(-2522.278\) & \(-2517.37\) & \(-2949.566\) & \(-2944.658\) & \(-2819.904\) & \(-2814.996\) \\ \(Z_{2}Z_{5}\) & \(-3142.166\) & \(-3137.258\) & \(-2540.273\) & \(-2535.365\) & \(-3010.377\) & \(-3005.469\) & \(-2867.096\) & \(-2862.188\) \\ \(Z_{3}Z_{4}\) & \(-3146.562\) & \(-3141.654\) & \(-2543.441\) & \(-2538.533\) & \(-2929.757\) & \(-2987.849\) & \(-2873.682\) & \(-2868.774\) \\ \(Z_{3}Z_{5}\) & \(-3093.642\) & \(-3088.734\) & \(-2522.063\) & \(-2517.155\) & \(-2923.267\) & \(-2918.359\) & \(-2803.811\) & \(-2798.903\) \\ \(Z_{4}Z_{5}\) & \(-3103.555\) & \(-3098.648\) & \(-2574.042\) & \(-2569.134\) & \(-2932.939\) & \(-2928.031\) & \(-2815.509\) & \(-2810.602\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The Akaike information criterion (AIC) and Bayesian information criterion (BIC) values for the candidate copulas. \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 \(Z_{1}Z_{2}\) & \(3.993\times 10^{-8}\) & \(Z_{2}Z_{4}\) & \(1.558\times 10^{-7}\) \\ \(Z_{1}Z_{3}\) & \(4.522\times 10^{-8}\) & \(Z_{2}Z_{5}\) & \(1.560\times 10^{-7}\) \\ \(Z_{1}Z_{4}\) & \(4.570\times 10^{-8}\) & \(Z_{3}Z_{4}\) & \(6.073\times 10^{-7}\) \\ \(Z_{1}Z_{5}\) & \(4.570\times 10^{-8}\) & \(Z_{3}Z_{5}\) & \(6.220\times 10^{-7}\) \\ \(Z_{2}Z_{3}\) & \(1.418\times 10^{-7}\) & \(Z_{4}Z_{5}\) & \(4.459\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: 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{3}{c}{**Copula Parameters \(\theta\)**} \\ \cline{2-6}  & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline \(Z_{1}Z_{2}\) & 0.9791 & 8.194 & 6.759 & 27.58 \\ \(Z_{1}Z_{3}\) & 0.9786 & 8.017 & 6.801 & 27.85 \\ \(Z_{1}Z_{4}\) & 0.9790 & 7.870 & 6.812 & 28.00 \\ \(Z_{1}Z_{5}\) & 0.9787 & 7.706 & 6.772 & 27.19 \\ \(Z_{2}Z_{3}\) & 0.9786 & 7.674 & 6.821 & 27.46 \\ \(Z_{2}Z_{4}\) & 0.9777 & 7.566 & 6.596 & 26.34 \\ \(Z_{2}Z_{5}\) & 0.9784 & 7.717 & 6.842 & 27.24 \\By comparing AIC and BIC values, it can be seen that the Gaussian copula was the best-fitting copula for the two performance functions under failure criterion II.