# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 3 Engineering Applications
### Calculation of System Reliability Indicators
#### 3.2.5 Calculation of the System Failure Probability  
Based on the results of the analysis in Section 3.2.4, the failure probabilities of the failure mode combinations are brought into the corresponding Copula functions, respectively, and the values of the joint failure probabilities of the two-dimensional failure modes can be calculated as shown in the following Table 7.  
From the failure probability values calculated in the above table, it can be seen that all the two-dimensional joint failure probabilities combined with the seventh failure mode have a large order of magnitude, all of which were \(10^{-17}\). It can be concluded that the correlation between the seventh failure mode and the other four failure modes is weak. The orders of magnitude of the failure probability values calculated by the other four failure \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(g_{i}\delta_{j}\)} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} \\  & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** \\ \hline \(g_{33}\xi_{4}\) & \(-2394.7\) & \(-2390.5\) & \(-1848.5\) & \(-1844.3\) & \(-211.3\) & \(-207.1\) & \(-2431.1\) & \(-2426.9\) \\ \(g_{33}\xi_{5}\) & \(-\mathbf{3092.1}\) & \(-\mathbf{3087.9}\) & \(-2309.7\) & \(-2305.5\) & \(-216.7\) & \(-212.4\) & \(-3064.3\) & \(-3060.1\) \\ \(g_{33}\xi_{6}\) & \(-\mathbf{3493.5}\) & \(-\mathbf{3489.2}\) & \(-2693.9\) & \(-2689.7\) & \(-218.4\) & \(-214.2\) & \(-3443.5\) & \(-3439.3\) \\ \(g_{337}\) & \(-\mathbf{3255.7}\) & \(-\mathbf{3251.5}\) & \(-2566.8\) & \(-2562.5\) & \(-217.4\) & \(-213.2\) & \(-3186.6\) & \(-3182.3\) \\ \(g_{48}\xi_{5}\) & \(-2444.2\) & \(-2440.0\) & \(-1871.0\) & \(-1866.8\) & \(-211.7\) & \(-207.5\) & \(-2502.7\) & \(-2498.5\) \\ \(g_{48}\xi_{6}\) & \(-2507.3\) & \(-2503.0\) & \(-1941.4\) & \(-1937.2\) & \(-212.3\) & \(-208.1\) & \(-2550.0\) & \(-2545.8\) \\ \(g_{487}\) & \(-2531.2\) & \(-2526.9\) & \(-1962.6\) & \(-1958.4\) & \(-212.6\) & \(-208.4\) & \(-2566.7\) & \(-2562.5\) \\ \(g_{586}\) & \(-\mathbf{3723.4}\) & \(-\mathbf{3719.2}\) & \(-2857.7\) & \(-2853.5\) & \(-218.9\) & \(-214.7\) & \(-3614.9\) & \(-3610.7\) \\ \(g_{587}\) & \(-\mathbf{3901.5}\) & \(-\mathbf{3897.3}\) & \(-2816.5\) & \(-2812.3\) & \(-219.5\) & \(-215.3\) & \(-3759.5\) & \(-3755.3\) \\ \(g_{687}\) & \(-\mathbf{3875.4}\) & \(-\mathbf{3871.2}\) & \(-2829.3\) & \(-2825.1\) & \(-219.3\) & \(-215.1\) & \(-3667.6\) & \(-3663.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: List of calculated results of AIC and BIC values (the bolded parts in the table are the minimum values of AIC and BIC for each failure mode combination). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(g_{i}\delta_{j}\) & **Joint Failure Probability** & \(g_{i}\delta_{j}\) & **Joint Failure Probability** \\ \hline \(g_{33}\xi_{4}\) & \(2.4196\times 10^{-7}\) & \(84\xi_{6}\) & \(2.5273\times 10^{-7}\) \\ \(g_{33}\xi_{5}\) & \(7.5465\times 10^{-7}\) & \(84\xi 7\) & \(5.5511\times 10^{-17}\) \\ \(g_{33}\xi_{6}\) & \(1.9187\times 10^{-6}\) & \(85\xi 6\) & \(7.5465\times 10^{-7}\) \\ \(g_{33}\xi_{7}\) & \(5.5511\times 10^{-17}\) & \(85\xi 7\) & \(5.5511\times 10^{-17}\) \\ \(g_{48}\xi_{5}\) & \(2.1418\times 10^{-7}\) & \(86\xi 7\) & \(5.5511\times 10^{-17}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Two-dimensional joint failure probabilities for each combination of failure modes.