## 4 Numerical Simulation
### System Reliability Analysis  
Because there are many failure modes in actual engineering, and there is a certain correlation among them, how to consider the correlation between failure modes is very important to the reliability theory of the structural system. Copula theory is introduced to analyze the correlation of failure modes of the structural system in this paper, and each Copula function has its own characteristics during calculation, so it is necessary to select the appropriate Copula function. First, the failure credibility of each member should be determined, as shown in Table 2.  
For the optimization of the Copula function in this paper, the Akaike information criterion (AIC) and Bayesian information criterion (BIC) commonly used in engineering are adopted, that is, the Copula function with minimum AIC and BIC values is used as the optimal Copula function to fit the original observation data [48]. The commonly used Gaussian Copula, Clayton Copula, Gumbel Copula, and Frank Copula are selected as the candidate copula functions in this paper, and the corresponding Copula parameters, as well as the AIC and BIC values, are calculated, as shown in Tables 3 and 4. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{Bel_{1}}\) & \(\mathbf{Bel_{2}}\) & \(\mathbf{Bel_{3}}\) & \(\mathbf{Bel_{4}}\) & \(\mathbf{Bel_{5}}\) \\ \hline
1.000 & 0.7793 & 0.7635 & 0.7793 & 1.000 \\ \hline \end{tabular}
\end{table} 
Table 2: Failure credibility _Beli_ of each member. Figure 4: Failure probability of 1# slab in different hinge joint damage intervals under different membership functions. (The maximum credibility is 0.4): (**a**) Hinge joint damage case 1; (**b**) hinge joint damage case 2; (**c**) hinge joint damage case 3; (**d**) hinge joint damage case 4. As can be seen from Table 4, the Gaussian Copula is determined as the most appropriate Copula function among the three failure criteria according to AIC and BIC minimum principles. Then, the combined failure probability of members considering the failure credibility under three failure criteria is calculated, combined with Equation (15) based on the Gaussian Copula, as demonstrated in Table 5.  
The system reliability of three different failure criteria considering member failure credibility is calculated. The above analysis process is adopted to calculate the system reliability of three different failure criteria without considering the member failure credibility, whose results are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} & \multicolumn{2}{c}{**[PERSON]**} \\ \cline{2-7}  & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** & **AIC** & **BIC** \\ \hline \(Z_{1}Z_{2}\) & \(-2630.94\) & \(-2626.03\) & \(-2060.63\) & \(-2055.73\) & \(-2504.95\) & \(-250  
It can be seen from Figure 5 that the failure credibility of members has an impact on the system reliability of the three failure criteria. Compared to not considering member failure credibility, the failure probability of failure criterion I increases while the failure probability of failure criterion II and III decreases when considering member credibility. Because the above analysis does not consider hinge joint damage, the system reliability that considers both the member failure credibility and the fuzzy detection information of hinge joint damage degree is analyzed according to the reliability evaluation method considering the fuzzy detection information of hinge joint damage degree proposed in this paper. The corresponding four damage degree intervals in Table 2 when the damage degree of hinge joints is 0.2 as the maximum reliability value are selected. Taking case 1 as an example, the system reliability of different failure criteria is analyzed based on Copula theory, whose results are shown in Figure 6.  
Figure 5: System reliability under different failure criteria considering failure credibility and not considering failure credibility.