## 3 Results and Discussion
### System Reliability  
According to the method proposed in this paper, the system reliability of the case study can be evaluated. Firstly, a hierarchical model shown in Figure 2 is established. Secondly, the judgment matrices of different damage cases of hinge joints are established. When the hinge joints are intact, the element \(a_{ij}\) is calculated according to Equation (7) and the judgment matrix \(A_{int\,act}\) is constructed. When the hinge joints are damaged, the element \(a_{ij}\) is calculated according to Equation (8) and the corresponding judgment matrices are established. \[ \begin{split} A_{int\,act}=\left[\begin{array}{ccccc}1.000&1.083 &1.089&1.083&1.000\\ 0.924&1.000&1.006&1.000&0.924\\ 0.918&0.994&1.000&0.994&0.918\\ 0.924&1.000&1.006&1.000&0.924\\ 1.000&1.083&1.089&1.083&1.000\end{array}\right]\end{split} \]  
\[ \begin{split} A_{case\,2}=\left[\begin{array}{ccccc}1.000&1.083 &1.089&1.083&1.000\\ 0.924&1.000&1.006&1.000&0.924\\ 0.918&0.994&1.000&0.994&0.918\\ 0.924&1.000&1.006&1.000&0.924\\ 1.000&1.083&1.089&1.083&1.000\end{array}\right]\end{split} \]  
\[ \begin{split} A_{case\,4}=\left[\begin{array}{ccccc}1.000&0.452 &0.375&1.840&1.696\\ 2.211&1.000&0.829&4.068&3.750\\ 2.667&1.207&1.000&4.908&4.525\\ 0.543&0.246&0.204&1.000&0.922\\ 0.589&0.267&0.221&1.085&1.000\end{array}\right]\end{split} \]  
\[ \begin{split} A_{case\,6}=\left[\begin{array}{ccccc}1.000&1.083 &0.169&1.083&1.000\\ 0.924&1.000&0.156&1.000&0.924\\ 5.907&6.395&1.000&6.395&5.907\\ 0.924&1.000&0.156&1.000&0.924\\ 1.000&1.083&0.169&1.083&1.000\end{array}\right]\end{split} \]  
\[ \begin{split} A_{case\,8}=\left[\begin{array}{ccccc}1.000&1.876 &0.305&0.368&0.782\\ 0.553&1.000&0.163&0.196&0.417\\ 3.275&6.142&1.000&1.207&2.559\\ 2.714&5.091&0.829&1.000&2.121\\ 1.279&2.400&0.391&0.471&1.000\end{array}\right]\end{split} \]  
Comparing Figure 4 with Figure 5, it can be seen that the variation tendency of the range analysis values of the reliability indexes and LLDF under different damage cases of hinge joints is basically the same. Different damaged hinge joints have different effects on the reliability index. ### System Reliability  
According to the method proposed in this paper, the system reliability of the case study can be evaluated. Firstly, a hierarchical model shown in Figure 2 is established. Secondly, the judgment matrices of different damage cases of hinge joints are established. When the hinge joints are intact, the element \(a_{ij}\) is calculated according to Equation (7) and the judgment matrix \(A_{int\,act}\) is constructed. When the hinge joints are damaged, the element \(a_{ij}\) is calculated according to Equation (8) and the corresponding judgment matrices are established. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Slab Segment Number**} & \multicolumn{8}{c}{**Case Number**} \\ \cline{2-9}  & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** & **Case 7** & **Case 8** & **Case 9** \\ \hline
1 & 0.0619 & 0.0406 & 0.0619 & 0.0760 & 0.0632 & 0.0603 & 0.0619 & 0.0619 & 0.0642 \\
2 & 0.0211 & 0.0374 & 0.0168 & 0.1680 & 0.0839 & 0.0557 & 0.0549 & 0.0330 & 0.0621 \\
3 & 0.0151 & 0.1678 & 0.1678 & 0.2027 & 0.1057 & 0.3562 & 0.3562 & 0.2027 & 0.3562 \\
4 & 0.0759 & 0.1279 & 0.1279 & 0.0413 & 0.0839 & 0.0557 & 0.0609 & 0.1680 & 0.0621 \\
5 & 0.0822 & 0.1384 & 0.1384 & 0.0448 & 0.0632 & 0.0603 & 0.0660 & 0.0792 & 0.0642 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Range analysis values of reliability indexes of each slab segment under different damage cases of hinge joints. Thirdly, the weight vector \(\mathbf{W}=(\omega_{1};\omega_{2}\cdots\omega_{n})\) and the largest eigenvalue \(\lambda_{max}\) are calculated according to Equations (9) and (10). Then, the performance consistency of the judgement matrix is checked according to Equations (11) and (12). The results are shown in Table 8.  
As can be seen in Table 8, the \(CR\) value of each matrix is less than 0.1, which means that all judgment matrices meet the consistency requirement.