## 3 Results and Discussion
### System Reliability
Lastly, the system reliability indexes are calculated according to Equation (13). When the hinge joints are intact, the reliability indexes of each slab segment are shown in Table 6, and the system reliability index can be calculated as follows:  
\[\beta_{S-int\,act}=0.2098\times 4.314\,+\,0.1938\times 4.671\,+\,0.1926\times 4.698 \,+\,0.1938\times 4.671\,+\,0.2098\times 4.314=4.526.\]  
When the hinge joint is damaged, in order to ensure a more secure system reliability evaluation result, the minimum reliability indexes of different damage degrees of each slab segment under different damage cases of hinge joints are selected. The minimum reliability indexes taken from Table 6 are shown in Table 9. Then, the system reliability indexes considering the hinge joint damage can be calculated according to Equation (13). \[\beta_{S-case1}=0.2416\times 4.253\,+\,0.0824\times 4.649\,+\,0.0589\times 4.673 \,+\,0.2963\times 4.574\,+\,0.3208\times 4.232\,=\,4.399.\]  
\[\beta_{S-case2}=0.0793\times 4.274\,+\,0.0730\times 4.612\,+\,0.3277\times 4.506\,+\,0.2 498\times 4.521\,+\,0.2703\times 4.176\,=\,4.409.\]  
\[\beta_{S-case3}=0.1207\times 4.253\,+\,0.0328\times 4.633\,+\,0.3272\times 4.505\,+\,0.2 494\times 4.521\,+\,0.2699\times 4.176\,=\,4.393.\]  
\[\beta_{S-case4}=0.1426\times 4.238\,+\,0.3153\times 4.481\,+\,0.3804\times 4.470\,+\,0.07 75\times 4.608\,+\,0.0841\times 4.270\,=\,4.434.\]  
\[\beta_{S-case5}=0.1580\times 4.251\,+\,0.2098\times 4.565\,+\,0.2643\times 4.567\,+\,0.2 098\times 4.565\,+\,0.1580\times 4.251\,=\,4.466.\]  
\[\beta_{S-case6}=0.1025\times 4.254\,+\,0.0947\times 4.594\,+\,0.6056\times 4.317 \,+\,0.0947\times 4.594\,+\,0.1025\times 4.254\,=\,4.356.\]  
Table 8.0 \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Matrix** & \(\lambda_{max}\) & **CI** & **CR** & **\(\mathbf{W}\)** \\ \hline \(A_{int\,act}\) & \(5.0004\) & \(1.0498\times 10^{-4}\) & \(9.3734\times 10^{-5}\) & (0.2098; 0.1938; 0.1926; 0.1938; 0.2098) \\ \(A_{case1}\) & \(4.9999\) & \(-2.7785\times 10^{-5}\) & \(-2.4808\times 10^{-5}\) & (0.2416; 0.0824; 0.0589; 0.2963; 0.3208) \\ \(A_{case2}\) & \(4.9994\) & \(-1.5615\times 10^{-4}\) & \(-1.3942\times 10^{-4}\) & (0.0793; 0.0730; 0.3277; 0.2498; 0.2703) \\ \(A_{case3}\) & \(4.9981\) & \(-4.6301\times 10^{-4}\) & \(-4.1340\times 10^{-4}\) & (0.1207; 0.0328; 0.3272; 0.2494; 0.2699) \\ \(A_{case4}\) & \(5.0005\) & \(1.1798\times 10^{-4}\) & \(1.0534\times 10^{-4}\) & (0.1426; 0.3153; 0.3804; 0.0775; 0.0841) \\ \(A_{case5}\) & \(5.0005\) & \(1.2918\times 10^{-4}\) & \(1.1534\times 10^{-4}\) & (0.1580; 0.2098; 0.2643; 0.2098; 0.1580) \\ \(A_{case6}\) & \(4.9987\) & \(-3.2423\times 10^{-4}\) & \(-2.8950\times 10^{-4}\) & (0.1025; 0.0947; 0.6056; 0.0947; 0.1025) \\ \(A_{case7}\) & \(5.0000\) & \(3.4477\times 10^{-6}\) & \(3.0783\times 10^{-6}\) & (0.1032; 0.0915; 0.5938; 0.1015; 0.1100) \\ \(A_{case8}\) & \(4.9999\) & \(-3.3564\times 10^{-5}\) & \(-2.9968\times 10^{-5}\) & (0.1136; 0.0606; 0.3721; 0.3084; 0.1454) \\ \(A_{case9}\) & \(4.9994\) & \(-1.3097\times 10^{-4}\) & \(-1.4669\times 10^{-4}\) & (0.1055; 0.1020; 0.5851; 0.1020; 0.1055) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The largest eigenvalue, the weight vector, the consistency index and the consistency ratio of each matrix. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Slab Segment Number**} & \multicolumn{8}{c}{**Case Number**} \\ \cline{2-10}  & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** & **Case 7** & **Case 8** & **Case 9** \\ \hline
1 & 4.253 & 4.274 & 4.253 & 4.238 & 4.251 & 4.254 & 4.253 & 4.253 & 4.250 \\
2 & 4.649 & 4.612 & 4.633 & 4.481 & 4.565 & 4.594 & 4.594 & 4.616 & 4.587 \\
3 & 4.673 & 4.505 & 4.505 & 4.470 & 4.567 & 4.317 & 4.317 & 4.470 & 4.317 \\
4 & 4.574 & 4.521 & 4.521 & 4.608 & 4.565 & 4.594 & 4.588 & 4.481 & 4.587 \\
5 & 4.232 & 4.176 & 4.176 & 4.270 & 4.251 & 4.254 & 4.248 & 4.235 & 4.250 \\ \hline \hline \end{tabular}
\end{table}