## 4 Numerical Simulation
### Member Reliability Analysis  
Four representative damage cases of hinge joints are selected on the grounds of the number of damaged hinge joints in this paper. Case 1: Hinge joint 1 damages; Case 2: Hinge joints 1 and 2 damage; Case 3: Hinge joints 1, 2, and 3 damage; Case 4: Hinge joints 1, 2, 3, and 4 damage. The difference between the cognition level and engineering experience of the detection personnel will lead to different fuzzy detection intervals; the smaller the fuzzy detection interval, the higher the confidence degree of hinge joints damage. In order to simulate the phenomenon of different descriptions of hinge joints damage degree by detection personnel, four different fuzzy detection intervals are given with damage degree 20% as the maximum credibility value, and eight different fuzzy detection intervals are given with damage degree 40% as the maximum reliability value. The division and numbering of the intervals are shown in Table 1.  
Triangular, normal-type, and rectangular membership functions are selected for calculation in order to analyze the difference in reliability assessment results generated by fuzzy detection information of hinge joints described by different membership functions. In the normal membership function, \(k=1\) and \(k=2\) are two situations and are considered to represent two different confidence levels. Taking 1# slab as an example, the member failure probability of considering the fuzzy detection information of hinge joints under four kinds of hinge joint damage cases at different hinge joint damage intervals in Table 2 is calculated, and the failure probability of hinge joints in deterministic damage degree with \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**The Maximum Credibility is 20\%** & \multicolumn{4}{c}{**The Maximum Credibility is 40\%**} \\ \hline
**Interval** & **Damage Range** & **Interval** & **Damage Range** & **Interval** & **Damage Range** \\
**Number** & **Damage Range** & **Number** & **Number** & **Damage Range** \\ \hline
1 & [15–25\%] & 1 & [35–45\%] & 5 & [15–65\%] \\
2 & [10–30\%] & 2 & [30–50\%] & 6 & [10–70\%] \\
3 & [5–35\%] & 3 & [25–55\%] & 7 & [5–75\%] \\
4 & [0–40\%] & 4 & [20–60\%] & 8 & [0–80\%] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Division of damage degree interval. Figure 2: Bridge profile diagram. 20% and 40% is calculated by the FOSM method based on the modified hinged-jointed plate method proposed in our research paper [47] and the response surface method. The member failure probability under the condition of considering fuzzy information and deterministic damage degree is compared, whose calculation results are shown in Figures 3 and 4. As can be seen from Figures 3 and 4, when the fuzzy detection information of hinge joint damage is expressed with different membership functions, the failure probability of #1 slab in all cases presents an increasing trend with the increase in the range of the damage interval, and the smaller the damage range is, the closer the failure probability is to failure probability corresponding to the degree of deterministic damage of hinge joints, which is in accordance with the objective law, so the accuracy of the proposed reliability analysis method considering the fuzzy detection of hinge joints damage is verified. In addition, the change rate of failure probability varies greatly with the increase in interval range when different membership functions are used to deal with the damage interval of hinge joints. The deviation between member failure probability and deterministic damage failure probability obtained by using different membership functions is the rectangle membership function, normal membership function (\(k=1\)), triangle membership function, and normal membership function (\(k=2\)), from large to small, which is mainly due to the different credibilities of different membership functions to the damage degree in the damage interval.