## 1 Introduction
### Fuzzy Detection Information Expression of Hinge Joints Damage Degree  
The results of a large number of hinged slab beam bridges show that hinge joints damage is very common; however, it is difficult for engineers and technicians to quantify the damage degree in practical engineering deterministically due to the complex stress state of hinge joints. Maybe it is easier to give a fuzzy interval of the hinge joints damage degree. Therefore, the membership function of fuzzy theory is suggested to express the damage degree interval in this paper, in order to describe the recognition of the damage degree of hinge joints. At present, there are many types of functions that can be used as membership functions, among which the most commonly used types are normal, triangle, trapezoid, rectangle, and so on. The confidence degree of the middle values in the confidence interval of normal and triangle membership functions is the highest, while that of the upper and lower limits of damage is the lowest, and the confidence degree of the values in the damage interval is the same for rectangle type. The credibility of engineers in the damage interval is often inconsistent according to the law of credibility in practical engineering, and sometimes similar for each value in the interval. Therefore, triangle, normal, and rectangle-type membership functions are selected to express the damage interval [44]. \([D_{L},D_{U}]\) is used to represent the damage degree interval given by the detection personnel, where \(D_{L}\) represents the lower limit of the damage degree and \(D_{U}\) denotes upper limit of damage. Therefore, triangle, normal-type, and rectangular membership functions are used to express the damage degree interval \([D_{L},D_{U}]\) according to the concept of membership degree in fuzzy mathematics, which are as follows:  
Figure 1: Flow chart of the reliability evaluation method proposed in this paper. Triangle membership function:  
\[u(\overline{x})=\left\{ \begin{array}{ll}\frac{\overline{x}-D_{L}}{(D_{L}+D_{U})/ 2-D_{L}},&D_{L}<\overline{x}\leq(D_{L}+D_{U})/2\\ \frac{D_{U}-\overline{x}}{D_{U}-(D_{L}+D_{U})/2},&(D_{L}+D_{U})/2<\overline{x} \leq D_{U}\\ 0,&Others\end{array} \right., \tag{1}\]  
where \(\overline{x}\) is a fuzzy variable. Normal-type membership function:  
\[u(\overline{x})=e^{-(\frac{\mu-\overline{x}}{\alpha})^{2}},\quad-\infty< \overline{x}<+\infty, \tag{2}\]  
where \(\mu=(D_{L}+D_{U})/2,\sigma=(D_{U}-\mu)/k=(\mu-D_{L})/k\), and \(k\) represents the credibility of the detection personnel to the damage degree in the interval, and the greater the value, the greater the reliability. Rectangular membership function:  
\[u(\overline{x})=\left\{ \begin{array}{ll}1,&D_{L}\leq\overline{x}\leq D_{U} \\ 0,&Others\end{array} \right., \tag{3}\]  
The above equation indicates that the credibility of detection personnel on damage degree in the interval is the same when the rectangular membership function is used. However, the variables used in the reliability evaluation are all random variables, so it is difficult to directly solve the fuzzy reliability of the structure by using the membership function. Therefore, the method of equivalent randomization of the membership function is innovatively proposed in this paper to convert the fuzzy reliability calculation into an ordinary reliability calculation, and the fuzzy variables in the membership function are further transformed into random variables by obtaining the expression form of the probability density function, whose expression is as follows:  
\[f(x)=\frac{u(\overline{x})}{\int_{u}^{b}u(\overline{x})d\overline{x}},a<x\leq b, \tag{4}\]  
where \(x\) is the random variable obtained from the transformation of fuzzy variable \(\overline{x}\), and \(f(x)\) is the probability density function of \(x\), which satisfies the following conditions based on Equation (4),  
\[\left\{ \begin{array}{ll}f(x)\geq 0\\ \int_{u}^{b}f(x)=1\end{array}