# Fuzzy Synthetic Evaluation of the Long-Term Health of Tunnel Structures
## 2 Determination of the Evaluation Methods and Ideas for the Tunnel Structural Health
### Application of Fuzzy Synthetic Evaluation of Long-Term Structural Health in Cang Ling Tunnel
#### 3.4.3 Determination of the Membership Function  
Another key problem in fuzzy synthetic evaluation is the determination of the membership function. The key to determining the fuzzy relation is to determine the membership degree of each factor, which means determining the quantitative relationship between the evaluation factors and the evaluation levels. The function to measure the membership degree between the evaluation factors and the evaluation levels is called a membership function. Considering the strong subjectivity of experts' opinion, the membership function is determined by mathematical functions. The influencing factors in the evaluation model include qualitative and quantitative indices. As for quantitative indices, they can be obtained directly from the membership function when determining the subordinate degree. As for qualitative indices, they should be properly quantified first. Then, the subordinate degree can be obtained from the membership function. The stress characteristics of a tunnel structure, through the analysis of the existing data, almost follow the normal distribution [36]. Therefore, it is expressed in the normal form when constructing the membership function. As for the engineering geological characteristics and the seismic intensity, the normal distribution is also commonly used when determining the membership function of each index because of their large discreteness [34; 35]. Considering that the types of the index functions are almost the same, the difference is only reflected in the parameters of membership function. This paper takes the structure type of the grade IV standard section whose lining thickness is 45 cm as an example. The membership functions of the contact pressure are:  
\[\mu_{\text{I}}=\left\{ \begin{array}{ll}1&x<270\\ e^{-(\frac{x-70}{292.5})^{2}}&x\geq 270\end{array} \right. \mu_{\text{II}}=\left\{ \begin{array}{ll}e^{-(\frac{x-776}{292.5} )^{2}}&x<776\\ e^{-(\frac{x-776}{292.5})^{2}}&x\geq 776\end{array} \right. \mu_{\text{III}}=\left\{ \begin{array}{ll}e^{-(\frac{x-1137.5}{209})^{2}}&x< 1137.5\\ e^{-(\frac{x-1137.5}{114.4})^{2}}&x\geq 1137.5\end{array} \right.\] \[\mu_{\text{IV}}=\left\{ \begin{array}{ll}e^{-(\frac{x-1335.4}{114.4 })^{2}}&x<1335.4\\ 1&x\geq 1335.4\end{array} \right..\]  
As for the determination of three quantitative index membership functions--the ground stress influencing factor (SRF), the span of the tunnel, and the service time of the tunnel--trapezoidal membership functions are constructed. The membership functions of the tunnel span are listed as follows: \begin{table}
\begin{tabular}{c c c c} \hline
**Control Layer** & **Weight of the Control Layer** & **Index Layer** & **Weight of the Index Layer** \\ \hline Stress characteristics of the main structure & 0.524 & \begin{tabular}{c} Contact pressure \\ between layers \\ Conventional safety evaluation \\ \end{tabular} & 0.5 \\ \hline Engineering geological characteristics & 0.244 & \begin{tabular}{c} Rock quality designation RQD \\ Rock compressive strength \\ Rock integrity coefficient \\ Situation of the \\ underground water \\ fracture zone \\ \end{tabular} & 0.206 \\ \hline Stress field characteristics & 0.107 & \begin{tabular}{c} Angle between the axis of the \\ tunnel and the direction of the \\ maximum principal stress \\ Initial geostress influencing factor, SRF \\ \end{tabular} & 0.333 \\ \hline Cavity characteristics & 0.063 & \
\begin{tabular}{c} Span of the tunnel \\ Influence of the shape of the \\ tunnel (height-span ratio) \\ \end{tabular} & 0.5 \\ \hline Other influencing factors & 0.063 &
\begin{tabular}{c} Service time of the tunnel \\ Design and construction level \\ Seismic intensity \\ \end{tabular} & 0.320 \\ \hline \end{tabular}
\end{table} 
Table 5: Weight of each factor and its relevant indices in the evaluation model. \[\mu_{\text{I}}= \begin{cases}1&x<5\\ 3-0.4x&5\leq x\leq 7.5\\ 0&x\geq 7.5\end{cases} \ \mu_{\text{II}}= \begin{cases}0&x<5\\ 0.4x-2&5\leq x\leq 7.5\\ 1&7.5\leq x\leq 10\\ 5-0.4x&10\leq x\leq 12.5\\ 0&x\geq 12.5\end{cases} \ \mu_{\text{III}}=