## 4 The Calculation of Reliability Based on Fuzzy Mathematics
### Fuzzy Seismic Spectra
#### 4.3.3 Fuzzy Comprehensive Evaluation of Site Classification  
According to item 4.1.6 in the Chinese seismic design specifications [21], mean shear wave velocity and overburden thickness are taken as the main evaluation factors of site classification, and the normal function shown in Equation (27) was assigned as the membership function of different evaluation factors:  
\[\upmu(x)=\exp\left[-\left(\frac{x-m_{j}}{b_{j}}\right)^{2}\right], \tag{27}\]  
where \(m_{j}\) and \(b_{j}\) are used to reflect that the membership function of two adjacent site classes has the same value at its demarcation point and \(m_{j}=0.5(x_{j}+x_{j+1})\), \(b_{j}=0.6(x_{j+1}-x_{j})\), \(x_{j}\), and \(x_{j+1}\) are threshold values of adjacent site classes. The evaluation factor set of mean shear wave velocity \(U_{1}\) and overburden thickness \(U_{2}\) corresponded to the evaluation grade domains \(V\) and \(W\) of the site, respectively. Due to the difference in the evaluation grade domain, this paper adopted the fuzzy operator \(M(\bullet,+)\) and the two-step evaluation method to carry on multi-factor comprehensive evaluation of site classification. (1) First-step Evaluation  
As discussed above, \(U_{1}=\{u_{1}\}\) is a single factor set, where \(\upmu_{1}\) is the evaluation factor for the equivalent shear wave velocity; \(V=\{v_{1},v_{2},v_{3},v_{4}\}\) is a fuzzy vector of site classification evaluation grade domain where \(\
u_{1}\), \(\
u_{2}\), \(\
u_{3}\), \(\
u_{4}\) are the membership of a site in Class I, II, III, or IV, respectively. \(A=\{a_{1},a_{2},a_{3},a_{4}\}\) is a corresponding single factor fuzzy relation vector, where \(a_{i}\) is the membership of the mean shear wave velocity to site classification. During first-step evaluation, there is only one evaluation factor, i.e., equivalent shear wave velocity, so the fuzzy relationship mentioned above is the membership degree of the equivalent shear wave velocity to the site classification. Thus, we can calculate the membership function of equivalent shear wave velocity from the Chinese seismic design specifications [21] and Equation (27). The mathematical and graphic expression of this function are shown in Equation (28) and Figure 2, respectively. \[A_{1}(v_{s})= \begin{cases}1&0<v_{s}\leq 110\\ e^{-(\frac{v_{s}-110}{48})^{2}}&v_{s}>110\end{cases}  \tag{28a}\] \[A_{2}(v_{s})=e^{-(\frac{v_{s}-200}{60})^{2}},v_{s}>0 \tag{28b}\]\[A_{3}(v_{s}) =e^{-(\frac{v_{3}-375}{150})^{2}},v_{s}>0 \tag{28c}\] \[A_{4}(v_{s}) =e^{-(\frac{v_{3}-650}{180})^{2}},v_{s}>0 \tag{28d}\]  
2. Second-step Evaluation \(W=\{w_{1},w_{2},w_{3},w_{4}\}\) is the evaluation grade fuzzy vector of site classification where \(w_{1}\), \(w_{2}\), \(w_{3}\), \(w_{4}\) are the membership of a factor to four site classes. The fuzzy relation matrix of overburden thickness is shown in Equation (29): \[B=\left[ \begin{array}{cccc}b_{11}&b_{12}&b_{13}&b_{14}\\ b_{21}&b_{22}&b_{23}&b_{24}\\ b_{31}&b_{32}&b_{33}&b_{34}\\ b_{41}&b_{42}&b_{43}&b_{44}\\ \end{array} \right],\] (29) where \(b_{rs}\) (\(r=1,\,2,\,\dots\,4\); \(s=1,\,2,\,\dots\,4\)) is the membership of the overburden thickness of the \(r\)th site category to the \(s\)th site category. The evaluation grade fuzzy quantity of site classification \(W\) can be obtained based on a comprehensive evaluation. \(W=VB=V(,+)B\), where, \(V\) is the measurement matrix of the evaluation factors. Based on the Chinese seismic design specifications [21] and Equation (27), the mathematical and graphic expression of the membership functions of overburden thickness are shown in Equations (30)-(33) and Figure 3a-d, respectively. When \(800\geq v_{s}>500\),  
\[\mu_{1}(d)= \begin{cases}1&d=0\\ 0&d>0\end{cases}  \tag{30a}\] \[\mu_{2}(d)=0,d\geq 0\] (30b) \[\mu_{3}(d)=0,d\geq 0\] (30c) \[\mu_{4}(d)=0,d\geq 0. \tag{30d}\]  
When \(500\geq v_{s}>250\),  
\[\mu_{1}(d)= \begin{cases}1&0<d\leq 3\\ e^{-(\frac{d-3}{24})^{2}}&d>3\end{cases}  \tag{31a}\] \[\mu_{2}(d)= \begin{cases}e^{-(\frac{d-7}{24})^{2}}&0<d<7\\ 1&d\geq 7\end{cases}