## 4 The Calculation of Reliability Based on Fuzzy Mathematics
### Fuzzy Seismic Spectra
#### 4.3.3 Fuzzy Comprehensive Evaluation of Site Classification
\] (31b) \[\mu_{3}(d)=0,d\geq 0\] (31c) \[\mu_{4}(d)=0,d\geq 0. \tag{31d}\]  
Figure 2: Membership function of mean shear wave velocity. When \(250\geq v_{s}>150\),  
\[\mu_{1}(d)=e^{-(\frac{d}{2\delta})^{2}},d>0 \tag{32a}\] \[\mu_{2}(d)= \begin{cases}e^{-(\frac{d-15}{2\delta})^{2}}&0<d<15\\ 1&15\leq d\leq 40\\ e^{-(\frac{d-9\delta}{12})^{2}}&d>40\end{cases} \] (32b) \[\mu_{3}(d)= \begin{cases}e^{-(\frac{d-80}{36})^{2}}&0<d\leq 80\\ 1&d\geq 80\end{cases} \] (32c) \[\mu_{4}(d)=0,d>0. \tag{32d}\]  
When \(v_{s}\leq 150\),  
\[\mu_{1}(d)=e^{-(\frac{d}{36})^{2}},d>0 \tag{33a}\] \[\mu_{2}(d)=e^{-(\frac{d-9}{72})^{2}},d>0\] (33b) \[\mu_{3}(d)= \begin{cases}e^{-(\frac{d-20}{6})^{2}}&0<d<20\\ 1&20\leq d\leq 50\\ e^{-(\frac{d-50}{36})^{2}}&d>50\end{cases} \] (33c) \[\mu_{4}(d)= \begin{cases}e^{-(\frac{d-100}{24})^{2}}&0<d<100\\ 1&d\geq 100\end{cases} . \tag{33d}\]  
(3) Comprehensive evaluation of site characteristic period \(Tg\)  
Figure 3: Membership function of thickness of covering layer. (**a**) \(800\geq v_{s}>500\); (**b**) \(500\geq v_{s}>250\); (**c**) \(250\geq v_{s}>150\); and (**d**) \(v_{s}\leq 150\). As per the site characteristic period value given in Table 5.1.4.2 of the Chinese seismic design specifications [21], the calculation method for site characteristic period \(T_{S}\) is shown in Equation (34):  
\[T_{S}=\sum_{i=1}^{4}w_{i}T_{S}/\sum_{i=1}^{4}w_{i}. \tag{34}\]  
Since Class I\({}_{0}\) is not necessary when carrying out the fuzzy comprehensive evaluation, the corresponding characteristic site period value of Class I\({}_{0}\) can be obtained directly from Table 5.1.4.2 of the Chinese seismic design specification [21]. (4) The fuzzy seismic response spectrum  
The steps for determining the fuzzy seismic response spectrum are as follows:  
(1) Obtain the maximum value of horizontal earthquake influence coefficient \(\alpha_{\max}\) from Equation (26). (2) Identify the depth of site overburden layer \(d\) based on the site drilling geological data, and calculate the equivalent shear wave velocity \(v_{s}\) according to formula \(v_{s}=d_{0}/\sum\limits_{i=1}^{n}\left(d_{i}/v_{si}\right)\). (3) Calculate single factor fuzzy relation vector \(A=(a_{1},a_{2},a_{3},a_{4})\) in accordance with the membership function of equivalent shear wave velocity in Equation (28). During first-step evaluation, there is just one evaluation factor, which is equivalent shear wave velocity, so the fuzzy relation mentioned above is also the membership degree of the mean shear wave velocity to every site classification evaluation grade, that is \(V=A\), \(v_{i}=a_{i}\). (4) According to the membership functions Equations (30)-(33) of overburden thickness, calculate the evaluation factors of the fuzzy relation matrix for \(B\). The calculation formula of the fuzzy vector of evaluation grade for site classification is \(W=VB=V(,+)B\). (5) Calculate the comprehensive evaluation value of site characteristic period \(T_{S}\) according to Equation (34). (6) After obtaining parameters \(\alpha_{\max}\) and \(T_{S}\), which are given full consideration of fuzziness, use the parameters of the earthquake response spectrum curve from the Chinese seismic design specifications [21] before the fuzzy earthquake response spectrum can be obtained.