# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 4 Non-Probabilistic Reliability Analysis Method for Slopes Based on Fuzzy Set Theory  
(1) Geotechnical parameters are described as certain specific types of fuzzy numbers according to their fuzzy uncertainty characteristics. By taking \(m\) different levels of a \(\lambda\)-cut set, the fuzzy numbers are transformed into the corresponding interval numbers for each \(\lambda\)-cut set, and the value range of each interval number is obtained. (2) On the cut set of \(\lambda_{i}\) (\(i=1,\ 2,\ \ldots,\ m\)), the corresponding ellipsoidal model is constructed according to the interval ranges of geotechnical parameters, and the center \(X_{io}\), the radius \(e_{i}\) and the covariance matrix \(C_{X}\) of the ellipsoidal model are calculated using Equations (6) and (7). Then, \(2n+1\) groups of sample points are selected according to the interval range of the geotechnical parameters by the Bucher design [29], and the slope safety factor corresponding to each group of sample points is calculated using the limit equilibrium method. Using Kriging-based or PCE-based surrogates [30,31] to approximate the slope performance function can improve prediction accuracy. However, too many basis functions tend to introduce more pending coefficient terms, which leads to a sharp increase in the frequency of calling the original analytical model. Given this consideration, a quadratic polynomial response surface function without cross terms is used in this paper. A quadratic polynomial response surface function without cross terms is established as the approximate slope performance function:  
\[M=G(X_{1},\ X_{2},\ \cdots,\ X_{n})=F_{\rm s}(X_{1},\ X_{2},\ \cdots,\ X_{n})-F_{ \rm s0}=a+\sum_{i=1}^{n}b_{i}X_{i}+\sum_{i=1}^{n}c_{i}X_{i}^{2} \tag{15}\]  
where \(F_{\rm s0}\) is the allowable safety factor of the slope and \(a,\ b_{i},\ c_{i}(i=1,\ 2,\ \cdots,\ n)\) are the factors of the polynomial that is to be determined. According to Equation (12), the above approximate slope performance function can be standardized by the ellipsoidal model:  
\[M=G(\mathbf{X})=G(\mathbf{u})=a+\sum_{i=1}^{n}b_{i}\Big{(}{C_{Xi}}^{0.5}u_{i}+X_{ic} \Big{)}+\sum_{i=1}^{n}c_{i}\Big{(}{C_{Xi}}^{0.5}u_{i}+X_{ic}\Big{)}^{2} \tag{16}\]  
Figure 3: Schematic diagrams of the non-probabilistic failure degree solution based on the ellipsoidal model (**a**) Two-dimensional case; (**b**) Three-dimensional case. Firstly, an initial value of the non-probabilistic reliability index \(\eta_{i,k}\) and a design point \(x_{i,k}\)* can be obtained by the HL-RF algorithm in the approximate limit state surface (\(G(\mathbf{u})=0\)). Secondly, a new run of the Bucher design is performed based on the design point \(x_{i,k}\)*, and a new response surface is constructed and standardized to obtain a new value of the non-probabilistic reliability index \(\eta_{i,k+1}\) and a new design point \(x_{i,k+1}\)*. Thirdly, if \(\eta_{i,k}\) and \(\eta_{i,k+1}\) satisfy the convergence condition  
\[\left|\frac{\eta_{i,k+1}-\eta_{i,k}}{\eta_{i,k}}\right|<10^{-4} \tag{17}\]  
take \(\eta_{i}=\eta_{i,k+1}\) as the slope non-probabilistic reliability index of this cut set. Otherwise, repeat the Bucher design (each time, a new Bucher design is conducted using the design points obtained from the previous step), response surface establishment and non-probabilistic reliability calculation process until the convergence condition is satisfied. Finally, calculate the slope failure degree \(F_{i}\) of this cut set using Equation (14).