# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 5 Example Analysis  
### Basic Information  
A two-layer slope from the literature [33] was studied, and the cross-sectional dimensions of the slope are shown in Figure 5. The weight \(\gamma\) of both the upper and lower soil layers is 20 \(\mathrm{kN}/\mathrm{m}^{3}\), the modulus of elasticity \(E\) is 100 MPa, and Poisson's ratio \(\mu\) is 0.3. No water table or external water was considered. So the total stress method is appropriate. The subsequent study uses the Bishop method in the limit equilibrium method as the basis for calculating the slope safety factor. The statistical characteristics of the shear strength parameters of each soil layer are detailed in Table 1.  
In practical engineering, the distribution types of the fuzzy affiliation function of geotechnical parameters are determined using various methods (e.g., fuzzy distribution fitting method) according to their fuzzy distribution characteristics. In this paper, in order to analyze the influence of different fuzzy number types on the calculation results, four types of fuzzy numbers (triangular fuzzy numbers, trapezoidal fuzzy numbers, normal fuzzy numbers and lognormal fuzzy numbers) for the soil property parameters are considered. Since the shape parameters (\(k\), \(k_{1}\), \(k_{2}\)) of the fuzzy affiliation function affect the upper and lower bounds of the parameter interval, they are also considered. When the shape parameters with different distribution forms of geotechnical parameter affiliation functions take smaller values, the reliability index calculated by the fuzzy point estimation method is large and the failure probability is small, which will overestimate the slope stability reliability [9]. Similarly, the shape parameter \(k\) is set as 2.0, 2.5 and 3.0 for triangular fuzzy numbers, normal fuzzy numbers, and lognormal fuzzy numbers, and \(k_{1}=1.5\) with \(k_{2}=2.0\), \(k_{1}=1.0\) with \(k_{2}=2.5\) and \(k_{1}=0.5\) with \(k_{2}=3.0\) are taken for trapezoidal fuzzy numbers. There are various options for the cut levels [32; 34]. To balance the efficiency and the accuracy of the calculation results, the cut levels of a \(\lambda\)-cut set were set as 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, and 0.95.  
Using the method in Section 2.2, the parameter value intervals corresponding to different \(\lambda\)-cut sets can be easily obtained. Due to the limited space, only the value intervals of soil parameters obtained from triangular fuzzy numbers (i.e., treating the soil parameters as triangular fuzzy numbers) are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Soil Layer Number**} & \multicolumn{2}{c}{**Cohesion, kPa**} & \multicolumn{2}{c}{**Internal Friction Angle, \({}^{\circ}\)**} \\ \cline{2-5}  & **Mean Value** & **Standard Deviation** & **Mean Value** & **Standard Deviation** \\ \hline
1 & 38.31 & 7.66 & 0 & 0 \\
2 & 23.94 & 4.79 & 12 & 1.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Statistical characteristics of shear strength parameters.  
Figure 5: The slope calculation cross-section.