# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 5 Example Analysis
### Calculation Results and Analysis  
2.1 Influences of Different Fuzzy Number Types and Shape Parameters on the Non-Probabilistic Reliability of the Slope  
Considering the case of the allowable safety factor \(F_{s0}=1.0\), the non-probabilistic reliability indexes and failure degrees of the slope obtained from different types of fuzzy numbers (i.e., the soil parameters are considered different types of fuzzy numbers) were calculated, and the results are shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(\lambda\)} & \multicolumn{3}{c}{\(k\) = 2.0} & \multicolumn{3}{c}{\(k\) = 2.5} & \multicolumn{3}{c}{\(k\) = 3.0} \\ \cline{2-10}  & \(c_{1}\),  
In Figure 6, with the increase in \(\lambda\)-cut set levels, the non-probabilistic reliability indexes of the slope increased while the failure degrees decreased, and the amplitude of fluctuation of non-probabilistic reliability indexes increased while the amplitude of fluctuation of failure degrees decreased. In particular, when the cut set level \(\lambda\) was more than 0.9, the changes in the non-probabilistic reliability indexes were significant. When considering the soil parameters as triangular fuzzy numbers, normal fuzzy numbers or log-normal fuzzy numbers, with the reduction of the \(k\) value, the non-probabilistic reliability indexes of the slope grew while the failure degrees decreased. Moreover, the difference between the non-probabilistic reliability indexes of the slope caused by different \(k\) values became more pronounced with the rise in the cut set level. When considering the soil parameters as trapezoidal fuzzy numbers, with the change in cut set level \(\lambda\), the larger the difference between the shape parameters \(k_{1}\) and \(k_{2}\) was, the larger the amplitudes of fluctuation of non-probabilistic reliability indexes and failure degrees were. To further compare the effects of different types of fuzzy numbers on the non-probabilistic reliability indexes and failure degrees of the slope, the calculation results in Figure 6 are organized by shape parameter in Figure 7.  
In Figure 7, the non-probabilistic reliability indexes based on different types of fuzzy numbers were relatively close when the cut set level \(\lambda\) was small, but the failure degrees were different. To be specific, with the same value of \(k\) (\(k_{2}\)), the slope failure degrees obtained from trapezoidal fuzzy numbers were larger, while those obtained from triangular fuzzy numbers and normal fuzzy numbers were medium, and those from lognormal fuzzy numbers were smaller. The slope failures based on different types of fuzzy numbers were close (except for the case of trapezoidal \(k_{2}\) = 2.0) when the cut set level \(\lambda\) was large, and they all converged or decreased to zero, while the non-probabilistic reliability indexes  
Figure 6: The non-probabilistic reliability of the slope obtained from different types of fuzzy numbers. (**a**) Geotechnical parameters as triangular fuzzy numbers; (**b**) geotechnical parameters as trapezoidal fuzzy numbers; (**c**) geotechnical parameters as normal fuzzy numbers; (**d**) geotechnical parameters as log-normal fuzzy numbers. were different. Specifically, with the same value of \(k\) (\(k_{2}\)), the non-probabilistic reliability indexes obtained from triangular fuzzy numbers were larger, while those obtained from log-normal fuzzy numbers and normal fuzzy numbers were medium (and the two results were consistent), and those from trapezoidal fuzzy numbers were smaller. 2.2 Influences of Different Allowable Safety Factors on the Non-Probabilistic Reliability of the Slope  
According to the slope failure degree corresponding to each cut set level, the overall slope failure degree can be calculated using Equation (18); the results are shown in Table 3. Since the allowable safety factor of the slope in practical engineering is usually greater than 1 for safety stock, the overall slope failure degree is also given in Table 3 when the slope allowable safety factor \(F_{s0}\) is 1.1 or 1.2.