# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 5 Example Analysis
### Calculation Results and Analysis
For comparison, the slope failure probabilities obtained by the fuzzy point estimation method (a reliability analysis approach that combines the fuzzy set theory and the statistical moment point estimation method) are also listed in Table 3.  
From Table 3, it can be found that for triangular fuzzy numbers, normal fuzzy numbers and log-normal fuzzy numbers, the overall failure degree of the slope grew with the increase in the shape parameter. For trapezoidal fuzzy numbers, the overall failure degree of the slope was not only affected by the value of \(k_{2}\), but also by the difference between \(k_{1}\) and \(k_{2}\) (which affects the shape of the trapezoidal affiliation function). When \(k\) and \(k_{2}\) had the same value, the overall failure degree of the slope from trapezoidal fuzzy numbers was the largest, while those from triangular fuzzy numbers and normal fuzzy numbers were in the middle, and that from log-normal fuzzy numbers was the smallest. Additionally, the overall failure degrees of the slope based on different types of fuzzy numbers rise with the increase in the allowable safety factor of the slope, which is consistent with engineering practice. Figure 7: Non-probabilistic reliability indexes and failure degrees of the slope with different shape parameters. (a) shape parameter \(k\) (\(k_{2}\)) = 2.0; (b) shape parameter \(k\) (\(k_{2}\)) = 2.5; (c) shape parameter \(k\) (\(k_{2}\)) = 3.0. Comparing the non-probabilistic failure degree of the slope obtained by the method proposed in this paper with the failure probability of the slope calculated by the fuzzy point estimation method, the overall failure degree and the failure probability were essentially consistent in order of magnitude, however, the influence of different types and shape parameters of fuzzy affiliation functions on the failure probability of the slope was greater. A Monte-Carlo-simulation-based reliability analysis for the slope without consideration of the fuzziness factors was also conducted, and the calculation results were 0.010350 (when the allowable safety factor is 1.0), 0.023279 (when the allowable safety factor is 1.1) and 0.050221 (when the allowable safety factor is 1.2). Comparing the above results with the data in Table 3, the non-probabilistic failure degree of the slope was closer to the Monte Carlo simulation result than the slope failure probability calculated by the fuzzy point estimation method. Part of the reason is that the non-probabilistic approach considers the whole geotechnical parameter interval while the fuzzy point estimation method uses only a finite combination of interval points. [PERSON] [35] pointed out that considering the fuzziness of geotechnical parameters will rise the failure probability of slopes. According to this view, it is suggested to use the triangular fuzzy number and take the shape parameter as 3.0 for this slope. The reason why the trapezoidal fuzzy number is not recommended is that the results obtained from this type of fuzzy number were more complex and varied greatly since there were two shape parameters (\(k_{1}\) and \(k_{2}\)) need to be determined. It should be noted that in the actual slope project, the selections of geotechnical fuzzy number type and shape parameter should be determined according to the test data. The shape parameter \(k\) determines the maximum possible calculation interval for geotechnical parameters, and it is necessary to consider the local averaging effect of parameter variability from the experimental values of points to the calculated values of units [8]. And the value of \(k\) is influenced by many factors, such as the variation and correlation of parameters and the sample size.