# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 1 Introduction  
Under the influence of long-term geological action and construction conditions, a certain degree of fuzzy uncertainty exists in slope engineering [1, 2, 3, 4], and this fuzzy uncertainty mainly comes from three aspects: the fuzziness of geotechnical parameters, the fuzziness of the slope instability damage state and the fuzziness of external actions. The fuzziness of geotechnical parameters refers to the influence of test errors, geotechnical group divisions, people's incomplete grasp of the regularity of things, and other factors, leading to a range of values for basic variables such as the physical and mechanical parameters of geotechnical properties; the fuzziness of the slope instability damage state refers to the inability to clearly demarcate between the slope in the stable state and the failure state caused by the cumulative progressive nature of the slope deformation damage process; the fuzziness of the external action refers to the fuzzy description of the external action strength (such as the impact of groundwater, seismic forces, human activities, etc.) caused by the limitation of cognition and the risk of external actions. For the fuzziness of geotechnical parameters and slope destabilization damage state, the fuzzy mathematics method can be used as a solution, and for the fuzziness of external action, it can be solved by taking the empirical value. In order to assess the stability state of the slope as accurately as possible, the influence of fuzzy uncertainty factors in slope engineering should be considered. The main research milestones include the following: (1) [PERSON] et al. [5], [PERSON] et al. [6], [PERSON] et al. [7], and [PERSON] et al. [8] described the geotechnical parameters as fuzzy numbers and transformed them into interval numbers by taking fuzzy cut sets, then obtained the fuzzy set of slope safety factors using the limit equilibrium method or calculated the reliability index of the slope using the statistical moment point estimation method. This type of method is widely used because of the clear principle, relatively simple operation process and more comprehensive calculation results. However, it also has some drawbacks, such as the calculation error from the statistical moment point estimation method, the interval expansion problems that exist in the calculation process, the inconsistent results from different fuzzy point estimation methods (e.g., [PERSON] et al. [8] and [PERSON] et al. [9]) and the large influence of different types of fuzzy affiliation functions on slope reliability. (2) [PERSON] et al. [10] transformed the fuzzy variables into random variables by equivalent transformation and calculated the fuzzy random reliability index of each finite unit in the slope profile. This type of method can directly transform fuzzy variables into random variables to facilitate the use of conventional reliability analysis methods; however, when the geotechnical parameter affiliation function is complex, the equivalence transformation process is difficult and the effectiveness of the method needs to be verified. (3) [PERSON] et al. [11], [PERSON] [12], [PERSON] et al. [13], [PERSON] et al. [14] and [PERSON] et al. [15] made fuzzy judgments on the slope stability influence factors, constructed the fuzzy matrix, and determined the risk level of the slope by fuzzy comprehensive evaluation. This type of method can obtain the classification index of slope stability grade, and it can fully consider a variety of environmental influencing factors, and there is no complicated calculation process. In the actual operation, the determination of the weight of the influencing factors is subjective and empirical, and the result can not contain as much information as reliability methods. In fact, after converting the geotechnical fuzzy numbers into interval numbers by taking fuzzy cut sets (like the first type of method), the non-probabilistic reliability method can also be used to analyze the stability of the slope.