# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 3 Non-Probabilistic Reliability Analysis Method Based on the Ellipsoidal Model
### Construction of the Ellipsoidal Model
From Figure 2, it can be seen that when the value of the non-probabilistic reliability index \(\eta\) is more than 1, the whole standardized ellipsoidal region is in the safe region, indicating that the structure is in a safe state; when the value of the non-probabilistic reliability index \(\eta\) is less than \(-1\), the whole standardized ellipsoidal region is in the failure region, indicating that the structure is in a failure state; when the value of non-probabilistic reliability index \(\eta\) is between \(-1\) and \(1\), part of the standardized ellipsoidal region is in the safety region and part of the standardized ellipsoidal region is in the failure region, indicating that the structure has some possibility of failure. In this case, the non-probabilistic failure degree \(F\) can be used to evaluate the possibility of failure. The failure degree is defined as the ratio of the area of the ellipsoidal region in the failure region to the total area of the ellipsoid (in three-dimensional space, the failure degree is defined as the volume ratio). When \(\eta>1\), the value of the failure degree of the structure is 0; when \(\eta<-1\), the value of the failure degree of the structure is 1; when \(-1<\eta<1\), the value of the failure degree of the structure is between 0 and 1, and it can be obtained by computing the area (volume) of the ellipsoidal region in the failure region using the integration method. When the expression of the structure performance function is very complicated, it is difficult to compute the failure degree using the integration method. To this end, a simplified method proposed by [PERSON] et al. [28] can be used: firstly, the limit state function of the structure is expanded into a Taylor series at the design point in the standard space and taken to the quadratic term. The design point is the point in the standard space with the minimum distance from the origin on the limit state surface. The origin of coordinates in standard space can be chosen as the initial design point, and the subsequent design points are determined by iterative calculation. Secondly, the standard space is rotated and changed from the standard space \(U\) to the space \(V\) (as shown in Figure 3) using matrix theory to obtain the average curvature at the design point. Finally, the area of the structure  
Figure 2: Implications of the non-probabilistic reliability indexes based on the ellipsoidal model. in the failure region (volume) is solved using the integration method. According to this method, the non-probabilistic failure degree can be calculated using the following equation:  
\[F=\frac{1-sgn({v_{n}}^{*})}{2}+sgn({v_{n}}^{*})\frac{1}{2}I_{1-({v_{n}}^{*})^{2}} \left(\frac{n+1}{2},\frac{1}{2}\right)+\frac{\kappa^{*}}{n+1}\frac{1}{B\left( \frac{n+1}{2},\frac{1}{2}\right)}\left[1-{({v_{n}}^{*})}^{2}\right]^{\frac{n+1 }{2}} \tag{14}\]  
where \(n\) is the dimension of the uncertain parameters, \(\kappa^{*}\) is the average curvature at the design point, \(v_{n}\) is the corresponding value of the feature plane position, \(B(\bullet,\ \bullet)\) is the Beta function, and \(I. (\bullet,\ \bullet)\) is the Beta function with incomplete regularization.