# Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory
## 5 Example Analysis
### Calculation Results and Analysis
#### 5.2.3 The Relationship between Non-Probabilistic Reliability Indexes and Failure Degrees  
In traditional non-probabilistic analysis, whether the non-probabilistic reliability index is greater than 1 is usually used as the criterion to judge whether the structure is safe. According to this method, if the non-probabilistic reliability index of the slope is greater than 1, the possibility of the slope instability is 0 and the slope is in a safe state, which is not controversial; if the non-probabilistic reliability index of the slope is less than 1, the slope is judged to be unstable, which may be too conservative under certain circumstances (such as when the actual possibility of instability is very small) and cause design waste. All the data for the non-probabilistic reliability indexes and failure degrees of the slope obtained in the calculation process of this paper were compiled, and the distribution range of non-probabilistic reliability indexes and their corresponding failure degrees were \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\
\begin{tabular}{c} **Allowable** \\ **Safety** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Shape** \\ **Parameter** \\ \end{tabular} } & \multicolumn{4}{c}{**Triangular**} & \multicolumn{4}{c}{**Trapezoidal**} & \multicolumn{4}{c}{**Normal**} & \multicolumn{2}{c}{**Log-Normal**} \\ \cline{3-10}  & & & **Failure** & **Failure** & **Failure** & **Failure** & **Failure** & **Failure** & **Failure** \\  & & & **Degree** & **Probability** & **Degree** & **Probability** & **Degree** & **Probability** & **Degree** & **Probability** \\ \hline \multirow{3}{*}{1.0} & 2.0 & 0.00238 & 0.00034 & 0.03576 & 0.07330 & 0.00167 & 0.00088 & 0.00061 & 0.00068 \\  & 2.5 & 0.01071 & 0.00451 & 0.029290 & 0.05410 & 0.00425 & 0.00225 & 0.00191 & 0.00166 \\  & 3.0 & 0.02385 & 0.01854 & 0.03437 & 0.04476 & 0.00561 & 0.00311 & 0.00265 & 0.00224 \\ \hline \multirow{3}{*}{1.1} & 2.0 & 0.00661 & 0.00253 & 0.07782 & 0.12211 & 0.00541 & 0.00496 & 0.00280 & 0.00420 \\  & 2.5 & 0.02003 & 0.01588 & 0.05582 & 0.09732 & 0.01005 & 0.00967 & 0.00557 & 0.00781 \\  & 3.0 & 0.03765 & 0.04411 & 0.05442 & 0.08431 & 0.01216 & 0.01218 & 0.00686 & 0.00967 \\ \hline \multirow{3}{*}{1.2} & 2.0 & 0.01654 & 0.01345 & 0.14011 & 0.19013 & 0.01570 & 0.02116 & 0.01038 & 0.01925 \\  & 2.5 & 0.03701 & 0.04616 & 0.10304 & 0.16166 & 0.02346 & 0.03311 & 0.01566 & 0.02885 \\ \cline{1-1}  & 3.0 & 0.06013 & 0.09260 & 0.08662 & 0.14561 & 0.02665 & 0.03866 & 0.01780 & 0.03319 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The overall failure degree and failure probability of the slope using different allowable safety factors. counted when the non-probabilistic reliability index \(\eta\) was less than 1. The results are shown in Table 4.  
As can be seen from Table 4, for this two-layer slope, in some cases where \(\eta<1\) (e.g., the failure degree of the slope is less than 0.01 for \(\eta>0.9\)), the failure degree of the slope was not large and the corresponding risk level was still acceptable. Referring to the design criteria for failure probability of the slope in reliability analysis [36] (since there are no design criteria for the non-probabilistic failure degree of the slope at present, in view of the similarity between failure degree in non-probabilistic reliability analysis and failure probability calculated from a Monte Carlo simulation in the reliability analysis, the non-probabilistic failure degree herein is used as an analogy to the failure probability), 0.9 can be taken as the evaluation criterion for the non-probabilistic reliability index of the slope. According to the traditional non-probabilistic reliability analysis method, the slope is judged to be unstable when \(0.9<\eta<1\); however, in this paper, this slope was considered to be stable with a small failure degree.