# Non-Probabilistic Reliability Analysis of Slopes Based on a Multidimensional Parallelepiped Model
## 4 Example Analysis
### Calculation Results and Analysis  
#### 4.2.1 Feasibility Validation of the Non-Probabilistic Reliability Analysis Method for the Slope  
In Table 1, the sample mean values of the cohesion \(c\) and the internal friction angle \(\varphi\) are 15.03 kPa and 29.86\({}^{\circ}\), respectively. The sample standard deviations are 3.02 kPa and 3.16\({}^{\circ}\), respectively. The correlation coefficient of the cohesion and the internal friction angle is \(-\)0.70. The Monte Carlo method was used, which is a more accurate method in probabilistic reliable analysis. And \(1\times 10^{7}\) simulations were performed while considering or not considering the parameter correlation (the correlation coefficient is 0). The failure probability of the slope was calculated to be 0 or 0.0032, respectively. When using the method proposed in this paper for non-probabilistic reliability analysis, according to the 3\(\sigma\) truncation law, the marginal intervals of the cohesion \(c\) and internal friction angle \(\varphi\) are [5.97, 24.09] and [20.38, 39.34], respectively. All the data presented in Table 1 are located in the range of the marginal intervals; thus, there was no need to exclude the sample points. The multidimensional parallelepiped model was constructed based on the sample data, and the global optimal solution method was used to calculate the non-probabilistic reliability index of the slope. The results are shown in Table 2.  
As shown in Table 2, under the consideration of the correlation between the cohesion and the internal friction angle, the non-probabilistic reliability indexes of the slope obtained based on the MP-1 model and the MP-2 model are very close to each other, and the values \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample Number** & **Cohesion, kPa** & **Internal Friction Angle, \({}^{\circ}\)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: Test data of shear strength parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Correlation Coefficient** & \begin{tabular}{c} **Non-Probabilistic Reliability** \\ **Index (MP-1 Model)** \\ \end{tabular} & \
\begin{tabular}{c} **Non-Probabilistic Reliability** \\ **Index (MP-2 Model)** \\ \end{tabular} &
\begin{tabular}{c} **Failure Probability (Monte** \\ **Carlo Method)** \\ \end{tabular} \\ \hline \(-\)0.7 & 1.1972 & 1.1973 & 0 \\
0 & 0.6023 & 0.6023 & 0.0032 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of results of non-probabilistic and probabilistic methods. are all greater than 1, which indicates that the slope is in a stable state. It is in agreement with the calculation results obtained using the Monte Carlo method in the probabilistic reliability analysis. Because both the MP-1 model and the MP-2 model degenerate into a traditional interval model when sample correlation is not taken into account, the non-probabilistic reliability indexes of the slope obtained based on the MP-1 model and the MP-2 model are consistent without considering the correlation between the cohesion and the internal friction angle. At this time, the calculated non-probabilistic reliability indexes are all less than 1, indicating that there is a risk of destabilization and damage of the slope, which is also consistent with the calculation results obtained using the Monte Carlo method. It can be seen that it is feasible to use the non-probabilistic reliability method based on the multidimensional parallelepiped model to analyze the stability of the slope, and it can solve the problem that a traditional interval model cannot consider the correlation between the uncertainty parameters. In this way, the calculation results are more in line with the actual situation of slope engineering.