# 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
2.2 Influence of Correlation of Shear Strength Parameters on the Non-Probabilistic Reliability of the Slope  
In order to study the influence of the correlation coefficients of the cohesion \(c\) and the internal friction angle \(\varphi\) on the non-probabilistic reliability indexes of the slope, the sample mean and standard deviation of the original data presented in Table 1 were used as the benchmark to generate 25 sets of random data with correlation coefficients of \(-0.7\), \(-0.5\), \(-0.3\), \(-0.1\), \(0\), \(0.1\), \(0.3\), \(0.5\), and \(0.7\), respectively. Then, the methodology proposed in this paper was used to calculate the non-probabilistic reliability indexes of the slope. The results are shown in Figure 6. In this figure, ME indicates that an ellipsoid model was used, and MI indicates that a traditional interval model (the special case where the correlation coefficient is 0 in the MP-1 model and the MP-2 model) was used. As shown in Figure 6, when there is a negative correlation between the cohesion \(c\) and the internal friction angle \(\varphi\), the non-probabilistic reliability indexes of the slope obtained based on the ellipsoid model and those based on the multidimensional parallelepiped model both decrease with an increase in the correlation coefficient. The magnitude of the change of the calculation results obtained based on the ellipsoid model is larger, and the magnitude of the change of the calculation results obtained based on the multidimensional parallelepiped model is relatively uniform. With correlation coefficients of \(-0.7\) and \(-0.5\), the failure probabilities of the slope calculated using the Monte Carlo method are,  
Figure 6: The curve of correlation coefficients of geotechnical parameters and non-probabilistic reliability indexes. respectively, 0 and 0.003% (close to 0). The non-probabilistic reliability indexes of the slope obtained based on the ellipsoid model and those based on the MP model are greater than 1, which is reasonable. However, the non-probabilistic reliability indexes of the slope obtained based on the interval model are less than 1, which underestimates the stability of the slope. With correlation coefficients of \(-0.3\) and \(-0.1\), the failure probabilities of the slope calculated using the Monte Carlo method are 0.061% and 0.203%, respectively. The non-probabilistic reliability indexes of the slope obtained using the MP model and the interval model are less than 1, which is also reasonable. But the non-probabilistic reliability indexes of the slope obtained using the ellipsoid model are greater than 1, which overestimates the stability of the slope. When there is a positive correlation between the cohesion \(c\) and the internal friction angle \(\varphi\), the non-probabilistic reliability indexes of the slope obtained based on the ellipsoid model decreases slightly with the increase in the correlation coefficient. The non-probabilistic reliability indexes of the slope obtained based on the multidimensional parallelepiped model remain unchanged and are the same as the calculation results obtained based on the interval model. It is consistent with the situation presented in the literature [21]. There are two reasons to explain this result. On the one hand, in actual slope engineering, the cohesion and the internal friction angle of soil are negatively correlated in most cases. And the influence of the negative correlation on slope stability is mainly considered. On the other hand, as shown in Figure 6, when the correlation coefficient is greater than 0, the difference between the non-probabilistic reliability indexes of the slope based on the ellipsoid model and those based on the MP model or the interval model is small. The values are all less than 1, which indicates that there is the possibility of instability of the slope. At this time, a change in the value of the non-probabilistic reliability indexes of the slope has no effect on the judgment of the stability of the slope.