# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 3 Comprehensive Safety Factor
### Comparison of Results  
In the traditional strength reduction method, the same reduction coefficient is applied synchronously, and the reduction coefficient of the critical slope is taken as the safety factor. The safety factor \(F_{1}\) can be obtained iteratively through numerical simulation software, as shown in Figure 12.  
When the dual-coefficient reduction method is applied, two safety factors are obtained for cohesion and internal friction angle, respectively. Some studies propose using the average of the two reduction coefficients as the comprehensive safety factor \(F_{2}\), which is shown as follows:  
\[F_{2}=\frac{k_{c}+k_{\varphi}}{2} \tag{29}\]  
Alternatively, the arithmetic square root of the product of the two reduction coefficients can be used as the safety factor \(F_{3}\), which is shown as follows:  
\[F_{3}=\sqrt{k_{c}\cdot k_{\varphi}} \tag{30}\]  
Other studies have proposed integrating statistical principles by analyzing large datasets of slopes under diverse conditions to derive the comprehensive safety factor \(F_{4}\) through the following mathematical fitting:  
\[F_{4}=\frac{\sqrt{2}k_{c}k_{\varphi}}{\sqrt{k_{c}^{2}+k_{\varphi}^{2}}} \tag{31}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Group No.** & \(\mathbf{c}\)**/kPa** & \(\mathbf{\varphi}\)**/(\({}^{\circ}\))** & \(\mathbf{\sigma_{1}}\)**/kPa** & \(\mathbf{\sigma_{3}}\)**/kPa** & \(\mathbf{\alpha}\)**/(\({}^{\circ}\))** & \(\mathbf{\sigma_{n}}\)**/kPa** & \(\mathbf{\tau}\)**/kPa** \\ \hline Critical Slope No. 4 & 74.45 & 22.98 & 377.54 & 72.24 & 56.49 & 164.55 & 144.45 \\ Critical Slope No. 5 & 76.69 & 22.51 & 383.87 & 74.34 & 56.25 & 169.11 & 146.98 \\ Critical Slope No. 6 & 73.91 & 23.10 & 376.26 & 71.76 & 56.55 & 163.53 & 143.88 \\ Critical Slope No. 7 & 74.02 & 23.07 & 376.49 & 71.91 & 56.54 & 163.78 & 143.99 \\ Critical Slope No. 8 & 74.56 & 22.95 & 378.36 & 72.51 & 56.48 & 165.06 & 144.67 \\ Critical Slope No. 9 & 74.67 & 22.92 & 378.73 & 72.71 & 56.46 & 165.39 & 144.81 \\ Critical Slope No. 10 & 74.13 & 23.04 & 377.82 & 72.47 & 56.52 & 164.65 & 144.37 \\ Critical Slope No. 11 & 74.78 & 22.89 & 380.02 & 73.26 & 56.45 & 166.23 & 145.18 \\ \hline \hline \end{tabular}
\end{table}
Table 7: _Cont._  
Figure 12: Contour map of plastic zone distribution in the critical slope.  
Other studies define the comprehensive safety factor \(F_{5}\) based on the shortest path of strength reduction, which is shown as follows:  
\[\begin{split} F_{5}&=\frac{1}{1-R/\sqrt{2}}\\ R&=\sqrt{\left(1-\frac{1}{k_{c}}\right)^{2}+\left(1- \frac{1}{k_{\varphi}}\right)^{2}}\end{split} \tag{32}\]  
The safety factors calculated using the methods described above are compiled in Table 8.  
The results in Table 8 indicate that the safety factor derived from the proposed method is 2.5% lower than that obtained using traditional methods. This method achieves higher precision in classifying slopes into stable and unstable states. By employing a Bayesian Gaussian Mixture Model to process dissipated energy data it demonstrates significant advantages in complex geological conditions, particularly for slopes near critical states where it provides more accurate stability assessments.  
This method pioneers a new paradigm in slope stability analysis. On the one hand, scholars have conducted extensive research on the reduction in cohesion and internal friction angle under various slope conditions, while the random reduction method proposed in this study can be practically applied by integrating specific reduction paths derived from these investigations. Furthermore, it establishes a novel direction for slope stability research. With continuous refinement and optimization of algorithms, this method will further enhance its ability to distinguish between stable and unstable slopes with greater precision.