# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 3 Comprehensive Safety Factor  
### Definition of Comprehensive Safety Factor  
The traditional strength reduction method defines the safety factor by continuously reducing the strength parameters of the slope until it reaches the critical state, using the  
Figure 1: Schematic diagram of the Bayesian Gaussian Mixture Model.  
reduction coefficient at that point as the safety factor. In the dual-parameter reduction method, since there are two reduction coefficients for cohesion and internal friction angle defining the comprehensive safety factor is the core issue of the dual-parameter reduction method [25]. If the comprehensive safety factor is simply defined as the average of the reduction coefficients for cohesion and internal friction angle from a mathematical perspective, it will lack physical meaning and will be difficult to gain recognition for guiding real-world engineering problems. In traditional safety factor definitions, the ratio of the slope's anti-sliding force to the sliding force is also used, which is shown as follows:  
\[\left\{\begin{array}{l}F_{\mathrm{s}}=\frac{f_{Anti-sliding}}{f_{sliding}} \\ \frac{f_{Anti-sliding}}{f_{sliding}}=1\end{array}\right. \tag{18}\]  
where \(F_{\mathrm{s}}\) is the slope safety factor; \(f_{Anti-sliding}\) and \(f_{sliding}\) are the anti-sliding force and sliding force of the initial slope, respectively; and \(f_{Anti-sliding}\)\({}^{\prime}\) and \(f_{sliding}\)\({}^{\prime}\) are the anti-sliding force and sliding force of the critical slope, respectively, at which point the slope safety factor is 1.  
Since the reduction in cohesion and internal friction angle does not change the sliding force, i.e., \(f_{sliding}=f_{sliding}\)\({}^{\prime}\), it follows that:  
\[F_{\mathrm{s}}=\frac{f_{Anti-sliding}}{f_{Anti-sliding}}\text{ }^{\prime}=\frac{cA+\sigma_{\mathrm{n}}A\tan\varphi}{cA/k_{c}+\sigma_{ \mathrm{n}}A\tan\varphi/k_{\varphi}} \tag{19}\]  
where \(c\) and \(\varphi\) are the initial cohesion and internal friction angle, respectively; \(\sigma_{\mathrm{n}}\) is the normal stress; \(A\) is the sliding surface area; and \(k_{c}\) and \(k_{\varphi}\) are the reduction coefficients for cohesion and internal friction angle, respectively.  
In the traditional strength reduction method, \(k_{c}=k_{\varphi}\). If it is assumed that the normal stress on the slope element remains approximately equal after reducing only cohesion and the internal friction angle, the definition of the safety factor in the traditional strength reduction method can be obtained as follows:  
\[F_{\mathrm{s}}=k_{c}=k_{\varphi} \tag{20}\]  
Thus, it can be considered that the comprehensive safety factor defined by Equation (19) holds equivalent reference significance to the safety factor defined by the traditional strength reduction method. Simplifying this equation yields the following:  
\[F_{\mathrm{s}}=\frac{cA+\sigma_{\mathrm{n}}A\tan\varphi}{cA/k_{c}+\sigma_{ \mathrm{n}}A\tan\varphi/k_{\varphi}}=\frac{c+\sigma_{\mathrm{n}}\tan\varphi}{ c/k_{c}+\sigma_{\mathrm{n}}\tan\varphi/k_{\varphi}} \tag{21}\]  
Introducing the following shear strength formula:  
\[\tau=c+\sigma_{\mathrm{n}}\tan\varphi \tag{22}\]  
The expression for the safety factor can be defined as follows:  
\[F_{\mathrm{s}}=\frac{\tau}{\tau^{\prime}} \tag{23}\]  
where \(\tau\) and \(\tau^{\prime}\) are the shear strengths of the rock mass in the initial state and critical state, respectively.