# Comprehensive Method to Test the Stability of High Bedding Rock Slope Subjected to Atomized Rain
## 3 Theoretical Research on the Comprehensive Method  
### Safety Factor Expression of the Comprehensive Method  
Based on the similarity theory of physical model and limit equilibrium method, the safety factor expression of rock slope by the comprehensive method can be derived. (1) Strength reduction method (the mechanical parameter reduction of structural planes)  
Schematic diagrams of the rock slope under natural condition and under loads are shown in Figure 1.  
Figure 1: Schematic diagram of rock slope in natural state and load analysis: (**a**) slope in natural state; (**b**) load analysis. According to the shear strength formula, the antislide force \(F\) and sliding force \(T\) of the slope block are expressed as follows:  
\[F=f^{\prime}N_{y}+c^{\prime}A=f^{\prime}N\cos\theta+c^{\prime}A \tag{1}\]  
\[T=N_{x}=N\sin\theta \tag{2}\]  
where \(f^{\prime}\) is the friction factor of the bottom surface, \(c^{\prime}\) is the cohesion force of the bottom surface, \(A\) is the contact area of block per unit width, \(N\) is the deadweight of block per unit width, \(N_{x}\) is the horizontal component of \(N\) on the bottom surface, \(N_{y}\) is the vertical component of \(N\) on the bottom surface, and \(\theta\) is the dip angle of the bottom surface in the natural state. The slope safety factor \(K\) in the natural state is defined as follows:  
\[K=\frac{F}{T}=\frac{f^{\prime}N\cos\theta+c^{\prime}A}{N\sin\theta} \tag{3}\]  
For the strength reduction method, the decrease in mechanical parameters of the structural planes is mainly achieved by reducing the values of \(f^{\prime}\) and \(c^{\prime}\). Here, global slope failure occurs when these values are reduced as the values of \(f^{\prime}\) and \(c^{\prime}\) are divided by \(K_{s}\). The \(K^{\prime}_{s}\) in the ultimate state is defined as follows:  
\[K^{\prime}_{s}=\frac{(f^{\prime}/K_{s})N\cos\theta+(c^{\prime}/K_{s})A}{N\sin \theta}\]  
\[=\frac{1}{K_{s}}\cdot\frac{f^{\prime}N\cos\theta+c^{\prime}A}{N\sin\theta}=1 \tag{4}\]  
Thus, the strength reserve factor \(K_{\text{ss}}\) can be derived by dividing the right side of Formula (3) by that of Formula (4):  
\[K_{\text{ss}}=\frac{K}{K^{\prime}_{s}}=\frac{\frac{f^{\prime}N\cos\theta+c^{ \prime}A}{N\sin\theta}}{\frac{1}{K_{s}}\cdot\frac{f^{\prime}N\cos\theta+c^{ \prime}A}{N\sin\theta}}=K_{s} \tag{5}\]  
Formula (5) clearly shows that \(K_{\text{ss}}\) equals the reduction ratios of \(f^{\prime}\) and \(c^{\prime}\), which is to say that \(K_{\text{ss}}\) depends on the reduction ratios of the structural plane's mechanical parameters. (2) Overloading method by steel frame lifting technique  
Schematic diagrams of the rock slope and the loads at the lifting state are shown in Figure 2.  
In the overloading method, the increased sliding forces of the slope rock mass are mainly achieved by lifting the steel frame. Here, global slope failure occurs when the uplift angle of the steel frame equals to \(\Phi\). The \(K^{\prime}_{p}\) at the ultimate state is defined as follows:  
\[K^{\prime}_{p}=\frac{f^{\prime}N\cos(\theta+\phi)+c^{\prime}A}{N\sin(\theta+ \phi)} \tag{6}\]  
Figure 1:0 Schematic diagram of rock slope in lifting state and load analysis: (**a**) slope in lifting state; (**b**) load analysis. Thus, the overload factor \(K_{sp}\) can be derived by dividing the right side of Formula (3) by that of Formula (6):  
\[ \begin{array}{ll}K_{sp}&=\frac{K}{K_{p}^{\prime}}=\frac{\frac{f^{\prime}N\cos \theta+c^{\prime}A}{N\sin\theta}}{f^{\prime}N\cos(\theta+\phi)+c^{\prime}A}\\ &=\frac{f^{\prime}N\cos\theta+c^{\prime}A}{f^{\prime}N\cos(\theta+\phi)+c^{ \prime}A}\cdot\frac{N\sin(\theta+\phi)}{N\sin\theta}\\ &=\frac{f^{\prime}+\frac{c^{\prime}A}{N\cos\theta}}{f^{\prime}+\frac{N\cos( \theta+\phi)}{N\cos(\theta+\phi)}}\cdot\frac{\tan(\theta+\phi)}{\tan\theta} \end{array}