## 3 Analysis Method and Assumptions
### Method for Solving the Factor of Safety  
In general, different contexts employ various definitions of the factor of safety. The limit equilibrium method mentioned in the introduction defines the factor of safety as the ratio of the resisting force to the uplifting force, with differences lying in the components of the resisting force. These differences arise from the shape of the overlying rock mass affected by uplift, leading to variations in the rock mass's weight or whether the rock mass's strength is considered. Once the accurate fracture surface is determined, the weight of the overlying rock mass can be ascertained. However, the strength of the rock mass resisting uplift remains an uncertain factor. Therefore, this paper adopts the concept of strength reserve and defines the safety factor, \(F_{S}\), as  
\[\tau=\frac{1}{F_{S}}(c+\sigma\tan\varphi), \tag{4}\]  
where \(\sigma\) is the positive stress on the crack surface, and \(\tau\) is the shear stress. The normal stress on a differential unit on the crack surface is \(\sigma\mathbf{d}S\), and the shear stress is \(\sigma\mathbf{d}S\), where \(\mathbf{n}\) and \(\mathbf{s}\) are, respectively, the unit normal vector on the differential unit (pointing inside the block) and the direction of slip resistance. Then, the reaction force of the differential unit \(\mathrm{d}S\) is  
\[\mathbf{d}\mathbf{f}=(\sigma\mathbf{n}+\tau\mathbf{s})\mathrm{d}S, \tag{5}\]  
the torques is  
\[\mathrm{d}\mathbf{m}=\mathbf{r}\times\mathrm{d}\mathbf{f}, \tag{6}\]  
Then, the overall equilibrium equation is organized as  
\[\int_{S}(F_{S}\mathbf{n}^{\prime}+\tan\varphi\mathbf{s}^{\prime})\sigma\mathrm{d}S+F_ {S}\mathbf{f}_{m}+\int_{S}\mathbf{c}\mathbf{s}^{\prime}\mathrm{d}S=0 \tag{7}\]  
where  
\[\begin{cases}\mathbf{n}^{\prime}=\binom{\mathbf{n}}{r\times n}\\ \mathbf{s}=\binom{\mathbf{n}}{r\times n}\\ \mathbf{f}_{m}=\binom{\mathbf{f}_{a}}{\mathbf{m}_{a}}\end{cases} \tag{8}\]  
where \(\mathbf{f}_{a}\) and \(\mathbf{m}_{a}\) are the active force moment, respectively. The active force includes the self-weight of the rock mass, the surface load, and the air internal pressure.  
The equilibrium equation for a differential element taken from the fracture surface to the ground using a plumb line is  
\[\sigma\mathbf{d}S+\tau\mathbf{d}S+\mathbf{d}\mathbf{w}+\mathrm{d}\mathbf{h}=\mathbf{0} \tag{9}\]  
Figure 6: Schematic diagram of potential fracture surface solution.  
where \(\mathbf{dw}\) is the computable active force on the element, including self-weight, ground load, air pressure, et al., and \(\mathbf{h}\) is the force between the elements that is not directly calculable. Multiplying the terms in Equation (9) by the vector \(\mathbf{n}\) yields  
\[\sigma=\sigma_{0}+\sigma_{h} \tag{10}\]  
where \(\sigma_{0}\) is computable, while \(\sigma_{h}\) is not directly computable. Equation (10) can be approximated by the following equation:  
\[\sigma=\sigma_{0}+\mathbf{f}(\mathbf{x};\mathbf{a}) \tag{11}\]  
where \(\mathbf{f}(\mathbf{x};\mathbf{a})\) is a function of \(\mathbf{x}\) containing a coefficient to be determined \(\mathbf{a}\). \(\mathbf{x}\) is the horizontal coordinates, and the number of coefficients \(\mathbf{a}\) depends on the number of equation sets. \(\mathbf{f}(\mathbf{x};\mathbf{a})\) is usually constructed as an interpolating function.  
\[\mathbf{f}(\mathbf{x};\mathbf{a})=\mathbf{I}^{T}\mathbf{a} \tag{12}\]  
where \(\mathbf{I}=(l_{1},l_{2},\dots,l_{5})\) satisfies the normalization condition. By substituting Equation (11) into Equation (7), a six-variable nonlinear equation system \(g(F_{S},\mathbf{a})\) can be obtained.  
\[g(F_{S},\mathbf{a})=0 \tag{13}\]  
The unknown can then be solved for by [PERSON]'s iterative method.