## 3 Slope Stability Calculation and Analysis Method
### Common Slope Stability Analysis Method  
In slope stability analysis, the theory of Mohr-Coulomb is used for the distribution of the saturated soil shear strength. In many cases, effective stress is considered the only stress variable, in particular in a detailed analysis. However, the traditional Mohr-Coulomb theory is not utilized in unsaturated soil slope engineering. In order to analyze the contributions of matric suction and pore air pressure to shear strength, the modified Mohr-Coulomb  
Figure 1: Hydraulic characteristics of the seepage analysis.  
failure criterion is written as follows, using two independent stress variables, including effective stress and matric suction [22]:  
\[\tau=c^{\prime}+(\sigma-p_{a})tan\varphi^{\prime}+(p_{a}-p_{w})tan\varphi^{b} \tag{6}\]  
where \(\tau\) is the shear strength of unsaturated soil, \(c^{\prime}\) is the effective cohesion, \(\sigma\) is the total normal stress, \(p_{a}\) is the pore air pressure, \(p_{w}\) is the pore water pressure, \(\varphi^{\prime}\) is the angle of internal friction associated with the net normal stress state variable (\(\sigma-p_{a}\)), and \(\varphi^{b}\) is an angle indicating the rate of increase in shear strength relative to the matric suction (\(p_{a}-p_{w}\)).  
In this research, the residual thrust method was adopted in order to perform the slope stability analysis, incorporating matric suction and pore gas pressure. The calculation process is as follows: first the safety factor was assumed, and then the thrust was calculated from the first slide at the top of the slope to the last slide. When the thrust was equal to zero, the result of the safety factor would be the final value. Figure 2 shows the calculation diagram of the residual thrust method. The equation could be written as follows [23]:  
\[\begin{split} P_{i}=W_{i}sin\alpha_{i}-\frac{\left[c_{i}^{\prime} l_{i+}+\left(W_{i}cos\alpha_{i}-p_{a}^{\prime}l_{i}\right)tan\varphi_{i}^{ \prime}+l_{i}\left(p_{a}^{\prime}-p_{w}^{\prime}\right)tan\varphi_{i}^{b} \right]}{\Gamma_{s}}+P_{i-1}\psi_{i}\\ \psi_{i}=\left[\cos(\alpha_{i-1}-\alpha_{i})-\frac{tan\varphi_{i }^{\prime}}{\Gamma_{s}}\sin(\alpha_{i-1}-\alpha_{i})\right.\end{split} \tag{7}\]  
where \(F_{s}\) is the safety factor of the slope, \(P_{i}\) is the sliding force of soil slice \(i\), \(c_{i}^{\prime}\) is the effective cohesive force of slice \(i\), \(\varphi_{i}^{\prime}\) is the effective internal friction angle of slice \(i\), \(l_{i}\) is the width of soil slice \(i\), \(W_{i}\) is the weight of soil slice \(i\), \(a_{i}\) is the angle of bottom soil slice \(i\), \(p_{a}^{\prime}\) and \(p_{w}^{i}\) are pore gas pressure and pore water pressure of slice \(i\), \(tan\varphi^{b}\) is the rate of shear strength that increases with an increase in matric suction, and \(\psi_{i}\) is the transfer coefficient of slice \(i\). The model recommended by [PERSON] [24] was that \(tan\varphi^{b}\) in the current study can be expressed as a function of effective saturation. The equation is as follows:  
\[tan\varphi^{b}=S^{*}*tan\varphi^{\prime} \tag{8}\]  
where the effective saturation is \(S^{*}=(S_{r}-S_{rw})/(1-S_{rw})\).