# Factors Influencing the Stability of a Slope Containing a Coal Seam in a Goal
## 2 Model Analysis of a Slope Containing a Coal Seam in a Goaf  
### Influencing Factors and the Rock Mass Mechanical Parameters  
To analyze the factors influencing the stability of a slope containing a coal seam in a goat, we designed a generalized model, based on an actual project: the K1641 + 890-K1641 + 950 section of the Jixi Section of the Dan-A-Highway reconstruction and expansion project (hereinafter referred to as the actual project), as shown in Figure 2, where i is the slope gradient, T is the coal seam thickness, \(\alpha\) is the coal seam dip angle, H is the coal seam position (that is, the distance between the coal seam exposed on the slope surface and the top of the slope), S is the roadway position (that is, the horizontal distance between the roadway center and the slope foot), and W is the roadway width. In addition to the influencing factors shown in Figure 2, we considered the coal seam cohesion c, the coal seam internal friction angle \(\varphi\), and the mining depth and mining thickness ratio R (R = D/d) in this study.  
The main rock strata of the slope included strongly weathered sandstone, a coal seam, and moderately weathered sandstone. We determined mechanical rock mass parameters of the slope model according to the laboratory test parameters of the rock mass mechanics of the actual project. The parameters of each rock stratum are shown in Table 1.  
Figure 2: Schematic diagram of the influencing factors. i is the slope gradient, T is the coal seam thickness, \(\alpha\) is the coal seam dip angle, H is the coal seam position (that is, the distance between the coal seam exposed on the slope surface and the top of the slope), S is the roadway position (that is, the horizontal distance between the roadway center and the slope foot), and W is the roadway width. Blue dotted line is the boundary line of the rock strata.  
### Influencing Factors and the Rock Mass Mechanical Parameters  
To calculate the slope stability coefficient, we used Midas GTS software, which uses the strength reduction method to continuously reduce the strength parameters of the rock and soil mass, namely the cohesion and internal friction angle, until the soil is destroyed. The final reduced ratio is the slope stability coefficient. The calculation formulae are as follows:  
\[\overline{c}=\frac{c}{F_{s}} \tag{1}\]  
\[\overline{\varphi}=\arctan\left(\frac{tan\varphi}{F_{s}}\right) \tag{2}\]  
where \(c\) and \(\overline{c}\) are the cohesive force before and after the reduction, respectively; \(\varphi\) and \(\overline{\varphi}\) are the internal friction angles before and after the reduction, respectively.  
During the calculation of the slope stability coefficient of the model, the Mohr-Coulomb yield criterion is adopted to judge the failure of the rock mass, that is, the cohesion and internal friction angle of the rock and soil mass are the factors that determine its shear failure. The calculation is shown in Formula (3). When the size of the shear stress exceeds the shear stress controlled by the cohesion and the internal friction angle of the rock and soil mass, the rock and soil mass experience shear failure.  
\[f_{s}=\sigma_{1}-\sigma_{3}\frac{1+sin\varphi}{1-sin\varphi}-2c\sqrt{\frac{1+ sin\varphi}{1-sin\varphi}} \tag{3}\]  
here, \(f_{s}\) is the shear stress, \(\sigma_{1}\) is the maximum principal stress, and \(\sigma_{3}\) is the minimum principal stress.