## 3 Analysis of Abutment Stress and Bearing Strength of Isolated Working Face
### Bearing Strength of Isolated Working Face  
The stress state of the coal varies across different areas of the isolated working face, transitioning from a uniaxial stress state near the coal wall to a biaxial stress state and finally to a triaxial stress state at greater depths [28], as illustrated in Figure 7.  
In the uniaxial stress state, the coal seam is subjected to stress in one direction, typically in exposed areas, resulting in large deformations and instability, especially near the excavation face. In the biaxial stress state, the coal seam experiences stress in two directions, offering higher stability, but localized instability may still occur. In the triaxial stress state, the coal seam is subjected to stress in three directions, providing stronger confinement, generally leading to greater stability and smaller deformations. For areas with uniaxial or biaxial stress, yielding support systems (such as D-bolts) are recommended to absorb dynamic energy. In triaxial stress areas, while the coal seam is more stable, localized support is still needed to address potential fractures and weak zones.  
The bearing strength of coal in the working face is approximately linear with its confining pressure before failure [29]. Therefore, the relationship between the bearing strength \(R_{\rm x}\) (MPa) of coal and the distance x from the coal wall can be expressed as [28]:  
\[R_{\rm x}=\lambda\sigma_{\rm c}=\left\{\begin{array}{ll}\frac{\sigma_{\rm c }\times}{\rm x_{p}}&\left[0,x_{\rm p}\right)\\ \sigma_{\rm c}+\frac{\sigma_{\rm c}(k-1)\left(x-x_{\rm p}\right)}{x_{\rm e}}& \left[x_{\rm p},x_{\rm p}+x_{\rm e}\right)\\ k\sigma_{\rm c}&\left[x_{\rm p}+x_{\rm e},L-x_{\rm p}-x_{\rm e}\right)\\ \sigma_{\rm c}+\frac{\sigma_{\rm c}(k-1)\left(L-x-x_{\rm p}\right)}{x_{\rm e}} &\left[L-x_{\rm p}-x_{\rm e},L-x_{\rm p}\right)\\ \frac{\sigma_{\rm c}}{x_{\rm p}}(L-x)&\left[L-x_{\rm p},L\right]\end{array}\right. \tag{7}\]  
Figure 6: Abutment stress curve diagram of the isolated working face.  
Figure 7: The stress state of coal in different areas of working face.  
In the equation, \(\lambda\) represents the confining pressure coefficient of coal in different regions. When the coal is in the free state, \(\lambda\) = 0; at the elastic-plastic boundary position, \(\lambda_{\rm min}\) = 1; and at the boundary between the elastic zone and the original rock stress, \(\lambda_{\rm max}\) = k, where k is related to the uniaxial compressive strength of the coal. The greater the strength, the smaller the value of k. \(\sigma_{\rm c}\) denotes the uniaxial compressive strength of coal (MPa), and \(\rm x_{\rm p}\) represents the width of the plastic zone (m). \(\rm x_{\rm p}=\frac{m8}{2\tan\varphi}\ln\left(1+\frac{m\gamma H\tan\varphi}{c}\right)\)[30], n denotes the stress concentration factor of the roadway surrounding rock, m represents the thickness of the coal seam (m), \(\delta\) is the lateral pressure coefficient of the roadway coal wall, \(\varphi\) is the internal friction angle of coal (\({}^{\circ}\)), and c represents the cohesion of coal (MPa). \(\rm x_{\rm e}\) denotes the width of the elastic zone (m). \(\rm x_{\rm e}=\frac{m}{2\delta t}\ln\frac{n}{1.05}\)[30], f represents the friction coefficient between the coal seam floor and the rock strata.  
Using Equation (7), the bearing strength curve for the isolated working face can be approximated, as shown in Figure 8.  
From the figure, the corresponding bearing strength \(\rm R_{i}\) at a specific point \(\rm x_{i}\) on the working face can be intuitively observed. Additionally, the total bearing strength R (MPa) on the working face can be obtained by integrating the curve \(\rm R_{x}\) over the interval [0, L].