## 3 Analysis of Abutment Stress and Bearing Strength of Isolated Working Face  
### 3.1. Abutment Stress of Isolated Working Face  
The abutment stress estimation model for the isolated working face is depicted in Figure 5. Here, \(\alpha\) represents the contact gangue angle (\({}^{\circ}\)), \(\beta\) denotes the rock movement angle (\({}^{\circ}\)), and \(H\) indicates the buried depth of the coal seam (m). The width of the isolated working face is denoted as \(L\) (m), and the width of the goat is denoted as \(l\) (m). The lengths of the contact gangue in the lowermost rock layer and the surface subsidence basin are denoted as \(d_{1}\) and \(d_{2}\) (m), respectively, and the caving height of the goat is denoted as \(h\) (m). The coordinate system is established, as illustrated in the figure. In this study, the width of the isolated working face at Wuyang Mine is \(L=295\,\mathrm{m}\), and the width of the goat on both sides is \(l=450\,\mathrm{m}\). The lengths of the contact gangue in the lowermost rock layer and the surface subsidence basin are \(d_{1}=280\,\mathrm{m}\) and \(d_{2}=470\,\mathrm{m}\), respectively, and the caving height of the goat is \(h=120\,\mathrm{m}\). These parameters are utilized in the abutment stress model to calculate the stress distribution across the working face. Figure 5: Overburden spatial structure of the isolated working face. The abutment stress \(\sigma_{\rm x}\) (MPa) at a specific point on the working face is composed of the self-weight stress \(\sigma_{\rm q}\) (MPa) from the overlying strata at this point, and the stresses \(\Delta\sigma_{1}\) (MPa) and \(\Delta\sigma_{2}\) (MPa) transferred from the goaf on both sides of the isolated working face to this point [26], which can be expressed as  
\[\sigma_{\rm x}=\sigma_{\rm q}+\Delta\sigma_{1}+\Delta\sigma_{2} \tag{1}\]  
In this context, \(\sigma_{\rm x}\) in Equation (1) specifically refers to the vertical component of the stress, which results primarily from the self-weight of the overlying strata, combined with the stress contributions transferred from the adjacent goafs. The vertical stress component plays a critical role in determining the stability of the coal seam and the surrounding rock mass. The self-weight stress \(\sigma_{\rm q}\) of the overlying strata can be expressed as [27]:  
\[\sigma_{\rm q}=\left\{ \begin{array}{ll}\gamma\rm xtan\beta&[0,Hcot\beta]\\ \gamma H&(Hcot\beta,L-Hcot\beta)\\ \gamma(L-x)\tan\beta&[L-Hcot\beta,L]\end{array} \right. \tag{2}\]  
In this equation, \(\gamma\) represents the average bulk density of the overlying strata (\(\rm kN/m^{3}\)). The stress transferred from the overlying key strata in the goaf to the coal of the isolated working face is assumed to follow an isosceles triangle distribution, which is a common assumption for stress modeling in mining mechanics. Therefore, \(\Delta\sigma_{1}\) can be expressed as [27]:  
\[\Delta\sigma_{1}=\left\{ \begin{array}{ll}\frac{\Delta\sigma\rm xtan\beta}{H }&[0,Hcot\beta]\\ \Delta\sigma\left(2-\frac{\rm xtan\beta}{H}\right)&(Hcot\beta,2 Hcot\beta)\\ 0&[2 Hcot\beta,L]\end{array} \right. \tag{3}\]  
In this equation, \(\Delta\sigma\) represents the maximum value of the stress increment generated by the overlying key strata on the coal of the isolated working face, which can be expressed as  
\[\Delta\sigma= \frac{\gamma[1-(d_{1}+d_{2})/2+(H+h)\cot\beta](H-h)}{4 Hcot\beta} \tag{4}\]  
Thus, \(\Delta\sigma_{2}\) is derived as:  
\[\Delta\sigma_{2}=\left\{ \begin{array}{ll}0&[0,L-2 Hcot\beta]\\ \Delta\sigma\left(2-\frac{(L-x)\rm tan\beta}{H}\right)&(L-2 Hcot\beta,L-Hcot \beta)\\ \frac{\Delta\sigma(L-x)\rm tan\beta}{H}&[L-Hcot\beta,L]\end{array} \right. \tag{5}\]  
By substituting Equation (2), Equation (3), and Equation (5) into Equation (1), the piecewise function expression for the abutment stress \(\sigma_{\rm x}\) at a specific point of the isolated working face and the distance \(x\) (m) from the left goaf is obtained:  
\[\sigma_{\rm x}=\left\{