## 3 Analysis of the Coal Pillar Width under the Influence of the Water-Logged Goal
### Calculation of the Width of the Plastic Zone of Waterproof Coal Pillars  
Since the coal pillar is long and its length significantly exceeds its height, a specific cross-section was taken along the width direction at a certain position along the length of the coal pillar. A microelement body within the cross-section was then considered, treating the problem as plane stress. The width of the plastic zone was calculated based on [PERSON]'s criterion. The calculation of coal pillar width using [PERSON]'s criterion was based on the following assumptions:  
1. The properties of the coal body are continuous, homogeneous, isotropic, and perfectly elastic. 2. Plastic damage occurs when the coal pillar is subjected to shear stress, and the damaged surface is parallel to the coal seam level. 3. The coal body undergoes plastic deformation, and the deformation is negligible. 4. The force on the coal pillar only considers the gravity of the overlying rock and does not consider the influence of its tectonic stress. 5. The stresses above the coal body are symmetrical about the neutral plane. 6. The influence of coal body force is not considered. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Area Number** & **1** & **2** & **3** & **4** \\ \hline
2305 Length of working face/m & 0–800 & 800–1015 & 1015–1084 & 1084–1540 \\ Width of coal pillar/m & 60 & 54 & 52 & 54 \\ Ponding pressure/MPa & 1.06–0.6 & 0.6–0.36 & 0.36–0.3 & 0.3–0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of water accumulation pressure in different areas of the 2305 working face. Figure 4: Schematic structure and force of the waterproof coal pillar by section. The width of the plastic zone of the coal column in the section was calculated according to the assumptions above. The mechanical calculation model of the plastic zone width is shown in Figure 5.  
The equilibrium equations are listed with the \(x\)-axis and \(y\)-axis as projection axes, and the plane stress equilibrium differential equations can be obtained as follows:  
\[\left\{ \begin{array}{l}\frac{\partial\sigma_{x}}{\partial x}+\frac{\partial \tau_{yx}}{\partial y}=0\\ \frac{\partial\sigma_{y}}{\partial y}+\frac{\partial\sigma_{yx}}{\partial x}=0 \end{array} \right. \tag{2}\]  
When critical damage occurs to the coal pillar along the seam level, the stresses between the seams satisfy the stress limit equilibrium condition, i.e., [PERSON]'s criterion [36]:  
\[\tau_{yx}=-(C_{0}+\sigma_{y}\tan\varphi_{0}) \tag{3}\]  
where \(C_{0}\) is the cohesive force between the coal body and roof and floor (in MPa) and \(\varphi_{0}\) is the friction angle between the coal seam and the roof and floor (in \({}^{\circ}\)). The stress boundary condition at the interface between the plastic and elastic regions is as follows:  
\[\left\{ \begin{array}{l}\left(\sigma_{y}\right)_{x=l_{1}}=K_{1}\gamma H\\ \left(\sigma_{x}\right)_{x=l_{1}}=\lambda\cdot K_{1}\gamma H\end{array} \right. \tag{4}\]  
where \(\lambda\) is the lateral pressure coefficient at the interface between plastic and elastic zones. Equations (3) and (4) can be obtained as follows:  
\[\frac{\partial\sigma_{y}}{\partial x}\tan\varphi_{0}=\frac{\partial\sigma_{y }}{\partial y} \tag{5}\]  
Assume that the function \(\sigma_{y}\) is:  
\[\sigma_{y}=f(x)g(y)+a \tag{6}\]  
The combination of Equations (5) and (6) yields:  
\[\frac{f^{\prime}(x)}{f(x)}\tan\varphi_{0}=\frac{g^{\prime}(y)}{g(y)} \tag{7}\]  
Figure 5: Mechanical calculation model of the width of the plastic zone. This can be obtained by making the left and right sides of the equal sign of Equation (7) equal to the constant \(b\):  
\[\left\{ \begin{array}{l}f(x)=c_{1}e^{\frac{b}{\tan\varphi_{0}}x}\\ g(y)=c_{2}e^{by}\end{array} \right. \tag{8}\]  
Equations (3), (6) and (8) are obtained by association:  
\[\left\{ \begin{array}{l}\sigma_{y}=ce^{\frac{bx}{\tan\varphi_{0}}}\\ \tau_{yx}=-\bigg{[}\bigg{(}ce^{\frac{bx}{\tan\varphi_{0}}}e^{bx}+a\bigg{)}\tan \varphi_{0}+C_{0}\bigg{]}\end{array} \right.