# Study on the Law of Mine Pressure Manifestation in Three-Soft Coal Seam Isolated Working Face
## 3 Analysis of Failure Mechanics Model of Basic Roof of Isolated Island Working Face
### Mechanical Model Analysis of Basic Roof Periodic Fracture of Gudao Working Face  
It can be seen from Figure 5 that the basic roof of the island working face can be regarded as a thin plate with three sides simply supported and one side firmly supported during the period of periodic pressure. Because the initial fracture step and periodic fracture step of the roof are often smaller than the stop length, the stop tendency direction, that is, the y-axis direction, is the long side of the plate, and the stop strike direction, that is, the x-axis direction, is the short side of the plate. The length of the long side is 2b. The short side length is the advance length, which is set to 2a using the Rayleigh-Ritz method.  
From the above analysis, we can understand the basic top boundary conditions:  
\[\left(\omega\right)_{x=\pm a}=0,\ \left(\omega\right)_{y=\pm b}=0,\ \left(\frac{ \partial\omega}{\partial x}\right)_{x=\pm a}=0 \tag{8}\]  
The first-order flexural plane equation conforming to the boundary case is constructed:  
\[\omega=A_{2}\omega_{2}=A_{2}(x+a)(x-a)^{2}(y^{2}-b^{2}) \tag{9}\]  
Among these, \(\omega_{2}\) is a first-order deflection function; \(A_{2}\) is a constant.  
Then, the total potential energy of sheet bending is calculated as  
\[\begin{split} E_{p}=\frac{D}{2}\iint\left\{\left[\frac{\partial^{2} \omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}}\right]^{2}-2( 1-\mu)\left[\frac{\partial^{2}\omega}{\partial x^{2}}\cdot\frac{\partial^{2} \omega}{\partial y^{2}}-\left[\frac{\partial^{2}\omega}{\partial x\partial y} \right]^{2}\right]\right\}dxdy\\ -\iint q\omega dxdy=\frac{768 DA_{2}{}^{2}{\rm a}^{3}{\rm b}}{105} \big{[}3{\rm a}^{4}+7a^{2}b^{2}+21b^{4}\big{]}+\frac{16a^{4}b^{3}}{9}A_{2}q \end{split} \tag{10}\]  
According to \(\frac{\partial E_{p}}{\partial A_{2}}=0\), we obtain  
\[A_{2}=\frac{35{\rm a}b^{2}q}{288(3a^{4}+7a^{2}b^{2}+21b^{4})} \tag{11}\]  
Then, we continue to deduce  
\[\omega=\frac{35 ab^{2}q}{288D(3a^{4}+7a^{2}b^{2}+21b^{4})^{2}}(x+{\rm a})(x-a)^ {2}\Big{(}y^{2}-b^{2}\Big{)} \tag{12}\]  
The basic top moment can be obtained:  
\[\begin{split} M_{\rm x}&=-2 DA_{2}\Big{[}\big{(}3{ \rm x}-{\rm a}\big{)}\big{(}y^{2}-b^{2}\big{)}+\mu(x+a)(x-a)^{2}\Big{]}\\ M_{\rm y}&=-2 DA_{2}\Big{[}\mu(3{\rm x}-{\rm a}) \big{(}y^{2}-b^{2}\big{)}+(x+a)(x-a)^{2}\Big{]}\end{split} \tag{13}\]  
The basic top stress can be pushed:  
\[\left\{\begin{array}{l}\sigma_{\rm x}=-\frac{2A_{2}Ez}{1-\mu^{2}}\Big{[}(3 {\rm x}-{\rm a})\big{(}y^{2}-b^{2}\big{)}+\mu(x+a)(x-a)^{2}\Big{]}\\ \sigma_{\rm y}=-\frac{2A_{2}Ez}{1-\mu^{2}}\Big{[}\mu(3{\rm x}-{\rm a})\big{(} y^{2}-b^{2}\big{)}+(x+a)(x-a)^{2}\Big{]}\\ \tau_{xy}=-\frac{2A_{2}Ez}{1-\mu^{2}}(x-a)(3x+a)y\end{array}\right. \tag{14}\]  
From Equations (13) and (14), it is known that the bending moment of the basic roof reaches its peak at the central point (0,0), where there are two stresses at this position:  
\[\left\{\begin{array}{l}\sigma_{x}=-\frac{2A_{2}Ez}{1-\mu^{2}}\Big{(}{\rm a }{\rm b}^{2}+\mu{\rm a}^{3}\Big{)}\\ \sigma_{y}=-\frac{2A_{2}Ez}{1-\mu^{2}}\Big{(}{\rm a}^{3}+\mu{\rm a}{\rm b}^{2} \Big{)}\end{array}\right. \tag{15}\]  
Figure 5: The basic roof mechanics model with three sides simply supported and one side fixed supported.  
It can be inferred that the basic roof begins to fail and develop cracks at the central point, which propagate continuously along the x-direction. At the edge of the fixed support, the crack splits into two, and the fixed support edge eventually fractures completely. Ultimately, the failure state of the basic roof forms a semi-\"X\"-shaped pattern.