# 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 Gudao Working Face Before Basic Roof Failure  
It can be seen from Figure 4 that the basic roof of Gudao stope can be regarded as an elastic thin plate with simple support on the upper and lower sides (goaf side) and fixed support on the left and right sides (coal wall and coal pillar) before the initial pressure. Because the initial fracture step and periodic fracture step of the roof are often smaller than the stope length, the stope tendency direction, that is, the y-axis direction, is the long side of the plate, and the stope 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. The Rayleigh-Ritz method can transform complex partial differential equations into algebraic equations through the variational principle and solve them using selected trial functions. It is suitable for solving thin plate problems involving complex boundary conditions and multiple forces, making it an effective computational approach. From the above analysis, we can understand the basic top boundary conditions:  
\[ \begin{array}{l}\left(\omega\right)_{x=\pm a}=0,\ \ \ \ \ \ \left(\omega \right)_{y=\pm b}=0\\ \left(\frac{\partial\omega}{\partial x}\right)_{x=\pm a}=0,\ \ \ \left(\frac{\partial^{2}\omega}{ \partial y^{2}}\right)_{x=\pm b}=0\end{array}  \tag{1}\]  
The first-order flexural plane equation conforming to the boundary case is constructed:  
\[\omega=A_{1}\omega_{1}=A_{1}(1-\cos\frac{2\pi x}{a})\cos\frac{\pi y}{2b} \tag{2}\]  
Among these, \(\omega_{1}\) is a first-order deflection function; \(A_{1}\) is a constant. Then, the total potential energy of sheet bending is calculated as  
\[ \begin{array}{l}E_{p}=\frac{D}{2}\iint\left\{\left[\frac{\partial^{2}\omega} {\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}}\right]^{2}-2(1- \mu)\left[\frac{\partial^{2}\omega}{\partial x^{2}}\cdot\frac{\partial^{2} \omega}{\partial y^{2}}-\left[\frac{\partial^{2}\omega}{\partial x\partial y }\right]^{2}\right]\right\}dxdy\\ -\iint q\omega dxdy=\frac{\pi^{4}D{A_{1}}^{2}}{2}\left[\frac{16b}{a^{3}}+ \frac{2}{ab}+\frac{3a}{16b^{3}}\right]-\frac{8 ab}{\pi}A_{1}q\end{array}  \tag{3}\]  
Figure 4: The basic roof mechanics model with two sides fixed and two sides simply supported. where \(D=\frac{EI^{3}}{12(1-v^{3})}\), where \(E\) is the elastic modulus, \(h\) is the thickness of the thin plate, and \(v\) is the displacement on the \(y\)-axis; \(\mu\) is [PERSON]'s ratio; \(q\) is the overburden load; According to \(\frac{\partial E_{p}}{\partial A_{1}}=0\), we obtain  
\[A_{1}=\frac{128a^{4}b^{4}q}{\pi^{5}D(3a^{4}+32a^{2}b^{2}+256b^{4})} \tag{4}\]  
Continue to deduce  
\[\omega=\frac{128a^{4}b^{4}q}{\pi^{5}D(3a^{4}+32a^{2}b^{2}+256b^{4})}\bigg{[}1- \cos\frac{2\pi x}{a}\bigg{]}\cos\frac{\pi y}{2b} \tag{5}\]  
The basic top moment can be obtained:  
\[ \begin{array}{l}M_{x}=\frac{-\pi^{2}DA_{1}}{2a^{2}b^{2}}\left[b^{2}\cos \frac{2\pi x}{a}\cos\frac{\pi y}{2b}-\mu a^{2}\sin^{2}\frac{\pi x}{a}\cos \frac{\pi y}{2b}\right]\\ M_{y}=\frac{-\pi^{2}DA_{1}}{2a^{2}b^{2}}\left[\mu b^{2}\cos\frac{2\pi x}{a} \cos\frac{\pi y}{2b}-a^{2}\sin^{2}\frac{\pi x}{a}\cos\frac{\pi y}{2b}\right] \end{array}  \tag{6}\]  
The basic top stress can be pushed:  
\[\left\{ \begin{array}{l}\sigma_{x}=\frac{-\pi^{2}A_{1}E_{2}}{2a^{2}b^{2}(1- \mu^{2})}\left[b^{2}\cos\frac{2\pi x}{a}\cos\frac{\pi y}{2b}-\mu a^{2}\sin^{2} \frac{\pi x}{a}\cos\frac{\pi y}{2b}\right]\\ \sigma_{y}=\frac{-\pi^{2}A_{1}E_{2}}{2a^{2}b^{2}(1-\mu^{2})}\left[b^{2}\cos \frac{2\pi x}{a}\cos\frac{\pi y}{2b}-a^{2}\sin^{2}\frac{\pi x}{a}\cos\frac{\pi y }{2b}\right]\\ \tau_{xy}=\frac{\pi^{2}A_{1}E_{2}}{ab(1+\mu)}\sin\frac{2\pi x}{a}\sin\frac{ \pi y}{2b}\end{array} \right. \tag{7}\]  
From Equations (6) and (7), it is calculated that the bending moment and stress at the center point (6a, 0) of the fixed support edge of the basic roof reach their maximum values.