## 2 Analysis of Overburden Load Transfer Mechanical Model
### Roof Breaking and Load Transfer Process
#### 2.1.3 Mechanical Analysis of Periodic Fracture of Main Roof  
As the working face continues to advance, the periodic weighting phenomenon causes a large number of cracks inside the 'plate' to begin to collapse periodically. When the clamped long side of the rectangular roof is broken, the boundary conditions will change accordingly. Therefore, it is necessary to reconstruct the mechanical model of simply supported on three sides and clamped supported on one side and solve the mechanics according to the new strength failure criterion(Figure 7).  
The boundary conditions of the thin plate are  
\[\begin{array}{l}(\omega)\\ x=0\\ x=a\end{array}=0,(\omega)_{y=\pm b}=0,\left(\frac{\partial\omega}{\partial x} \right)_{x=a}=0 \tag{16}\]  
Figure 7: ‘Plate’ mechanical model of periodic fracture of roof.  
Construction of First Order deflection surface equation satisfying boundary conditions by Ritz Method:  
\[\omega=C_{2}\omega_{2}=C_{2}x^{2}(x-a)\sin\frac{\pi y}{b} \tag{17}\]  
The total potential energy of the thin plate is  
\[\begin{array}{l}E_{p}=V_{\varepsilon}-\iint q\omega dxdy=\frac{D}{2}\iint\left\{ \left(\
abla^{2}\omega\right)^{2}-2(1-\mu)\left[\frac{\partial^{2}\omega}{ \partial x^{2}}\frac{\partial^{2}\omega}{\partial y^{2}}-\left(\frac{\partial ^{2}\omega}{\partial x\partial y}\right)^{2}\right]\right\}dxdy-\iint q\omega dxdy =\\ \frac{Da^{2}C_{2}^{2}}{840b^{4}}(840b^{4}+2a^{4}\pi^{4}+168a^{2}b^{2}\pi^{2}- 112a^{2}b^{2}\pi^{2}\mu)+\frac{qa^{4}b}{6\pi}C_{2}\end{array} \tag{18}\]  
From \(\frac{\partial E_{p}}{\partial C_{2}}=0\) to:  
\[C_{2}=-\frac{70 qa^{4}b^{4}}{\pi^{2}(840b^{4}\pi+2a^{4}\pi^{4}+168a^{2}b^{2}\pi ^{2}-112a^{2}b^{2}\pi^{2}\mu)} \tag{19}\]  
Further calculated:  
\[\omega=-\frac{70 qab^{4}}{\pi^{2}(840b^{4}\pi+2a^{4}\pi^{4}+168a^{2}b^{2}\pi^{ 2}-112a^{2}b^{2}\pi^{2}\mu)D}x^{2}(x-a)\sin\frac{\pi y}{b} \tag{20}\]  
Using the internal force bending moment formula of elasticity, the bending moment of the thin plate is obtained as  
\[M_{x}=-D(\frac{\partial^{2}\omega}{\partial x^{2}}+\mu\frac{\partial^{2} \omega}{\partial y^{2}})=-DC_{2}[(6x-2a)\sin\frac{\pi y}{b}-\mu\frac{\pi^{2}}{ b^{2}}x^{2}(x-a)\sin\frac{\pi y}{b}] \tag{21}\]  
\[M_{y}=-D(\mu\frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2} \omega}{\partial y^{2}})=-DC_{2}[\mu(6x-2a)\sin\frac{\pi y}{b}-\frac{\pi^{2}}{ b^{2}}x^{2}(x-a)\sin\frac{\pi y}{b}] \tag{22}\]  
Then the stress of a rectangular roof is  
\[\begin{array}{l}\sigma_{x}=-\frac{Ez}{1-\mu^{2}}(\frac{\partial^{2}\omega} {\partial x^{2}}+\mu\frac{\partial^{2}\omega}{\partial y^{2}})=-\frac{EzC_{2} }{1-\mu^{2}}[(6x-2a)\sin\frac{\pi y}{b}-\mu\frac{\pi^{2}}{b^{2}}x^{2}(x-a) \sin\frac{\pi y}{b}]\\ \sigma_{y}=-\frac{Ez}{1-\mu^{2}}(\frac{\partial^{2}\omega}{\partial y^{2}}+ \mu\frac{\partial^{2}\omega}{\partial x^{2}})=-\frac{EzC_{2}}{1-\mu^{2}}[\mu(6 x-2a)\sin\frac{\pi y}{b}-\frac{\pi^{2}}{b^{2}}x^{2}(x-a)\sin\frac{\pi y}{b}]\\ \tau_{xy}=-\frac{Ez}{1+\mu}\frac{\partial^{2}\omega}{\partial x\partial y}=- \frac{EzC_{2}}{1+\mu}\frac{\pi}{b}(3x^{2}-2 xa)\cos\frac{\pi y}{b}\end{array} \tag{23}\]  
According to the bending moment Formulas (21) and (22), the maximum bending moment of the roof is located at the center of the plate near the simply supported short side. The bending moment decreases in turn from 'plate surface-simple supported short side-simple supported long side-fixed supported short side'. At this time, the load on the plate surface is relatively large. Therefore, the rectangular roof will first break at the position near the simply supported short side of the thin plate surface, resulting in transverse cracks along the x direction, and continue to extend to both ends with the mining progress. Finally, the fracture cracks at the proximal end of the short side are bifurcated, forming a horizontal 'X' type fracture of the plane, and the overall fracture process is 'X-O' type.