## 2 Analysis of Overburden Load Transfer Mechanical Model
### Roof Breaking and Load Transfer Process
#### 2.1.2 Basic Mechanical Analysis of Initial Fracture  
With the expansion of the working face advancing the goat, the load of the rock layer under the basic roof gradually increases, the cracks in the beam gradually develop and expand, the stable structure of the beam is broken, the load cannot be transmitted to the surrounding rock mass through the beam, and the load of the overlying rock layer begins to transfer from the beam to the plate. Because the lower coal seam of the initial edge of the basic roof of the stope has been mined in order to simplify the mechanical structure, it can be regarded as a simply supported edge, and the plate can be regarded as a thin rectangular plate with one side simply supported, and three sides clamped support. Therefore, the mechanical model of one side simply supported and three sides clamped supported is constructed for the roof of the stop(Figure 6). The boundary conditions of the thin plate are  
\[ \begin{array}{c}\left(\omega\right)\ \ x=a\ \ =0,\left(\omega\right)_{y=\pm b}=0\\ x=0\\ \left(\frac{\partial\omega}{\partial x}\right)_{x=a}=0,\left(\frac{\partial \omega}{\partial y}\right)_{y=\pm b}=0,\left(\frac{\partial^{2}\omega}{ \partial x^{2}}\right)_{x=0}=0\end{array}  \tag{7}\]  
Set the deflection expression to  
\[\omega=C_{1}\omega_{1}=C_{1}\frac{x}{a}\left(x^{2}-a^{2}\right)^{2}\left(y^{2 }-b^{2}\right)^{2} \tag{8}\]  
In the formula: \(\omega\) is deflection, \(C_{1}\) is constant. So  
\[\omega_{1}=\frac{x}{a}\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{2}\right)^{2}\]  
\[\
abla^{4}\omega=\frac{\partial^{4}\omega}{\partial x^{4}}+\frac{\partial^{4} \omega}{\partial y^{4}}+2\frac{\partial^{4}\omega}{\partial x^{2}\partial y^ {2}}=C_{1}[\frac{120x}{a}\left(y^{2}-b^{2}\right)^{2}+\frac{24x}{a}\left(x^{2 }-a^{2}\right)^{2}+(\frac{20}{a}x^{3}-12 xa)(12y^{2}-b^{2})] \tag{9}\]  
In the formula, \(\
abla^{4}\) is gradient operator. Bring into  
\[\iint D(\
abla^{4}\omega)\omega_{m}dxdy=\iint q\omega_{m}dxdy\]  
Get the  
\[ \begin{array}{c}4\int_{0}^{a}\int_{0}^{b}DC_{1}[\frac{120x}{a}\left(y^{2}-b^ {2}\right)^{2}+\frac{24x}{a}\left(x^{2}-a^{2}\right)^{2}+(\frac{20}{a}x^{3}-1 2 xa)(12y^{2}-b^{2})]\times\\ \frac{x}{a}\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{2}\right)^{2}dxdy=q\int _{-a}^{a}\int_{-b}^{b}\frac{x}{a}\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{ 2}\right)^{2}dxdy\end{array}  \tag{10}\]  
The calculation:  
\[C_{1}=\frac{1617q}{4(168a^{5}-55a^{3}b^{2}+1320 ab^{4})D} \tag{11}\]  
Further calculated:  
\[\omega=\frac{1617q\frac{x}{a}\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{2} \right)^{2}}{4a(168a^{4}-55a^{2}b^{2}+1320b^{4})D} \tag{12}\]  
In the formula: \(D\) is flexural rigidity of thin plate, GPa-m; \(\omega_{1}\) is the First-order deflection function; \(q\) is load, Mpa. Figure 6: 'Plate’ mechanical model of initial fracture of roof. Using the internal force bending moment formula of elasticity, the bending moment of the thin plate is obtained as  
\[M_{x}=-D(\frac{\partial^{2}\omega}{\partial x^{2}}+\mu\frac{\partial^{2}\omega}{ \partial y^{2}})=-DC_{1}[\frac{1}{a}(20x^{3}-12 xa^{2})(y^{2}-b^{2})^{2}+\mu\frac {x}{a}(x^{2}-a^{2})^{2}(12y^{2}-4b^{2})] \tag{13}\]  
\[M_{y}=-D(\mu\frac{\partial^{2}\omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{ \partial y^{2}})=-DC_{1}[\mu\frac{1}{a}(20x^{3}-12 xa^{2})(y^{2}-b^{2})^{2}+ \frac{x}{a}(x^{2}-a^{2})^{2}(12y^{2}-4b^{2})] \tag{14}\]  
In the formula: \(M_{x}\), \(M_{y}\) are Moment in \(x\) and \(y\) directions, MN\(\cdot\)m; \(\mu\) is poisson ratio. Then the stress of a rectangular roof is  
\[