## 4 Theoretical Analysis of Roof Influence on Underlying Coal and Rock Strata After Coal Seam Mining
### Analysis of the Mechanical Model
#### 4.1.2 Establishment and Solution of the Mechanical Model  
According to the mining and geological conditions of the 22,031 working face of XinDeng Coal Mine, in order to facilitate the calculation, the load above the main roof is simplified as a uniform load, and the influence of the plastic failure of the underlying strata on the elastic foundation parameters is ignored. Selecting the section at a certain distance behind the working face as the research object, a mechanical model of the action of the roof on the underlying coal and rock strata after coal seam mining is established, as shown in Figure 3. In the figure, \(q\) is the load on the overlying strata of the main roof; \(k\) is the Winkler elastic foundation coefficient; and \(h_{j}\) is the thickness of the main roof. \(L\) is the suspended length of the main roof along the working face direction. (1) Differential equation of main roof deflection  
For the main roof above the goaf, the differential equation of roof deflection is:  
\[\frac{d^{4}\omega(x)}{dx^{4}}=\frac{12q\big{(}1-{\
u_{j}}^{2}\big{)}}{E_{j}{h_ {j}}^{3}} \tag{1}\]  
In the equation: \(\omega(x)\) is the deflection of the main roof; \(E_{j}\) is the elastic modulus of the main roof rock layer; and \(\
u_{j}\) is the Poisson's ratio of the main roof rock layer. For the main roof on both sides of the goaf, the differential equation of roof deflection is:  
\[\frac{d^{4}\omega(x)}{dx^{4}}+\frac{12k\big{(}1-{\
u_{j}}^{2}\big{)}}{E_{j}{h_ {j}}^{3}}=\frac{12q\big{(}1-{\
u_{j}}^{2}\big{)}}{E_{j}{h_{j}}^{3}} \tag{2}\]  
Figure 3: Roof stress analysis diagram. In the equation: \(k=\left(E_{z}h_{z}+E_{c}h_{c}\right)/\left(h_{z}+h_{c}\right)^{2}\), \(E_{z}\) and \(E_{c}\) are the elastic moduli of the direct roof and the coal seam, respectively; and \(h_{z}\) and \(h_{c}\) are the thicknesses of the direct roof and the coal seam, respectively. (2) Model boundary conditions  
At an infinite distance from the goaf, the main roof is in a clamped state between the overlying and underlying strata, and the deflection and rotation angle of the main roof are always zero. The deflection, rotation angle, bending moment, and shear force of the main roof at the boundary of the goaf are continuous. Next, the model is solved. The model shown in Figure 2 is axisymmetric about the o-z axis, and only the left side of the main roof deflection equation needs to be solved. The general solutions of Equations (1) and (2) are obtained as follows:  
\[\omega_{1}(x)=\frac{qx^{4}\Big{(}1-v_{j}^{2}\Big{)}}{2 Ejh_{j}^{3}}+A_{1}\frac{ x^{3}}{6}+A_{2}\frac{x^{2}}{2}+A_{3}x+A_{4} \tag{3}\]  
\[ \begin{array}{c}\omega_{2}(x)=e^{-ax}(A_{5}\sin(ax)+A_{6}\cos(ax))+\\ e^{ax}(A_{7}\sin(ax)+A_{8}\cos(ax))\end{array}  \tag{4}\]  
In the equation: \(A_{1}\)-\(A_{8}\) are unknown coefficients; \(\alpha=\sqrt[4]{3k\Big{(}1-v_{j}^{2}\Big{)}/E_{j}h_{j}^{3}}\). According to the boundary conditions of the model, there are the following equations:  
\[\left\{ \begin{array}{c}\omega_{2}(x)|_{x\rightarrow-\infty}=0\\ \omega_{2}^{\prime}(x)|_{x\rightarrow-\infty}=0\end{array} \right. \tag{5}\]  
\[\left\{ \begin{array}{c}\omega_{1}\Big{(}-\frac{L}{2}\Big{)}=\omega_{2}\Big{(} -\frac{L}{2}\Big{)}\\ \omega_{1}^{\prime}\Big{(}-\frac{L}{2}\Big{)}=\omega_{2}^{\prime}\Big{(}- \frac{L}{2}\Big{)}\\ \omega_{1}^{\prime}\Big{(}-\frac{L}{2}\Big{)}=\omega_{2}^{\prime\prime}\Big{(} -\frac{L}{2}\Big{)}\\ \omega_{1}^{\prime\prime}\Big{(}-\frac{L}{2}\Big{)}=\omega_{2}^{\prime\prime \prime}\Big{(}-\frac{L}{2}\Big{)}\end{array} \right. \tag{6}\]  
The rotation angle and shear force at the midline of the model are zero, so there are the following equations:  
\[\left\{ \begin{array}{c}\omega_{1}^{\prime}(0)=0\\ \omega_{1}^{\prime\prime\prime}(0)=0\end{array} \right.