## 3 Mathematical Model Construction Based on Theoretical Mechanics Methods
### Analysis of Mechanical Properties of Roof and Floor  
The mechanical properties of the roof and floor are analyzed to obtain the minimum support strength required for the TSR to support the roof stably and the maximum allowable value of the specific pressure between the base and the floor. To ensure that the roof does not fracture, the support strength of the TSR must ensure that the deformation of the roof is maintained within a certain range, ensuring that the tensile stress of the roof does not exceed the tensile strength limit of the roof. The thin plate theory [27] is used to study the support strength. Assume that the roof is a continuous layered elastic medium, the roof rock mass is an isotropic homogeneous body, and there is no tie force between the roof layers. The roof model of the section coal pillar excavation and mining working face is shown in Figure 7. The TSR is responsible for supporting the empty top area; the two sides of the empty top area are sidewalls with a length of \(a\) (m), the area to be excavated is located in front of the direct roof with a width of \(b\) (m), the thickness of the roof is \(\delta\) (m), and the plane of the coordinate axis \(z=0\) is located at \(\delta/2\) (m) of the roof thickness. The anchoring area comprises anchor bolts and bolt nets behind the roof. Therefore, the boundary conditions of the roof can be regarded as three sides with a clamped edge and one side with a simply supported edge. The boundary conditions of the clamped edge OC, OD, and DE are as follows [28]:  
\[\left\{ \begin{array}{l}\left(w\right)_{y=0}=\left(w\right)_{x=0}=\left(w \right)_{y=b}=0\\ \left(\frac{2\alpha y}{\alpha y}\right)_{y=0}=\left(\frac{\alpha w}{\alpha y} \right)_{x=0}=\left(\frac{\alpha w}{\alpha y}\right)_{y=b}=0\\ \end{array} \right., \tag{15}\]where \(w\) is the deflection equation of the roof. The boundary conditions of the simply supported edge CE are as follows:  
\[\left\{ \begin{array}{l}(w)_{x=a}=0\\ \left(\frac{\partial^{2}w}{\partial y}\right)_{x=a}=0\end{array} \right.. \tag{16}\]  
Based on the assumption, after the thin plate is deformed, the straight line originally perpendicular to the middle plane remains straight, and the length remains unchanged and is always perpendicular to the deformed middle plane. The strain in the thickness direction can be neglected, and the deflection w is only related to the plane coordinates \((x,y)\) and has nothing to do with the thickness direction coordinate \(z\). The stress state of the thin plate can be approximately regarded as a plane stress state; that is, the normal stress and shear stress perpendicular to the middle plane are much smaller than the normal stress and shear stress in the middle plane and can be ignored. The deformation of the middle plane of the thin plate is very small, and it can be considered that there is no tensile or compressive deformation of the middle plane. The external loading on the roof is \(q\). According to the thin plate theory, the differential equation representing the deflection \(w\) of the roof can be obtained as follows:  
\[\left\{ \begin{array}{l}\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial ^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}= \frac{q}{D}\\ D=\frac{\varepsilon\delta^{2}}{12(1-\mu^{2})}\end{array} \right., \tag{17}\]  
where \(D\) is the bending stiffness of the roof, N/m; \(E\) is the elastic modulus of the roof, MPa; and \(\mu\) is the Poisson's ratio. Then, the tensile stresses \(\sigma_{x}\) and \(\sigma_{y}\) in the \(x\) and \(y\) directions of the roof are as follows:  
\[\left\{