## 4 Discussion
### Energy Calculation Model for Hard Roof Considering Mining Speed and Analysis of Influencing Factors
Substituting the above relationship into Equation (3), the maximum tensile stress in the rock beam can be expressed as:  
\[\sigma_{\text{max}}=\frac{q\left(V_{\text{m}}t_{\text{m}}\right)^{2}}{2h^{2}} =\frac{q\left(\beta V_{\text{p}}t_{\text{m}}\right)^{2}}{2h^{2}} \tag{5}\]  
To find the partial derivative of \(\sigma_{\text{max}}\) with respect to time \(t_{\text{m}}\), we can derive:  
\[\overset{\bullet}{\sigma}_{\text{max}}=\frac{\partial\sigma_{\text{max}}}{ \partial t_{\text{m}}}=\frac{\partial\sigma_{\text{max}}}{\partial d}\frac{ \partial d}{\partial t_{\text{m}}}=\frac{\beta qd}{h^{2}}V_{\text{p}} \tag{6}\]  
Equation (6) shows a linear positive correlation between the change rate of maximum tensile stress in the hard roof and mining speed, indicating that under a specific roof span, a higher mining speed results in a greater change rate of maximum tensile stress. Research indicates that the tensile strength of coal series sandstone and the change rate of tensile stress approximately exhibit a linear relationship [27]:  
\[R_{\text{T}}=a\overset{\bullet}{\sigma}+b \tag{7}\]  
Figure 9: Stress analysis model of roof during initial fracture. where \(\overset{\bullet}{\sigma}\) represents the change rate of tensile stress, \(a\) and \(b\) are constants related to the rock type, which, based on previous laboratory results, are taken to be 0.10151 and 7.70138, respectively. Based on Equations (6) and (7), the tensile strength can be expressed as:  
\[R_{\mathrm{T}}=\frac{a\beta qd}{h^{2}}V_{\mathrm{P}}+b \tag{8}\]  
Based on knowledge from material mechanics, the maximum span of a rock beam with fixed supports at both ends at the time of initial fracture is given by \(d_{\mathrm{max}}=h\left(2R_{\mathrm{T}}/q\right)^{1/2}\). By combining Equation (8) to eliminate \(R_{\mathrm{T}}\) and then rearranging, we can obtain a quadratic equation in terms of \(d_{\mathrm{max}}\):  
\[d_{\mathrm{max}}^{2}-2a\beta V_{\mathrm{P}}d_{\mathrm{max}}-\frac{2 bh^{2}}{q}=0 \tag{9}\]  
To find the real roots of this equation, we first calculate the discriminant \(\Delta\) and find that \(\Delta>0\), indicating that the equation has two distinct real roots. Solving the equation, we can obtain the two real roots as follows:  
\[d_{\mathrm{max}}=\left\{ \begin{array}{l}a\beta V_{\mathrm{P}}+\sqrt{a^{2} \beta^{2}V_{\mathrm{P}}^{2}+\frac{2 bh^{2}}{q}}\\ \mathrm{or}\\ a\beta V_{\mathrm{P}}-\sqrt{a^{2}\beta^{2}V_{\mathrm{P}}^{2}+\frac{2 bh^{2}}{q }}\end{array} \right. \tag{10}\]  
Clearly, the second root being negative is not practically meaningful, so we analyze the first root with a positive sign. According to Prof. [PERSON]'s viewpoint, the elastic energy of the coal-rock layer system primarily consists of the volumetric elastic energy of the coal (\(U_{V}\)), the deformation elastic energy of the coal (\(U_{\mathrm{I}}\)), and the bending elastic energy of the roof (\(U_{\mathrm{W}}\)) [26]. The bending elastic energy of the roof at the time of initial fracture is:  
\[U_{\mathrm{w}}=\frac{q^{2}d^{5}}{576 EJ_{z}} \tag{11}\]  
By substituting the positive root from Equation (10) into Equation (11), and incorporating the values of the coefficients \(a\), \(b\), and \(\beta\), we can derive an equation as follows:  
\[U_{\mathrm{w}}=\frac{q^{2}}{48 Eh^{3}}(19.4\cdot V_{\mathrm{P}}+\sqrt{376.36 \cdot V_{\mathrm{P}}^{2}+\frac{15.4\cdot h^{2}}{q}})^{5} \tag{12}\]  
which describes the relationship between the accumulated bending elastic energy at the limit span during the initial fracture process of the roof and the mining speed of the working face.