## 2 Theoretical Analysis
### Hydraulic-Mechanical (HM) Model for Seam Floors
The vertical force (\(F\)) can be decomposed into a normal force (\(F_{\mathrm{n}}\)) and a tangential force (\(F_{\mathrm{t}}\)) on the semi-infinite boundary. According to the principle of elasticity [39], the stress components induced by normal or tangential loads are given in Equations (2) and (3), respectively. \[\left. \begin{array}{rcl}\sigma_{z}^{(l)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{ \beta_{i+1}}F_{\mathrm{n}}(x)\cos^{2}\beta\mathrm{d}\beta\\ \sigma_{x}^{(i)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{\beta_{i+1}}F_{\mathrm{n}}( x)\sin^{2}\beta\mathrm{d}\beta\\ \tau_{zx}^{(i)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{\beta_{i+1}}F_{\mathrm{n}}( x)\sin\beta\cos\beta\mathrm{d}\beta\end{array} \right\} \tag{2}\]  
\[\left. \begin{array}{rcl}\sigma_{zt}^{(i)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{ \beta_{i+1}}F_{\mathrm{t}}(x)\sin\beta\cos\beta\mathrm{d}\beta\\ \sigma_{xt}^{(i)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{\beta_{i+1}}F_{\mathrm{t} }(x)\frac{\sin^{3}\beta}{\cos\beta}\mathrm{d}\beta\\ \tau_{zx}^{(i)}&=&-\frac{2}{\pi}\int_{\beta_{i}}^{\beta_{i+1}}F_{\mathrm{t} }(x)\sin^{2}\beta\mathrm{d}\beta\end{array} \right\} \tag{3}\]  
The total stress components of \(G(x,z)\) can then be expressed by Equation (4). \[\left. \begin{array}{rcl}\sigma_{z}&=&\sum\limits_{i=1}^{\mathrm{n}}\left( \sigma_{z}^{(i)}+\sigma_{zt}^{(i)}\right)\\ \sigma_{x}&=&\sum\limits_{i=1}^{\mathrm{n}}\left(\sigma_{x}^{(i)}+\sigma_{xt }^{(i)}\right)\\ \tau_{zx}&=&\sum\limits_{i=1}^{\mathrm{n}}\left(\tau_{zx}^{(i)}+\tau_{zxt}^{( i)}\right)\end{array} \right\} \tag{4}\]  
In Equation (4), \(\sigma_{z}\), \(\sigma_{x}\) and \(\tau_{zx}\) represent vertical stress, horizontal stress and shear stress, respectively. According to the previous literature [16; 40; 41], the floor rock mass in underground engineering is indicated to shear deformation after excavation. To our general knowledge, the Mohr-Coulomb criterion is suitable for characterizing the yield failure of rocks in terms of engineering stability, with the most generalized form represented by Equation (5):  
\[\sigma_{1}\ =\ \frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}+\frac{2c\cos \varphi}{1-\sin\varphi} \tag{5}\]  
where the maximum principal stress \(\sigma_{1}\) and the minimum principal stress \(\sigma_{3}\) can be defined as Equation (6) [42]:  
\[\left. \begin{array}{rcl}\sigma_{1}&=&\frac{\sigma_{z}+\sigma_{x}}{2}\pm \sqrt{\left(\frac{\sigma_{z}-\sigma_{x}}{2}\right)^{2}+\tau_{zx}^{2}}\end{array} \right. \tag{6}\]  
Consequently, Equation (7) can be used as a discriminant function to identify the vertical range of the damage zone:  
Figure 4: Decomposition of vertical load acting on the upper boundary in Figure 3b. \[F(x,\ z)\ =\ \frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}+\frac{2c\cos\varphi}{1- \sin\varphi}-\sigma_{1} \tag{7}\]  
where \(\varphi\) denotes the frictional angle and \(c\) indicates the cohesive strength. Values of \(F(x,z)<0\) and \(F(x,z)\geq 0\) represent a failure and non-failure of the point \(G(x,z)\) of floor strata, respectively. The configuration parameters of the models used in the present study and shown in Figure 3 were based on the geological records and routine monitoring of the Huafeng coal mine. The coal seam is exploited by longwall mining on the strike, and the full caving method is suitable for roof management. The suggested parameters of the models are: \(k_{1}=1.9\), \(k_{2}=2.2\), \(k_{3}=2.1\), \(L=100\) m, \(\alpha=30^{\circ}\), \(H=450\) m, \(H_{\text{m}}=28\) m, \(\rho=1000\) kg\(\cdot\)m\({}^{-3}\), \(\gamma=25\) kN\(\cdot\)m\({}^{-3}\), g = 10 N\(\cdot\)kg \({}^{-1}\), \(P_{0}=3.2\) MPa, \(\varphi=28^{\circ}\), \(c=2\) MPa, \(l_{1}=20\) m, \(l_{2}=10\) m, \(l_{3}=7\) m, \(l_{4}=15\) m, \(l^{\prime}_{1}=l^{\prime}_{4}=18\) m and \(l^{\prime}_{2}=l^{\prime}_{3}=9\) m. As shown in Figure 5, the theoretical failure ranges of the horizontal and inclined floor strata resulting from mining and hydraulic pressures can be calculated by substituting the relevant parameters into Equations (1)-(7). The failure boundaries are calibrated with yellow marks.