## 3 Theoretical Analysis of Rational Layout Pattern of Small Panel and Large Coal Pillar
### Analysis of Mining Scale in Panel  
Under the action of the self-weight of the overlying rock strata and the stress transmitted from the overlying rock strata to both sides of the goat, the panel-isolated coal pillar is compressed. The main key strata were flexed under the load of the overlying strata. As the scale of the goat on both sides of the panel-isolated coal pillar increases, the transmitted stress from the goat on both sides also increases, leading to an increase in the compression of the panel-isolated coal pillar. When the compression of the coal pillar exceeds the deflection limit of the main key strata, the main key strata and the lower strata are separated. At this point, the central part of the main key strata is unsupported, and the thick hard sandstone main key layer is prone to fracturing, triggering large-energy microseismic events and rockburst. Based on the mechanical characteristics of the key strata, the main key strata after delamination from the lower rock strata can be considered as a suspended beam fixed at both ends. The stress from the overlying rock layer can be simplified as a uniformly distributed load, with a concentrated reaction force acting on both sides of the beam. Taking the center of the left end of the suspended beam as the origin, the direction along the axis of the suspended beam as the \(x\)-axis, and the direction along the thickness of the suspended beam as the \(y\)-axis, an analysis model of the flexural deformation of the main key strata is established, as shown in Figure 7.  
In Figure 7, \(q_{0}\) is the uniformly distributed load of the overlying rock above the main key strata (including the self-weight of the main key strata), \(h\) is the thickness of the main key strata, \(l_{1}\) is the width of the third panel, \(l_{2}\) is the width of the second panel, \(d\) is the width of the panel-isolated coal pillar, \(F\) is the reaction force at the fixed support end of the main key strata, and \(M\) is the bending moment at the fixed support ends of the main key strata. The ultimate span \(l\) of the key strata can be expressed as follows:  
\[l=h\sqrt{\frac{\delta}{q_{0}}+1+\frac{5}{4}\mu} \tag{7}\]  
Figure 6: Analysis model of bearing stress in elastic zone of panel-isolated coal pillar. In the equation, \(\delta\) is the uniaxial tensile strength of the main key strata, \(\mu\) is the Poisson's ratio of the main key strata. Simultaneously, after the separation of the main key strata and the lower rock layer, the deflection equation of the beam axis is given by:  
\[z=\frac{q_{0}x^{3}}{2 Eh^{3}}(2l+x) \tag{8}\]  
Under the condition of fixed support at both ends, the maximum deflection occurs at the midpoint of the beam, that is, \(x=l/2\). Therefore, the maximum deflection of the beam is given by:  
\[\omega_{max}=z_{(x=l/2)}=\frac{5q_{0}t^{4}}{32 Eh^{3}} \tag{9}\]  
Before mining the second and third panels, the panel-isolated coal pillar was only affected by the original overlying rock stress. Based on the stress analysis model of the coal pillar in Figure 4, the compression amount \(\
u\) of the isolated coal pillar during mining can be calculated as follows:  
\[v=em=\frac{Q_{\rm t}-Q_{0}}{E_{0}d_{0}}m \tag{10}\]  
In the equations, \(m\) represents the mining height, \(\epsilon\) represents the strain of the isolated coal pillar, \(E_{0}\) is the elastic modulus of the coal, \(d_{0}\) is the elastic area width of the panel-isolated coal pillar. According to the actual conditions of the mine, \(h=200\) m, \(\delta=2.01\) MPa, \(\mu=0.24\), \(E=8\times 10^{3}\) MPa, \(E_{0}=2.67\times 10^{3}\) MPa, \(d_{0}=160\) m, the uniformly distributed load above the key strata \(q_{0}=12.50\) MPa. Substituting these values into the calculations, the limit span \(l\) of the key strata is determined to be 241 m, which is greater than \(D\). The maximum deflection of the main key layer \(\omega_{max}\) is 0.104, before mining in the second and third panels, the initial load on the isolated coal pillar \(Q_{0}=\gamma Hd\).