## 3 Theoretical Analysis of Rational Layout Pattern of Small Panel and Large Coal Pillar
### Analysis of the Width of Panel Isolation Coal Pillar  
The load borne by the isolated coal pillar comes from two sources. One is the self-weight of the overlying rock layer above the coal pillar, and the other is the transfer stress of the overlying rock layer on both sides of the goat. Based on this, an analysis model for the force on the panel-isolated coal pillar is established, as shown in Figure 5. In Figure 5, \(L_{1}\) represents the goat width of the third panel, \(L_{2}\) represents the goat width of the second panel, \(D\) represents the width of the panel-isolated coal pillar, and \(H\) represents the burial depth of the two panels.  
From Figure 5, it can be observed that the load borne by the panel-isolated coal pillar \(Q_{t}\) includes the self-weight of the coal pillar \(Q_{m}\), the load transmitted by the overlying rock layer of the goat in the third panel \(Q_{1}\), and the load transmitted by the overlying rock layer of the goat in the second panel \(Q_{2}\), namely  
\[Q_{t}=Q_{m}+Q_{1}+Q_{2} \tag{1}\]  
Based on the geometric relationship, \(Q_{t}\), \(Q_{1}\), and \(Q_{2}\) can be expressed as follows:  
\[Q_{m}=\gamma HD \tag{2}\]  
\[Q_{1}=\frac{\gamma}{2}\left(HL_{1}-\frac{\pi{L_{1}}^{2}}{8}\right) \tag{3}\]  
\[Q_{2}=\frac{\gamma}{2}\left(HL_{2}-\frac{\pi{L_{2}}^{2}}{8}\right) \tag{4}\]  
In the equation, \(\gamma\) represents the bulk density of overlying rock strata.  
As shown in Figure 6, in the layout pattern of a small panel and a large coal pillar, the stress distribution in the elastic area of the panel-isolated coal pillar is uniform. When the bearing stress of the panel-isolated coal pillar approaches bearing capacity, the coal pillars tend to reach a critical state of high stress instability. Under mining disturbance, this can easily induce rockburst. The calculation of the bearing stress in the elastic zone of the panel-isolated coal pillars is as follows:  
\[\sigma_{a}=\frac{Q_{t}-2Q_{s}\sigma_{s}}{D_{t}} \tag{5}\]  
Figure 5: Stress analysis model of panel-isolated coal pillar.  
In the equation, \(\sigma_{a}\) represents the bearing stress in the elastic area; \(D_{\rm t}\) represents the width of the elastic area, \(D_{\rm t}=D-2D_{\rm s}\); \(D_{\rm s}\) represents the width of the plastic area; \(\sigma_{\rm s}\) represents the bearing stress in the plastic area, which is equal to the uniaxial compressive strength of coal.  
The ratio of the bearing stress in the elastic area of the panel-isolated coal pillar to its bearing capacity is given by:  
\[\mu=\frac{\sigma_{a}}{\sigma_{\rm 3c}} \tag{6}\]  
In the equation, \(\sigma_{\rm 3c}\) represents the bearing capacity of the coal, taken as three times the uniaxial compressive strength of coal.  
Based on the actual conditions of the mine, \(\gamma=25\) kN/m\({}^{3}\), \(H=950\) m, \(L_{1}=550\) m, \(L_{2}=900\) m, \(D=200\) m, \(D_{\rm s}=20\) m, the uniaxial compressive strength of coal \(\sigma_{\rm c}=18\) MPa. Substituting these parameters into Equations (1) to (6), \(\mu=1.77>1\). If \(\mu=1\), \(D=257\) m can be obtained under the condition that the inclined width of the goat on both sides of the panel-isolated coal pillar is constant. Therefore, under the current layout pattern, there is a risk of rockburst induced by instability of the panel-isolated coal pillar, and the critical width of instability is 257 m.