## 4 Construction of the Dynamic and Static Abutment Pressure Calculation Model
### Static Abutment Pressure Calculation Model  
Before and after the goaf is formed, the load of the roof strata to be borne by the coal seam is constant. As shown in Figure 8, before the goaf is formed, the load of the roof strata is evenly distributed on the coal seam; after the goaf is formed, the load of the roof strata will be transferred to the coal body outside the goaf and the gangue in the goaf.  
In the formula, \(\Delta\sigma\) is the common sum (MPa) of the stress increment formed by the load of the overhanging part of each key stratum of the stop or the load of the broken block segment transferred to the coal body on one side, namely, \(\Delta\sigma=\sum\Delta\sigma_{i}\); \(\Delta\sigma_{i}\) is the stress increment (MPa) caused by the load transferred to the coal body on one side by the overhanging part or the broken block section of the i th key stratum, \(i=1\)-m.  
(1) Gravity stress  
The critical stratum in roof strata is the medium of load transfer in roof strata. The static abutment pressure calculation model can be established according to the critical stratum theory and the load transfer characteristics of roof strata, as shown in Figure 9. Before the critical stratum breaks, half of the load in the overhanging part is transferred to the coal body on one side of the goaf.  
Let the load of the hanging part of key stratum i be \(q_{i}\), then the load transferred to the coal body on one side of the goaf is \(Q_{i}=q_{i}/2\). \(q_{i}\) is calculated as shown in Equation (2).  
Suppose that the distance from the ith key stratum to the coal seam is \(H_{i}\), and the rock formation controlled by it has \(m_{i}\) rock layers, denoted as \(c_{i\parallel j}(j=1,2,\ldots,m_{i})\) from bottom to top, then  
\[q_{i}=\gamma\left[2\left(H_{i}+\frac{1}{2}\sum_{j=1}^{m_{i}}h_{i\parallel j} \right)\cot\theta+I\right]\sum_{j=1}^{m_{i}}h_{i\parallel j}, \tag{2}\]  
where, \(h_{i\parallel j}\) is the thickness of rock layer (m), \(I\) is the width of working face (m), \(\theta\) is the fracture angle, and \(\gamma\) is the bulk density.  
After the key stratum is broken, half of the load of the broken block section is transferred to the coal body on one side of the goaf. Assuming that the length of the fault block section of the \(i\) th key stratum is \(L_{i}\) and the load is \(q_{i}^{\prime}\), then the load \(Q_{i}=q_{i}^{\prime}/2\) transferred to the coal body on one side of the goaf is transferred. \(q_{i}^{\prime}\) is calculated as follows:  
\[q_{i}^{\prime}=\gamma\left(L_{i}+\frac{1}{2}\cot\theta\sum_{j=1}^{m_{i}}h_{i \parallel j}\right)\sum_{j=1}^{m_{i}}h_{i\parallel j}, \tag{3}\]  
(2) Stress increment  
When there are multiple key layers in the stope roof strata, the stress increment generated by each key layer is superimposed, as shown in Equation (4), and \(\Delta\sigma\) is obtained. The formula for calculating \(\sigma_{q}\) is:  
\[\sigma_{q}=\begin{cases}\gamma I&(x=0\to I\cot\theta)\\ \gamma x\tan\theta&(x=I\cot\theta H\cot\theta),\\ \gamma H&(x=H\cot\theta\infty)\end{cases} \tag{4}\]  
Figure 9: Static abutment pressure calculation model [45]. (**a**) Calculation model of the load on the overhanging part of the critical layer before breaking. (**b**) Load calculation model of the critical layer fault block segment.  
It can be seen that the calculation formula of static abutment pressure is a group of piecework functions. The distribution characteristics of static abutment pressure are related to the fracture angle, the number of key layers, the position of key layers, the thickness of each rock group, the length of the broken block, and the buried depth.