## 1 Introduction
### Single Key Layer Category 2 Direct Top
#### 3.2.2 Instance Validation  
For the Daliuta 21305 working face mine, regarding the 1-2 coal seam, by arranging drill holes in the working face and the surrounding areas of the coal mine, detailed information regarding the coal and rock seams is obtained, and a comprehensive bar chart of the working face is drawn. The average mining height is 4.3 m, the dip angle is \(0^{0}\sim 5^{0}\), the burial depth is 120-130 m, the upper 1-2 coal has been mined, the average mining height is 3.2 m, the average spacing of the layers is 20 m, and the inter-mining ratio is \(G=4.7\). There is only one key stratum above the coal seam, and the thickness of the key stratum is 8.4 m. Under the key stratum is the immediate roof, and the thickness of the immediate roof is 11.7 m. There is a thick rock layer with high strength and undeveloped fissures in the immediate roof, which conforms to the second type of immediate roof structure of a single key layer. The comprehensive histogram of the working face is shown in Figure 8.  
According to the comprehensive column diagram of the Daliuta 21305 working face, the mining conditions of the working face, and the breaking characteristics of the overlying rocks of the shallow buried coal seam, it can be seen that the support efficiency \(\mu=0.9\), the width of the support \(b=1.75\) m, the distance of the top control \(l_{k}=5.0\) m, the depth of the flake gangs \(l_{p}=0.9\) m, the cycle of the coming pressure step \(l=11\) m, and the length of the non-suspended top \(l_{1}=2.5\) m. The suspended roof length \(l_{2}=4.5\) m, the direct top hard layer below the rock layer height \(ha=1.4\) m, the direct top hard layer height \(hb=3.7\) m, the direct top hard layer above the rock layer height \(hc=6.6\) m, the direct top capacity \(\gamma=23\) kN\(\cdot\)m\({}^{-3}\), the step rock beam key block length \(L_{1}=10\) m, the key layer thickness \(h=8.4\) m, the step rock beam block degree \(i=0.84\), the step rock beam M block turning angle \(\theta_{1}=3^{\circ}\), the maximum turning angle \(\theta_{1}max=6^{\circ}\), the mining height \(m2=4.3\) m, the breaking angle \(\alpha=75^{\circ}\), the height of loading layer \(h_{3}=1.0\) m, the bulk weight \(\rho_{1}g=\rho g=25\) kN\(\cdot\)m\({}^{-3}\), and the mean spreading load \(q=559\) kN\(\cdot\)m\({}^{-2}\) of the collapsed roof plate of the upper coal seam.  
According to Equation (22), when the depth of the sheet gang \(l_{p}=0\) m,  
\[P_{\rm 2m}=10,318\ {\rm kN} \tag{23}\]  
When the depth of the sheet gang \(l_{p}=0.9\) m,  
\[P_{\rm 2m}=11,349\ {\rm kN} \tag{24}\]  
Figure 8: Comprehensive histogram of the Daliuta 21305 working face.  
According to the mining practice of the 21305 working face, the rated working resistance of the stent chosen for the working face is 12,000 \(\mathrm{kN}/\mathrm{stent}\), and the working resistance of the stent during the period of coming pressure is at least 10,446 kN and at most 11,992 kN, which is larger than the rated working resistance of the stent. The average load is 11,219 kN. The stent has better adaptability, and the theoretical calculations are in line with the results of engineering practice.