# Study on the \"Two-Zone\" Heights in Lower Slice Mining Under Thick Al Alluvium and Thin Bedrock
## 3 Analysis of the Development Height of 'Two Zones' of Overburden Rock
### Analysis of the Failure Transfer Mechanical Model of 'Two Zones' in Overburden Rock  
When the initial layer of rock strata reaches its maximum \(L_{k}\) and \(L_{s}\) and fractures to create penetrating cracks due to transmission failure, it is classified as a water-conducting fracture zone. By analogy, when \(\Delta_{i,i+1}\leq 0\), the failure transfer of overlying strata ceases, and the development of transfixion water-conducting fractures stops. Currently, the distance  
Figure 5: The transfer state of overlying strata on the working face.  
Figure 6: Overburden rock failure transfer process and its stage division.  
from the first layer of rock above the coal seam to the height after the destruction of the rock layer represents the height of the water-conducting fractured zone, while the upper part of the rock layer forms the bending subsidence zone.  
Judging the development height of the water-conducting fracture zone can be transformed into analyzing whether the strata above the coal seam reach their \(L_{k\_max}\) and \(L_{s\_max}\). Two structural mechanics models of rock suspension integrity and rock suspension stability are constructed, as shown in Figure 7. The model assumes that the overlying strata load is uniformly distributed and the beam has a unit width.  
In the diagram, Fs is shear force, M is bending moment, \(G_{i}\) is the weight of the suspended part of the rock layer, \(Q_{i}\) is the weight of the suspended part of the rock layer, \(q_{i}\) is the load of the upper rock layer acting on the \(i\)-th rock layer, and \(x\) is the suspension distance of the \(i\)-th rock layer falling behind the \(i+1\)-th rock layer.  
When the \(i\)-th stratum is completely suspended, both ends of the stratum are within the rock mass. Suppose there are \(n\) layers of overlying strata on the working face, and the load set of the suspended part of the \(i\)-th layer is \(q_{i}\).  
\[q_{i}=\frac{E_{i}h_{i}^{3}(\gamma_{i}h_{i}+\gamma_{i+1}h_{i+1}+\cdots+\gamma_{ n}h_{n})}{E_{i}h_{i}^{3}+E_{i+1}h_{i+1}^{3}+E_{n}h_{n}^{3}} \tag{3}\]  
In this formula, \(E_{i}\) represents the elastic modulus of the \(i\)-th layer, and \(h_{i}\) represents the thickness of the \(i\)-th layer.  
When the maximum normal stress of the first layer of rock exceeds its ultimate tensile strength \(R_{T}\), the rock fractures. Therefore, its ultimate suspension distance can be expressed as follows:  
\[L_{\text{kimax}}=h_{i}\sqrt{\frac{2R_{T}}{q_{i}+\gamma_{i}h_{i}}} \tag{4}\]  
The rock layer breaks and collapses when the suspension distance exceeds its limit, causing water to flow through the fracture. When the \(i\)-th stratum is suspended and broken, one end is fixed in the rock mass, and the other is suspended, forming a 'cantilever beam' structure.  
\[L_{\text{simax}}=h_{i}\sqrt{\frac{2R_{T}}{3(q_{i}^{\prime}+\gamma_{i}h_{i})}} \tag{5}\]  
where \(q_{i}\) represents the load concentration of the overhanging part of the rock stratum.  
Based on the above analysis, the formula for the criteria of hanging collapse and hanging fracture of a rock stratum above the coal seam can be obtained as follows:  
\[L_{k}\geq h_{i}\sqrt{\frac{2R_{T}}{q_{i}+\gamma_{i}h_{i}}} \tag{6}\]  
\[L_{s}\geq h_{i}\sqrt{\frac{2R_{T}}{3(q_{i}^{\prime}+\gamma_{i}h_{i})}} \tag{7}\]  
The calculation model is appropriate for determining the overburden failure height in near-horizontal coal seam mining. When dealing with inclined coal seams, it is essential to take into account the impact of the coal seam dip angle on the overburden failure height.  
Figure 7: Different-state-structure rock mechanics model.