## 4 Discussion  
### Mechanism of the \"Square effect\" of Thick-Hard Roof  
In longwall mining, the thick-hard roof usually has a large caving interval according to the key strata key theory, and when the weighting interval equals the length of the working face, it means that the gob is a square at this time, the underground pressure reaches its maximum, causing severe ground pressure behavior even coal burst sometimes, called the \"square effect\" [38]. Actually, the \"square effect\" is the result of the breaking and movement of a thick-hard roof. We can use the plate model to explain this phenomenon, as shown in Figure 2. It is known that most of the overlying coal seams are sedimentary strata that are characterized by obvious layers and weak planes, which separate easily under shearing stress; therefore, even the thick-hard roof can be regarded as a thin plate to analyze the stress distribution under different boundary conditions, as shown in Figure 9,  
Figure 8: Design of the deep-hole blasting of the hard–thick roof. (**a**) Plane sketch of deep-hole blasting program. (**b**) Profile layout of deep-hole blasting borehole.  
and we can obtain the maximum normal stress both on the strike (\(x\)-axis,) and (\(y\)-axis) dip direction, as Equation (4).  
\[\left\{\begin{array}{l}\sigma_{\text{xmax}}=\frac{12 uqa^{4}b^{2}}{\pi^{2}h^{2} \left(3a^{4}+2a^{2}b^{2}+3b^{4}\right)},x=0,a;y=b/2\\ \sigma_{\text{ymax}}=\frac{12 uqa^{2}b^{4}}{\pi^{2}h^{2}\left(3a^{4}+2a^{2}b^{2 }+3b^{4}\right)},x=a/2;y=0,b\end{array}\right. \tag{4}\]  
where \(\mu\) is the Poisson's ratio; \(q\) is the overlying load on the roof; \(a\) and \(b\) are the overhang length and width of the roof, can be simplified as the advancing distance and width of the working face, respectively, and \(h\) is the thickness of the roof plate.  
Based on Equation (3), at the early stage of mining excavation, that is \(a\leq b\), \(\sigma_{x\text{max}}\) is always greater than \(\sigma_{\text{ymax}}\); therefore, \(\sigma_{x\text{max}}\) plays a vital role in the breaking of the roof. Although the thickness of the thick-hard roof can be tens of meters, the fracture and cave of the roof under longwall mining is usually in layers, and statistics showed that when the thickness of a single layer is more than 5 m, it can be referred to as hard-collapsed roof and would form a large area of a suspended roof, so we assumed the plate thickness h = 5 m in the simplified model. Assuming \(\mu\)= 0.25, \(h\)= 5.0 m, we can obtain a trend graph of \(\sigma_{x\text{max}}/q\) with the advancing of mining in different widths of working face, as shown in Figures 10 and 11. Figure 10 shows that the normal stress always reaches the maximum stress at the midpoint of the long side first under uniform load and has a positive correlation with the length of the working face. Figure 11 shows the relation of the \(\sigma_{x\text{max}}/q\) with the advancing distance of that \(\sigma_{x\text{max}}/q\) increased with the advancing distance rapidly to its maximum when \(a=b\), which means that the roof plate was a square now, and at this moment, the roof was most likely to break and cave causing much more violent ground pressure behavior than usual, that was figuratively called the \"square effect\" by the mining engineers in China. Equation (4) and Figure 11 explain this phenomenon by a simplified mechanical model.  
Figure 10: Normal stress of \(x\)-axis with the advancing of mining in different widths of working face.  
Figure 9: Elastic thin plate model of the hard roof.