## 4 Discrete Element Numerical Model and Analysis Index
### Comparative Numerical Analysis on Roof-Cutting  
Figure 12 vividly depicts the roof collapse conditions of the lower seam's working face with non-roof-cutting and roof-cutting. In the case of non-roof-cutting, the roof is laterally suspended, the overlying rock block exerts pressure on the coal pillar. However, once the roof-cutting is carried out, the roof collapses completely. As a result, the pillar is shielded from the roof's lateral pressure, and the pressure it bears is remarkably reduced.  
The illustration in Figure 13 compares the changes in displacement and stress indexes between roof-cutting and non-roof-cutting. It can be seen from the displacement stress condition that when the roof is not cut, the roof's large area is suspended above the wall, which aggravates the stress concentration. After the roof-cutting, the roof falls in time with the mining. Further analysis of the vertical stress, the maximum shear stress, and the second invariant of deviatoric stress shows that roof-cutting effectively alleviates the pillar's stress concentration.  
Figure 12: Roof-caving situation under non-roof-cutting and roof-cutting: (**a**) non-roof-cutting; (**b**) roof-cutting.  
Figure 13: Comparison of each index under non-roof-cutting and roof-cutting: (**a**) displacement; (**b**) vertical stress; (**c**) maximum shear stress; (**d**) the second invariant of deviatoric stress.  
A monitoring line is set up within the model presented in Figure 13, and the variation in each index value is acquired, as demonstrated in Figure 14.  
Through a comprehensive analysis of the cloud image in conjunction with the curve, the following conclusions are drawn: (1) The displacement curve analysis reveals that without roof-cutting, the roof subsidence gradually diminishes from the goat towards the pillar side, and an evident suspended roof structure is formed. After the roof-cutting operation is executed, the amount of roof caving from the goat to the pillar side increases significantly, and the values are relatively close. This indicates that the roof has achieved full and uniform caving, effectively preventing the coal pillar from being laterally compressed by the roof. (2) The vertical stress and shear stress curves vividly demonstrate that both the peak values and the ranges are remarkably diminished after roof-cutting. Specifically, the vertical stress's peak value dropped from 28.0 MPa to 16.46 MPa, and the maximum shear stress's peak value decreased from 10.50 MPa to 5.93 MPa. This clearly indicates that roof-cutting weakens the roof's stress-transfer path and effectively releases the stress concentration in the coal pillar. (3) Post roof-cutting, the concentration range of \(J_{2}\) is evidently reduced. Moreover, its peak value decreased from 2.71 \(\times\) 10\({}^{14}\) Pa\({}^{2}\) to 7.23 \(\times\) 10\({}^{13}\) Pa\({}^{2}\). This implies that the degree of the pillar's plastic deformation is alleviated, and the pillar's bearing capacity is further safeguarded.  
Figure 14: Monitoring value of each index under non-roof-cutting and roof-cutting: (**a**) displacement; (**b**) vertical stress; (**c**) maximum shear stress; (**d**) the second invariant of deviatoric stress.