## 4 Effect Analysis of Roof Self-Stability Factors
### Effect Analysis of Roof Self-Stability Factors along Dip  
1. Preliminary setting of parameters  
It is denoted that coal-body length \(l_{1}=50\) m and \(l_{6}=50\) m at the left and right ends along dip; stress concentration coefficient \(K_{3}=1\) in the middle of the coal pillar. The stress concentration coefficient of the coal pillar is affected by the working-face length, coal-pillar length, and elastic-foundation coefficient of the backfilling body. Figure 5: Relationship among the unsupported-roof distances, roof subsidence, and bending moments. Figure 6: Relationship among advancing distances of the working face, roof subsidence, and bending moments. The functional relationship between the stress concentration factor and backfilling working face is expressed as Equation (23) when the coal-pillar length is 100 m, and the working-face length is within a certain range [23]. \[K=ae^{bl} \tag{23}\]  
where \(K\) is the stress concentration coefficient of the coal pillar; \(l\) the length of the backfilling working face, m; \(a\) and \(b\) are coefficients. When lengths of the backfilling working face \(l_{2}\) = 87 m and \(l_{5}\) = 87 m, the coal-pillar length and stress concentration coefficient are fitted [24] (see Figure 7). Other parameters along the dip are consistent with those along the strike. The effect law of the coal-pillar length and working face length on roof stability is analyzed along dip. (2) Effect analysis of working face size  
The working face length is set to be 50, 60, and 70 m, respectively. Figure 8 shows the correlation curves among the working face length, roof deflection, and bending moment through mechanical calculations. An M-shaped curve relationship exists between the working face length and roof subsidence along the dip of coal seams. Roof subsidence increases with the increased working-face length, and the increase rate becomes smaller. When the working-face length increases from 50 to 70 m, maximum roof subsidence is increased by 6.82% (from 0.22 to  
Figure 8: Relationship among the working face lengths, roof subsidence, and bending moments. Figure 7: Relationship between coal-pillar lengths and stress concentration coefficients. 0.235 m). Roof subsidence is restrained by adopting dense filling and retaining a long coal pillar between backfilling working faces. The roof bending moment curve above the goat is transformed from M-shaped to parabolic with increased working face length, and the maximum bending moment gradually increases. When the working-face length increases from 50 to 70 m, the maximum roof bending moment increases by 41.86% (from 29.93 to 42.46 MN\(\cdot\)m). The working face length impacts roof failure. In the process of working face mining, the increase of face length causes the increase of roof exposed in the goat area, which is not conducive to the formation of self-stabilizing structure of roof. In order to improve the stability of roof, the length of working face can be appropriately reduced. Effect analysis of section coal-pillar size  
The section coal-pillar length is set to be 60, 70, and 80 m, respectively. Figure 9 shows the correlation curves among the coal-pillar length, roof deflection, and bending moment through mechanical calculations. The coal-pillar length is negatively correlated with roof subsidence. The coal pillar has no obvious effect on roof subsidence under dense filling. The overall roof stability depends on the elastic-foundation coefficient and roof load concentration. The possible failure position changes with increased coal-pillar length, and the maximum roof bending moment gradually decreases. Therefore, the coal-pillar length significantly impacts roof failure. When the coal-pillar length increases from 60 to 80 m, the roof maximum bending moment is decreased by 32.05% (from 39.91 to 27.12 MN\(\cdot\)m).