## 3 Calculation of Coal Pillar Width of Gob-Side Entry Driving
### Study on the Reasonable Coal Pillar Width  
After the recovery of the working face in the upper section, the gob-edge immediate roof collapses, and the main roof rotates. The gangue supports the rock block \(B\) of the main roof; the rock block \(B\) and adjacent blocks squeeze each other to form a relatively stable structure. The fracture of the main roof divides the lateral abutment pressure into the obvious internal stress field S\({}_{1}\) (stress falling area) and the external stress field S\({}_{2}\) (stress rising area). The location of the main roof fracture is the boundary of the two stress fields (Figure 5a). To provide a good stress environment for the roadway and to facilitate roadway maintenance, roadways are generally arranged in the stress falling area in coal pillar width selection ([PERSON] et al., 1986 [25]). In the internal stress field, the abutment pressure \(x\) m away from the coal wall can be expressed as:  
\[\sigma_{x}=G_{x}y_{x} \tag{6}\]  
where \(G_{x}\) is the stiffness of the coal body \(x\) m away from the coal wall, Pa; \(y_{x}\) is the deformation of the coal body \(x\) away from the coal wall, m.  
Figure 5: Mechanical model. (**a**) Distribution diagram of the internal and external stress fields. (**b**) Breaking structure model of the main roof. As the abutment pressure develops into the deep part of the coal wall, the stiffness of the coal body gradually rises while the compression gradually decreases. This is simplified as a process of linear variation. According to the geometric relationship, then:  
\[y_{x}=\frac{y_{0}}{x_{0}}(x_{0}-x),G_{x}=\frac{G_{0}}{x_{0}}x \tag{7}\]  
where \(G_{0}\) is the maximum stiffness of the coal body within the internal stress field, Pa; \(y_{0}\) is the compression of the gob-edge coal body, m; \(x_{0}\) is the distribution range of the internal stress field, m. According to Equations (6) and (7), the abutment pressure of the internal stress field is  
\[\int_{0}^{x_{0}}\sigma_{v}dx=\frac{G_{0}y_{0}x_{0}}{6} \tag{8}\]  
According to the theory of internal and external stress fields, the abutment pressure within the internal stress field on the coal body around the slope is equal to the self-weight of the main roof strata during the first weighting of the working face; thus,  
\[\frac{G_{0}y_{0}x_{0}}{6}=abC_{0}\gamma \tag{9}\]  
where \(a\) is the inclined length of the working face, m; \(C_{0}\) is the first weighting step distance of the adjacent working face, m; \(b\) is the thickness of the main roof strata, m; and \(\gamma\) is the bulk density of the main roof rock, kN/m\({}^{3}\). According to Figure 5b, then:  
\[\frac{y_{0}}{x_{0}}=\frac{\Delta h}{L}=\frac{M-h(K_{c}-1)}{L} \tag{10}\]  
\[y_{0}=\frac{x_{0}}{L}[M-h(K_{c}-1)] \tag{11}\]  
where \(M\) is the mining thickness of the coal seam, m; \(h\) is the thickness of the immediate roof rock strata, m; \(\Delta h\) is the maximum subsidence of the key block, m; \(L\) is the span of the hinged rock beams, which is nearly equal to the periodic weighting step distance of the fully mechanized caving face, m; and \(K_{c}\) is the dilatancy coefficient of the immediate roof rock. According to the inclusion theory, the stiffness of the coal body \(G_{0}\) ([PERSON] et al., 2016 [26]) in the plastic state is  
\[G_{0}=\frac{E}{2(1+v)\xi} \tag{12}\]  
where \(E\) is the elastic modulus of the coal body, \(Pa\); \(v\) is Poisson's ratio; and \(\xi\) is the influence coefficient, which is related to the development of fractures in the coal body. The range of the internal stress field \(x_{0}\) can be obtained based on Equations (9)-(12):  
\[x_{0}=\sqrt{\frac{12 abC_{0}\gamma\xi L(1+v)}{E[M-h(K_{c}-1)]}} \tag{13}\]  
According to geological data from the mine and on-site measurements, \(a=120\) m, \(b=12\) m, \(C_{0}=35\) m, \(\gamma=27\) kN/m\({}^{3}\), \(\xi=0.85\), \(L=16\) m, \(v=0.3\), \(E=24\) GPa, \(M=3\) m, \(h=7\) m, and \(K_{c}=1.32\).