## 3 Theoretical Calculation of the Narrow Coal Pillar Width for Double-Roadway Excavation
### Mechanical and Structural Modeling of the Immediate Roof  
According to the overburden rock structure of the working face in the upper section, the mechanical structure model of the immediate roof of this section was established, as shown in Figure 3 below. The overburden rock cannot completely fill the goat when the roof is not cut; therefore, the force of the goat gangue on the roof was ignored in this model. Figure 2: NCPDRE layout: (**a**) narrow coal pillar double-lane excavation layout; (**b**) A-A section (1—this section’s return-air roadway; 2—this section’s air-inlet roadway; 3—coal pillar; 4—next section’s return-air roadway). Here, \(q_{0}\) is the load of the overlying rock strata; \(x_{0}\) is the plastic zone width of the roadway, and \(\sigma_{y}\) is the supporting pressure of the coal rib. \(d\) is the width of the roadway, and \(q_{1}\) is the support resistance; \(B_{c}\) is the coal pillar width, and \(q_{2}\) is the bearing load of the coal pillar to the roof; \(l\) is the immediate roof length in the goal. According to the static equilibrium conditions, \(\sum F_{x}=0\); \(\sum F_{y}=0\); \(\sum M_{o}=0\). The equilibrium equation is as follows:  
\[\left\{ \begin{array}{l}F_{x}=F_{ox}=0\\ F_{y}=q_{0}(x_{0}+d+B_{c}+l)-F_{oy}-\int_{0}^{x_{0}}\sigma_{y}dx-q_{1}d-q_{2} B_{c}=0\\ M_{o}=\frac{1}{2}q_{0}(x_{0}+d+B_{c}+L)^{2}+M-\int_{0}^{x_{0}}\sigma_{y}xdx-q_{1} d\Big{(}x_{0}+\frac{1}{2}d\Big{)}-q_{2}B_{c}\Big{(}x_{0}+d+\frac{1}{2}B_{c} \Big{)}=0\end{array} \right. \tag{1}\]  
where \(F_{x}\) is the resultant force of the mechanical model in the x-direction, \(F_{y}\) is the resultant force of the mechanical model in the y-direction, and \(M_{o}\) is the resultant moment of the mechanical model at the origin O. \(F_{ox}\) and \(F_{oy}\) are the binding forces of the fixed end constraint along the x and y directions; \(M\) is the binding force couple of the fixed end constraint. The \(x_{0}\) can be calculated as follows:  
\[x_{0}=\frac{mA}{2 tan\varphi_{0}}ln\frac{k\gamma H+C_{0}/tan\varphi_{0}}{C_{0} /tan\varphi_{0}+P_{x}/A} \tag{2}\]  
where \(m\) is the mining height of the coal seam, m; \(A\) is the lateral pressure coefficient of the mine; \(\varphi_{0}\) is the internal friction angle of the contact surface between the coal seam and the upper and lower slabs, \({}^{\circ}\); \(k\) is the stress concentration coefficient in the coal body; \(\gamma\) is the average capacity of the overlying rock layer, N/m\({}^{3}\); \(H\) is the depth of burial of the coal seam, m; \(C_{0}\) is the adhesion of the contact surface between the coal seam and the top and bottom slabs, MPa; and \(P_{x}\) is the load-bearing capacity of the coal wall, MPa. In this model, the roadway support pressure was assumed as linearly distributed, and the immediate roof was regarded as an elastomer. The linear distribution of stress was more consistent. This was also beneficial to the integral calculation in this study. The supporting stress function of the constructed coal rib on the immediate roof was determined as follows [31]:  
\[\sigma_{y}=k\gamma H\bigg{(}1-\frac{x}{x_{0}}\bigg{)},\ (0\leq x\leq x_{0}) \tag{3}\]  
For the sake of subsequent simple calculations, the integral part of the mechanical equilibrium equation was calculated in advance:  
\[F_{1}=\int_{0}^{x_{0}}\sigma_{y}dx=\frac{1}{2}k\gamma Hx_{0};\ M_{1}=\int_{0}^ {x_{0}}\sigma_{y}xdx=\frac{1}{6}k\gamma H{x_{0}}^{2} \tag{4}\]  
Figure 3: The mechanical model of the immediate roof after upper section mining. According to the equilibrium equation, the constrained reaction force at the fixed end can be calculated as follows:  
\[\left\{ \begin{array}{l}F_{ox}=0\\ F_{oy}=q_{0}(x_{0}+d+B_{c}+l)-F_{1}-q_{1}d-q_{2}B_{c}\\ M=M_{1}+q_{1}d\Big{(}x_{0}+\frac{1}{2}d\Big{)}+q_{2}B_{c}\Big{(}x_{0}+d+\frac{1 }{2}B_{c}\Big{)}-\frac{1}{2}q_{0}(x_{0}+d+B_{c}+l)^{2}\end{array} \right.