## 3 Theoretical Analysis
### Weakly Cemented Return Mining Roadway Width Retention Design  
1. Mechanical model of weak cementation return mining roadway failure  
In the elastoplastic deformation state, the mechanical model of the ultimate equilibrium of the coal column [(20; 21)] is as shown in Figure 5. The stress is concentrated in the coal column on the side of the goat area. Close to the edge of the coal column, the vertical stress gradually increases, and within a certain distance from the edge, the supporting pressure will be in a state of balance with the bearing capacity of the coal column.  
Figure 4: Deformation of the roadway in the peak area of the support pressure of the large coal column roadway in the adjacent mining area: (**a**) bending of steel beam; (**b**) anchor cable failure; (**c**) pillars sinking; (**d**) serious deformation of coal pillar side.  
Figure 5: Mechanical model of the ultimate equilibrium of the coal column [(20; 21)].  
This can be seen from the mechanical model in Figure 5.  
\[\left\{\begin{aligned} \sigma_{y}=\frac{1}{A}\bigg{(}P_{0}+\frac{C_{0}}{ \tan\varphi_{0}}\bigg{)}e^{\frac{2\tan\varphi_{0}}{mA}+\frac{\tan^{2}\varphi_{0 }}{A}}\\ \tau_{xy}=-\big{(}C_{0}+\sigma_{y}\tan\varphi_{0}\big{)}\\ X_{0}=\frac{mA}{2\tan\varphi_{0}}\ln\bigg{(}\frac{K\gamma H\tan \varphi_{0}+C_{0}}{C_{0}+P_{0}}\bigg{)}\end{aligned}\right. \tag{1}\]  
In the formula, \(\sigma_{y}\)--vertical stress of the coal body; \(A\)--lateral pressure coefficient; \(P_{0}\)--roadway support strength; \(C_{0}\)--coal body cohesion; \(\varphi_{0}\)--friction angle in coal; \(m\)--coal seam thickness; \(\tau_{xy}\)--shear stress between the coal body and top bottom plate; \(K\)--goaf-side stress concentration coefficient; \(H\)--roadway burial depth; \(\gamma\)--average bulk density of the weakly cemented rock layer.  
The minimum theoretical width of the section coal column can be written as follows:  
\[B=k(X_{0}+X_{1}) \tag{2}\]  
where \(X_{1}\) is the effective length of the anchor rod of the narrow coal column and \(k\) is the surplus coefficient with a value from 1.15 to 1.45.  
Here, \(m=8.4\) m; \(A=1.3\); \(K=4\); \(H=650\) m; \(\varphi_{0}=32\); \(C_{0}=5\) MPa; \(P_{0}=0.3\) MPa; and \(\gamma=0.025\) MN/m\({}^{3}\). These can be substituted into (1), to obtain \(X_{0}=5.6\) m. \(X_{1}=1.8\) m and \(k=1.15\) can be substituted into Formula (2) to obtain \(B=8.51\) m, and the width of the actual small coal column is approximately 5 m.  
2. Analysis of mine pressure monitoring results on adjacent working surfaces  
According to the mine pressure monitoring results of working surface 130,203, the stresses affected by different-width coal columns are analyzed. The support pressure first decreases and then increases with the increase in the width of the coal column, and finally tends to the original rock stress level. The stress when the coal column is 5 m in width is only 15 MPa, the stress in the coal column with a width of 5-8 m does not change much, and when the width of the coal column reaches 20 m, the stress reaches the peak 50 MPa; when the width of the coal column is 35 m, the stress is 33 MPa, which is still a high stress level.  
Considering the comprehensive placement of working surface 130,205 and the geological conditions, the optimal width of the coal column is determined to be 5 m.