## 2 Lateral Load-Bearing Behavior of the Roadway-Side Support
### Stability of Lateral Sliding of Supporting Body Beside Gob-Side Entry  
From the perspective of the overall stability of the side support of the gob-side entry, during the entire process of \"given deformation\" and compaction bearing in the interaction between the key block and the gangue in the gob, the roadway-side support exhibits a tendency to rotate and slide inward due to the effects of roof rotation and lateral compression from the gangue pile [32]. Based on the analysis of the ultimate equilibrium conditions of the support structure, a self-stability model for the coal gangue roadway-side support can be obtained, as shown in Figure 9.  
From Figure 9, it can be seen that assuming the unit length of the roadway-side filling wall is \(\mathbf{L_{m}}\), the external forces acting on it are uniformly distributed and applied at \(\mathbf{L_{m}/2}\). When the roadway-side support is in a critical sliding state laterally, the lateral thrust exerted via the filling area on the filling wall must be less than or equal to the sum of the frictional force at the interface between the roof, floor, and the wall and surrounding rock, as well as the internal shear strength of the wall, that is,  
\[F_{2}\sin\phi_{2}+P_{a}\leq f_{1}+f_{2}+2F_{lb}+F_{Jx}\cos\phi_{7} \tag{19}\]  
where \(\mathbf{\phi_{7}}\) is the angle between the slip surface inside the roadway-side support and the horizontal, \(\mathbf{F_{Jy}}\) is the compressive resistance of the internal slip surface, \(\mathbf{G_{c}}\) is the self  
Figure 9: Analysis of limit equilibrium conditions of roadside support body: (**a**) self-stabilization model of roadside support body; (**b**) force analysis model of unit width range of roadside support body. weight of the roadway-side support per unit length, \(F_{z}\) is the deformation resistance of the overlying rock layer, and \(F_{lb}\) is the shear resistance of the internal wall surface of the roadway-side support per unit length. \(f_{1}\) and \(f_{2}\), are the static frictional forces generated via the resistance to lateral pressure at the upper and lower ends. According to [PERSON]'s circle theory, when the angle of the slip surface \(\varphi=45^{\circ}+\varphi_{7}/2\), the shear resistance is maximized. Therefore, the following condition holds:  
\[\left\{ \begin{array}{l}f_{1}=\mu_{1}(F_{z}\cos\phi_{2})\\ f_{2}=\mu_{2}(F_{z}\cos\phi_{2}+G_{c})\\ G_{c}=\gamma_{3}L_{m}L_{n}h_{m}\\ F_{lb}=c_{w}L_{n}h_{m}\\ F_{jx}=\frac{L_{m}L_{n}}{\cos\phi_{7}}\big{(}c_{w}+\sigma_{jy}\tan\phi_{7} \big{)}\end{array} \right. \tag{20}\]  
In the equation, \(\mathbf{n_{1}}\) and \(\mathbf{n_{2}}\) represent the friction coefficients between the filling wall and the roof and floor, respectively. \(\gamma_{3}\) is the unit weight of the support structure. \(c_{w}\) is the cohesion of the roadway-side support structure, and \(\varphi_{w}\) is the internal friction angle of the slip surface. By solving the system of Equations (19) and (20), the following can be obtained:  
\[P_{a}\leq(\mu_{1}+\mu_{2})F_{z}\cos\phi_{2}+(\mu_{2}+\tan\phi_{7})\gamma_{3}L _{m}L_{n}h_{m}+(2c_{w}h_{m}+L_{m}c_{w})L_{n} \tag{21}\]  
From Equation (21), it can be seen that the lateral sliding stability of the roadway-side support wall is influenced by factors such as the mining height, \(\mathbf{h_{m}}\), and the support structure width, \(\mathbf{L_{u}}\), as shown in the figure. Overall, the degree of lateral loading on the roadway-side support is positively correlated with \(\mathbf{h_{m}}\) and \(\mathbf{L_{u}}\). When \(\mathbf{h_{m}}\) or \(\mathbf{L_{u}}\) is smaller, the lateral bearing capacity of the roadway-side support is weaker, making it more prone to sliding failure. Conversely, when \(\mathbf{h_{m}}\) or \(\mathbf{L_{u}}\) is larger, the lateral bearing capacity of the roadway-side support increases, making it less likely to experience sliding failure and thus making the roadway-side support more stable.