## 2 Lateral Load-Bearing Behavior of the Roadway-Side Support
### The Lateral Loading Characteristics of Gangue
#### 2.2.2 Study on the Lateral Compression Characteristics of Gangue
\(\mathbf{\phi_{6}}\) is the wedge angle, which is the angle between the hypothetical sliding surface of the wedge-shaped body and the horizontal plane. \(\mathbf{F_{n}}\) is the support force exerted via the gangue pile at the bottom of gangue zone B on the wedge-shaped triangular body. \(\mathbf{\beta}\) is the internal friction angle of the gangue. \(\mathbf{G_{MB}}\) is the self-weight of the wedge-shaped triangular body. \(\mathbf{h_{m}}\) is the mining height of the coal seam. If the equilibrium of the wedge-shaped body is to be maintained, the following condition holds:  
1. The self-weight, \(\mathbf{G_{MB}}\), of the wedge-shaped triangular body. Figure 7: Coulomb Earth pressure theory model. (**a**) Basic model of the Coulomb soil pressure distribution; (**b**) the force condition of the wedge-shaped triangular block ABC during the compaction of the gangue; (**c**) the simplified force model of the wedge triangular block. Under the influence of the wedge-shaped sliding surface and the constraints of the roadway-side support structure, the self-weight, \(G_{MB}\), of the wedge-shaped triangular body can be calculated using the following equation:  
\[G_{MB}=\gamma_{2}V_{MB}=\frac{\gamma_{2}{h_{m}}^{2}\cos\phi_{4}\times\cos\phi_{6 }}{2\sin(\phi_{6}-\phi_{4})} \tag{12}\]  
In the equation, \(\mathbf{h_{m}}\) represents the length of the wedge-shaped triangular body in the vertical direction. * The calculation of the overlying load, \(\mathbf{F_{D}}\), on the triangular block. The overlying load, \(\mathbf{F_{D}}\), on the triangular block is composed of two parts: the self-weight, \(G_{\mathbf{AM}}\), of the gauge above the triangular block and the load, \(\mathbf{F_{d}}\), from the roof. The relationship is as follows:  
\[\left\{ \begin{array}{l}F_{d}=F_{gj}+F_{gsm}\times\sin\phi_{3}\\ G_{MA}=\frac{\gamma_{1}(h_{s}+H_{m}-h_{m})h_{m}\cos\phi_{4}\times\cos\phi_{5}} {\sin(\phi_{6}-\phi_{4})}\end{array} \right. \tag{13}\]  
* The lateral support resistance, \(\mathbf{P_{a}}\), of the roadway-side filling wall. * The support force, \(\mathbf{F_{n}}\), exerted via the lower gangue pile. By solving the system of Equations (12) and (13), the solution can be obtained as follows:  
In the horizontal direction,  
\[P_{a}\cos\phi_{5}=F_{n}\sin(\beta-\phi_{6}) \tag{14}\]  
In the horizontal direction,  
\[G_{MB}+F_{D}=P_{a}\sin\phi_{5}+F_{n}\cos(\beta-\phi_{6}) \tag{15}\]  
When \(\mathbf{\beta}=\mathbf{\phi_{6}}\), i.e., when the support force \(\mathbf{F}\) exerted via the lower gangue pile is directed vertically upwards, the following Equation (14) can be obtained:  
\[P_{a}=0 \tag{16}\]  
When \(\mathbf{\beta}\
eq\mathbf{\phi_{6}}\), solving the system of Equations (12)-(15) yields the following:  
\[P_{a}=\frac{\tan\phi_{5}\times\tan(\beta-\phi_{6})}{2\times[\tan\phi_{5} \times\tan(\beta-\phi_{6})+1]}\times\left[ \begin{array}{c}\frac{h_{m}\cos \phi_{4}\times\cos\phi_{5}\left[2\gamma_{1}(h_{s}+H_{m}-h_{m})+\gamma_{2}h_{m} \right]}{\sin(\phi_{6}-\phi_{4})}\\ +(f_{A}+f_{B})+\frac{(f_{C}+f_{D})h_{s}\cos(\phi_{2}-\phi_{3})}{\sin\phi_{3}} \end{array} \right] \tag{17}\]  
For the convenience of calculation, assume \(\mathbf{\phi_{4}}=\mathbf{\phi_{5}}=0\), meaning that the lateral support resistance, \(\mathbf{P_{a}}\), exerted via the roadway-side filling wall acts vertically on the edge (AB), and the gangue wedge edge (AC) is horizontal. Equation (17) simplifies to the following:  
\[\left\{\begin{array}{l}P_{a}==\frac{\tan(\beta-\phi_{6})}{2}\times\left[ \frac{h_{m}\left[2\gamma_{1}(h_{s}+H_{m}-h_{m})+\gamma_{2}h_{m}\right]}{\tan \phi_{6}}+\frac{10.39\times(K_{g}-1)c_{c}^{1042}\sin\phi_{2}}{K_{g}^{2}}\times Q  
Based on the single influencing factor control variable method, with other parameters held constant, the impact of the roadway-side support structure's lateral compression of the gangue pile on the lateral resistance is analyzed.