## 2 Lateral Load-Bearing Behavior of the Roadway-Side Support
### Roof Rotation Side Loading Characteristics
\begin{array}{l}M_{F_{zj}}=\frac{F_{zj}d_{z1}}{2}\\ M_{T_{ab}}=\frac{T_{cb}s}{2}\sin\phi_{1}\\ M_{T_{cb}}=T_{cb}\big{(}h_{j}-L_{b2}\sin\phi_{1}-\frac{s}{2}\cos\phi_{1}\big{)} \\ M_{b}=G_{b}L_{b}\cos\phi_{1}\\ M_{q}=\frac{dL_{b}^{2}\cos^{2}\phi_{1}}{2}\\ M_{F_{ab}}=F_{cb}\Big{(}\frac{L_{b}}{\cos\phi_{1}}+\big{(}h_{j}-L_{b}\sin\phi_{ 1}-\frac{s}{2}\cos\phi_{1}\big{)}\tan\phi_{1}\Big{)}\\ \end{array} \right. \tag{4}\]Combining Formulas (1)-(4), we can get the following:  
\[\left\{ \begin{array}{l}T_{ab}=T_{cb}=\frac{G_{b}L_{b}\cos\phi_{1}+\frac{1}{2}qL_{ b}^{2}\cos^{2}\phi_{1}-\frac{1}{2}\left(f_{A}+2f_{B}\right)l_{0}^{2}-\frac{1}{2}F_{z}l_{ \mu 1}}{2\sin\phi_{1}-\mu_{B}L_{c}-\left(1+\mu_{B}\tan\phi_{1}\right)\left(h_{j}-L _{b}\sin\phi_{1}-\frac{1}{2}\cos\phi_{1}\right)}\\ F_{zj}=G_{b}+q_{s}L_{b}\cos\phi_{1}-\frac{1}{2}\left(f_{A}+f_{B}\right)l_{2} \end{array} \right. \tag{5}\]  
During the fracture and rotation of the basic roof, the underlying direct roof beam is also squeezed and rotated synchronously. The mutual compression and interlocking between the broken blocks of the articulated roof have a certain bearing effect on the overlying rock strata [24]. At the same time, due to the bending and sinking of the roof, the articulated roof structure will also generate a certain force on the lower non-articulated roof structure. Taking point \(\mathbf{O_{2}}\) as the coordinate origin, according to mechanical equilibrium, we can get the following:  
\[\left\{ \begin{array}{l}\frac{T_{el}}{\sin\phi_{2}}+F_{zs}+G_{s}=F_{c}+F_{z}+ F_{gsm}\\ \frac{F_{gsm}}{\cos\phi_{2}}+F_{c}+F_{z}+F_{gsm}=F_{zs}+G_{s}\\ M_{s}+F_{zj}\left(\frac{l_{1}\cos\phi_{2}}{2}+h_{s}\sin\phi_{2}\right)+G_{s}x_{ 2}=\frac{F_{c}\left(x_{0}-h_{s}\tan\phi_{2}\right)\cos\phi_{2}}{2}+F_{z}\left( x_{0}+a+\frac{b}{2}-\frac{h_{s}\tan\phi_{2}}{2}\right)\cos\phi_{2}+F_{gsm}x_{3} \end{array} \right. \tag{6}\]  
Take segment CD to calculate the gauge support resistance \(F_{gsm}\), which is as follows:  
\[\left\{ \begin{array}{l}F_{gsm}=\frac{1}{2}(f_{C}+f_{D})\frac{h_{s}\cos(\phi_ {2}-\phi_{3})}{\sin\phi_{3}}\\ x_{3}=\frac{\left(f_{C}+2f_{D}\right)}{3\left(f_{C}+f_{D}\right)}\left(\frac{ h_{s}\sin(\phi_{3}-\phi_{2})}{\sin\phi_{3}}\right)+\left(l_{s}-\frac{h_{s}}{\tan \phi_{3}}\right)\sin\phi_{2}\end{array} \right. \tag{7}\]  
Taking \(O_{1}\) as the origin, we can find the bending moment:  
\[T_{sl}=\frac{1}{h_{s}}\left\{ \begin{array}{l}F_{zj}l_{s}\cos\phi_{2}+2G_{s} (x_{2}-h_{s}\tan\phi_{2})-F_{c}\left(\frac{x_{0}}{2}-h_{s}\tan\phi_{2}\right) \cos\phi_{2}\\ -F_{gsm}(x_{3}-h_{s}\tan\phi_{2})-F_{z}\left(x_{0}+a+\frac{b}{2}-h_{s}\tan\phi _{2}\right)\cos\phi_{2}\end{array} \right\} \tag{8}\]  
The self-weight of the direct top block \(\mathbf{G_{s}}\) is as follows:  
\[G_{s}=\frac{\gamma_{s}L_{b\mathbf{1}}h_{s}l_{s}(2-\sin\phi_{3})}{2} \tag{9}\]  
In the formula, \(\gamma_{s}\) is the basic top rock beam density; \(L_{b\mathbf{1}}\) is the basic top block length. Therefore, the lateral horizontal force \(\mathbf{F_{z-x}}\) exerted via the direct top block on the roadside support is as follows:  
\[F_{z-x}=\frac{h_{s}\sin\phi_{2}}{2h_{s}\tan\phi_{2}-\left(x_{0}+a+\frac{b}{2} \right)}\left\{ \begin{array}{l}F_{c}\left(\frac{x_{0}}{2h_{s}\tan\phi_{2}} \right)-F_{zj}\left(1+\frac{l_{s}}{h_{s}\tan\phi_{2}}\right)\\ -G_{s}\left[1+\frac{2\left(x_{2}-h_{s}\tan\phi_{2}\right)}{h_{s}\sin\phi_{2}} \right]+F_{gsm}\frac{\left(x_{3}-h_{s}\tan\phi_{2}\right)}{h_{s}\sin\phi_{2}} \end{array} \right\} \tag{10}\]  
Figure 4: The support resistance curve of the gangue pile in the gob.