## 6 Support Span Reduction Effect of Roadside Filling Body
### Analysis of Supporting Span Reduction Effect  
Based on the deflection curve of the simply supported beam under uniform load [33; 34; 35], the rotation angle equation and deflection (deformation) curve equation of the model shown in Figure 4a of Section 2.3 are obtained as given in Equations (9)-(11).  
\[EIw^{\prime}=\frac{ql}{2}\cos\theta x-\frac{q}{2}\cos\theta x^{2}, \tag{9}\]  
\[EIw^{\prime}=\frac{ql}{4}\cos\theta x^{2}-\frac{q}{6}\cos\theta x^{3}-\frac{ qL^{3}}{24}\cos\theta, \tag{10}\]  
\[EIw=\frac{ql}{12}\cos\theta x^{3}-\frac{q}{24}\cos\theta x^{4}-\frac{qL^{3}}{24 }\cos\theta x, \tag{11}\]  
The equations for the rotation angle and deflection (deformation) of the model, as shown in Figure 4b of Section 2.3, are given as Equations (12)-(20).  
\[EIw_{1}{}^{\prime}=qa\cos\theta x-\frac{q}{2}\cos\theta x^{2}, \tag{12}\]  
\[EIw_{1}{}^{\prime}=\frac{qa}{2}\cos\theta x^{2}-\frac{q}{6}\cos\theta x^{3}- \frac{qa^{3}}{12}\cos\theta, \tag{13}\]  
\[EIw_{1}=\frac{qa}{6}\cos\theta x^{3}-\frac{q}{12}\cos\theta x^{4}-\frac{qa^{3 }}{12}\cos\theta x, \tag{14}\]  
\[EIw_{2}{}^{\prime\prime}=\frac{qa}{2}\cos\theta(x-a)-\frac{q}{2}\cos\theta(x-a )^{2}, \tag{15}\]  
\[EIw_{2}{}^{\prime}=\frac{qa}{4}\cos\theta(x-a)^{2}-\frac{q}{6}\cos\theta(x-a)^ {3}+\frac{q\cos\theta b^{2}}{24}(b-2a), \tag{16}\]  
\[EIw_{2}=\frac{qa}{12}\cos\theta(x-a)^{3}-\frac{q}{24}\cos\theta(x-a)^{4}+ \frac{q\cos\theta b^{2}}{24}(b-2a)(x-a), \tag{17}\]  
\[\omega(x)=\frac{qlx^{3}\cos\theta}{12 EI}-\frac{qx^{4}\cos\theta}{24 EI}-\frac{ ql^{3}x\cos\theta}{24 EI}, \tag{18}\]  
\[\omega_{1}(x)=\frac{qax^{3}\cos\theta}{6 EI}-\frac{qx^{4}\cos\theta}{12 EI}- \frac{qa^{3}x\cos\theta}{12 EI}\ 0\ \leq\ x\ \leq\ a, \tag{19}\]  
\[\omega_{2}(x)=\frac{qa(x-a)^{3}\cos\theta}{12 EI}-\frac{q(x-a)^{4}\cos\theta}{ 24 EI}+\frac{qb^{2}(b-2a)(x-a)\cos\theta}{24 EI}\ a\ \leq\ x\ \leq\ l, \tag{20}\]  
In Equations (12)-(20), \(q\) is the load acting on the simplified beam, kN/m; \(l\) is the roadway roof span, m; \(\theta\) is the dip angle of the coal seam, degrees; \(x\) is the distance from point A along the dip of the coal seam, m;\(f\) is the support force of the roadside filling wall, kN; \(a\) is the distance between the action point of roadside support force and point A, m; and \(b\) is the distance between the action point of roadside filling body and point B, m.  
In order to quantitatively study the supporting role of the roadside support wall of the GER, the actual parameters of 3131 roadway were substituted into the Equations (9)-(11);the deformation of roadway roof with and without roadside support is calculated and compared. The results are shown in Figure 15.  
Figure 15 shows that the overall subsidence (deformation) of the roof is significantly reduced when there is a roadside support body, and the maximum subsidence is reduced by about 85% in the roof compared with that without roadside support. For the inclined coal seam GER, when using the roadside filling to support the roof, the strength of the filling body can effectively support the roof. It also reduces the effective span of the roof, thus enhancing the stability of the roof. Therefore, compared with the horizontal coal seam, when the roadway side support is used in the GER of the inclined coal seam, the roadside filling fully supports and reduces the span.