## 3 Mechanical Analysis of Overlying Rock Movement of the Filled Working Face
### Analysis of the Roof Subsidence Law of the Filled Working Face
The rock beam bends due to its volume weight, and the deflection curve formula of the beam is obtained by the integral method, as follows: \[w_{q}(x)=\frac{qx^{2}}{24 EI}(x-l)^{2}\] (4)2. The supporting load of the filling hydraulic support on the roof is based on the triangular load in the filling working face. The front of the triangular load has an appropriate unsupported roof distance. Figure 4b shows the mechanical model. Actual paste-filling working face support has a supporting function, and the overlying roof no longer sinks. Errors will occur if the actual support load on the roof is used to calculate the roof deflection curve. Therefore, the equivalent supporting force of the self-weight load of the overlying strata is set in the supporting area. This is to simulate the supporting force of the support on the roof. The two can offset the influence of force relatively without changing the structural form of the beam. Figure 4c presents a mechanical model. The mechanical deflection of the roof in the filling area can be calculated as follows: \[w_{q_{a}}(x)=\left\{ \begin{array}{l}\frac{1}{6L^{2}}\big{(}R_{A}x^{3}+3M_{A} x^{2}-\frac{qx^{4}}{4}\big{)},\ 0\leq x\leq\text{a}\\ \frac{1}{6L^{2}}\big{[}R_{A}x^{3}+3M_{A}x^{2}+\frac{q^{2}}{4}\big{(}a^{3}-4a^{ 2}x+6 ax^{2}-4x^{3}\big{)}\big{]},\ a<x\leq l\end{array} \right.\] (5)
3. The supporting force of the filling body on the roof strata is considered a uniform load. Figure 4d lists the mechanical model and \(a=d+0.5c\). The supporting force generated by the two ends of the beam is as follows. \[R_{A}=-\frac{qc}{4I3}(12b^{2}l-8b^{3}+c^{2}l-2 bc^{2})\] (6) and \[R_{B}=q_{c}-R_{A}\] (7) The mechanical deflection of the roof in section A-C is calculated as follows: \[w(x)=\frac{qc}{24 EI^{3}}\Big{[}\Big{(}12 ab^{2}-3 bc^{2}+c^{2}l\Big{)}lx^{2}- \Big{(}12b^{2}l-8b^{3}+c^{2}l-2 bc^{2}\Big{)}x^{3}\Big{]}\] (8) The mechanical deflection of the roof in section C-D is calculated as follows: \[w(x)=\frac{q}{24 EI^{3}}\Big{[}\Big{(}12 ab^{2}-3 bc^{2}+c^{2}l\Big{)}clx^{2}- \Big{(}12b^{2}l-8b^{3}+c^{2}l-2 bc^{2}\Big{)}cx^{3}+l^{3}(x-d)^{4}\Big{]}\] (9) The mechanical deflection of the roof in section D-B is calculated as follows: \[w(x)=\frac{qc}{24 EI^{3}}\Big{[}\Big{(}12 ab^{2}-3 bc^{2}+c^{2}l\Big{)}lx^{2}- \Big{(}12b^{2}l-8b^{3}+c^{2}l-2 bc^{2}\Big{)}x^{3}+l^{3}(x-a)^{3}+\frac{qc^{2}l ^{3}(x-a)}{4}\Big{]}\] (10) The model can be simplified when the paste-filling working face advances for a long distance. The supporting force of each filling body on the roof is assumed to be the  
Figure 4: Simplified models of roof stress. concentrated force from the uniform load set previously. Figure 4e presents the force model. The equation of the beam deflection curve obtained by the integral method is as follows:  
\[w_{q}(x)=\left\{ \begin{array}{l}\frac{q\epsilon b^{2}x^{2}}{6 EI^{2}}(3 al-lx-2 ax),\;0\leq x\leq\text{a}\\ \frac{q\alpha^{2}(l-x)^{2}}{6 EI^{3}}(lx+2 bx-al),\;a<x\leq l\end{array} \right. \tag{11}\]  
The deflection formula behind the paste-filling working face can be obtained through the calculation of the mechanical formula group. The deflection formula of the roof behind the filling working face is a quadric equation with one variable after analyzing the results of each calculation formula. That is,  
\[\text{z}=\alpha_{4}x^{4}+\alpha_{3}x^{3}+\alpha_{2}x^{2}+\alpha_{1}x+\beta \tag{12}\]  
Then,  
\[ \begin{array}{l}z_{1}=-\frac{p_{z}}{k_{\text{s}}}-\frac{\beta^{2}}{a^{2}} \frac{p_{z}}{k_{\text{s}}}-\frac{q\pi^{2}}{2k1}(x-l)^{2}-\frac{E_{\text{tot}}( 4\vec{x}^{2}-20k^{4}+32k^{3}\vec{p}-17x^{4}\vec{p}-8k^{2}\vec{p}+16x^{4}-18 \vec{p}^{2})}{144 EI^{2}[128 EI^{2}]-2E_{\text{tot}}(x^{2}-x)]}\times\\ \left\{\begin{array}{l}x^{2}(c-2l)^{2}(2 cx-3 cI+2 lx)\\,\;0\leq x\leq 0.5c\\ \vec{x}^{2}(l-x)^{2}(d+2 cx-6 lx)\\,\;0.5c<x\leq l\end{array}\right.&+\frac{1}{54 EI^{2}}q(l-x)\\ \end{array}