## 3 Mechanical Analysis of Overlying Rock Movement of the Filled Working Face
### Analysis of the Roof Subsidence Law of the Filled Working Face  
[PERSON] of Poland first studied the problems of roof bending, subsidence, and stress distribution in filling mining. He also theoretically proved the correctness of the pressure wave theory. [PERSON] analyzed the roof deflection and the vertically stressed distribution before and after the filled mining working face on this basis. The strength change in the filling is not analyzed when the filling body behind the working face is discussed. Therefore, the roof deflection curve behind the working face  
Figure 3: Stress diagram of the paste-filling working face. Figure 2: Isolated filling area. should be determined according to the strength of the filling material. The roof deflection curve in front of the paste-filling working face should be analyzed according to [PERSON]'s formula. The law of the roof subsidence curve of the paste-filling working face is composed of two parts. One is the roof subsidence at the coal wall (\(w_{1}\)). It is only a relative quantity and does not affect the shape of the deflection curve behind the working face. The other is the stable subsidence part (\(w_{2}\)), which is obtained by the interaction of the volume weight of the roof strata and the supporting force of supports and the filling body. This part is accumulated by each step distance during filling. That is,  
\[z_{2}=w_{1}+w_{2} \tag{1}\]  
The roof subsidence at the coal wall (\(w_{1}\)) is equal to the value of the roof deflection curve in front of the working face at \(x\) = 0, namely,  
\[w_{1}=-\frac{p_{z}}{k_{v}}-\frac{\beta^{2}}{\alpha^{2}}\frac{p_{z}}{k_{s}} \tag{2}\]  
The solution of the part (\(w_{2}\)) interacting with the volume weight of the roof strata, the supporting force of the support, and the supporting force of the filling body is as follows. The overlying strata roof on the working surface is regarded as the fixed support beam at both ends, and the pressure comes from the roof strata at the initial stage of paste filling. It is only subject to the weight of the overlying direct roof at the initial stage of mining since the roof of backfill mining does not collapse. Then, the weight is multiplied by the pressure coefficient of the upper roof against the immediate roof. The strength of the backfill body gradually increases. The span of the immediate roof increases, and the subsidence of the immediate roof increases with the advance of the working face. The subsidence curve of the roof changes with each step of the advance because the compression of the backfill body is irreversible. The solution of the second part, \(w_{2}\), of the roof subsidence deflection curve behind the paste-filling working face is as follows. The total deflection of the fixed beam at both ends can be generated by the separate action of various forces at each step and superimposed on each other to obtain the deflection curve under the same coordinates. The boundary condition of the beam is that the angles generated by various forces at the left and right ends of the beam are 0. Therefore, the deflection formula of the beam can be expressed as  
\[w_{2}=\sum_{n=1}^{n}f(x_{n})=\sum_{n=1}^{n}\left[w(q_{rock})+w(x_{a})+w(x_{b})+ w(x_{1})+w(x_{2})+\cdots\cdots+w(x_{n})\right] \tag{3}\]  
where \(f(x_{n})\) is the deflection of roof strata at each advance step; \(w(q_{rock\ mass})\) is the deflection caused by the volume weight of the roof strata; \(w(x_{a})\) is the deflection of the roof strata in the coal mining area generated by the filling hydraulic support; and \(w(x_{b})\) is the deflection generated by the filling area of hydraulic support on the roof strata. The area has no supporting function, and the value is 0. \(w(x_{n})\) is the deflection caused by the support of each filling body on the roof strata. 1. The calculation model of the deflection generated by the volume weight of the roof strata can be simplified (see Figure 4a).