## 3 Construction of Roof Self-Stability Model in Backfilling Mining  
### Self-Stability Model of Strike Roof  
The working face is advanced along strike. The strike roof support system consists of the coal body, backfilling body, and hydraulic support, which cooperate to control the roof. A mechanical model is established to analyze the effect laws of elastic foundation coefficients, support peak loads, and unsupported-roof distances on roof self-stability. It is simplified as a beam model composed of the coal body, backfilling body supported by the elastic foundation, backfilling hydraulic support in the form of linear loads, and unsupported-roof areas (see Figure 1). Figure 1 shows that \(q_{0}\) is the initial rock stress; \(k_{1}\) and \(k_{2}\) are the stress concentration coefficients of the front and rear coal bodies; \(q_{c}\) is the assumed uniform load on the immediate roof above the backfilling body (the load concentration caused by the potential saving height of overlying strata); \(k_{c}\) is the elastic foundation coefficient of coal body; \(k_{g}\) the elastic foundation coefficient of backfilling body; \(q_{3}(x)\) and \(q_{4}(x)\) are the simplified linear functions of loads above coal body, respectively; \(q_{1}(x)\) and \(q_{2}(x)\) are loads of the top beam before and after backfilling hydraulic support; \(q\) is the support load concentration; \(L_{1}\) and \(L_{6}\) are the coal body lengths; \(L_{2}\) is the distance from the support top to the rear column; \(L_{3}\) the distance from the rear column to the end of the support; \(L_{4}\) the unsupported-roof distance; \(L_{5}\) the backfilling body length. When \(-L_{1}<x<0\), the deflection differential equation of the roof is expressed as  
\[EI\frac{d^{4}w_{1}(x)}{dx^{4}}+k_{c}w_{1}(x)=q_{3}(x) \tag{1}\]  
Figure 1: Mechanical model of roof strike. where \(E\) is the elastic modulus of the roof, \(GPa\); \(I\) the inertia moment of the beam; \(q_{3}(x)\) the load above the front coal body. \[q_{3}(x)=\frac{(k_{1}-1)q_{0}}{L_{1}}x+k_{1}q_{0} \tag{2}\]  
The deflection equation of the roof in segment \(-L_{1}<x<0\) can be obtained. \[ \begin{array}{l} w_{1}(x)=\frac{\frac{(k_{1}-1)q_{0}}{L_{1}}x+ k_{1}q_{0}}{k_{c}}+e^{-\alpha x}(A_{1}\cos(\alpha x)+B_{1}\sin(\alpha x))\\ +e^{\alpha x}(C_{1}\cos(\alpha x)+D_{1}\sin(\alpha x))\end{array}  \tag{3}\]  
where characteristic coefficient \(\alpha=\sqrt[4]{\frac{k_{c}}{4 EI}}\). When \(0<x<L_{2}\), the deflection differential equation of the roof is expressed as follows. \[EI\frac{d^{4}w_{2}(x)}{dx^{4}}+q_{1}(x)=q_{c} \tag{4}\]  
where \(q_{1}(x)\) is the front top beam load of the support. \[q_{1}(x)=\frac{qx}{L_{2}} \tag{5}\]  
The deflection equation of the roof in segment \(0<x<L_{2}\) can be obtained. \[w_{2}(x)=\frac{-qx^{5}}{120 EIL_{2}}+\frac{q_{c}x^{4}}{24 EI}+\frac{A_{2}x^{3}}{ 6}+\frac{B_{2}x^{2}}{2}+C_{2}x+D_{2} \tag{6}\]  
When \(L_{2}<x<L_{2}+L_{3}\), the deflection differential equation of the roof is expressed as follows. \[EI\frac{d^{4}w_{3}(x)}{dx^{4}}+q_{2}(x)=q_{c} \tag{7}\]  
where \(q_{2}(x)\) is the front top beam load of the support. \[q_{2}(x)=\frac{-q}{L_{3}}x+\frac{L_{2}+L_{3}}{L_{3}}q \tag{8}\]  
The deflection equation of the roof in the segment \(L_{2}<x<L_{2}+L_{3}\) can be obtained. \[ \begin{array}{l} w_{3}(x)=\frac{qx^{5}}{120}-\frac{qx^{4}L_{2}}{ 24}-\frac{qL_{3}x^{4}}{24}+\frac{q_{c}L_{3}x^{4}}{24}\\ +\frac{A_{3}x^{3}}{6}+\frac{B_{3}x^{2}}{2}+C_{3}x+D_{3}\end{array}  \tag{9}\]  
When \(L_{2}+L_{3}<x<L_{2}+L_{3}+L_{4}\), the deflection differential equation of the roof is expressed as follows. \[EI\frac{d^{4}w_{4}(x)}{dx^{4}}=q_{c} \tag{10}\]  
The deflection equation of the roof in segment \(L_{2}+L_{3}<x<L_{2}+L_{3}+L_{4}\) can be obtained. \[w_{4}(x)=\frac{q_{c}x^{4}}{24 EI}+\frac{A_{4}x^{3}}{6}+\frac{B_{4}x^{2}}{2}+C_{ 4}x+D_{4} \tag{11}\]When \(L_{2}+L_{3}+L_{4}<x<L_{2}+L_{3}+L_{4}+L_{5}\), the deflection differential equation of the roof is expressed as follows.