# Mechanics Model of Floor Heave: Case Study on Thin Coal Seam with Soft Roof and Floor
## 3 Mechanical Model of Floor Heave in Gob-Side Entry Retaining
### Stress Analysis of Gob-Side Entry Retaining  
A stress study of the floor of the gob-side entry retaining roadway can be established based on the analysis mentioned above, as shown in Figure 6. The working face's advanced support pressure is shown as \(k_{1}\), the high support pressure zone's stress concentration coefficient is shown as \(k_{2}\), the filling body next to the roadway's stress concentration coefficient is shown as \(k_{3}\), the stress concentration coefficient after compaction of goaf is shown as \(k_{4}\), \(\gamma\) reflects the average rock layer's weight in units (N/m\({}^{3}\)), and h reflects the depth of the mine. The theoretical method of elastic foundation beams is adopted in this paper. The elastic foundation beam model is widely used to solve the deformation problems of the roof and floor of working faces. The mechanical model has the following assumptions: (1) The foundation is simplified as an elastic body. (2) The settlement at any point on the foundation is directly proportional to the pressure on that unit. (3) The floor is assumed to be a beam, and the deformation conforms to the solution assumption of the beam. (4) The deformation of the floor is continuous. The high support pressure area formed by the superposition of lateral support pressure and advanced support pressure on the left side of the roadway and the stress concentration area of the filling body next to the roadway on the right side are simplified as uniformly distributed loads in order to simplify the study of the mechanical behavior of the floor heave of the goaf retaining roadway without losing its essence. As an elastic foundation, the fundamental floor's force on the nearby floor is distilled. In Figure 7, the model is displayed. Figure 6: Diagram of the floor of the gob-side entry retaining highway after stress analysis. Figure 7: Mechanical model of elastic foundation floor and gob-side entry retaining highway. The Winkle assumption of an elastic foundation states that any point's settlement on its surface is directly proportionate to the pressure applied to that location per unit area [37; 38; 39], i.e.,  
\[q=k\omega(x) \tag{1}\]  
The pressure per unit area is denoted by \(q\) in the formula: \(\omega(x)\) represents the foundation's bending; \(k\) shows the foundation coefficient, which is described as,  
\[k=\frac{E}{H} \tag{2}\]  
\(E\) and \(H\) in the formula stand for the rock layer's elastic modulus and thickness, respectively. The basic differential equation of the foundation beam deflection curve should be satisfied by the relationship between the deflection of the roof and the load on the roof rock beam, according to the theory of elastic foundation rock beams:  
\[EI\frac{d^{4}\omega(x)}{dx^{4}}=q_{0}-q \tag{3}\]  
The differential equation for the roof rock beam's deflection curve is as follows:  
\[EI\frac{d^{4}\omega_{1}(x)}{dx^{4}}+k\omega_{1}(x)=k_{2}\gamma H,x<0 \tag{4}\]  
\[EI\frac{d^{4}\omega_{2}(x)}{dx^{4}}=-\gamma H,0<x<x_{1} \tag{5}\]  
\[EI\frac{d^{4}\  
Similarly, when \(x_{2}<x\), the differential equation describing the roof deflection curve in the roadway roof control region has the following general solution:  
\[\omega_{3}(x)=e^{-\gamma x}(C_{1}\cos\gamma x+C_{2}\sin\gamma x)+\frac{k_{3} \gamma H}{k} \tag{12}\]  
In the equation, \(\gamma=\sqrt[4]{\frac{k}{AE}}\). When \(0<x<x_{1}\), in the roadway roof control area, the differential equation describing the roof deflection curve has the following general solution:  
\[\omega_{2}(x)=-\frac{\gamma H}{24 EI}X^{4}+B_{1}X^{3}+B_{2}X^{2}+B_{3}x+B_{4} \tag{13}\]  
In conclusion, the differential equation for the roof beam's deflection curve is:  
\[\left\{ \begin{array}{l}\omega_{1}(x)=e^{\alpha x}(A_{1}\cos\alpha x+A_{2} \sin\alpha x)+\frac{k_{3}\gamma H}{k},\\ x<0\\ \omega_{2}(x)=-\frac{\gamma H}{24 EI}X^{4}+B_{1}X^{3}+B_{2}X^{2}+B_{3}x+B_{4}, \\ 0<x<x_{1}\\ \omega_{3}(x)=e^{-\gamma x}(C_{1}\cos\gamma x+C_{2}\sin\gamma x)+\frac{k_{3} \gamma H}{k},\\ x_{1}<x\end{array} \right.