## 2 Deformation and Fracture Mechanisms of THRs in UMCs
### Mechanical Analysis of Individual Zones  
1. Free overhang According to the bending theories for beams with single generalized displacement [19], we can write the mechanical equation for the free overhang \((-l,0)\) as follows: \[\left\{ \begin{array}{l}\theta_{1}=\frac{\mathrm{d}\omega_{1}}{\mathrm{d}x}+ \frac{EI}{\mathrm{d}^{3}\omega_{1}}\\ M_{1}=-EI(\frac{\mathrm{d}^{2}\omega_{1}}{\mathrm{d}x^{2}}+\frac{EI}{\mathrm{d} }\frac{\mathrm{d}^{4}\omega_{1}}{\mathrm{d}x^{4}})\\ F_{1}=-EI\frac{\mathrm{d}^{3}\omega_{1}}{\mathrm{d}x^{3}}\\ -EI\frac{\mathrm{d}^{4}\omega_{1}}{\mathrm{d}x^{4}}=-q\cos\alpha\end{array} \right.\] (1) where \(\omega_{1}\) is the deflection of the THR; \(\theta_{1}\) is the rotational angle of the THR; \(M_{1}\) is the bending moment of the THR; \(F_{1}\) is the shear force in the THR; \(q\) is the overburden load above the mining-induced roof; \(E\) is the elastic modulus of the THR; \(\
u\) is the Poisson's ratio of the THR; \(I\) is the moment of inertia of the THR, \(I=bh^{3}/12\); \(b\) is the width of the THR, the  
Figure 1: Mechanical model of the bearing capacity of the THR in the UMC. unit width being \(b=1\) m; \(h\) is the thickness of the THR; \(T\) is the shear stiffness of the THR, conceived as a beam with a rectangular cross-section, \(T=5 Ebh/[12(1+\
u)]\). The general solution to the deflection curve equation is found using the last term in Equation (1):  
\[\omega_{1}=\frac{q\cos\alpha}{24 EI}x^{4}+C_{1}x^{3}+C_{2}x^{2}+C_{3}x+C_{4} \tag{2}\]  
where \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) are all undetermined coefficients. By substituting Equation (2) into (1), we have  
\[\left\{ \begin{array}{l}\theta_{1}=\frac{q\cos\alpha}{6 EI}x^{3}+3C_{1}x^{2}+ (2C_{2}+\frac{q\cos\alpha}{T})x+\frac{6 EI}{T}C_{1}+C_{3}\\ M_{1}=-\frac{q\cos\alpha}{2}x^{2}-6 EI_{1}x-2 EI_{2}-\frac{Elg\cos\alpha}{T}\\ F_{1}=-\frac{q\cos\alpha}{24 EI}x^{4}+C_{1}x^{3}+C_{2}x^{2}+C_{3}x+C_{4}\\ \end{array} \right. \tag{3}\]  
The following boundary conditions are applied to the two ends of the free overhang in the THR, as follows  
\[\left\{ \begin{array}{l}M_{1}\mid_{x=-l}=0\\ F_{1}\mid_{x=-l}=0\\ \end{array} \right. \tag{4}\]  
\[\left\{ \begin{array}{l}\theta_{1}\mid_{x=0}=\frac{6 EI}{T}C_{1}+C_{3}\\ M_{1}\mid_{x=0}=-2 EI_{2}-\frac{Elg\cos\alpha}{T}\\ F_{1}\mid_{x=0}=-6 EI_{1}\\ \omega_{1}\mid_{x=0}=C_{4}\\ \end{array} \right. \tag{5}\]  
By substituting Equation (4) into (3), we have  
\[\left\{ \begin{array}{l}C_{1}=\frac{q\cos\alpha}{6 EI}l\\ C_{2}=\frac{q\cos\alpha}{4 EI}l^{2}-\frac{q\cos\alpha}{2T}\\ \end{array} \right. \tag{6}\]  
By substituting Equation (6) into (5), we have  
\[\left\{ \begin{array}{l}\theta_{1}\mid_{x=0}=\frac{q\cos\alpha}{T}l+C_{3}\\ M_{1}\mid_{x=0}=-\frac{q\cos\alpha}{2}l^{2}\\ F_{1}\mid_{x=0}=-q\cos\alpha l\\ \omega_{1}\mid_{x=0}=C_{4}\\ \end{array} \right. \tag{7}\]  
2. Coal-bearing zone  
According to the elastic foundation beam theory [20], the coal-bearing zone in the THR is treated as a semi-infinite long beam. Thus, we can write the mechanical equation of the coal-bearing zone (0, +\(\infty\)) as follows:  
\[\left\{ \begin{array}{l}\theta_{2}=\frac{\mathrm{d}\omega_{2}}{\mathrm{d}x} \\ M_{2}=-EI\frac{\mathrm{d}^{2}\omega_{2}}{\mathrm{d}x^{2}}\\ F_{2}=-EI\frac{\mathrm{d}^{2}\omega_{2}}{\mathrm{d}x^{3}}\\ E\frac{\mathrm{d}^{4}\omega_{2}}{\mathrm{d}x^{4}}+k\omega_{2}=q\cos\alpha\\ \end{array} \right. \tag{8}\]  
where \(\omega_{2}\) is the deflection of the THR; \(\theta_{2}\) is the rotational angle of the THR; \(M_{2}\) is the bending moment of the THR; \(F_{2}\) is the shear force in the THR; \(k\) is the elastic foundation coefficient. The general solution to the deflection curve equation for the coal-bearing zone (0, +\(\infty\)) is found using the fourth term in the Equation (8):  
\[\omega_{2}=\left( \begin{array}{l}\mathrm{e}^{\mathrm{d}x}(C_{5}\sin Ax+C_{6} \cos Ax)\\ +\mathrm{e}^{-Ax}(C_{7}\sin Ax+C_{8}\cos Ax)+\frac{q\cos\alpha}{k}\\ \end{array}