## 2 Deformation and Fracture Mechanisms of THRs in UMCs
### Mechanical Analysis of Individual Zones
\right) \tag{9}\]  
where \(A=\sqrt[4]{\frac{k}{4 EI}}\); \(C_{5}\), \(C_{6}\), \(C_{7}\), and \(C_{8}\) are undetermined coefficients. When \(x{\rightarrow}+\infty\), the roof subsidence approaches a constant value, and Equation (8) is simplified as follows:  
\[\omega_{2}=\mathrm{e}^{-Ax}(C_{5}\sin Ax+C_{6}\cos Ax)+\frac{q\cos\alpha}{k} \tag{10}\]  
By substituting Equation (10) into (8), we obtain the equations of the rotational angle, bending moment, and shear force:  
\[\left\{ \begin{array}{l}\theta_{2}=A\mathrm{e}^{-Ax}((C_{5}-C_{6})\cos Ax-(C _{5}+C_{6})\sin Ax)\\ M_{2}=2A^{2}\mathrm{e}^{-Ax}EI(C_{5}\cos Ax-C_{6}\sin Ax)\\ F_{2}=-2A^{3}\mathrm{e}^{-Ax}EI((C_{5}+C_{6})\cos Ax+(C_{5}-C_{6})\sin Ax)\\ \end{array} \right. \tag{11}\]  
The following boundary conditions are applied to the left end of the coal-bearing zone in the THR:  
\[\left\{ \begin{array}{l}\theta_{2}\mid_{x=0}=A(C_{5}-C_{6})\\ M_{2}\mid_{x=0}=2A^{2}C_{5}EI\\ F_{2}\mid_{x=0}=-2A^{3}(C_{5}+C_{6})EI\\ \omega_{2}\mid_{x=0}=C_{6}+\frac{q\cos\alpha}{k}\end{array} \right. \tag{12}\]  
(3) Junction between different zones  
At the junction between different zones (\(x=0\), that is, at the coal wall), the continuity of stress-induced deformation is satisfied. Hence,  
\[\left\{ \begin{array}{l}\omega_{1}\mid_{x=0}=\omega_{2}\mid_{x=0}\\ \theta_{1}\mid_{x=0}=\theta_{2}\mid_{x=0}\\ M_{1}\mid_{x=0}=M_{2}\mid_{x=0}\\ F_{1}\mid_{x=0}=F_{2}\mid_{x=0}\\ \end{array} \right. \tag{13}\]  
By solving Equation (13), we have  
\[\left\{ \begin{array}{l}C_{3}=-\frac{lq\cos a\left(2A^{2}EI+AIT+T\right)}{2A^ {2}EIT}\\ C_{4}=\frac{q\cos a\left(4A^{3}EI+AkT^{2}+2 kl\right)}{4A^{3}EIk}\\ C_{5}=-\frac{p_{2}\cos\alpha}{4A^{2}EI}\\ C_{6}=\frac{lq\cos a(l+2)}{4A^{3}EI}\\ \end{array} \right. \tag{14}\]  
By substituting \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) into Equation (3), we obtain the equations of the rotational angle, bending moment, shear force, and deflection of the free overhang (\(-l\), \(0\)) in the THR:  
\[\left\{ \begin{array}{l}\theta_{1}=\frac{q\cos a\left(3\left(\left(A^{2}x^{2} -1\right)+A^{2}x^{3}+3A^{2}(Ax-1)\right)\right.}{6A^{2}EI}\\ M_{1}=-\frac{1}{2}q\cos\alpha(l+x)\\ F_{1}=-q\cos\alpha(l+x)\\ \omega_{1}=\frac{q\cos a\left(\begin{array}{c}A^{3}\left(\begin{array}{c}- 12 EIkx(2l+x)+24 EIT\\ +kTx^{2}\left(6l^{2}+4 lx+x^{2}\right)\\ -12A^{2}kl^{2}Tx+6 AklT(1-2x)+12 klT\end{array}\right)\\ \end{array}\right)}{24A^{3}EIkT}\\ \end{array} \right. \tag{15}\]By substituting \(C_{5}\) and \(C_{6}\) into Equations (10) and (11), we obtain the equations of the rotational angle, bending moment, shear force, and deflection of the coal-bearing zone (0, +\(\infty\)) in the THR:  
\[\left\{ \begin{array}{l}\theta_{2}=-\frac{lqe^{-Ax}((Al+1)\cos Ax+\sin Ax)\cos \alpha}{2A^{2}EI}\\ M_{2}=-\frac{lqe^{-Ax}((Al+2)\sin Ax+Al\cos Ax)\cos\alpha}{2A}\\ F_{2}=lqe^{-Ax}((Al+1)\sin Ax-\cos Ax)\cos\alpha\\ \omega_{2}=\frac{1}{4}q\cos\alpha\left(\frac{le^{-Ax}((Al+2)\cos Ax-Al\sin Ax)} {A^{3}EI}+\frac{4}{k}\right)\end{array} \right. \tag{16}\]  
Given the above, we obtain the bending moment equation of the THR in the UMC throughout the mining process:  
\[M=\left\{ \begin{array}{l}-\frac{1}{2}q\cos\alpha(l+x)^{2}\to x\in[-l,0]\\ -\frac{lqe^{-Ax}((Al+2)\sin Ax+Al\cos Ax)\cos\alpha}{2A}\to x\in[0,+\infty] \end{array} \right. \tag{17}\]  
Therefore, the maximum bending moment of the THR ahead of the coal wall distance \(x\) is:  
\[x=-\frac{2\arctan\left(1+Al-\sqrt{2+2 Al+A^{2}l^{2}}\right)}{A} \tag{18}\]  
According to the maximum tensile stress criterion and previous studies [21], coal-measure formation rocks are most susceptible to tensile fracture. The deep beam is usually fractured in the upper boundary of the roof in front of the coal wall.