# Study on Asymmetric Support of Anchor Cable with C-Shaped Tube in Inclined Coal Seam Roadway
## 3 Roadway Deformation Law in Inclined Coal Seam
### Brief Analysis of Roadway Deformation  
The original condition of stress in coal rock is shattered during roadway excavation, and the stress is redistributed. The tangential stress increases gradually as the radial stress decreases. During the secondary stress distribution process, some surrounding rocks experience concentrated stress. When the concentrated stress exceeds the strength of the surrounding rock, deformation or even failure occurs, which forms a circular or elliptical distribution area consisting of a plastic zone, an elastic zone, and an original stress field zone from the inside out. The stress state of the unit in the surrounding rock of the rectangular roadway is shown in Figure 13.  
Figure 12: Vertical stress of roadway stress monitoring points under the initial support scheme. Figure 13: Stress state of unit in surrounding rock of rectangular roadway. (1) The rock mass in the elastic zone satisfies Hooke's law. According to the theory of elastic mechanics, the stress at a point in the elastic zone is given by [42]:  
\[\left\{ \begin{array}{l}\sigma_{\rm r}=p-[\frac{2p(2\sin\varphi-1)+\sigma_{\rm c }(1-\sin\varphi)}{2\sin\varphi}].(\frac{R_{\rm P}^{2}}{r^{2}})\\ \sigma_{\rm\theta}=p+[\frac{2p(2\sin\varphi-1)+\sigma_{\rm c}(1-\sin\varphi)}{ 2\sin\varphi}].(\frac{R_{\rm P}^{2}}{r^{2}})\end{array} \right. \tag{11}\]  
where \(\sigma_{\rm r}\) is the radial stress at the point, \(\sigma_{\rm\theta}\) is the tangential stress at the point, \(r\) is the distance from the point to the center of the circle, \(p\) is the stress before the excavation of the roadway at the point, \(\varphi\) is the angle of the rock internal friction,\(R_{p}\) is the radius of the plastic zone, and \(\sigma_{\rm c}\) is the uniaxial compressive strength of the rock. The displacement of the surrounding rock mass caused by the excavation of the roadway in the elastic zone is given by  
\[u_{1}=\frac{1+\mu}{2E}.[2p\sin\varphi+\sigma_{\rm c}(1-\sin\varphi)]-\frac{R_ {\rm P}^{2}}{r} \tag{12}\]  
(2) Assuming that the rock mass follows the Mohr-Coulomb criterion at yielding, the plastic criterion can be written as  
\[\sigma_{\rm\theta}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{\rm r}+\sigma_{ \rm c} \tag{13}\]  
Assuming that the rock mass in the plastic zone satisfies the condition of volume conservation under small deformation, it can be expressed as  
\[\left\{ \begin{array}{l}\varepsilon_{\rm\theta}=\frac{u_{2}}{r}\\ \varepsilon_{\rm r}=\frac{d\mu}{dr}\\ \varepsilon_{\rm\theta}+\varepsilon_{\rm r}=0\end{array} \right. \tag{14}\]  
where \(\varepsilon_{\rm\theta}\) is tangential strain at any point, \(\varepsilon_{\rm r}\) is radial strain at any point, and \(u_{2}\) is displacement at any point in the plastic zone. By combining Equations (12) and (14) and considering the displacement boundary continuity at the intersection of the elastic zones and plastic zones, the displacement of the surrounding rock at any point in the plastic zone can be obtained by solving the system of equations:  
\[u_{2}=\frac{(1+\mu)R_{\rm P}}{2 Er}.[2\;p\sin\varphi+\sigma_{\rm c}(1-\sin \varphi)]-\frac{R_{\rm P}^{2}}{r} \tag{15}\]  
Therefore, it can be seen from Equations (12) and (15) that after the excavation of the roadway, the displacement of a point in the surrounding rock not only depends on the mechanical properties of surrounding rock, such as elastic modulus, Poisson's ratio, uniaxial compressive strength, and internal friction angle, but also on the original stress at that point before the excavation of the roadway. The greater the stress in the stress field at a certain point before the excavation of the roadway, the greater the deformation of the surrounding rock.