# Study on Asymmetric Support of Anchor Cable with C-Shaped Tube in Inclined Coal Seam Roadway
## 2 Research on the Mechanical Characteristics of Anchor Cable with C-Shaped Tube
### A Brief Analysis of the Shear Strength of ACC Anchored Joint Plane  
When there is relative displacement occurring in the rock mass near a rock joint plane, the ability of the joint plane to resist rock mass movement is known as the shear strength of the joint plane itself. Extensive research has shown that the shear strength of the joint plane follows the Mohr-Coulomb criterion [37].  
\[\tau=\sigma\tan\varphi+c \tag{1}\]  
where \(\tau\) represents the shear strength of the joint plane; \(\sigma\) represents the normal stress acting on the joint plane; \(\varphi\) represents the internal friction angle of the joint plane; and \(c\) represents the cohesion strength of the joint plane.  
When an \(ACC\) is employed as a support system, during sliding displacement along the joint plane, the C-shaped tube undergoes compression, generating radial resistance to counteract the deformation of the joint plane. This resistance continues until the C-shaped tube fully encloses the anchor cable. The resulting circumferential load \(q_{c}\) provided by the C-shaped tube can be determined [38].  
\[q_{c}=\frac{E_{c}I_{c}}{3\pi r_{c}^{4}}u \tag{2}\]  
Figure 1: \(ACC\) structure diagram.  
Figure 2: \(ACC\) deformation under force schematic picture.  
In the equation, \(E_{c}\) denotes the elastic modulus of the C-shaped tube, \(I_{c}\) represents the moment of inertia of the cross-section of the C-shaped tube, \(r_{c}\) represents the outer diameter of the C-shaped tube, and \(u\) represents the closure displacement of the C-shaped tube.  
Consequently, the resistance \(Q_{c}\) provided by the C-shaped tube per unit length is  
\[Q_{c}=\int_{0}^{\pi}q_{c}\sin\theta\cdot d\theta=\int_{0}^{\pi}\frac{E_{c}I_{c} }{3\pi r_{c}^{4}}u\sin\theta\cdot d\theta=\frac{2E_{c}I_{c}}{3\pi r_{c}^{4}}u \tag{3}\]  
Since the contact area between the rock mass on either side of the joint plane and the C-shaped tube is limited, an influence range coefficient \(f\) is introduced to represent the extent of the joint plane's impact on the C-shaped tube. This allows us to determine the shear strength generated by the closure of the C-shaped tube within this specific influence range.  
\[\tau_{c}=\frac{2E_{c}I_{c}f}{3\pi r_{c}^{4}A}u \tag{4}\]  
where \(A\) is the area of the joint surface being anchored.  
The shear strength of the joint plane with anchor reinforcement consists of three components: the inherent shear strength of the joint plane, the shear strength induced by the \"nail\" action of the support component, and the shear strength caused by the effect of the axial force of the support component on the joint plane [39; 40]. By integrating the aforementioned analysis and incorporating further improvements, the shear strength of the \(ACC\) anchored joint plane can be determined.  
\[\tau_{ACC}=\tau_{0}+\tau_{c}+\tau_{aj}+\tau_{az} \tag{5}\]  
\[\tau_{0}=\sigma_{0}\tan\varphi+c \tag{6}\]  
\[\tau_{c}=\frac{2E_{c}I_{c}f}{3\pi r_{c}^{4}A}u \tag{7}\]  
\[\tau_{aj}=\frac{N_{ACC}}{A}\tan\varphi \tag{8}\]  
\[\tau_{az}=\frac{T_{ACC}}{A} \tag{9}\]  
where \(N_{ACC}\) is the normal force applied by the \(ACC\) to the joint plane perpendicular and \(T_{ACC}\) is the component of the resultant force of the axial force and shear force of the \(ACC\) in the direction parallel to the joint plane.