## 4 Strength Model of Inclined Pillar  
### Establishment of the Strength Model of Inclined Pillar  
An inclined pillar is loaded with compression and shear stress; the Mohr-Coulomb criterion is used to derive shear strength models for inclined pillars. According to the relationship between Mohr's circle and the Mohr-Coulomb strength envelope and considering the symmetry requirement of the strength criterion ([PERSON], 1975), as shown in Figure 8, point \(A\) is the intersection of the strength envelope and the \(\sigma\)-axis. Point \(B\) is the center of the average stress circle of the vertical pillar. Point \(C\) is the tangent point between the strength envelope and the average stress circle of the vertical pillar that is in the limit stress state. Point \(E\) is the center of the average stress circle of the inclined pillar. Point \(F\) is the tangent point between the strength envelope and the average stress circle of the inclined pillar that is in the limit stress state. Point \(D\) is the point of intersection that passes through the \(E\) point as the vertical line of the \(\sigma\)-axis. Then, some geometric relations are obtained.  
\[OE=EF=EG+GF=EG+HC \tag{11}\]  
where \(EG\) and \(HC\) are as follows:  
\[\left.\begin{array}{l}EG=OE\sin(\varphi_{m}-\beta)=\frac{DE}{\sin\beta}\sin (\varphi_{m}-\beta)\\ HC=BC-BH=BC(1-\sin\varphi_{m})\end{array}\right\} \tag{12}\]  
Figure 7: Stress path controlled by the in-situ stress ratio.  
Substituting Equation (12) into (11),  
\[DE=DE\sin(\varphi_{m}-\beta)+BC(1-\sin\varphi_{m})\sin\beta \tag{13}\]  
Where both \(DE=0.5\tau_{\theta}\) and \(BC=0.5S_{p0}\) are substituted into Equation (13), the limit equilibrium equation of shear stress on the pillar is obtained.  
\[\left[\tau_{\theta}\right]_{peak}=\frac{(1-\sin\varphi_{m})\sin\beta}{1-\sin( \varphi_{m}-\beta)}S_{p0} \tag{14}\]  
In Equation (14), when the right side is smaller than the left side, the pillar fails. Therefore, the shear strength of the pillar is obtained.  
\[T_{p}=\frac{(1-\sin\varphi_{m})\sin\beta}{1-\sin(\varphi_{m}-\beta)}S_{p0} \tag{15}\]  
where \(\beta\) is the inclination of a straight line, and the center of the stress circle is on the straight line.  
In this paper, \(\beta\) is called the equivalent internal friction angle of the pillar, where \(\tan\beta\)\(=0.5\tau_{\theta}/0.5\sigma_{\theta}\), and \(\tan\beta\) can be expressed by the in-situ stress ratio and dip angle.  
\[\tan\beta=\frac{\tau_{\theta}}{\sigma_{\theta}}=\frac{(1-k)\sin 2\theta}{(1+k) +(1-k)\cos 2\theta} \tag{16}\]  
Compared with the vertical pillar, the inclined pillar is additionally affected by shear load. The ratio of the shear strength of the inclined pillar to the strength of the vertical pillar is called the contribution factor of shear load (\(CFSL\)), which is represented by the symbol \(f\).  
\[f=\frac{T_{p}}{S_{p0}}=\frac{(1-\sin\varphi_{m})\sin\beta}{1-\sin(\varphi_{m}- \beta)} \tag{17}\]  
Figure 8: The relationship between the generalized Mohr circle and the strength envelope.  
The _CFSL_ characterizes the contribution of shear load to the total load and bearing capacity of the pillar. The larger the value of _CFSL_, the greater is the contribution of the shear load to the total load of the pillar and the smaller is the bearing capacity of the pillar. In particular, when \(f=0\), the shear load does not contribute to the total load; that is, the shear load is zero, and the bearing capacity of the pillar is the largest.