## 5 Analysis of Factors Affecting Bearing Capacity of Inclined Pillar
### Dip Effect on the Bearing Capacity of Pillar  
Similarly, when the in-situ stress ratio \(k\) is given, the variation of _CFSL_ with dip angle is analyzed. The partial derivatives of \(\theta\) on both sides of Equation (16) are obtained. \[\frac{\partial(\tan\beta)}{\partial\theta}=\frac{2(1-k)\big{[}(1+k)\cos 2 \theta+(1-k)\big{]}}{\big{[}(1+k)+(1-k)\cos 2\theta\big{]}^{2}} \tag{27}\]  
From Equation (21), when \(\tan\beta\) is a function of \(\theta\), its monotonicity is determined by \(k\). When \(k\) = 1.0, then \(\frac{\partial(\tan\beta)}{\partial\theta}=0\), which indicates that \(\tan\beta\) is constant. When \(k\) \(\
eq\) 1.0, the two cases of \(k\)\(\in\)[0.3,1.0) and \(k\)\(\in\)(1.0, 3.5] are analyzed, respectively. Let \(\theta_{0}\) be obtained. \[\theta_{0}=\frac{1}{2}\text{arccos}\frac{k-1}{k+1} \tag{28}\]  
When \(k\)\(\in\)[0.3,1.0), the monotonicity of function \(f\) is opposite to that of function \(\tan\beta\). From differential knowledge, let \(\frac{\partial(\tan\beta)}{\partial\theta}\geq 0\); it shows that the monotone increasing interval of function \(\tan\beta\) is \(\theta\)\(\in\)[\(\theta_{0}\), 90\({}^{\circ}\)], and the monotone decreasing interval is \(\theta\)\(\in\)[0\({}^{\circ}\), \(\theta_{0}\)]. Thus, the monotone increasing interval of function \(f\) is \(\theta\)\(\in\)[0\({}^{\circ}\), \(\theta_{0}\)], and the monotone decreasing interval is \(\theta\)\(\in\)[\(\theta_{0}\), 90\({}^{\circ}\)]. Therefore, when \(k\)\(\in\)[0.3,1.0), the _CFSL_ first increases and then decreases with the increase of inclination \(\theta\), and the _CFSL_ achieves the maximum value at \(\theta=\theta_{0}\). In the same way, when \(k\)\(\in\)(1.0, 3.5], the monotonicity of function \(f\) is also opposite to that of function \(\tan\beta\). Let \(\frac{\partial(\tan\beta)}{\partial\theta}\geq 0\), the monotone increasing interval is \(\theta\)\(\in\)[0\({}^{\circ}\), \(\theta_{0}\)], and the monotone decreasing interval is \(\theta\)\(\in\)[\(\theta_{0}\), 90\({}^{\circ}\)]. Therefore, the monotone increasing interval of function \(f\) is \(\theta\)\(\in\)[\(\theta_{0}\), 90\({}^{\circ}\)], and the monotone decreasing interval is \(\theta\)\(\in\)[0\({}^{\circ}\), \(\theta_{0}\)]. It can be seen that the _CFSL_ decreases at first and then increases with the increase of inclination \(\theta\), and the _CFSL_ reaches the minimum value at \(\theta=\theta_{0}\). To further illustrate the correctness of the above two conclusions, when \(k\)\(\in\)[0.3,1.0], the in-situ stress ratio \(k\) increases with the increment of 0.1 from 0.3, the variation law of the _CFSL_ with \(\theta\) is calculated at each value of \(k\), as shown in Figure 13a, which shows that at each value of \(k\), the _CFSL_ decreases at first and then increases with the increase of inclination angle \(\theta\). At each value of \(k\), when the _CFSL_ obtains the maximum value, the corresponding angles are 61.29\({}^{\circ}\), 57.69\({}^{\circ}\), 54.74\({}^{\circ}\), 52.24\({}^{\circ}\), 50.08\({}^{\circ}\), 48.19\({}^{\circ}\), and 46.51\({}^{\circ}\), respectively. Similarly, when \(k\)\(\in\)[1.0, 3.5], the in-situ stress ratio \(k\) increases with an increment of 0.4 from 1.1; at this  
Figure 12: Relationship between the contribution factor of shear load (_CFSL_) and the in-situ stress ratio. time, the variation of \(CFSL\) with \(\theta\) is shown in Figure 13b, and the \(CFSL\) decreases at first and then increases with the increase in inclination \(\theta\). At each value of \(k\), when the \(CFSL\) obtains the minimum value, the corresponding angles are \(43.64^{\circ}\), \(39.23^{\circ}\), \(35.96^{\circ}\), \(33.40^{\circ}\), \(31.32^{\circ}\), \(29.59^{\circ}\), and \(28.13^{\circ}\), respectively. As shown by Figure 13, the variation of \(CFSL\) with inclination angle is consistent with that of theoretical analysis.