## 5 Analysis of Factors Affecting Bearing Capacity of Inclined Pillar  
### Stress Path Effect on the Bearing Capacity of Pillar  
The main factors affecting the stress path of the inclined pillar are the area extraction ratio, inclination, and the in-situ stress ratio and their changes in stress state and stress path. If the pillar failure obeys the Mohr-Coulomb strength theory, the relationship between the stress path under the control of three factors and the Mohr-Coulomb strength envelope is presented in Figure 10. Point \(A\) represents the average stress state of the vertical pillar controlled by the area extraction ratio, and Point \(B\) represents the ultimate stress state at the same position as Point \(A\). Point \(C\) represents the average stress state of the inclined pillar controlled by the area extraction ratio, and Point \(D\) represents the ultimate stress state at the same position as Point \(C\). Point \(E\) represents the average stress state of the vertical pillar controlled by the inclination angle, and Point \(D\) represents the ultimate stress state of the inclined pillar at the same position as Point \(E\). Point \(C\) represents the average stress state of the vertical pillar controlled by the in-situ stress ratio, and Point \(F\) represents the ultimate stress state of the inclined pillar at the same position as Point \(C\). From Figure 10, the straight arrow from Point \(A\) to Point \(B\) (\(\overset{\rightarrow}{AB}\)) and the straight arrow from Point \(C\) to Point \(D\) (\(\overset{\rightarrow}{CD}\)) represent the stress path controlled by the area extraction ratio, the arc arrow from Point \(E\) to Point \(D\) (\(\overset{\rightarrow}{ED}\)) and the arc arrow from Point \(E\) to Point \(F(\overset{\rightarrow}{EF}\)) represent the stress path controlled by the inclination angle, and the curved arrow from Point \(C\) to Point \(F\) (\(\overset{\rightarrow}{CT}\)) represent the stress path controlled by the in-situ stress ratio. The straight arrows \(\overset{\rightarrow}{AB}\) and \(\overset{\rightarrow}{CD}\) represent the stress path controlled by the area extraction ratio, where the straight arrow \(\overset{\rightarrow}{AB}\) represents the stress evolution path on the vertical pillar, and the straight arrow \(\overset{\rightarrow}{CD}\) represents the stress evolution path on the inclined pillar. With the increase of area extraction ratio, the load on the pillar increases gradually, and the generalized Mohr circle radius increases gradually. The generalized Mohr circle moves outward on a straight line with the slope of \(\tan\beta\), and the closer the distance to the Mohr-Coulomb strength envelope, the easier it is for the pillar to reach the ultimate stress state. As a result, it demonstrates that the greater the area extraction ratio and the larger the load on the pillar, the greater the risk of pillar instability. Compared with the vertical  
Figure 10: Stress evolution path of inclined pillars. pillar, it is easier for the stress path of the inclined pillar to reach the ultimate stress state and, hence, the probability of instability is greater. The arc arrows \(\overset{\rightarrow}{ED}\) and \(\overset{\rightarrow}{EF}\) represent the stress path controlled by the inclination angle, where the arc arrow \(\overset{\rightarrow}{ED}\) represents the stress evolution path when the shear stress component is positive and the arc arrow \(\overset{\rightarrow}{EF}\) represents the stress evolution path when the shear stress component is negative. The results caused by the two stress paths are the same but in different directions. In this situation, the radius of the generalized Mohr circle is invariant. With the increase of dip angle, the compressive stress component on the pillar decreases, the shear stress component increases, and the distance between the generalized Mohr circle and the strength envelope gradually decreases until it is tangent to the envelope.