## 4 Computational Results and Analysis
### Influence of the Fracture Angle in Surrounding Rocks on the Mechanical Properties of the Bolts  
Fractures, as an intrinsic component of rock mass, also affect the mechanical properties of the bolts. The fracture angle (the angle between a fracture and a bolt) was set to 90\({}^{\circ}\), 60\({}^{\circ}\), 45\({}^{\circ}\), and 30\({}^{\circ}\), of which the computations are represented by Schemes 8 to 11 in Table 1. In the computation, fractures were formed by directly deleting balls. The fractures were 90 mm long and 3 mm thick, with centers at the same location on the axis and 50 mm from the free face. Due to the different fracture angles, bolts are intersected by fractures within different ranges, in which the bolt stress is always affected by fracturing; this range is called the fracture zone, as shown in Figure 17.  
The stress distributions on bolts under a fracture angle of 90\({}^{\circ}\) are illustrated in Figure 18a. In the initial stage of loading (0.5 mm and 1.5 mm), the stress distribution is on the bolts similar to that in homogeneous surrounding rocks, both in a first increase and then decrease trend. In the middle stage of loading (3.5 mm and 4.5 mm), the stress on the bolts tends to decrease in the fracture zone, thus forming a valley. Meanwhile, the peak stress changes from the original one into two peaks that are distributed on both sides of the fracture, thus rendering the stress distribution curves M-shaped. In the final stage of loading, that is, failure of the anchored rock mass (4.6 mm), the overall bolt stress increases while still decreasing in the fracture zone, while the peak stress in the deeper part of the bolts exceeds that in the shallow part. Figure 16: Values of evaluation indices for the support capacity of bolts in rock mass with different strengths. Figure 17: Numerical model of an anchored rock mass with fractures. The stress distribution curves of bolts under a fracture angle of 60\({}^{\circ}\) are shown in Figure 19a. As the rock mass is compressed, the overall stress on the bolts gradually increases, while that in the fracture zone increases abruptly, forming a peak. In the initial and middle stages of loading (1.5 mm and 2.5 mm), there are three peaks on the stress distribution curves, in which two peaks are distributed on both sides of the fracture, and their values reduce successively with the increasing embedded depth of the bolts. In the middle and late stages of loading (3.5 mm and 4.4 mm), the peak stress in the shallow part gradually shifts to the fracture zone at greater depth. Under the condition, there are only two peaks in the stress distribution, and that in the fracture zone is larger. The stress distributions along bolts when the fracture angle is 45\({}^{\circ}\) are depicted in Figure 20a. Throughout the loading process, the bolt stress first increases, then decreases, and there is only one peak, which is located in the fracture zone. The stress distributions along bolts under a fracture angle of 30\({}^{\circ}\) are illustrated in Figure 21a. It can be seen that the stress distribution is similar to that in an intact rock mass; that is, the stress increases and then decreases. The bolt stress in the fracture zone does not decrease or increase abruptly. Throughout the loading process, the bolt stress has the only peak value, which is located in front of the fracture zone, and it does not shift to greater depth. Figure 19: Stress distributions and fracture distribution on bolts under a fracture angle of 60\({}^{\circ}\). (**a**) Stress distribution of bolts; (**b**) Fracture distribution at failure. Figure 18: Stress distributions and fracture distribution on bolts under a fracture angle of 90\({}^{\circ}\). (**a**) Stress distribution of bolts; (**b**) Fracture distribution at failure. The distributions of secondary fractures at the failure of anchored rock mass containing fractures with different angles are shown in Figures 18b, 19b, 20b and 21b.