## 2 Numerical Simulation of Anchored Structural Surface
### Deformation and Stress Characteristics of the Bolt  
The Von Mises stress intensity of the bolt when it yielded in the coal, sandstone, and granite structural surfaces is shown in Figure 9. The maximum shear stress is shown in Figure 10, and the axial stress is shown in Figure 11. It can be seen that the areas where the bolt deformation occurred were mainly distributed near the structural surfaces. From the simulation results, when the bolt yielded, the lengths of the bolt deformation areas in the structural surfaces of the coal, sandstone, and granite were 12.3193 cm, 11.9166 cm, and 11.4171 cm, respectively, which are approximately the same as the damage ranges of the anchoring agents. The deformation areas of the bolt were mainly concentrated near the structural surfaces, and the areas of the bolt far away from the structural surfaces were almost not deformed. The local deformation of the bolt was roughly \"S\" shaped.  
Figure 8: The change trend of the damage range of the anchoring agent under structural surface dislocation.  
Figure 9: Von Mises stress distribution at bolt yield (Pa).  
Figure 10: Maximum shear stress distribution at bolt yield (Pa).  
Figure 11: Axial stress distribution at bolt yield (Pa).  
Figure 12: Stress distribution curve along the length of the bolt when it yielded.  
The stress state of each part of the bolt was monitored, and the stress distribution diagrams of the bolt are shown in Figures 10 and 11. It can be seen that the shear stress on the bolt was mainly concentrated near the structural surface and the angle between the maximum shear stress and the structural surface is approximately 45\({}^{\circ}\). The axial stress of the bolt shows an approximately origin-symmetric stress distribution on both sides of the structural surface, where the tensile stress is positive, and the compressive stress is negative. With the shear displacement of the structural surface, the bolt deformation appears as a tensile stress zone and a compressive stress zone. The simulation results show that when the bolt yielded, the average axial stresses on the bolt near the structural surfaces of the coal, sandstone, and granite were 221.36 MPa, 201.78 MPa, and 188.69 MPa, respectively, and the average shear stresses were 114.01 MPa, 194.56 MPa, and 230.63 MPa. It was found that the weaker the lithology of the structural surface rock was, the larger the deformation area of the bolt, the larger the tensile stress of the bolt, and the smaller the corresponding shear stress.  
The distributions of the average axial stress and average shear stress along the length of the bolt when it yielded in the structural surfaces of coal, sandstone, and granite rock are shown in Figure 12. It can be seen that the areas where deformation occurred on the bolt were mainly concentrated near the structural surface, and the maximum value of stress was at the position to the right of the structural surface. This mild asymmetry was due to the fact that the loading was applied to the rock below the structural surface, which made the deformation of the bolt at the slightly lower position near the structural surface somewhat larger. When the bolt yielded, the contribution of the axial stress and shear stress to the yielding varied among the different lithologies. According to Equation (2), the contribution of the axial stress was 30.63% when the bolt yielded in the coal structural surface, 25.25% when the bolt yielded in the sandstone structural surface, and 22.25% when the bolt yielded in the granite structural surface. At this time, the ratios of axial stress to shear stress at the structural surface of the bolt were 1.94, 1.04, and 0.82, respectively, which shows that the weaker the lithology of the structural face was, the higher the axial stress generated by tensile deformation, and the lower the shear stress generated by the shear action of the surrounding rock.