## 2 Numerical Simulation of Anchored Structural Surface
### Assessment of Bol Yield  
With the increase in the shear displacement of the structural surface, for the bolt in the area near the structural surface, the axial force increased due to its tensile action. In addition, the shear force of the bolt increased due to the shear action of the surrounding rock on the bolt. For this kind of bolt in the tension-shear coupled state, the [PERSON] criterion can be used to judge whether the bolt will yield. The expression of the Von Mises criterion is:  
\[\frac{1}{2}\left[\left(\sigma_{x}-\sigma_{y}\right)^{2}+\left(\sigma_{y}- \sigma_{z}\right)^{2}+\left(\sigma_{z}-\sigma_{x}\right)^{2}+6\left(\tau_{xy}^ {2}+\tau_{yz}^{2}+\tau_{zx}^{2}\right)\right]=\sigma_{s}^{2} \tag{1}\]  
For the bolt in the rock mass of the structural surface, it is mainly affected by the axial stress \(\sigma_{z}\) and shear stress \(\tau_{xy}\), so the [PERSON] criterion can be simplified to:  
\[\sqrt{\sigma^{2}+3\tau^{2}}=\sigma_{s} \tag{2}\]  
where \(\sigma\) is the axial stress of the bolt, \(\tau\) is the shear stress of the bolt, and \(\sigma_{s}\) is the yield strength of the bolt. When the structural surface is dislocated, the bolt near the structural surface is deformed, and if the internal stress at a point in the bolt satisfies Equation (2), the point yields and begins \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Name**} & **Bulk Modu-** & **Shear Modu-** & **Cohesion/MPa** & **Friction Angle/\({}^{s}\)** & **Tensile Strength/MPa** \\ \hline Coal & 9.9 & 6.5 & 8 & 32 & 2 \\ Sandstone & 15.6 & 12.7 & 11 & 43 & 5 \\ Granite & 43.9 & 30.2 & 14 & 52 & 8 \\ Anchoring & 6.9 & 5.6 & 12 & 30 & 6 \\ agent & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of surrounding rock mass and anchoring agent. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Name**} & **Elastic Modulus /GPa** & **Poisson’s Ratio** & **Yield Strength /MPa** \\ \hline Bolt & 210 & 0.32 & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical parameters of bolt. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Name**} & **Normal Stiffness/GPa \(\cdot\) m\({}^{-1}\)** & **Shear Stiffness/GPa \(\cdot\) m\({}^{-1}\)** & **Cohesion /MPa** & **Friction Angle/\({}^{c}\)** \\ \hline Upper rock mass–lower & 2000 & 2000 & 0.1 & 20 \\ rock mass & & & & \\ Anchoring & & & & \\ agent–surrounding & 10,000 & 10,000 & 1 & 30 \\ rock mass & & & & \\ Bol–anchoring agent & 70,000 & 70,000 & 10 & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of contact surface. to enter the plastic state. According to Equation (2), how much the axial stress and shear stress each account for when the bolt yields can be calculated, and thus, the degree of contribution of the axial stress and shear stress to the yielding of the bolt can be determined. With the increase in the shear displacement of the structural surface, the stress on the bolt is the greatest near the structural surface. Taking the structural surface of sandstone as an example, the Von Mises stress of the cross-section at the location of the structural surface on the bolt when it yielded is shown in Figure 4, which shows that the stress distribution on the cross-section was not uniform. The Von Mises stress at each node on the cross-section was determined and averaged. When the average Von Mises stress reached 400 MPa, the bolt was considered to have yielded. Figure 5 shows the average Von Mises stress of the bolt versus the shear displacement at the bolt-structural surface intersection. When the bolt yielded in the anchored structural surface, the shear displacements of the structural surfaces of the coal, sandstone, and granite were 5.363 mm, 5.030 mm, and 4.624 mm, respectively. With the structural surface dislocation, it can be seen that the trends of the Von Mises stress of the bolt in the different lithologies of structural surfaces were roughly the same.