## 2 Methodology  
### Simulation Theory  
Numerical simulation of the bolt mechanical properties was conducted using the finite difference software FLAC3D 3.00. Rock bolt elements were employed to simulate the bolts, and these were defined by geometric parameters, material properties, and coupled spring parameters. A line segment between two structural nodes is represented as a single element component, with the components between two nodes sharing the same symmetric cross-section parameters. The rock bolt element effectively combines the functions of beam elements and bolt elements, making it suitable for simulating bolts subjected to both normal and axial frictional forces [22]. Each structural element component possesses its own local coordinate system, and it is used to specify inertial moments, distributed loads, and defined force and moment signs. As shown in Figure 1, the local coordinate system of the components of the simulated bolts was defined by the positions of its two nodes (1 and 2) and the vector Y with the following rules: the central axis coincides with the X axis, the X axis direction is from Node 1 to Node 2, and the Y axis lies in the cross-sectional plane. The interaction between the bolt rod and the rock mass was simulated using coupling springs. These nonlinear, sliding coupling springs transmit forces and moments between structural element nodes and solid elements. The sign conventions for forces and moments on components are illustrated in Figure 2.  
Figure 1: The local coordinate system and 12-degree-freedom component. For Node 1, \(u_{1}\), \(v_{1}\), and \(w_{1}\) are translational displacements along the X axis, Y axis, and Z axis, respectively; and \(\theta_{x1}\), \(\theta_{y1}\), and \(\theta_{x1}\) are rotation angles around the X axis, Y axis, and Z axis, respectively. The symbols of the degrees of freedom of Node 2 are consistent with that of Node 1. During the process of bolt pullout, the axial behavior of the bolt can be described using a one-dimensional constitutive model, as expressed in the following equation:  
\[K=\frac{AE}{L}, \tag{1}\]  
where \(K\) represents the axial stiffness of the bolt, \(A\) denotes the cross-sectional area of the bolt, \(E\) stands for the elastic modulus, and \(L\) signifies the length of the bolt. The tensile yield strength \(F_{t}\) and compressive yield strength \(F_{c}\) of the bolt were determined, and these two limits must not be exceeded during the utilization of the bolt. When subjected to tensile loading, the bolt's interaction with the surrounding rock is simulated through tangential coupling springs. The function of these springs is analogous to the tangential action mechanism of grouted bolts that is depicted in Figure 3; therefore, the properties of the tangential coupling springs can represent the grouting properties. The shear stress on the contact surface between the bolt and the rock mass primarily considers its cohesive force and frictional force. The influence of tangential coupling springs around the bolt is mainly reflected through parameters such as the stiffness, cohesive force, internal friction angle, outer boundary radius of the bolt, and effective stress around the bolt. The relationship between them can be expressed by the following equation [26]:  
\[F_{s}/L=K_{s}\big{(}u_{p}-u_{m}\big{)}, \tag{2}\]  
where \(F_{s}\) is the shear force, \(K_{s}\) is the shear stiffness of the coupling spring, and \(u_{p}\) and \(u_{m}\) are the axial displacements of the rock bolt element and the surrounding medium, respectively. Figure 3: Mechanical model of the bolt element. Figure 2: The symbol protocol of the endpoint force and moment of the component. For Node 1, Fx1, Fy1, and Fz21 are forces along the X axis, Y axis, and Z axis, respectively; and Mx1, My1, and Mz1 are moments along the X axis, Y axis, and Z axis, respectively. The symbols of Node 2 are consistent with that of Node 1.