# Numerical Simulation Analysis Method for Rockburst Control in Deeply Buried Caverns
## 3 Numerical Analysis Method for Rockburst Control with Anchor Rod Reinforcement
### Mathematical Model of Implicit Anchor Rod Element  
The role of anchor rod support can be summarized as consolidation combination and arch suspension [31]. When an anchor rod is inserted into the rock mass it will be combined with the surrounding rock mass, improving the bending and shear strength and increasing the internal resistance in the surrounding rock against rockburst. When the number of anchor rods increases to a certain extent, a bearing ring will be formed which can improve the integrity of the surrounding rock, effectively reducing the stress concentration, and increase the internal factors of rock mass resistance to rockburst. Assuming that there is no relative sliding between the anchor rod and the surrounding rock and that the support effect of the anchor is axisymmetric, the support effect of the anchor can be simulated by using an implicit cylindrical element. Such cylindrical elements can be simplified as an axisymmetric problem with anisotropic layer occurrence. Its stress-strain relationship can be expressed as follows:  
\[\begin{bmatrix}\sigma_{r}\\ \sigma_{\theta}\\ \tau_{\theta\theta}\end{bmatrix}=\frac{E_{r}}{m}\begin{bmatrix}1-n\mu_{r}^{2 }&(1+\mu_{r})\mu_{\theta}&0\\ (1+\mu_{r})\mu_{\theta}&(1-\mu_{r}^{2})/n&0\\ 0&0&b/2n\end{bmatrix}\begin{bmatrix}\varepsilon_{r}\\ \varepsilon_{\theta}\\ \varepsilon_{r\theta}\end{bmatrix}=[D_{e}][\varepsilon] \tag{5}\]  
where \(m=(1+\mu_{r})b\), \(b=1-\mu_{r}-2n\mu_{\theta\theta}^{2}\), and \(n=E_{r}/E_{\theta}\), \(E_{r}\) represent the radial elastic modulus, \(\mu_{r}\) represents radial Poisson's ratio, \(E_{\theta}\) represents the axial elastic modulus, and \(\mu_{\theta}\) represents [PERSON]'s axial ratio. The radial elastic modulus is taken based on the combination of anchor rods and rock masses. The axial elastic modulus is taken based on the shear stiffness of anchor rods and reinforced rock masses. Given that the anchor rod element is implicit in the three-dimensional rock mass element, we can establish the cylindrical isoparametric elements shown in Figure 1. According to the virtual work principle of the finite element, we can calculate the stiffness matrix of the anchor rod element by integrating the isoparametric elements as follows:  
\[\begin{bmatrix}K_{g}\end{bmatrix}=\int{[B]^{T}[D_{e}][B]d_{c}}=\int_{0}^{1} \int_{-1}^{1}[B]^{T}[D_{e}][B]2\pi r|J|d_{r}d_{z} \tag{6}\]  
where \([B]\) is the strain matrix of the column element and \(|J|\) is the value of the Jacobian matrix of the anchor column element. Based on the basic theory and the interpolation principle of finite element stiffness matrix calculation [32], the additional stiffness \([K_{R}]\) generated by the anchor rod in the rock mass element can be added to the stiffness matrix of the rock mass element according to the principle of stress-strain equivalence as follows:  
\[\begin{bmatrix}K_{R}\end{bmatrix}=[N]^{T}[K_{g}][N] \tag{7}\]  
where \([N]\) is the conversion matrix between the anchor rod element and the rock mass element and the specific calculation can be found in reference [32]. Superimposing the additional stiffness of Equation (7) into the rock mass element can reasonably and effectively reflect the effect of the anchor on rockburst containment.  
Using implicit anchor rods, we can model the anchor rod elements and rock mass elements separately, which facilitates the analysis and calculation of various anchor rod arrangements, making modeling simple and effectively improving the calculation speed of anchor rod support. Using the equivalence principle of finite element interpolation to add the anchor rod element stiffness into the rock mass element stiffness can not only reflect the reinforcement and support effect of the anchor itself but also fully reflect the joint action of the anchor and the surrounding rock.