# Numerical Simulation Analysis Method for Rockburst Control in Deeply Buried Caverns
## 3 Numerical Analysis Method for Rockburst Control with Anchor Rod Reinforcement
### Analysis of the Rockburst Control Effect of Anchor Rod Support  
According to the mechanism of anchor reinforcement of surrounding rock, anchor support mainly controls the rockburst in two ways. On the one hand, after the anchor rod is inserted into the rock mass, the strength of the rock mass is improved through consolidation combination and arch suspension to resist rockburst. On the other hand, the cracked rock mass and the complete rock mass are combined to form a complement through the system anchors which limits the deformation of the surrounding rock, reduces the stress concentration and energy accumulation, and curbs the occurrence of rockburst. After the anchor rod is inserted into the rock mass element, it forms a cohesive bearing body with the surrounding rock, resulting in an increase in the stiffness of the combined element. Therefore, the elastic modulus \(E_{Y}\) and compressive strength \(R_{Y}\) of the combined rock mass can be increased based on the stiffness ratio between the rock mass element and the anchor rod element. \[ \begin{array}{c}E_{Y}=\frac{E_{s}V_{s}+mE_{s}V_{s}}{V_{s}+mF_{s}}\\ R_{Y}=\frac{R_{s}V_{s}+mR_{s}^{2}V_{s}}{V_{s}+mF_{s}}\end{array}  \tag{8}\]  
where \(E_{s}\) is the elastic modulus of the rock mass element, \(R_{s}\) is the compressive strength of the rock mass element, \(V_{s}\) is the volume of the rock mass element, \(E_{g}\) is the elastic modulus of the anchor rod element, \(R_{g}\) is the compressive strength of the anchor rod element, \(V_{g}\) is the element volume of the anchor rod element, and n represents the number of anchor rods inserted into the rock mass element. The elastic load part (\(F_{e}\)) of the total load \(F\) released during excavation will be quickly released. If the anchor rod support is provided promptly and the anchor rod passes through the cracked rock mass and connects with the stable rock mass to form a whole, the anchor rod and surrounding rock will jointly bear part of the plastic excavation release load \(Fp\), as follows:  
\[F_{p}=\alpha F=c(1-S)F \tag{9}\]  
where \(S\) is the elastic load release coefficient and \(\alpha\) is the plastic load coefficient jointly borne by the anchor rod and surrounding rock. \(c\) is the coefficient of partial plastic load  
Figure 1: Mathematical Model of the Anchor Rod Element. (1–4 are the four node numbers of the anchor rod element). release during anchor rod construction which can be determined based on the anchor rod construction plan and surrounding rock conditions. After applying anchor rod support, the incremental variable plastic damage stiffness method is used to calculate _Fp_. Iterative methods for the determination of various coefficients can be found in reference [33]. \[[K_{e}]\{\Delta\delta_{i}\}_{j}=\big{\{}\Delta F_{pi}\}+[K_{D}]\{\Delta\delta_{ i}\}_{j-1} \tag{10}\]  
where \([K_{D}]\) is the stiffness matrix of the damaged element. After the iteration of each load, the damage stiffness is corrected. The rockburst grade is judged according to Equation (1) or Equation (3). The stiffness matrix of the rockburst element is reduced by the power exponential function according to the grade until it can be replaced by the air element. During the iterative calculation process, for the anchor rod element that has not yielded, the additional load \(\{f\}\) generated by the anchor stress increment \(\Delta\sigma\) in this iteration process of this level can be inversely superimposed on the rock mass and then substituted into Equation (10) for the next iteration, as follows:  
\[\{f\}=A_{g}\Delta\sigma[L,M,N,-L,-M,-N]^{T} \tag{11}\]  
where \(A_{g}\) is the area of the anchor rod and \(L\), \(M\), and \(N\) are the cosines of the anchor rod axis and the calculation coordinates. When the anchor rod stress exceeds the yield strength, the anchor rod no longer provides the support reaction. Due to the passive force on the anchor rod during the iteration process, the deformation stress continues to increase.