# Mechanisms of Tunnel Rockburst Development Under Complex Geostress Conditions in Plateau Regions
## 3 FLAC3D Secondary Development and Validation
### Validation of Russense Rockburst Criterion Visualization Development  
The key to visualizing the Russense rockburst criterion in numerical models lies in the calculation of the tangential stress of the surrounding rock; thus, validating the FISH program primarily depends on verifying the accuracy of tangential stress calculations. Generally, there are two main methods for determining the tangential stress of the surrounding rock: analytical and numerical approaches. Among these, the Kirsch analytical solution based on elastic mechanics (Equation (4)) can accurately calculate the distribution of tangential stress under elastic conditions and is therefore suitable for validating the accuracy of the FISH-based tangential stress calculation program developed in this study. Since the Kirsch analytical solution is limited by the assumption that the surrounding rock is continuous, homogeneous, and isotropic, the FLAC3D numerical model was implemented using an elastic model for comparison. \[\sigma_{\theta}=\frac{1}{2}(1+\lambda)p(1+\frac{a^{2}}{r^{2}})-\frac{1}{2}( \lambda-1)p(1+3\frac{a^{4}}{r^{4}})cos2\theta \tag{4}\]  
where \(\theta\) is the angle between the line connecting the element and the tunnel center and the horizontal axis; \(a\) is the tunnel radius; \(r\) is the distance between the element and the tunnel center; \(p\) is the initial vertical stress acting on the rock mass; and \(\lambda\) is the lateral pressure coefficient. Taking a tunnel with a burial depth of 500 m as an example, the tangential stress was calculated using the Kirsch solution based on the elastic physical and mechanical parameters listed in Table 1, and the corresponding contour map was generated using MATLAB 2022a, as shown in Figure 3a. The calculation results obtained under the same conditions using the FISH program developed in Section 3.2 are shown in Figure 3b. The tangential stress contour map obtained from the numerical solution (Figure 3b) shows good agreement with that from the analytical solution (Figure 3a), with both maps exhibiting nearly identical ranges. To further compare the accuracy of the results, six monitoring points were placed sequentially at the tunnel crown, representing the stress concentration area, with each point spaced 1 m apart from the tunnel wall (Figure 4). The results from both the numerical and analytical solutions are detailed in Table 3. Comparison reveals that the numerical solution closely matches the analytical solution, with an average error of only 2.47% across the monitoring points, a minimum error of just 0.3%, and a maximum error not exceeding 5%. In FLAC3D, the actual monitoring points correspond to the nearest grid nodes to the specified locations, so slight discrepancies between the monitored results and theoretical values are inevitable due to mesh discretization. Therefore, it can be concluded that the numerical results are highly consistent with the theoretical solutions, validating the correctness of the FISH program. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Monitoring Point**} & **Analytical** & **Numerical** & \multirow{2}{*}{**Error (\%)**} \\  & **Solution (\(\times\)10\({}^{7}\) Pa)** & & \\ \hline \#1 & 2.9485 & 2.9815 & 3.3\% \\ \#2 & 2.5683 & 2.5970 & 2.9\% \\ \#3 & 2.3385 & 2.3241 & 1.4\% \\ \#4 & 2.1876 & 2.2322 & 4.5\% \\ \#5 & 2.0812 & 2.0837 & 0.3\% \\ \#6 & 2.0113 & 2.0349 & 2.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of tangential stress between theoretical results and numerical results. Figure 3: Tangential stress cloud of analytical and numerical solutions. (**a**) Tangential stress contour map from analytical solution. (**b**) Tangential stress contour map from numerical solution.