## 4 Results
### FEM Validation Results and Identification of Rockburst-Prone Areas
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**No.**} & \multicolumn{3}{c}{**Proposed LEM Approach**} & \multicolumn{3}{c}{**FEM Solution**} & \multicolumn{2}{c}{**Correlation**} \\ \cline{2-9}  & – & – & \(\alpha_{\text{1 LEM}}\) & \(\alpha_{\text{2 LEM}}\) & \(\alpha_{\text{1 FEM}}\) & \(\alpha_{\text{2 FEM}}\) & \(\alpha_{\text{1 fem}}\) \\  & \(\mathbf{F}_{min}\) & \(\mathbf{F}_{av}\) & **(deg.)** & **(deg.)** & **(deg.)** & **(deg.)** & **(deg.)** & **(deg.)** \\ \hline
1 & 0.84 & 0.89 & 56.9 & 78.8 & 51 & 79 & \\
2 & 0.55 & 0.69 & 49.3 & 85.4 & 47 & 84 & \\
3 & 0.73 & 0.81 & 47.8 & 79.8 & 50 & 79 & \\
4 & 0.47 & 0.62 & 40.9 & 85.5 & 38 & 86 & \\ \hline \end{tabular}
\end{table} 
Table 3: Validation of the proposed limit equilibrium method with FEM-based numerical simulations. The presented results, obtained using both independent approaches, indicate their high correlation, particularly in terms of the shapes of the ejected rock wedge and the envelope of maximum shear strain in a function of the angle of internal friction and cohesion of the rock mass. Also, the presence of \"locally\" stable triangular areas within the sidewall as the cohesion effect is visible. It should be highlighted that when analyzing the results presented in Table 3, one may notice that the correlation between the proposed approach and numerical solutions is relatively high and reach values of 0.86 in the case of \(\alpha_{1}\) and 0.97 in the case of \(\alpha_{2}\). The abovementioned results permit accepting the LEM as an appropriate and sufficiently accurate approach in preliminary modeling rockbursts in deep mines. Areas of potential rockburst were identified [42] based on the calculated 3D distribution of the safety margin, formulated based on the Mohr-Coulomb theory of shearing failure, the mathematical notation of which has the following form:  
\[S_{m}=-\sigma_{3}\frac{1+sin\phi}{1-sin\phi}+\frac{2c\ cos\phi}{1-sin\phi}+ \sigma_{1} \tag{11}\]  
where \(\sigma_{3}\) and \(\sigma_{1}\)--minor and major principal stresses. The calculation results refer to two alternative cases:  
* a 2D case relating to the gallery point, far enough from the intersections (cross-cuts, ramps, etc. of the cavities excavated from the galleries)--(Figure 10 left);
* a 3D case, relating exclusively to intersections (Figure 10 right). The obtained distribution of the value of the safety margin for the 2D case allows us to conclude that the shape of rock volume ejected from the sidewalls (\(S_{m}<0\)) is a longitudinal segment of an elliptic cylinder (Figure 10 left) supported at the corners of the excavation. The straight line tangent to the surface of this solid drawn from these corners is inclined in relation to the horizontal plane at the angle of \(\alpha_{1}\cong 57.\deg\), so in a similar way to that calculated for Scenario 1 (Table 3), but with the use of the FEM numerical model formulated strictly for a plane deformation state. The geometry of the ejected three-dimensional solid at the intersection corner (Figure 10 right) is more complex, since it may be approximately described as a kind of hexahedron--a geometric solid with six faces (slip planes), which are formed by the interpenetration of two mutually perpendicular triangular prisms (2D LEM solutions). This statement is valid only if the local geomechanical conditions permit rockburst development in the galleries sufficiently far away from the intersection. Otherwise, only 3D FEM or similar computer codes can be utilized, since 3D analytical extension of [PERSON]'s solution (Equation (4)) to three dimensions is not available so far. Despite the mentioned differences in the problem  
Figure 9: Maximum shear strain distribution obtained numerically (model 1, top; model 4, bottom). formulation, the similarities and compatibility proved between the two approaches, FEM and LEM, permit accepting the LEM as an appropriate and sufficiently accurate approach in modeling rockbursts in deep mines.