## 4 Results
### FEM Validation Results and Identification of Rockburst-Prone Areas  
Because the developed approach is very promising, the FEM-based validation of the LEM-based stability analyses was performed to ensure the reliability of the proposed solution. The 2D FEM-based calculation was performed with the use of RS2 software developed by RocScience [40]. Simulations were conducted for the four scenarios differing in the values of cohesion and angles of internal friction that characterize the rock mass surrounding the considered underground excavation. These models were presented in Table 2.  
To increase the reliability of the numerical calculations modulus of intact rocks deformation, \(E_{o}\) = 40 GPa was reduced to rock mass modulus acc. to as follows: \(E_{m}\) = 36.0 \(\div\) 40.0 GPa, depending on the nodes' distance from the opening walls envelope (Figure 8). The Poisson's ratio value was equal to \(\upsilon\) = 0.2 (\(p_{x}\) = 5 MPa). Then, based on [41], it was assumed that the zone of disturbance around the excavation ranges to the depth of 1.25 \(\times\) H from the sidewall and therefore the rock mass surrounding the opening was divided into several zones of different values of Geological Strength Index (GSI), as shown in Figure 7. At the same time, it was assumed that:  
\[GSI=100\,\left(1-\frac{D}{2}\right) \tag{9}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Reference** & \multicolumn{4}{c}{**Increment of the Basic Parameters**} \\
**Result Values** & \(\Delta B\) = 1 m & \(\Delta p_{x}\) = 1 MPa & \(\Delta p_{z}\) = 1 MPa & \(\Delta\varphi\) = 1 deg. \\ \hline \(P_{x}\) = 4.64 MN & 0.375 MN/m & \(-\)0.167 MN/MPa & 1.13 MN/MPa & \(-\)0.42 MN/deg. \\ \(\alpha_{1}\) = 64.1 deg. & 0.45 deg./m & 0.260 deg./MPa & \(-\)0.52 deg./MPa & 0.82 deg./deg. \\ \(\alpha_{2}\) = 86.1 deg. & 0.35 deg./m & \(-\)0.12/deg./MPa & 0.32 deg./MPa & \(-\)0.02 deg./deg. \\ \hline \end{tabular}
\end{table} 
Table 1: The unitary increment of values of the model’s basic parameters. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & **Cohesion c (MPa)** & **Angle of Internal Friction \(\varphi\) (deg)** \\ \hline
1 & 8 & 35 \\
2 & 4 & 35 \\
3 & 8 & 25 \\
4 & 4 & 25 \\ \hline \end{tabular}
\end{table} 
Table 2: Strength parameters for the considered models. where \(D\)--the disturbance factor [41], the value for which the analyzed case changes from zero (intact rocks) to 0.8 (rocks slightly damaged by the effect of the production blasting), 100--GSI for intact rocks. Finally, the rock mass modulus of deformation \(E_{m}\), appropriate for the distinguished zones, was assessed based on the following relationship:  
\[E_{m}=E_{0}\left\{0.025672+\frac{1-\frac{D}{2}}{1+exp^{\frac{6\pi 150-GSI}{ \Pi}}}\right\} \tag{10}\]  
It was assumed that the most appropriate criteria for the comparison of the developed LEM-based solution and FEM numerical model are the degree of matching of the expected shape of the critical wedges identified by angles \(\alpha_{1}\) and \(\alpha_{2}\), and the bands of the maximum shear strain's values distributed within the excavation sidewalls. The shear strain is treated here as a fundamental factor that induces the rockburst phenomena. It is also assumed that the shape of the shear stress envelope does not change itself significantly with the stress increment. The qualitative representation of results obtained with the use of FEM-based numerical simulations for Model 1 and Model 4 are presented in Figure 9, whereas the compliance of the results obtained with the numerical and analytical solutions are presented in Table 3.  
Figure 8: Modulus of deformation of surrounding rock mass, scaled down using the Hoek–Diederichs approach [41].