## 4 Results
### Identification of Rockburst-Prone Mine Areas  
Knowing the characteristics of the rockburst mechanism, the location of areas prone to instability was determined. Therefore, the general equation relating the rock's strength parameters (\(c\
eq 0\) and \(\Phi\
eq 0\)) to the load that is acting in conditions where the limit equilibrium may be obtained only in one location is presented below:  
\[sina[(cosatan\phi+sina)\int_{\frac{\Phi}{2}}^{\frac{\Phi}{2}+d}\sigma_{z}(x)dx+ (sinatan\phi-cosa)\int_{\frac{\Phi}{2}}^{\frac{\Phi}{2}+d}\sigma_{x}(x)dx]+ \frac{cH}{2}=0 \tag{17}\]  
where \(\phi\)--the angle of internal friction; \(\alpha\)--the angle of inclination of the external plane of the block of solid rock ejection; B--opening width; H--opening height; \(d=\frac{h}{2 atan\alpha}\); and c--cohesion. Gradually increasing the value of \(p_{z}=H_{0}\gamma_{0,\alpha\
u}\) (where \(H_{o}\)--overburden thickness, \(\gamma_{0,\alpha\
u}=0.026\frac{MN}{m^{3}}\))--the average bulk weight of the rock mass), the value of the critical depth \(H_{\alpha}\), at which the beginning of the rockburst process may be observed, has been assessed within the parameters' space and is characterized in Section 3. Simultaneously, searching for the critical value F = 1, the angle \(\alpha\) value has been scanned from zero to \(\frac{\pi}{2}\). Based on the available literature concerned with the strain-strength parameters of the rock mass in the area of Polish copper mines [38], the values of critical depths, from the point of view of rockburst hazards, have been determined. The laboratory-obtained values  
Figure 11: Predicted changes in the acceleration (**left**) and velocity (**right**) of the detached rock wedge movement (t\({}_{0}\) = 0.0131 s). of rock mass parameters have been scaled down to in situ (rock mass) values using the approach given by [45] (Table 4). These estimates permitted the identification of the critical depths from which a serious rockburst may occur. This knowledge has proved that copper ore exploitation at depths greater than 1100 m will be associated with an increased level of rockburst risk, indicating that at this depth, the extensive use of ductile rock bolts is highly recommended as basic ground support, at least in the long-life workings such as machine chambers, etc. In each case of well-known mining-geology conditions, the development of individual preliminary analyses using data shown in Figure 12 is strongly recommended. As a result, the rockburst factor of safety over the entire mine may be determined according to the following formula:  
\[F_{srb}=\frac{H_{cr}(x,y)}{H_{o}(x,y)} \tag{18}\]  
where \(H_{cr}\)--critical depth; \(H_{o}\)--depth of mining. This type of map may be associated with the x-y maps of ductile sidewall support distribution with detailed data concerning the unbalanced force \(P_{k}\) as well as the number, length, and capacity of the utilized rock bolts. To illustrate the advantages of the proposed approach, the rockburst factor of safety distribution (\(F_{srb}\)) over one of the mining panels of the Polish copper mine has been analyzed. The values of angles of internal friction and cohesion that are adequate for different types of rock have been assessed using the method based on tangents to the parabolic envelope of Mohr's circles [41]. At the same time, the \(H_{cr}\) value has been calculated, which permitted us to find the critical rock wedge that is \begin{table}
\begin{tabular}{c c c c} \hline
**Type of Rock** & **Sandstones** & **Anhydrites** & **Grainy Dolomites and Limestones** \\ \hline UCS (MPa) & 71–142 & 100–141 & 151–224 \\ GSI & 45–70 & 60–75 & 75–90 \\ Angle of & Triaxial tests & 64–67 & 61–65 & 64–69 \\ internal friction (\({}^{\circ}\)) & Reduced using GSI & 31.5–44.0 & 36.4–44.2 & 46.2–62.7 \\ Cohesion \(c\) (MPa) & Triaxial tests & 17.5–20.5 & 15.8–18.8 & 18.0–22.5 \\ Reduced using GSI & 2.8–11.6 & 5.5–11.4 & 13.2–30.1 \\ Stress ratio & 0.14–0.60 & 0.27–0.80 & 0.33–1.39 \\ Horizontal stress \(p_{x}\) (MPa) & 4–12 & 6–30 & 8–40 \\ \hline \end{tabular}
\end{table}