## 3 Material and Methods
### Analytical Solution  
The general closed-form solution of Maugis [36] concerning stress distribution around the elliptical excavation within the homogeneous and infinite elastic space was the base for the analytical LEM formulation developed with respect to mine opening safety. This solution has several undisputable advantages such as conciseness, simplicity, and ease of coding in computer programs, which led it to be used willingly in many technical applications until today. Newer studies in this field (e.g., [37; 38; 39]), based mainly on complex potential functions and conformal mapping representation, are characterized by considerable complexity, which means that they so far are rarely used in practical applications. Using the Maugis formulation, the relationships describing normal stress \(\sigma_{z}\) and \(\sigma_{x}\) distributed at the plane \(z=b=0.5\times H\), which are the result of the presence of the uniformly distributed horizontal pressure \(p_{x}\) and the vertical pressure \(p_{z}=\gamma_{av}H_{0}\), have been obtained. For the normal stresses acting in the \(z=b=0.5\times H\), the following relationships were obtained:  
\[\sigma_{z}=\frac{1}{2}\times\left(\overline{A}+\overline{B}+\overline{C}+ \overline{D}\right) \tag{2}\]  
\[\sigma_{x}=\frac{1}{2}\times\left(\overline{A}+\overline{B}-\overline{C}- \overline{D}\right) \tag{3}\]  
where \(\overline{A},\ \overline{B},\ \overline{C}\), and \(\overline{D}\), the algebraic types of functions of the elliptical opening size (half-axis \(a\) and \(b\)), coordinate x and the external load, \(\mathrm{p_{x}}\) and \(\mathrm{p_{z}}\). A formulated stability condition for the sidewalls of rectangular excavations that are prone to rockburst occurrence introduces the assumptions that normal stress \(\sigma_{x}\) and \(\sigma_{z}\) distribution at height \(z=0.5\times H\) in the sidewalls of the excavation of width \(B\) and height \(H\) are approximately equal to the corresponding values of the principal stresses obtained for the excavation of the elliptical cross-section with the half-axes equal to \(a\) and \(b\) (\(a>b\)). Also, the ejected rock wedge is of the shape of a triangle defined by the angles of \(\alpha\) and \(\beta\) that identify the lines AB and CB, i.e., the envelope of the area where the stability index value is less than \(1\) (Figure 4):  
Figure 3: General flowchart diagram explaining the proposed methodology. The stability index for the rock wedge ABC (Figure 3) and unbalanced ejecting force may be determined using the relationship presented below, based on the limit equilibrium principles:  
\[\text{stability index}:\quad\overline{F}=\frac{F_{p}}{F_{a}}=A_{f}+A_{c}+A_{b} \tag{4a}\]  
\[\text{or unbalanced force}:\ P_{k}=F_{a}-F_{p} \tag{4b}\]  
where \(A_{f}\)--relative resistance of internal friction forces; \(A_{c}\)--relative resistance of cohesion forces; and \(A_{b}\)--relative resistance of the forces mobilized within \(n\) rods of rock bolts, each of the capacity \(N_{i}\) and inclination angle \(\delta_{i}\). The safety index requires maintaining the following limit conditions: \(\overline{F}\geq 1.0\) or \(P_{k}\geq 0.0\ (MN)\). The resistance and the driving potentials are described as follows:  
\[F_{p}=tg(\phi+dil)[(cos\beta+cos\alpha)\int_{1}^{1+\frac{d}{a}}\sigma_{z}(t) dt+(sin\beta+sin\alpha)\int_{1}^{1+\frac{d}{a}}\sigma_{x}(t)dt]+cd\bigg{(}\frac{1}{ cos\alpha}+\frac{1}{cos\beta}\bigg{)}+\sum_{i=1}^{n}(N_{i}cos\delta_{i}) \tag{5}\]  
\[F_{a}=(sin\beta+sin\alpha)\int_{1}^{1+\frac{d}{a}}\sigma_{z}(t)dt-(cos\beta+ cos\alpha)\int_{1}^{1+\frac{d}{a}}\sigma_{x}(t)dt \tag{6}\]  
where \(t=x/a\), \(dil\)--dilation angle on the joint's surface; \(c\)--cohesion at the discontinuity surface, i.e., cohesion of the continuous (intact) rock mass; \(\phi\)--average angle of friction at the discontinuity surface; \(n\)--number of rock bolts installed within 1 m of the excavation length; \(N_{i}\)--rock bolt capacity (MN); and \(\delta_{i}\)--inclination angle from the horizontal plane.