## 2 Methodology
### Analysis of Instability around the Contour of an Underground Exacuation Leading to Overbreak
The sliding direction of the wedge is determined from the result of the active forces (vector \(\mathbf{A}\), Equation (1)). For a tetrahedron-shaped wedge (bordered by three sets of joints and the excavation contour surface), there are seven possible directions along which slide failure can occur. Sliding can be a consequence of gravitational fallout or movement along one of the three sets of joints or along the intersection of two sets of joints. In the first step of calculating the sliding direction, all seven possible directions are considered, after which, in the second step, it is determined which of the seven directions is valid. This calculation is conducted by establishing vector inequalities for each individual case. If a certain direction meets all vector inequalities, it represents the sliding direction; otherwise, if all seven possible sliding directions are not directions along which sliding can occur, the wedge is regarded as unconditionally stable. The normal forces acting on the planes along which wedge sliding can occur (one in the case of sliding along the joint and two in the case of sliding along the section line) are determined depending on the previously defined sliding direction (Figure 1). These are necessary to calculate the normal stress (force/area) on the slide plane and the shear strength along a discontinuity. To define the shear strength of the discontinuity, the Barton-Bandis (BB) failure criterion [19], which is defined by the following equation, is used:  
\[\tau=\sigma_{\mathrm{n}}\tan\left[\mathrm{JRC}\cdot\log_{10}\left(\frac{ \mathrm{JCS}}{\sigma_{\mathrm{n}}}\right)+\vartheta_{\mathrm{r}}\right] \tag{3}\]  
where JRC is the joint roughness coefficient, JCS is the joint wall compressive strength, \(\sigma_{\mathrm{n}}\) is the normal stress acting on the joint wall, and \(\vartheta_{\mathrm{r}}\) is the residual friction angle. Figure 1: Example of a wedge in two and three dimensions around contour of underground excavation with forces acting on it. N\({}_{\mathrm{i}}\)—normal component of force; J\({}_{\mathrm{i}}\)—shear component of force; W—weight of wedge. The residual friction angle \(\varphi_{\rm r}\) can be determined from the following equation:  
\[\varphi_{\rm r}=(\varphi_{\rm b}-20)+20(\rm r/R) \tag{4}\]  
where r is the rebound number, which is measured using a Schmidt hammer for natural (altered and/or wet) discontinuity; R is the rebound number, which is measured with a Schmidt hammer for unaltered and dry joint surface; and \(\varphi_{\rm b}\) is the base angle of friction. The roughness coefficient and compressive strength of the joint walls are affected by scale and are dependent on the size of the considered joint plane. According to [PERSON] [20], the dependencies on which the correction of JRC and JCS is based, with consideration of the scale effect, are as follows:  
\[\rm JRC_{n} =\rm JRC_{0}\left[\frac{L_{n}}{L_{0}}\right]^{-0.02:\rm JRC_{0}}JCS_{ n} =JCS_{0}\left[\frac{L_{n}}{L_{0}}\right]^{-0.03:\rm JRC_{0}} \tag{5}\]  
where coefficients with the index 0 refer to samples of reference length (L\({}_{0}\) = 10 cm), and coefficients with the index n correspond to the natural size of the observed joint of the rock mass (in the analysed case, the adopted block joint length was 3 m). The resistance force (J\({}_{\rm j}\)), which is a consequence of the engagement of shear strength along the discontinuity and acts in the direction opposite to the sliding direction, can be determined by the following equation:  
\[\rm J_{i}=\tau_{i}\alpha_{i}cos\Theta_{i} \tag{6}\]  
where \(\tau_{i}\) is the shear strength of the ith discontinuity, \(\alpha_{i}\) is the surface of the ith discontinuity, and \(\theta_{i}\) is the angle between the sliding direction and the ith discontinuity. The resisting forces are determined as an integral of shear strength along the discontinuity planes. In the last step, the size of the safety factor is determined as the ratio of resisting and driving forces.