## 4 Results and Discussion
### Numerical Approach to Evaluation of FoS  
We have noted in the Introduction that the general formula for the determination of the Factor of Safety is poorly applicable in the case of complex stress states. The complexity of a stress state is attributed to the enhanced influence of tangential stresses. In the considered case study site, the stress state complexity is due to the coal seam inclination. For this reason, the ratio of the uniaxial compressive strength and vertical stress must be replaced by a formula suitable for the generalized stress state. Since we employ the Drucker-Prager yield surface as a part of the constitutive response, we can extend the classical approach formula for a stability analysis to the case of the generalized stress state. Notably, the shear strength of a medium has the following expression within the Drucker-Prager model as an extension of the classical Coulomb-Mohr model  
\[\tau_{\text{s}}=Y+\alpha P \tag{1}\]  
We can also calculate the level of an equivalent stress acting on the medium element using the following expression:  
\[\tau=\sqrt{\frac{1}{2}S_{ij}S_{ij}} \tag{2}\]  
Using Equations (1) and (2), we can find the ratio that is represented by the factor of safety formula for the case of a generalized stress state  
\[F_{\text{s}}=\frac{\tau_{\text{s}}}{\tau}=\frac{Y+\alpha P}{\sqrt{\frac{1}{2} S_{ij}S_{ij}}} \tag{3}\]  
Figure 8: Distribution of vertical stress along the line drawn through the center of the model in the dip direction of seam 3. Let us consider the distribution of the stress-strain parameters in the vicinity of the protective pillar. Figure 9 illustrates the combined distributions of vertical stress \(\sigma_{yy}\), equivalent plastic strain \(\gamma^{P}\), and the factor of safety \(F_{s}\) along the line drawn through the center of the model in the dip direction of seam 3. Judging by the equivalent plastic strain distribution, the edge sections of the protective pillar have accumulated 1-2% of plastic strain in the course of the stress-strain evolution. This is quite enough for the rocks to become unstable. It can be noted that the stronger stress amplification on the right side of the pillar reduces the factor of safety \(F_{s}\) in that part of the pillar down to the range of 1-1.1, which is lower than in the left side. The latter suggests that roadway III is probably more prone to the violation of its functional properties. The same qualitative distributions have been obtained for other pillars in the width range of 5-50 m; meanwhile, the width of panels remained constant. Let us also consider the values of the factor of safety averaged over the entire volume of the protective pillars for various \(\mathrm{W/H}\) ratios. Figure 10 shows the dependence of the average \(F_{s}\) on the \(\mathrm{W/H}\) ratio. Note that dependence is a non-linear asymptotic function, approaching its asymptote where the pillar width is about 40 m. Considering the additional factor of 10% used in the engineering practice in order to account for the random factors disregarded in the stability calculations, we find that the pillar can be considered to be stable in the cases where the average \(F_{s}\) exceeds 1.1. Obviously, this criterion is met by a pillar with the ratio \(\mathrm{W/H}=11\). A further increase in the pillar width is not economically feasible, since there is no significant change in \(F_{s}\), according to the asymptotic dependence (see Figure 10). At the same time, a further increase in the pillar width results in a smaller coal recovery ratio. The pillar width in that case is 39.6 m. The field data suggest that the width of the protective pillars used in this particular mine at the given depth varies in the range of 40-60 m. Therefore, the model results satisfy the lower bound of this parameter. The resulting dependence of the factor of safety can be explained as follows. The failure of the protective pillars during mining starts from the surface of the pillars at the junction of the walls to the roof and floor.