## 1 Introduction  
The stability of coal pillars has been among the most important practical problems for ensuring the geodynamic safety of underground mining. Underground openings strongly disturb the virgin state of stress and strain and cause their redistribution [1]. Stress amplification in the vicinity of openings is responsible for the manifestation of rock, coal, and gas bursting; immediate roof collapses into the working zone; and the failure of protective pillars [2]. The stress-induced damage of pillars occasionally interrupts mining due to outbursts [3]. Clogging of the working sections of the air or conveyor roadways by rock fragments as a result of the pillars' failure leads to a shutdown of the mine equipment and can give rise to casualties among the personnel. Each emergency case demands extra measures towards the unscheduled restoration of the working zone and equipment to a safe state according to the project documentation and safety instructions. The emergency downtime of mining facilities results in the worsening of the mine's production efficiency. For this reason, the development of a scientific background for maintaining the pillars' stability is one of the most important challenges facing geomechanics. Alongside the development of technologies, new control methods and innovative solutions for maintaining the stability of protective pillars between the panels are continuously being developed. Several approaches to stability assessments of the pillars and edges of coal seams have been discussed in the literature. When designing protective pillars, one of the main tasks is to determine their optimal size, which will ensure mining safety while the coal recovery ratio is maximized [4]. One of the key parameters in various pillar design methods is the width-to-height ratio (W/H). The methods for calculating this parameter use both analytical [5] and empirical formulas [6; 7; 8; 9]. The analytical size estimates have been obtained on the assumption that the rocks are elastic, i.e., the formula is valid only for very hard rocks, where such an assumption is possible. The empirical formulas also have a key drawback in that they are applicable only for specific geotechnical and geological conditions that they have been derived for. In addition, these formulas refer mainly to the room-and-pillar-mining system and cannot be fully applied to assess the pillars' stability when a longwall is employed (an exception is [PERSON]'s method, designed entirely for the longwall). Another problem of coal pillar stability analysis is that one also needs to estimate the edge and core pillar stress to calculate the factor of safety (\(F_{\text{s}}\)). The coal pillar is generally treated to be stable when \(F_{\text{s}}\) is greater than unity. The latter, however, does not guarantee the complete safety of mining since pillar instability cases occur even when the safety condition is met [10]. This is not surprising, since if we consider the features of a rock sample's behavior under compression in laboratory conditions (e.g., [11]), we can find that damage is initiated long before the peak stress point. This suggests that the uniaxial compressive strength (UCS) might be better replaced by the crack damage threshold (0.7-0.9 of UCS) or even the crack initiation threshold (0.4-0.6 of UCS) in the formula for calculating \(F_{\text{s}}\). This might result in a low coal recovery ratio due to an excessive amount of the unextracted (leftover) coal reserves. Therefore, the trade-off between the safety and economic efficiency of mining is an acute and challenging problem associated with the pillars' design. Obviously, the classical methods, assuming a static load equally distributed across the width of the pillars, do not consider the patterns of their actual loading. A nonstationary behavior of stress and strain of the rock mass elements caused by the mining operations creates a unique history of pillar loading, which can be taken into account by means of three-dimensional geomechanical modeling.