## 3 Results of Modeling of Stress-Strain State and Rock Mass-Support Interaction  
The stress state of the rock mass and the mine working support were assessed via stress-strain modeling using the RS2 software ([[https://www.rocscience.com/software/rs2](https://www.rocscience.com/software/rs2)]([https://www.rocscience.com/software/rs2](https://www.rocscience.com/software/rs2)), accessed date 30 August 2024) package (Rockscience package), a finite element analysis (FEA) system that was developed to solve linear and nonlinear, stationary and non-stationary spatial problems involving solid mechanics, structural mechanics, and the mechanics of coupled fields [28; 29]. This software allows for the determination of three-dimensional stresses and loads generated in multiple directions, which are summed to obtain an equivalent stress known as the von Mises stress [30; 31]. The results of these calculations are presented as indicators of the stability coefficient of both the rock mass and the support system. RS2 also supports a range of constitutive models, such as the Mohr-Coulomb and Drucker-Prager models, as well as elastic behavior models. However, the focus of this  
Figure 2: Installation diagram of retractable supports at the mine. study was not on applying these models directly but rather on analyzing the interaction between the rock mass and the support system over time. This analysis was conducted using diagrams that reflect how the rock stability strength factor evolves under various reinforcement technologies. These diagrams provide insights into stress redistribution patterns, deformation, and load transfer between the rock mass and the support structure, helping to identify critical stress points and guide reinforcement strategies. This approach allows for a comprehensive understanding of the time-dependent behavior of the rock mass and support interaction, ensuring that the selected technologies can maintain long-term stability under complex geological conditions. The calculation results are presented as indicators of the factor of safety for rock stability and support. The calculation pattern is presented in Figure 3. The value of rock pressure is accepted for the mining depth of 550 m at stresses \(\sigma_{1}=\sigma_{2}=\sigma_{3}=16.2\) MPa. The physical and mechanical properties of rocks used during modeling are presented in Table 1.  
The fractures comprised two mutually perpendicular systems of fractures with an average distance between fractures of 500 mm. For modeling, the following supporting options were adopted:  
* Shotcrete, 50 mm thick;
* Friction anchors, SZA, 48 mm + reinforced support;
* Friction anchors, SZA, with a diameter of 48 mm + reinforced support + shotcrete with a thickness of 50 mm;
* Friction anchors, SZA, with a diameter of 48 mm + reinforced support + shotcrete with a thickness of 100 mm; \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Ultimate Tensile Strength** & **Internal Friction** & \multirow{2}{*}{**Adhesion C, MPa**} & \multirow{2}{*}{**Dynamic Modulus of Elasticity**} & \multirow{2}{*}{**Poisson**} & \multirow{2}{*}{**Hours**} \\
**\%,**\(\lambda\)**MPa** & & & & \(\text{E}^{\prime}\text{10-4},\text{MPa}\) & & \\ \hline
7 & 33 & 17 & 9.9 & 0.26 & 0 \\ \hline
5.3 & 30.0 & 12.8 & 7.4 & 0.25 & 6 \\ \hline
3.9 & 27.0 & 9.6 & 5.6 & 0.24 & 12 \\ \hline
3.0 & 24.0 & 7.2 & 4.2 & 0.23 & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical properties of rocks. Figure 3: The calculation pattern. SZA, friction anchors, with a diameter of 48 mm + reinforced support + shotcrete with a thickness of 150 mm; - Insulating layer of shotcrete-SZA friction anchors with a diameter of 48 mm + reinforced support + shotcrete with a thickness of 150 mm. The distribution of the factor of safety for rock stability immediately and 24 h after the excavation is exposed and subsequently reinforced with a 50 mm thick layer of shotcrete around the excavation is shown in Figures 4 and 5.