## 3 Analytical Method of Support Design  
### Calculation of the Support Pressure  
The post-mining state of stress resulting from an underground excavation can be calculated using analytical, elastic [34] or elasto-plastic [35; 36; 37], closed-form solutions. Analytical closed-form solutions are typically limited to simple geometries and material models; therefore, they should always be applied with appropriate caution. Nevertheless, an analytical approach to stress state calculation can serve as a valuable tool for a \"sanity check\" of the results obtained from more advanced numerical analyses. In this study, a closed-form solution based on an elasto-plastic strain-softening rock mass model was employed in accordance with the guidelines outlined in the Polish National Standards PN-G-05020 [38] and PN-G-05600 [39], as well as the additional recommendations provided by [PERSON] et al. [40; 41]. For excavation shapes other than circular, an equivalent excavation radius was calculated using the following equation:  
\[r_{eq}=\sqrt{\frac{A}{\pi}}, \tag{13}\]  
where \(A\) represents the excavation cross-sectional area. Peak and post-peak compressive strength of the rock mass were determined by the following relationships:  
\[R_{cg}=\frac{2\cdot c\cdot cos\varphi}{1-sin\varphi}, \tag{14}\]  
\[R^{\prime}_{cg}=\frac{2\cdot c^{\prime}\cdot cos\varphi^{\prime}}{1-sin \varphi^{\prime}}, \tag{15}\]  
where \(R_{cg}\) and \(R_{cg}{}^{\prime}\) represent the peak and post-peak compressive strength of the rock mass, respectively, \(c\) and \(c^{\prime}\) denote the peak and post-peak cohesive strength of the rock mass, while \(\varphi\) and \(\varphi^{\prime}\) are the peak and post-peak internal friction angles. To simplify further calculation, a factor \(\beta\), defined by Equation (8) was introduced as follows:  
\[\beta=\frac{2\cdot sin\varphi}{1-sin\varphi}. \tag{16}\]  
The radial stress at the boundary between the elastic and plastic zones (synonymous with the critical support pressure required to prevent rock mass failure), \(p_{g}\) was calculated as follows:  
\[p_{g}=\frac{2\cdot p_{z}-R_{cg}}{2+\beta}, \tag{17}\]  
where \(p_{z}\) represents the far-field (in situ) stress. The radius of the plastic zone is a function of an active support pressure. For a support system consisting of yielding steel sets equipped with sliding joints, the active support pressure can be estimated using the following equation:  
\[p_{a}=\frac{N_{j}}{r_{eq}\cdot e}, \tag{18}\]  
where \(N_{j}\) is the yielding force of the sliding joints, and \(e\) denotes the spacing between steel sets. The radius of the plastic zone, \(r_{l}\), was determined based on the estimated value of the active support pressure using the following equation:  
\[r_{l}=r_{eq}\cdot\Bigg{(}\frac{p_{g}\cdot\beta+R^{\prime}_{cg}}{p_{a}\cdot\beta+ R^{\prime}_{cg}}\Bigg{)}^{\frac{1}{\beta}}. \tag{19}\]  
The development of the plastic zone around the underground excavation corresponds to the occurrence of static rock mass pressure, which results from the gravitational load of the plasticized and fractured roof rocks [42]. The characteristic value of the static support pressure of the plasticized rock mass, \(q_{2N}\), was calculated using the following equation:  
\[q_{2N}=\gamma_{r}\cdot(r_{l}-r_{z}), \tag{20}\]  
where \(\gamma_{r}\) is the bulk weight of the roof rocks. The design value of the static support pressure of the plasticized rock mass, \(q_{20}\), was calculated as follows:  
\[q_{20}=n\cdot k_{s}\cdot q_{2N}, \tag{21}\]  
where \(n\) is the partial load factor according to Table 1 in the Polish National Standard PN-G-05600 [39], and \(k_{s}\) is the impact factor of the adjacent excavation, calculated using the following formula proposed by [PERSON] et al. [43]:  
\[k_{s}=1+\frac{1}{\Big{(}1+\frac{x_{s}}{w_{a}}\Big{)}^{2}}, \tag{22}\]  
where \(x_{s}\) is the distance between neighboring excavations, and \(w_{a}\) is the width of the adjacent excavation.