## 3 Analytical Method of Support Design
### Determination of Support Parameters  
Given the significant rock mass deformation anticipated based on the results of the initial analysis, yielding steel arches equipped with sliding joints were employed as the primary support for both the hoisting machine chamber and the technological shaft inset. The maximum spacing of the steel sets was determined using the relationship proposed by [PERSON] et al. for steel sets constructed from V-profiles [40; 41; 44]:  
\[e_{max}=min\left\{ \begin{array}{l}\frac{f_{sat}\cdot(m+n_{1})}{\left(\frac{M_ {max}}{N_{c}}+\frac{N_{cor}}{N_{c}}\right)\cdot m_{1}},\\ \frac{N_{i}}{N}\end{array} \right. \tag{23}\]  
where \(M_{max}\) is the maximum value of the bending moment, \(N_{cor}\) is the corresponding value of the axial force, \(N\) is the value of the axial force in the sliding joint, \(W_{x}\) is the elastic section modulus of the V-profile, A is the cross-sectional area of the V-profile,\(f_{yd}\) is the design yield strength of the steel, \(\phi\) is the buckling factor according to the Polish National Standard PN-B-03200 [45], while \(m\), \(n_{1}\), and \(m_{1}\) are the shape factor, material plasticity coefficient, and support working condition factor, respectively. To determine the internal forces in the steel arches subjected to the static support pressure, Autodesk Robot Structural Analysis software, based on the finite element method, was used. The steel arches were modeled as a 2D frame composed of bar elements. Additionally, to account for the ground-structure interaction between the rock mass and the steel sets, spring supports characterized by a stiffness coefficient \(k\) were introduced in the model. For the primary support design, a characteristic value of the static support pressure was considered in the frame models. The static schemes of the frame models for the hoisting machine chamber and the technological shaft inset are illustrated in Figure 2.  
The calculation results for the maximum steel set spacing in the hoisting machine chamber (excavation A) and the technological shaft inset (excavation B), based on Equation (23) and the internal force values in the steel arches obtained from the frame models, are summarized in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Symbol** & **Unit** & **A** & **B** \\ \hline \multicolumn{5}{c}{**2. Calculation results**} \\ \hline Far field (in situ) stress & \(p_{z}\) & MPa & 23.68 & 24.08 \\ Equivalent excavation radius & \(r_{eq}\) & m & 3.55 & 4.29 \\ Peak compressive strength of the rock mass & \(R_{c}\) & MPa & 10.30 & 10.81 \\ Post-peak compressive strength of the rock mass & \({R_{c}}^{\prime}\) & MPa & 4.11 & 4.77 \\ Computational factor & \(\theta\) & - & 2.48 & 2.55 \\ Radial stress at the boundary between plastic and elastic zone & \(p_{g}\) & MPa & 8.26 & 8.21 \\ Active support pressure & \(p_{a}\) & MPa & 0.14 & 0.16 \\ Radius of the plastic zone & \(r_{l}\) & m & 7.06 & 8.03 \\ Impact factor of the adjacent excavation & \(k_{s}\) & m & 1.08 & 1.04 \\ Charasteristic value of the static support pressure & \(q_{zN}\) & MPa & 0.085 & 0.090 \\ Design value of the static support pressure & \(q_{zo}\) & MPa & 0.109 & 0.113 \\ \hline \end{tabular}
\end{table} 
Table 6: _Cont._The composite support system, consisting of steel sets embedded in fiber-reinforced shotcrete, was employed as a secondary support for both the hoisting machine chamber and the technological shaft inset. The installation of secondary support in both excavations