## 4 Numerical Method of Support Design
### Evaluation of the Secondary Support Performance  
The installation of secondary support for both the hoisting machine chamber and the technological shaft inset should be carried out only after the deformations of the excavation opening have fully stabilized. In numerical models, the secondary support is initially introduced in a stress-free state to reflect this condition. The secondary support is typically subjected to long-term loading due to factors such as the progressive degradation of the primary support, weakening of the surrounding rock mass, rheological effects, and potential groundwater interactions. Since no significant groundwater-related risks were identified in the area of Shaft II, and the adjacent sandstone formation does not exhibit notable rheological behavior, it has been assumed that the secondary lining becomes loaded predominantly as a result of the gradual deterioration of the surrounding rock. This study assumes that the long-term loading of the secondary support results from a decrease in rock mass cohesion, while the internal friction angle remains unchanged [50]. Therefore, in  
Figure 6: Bending moment diagram for the yielding steel set used as a primary support of the hoisting machine chamber. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Symbol** & **Unit** & **A** & **B** \\ \hline \multicolumn{5}{c}{**1. Input parameters**} \\ \hline Section type & - & - & V29 & V36 \\ Steel grade & - & - & S480W & S480W \\ Maximum value of the bending moment & \(M_{max}\) & MNm & 0.0045 & 0.0119 \\ Steel set spacing & \(e\) & m & 0.600 & 0.500 \\ Cross-sectional area of the V-profile & \(A\) & m\({}^{2}\) & 0.000363 & 0.00452 \\ Elastic section modulus of the V-profile & \(W_{x}\) & m\({}^{3}\) & 0.0000875 & 0.0001276 \\ Radius of gyration of the V-profile & \(i_{x}\) & m & 0.0400 & 0.0452 \\ Characteristic yield strength of steel & \(f_{yk}\) & MPa & 480 & 480 \\ Design yield strength of steel & \(f_{yd}\) & MPa & 384 & 384 \\ Characteristic ultimate tensile strength of steel & \(f_{t}\) & MPa & 650 & 650 \\ Shape factor & \(m\) & - & 1.40 & 1.40 \\ Support working condition factor & \(m_{1}\) & - & 1.50 & 1.30 \\ Material plasticity coefficient & \(n_{1}\) & - & 0.35 & 0.35 \\ \hline \multicolumn{5}{c}{**2. Calculation results**} \\ \hline Strain ratio of the steel set & \(k\) & - & 0.687 & 0.900 \\ \hline \end{tabular}
\end{table} 
Table 9: Input data and calculation results of the strain ratio of the steel sets used as a primary support of the hoisting machine chamber (excavation A) and the technological shaft inset (excavation B). the final phase of the numerical simulations, the post-peak rock mass cohesion was reduced by 10%. The secondary support system, consisting of steel arches embedded in fiber-reinforced shotcrete, was modeled using the equivalent section approach proposed by [PERSON] and [PERSON] [51]. This method, widely recognized in both academic research and engineering practice, is extensively used by researchers and designers to evaluate the performance of support structures in tunnels and underground mine workings [19; 20; 22; 52; 53]. The equivalent section approach involves representing the composite support system--comprising steel sets and shotcrete lining--as an \"equivalent\" rectangular section with a width \(b\), thickness \(t_{eq}\), and modulus of elasticity \(E_{eq}\). First, the mechanical properties of this equivalent section are determined. Next, the load-bearing capacity of the steel sets and shotcrete lining are assessed separately. A numerical model of the tunnel is then developed, where beam elements representing the equivalent section are applied along the tunnel perimeter. Finally, bending moments and axial thrusts computed in the numerical model are redistributed back onto the individual components--steel sets and shotcrete lining--allowing for a more precise evaluation of their structural response under different loading conditions.