## 3 Analytical Method of Support Design
### Determination of Support Parameters
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Symbol** & **Unit** & **A** & **B** \\ \hline \multicolumn{4}{c}{**1. Input parameters**} \\ \hline Section type & - & - & V29 & V36 \\ Steel grade & - & - & S480W & S480W \\ Maximum value of the bending moment & \(M_{max}\) & MNm & 0.0168 & 0.0147 \\ Corresponding value of the axial force & \(N_{cor}\) & MN & 0.2962 & 0.2851 \\ Value of the axial force in the sliding joint & \(N\) & MN & 0.4780 & 0.5147 \\ Cross-sectional area of the V-profile & \(A\) & m\({}^{2}\) & 0.00363 & 0.00452 \\ Elastic section modulus of the V-profile & \(W_{x}\) & m\({}^{3}\) & 0.0000875 & 0.0001276 \\ Yielding force of the sliding joint & \(N_{j}\) & MN & 0.29 & 0.35 \\ 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 \\ Unsupported length of the steel set & \(l\) & m & 6.72 & 6.60 \\ Shape factor & \(m\) & - & 1.40 & 1.40 \\ Support working condition factor & \(m_{1}\) & - & 1.50 & 1.50 \\ Material plasticity coefficient & \(n_{1}\) & - & 0.35 & 0.35 \\ \hline \multicolumn{4}{c}{**2. Calculation results**} \\ \hline Senderness ratio according to PN-B-03200 & \(\lambda\) & - & 84.0 & 74.4 \\ Reference slenderness ratio according to PN-B-03200 & \(\lambda_{p}\) & - & 48.3 & 48.3 \\ Reduction factor according to PN-B-03200 & \(\phi\) & - & 0.27 & 0.33 \\ Max. steel set spacing due to profile strength & \(e_{1}\) & m & 0.913 & 1.117 \\ Max. steel set spacing due to sliding joint capacity & \(e_{2}\) & m & 0.607 & 0.680 \\ Maximum steel set spacing & \(e_{max}\) & m & 0.607 & 0.680 \\ Steel set spacing adopted for further calculations & \(e\) & m & 0.600 & 0.500 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Input data and calculation results of maximum steel set spacing for the hoisting machine chamber (excavation A) and the technological shaft inset (excavation B). Figure 2: Static schemes of the frame models ((**a**)—hoisting machine chamber; (**b**)—technological shaft inset). should only take place after all deformations of the excavation opening have ceased. Therefore, the determination of secondary support properties was carried out based solely on the value of the static support pressure. The following relationships, based on the guidelines provided Polish National Standard PN-G-05600:1998 [39] and [PERSON] et al. [40,41], were used to calculate the theoretical value of composite support capacity due to compression and shear failure, respectively:  
\[p_{sup1}=\frac{g\cdot f_{cd}\cdot e+A\cdot f_{d}}{r_{w}\cdot e}, \tag{24}\]  
\[p_{sup2}=\frac{4\cdot g\cdot f_{ctd}\cdot e+0.6\cdot A\cdot f_{d}}{r_{w}\cdot e}, \tag{25}\]  
where \(p_{sup1}\) and \(p_{sup2}\) denote the composite support capacity due to compression and shear failure, respectively, \(f_{cd}\) and \(f_{ctd}\) are the design compressive and tensile strengths of fiber-reinforced shotcrete, g is the shotcrete thickness, \(A\) is the cross-sectional area of the steel set profile, \(f_{d}\) is the design value of the ultimate strength of steel, \(e\) is the steel set spacing, and \(r_{w}\) is the contour radius of the excavation opening (outer radius of the shotcrete layer). The calculation results of the composite support capacity for the hoisting machine chamber (excavation A) and the technological shaft inset (excavation B), based on Equations (24) and (25), are summarized in Table 8.  
As indicated in Table 8, the values of support capacity, \(p_{sup1}\) and \(p_{sup2}\), for both the hoisting machine chamber and the technological shaft inset exceed the values of static support pressure presented in Table 6. Therefore, based on the analytical calculation method, it can be concluded that the secondary support systems should ensure long-term excavation stability, while maintaining the required safety factors.