## 3 Results  
### Schwedler Dome  
The Schwedler dome [9] with diameter of 25 m and height of 1 m is situated in the Beskid Nitski Mountains (Poland). A reinforced concrete ring supported by 16 reinforced concrete columns, 5 m high, constitutes a rigid support for the dome. The structure can cover the sports and entertainment hall. The dome consists of 81 nodes and 224 bars (Figures 2 and 3). Each of the bars was divided into 10 finite elements. The elements of the structure were made of steel tubes with yield stress fy = 355 MPa, Young's modulus E = 210 GPa, and Poisson's ratio v = 0.3. In order to dimension the designed structure, three groups of rods were modeled: meridians, parallels, and diagonals. Each meridian has a length of 2.511 m. The lengths of the parallels are, respectively: 4.877 m, 3.908 m, 2.934 m, 1.958 m, and 0.979 m. The roofing consists of tubular steel bars and glass panels. The loads were collected in accordance with applicable standards [N.1-N.3]. The static-strength analysis was performed according to EN-1993-1-1 [N.4] using Autodesk Robot Structure Professional 2022 software for the spatial frame structure, intended to produce the dimensioning of the steel elements. This structure was designed according to geometrically nonlinear analysis. Cross sections of individual groups of members are as follows: parallel (RO 273 \(\times\) 8), meridian (RO 127 \(\times\) 8.8), diagonal (RO 82.5 \(\times\) 6.3). Figure 3: Schwedler dome side view. Figure 2: Schwedler dome geometry plan view. The mass of the Schwedler dome is 40,953 kg. The mass of the bars of the supporting structures was modeled as evenly distributed, while the mass of the covering and roof equipment was modeled as concentrated in the nodes. Tables 1-4 summarize the results of analyzes for the Schwedler dome. Table 1 shows the critical load multipliers for linear buckling analysis and non-linear buckling analysis. Table 2 lists the vibration frequencies of the Schwedler dome in the case of standard modal analysis and modal analysis with normal forces taken into account. A graphical summary of selected eigenvectors of the standard modal analysis of the Schwedler dome is shown in Figure 4. Tables 3 and 4 show the current masses and effective masses for the first 10 forms, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Standard Modal Analysis**} \\ \cline{2-3}
**Form** & **Effective Mass UX** & **Effective Mass UY** & **Effective Mass UZ** \\  & **\%** & **\%** & **\%** \\ \hline
1 & 2.13 & 0.20 & 0.00 \\
2 & 2.33 & 2.33 & 0.00 \\
3 & 2.33 & 2.33 & 80.33 \\
4 & 2.33 & 2.33 & 80.33 \\
5 & 2.33 & 2.33 & 80.33 \\
6 & 2.33 & 2.33 & 80.33 \\
7 & 2.33 & 2.33 & 80.33 \\
8 & 2.33 & 2.33 & 88.58 \\
9 & 2.33 & 2.33 & 88.58 \\
10 & 2.33 & 2.33 & 88.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Effective modal masses for low-rise Schwedler dome (standard modal analysis). Figure 4: _Cont_. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Modal Analysis Taking into Account Normal Forces**} \\ \cline{2-4}
**Form** & **Current Mass UX** & **Current Mass UY** & **Current Mass UZ** \\  & **\%** & **\%** & **\%** \\ \hline
1 & 0.23 & 1.76 & 0.00 \\
2 & 1.76 & 0.23 & 0.00 \\
3 & 0.00 & 0.00 & \(2\times 10^{-09}\) \\
4 & \(5\times 10^{-08}\) & \(5\times 10^{-08}\) & \(8\times 10^{-09}\) \\
5 & \(3\times 10^{-08}\) & \(2\times 10^{-08}\) & \(88.75\) \\
6 & 0.00 & \(7\times 10^{-09}\) & \(1\times 10^{-08}\) \\
7 & \(1\times 10^{-07}\) & \(1\times 10^{-09}\) & \(4\times 10^{-06}\) \\
8 & \(1\times 10^{-08}\) & \(3\times 10^{-09}\) & \(1\times 10^{-06}\) \\
9 & \(2\times 10^{-08}\) & \(8\times 10^{-09}\) & \(8\times 10^{-09}\) \\
10 & \(5\times 10^{-09}\) & \(2\times 10^{-09}\) & \(0.07\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Current masses for low-rise Schwedler dome (modal analysis taking into account normal forces). Figure 4: _Cont_. Figure 4: Graphic illustration of eigenvectors for low-rise Schwedler dome (standard modal analysis). Figure 5: _Cont_.