## 3 Results
### Geodesic Dome  
The geodesic dome [33] with the span equal to 21 m and rise equal to 6.5 m lies in the Beskid Niski Mountains in Poland. The dome is based on 20 reinforced concrete columns, 5 m high. The whole covering is closed with a reinforced concrete ring forming a rigid structure, which simulates boundary conditions. The structure can cover the sports and entertainment hall. The dome consists of 51 nodes and 130 bars (Figures 6 and 7). Each of the dome bars was divided into 5 finite elements. The elements of the structure were assumed to be made of steel tubes with yield point fy = 235 MPa, Young's modulus E = 210 GPa, and Poisson's ratio v = 0.3. The loads were collected in accordance with applicable standards [N.1-N.3]. The static-strength analysis, intended to produce the dimensioning of the steel elements, was performed acc. To EN-1993-1-1 [N.4] using Autodesk Robot Structure Professional 2022 software for spatial frame structure. This structure was designed according to linear analysis. The bars of the structure were made with RO 101.6 \(\times\) 8. The mass of the geodesic dome is 25,350.7 kg. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Standard Modal Analysis**} & \multicolumn{4}{c}{**Modal Analysis Taking into Account Normal Forces**} \\ \cline{2-7}
**Form** & \begin{tabular}{c} **Effective** \\ **Mass UX** \\ **\%** \\ \end{tabular}  & \begin{tabular}{c} **Effective** \\ **Mass UY** \\ **\%** \\ \end{tabular}  & \begin{tabular}{c} **Effective** \\ **Mass UZ** \\ **\%** \\ \end{tabular}  & \ \begin{tabular}{c} **Effective** \\ **Mass UY** \\ **\%** \\ \end{tabular}  & \begin{tabular}{c} **Effective** \\ **Mass UZ** \\ **\%** \\ \end{tabular}  \\ \hline
1 & 0.00 & 0.00 & 5.31 & 0.00 & 0.00 & 5.15 \\
2 & 23.25 & 33.08 & 5.31 & 23.32 & 32.95 & 5.15 \\
3  
Figure 8: _Cont._  
Figure 8: \(Cont\). vibrations is determined by the stiffness of the structure, which in FEM is represented by the stiffness matrix. If, for some reason, the stiffness of the structure decreases, the natural frequency values decrease. After this short introduction, let us return to the two domes analyzed in the paper. The first is the geodesic dome. The mass of the dome is 25,350.7 kg. The critical load multiplier in both linear and nonlinear buckling analysis is quite high, 12.42 and 11.45, respectively. The relative error is \((12.42\,-\,11.45)/11.45=0.08=8\%\). Such values of the critical load multiplier, in accordance with the standard guidelines, entitle us to design structures in accordance with the first order theory. In the case of standard modal analysis, the first eigenvalue is 17.41 Hz, and after taking into account normal forces, it is 17.35 Hz. The relative error is \((17.41\,\,\mathrm{Hz}\,-\,17.35\,\,\mathrm{Hz})/17.35\,\,\mathrm{Hz}=0.0035=0.35\%\). The influence of normal forces on the obtained values of structure characteristics in the analysis of stability and dynamics is negligibly small. We did not observe a significant change in the stiffness of the structure after taking into account the normal forces. The presented design situation applies to most building structures, but not all. There is a group of structures where linear analysis is insufficient. The aim of this work was to draw attention to certain groups of structures in which failure to take into account the influence of normal forces may lead to large errors. An example of such a structure is the Schwedler dome with a mass of 40 953 kg. Already, in the buckling analysis, we can see a clear difference between the values of the critical load multipliers. Using the linear buckling analysis, we obtained a multiplier of 3.78, while in the nonlinear, it was 2.42. The relative error is \((3.78\,-\,2.42)/2.42=0.56=56\%\). In the case of standard modal analysis, the first eigenvalue is 3.57 Hz, while after taking into account normal forces, it is 2.85 Hz. The relative error is \((3.57\,-\,2.85)/2.85=0.25=25\%\). Differences are observed not only in eigenvalues but also in eigenvectors.