# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 3 Numerical Results and Discussion
Values of the reliability index and failure probability for selected computational steps are collated in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Random Variable**} & \multirow{2}{*}{**Mean Value \(\overline{\mathbf{X_{i}}}\)**} & \multirow{2}{*}{**Probability**} & \multicolumn{2}{c}{**Case 1**} & \multicolumn{2}{c}{**Case 2**} \\ \cline{3-8}  & & & **Standard** & & **Standard** & **Standard** & **Variation** & **Variation** \\  & & & **Type** & **Deviation** & **Coefficient** & **\(\sigma_{2,i}\)** & **Coefficient** \\ \hline Load \(P\) & 1 & Normal & 0.1 & 10\% & 0.1 & 10\% \\ Coordinate \({}^{2}\)” of node 1–\(Z_{1}\) & 1.6465 & Normal & 0.16465 & 10\% & 0.082325 & 5\% \\ Coordinate \({}^{2}\)” of node 2–\(Z_{2}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 3–\(Z_{3}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 4–\(Z_{4}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 5–\(Z_{5}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 6–\(Z_{6}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 7–\(Z_{7}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 8–\(Z_{8}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 9–\(Z_{9}\) & 1.432 & Normal & 0.1432 & 10\% & 0.0716 & 5\% \\ Coordinate \({}^{2}\)” of node 10–\(Z_{10}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 11–\(Z_{11}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 12–\(Z_{12}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 13–\(Z_{13}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 14–\(Z_{14}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 15–\(Z_{15}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 16–\(Z_{16}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ Coordinate \({}^{2}\)” of node 17–\(Z_{17}\) & 1.14 & Normal & 0.114 & 10\% & 0.057 & 5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of random variables. Figure 6: Procedure of the analysis. Additionally, this paper examines the elasticity index \(\mathrm{E}_{\beta}(\mathrm{p}_{\mathrm{i}})\). Figure 8 shows the deformed form of the considered structure. It is necessary to pay attention to the direction of displacement of individual nodes. The upper node, No. 1, and the middle nodes, Nos. 2 to 9, are moving downwards, relative to the undeformed form of the structure, while the lower nodes, Nos. 10 to 17, are moving upwards. Figures 9-12 show the values of the elasticity index \(\mathrm{E}_{\beta}(\mathrm{p}_{\mathrm{i}})\) with respect to the mean value in the first and the last computational step for all random variables. For the cases of concern, changes in the values of the elasticity index \(\mathrm{E}_{\beta}(\mathrm{p}_{\mathrm{i}})\) for variables \(P\) and \(Zi\), depending on the load value, are shown in Table 4. We can observe the occurrence of positive and negative values of the elasticity index, which depends on the direction of the node's displacement.