# The Impact of Node Location Imperfections on the Reliability of Single-Layer Steel Domes
## 3 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Case 1**} & \multicolumn{2}{c}{**Case 2**} \\ \cline{3-6}
**Load Multiplier \(\mu\)** & **Displacement \(q_{1}\)** & **Failure Probability** & **Reliability Index** & **Failure Probability** & **Reliability Index** \\  & & \(P_{\mathrm{f}}\) & \(\beta\) & \(P_{\mathrm{f}}\) & \(\beta\) \\ \hline
7.07 & 18.54 & 0.000248568 & 3.48229 & 6.30709 \(\times\) 10\({}^{-9}\) & 5.69121 \\
14.19 & 40.40 & 0.0807931 & 1.39976 & 0.0058181 & 2.52299 \\
16.31 & 47.80 & 0.148813 & 1.04154 & 0.0286027 & 1.90174 \\
18.16 & 54.77 & 0.217846 & 0.77949 & 0.0755421 & 1.43571 \\
22.87 & 75.62 & 0.393263 & 0.270823 & 0.305856 & 0.507631 \\
23.85 & 81.00 & 0.42604 & 0.186542 & 0.362944 & 0.3506 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of the reliability index \(\beta\) and failure probability \(P_{\mathrm{f}}\). Figure 7: The dependence between the reliability index \(\beta\) and the load multiplier \(\mu\). Figure 8: Deformed form of the structure. Figure 10: Elasticity index \(\mathrm{E_{\beta}(p_{i})}\) with respect to the mean value (\(\mathrm{p_{i}=\overline{X_{i}}}\)) for Case 1 and the load multiplier \(\mu\) = 23.85. Figure 9: Elasticity index \(\mathrm{E_{\beta}(p_{i})}\) with respect to the mean value (\(\mathrm{p_{i}=\overline{X_{i}}}\)) for Case 1 and the load multiplier \(\mu\) = 7.07. Figure 11: Elasticity index \(\mathrm{E_{\beta}(p_{i})}\) with respect to the mean value (\(\mathrm{p_{i}=\overline{X_{i}}}\)) for Case 2 and the load multiplier \(\mu=7.07\). \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Case 1**} & \multicolumn{2}{c}{**Case 2**} \\ \hline  & \(\mu=7.07\) & \(\mu=23,85\) & \(\mu=7.07\) & \(\mu=23.85\) \\ \hline p & \(-0.960682\) & \(-11,3796\) & \(-0.94165\) & \(-11,295\) \\ \(\mathrm{Z_{1}}\) & \(-11,22087\) & \(-20,5527\) & \(-1,04297\) & \(-20,428\) \\ \(\mathrm{Z_{2}}\) & \(-0.633033\) & \(-12,1532\) & \(-0,55432\) & \(-12,1929\) \\ \(\mathrm{Z_{3}}\) & \(-0.636345\) & \(-12,248\) & \(-0,63249\) & \(-12,2128\) \\ \(\mathrm{Z_{4}}\) & \(-0,627177\) & \(-12,2261\) & \(-0,57223\) & \(-12,1834\) \\ \(\mathrm{Z_{5}}\) & \(-0,634882\) & \(-12,1534\) & \(-0,60514\) & \(-12,2013\) \\ \(\mathrm{Z_{6}}\) & \(-0,63134\) & \(-12,2363\) & \(-0,59676\) & \(-12,1821\) \\ \(\mathrm{Z_{7}}\) & \(-0,643971\) & \(-12,2566\) & \(-0,60399\) & \(-12,2017\) \\ \(\mathrm{Z_{8}}\) & \(-0,630684\) & \(-12,2457\) & \(-0,57788\) & \(-12,1823\) \\ \(\mathrm{Z_{9}}\) & \(-0,638158\) & \(-12,2553\) & \(-0,62899\) & \(-12,1231\) \\ \(\mathrm{Z_{10}}\) & \(0,583069\) & \(11,8805\) & \(0,546972\) & \(11,9319\) \\ \(\mathrm{Z_{11}}\) & \(0,578446\) & \(11,8845\) & \(0,551486\) & \(11,8314\) \\ \(\mathrm{Z_{12}}\) & \(0,580928\) & \(11,8773\) & \(0,552808\) & \(11,8252\) \\ \(\mathrm{Z_{13}}\) & \(0,580211\) & \(11,9773\) & \(0,551094\) & \(11,9262\) \\ \(\mathrm{Z_{14}}\) & \(0,58106\) & \(11,8772\) & \(0,548862\) & \(11,9283\) \\ \(\mathrm{Z_{15}}\) & \(0,578546\) & \(11,8836\) & \(0,548516\) & \(11,9291\) \\ \(\mathrm{Z_{16}}\) & \(0,578429\) & \(11,8875\) & \(0,55195\) & \(11,9279\) \\ \(\mathrm{Z_{17}}\) & \(0,58093\) & \(11,8778\) & \(0,553645\) & \(11,8224\) \\ \hline \end{tabular}
\end{table} 
Table 4: Values of the elasticity index \(\mathrm{E_{\beta}(p_{i})}\) with respect to the mean value depending on the load multiplier applied. Figure 12: Elasticity index \(\mathrm{E_{\beta}(p_{i})}\) with respect to the mean value (\(\mathrm{p_{i}=\overline{X_{i}}}\)) for Case 2 and the load multiplier \(\mu=23.85\). In order to better illustrate the influence of random variables \(P\) and \(Z_{i}\) on the values of the reliability index \(\beta\), the values of the elasticity index were presented in relation to the standard deviation (Table 5). For the cases of concern, a major difference in the reliability index values was observed (Figure 7). For Case 1, the maximum value of the reliability index was 3.48229, which constitutes 61.19% of the value of Case 2.