## 4 Observations through a Numerical Example of Steel Diagrid Grillages
### Using the RMSE\({}_{\text{mad}}\) Criterion  
For all the FEM data series, each of the 480 previously mentioned approximations were adopted. However, those whose domains did not coincide with the examined interval of the random variables were rejected. In four out of eleven cases, the average of the criterion values from all the models exceeded the limit value of 36 (Table 7). In the remaining cases, several approximations with the maximum \(RMSE_{\text{mod}}\) values were compared visually. In the range of the FEM tests, the difference was imperceptible (Figure 17a). However, when considering them in the full range of the assumed variability of the random parameter, the diagrams of some of them significantly differed from one common course for the majority (Figure 17b). Such cases occurred even for the approximation with the maximum mean of the \(RMSE_{\text{mod}}\) values among those obtained for the given series of discrete data (Table 7). Therefore, all of them were omitted when selecting the response function in favor of the next ones sorted in the descending order according to the criterion value. Thus, the diagram of the dependence selected in this way--in Table 7, the corresponding \(RMSE_{\text{mod}}\) value is marked with an asterisk--coincided with those obtained for many unselected approximations having significantly different forms of formulas. In the authors' opinion, this indicates correctness of the solution. The forms of some of the final response function formulas additionally confirmed correctness of the response function selection according to the \(RMSE_{\text{mod}}\) criterion. When considering the variability of [PERSON]'s modulus, we obtained inverse proportionality to the first power for deflections and dependence on the root of \(e\) for the basic eigenfrequency \(\omega\). This is justified by well-known dependences of the linear elastic analysis:  
* for elastic displacements (with the assumption of small deformations): \[u_{\text{elastic}}=\frac{C}{e}\] (31)
* and for the first eigenfrequency: \[\omega=C\cdot\sqrt{e},\] (32) where \(C\) is a constant depending on the parameters of the structure and the considered performance function. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Random** & **Performance** & **Mean \(RMSE_{\text{mod}}\)** & **Approximation** & **Approximation Group** & **Response** \\
**Variable** & **Function** & **Mean \(RMSE_{\text{mod}}\)** & **Base \(POa\)** & **Formula** & **Function Group** \\ \hline \multirow{3}{*}{\(e\)} & \(u_{\text{g}}\) & 38.74 & 1 & \(y_{2}=w(1/x)\) & 1 \\ \cline{2-6}  & \(u_{1}\) & 37.13 & 1 & \(y_{7}=1/w(x)\) & 2 \\ \cline{2-6}  & \(\omega\) & 39.10 & 1 & \(y_{13}=\sqrt{w(x)}\) & 3 \\ \hline \multirow{6}{*}{\(t\)} & \multirow{3}{*}{\(u_{\text{g}}\)} & 26.22 & 10 & \(y_{41}=\text{ar}\sinh(w(\ln(x)))\) & – \\ \cline{2-6}  & & 25.62 * & 4 & \(y_{14}=\sqrt{w(1/x)}\) & 4 \\ \cline{2-6}  & & 25.51 & 4 & \(y_{7}=1/w(x)\) & – \\ \cline{2-6}  & & 25.40 & 4 & \(y_{16}=\sqrt{w(1/\sqrt{x})}\) & – \\ \cline{2-6}  & \(u_{1}\) & 28.02 * & 2 & \(y_{14}=\sqrt{w(1/x)}\) & 5 \\ \cline{2-6}  & \(\omega\) & 25.66 * & 4 & \(y_{11}=1/w(\ln(x))\) & 6 \\ \cline{2-6}  & \(\sigma_{\text{red}}\) & 20.16 * & 2 & \(y_{14}=\sqrt{w(1/x)}\) & 7 \\ \hline \multirow{6}{*}{\(l\)} & \(u_{\text{g}}\) & 22.52 * & 3 & \(y_{29}=\text{exp}(w(\ln(x)))\) & 8 \\ \cline{2-6}  & \(u_{1}\) & 25.58 * & 3 & \(y_{29}=\text{exp}(w(\ln(x)))\) & 9 \\ \cline{1-1} \cline{2-6}  & \(\omega\) & 37.42 & 1 & \(y_{29}=\text{exp}(w(\ln(x)))\) & 10 \\ \cline{1-1} \cline{2-6}  & \(\sigma_{\text{red}}\) & 23.11 * & 3 & \(y_{1}=w(x)\) & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Data on the response function selection for grillages according to the \(RMSE_{\text{mod}}\) criterion (the final selection is underlined or asterisk).