## 4 Results and Discussion
### Influence of Initial Geometric Imperfections  
The influence of initial geometric imperfections on the prediction of critical buckling load was studied. The selected model for the investigation was the ANFIS-BBO algorithm, as its prediction  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Technique** & **R** & **RMSE** & **MAE** & **StDerror** \\ \hline SA & 5.03 & 26.68 & 20.40 & 27.90 \\ BBO & 7.15 & 40.48 & 38.45 & 40.39 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Increase of performance index _Inc_ (in %) using hybrid models.  
Figure 6: Graphs of N\({}_{\text{u}}\) in function of sample index for (**a**) the training part and (**b**) the testing part.  
Figure 7: Graphs of normalized error histograms for (**a**) the training part and (**b**) the testing part.  
capability was proven in the previous section. In order to explore the sensitivity of each imperfection parameter, all inputs (i.e., \(\text{SLD}_{\text{x}}\), \(\text{SLD}_{\text{y}}\) and \(\text{I}_{\text{1}}\) to \(\text{I}_{\text{9}}\), as indicated in Section 3.2) were used to deduce the intensity of variables, such as min, max, and quantiles Q10, Q25, Q50, Q75, Q90. The values of these parameters in the normalized space are presented in Table 5.  
In the sensitivity analysis, each input was varied from its lowest to highest intensity values, while the median Q50 values were applied to all other variables. Thereby, the degree of sensitivity, denoted as \(DoS\) (in %), of the considered input variable could be calculated as follows:  
\[DoS_{i}^{j}=\frac{N_{i}^{j}-N_{Q50}^{all}}{N_{Q50}^{all}}\times 100, \tag{15}\]  
where \(N_{Q50}^{all}\) was the output critical buckling load when all input variables remained at their median values. \(N_{i}^{j}\) was the output critical buckling load when applying input number \(j\) at its intensity \(i\), and \(DoS_{i}^{j}\) was the associated degree of sensitivity of the input \(j\) (\(I=1,\ldots,7\) and \(j=1,\ldots,9\)). Figure 8 presents the calculated degrees of sensitivity of all imperfection parameters. It can be clearly seen that the initial in-plane loading eccentricity at the bottom end (\(\text{I}_{\text{9}}\)) had the highest impact on the prediction of critical buckling load. The degree of sensitivity of \(\text{I}_{\text{9}}\) ranged from 4 to 6.5%, when varying \(\text{I}_{\text{9}}\) from the highest to lowest level of values. The second input that exhibited a significant effect on the critical buckling load prediction was the initial in-plane loading eccentricity at the top end (\(\text{I}_{\text{8}}\)). Its degree of sensitivity varied from \(-1.5\) to 1%, when ranging \(\text{I}_{\text{8}}\) from its highest to lowest values. All other imperfections had a similar effect and varied in ranges so small that the influences could be neglected. A summary of sensitivity analysis is indicated in Table 6. From the original data, it was observed that the critical buckling load factor (\(\varphi\)) ranged from 0.58 to 1.02, exhibited an average value of 0.86, and a standard deviation of 0.09 (Table 1). This indicated that the instability was almost predicted when applying compression loading to columns. The actual buckling capacity of columns was reduced with an average value of 14%. It was also noticed that, when the slenderness of columns increases, the values of the critical buckling load factor decrease (e.g., for specimens from N\({}^{\circ}\)55 to 57). This meant that initial imperfections, by largely changing the capacity of columns under compression, played an important role. They needed to be taken into account in the modeling as their influence became relevant. The next section is dedicated to an analysis of the influence of slenderness ratios on the predicted results.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Variables** & **Min** & **Q10** & **Q20** & **Q50** & **Q75** & **Q90** & **Max** \\ \hline \(\text{SLD}_{\text{x}}\) & \(-1.00\) & \(-0.86\) & \(-0.76\) & \(-0.36\) & 0.23 & 0.63 & 1.00 \\ \(\text{SLD}_{\text{y}}\) & \(-1.00\) & \(-0.78\)