## 3 Results and Discussions
### Sensitivity Analysis  
In this section, the sensitivity analysis was performed in favor of quantifying the influence of these inputs to the predicted critical buckling load using the best-found model, namely ANFIS-SFLA. The procedure of the sensitivity analysis is detailed in the following:  
A new input space of eight dimensions was constructed based on the probability density function of each variable for evaluating the sensitivity analysis. The new input space composes of only eleven values for each input variable ranging from its smallest to highest quantile with a resolution of 10 (for instance, [0,10,20,30,40,50,60,70,80,90,100]). When calculating the sensitivity for a considered input variable, the values were taken successively from the eleven previously mentioned quantiles, whereas other inputs remained at their median. Therefore, the method obtains information on the deviation of output solution in varying statistically the considered input variable. In this work, the reference case refers to the one where all input variables stayed at their 50% quantile level. For illustration purpose, the more changes on the output solution as compared with the reference case, the more influence the considered input is. Deviation in output solution when evaluating the sensitivity of the \(j\)th input variable can be written as the following [48]:  
\[\theta_{i}^{j}=\frac{Y_{i}^{j}-Y_{ref}}{Y_{ref}}, \tag{5}\]  
where \(Y_{ref}\) is the output of the reference case, \(Y_{i}^{j}\) is the output solution using \(j\)th input at its \(i\)th quantile. Figure 9 shows the evaluation of deviation for eight input variables used in this study. Obviously, all considered input variables govern the buckling capacity of the cellular beam. The critical buckling  
Figure 8: Graphs of error histogram for (**a**) the training dataset and (**b**) the testing dataset. Figure 7: Graphs of histogram between actual and predicted data for (**a**) the training dataset and (**b**) the testing dataset. capacity of the beam decreases when the length, L, and the opening diameter, D, increases, but it increases when the inter-opening distance d, section height H, web thickness t\({}_{\text{web}}\), flange width w\({}_{\text{flange}}\), and flange thickness t\({}_{\text{flange}}\) increase. In addition, the beam end-opening distance, d\({}_{0}\), does not significantly affect the buckling capacity of the beam. However, it could be seen that the deviation in output solution is nonlinear in function of input variables, especially L, t\({}_{\text{web}}\), t\({}_{\text{flange}}\), and w\({}_{\text{flange}}\). In case of the length L, a high order of amplitude is observed as compared with other input variables. Using the deviation previously presented, the final influence level of each input is calculated by summing all deviations based on the following expression:  
\[\Theta^{j}=\sum_{i=1}^{11}\left|\theta_{i}^{j}\right| \tag{6}\]  
The influence level of all input variables obtained by the ANFIS-SFLA model was scaled into the range of [0, 1] as plotted in Figure 10b. It is seen that the most influenced variable is the beam length L (\(\Theta^{\text{L}}=0.429\)), following by the flange width w\({}_{\text{flange}}\) (\(\Theta^{\text{w}_{\text{flange}}}=0.215\)), the flange thickness t\({}_{\text{flange}}\) (\(\Theta^{\text{t}_{\text{flange}}}=0.130\)), the web thickness t\({}_{\text{web}}\) (\(\Theta^{\text{t}_{\text{web}}}=0.121\)), the inter-opening distance d (\(\Theta^{\text{d}}=0.047\)), the opening diameter D (\(\Theta^{\text{D}}=0.046\)), the section height H (\(\Theta^{\text{H}}=0.008\)), and the beam end-opening distance d\({}_{0}\) (\(\Theta^{\text{d}_{0}}=0.003\)), respectively. In parallel, linear statistical correlation between each input and output was also calculated and compared with the results of sensitivity analysis obtained by using ANFIS-SFLA (Figure 10a).