## 2 Materials and Methods  
### Description of Cellular Beams and Selection of Variables for Training ML Models  
The data used in this work was extracted from a validated finite element (FE) model, previously introduced in the literature by [PERSON] et al. [31]. The goodness of using data from a validated FE solver instead of experiments for training ML models has been proved in various investigations such as [PERSON] and [PERSON] [32] and [PERSON] and [PERSON] [33]. In this study, a total number of 3645 FE configurations on buckling behavior of I-section beams with circular openings were extracted to form the dataset for training the ML models, involving eight inputs and one output, detailed as follows:  
The cross section of the beams is I-shaped under the four controlled parameters, section height, flange width, thickness, and web thickness (Figure 1). As indicated in Table 1, the section height varies from 420 mm to 700 mm (560 mm of mean value and 114.33 mm of standard deviation). The web thickness ranges from 9 mm to 15 mm (mean value of 12 mm and standard deviation of 2.45 mm). The flange width varies from 162 mm to 270 mm (216 mm of mean value and 44.10 mm of standard deviation). The flange thickness ranges from 15 mm to 25 mm (mean value of 20 mm and standard deviation of 4.08 mm). For a beam with given length, various circular openings had been cut regularly along the beam axis, under three controlled parameters such as beam end-opening distance, opening diameter, and  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{ \begin{tabular}{c} **Beam** \\ **Length** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Beam** \\ **End-Opening** \\ **Distance** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Opening** \\ **Diameter** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Inter-Opening** \\ **Distance** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Section** \\ **Height** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Web** \\ **Thickness** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Flange** \\ **Width** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **Flange** \\ **Thickness** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Flange** \\ **Backline** \\ **Load** \\ \end{tabular}  } \\ \hline Notation & L & d\({}_{0}\) & D & d & H & t\({}_{\text{mb}}\) & w\({}_{\text{flange}}\) & t\({}_{\text{flange}}\) & q\({}_{\text{c}}\) \\ Unit & m & mm & mm & mm & mm & mm & Mm & mm & N/m \\ Role & Input &  
inter-opening distance (Figure 1). The beam end-opening distance ranges from 12 mm to 718 mm (mean value of 265.36 mm and standard deviation of 157.46 mm). The opening diameter varies from 247 mm to 560 mm (383.56 mm of mean value and 92.98 mm of standard deviation). The inter-opening distance ranges from 24.70 mm to 274.40 mm (mean value of 112.51 mm and standard deviation of 68.51 mm). Finally, the beam length varies from 4000 mm to 8000 mm (6000 mm of mean value and 1410 mm of standard deviation). The cellular beams were subjected to a uniformly distributed load, as schematized in Figure 1. The pinned-pinned boundary condition was applied for the beams. It should be noticed that twisting rotation at both ends of the beams was not allowed to expose the in-plane's rotation and translation influence. The constituent material of the beams is steel under two controlled elastic constants such as the Young's modulus of 210 GPa and the Poisson's coefficient of 0.30. Quad shell elements containing 4 nodes and 6 degrees of freedom at each node (3 for translations and 3 for rotations) were employed to discretize the beams in the FE scheme. The FE simulations and numerical results are obtained using ABAQUS [34]. It has been proved that such element can correctly model thin and moderate thickness shell structures with elastic and elasto-plastic behaviors of materials [35].