## 3 Materials and Methods
### Data Preparation

Table 1: Inputs and output of the present study. Based on the mentioned hypotheses above, input parameters for AI algorithms for predicting the buckling capacity of columns were divided into two main groups: slenderness ratios and initial geometric imperfections. Slenderness ratios (SLD) of columns containing two inputs around the x (out-of-plane) and y (in-plane) axes of the Y-section were denoted as \(\mathrm{SLD_{x}}\) and \(\mathrm{SLD_{y}}\), respectively (Table 1). The out-of-plane slenderness ratio (\(\mathrm{SLD_{x}}\)) varied from 25.8 to 79.50, whereas the in-plane slenderness ratio (\(\mathrm{SLD_{y}}\)) varied from 23.80 to 80.90. The corresponding mean values and standard deviations were calculated and are summarized in Table 1. On the other hand, the initial geometric imperfections consisted of nine inputs. The deviation in the u-direction (Figure 2d) of the mid-length cross section (denoted by I\({}_{1}\) where I stands for imperfections) ranged from \(-\)1.40 to 1.79 mm. The deviation in the v-direction (Figure 2d) of the mid-length cross section (I\({}_{2}\)) ranged from \(-\)1.34 to 1.78 mm. The rotation angle of the mid-length cross section (I\({}_{3}\)) varied from -0.0044 to 0.00514 rad. The initial out-of-plane bending (I\({}_{4}\)) ranged from \(-\)0.75 to 0.67 mm. The initial out-of-plane loading eccentricity at the top end (I\({}_{5}\)) ranged from \(-\)10.78 to 9.17 mm. The initial out-of-plane loading eccentricity at the bottom end (I\({}_{6}\)) ranged from \(-\)8.25 to 9.81 mm. The initial in-plane bending (I\({}_{7}\)) varied from 0.42 to 2.15 mm. The initial in-plane loading eccentricity at the top end (I\({}_{8}\)) ranged from \(-\)9.24 to 8.34 mm. The initial in-plane loading eccentricity at the bottom end (I\({}_{9}\)) ranged from \(-\)9.69 to 10.62 mm. It is noteworthy that the initial out-of-straightness imperfections were estimated by measuring the deflections and rotation of the mid-length cross section, while assuming that the latter exposed maximum displacement for pinned-pinned columns under compression. By making these measurements, initial in-plane and out-of-plane bendings were deduced. Initial in-plane and out-of-plane loading eccentricities at the columns' top and bottom ends were also found. The output  
Figure 2: Description of column: (**a**) schematization of pinned–pinned column under axial loading; (**b**) 3D view of the initial Y-section column; (**c**) deformed column under compression; (**d**) geometry of the Y-form cross section. of the prediction tools, the critical buckling load (denoted by N\({}_{\text{u}}\)), ranged from 735 to 1631 kN with a mean value of 1247.51 kN and a standard deviation of 221.01 kN. It can be observed that the critical buckling load factor (denoted as \(\varphi\)), ranging from 0.58 to 1.02, exhibited an average value of 0.86 and a standard deviation of 0.09. Complete values of the dataset could be found in the work of [PERSON] et al. [83]. In this work, the collected dataset was divided into two sets: training and testing. Training data (approximately 70% of the total data) were used to train the AI models, whereas testing data (approximately 30% of the remaining dataset) were used to quantify the performance of the AI models. Different from the original work, the training dataset (including 11 inputs and 1 output) was normalized in the [\(-1\); \(1\)] range. By considering all variables in a uniform range, bias within the dataset between inputs could be minimized. In this study, the range [\(-1\); \(1\)] was selected to better capture the non-Gaussian probability density of variables. A normalization process of parameters, such as the minimum and maximum values of the training data were performed to scale the testing dataset, as indicated in Table 2. The latter aims at preventing statistical correlation between training and testing dataset when normalizing. After the simulation process, a reverse transformation was applied to convert data from the normalization space to the real one.