## 3 Materials and Methods
### Data Preparation  
In this work, axial compression tests resulted from 57 Y-section columns, with steel with a yield strength of 420 MPa, collected by [PERSON] et al. [83], and these tests are briefly summarized in Table 1. In the data, all measurements of initial geometric imperfections are included. The Y-section columns were made by welding equal-leg steel angles and steel plates (Figure 2), which were manufactured from high-strength steel of 210 MPa Young's modulus. Fifty-seven columns were fabricated with slenderness ratios (SLDs) uniformly ranging from 30 to 80 and with different width-to-thickness ratios of steel angles and plates. The columns were tested under pinned-pinned boundary condition (Figure 2). More details on the fabrication process, the geometry of the columns, and the compression test setup can be found in the original work of [PERSON] et al. [83]. Initial imperfections of columns, such as fluctuations in mechanical properties of materials, residual stress, and initial geometric imperfections (out-of-straightness and loading eccentricity), were measured. First, the mechanical properties of steels were investigated. Tensile tests were conducted to show that the Young's modulus varied from 210 to 211 GPa for equal angle steels and from 209 to 214 GPa for plates, whereas the yield strength ranged from 450 to 455 MPa for equal angle steels and from 485 to 496 MPa for plates. From the obtained results, it could be concluded that the fluctuations in mechanical properties are small. Second, as regards the residual stress imperfection, the results from experimental measurements showed about zero resultant force and residual stresses moments over the measured cross sections. Last but not least, initial geometric imperfections, i.e., out-of-straightness at mid-length and loading eccentricities at both ends, exhibited a significant variation with respect to the effective length of columns. From these remarks, two hypotheses could be formulated: (i) the material constituting 57 columns was assumed to be homogeneous in terms of mechanical properties and (ii) residual stress had a negligible effect on the behavior of columns under compression. The advantage of high-strength steel over normal-strength steel in decreasing the impact of residual stress over yield strength can also be found in the literature [3; 11; 14]. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline N\({}^{\circ}\) & **SLD\({}_{\mathbf{x}}\)** & **SLD\({}_{\mathbf{y}}\)** & **I\({}_{\mathbf{1}}\)** & **I\({}_{\mathbf{2}}\)** & **I\({}_{\mathbf{3}}\)** & **I\({}_{\mathbf{4}}\)** & **I\({}_{\mathbf{5}}\)** & **I\({}_{\mathbf{6}}\)** & **I\({}_{\mathbf{7}}\)** & **I\({}_{\mathbf{8}}\)** & **I\({}_{\mathbf{9}}\)** & **N\({}_{\mathbf{u}}\)** & \(\boldsymbol{\varphi}\)** \\ \hline
1 & 25.8 & 30.2 & \(-\)0.28 & 0.32 & \(-\)3.7 & 0.03 & 1.67 & 2.89 & 0.42 & 3.46 & 2.47 & 1523 & 0.906 \\
2 & 26 & 30.2 & 0.42 & 0.49 & 4.02 & 0.05 & 2.15 & 2.73 & 0.64 & 5.43 & 2.14 & 1483 & 0.882 \\
3 & 25.8 & 30.4 & \(-\)0.39 & 0.57 & 2.35 & 0.12 & \(-\)3.44 & 4.25 & 0.68 & \(-\)3 & 5.36 & 1631 & 0.97 \\ - &. &. &. &. &. &. &. &. &. &. &. &. \\ - &. &. &. &. &. &. &. &. &. &. &. &. \\
55 & 78.5 & 62.6 & 1.61 & 1.44 & \(-\)0.96 & \(-\)0.12 & 1 & 1.21 & 2.15 & \(-\)2.09 & 5.92 & 760 & 0.604 \\
56 & 78.9 & 63.6 & \(-\)1.4 & 1.45 & 3.37 & 0.04 & 7.03 & \(-\)6.06 & 2.02 & \(-\)0.68 & \(-\)9.69 & 842 & 0.669 \\
57 & 78.1 & 62.6 & 1.09 & 1.52 & 1.97 & 0.31 & 6.1 & 2.53 & 1.85 & \(-\)0.75 & 4.6 & 735 & 0.584 \\ \hline Min & 25.80 & 23.80 & \(-\)1.40 & \(-\)1.34 & \(-\)4.40 & \(-\)0.75 & \(-\)10.78 & \(-\)8.25 & 0.42 & \(-\)9.24 & \(-\)9.69 & 735.00 & 0.58 \\ Average & 48.56 & 48.07 & 0.27 & 0.45 & 0.24 & 0.05 & \(-\)0.67 & 1.67 & 1.14 & \(-\)0.69 & 1.72 & 1247.51 & 0.86 \\ Max & 79.50 & 80.90 & 1.79 & 1.78 & 5.14 & 0.67 & 9.17 & 9.81 & 2.15 & 8.34 & 10.62 & 1631.00 & 1.02 \\ SD\({}^{*}\) & 16.28 & 14.83 & 0.84 & 0.84 & 3.16 & 0.29 & 4.33 & 4.54 & 0.53 & 4.76 & 5.23 & 221.01 & 0.09 \\ \hline \hline \end{tabular} \({}^{*}\)Standard deviation.  
\end{table}