## 5 Numerical Validation with Analytical Method  
Critical buckling and resistance loads were established by applying the ANSYS (r) 2022 R2 program [24]. Figures 6 and 7 represent the comparison of all results between the numerical and the analytical methods, each one correspondent to the end column applied boundary condition. The results were obtained with the analytical variation of \(N_{cr}\), \(N_{b,\leavevmode\
obreak\ kd}\) and the numerical solution \(N_{cr}\_{A}\), \(N_{b,\leavevmode\
obreak\ kd}\_{A}\) with the ANSYS (r) 2022 R2 program [24]. In Figure 6, the curve outlined by blue dots represents the limiting condition for elastic buckling results. The curve outlined by the orange dots represents the buckling resistance results. In both situations, there is an apparent convergence between numerical and analytical solutions. The results are presented as a ratio between the Euler elastic buckling load or the design buckling resistance and the yield load. The trend is always the same; the highest values correspond to the largest cross-section B80 until to the smallest B50. Furthermore, columns with shorter lengths achieve greater loads. Regarding the effect of boundary conditions, fixed-fixed columns have the highest critical load value and the lowest slenderness ratio. In fixed-pinned columns, the critical load level is lower, increasing the slenderness index. Pinned-pinned columns continue to record low values of critical load and increased slenderness, with fixed-free columns being those that present the lowest loads and highest slenderness indexes. Figure 7 shows the same results, but now only compares the solution of the design buckling resistance. It is noticeable that the numerical values present some differences relative to the analytical calculation. Generally, the results agree between analytical and numerical procedures, but some calculations are more conservative when using numerical tests. For numerical results, it was necessary to include the boundary conditions that should have the same real effect. The analytical method is based on some constant empirical factors, such as the imperfection factor, which is related to the buckling curve and cross-section. In the numerical solution, the imperfections were allocated to all geometry, with the previous values of the buckling displacements not being a constant value. Figure 5: Design buckling resistance and different constraints (B50 and L = 1500 mm), MPa. The design buckling resistance is not the same across the slenderness range. These values represent a failure for very slender or high columns with all different boundary conditions and slenderness greater than 0.2. Plastic damage is the critical load supported in low slender or thick columns, with slenderness lesser than 0.2. The exception occurs in fixed-free columns, where the damage occurs only due to buckling, without a plastic effect. According to the previous values, a relative error was obtained to compare the numerical with analytical values (Table 2). Equation (9) calculates the relative error according to the Euler critical load \(E_{\sigma}\), and Equation (10) calculates the relative error according to the design buckling resistance \(E_{b,Rd}\). \[E_{\sigma}=\left(\frac{N_{cr}-N_{cr,A}}{N_{cr}}\right)\times 100\% \tag{9}\]  
\[E_{b,Rd}=\left(\frac{N_{b,Rd}-N_{b,Rd,A}}{N_{b,Rd}}\right)\times 100\% \tag{10}\]  
Table 2 gives, in detail, the calculated relative error using Equations (9) and (10). Figure 6: Analytical and numerical solution of elastic buckling and design buckling resistance loads, where for analytical solution the variables are \(N_{cr}\) and \(N_{b,\,Rd}\), and for numerical results are \(N_{cr,A}\) and \(N_{b,\,Rd,A}\). The solution proved that the finite element method resulted in a close calculation of the analytical values for the elastic buckling of the SHS columns. The variation concerning the finite element method, and the analytical solution is between 0 and 6.8%.