## 3 Analytical Solution of Different SHS Columns, Cross-Section Sizes, Lengths, and End Constraints
### Plastic Resistance in Columns without Imperfections and with Nonlinear Material Properties  
To obtain the plastic load, a nonlinear elastic analysis was used in the ANSYS(r) 2022 R2 program [24] with the nonlinear steel material behavior of an ideal column. The equations are nonlinear and solved iteratively using the Newton-Raphson method, where the applied load is divided into small increments to have a better numerical performance. The maximum applied incrementally load gives the plastic collapse in the column without any imperfections, based on the achieved elastic-perfectly plastic material behavior. In this process, a convergence criterion was based on displacement with a tolerance of 9%, and the solution is obtained when the difference between the displacements is less than or equal to this tolerance. Figure 4 represents the equivalent stresses obtained for the applied maximum incremental load that achieves the plasticity failure of the material. The columns reach the same value maximum yield stress value in the length (235 MPa) for any SHS column, regardless of the applied boundary conditions. Only slightly higher values are located near the load application, which is insignificant, and lower values in the restrictions. The range of these values is automatically obtained in the finite element program in all columns. The corresponding plastic load in each column is 67.4, 61.6, 61.9, and 65.8 kN for fixed-fixed, fixed-pinned, pinned-pinned and fixed-free columns, close to the analytical value equal to 68.4 kN (Table 1). Figure 3: Elastic deformation (Euler load) and different constraints (B50 and L = 1500 mm). Design Buckling Resistance in Columns with Initial Imperfections and with Nonlinear Material Properties  
To obtain the design buckling resistance, a nonlinear inelastic buckling analysis in ANSYS (r) 2022 R2 program [24] was applied, including geometric imperfections and nonlinear properties of the material (Figure 2c). The shape of the global imperfections may be obtained from the column elastic buckling [23]. The longitudinal curvature, which is normally present in columns, is an imperfection due to manufacturing processes, handling, and transporting of the components. According to the authors, initial imperfections can be described by a harmonic function, among other solutions [19; 25; 26], where \(L/1000\) is the maximum measured amplitude imperfection. \[u(x)=\bigg{(}\frac{L}{1000}\bigg{)}sin\Big{(}\frac{\pi\,x}{L}\Big{)} \tag{7}\]  
To update the geometry of the straight column and insert the initial imperfection, the UPGEOM command from ANSYS (r) 2022 R2 program [24] was used. The process is automatically executed according to the following command: \(<\)/prep7\(>\) UPGEOM, maximum deflection, Load step, Sub step, File name \(>\). This command adds displacements from the previous analysis (from the elastic buckling analysis eigenvectors) and updates the model geometry to the deformed configuration [24] (Figure 2c). With this command, it is possible to establish the amplitude of the displacement for each nodal point. To implement it, it is necessary to calculate the FACT factor, achieved with the lateral displacement \(u(x)\) of linear elastic buckling, which is multiplied by the displacements being included into the coordinates. \[FACT=\bigg{(}\frac{L}{1000}\bigg{)}\,u(x) \tag{8}\]  
For this model, an increased incrementally compressive load is applied until the beginning of buckling in the column (Figure 5). The nonlinear buckling solution requires the use of Newton-Raphson to calculate a system of equations iteratively, concerning the nonlinear material and geometry [27]. Therefore, the solution requires running of the model with incremental load until the column fails by convergence, and this is the point of buckling strength. Figure 4: Plastic resistance and different constraints (B50 and L = 1500 mm), MPa. The results show the calculated design buckling resistance, as the last incremental load applied, obtained in a column with imperfections, which represents the deformation tendency.