## 5 Case Study/Parametric Study/Computational Examples
The results of the iterative optimization phase are presented in Table 8 and the graphical representation of the data from Table 8 is presented in Figure 20.  
Figure 19: The hinged-clamped beam-type element: **(a)** Schematic representation of the kinematic boundary conditions, where \(v\)—denotes the transversal displacement; \(Q\)—the axial force; **(b)** The variable cross-section beam-type element concerning the imposed boundary conditions; **(c)** Longitudinal section of the modeled beam-type element which highlights the variation of the cross-section. Figure 18: The critical buckling force variation as against the resistance of the cross-section to uniform compression based on the data from Table 7.  
In Table 8 as a function of the variation of the moment of inertia the resistance of the cross-section for uniform compression, \(F_{c,Rd}\), and the buckling resistance, \(F_{b,Rd}\), is computed and in the last column is presented the estimated error of the iteration cycle. The modeled hinged-clamped beam-type element is presented in Figure 19b,c. In the case of the hinged-clamped beam-type element through the variation of the cross-section, without material addition, a significant increase of **62.13%** for the critical buckling load is obtained and ensuring a proper failure condition as presented in Figure 20.  
**f)**: **The clamped-clamped beam-type element**  
The clamped-clamped beam-type element is presented in Figure 21a, which highlights the kinematic boundary conditions. The type of profile imposed on the reference cross-section is CHS101.6x6-6000-S355J0-EN10219, respectively values for the stiffness of the support connections are [_inf inf inf inf_]. The results of the iterative optimization phase are presented in Table 9 and the graphical representation of the data from Table 9 is presented in Figure 22.  
In Table 9 as a function of the variation of the moment of inertia the resistance of the cross-section for uniform compression, \(F_{c,Rd}\), and the buckling resistance, \(F_{b,Rd}\), is computed and in the last column is presented the estimated error of the iteration cycle. The modeled clamped-clamped beam-type element is presented in Figure 21b,c. In the case of the clamped-clamped beam-type element through the variation of the cross-section, without material addition, a significant increase of **28.25%** for the critical buckling load is obtained and ensuring a proper failure condition as presented in Figure 22.  
Figure 20: The critical buckling force variation as against the resistance of the cross-section to uniform compression based on the data from Figure 17:0 \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Id.** & **\(i\)1** & **\(i\)2** & **\(i\)3** & **\(i\)4** & **\(i\)5** & \(\begin{array}{c}F_{c,Rd}\) [N] \\ (\(\cdot\)10\({}^{3}\)) \\ \end{tabular} & \(F_{b,Rd}\) [N] & \(\Delta F_{b,Rd}\) \% &
\begin{tabular}[]{c} _Error\%_ \\ (\(\cdot\)10\({}^{-5}\)) \\ \end{tabular} \\ \hline
1. & 1 & 1 & 1 & 1 & 1 & 618.5 & 420.6 & - & - \\
2. & 1 & 1.7 & 1.2 & 1.7 & 1 & 618.5 & 587.5 & 39.68 & 9.8526 \\
3. & 1 & 1.5 & 1.6 & 1.5 & 1 & 618.5 & 600.7 & 42.82 & 0.0611 \\
4. & 1 & 1.9 & 1.6 & 1.9 & 1 & 618.5 & 681.9 & 62.13 & 19.1213 \\ \hline \end{tabular}
\end{table} 
Table 8: The data from the modelling process of the hinged-clamped beam-type element. Figure 21: The clamped-clamped beam-type element: (**a**) Schematic representation of the kinematic boundary conditions, where \(v\)—denotes the transversal displacement; \(Q\)—the axial force; (**b**) The variable cross-section beam-type element concerning the imposed boundary conditions; (**c**) Longitudinal section of the modeled beam-type element which highlights the variation of the cross-section.