## 5 Case Study/Parametric Study/Computational Examples
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Id.**} & \multirow{2}{*}{\(i\)**1**} & \multirow{2}{*}{\(i\)**2**} & \multirow{2}{*}{\(i\)**3**} & \multirow{2}{*}{\(i\)**4**} & \multirow{2}{*}{\(i\)**5**} & \(F_{c,Rd}\) [N] & \(F_{b,Rd}\) [N] (\(\cdot\)10\({}^{3}\)) & \multirow{2}{*}{\(\Delta F_{b,Rd}\) \%} & \multirow{2}{*}{\(Error\)\%} (\(\cdot\)10\({}^{-5}\)) \\ \hline
1. & 1 & 1 & 1 & 1 & 1 & 1263.7 & 685.8 & - & - \\
2. & 1 & 1.3 & 1.5 & 1.3 & 1 & 1263.7 & 933.1 & 36.06 & 0.054 \\
3. & 1 & 1.8  
The clamped-free beam-type element is presented in Figure 13a, which highlights the kinematic boundary conditions. The type of profile imposed on the reference cross-section is CHS3**23.9x12.5**-6000-S355 JO-EN10219, respectively values for the stiffness of the support connections are [_inf inf_ 0 ]. The results of the iterative optimization phase are presented in Table 4 and the graphical representation of the data from Table 5 is presented in Figure 14.  
Figure 12: The critical buckling force variation as against the resistance of the cross-section to uniform compression based on the data from Table 4.  
Figure 13: The clamped-free 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.  
In Table 5 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-free beam-type element is presented in Figure 13b,c.  
In the case of the clamped-free beam-type element through the variation of the cross-section, without material addition, a significant increase of **95.79%** for the critical buckling load is obtained and ensuring a proper failure condition as presented in Figure 14.  
**c) The guided-hinged beam-type element**  
The guided-hinged beam-type element is presented in Figure 15a, which highlights the kinematic boundary conditions. The type of profile imposed on the reference cross-section is CHS3**23.9x12.5**-6000-S355 JO-EN10219, respectively values for the stiffness of the support connections are [0 _inf inf_ 0]. The results of the iterative optimization phase are presented in Table 6 and the graphical representation of the data from Table 6 is presented in Figure 16.  
In Table 6 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 guided-hinged beam-type element is presented in Figure 15b,c.  
In the case of the guided-hinged beam-type element through the variation of the cross-section, without material addition, a significant increase of **94.81%** for the critical buckling load is obtained and ensuring a proper failure condition as presented in Figure 16.  
Figure 14: The critical buckling force variation against the resistance of the cross-section to uniform compression based on the data from Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Id.** & **\(i\)1** & **\(i\)2** & **\(i\)3** & **\(i\)4** & \(i\)5** & \(\begin{array}{c}F_{c,Rd}\) **[N]** \\ (\(\cdot\)10\({}^{3}\))** \\ \end{array}\) & \(F_{b,Rd}\) **[N]** & \(\Delta F_{b,Rd}\) **\%** & \(\begin{array}{c}Error\%\\ (\cdot\)10\({}^{-5}\))** \\ \end{tabular} \\ \hline
1. & 1 & 1 & 1 & 1 & 1 & 3946.5 & 2136.9 & - & - \\
2. & 2 & 1.8 & 1.5 & 1.3 & 1 & 3946.5 & 3632.2 & 69.97 & 0.015 \\
3. & 2 & 1.8 & 1.7 & 1.6 & 1 & 3946.5 & 3846.5 & 80.03 & 0.016 \\
4. & 2 & 2 & 1.9 & 1.8 & 1 & 3946.5 & 4183.9 & 95.79 & 0.043 \\ \hline \hline \end{tabular}