## 5 Case Study/Parametric Study/Computational Examples

\end{table}
Table 5: The data from the modelling process of the clamped-free beam-type element. Figure 16: The critical buckling force variation as against the resistance of the cross-section to uniform compression based on the data from Table 6. \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** & \(F_{c,Rd}\) [**N**] & \(F_{b,Rd}\) [**N**] & \(\Delta F_{b,Rd}\) \% &
\begin{tabular}{c} _Error\%_ \\ (\(\cdot\)10\({}^{-5}\)) \\ \end{tabular} \\ \hline
1. & 1 & 1 & 1 & 1 & 1 & 3946.5 & 2136.9 & - & - \\
2. & 2 & 1.9 & 2 & 1.8 & 1 & 3946.5 & 3828.2 & 79.14 & 0.318 \\
3. & 2 & 1.9 & 1.7 & 1.5 & 1 & 3946.5 & 3906.3 & 82.81 & 0.014 \\
4. & 2 & 2 & 1.9 & 1.7 & 1 & 3946.5 & 4163.0 & 94.81 & 0.008 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The data from the modelling process of the guided-hinged beam-type element. Figure 15: The guided-hinged 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. d) The clamped-guided beam-type element  
The clamped-guided beam-type element is presented in Figure 17a, which highlights the kinematic boundary conditions. The type of profile imposed on the reference cross-section is CHS193.7x8-6000-S355 JO-EN10219, respectively values for the stiffness of the support connections are [_inf_ _inf_ _0 inf_]. The results of the iterative optimization phase are presented in Table 7 and the graphical representation of the data from Table 7 is presented in Figure 18.  
In Table 7 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-guided beam-type element is presented in Figure 17b,c. In the case of the clamped-guided beam-type element through the variation of the cross-section, without material addition, a significant increase of **46.91%** for the critical buckling load is obtained and ensuring a proper failure condition as presented in Figure 18. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{Id.}\) & \(\mathbf{i1}\) & \(\mathbf{i2}\) & \(\mathbf{i3}\) & \(\mathbf{i4}\) & \(\mathbf{i5}\) & \(\begin{array}{c}F_{c,Rd}\ [\mathbf{N}]\\ (\mathbf{\cdot 10^{3}})\end{array}\) & \(\mathbf{F_{b,Rd}}\ [\mathbf{N}]\) & \(\mathbf{\Delta}F_{b,Rd}\ \%\) & \(\begin{array}{c}Error\%\\ (\mathbf{\cdot 10^{-5}})\end{array}\) \\ \hline
1. & 1 & 1 & 1 & 1 & 1506.2 & 1160.4 & - & - \\
2. & 1 & 1.3 & 1.5 & 1.3 & 1 & 1506.2 & 1350.9 & 16.42 & 0.0089 \\
3. & 1 & 1.5 & 1.2 & 1.5 & 1 & 1506.2 & 1503.8 & 29.59 & 10.7241 \\
4. & 1 & 1.8 & 1 & 1.8 & 1 & 1506.2 & 1704.7 & 46.91 & 157.371 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The data from the modelling process of the clamped-guided beam-type element. Figure 17: The clamped-guided 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. **e)** **The hinged-clamped beam-type element**  
The hinged-clamped beam-type element is presented in Figure 19a, which highlights the kinematic boundary conditions. The type of profile imposed on the reference cross-section is CHS127x5-6000-S355J0-EN10219, respectively values for the stiffness of the support connections are [_inf_ 0 _inf_ inf].