## 5 Case Study/Parametric Study/Computational Examples
\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** & \(\begin{array}{c}\mathbf{F_{c,Rd}}\ [\mathbf{N}]\\ \mathbf{(\cdot 10^{3})}\end{array}\) & \(\mathbf{F_{b,Rd}}\ [\mathbf{N}]\\ \mathbf{(\cdot 10^{3})}\end{array}\) & \(\mathbf{\Delta F_{b,Rd}}\) & \(\mathbf{\Delta F_{b,Rd}}\) & \(\mathbf{\Delta F_{b,Rd}}\) & \(\mathbf{\Delta F_{b,Rd}}\) \\ \hline
1. & 1 & 1 & 1 & 1 & 1 & 581.6 & 476 & - & - \\
2. & 1 & 1.4 & 1 & 1.4 & 1 & 581.6 & 536.6 & 12.73 & 142.042 \\
3. & 1 & 1.1 & 1.4 & 1.1 & 1 & 581.6 & 553.5 & 16.28 & 0.1849 \\
4. & 1 & 1.2 & 1.7 & 1.2 & 1 & 581.6 & 610.5 & 28.25 & 2.3417 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The data from the modelling process of the clamped-clamped beam-type element. Figure 22: The critical buckling force variation as against the resistance of the cross-section to uniform compression based on the data from Table 9.  
The clamped-clamped beam-type element  
The beam-type element with elastic supports is presented in Figure 23a, which highlights the kinematic boundary conditions. The type of profile imposed to the reference cross-section is CHS**101.6x6**-6000-S355 JO-EN10219, respectively values for the stiffness of the support connections are [0.002 [kN/m] 28.935 [kNm/m] 0.002 [kN/m] 28.935 [kNm/m]] 104 established according to Expression (64). The results of the iterative optimization phase are presented in Table 10 for ideal and elastic restraints. In Table 10, the symbol _fin_ represents the finite numerical value for the corresponding stiffness of the elastic support, determined according to Expression (64). The symbol _inf_ represents the ideal beam-type element support connections stiffness. In Table 10 it is highlighted that considering finite numerical values for the stiffness of the support connections the critical buckling load decreases radically, for example in case 1b the buckling force **decreases** by **84.07%** and in case 2b buckling force **decreases** with **55.31%**. To avoid such sudden changes, which influence negatively the failure condition, another type of cross-section is chosen, with superior geometric and inertial characteristics. In Table 11 the data from modelling a beam-type element with elastic support is presented with the reference cross-section CHS**139.7x6**-6000-S355 JO-EN10219, and the numerical \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Id.** & _i_**1** & _i_**2** & _i_**3** & _i_**4** & _i_**5** & _r_**1** & \(r\)2** & \(r\)3** & _r_**4** \\ \hline
1a. & 1 & 1.2 & 1.7 & 1.2 & 1 & _inf_ & _inf_ & _inf_ & _inf_ \\
1b. & 1 & 1.2 & 1.7 & 1.2 & 1 & _fin_ & _fin_ & _fin_ & _fin_ \\
2a. & 1 & 1.9 & 2 & 1.9 & 1 & _inf_ & _inf_ & _inf_ \\
2b. & 1 & 1.9 & 2 & 1.9 & 1 & _fin_ & _fin_ & _fin_ \\ \hline \end{tabular}
\end{table} 
Table 10: The data from the modelling process of the beam-type element with elastic supports. Figure 23: The beam-type element with elastic supports: (a) Schematic representation of the kinematic boundary conditions, where _L_—is the length of the beam-type element; _Q_—is the axial force; \(r\)1_—denotes the stiffness for a hindered transversal displacement for joint \(i\); \(r\)2—denotes the stiffness for a hindered rotation for joint \(i\); \(r\)3—denotes the stiffness for a hindered rotatron for joint \(j\); (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. values for the stiffness of the support connections [0.007 [kN/m] 78.996 [kNm/m] 0.007 [kN/m] 78.996 [kNm/m]]\(\cdot 10^{4}\) determined according to Expression (64). In Table 11 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.