## 5 Case Study/Parametric Study/Computational Examples
In the case of the beam-type element with elastic supports through the modification of the cross-section, without material addition, a significant increase of \(\mathbf{1725.05\%}\) for the critical buckling load is obtained as against the new reference cross-section beam-type element, and it is ensured a propper failure condition as presented in Figure 24. Figure 23b,c are presented the beam-type element with elastic supports. In Table 12 a comparison is presented of the critical buckling forces without and with elastic support, with the variation coefficients of the moment of inertial extracted from Table 10.  
Case 1b from Table 12 highlights the fact that by adding elastic supports the failure occurs due to the loss of stability and variation modelling a variable cross-section for \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Id.** & \(i\mathbf{1}\) & \(i\mathbf{2}\) & \(i\mathbf{3}\) & \(i\mathbf{4}\) & \(i\mathbf{5}\) & \(\mathbf{r1}\) & \(\mathbf{r2}\) & \(\mathbf{r3}\) & \(\mathbf{r4}\) \\ \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 \hline \end{tabular}
\end{table} 
Table 12: The data comparison from the modelling process of the beam-type element without and with elastic supports. Figure 24: The critical buckling force variation as against the resistance of the cross-section to uniform compression with respect to the data from Table 11.  
the beam-type element is not appropriate and the reference cross-section with superior geometric and inertial properties is required. Case 2b from Table 12 highlights that by adding elastic supports for a reference cross-section with superior geometric and inertial properties the failure occurs due to exceeding the resistance of the cross-section under uniform compression and variation modelling a variable cross-section for the beam-type element is necessary. Adopting elastic support instead of ideal support connections **at least a 55% decrease** is noticed in case of the critical buckling force. The explanation for the radical decrease of the critical buckling force in the case of elastic support connections may be found in an analysis of the displacement field of a constant cross-section beam-type element loaded with axial forces and bending moments at the joints. In the case of the hinged-hinged beam-type element the displacements associated with the bending moment \(M\), are proportional with \(M\cdot L^{2}\) divided by the stiffness coefficient \(E\cdot I\), expressed in Relation (90a). The displacements associated with the axial force \(Q\), are proportional with \(Q\cdot L\) divided by the stiffness coefficient \(E\cdot A\), presented in Relation (90b) [29]. \[a_{M}(x)\,\sim\,\frac{M(x)\cdot L^{2}}{E\cdot I},\ \forall\ xe[0:L] \tag{90a}\]  
\[a_{N}(x)\,\sim\,\frac{Q(x)\cdot L}{E\cdot A},\ \forall\ xe[0:L] \tag{90b}\]  
The quantities expressed in Relation (90a,b) have the same order of magnitude if conditions from Relation (91a,b) are satisfied, with respect to Brachmann-Landau notation [30]. \[a_{M}(x)=O(a_{N}(x))\langle=\rangle\exists\ c_{1}\in R^{*}as\ if\ |a_{M}(x)|<c_{1}\cdot a_{N}(x),\ \forall\ xe[0:L] \tag{91a}\]  
\[a_{N}(x)=O(a_{M}(x))\langle=\rangle\exists\ c_{2}\in R^{*}as\ if\ |a_{N}(x)|<c_{2}\cdot a_{M}(x),\ \forall\ xe[0:L] \tag{91b}\]  
For the quantities defined in Relation (91a,b), Relation (92) can be expressed, for a node in \([0:L]\) interval [29]. \[\frac{M\cdot L^{2}}{E\cdot I}\sim\frac{Q\cdot L}{E\cdot A} \tag{92}\]  
By expressing ratio \(M/Q\) from Relation (92) with respect to [PERSON]'s law and Navier's relation, Expression (93) is obtained [31].