## 3 Numerical and Analytical Analysis  
In order to compare the experimentally obtained results, a numerical and analytical analysis of the ratio of moment and rotation of the end section was carried out. To obtain the best possible comparison, the properties of the material obtained experimentally on the samples taken during the section of the columns were used. The numerical analysis was performed with a commercial software professionally used for bridge structures using FEM. A non-linear numerical analysis was used, which included material and geometric non-linearity. The material properties were taken as material working law diagrams (stress/deformation ratio) for concrete (Equation (2)) and for reinforcement (Equations (3)-(5)). These expressions resulted in a good match of the constitutive laws with the experimentally obtained ones. The quantities and the position of the reinforcement in each cross-section of the column were taken in the same way as in the experimentally tested specimens. Numerical column models comprise cantilever 3D beam finite elements (mesh size 2 cm) with 6 degrees of freedom which are fixed for all translations and rotations at the point corresponding to the connection of the column and the anchor block. The beam axis was assigned the column cross-section in its center of gravity. Model of the column was then loaded horizontally in steps corresponding to those performed in the experimental tests. The load increase was carried out until the load-bearing capacity of the element was reached. The vertical axial load was entered immediately at the beginning of the calculation and remained unchanged until the end of the calculation. The amount of the axial load changes for different columns (Table 1). Due to the properties of the software package used, it was possible to perform the calculation until the maximum concrete stress value and the corresponding relative deformation was reached. Then the ratio of moment and rotation of the final cross-section was observed. Points of interest, such as the yield strength, are marked to compare the results. Only  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Series**} & \multirow{2}{*}{ \begin{tabular}{c} **f\({}_{\text{cm,cube}}\)** \\ **Derive from Testing** \\ **(MPa)** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **f\({}_{\text{cm}}\)=** \\ **0.86\({}_{\text{cm,cube}}\)/0.92** \\ **(MPa)** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **f\({}_{\text{ck}}\)=** \\ **f\({}_{\text{cm}}\)=** \\ **(MPa)** \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} **f\({}_{\text{cm}}\)=** \\ **f\({}_{\text{cm}}\)=** \\ **f\({}_{\text{cm  
the monotonic load was applied in the numerical analysis and the results obtained were processed for comparison with experimental and analytical results. The analytically obtained ratio of moment and rotation of the end section is presented. This ratio is described by three characteristic points in the curve, which represent the cross-sectional state due to the load increase. The first domain is up to the point where the first crack occurs, followed by the domain until the yield strength of the reinforcement is reached (or concrete compressive failure), and finally the domain until the load bearing capacity of the cross-section is reached (concrete tensile strength or compression zone failure). The domain between the characteristic points was achieved by gradually increasing the load. Non-linear material properties were used. The non-linear stress-strain ratio specified in [46] was used for concrete, while the ratio specified in [47] was used for reinforcement. \[\frac{\sigma_{c}}{f_{cm}}=\frac{k\cdot\eta-\eta^{2}}{1+(k-2)\cdot\eta} \tag{2}\]  
where \(\sigma_{c}\) and \(\varepsilon_{c}\) are compressive stress and strain respectively. The coefficients \(\eta\) and \(k\) are given in [32].