# Numerical Analysis of the Beam-Column Resistance Compared to Methods by European Standards
## 3 Verification According to European Standards
### Resistance of Beam-Columns with IPE 100 Type of Cross-Section
#### 3.1.2 Axial Compression and Major-Axis Bending without Lateral Restraints
This method also provided more accurate results in the case of pinned-fixed beam-columns, especially for the lower values of bending moments. This is because the coefficients \(\omega_{x}\) and \(\omega_{xLT}\) modified the effect of reduction factors for flexural and lateral-torsional buckling at the places, where they do not have reasonable impact and, therefore, the formulas for stability verification change to cross-sectional verification formulas. Finally, it has to be mentioned that the accuracy of the compared design methods was highly dependent on the accurate determination of the limiting cases for the pure compression and bending. Whereas the reduction factors for flexural bending were found to be in a good agreement with the FEM simulations, the values of reduction factors for lateral-torsional buckling were, for most cases, more conservative. Findings were identical to results presented in [23; 33]. However, this issue is assumed to be solved in the new version of Eurocodes [34]. The analysis of beam-columns with free warping of end cross-sections is further presented. As was mentioned, to allow a free warping of the end cross-sections the modelling of boundary conditions had to be modified. Therefore, a similar approach to the modelling concept applied in the studies performed by [PERSON] or [PERSON] was adopted. [2; 24]. The deformation of such restrained member can be seen in Figure 19.  
Figure 18: Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with restrained warping. The displacements _Ux_ and _Uz_ of the end nodes of the flanges were coupled by constraint equations to the \"master\" nodes at the web-flange intersections. This means that the nodes remained on a straight line and the displacements of the slave nodes were dependent on the displacements of the chosen master node (Figure 20). A similar approach was used to couple the displacements _Ux_, _Uy_, _Uz_ and _Rx_ of the web nodes to the master node, which was located at the centre of gravity and shifted 1 mm away from the end cross-section. In the case of a pinned-fixed beam-column, such boundary conditions were modelled only at the pinned end, whereas at the fixed end no adjustment was needed. Comparison of the results using design methods and results of the numerical analysis for these loading conditions is presented on the Figures 21-25. Figure 19: The deformation of the beam-column without warping restraints at failure (scale factor = 12). Figure 20: Modelling technique of boundary conditions. Figure 14:0 Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with free warping. Figure 14:1 Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with free warping. A noticeable reduction of bending resistance can be seen compared to beam-columns with restrained warping in the member ends. This is due to the bigger impact of warping torsion at the most stressed cross-section. This difference increases with higher slenderness and is almost 13% in the case of a member with a parabolic bending moment diagram and relative slenderness \(\overline{\lambda}_{z}=2.0\). Moreover, the plasticity effects due to the higher impact of warping torsion were less significant, as the curves of members with lower slenderness had more linear behaviour. It should be noted that the method of Eurocode 9 doesn't consider this effect explicitly, as was not accounted in the calculation of exponents in the interaction formulae. Therefore, the methods of Eurocode 3 were more accurate for lower slenderness ratios, as they are based on the linear interaction formula.