# Numerical Analysis of the Beam-Column Resistance Compared to Methods by European Standards
## 4 Results and Discussion  
The comparison of design methods according to European standards for steel beam-columns with IPE and RHS types of cross-sections is presented in the previous sections. In and out-of plane buckling behaviour was the objective of the study. As was presented on the interaction diagrams, the accuracy of Methods 1 and 2 of Eurocode 3 was generally sufficient for loading conditions where the bending plane was the same as the buckling plane for both cross-sections. On the other hand, the accuracy of these design methods was lower for the combination of strong axis bending moment and out-of-plane buckling mode, especially for the IPE cross-section with warping restraints on the member ends. The reason seems to be in the linear interaction formulae and interaction factors \(kij\), whose values don't allow development of full plastic resistance. Moreover, the different values of the \(kij\) interaction factors were the reason of different resistance values between these two methods, as they were developed on different principles. The interaction factors according to Method 1 were derived, as far as possible, on theoretical aspects to provide a general and transparent design procedure where each coefficient represented a single phenomenon. On the contrary, Method 2 provided a design procedure with a reduced number of coefficients to minimize the amount of calculation work. The interaction factors were, therefore, determined on the basis of numerical simulations, although they were still dependent on physical parameters [4]. The example of the pinned-fixed beam column illustrates that the accuracy of these methods was lower in cases of loading conditions where the maximum values of primary and secondary moments did not occur in the same place. This was due to the fact that the resistance of cross-sections is the governing factor, and these methods can't handle such situations. Therefore, a cross-section check is also needed. The accuracy of these methods changed with the buckling modes, loading cases, or slenderness ratio; therefore, it was difficult to decide which method was more accurate. The design method according to Eurocode 9 for aluminium structures was also compared to the numerical analysis. This method also provides various degree of accuracy  
Figure 47: Interaction curves for weak axis flexural buckling of beam-columns with RHS cross-section. for different loading conditions. The resistances were mostly higher for lower slenderness ratios but, on the other hand, the resistances were lower than the resistances acquired by numerical simulations for higher slenderness ratios. This similarity was seen for all loading conditions and for both types of cross-section. In of this, in the case of lateral-torsional buckling of beam-columns with an IPE type of cross-section (strong axis bending and out-of-plane buckling mode) with restrained warping at the member ends, the accuracy seemed to be higher compared to the methods of Eurocode 3, although the resistances were found to be also on unsafe side for slenderness \(\overline{\lambda}_{z}=0.5\). However, the most specific was the out-of-plane buckling case with strong axis bending moment and an RHS cross-section, where the resistances were lower than the FEM resistances for all observed slenderness ratios. With rising slenderness, the differences were in order of tens of percents. Nevertheless, as was already mentioned, the interaction curves provided by this method had the same parabolic behaviour as the interaction curves gathered by GMNIA analysis for all slendernesses and both types of cross-section. This was especially visible in the example of lateral-torsional buckling of an IPE member subjected to constant bending moment diagram with relative slenderness ratio \(\overline{\lambda}_{z}=1.0\) (Figure 15), as the interaction curve was almost identical to the curve acquired by numerical analysis. This is possibly due to the exponential form of the interaction formula, which was based on the plastic cross-sectional criterion.