# 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  
In the case of beam-columns without sufficient lateral restraints, lateral deflections may occur due to the action of an axial compressive force; therefore, the behaviour of such restrained members is different compared to previous loading conditions. Moreover, twisting can also occur to members with cross-sections prone to torsional deformations subjected to bending moments, which generates additional normal stresses in the flanges. However, warping is often restrained in design practice by end plates, which affect the distribution of normal stresses and warping torsion. Therefore, two loading conditions were observed in this section, with restrained and free warping member ends. A loading condition with restrained warping is presented first, as the modelling technique of end  
Figure 12: Interaction curves for laterally restrained beam-columns with IPE 100 cross-section. Nomenclature: \(\overline{\lambda}\)—sslenderness ratio for the appropriate buckling axis, \(N_{Rd}\), \(M_{y,Rd}\), \(M_{z,Rd}\)—design values of compression and plastic bending moment resistances, \(N_{Ed}\), \(M_{y,Ed}\), \(M_{x,Ed}\)—design values of axial compressive force and bending moments. cross-sections remains the same. Reduction factors for lateral-torsional buckling for all methods were calculated according to clause 6.3.2.2 of Eurocode 3 [8]. However, another adjustment was needed in the case of members loaded by concentrated force in the middle of the span with relative slenderness \(\overline{\lambda}_{z}=0.5\). Due to the member lengths, the forces which created high bending moments had to be so high that they also trigger local instabilities in the web beneath. Therefore, one approach about how to treat this effect is to consider a web stiffener (Figure 13), which was also used in the study performed by [PERSON] [32]. Both concentrated transverse load and uniformly distributed load acted at a location where the upper flange and web were connected. The resistances of beam-columns with rigid warping restraints in the end cross sections are presented on the Figures 14-18. Figure 14: Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with restrained warping. Figure 13: Detail of the web stiffener. Figure 16: Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with restrained warping. Figure 17: Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with restrained warping. Figure 15: Interaction curves for laterally unrestrained beam-columns with IPE 100 cross-section with restrained warping. In the case of the pin ended beam-column, the resistances according to Methods 1 and 2 were significantly lower than resistances predicted by the FEM model, especially for the lower slenderness ratios where the differences were up to 20%. For these loading conditions Method 1 seems to be more accurate than Method 2, although the relevant differences were visible only for slenderness \(\overline{\lambda}_{z}=0.5\). In the case of the pinned-fixed beam-column, Method 1 provided far more accurate results, even though the accuracy was lower with the rising value of bending moment. This is because in this distribution of first-order bending moment the resistance of the cross-section governs, and these methods cannot handle such situations. It can be seen that the accuracy of Eurocode 9 method was significantly better than for methods of Eurocode 3. In cases of pin-ended beam columns, the method according to Eurocode 9 provided generally lower resistances compared to the numerical simulations. The unsafe values of resistances are provided for members with relative slenderness \(\overline{\lambda}_{z}=0.5\). However, it can be seen that the curves according to method of Eurocode 9 have the same behaviour compared to the FEM curves. This is most visible for members with constant bending moment and slenderness \(\overline{\lambda}_{z}=1.0\).