# Numerical Analysis of the Beam-Column Resistance Compared to Methods by European Standards
## 2 Numerical Analysis
### Finite Element Modelling  
The assumptions for finite element modeling (FEM) are presented. However, as the concept of modelling is similar for both IPE and RHS cross-sections, the assumptions for members with an IPE cross-section are presented, mentioning only differences for members with RHS cross-section in the Section 3.2. The shell element SHELL181 was used for the FE model. This is a four-node element with six degrees of freedom in every node and is applicable for nonlinear analysis including large strain and/or large rotation problems. Five integration points through the element layer were used, as only one layer through element thickness was assumed in the model. Since the objective of the numerical analysis was to determine ultimate beam-column resistances, the Newton-Raphson iteration method was chosen for the calculation. The cross-section parts consisted of twelve elements per flange width and twenty elements for web height. The connection between web and flange elements of hot-rolled I sections can be accomplished in various ways. A small material overlap is generated by the direct connection of the web and flange elements in their common nodes. One of the few approaches to avoid this material overlap is the use of a modified element in the location of the web-flange connection with dimensions such that area of this modified element is the same as the difference between radius zones and the overlapped area [22; 23]. Another method is to offset the web boundary elements by half of a flange thickness and connect the web and flange nodes by rigid constraint elements [24]. Finally, the direct connection of web and flange elements in their common nodes with a small material overlap was used in this analysis (Figure 3), although the tabular cross-sectional dimensions were used for analytically derived beam-columns resistances. These assumptions were chosen to simplify  
Figure 1: Definition of the internal forces and cross-sectional axes of the members. Figure 2: Load cases investigated in the numerical analysis. modelling in case of members with an RHS cross-section. Moreover, it was also observed that the radius zones had negligible influence in the case of lateral-torsional buckling [25]. This is also valid because the material overlap is almost at the same location as the radius zones, so the material overlap partly compensates the effect of the radius zones on the member resistance. The material properties of steel grade S235 were considered in the analysis, with a yield stress _fy_ = 235 MPa and modulus of elasticity \(E\) = 210,000 MPa. According to Annex C of Eurocode 3 part 1-5 [26], the following material models can be used while modelling plated structures:  
* elastic-plastic without strain hardening,
* elastic-plastic with a nominal plateau slope of 1 MPa or a similar low value,
* elastic-plastic with a strain hardening slope of _E_/100,
* true stress-strain curve modified from test results. The elastic-plastic material model with insignificant hardening slope was used because the strains at ultimate resistance of slender structures are small, so strain hardening has little effect on the ultimate resistance. On the other hand, the bilinear material model without strain-hardening could be the cause of convergence issues during numerical analysis [27]. Strain-hardening gains higher importance for stocky members, where the strains reach higher magnitudes at ultimate limit states. Imperfections are other phenomena which affect the resistance and behaviour of real members under combination of loading. In the case of numerical analysis of beam-columns, it is typical to consider geometric imperfections in the form of member bow curvature, and structural imperfections in the form of residual stresses. The initial imperfections have to be included in the nonlinear numerical analysis, not only due to the effect on the member resistance but also because buckling may not occur in the case of perfect straight members without any imperfections under axial compressive force and symmetric element distribution.