# Numerical Analysis of the Beam-Column Resistance Compared to Methods by European Standards
## 3 Verification According to European Standards
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Specimen** & **Section** & **Load Case** & \(\frac{-}{\lambda}\) & **N\({}_{\text{ed,rep}}\)/N\({}_{\text{pl}}\)** & **N\({}_{\text{ed,FEM}}\)/N\({}_{\text{pl}}\)** & **M\({}_{\text{ul,exp}}\)/M\({}_{\text{pl}}\)** & **M\({}_{\text{ul,FEM}}\)/M\({}_{\text{pl}}\)** & \(\xi\) \\ \hline T1 & 8 WF 31 & d & 0.4 & 0.130 & 0.130 & 1.03 & 1.01 & 0.98 \\ T2 & 8 WF 40 & d & 0.23 & 0.148 & 0.148 & 1.13 & 1.03 & 0.91 \\ T3 & 8 WF 31 & b & 0.45 & 0.496 & 0.496 & 0.59 & 0.58 & 0.98 \\ T8 * & 8 WF 31 & c & 0.63 & 0.588 & 0.665 & 0.16 & 0.16 & 1.13 \\ T11 * & 8 WF 31 & e & 0.64 & 0.862 & 0.836 & - & - & 0.97 \\ T12 & 8 WF 31 & c & 0.64 & 0.122 & 0.122 & 0.76 & 0.79 & 1.04 \\ T13 & 8 WF 31 & d & 0.63 & 0.122 & 0.122 & 1.02 & 0.97 & 0.95 \\ T14 & 8 WF 31 & a & 0.64 & 0.230 & 0.230 & 0.88 & 0.93 & 1.06 \\ T16 & 8 WF 31 & c & 0.48 & 0.123 & 0.123 & 0.75 & 0.85 & 1.13 \\ T18 * & 8 WF 31 & e & 0.32 & 0.882 & 0.933 & - & - & 1.06 \\ T19 & 8 WF 31 & c & 0.32 & 0.121 & 0.121 & 0.78 & 0.88 & 1.13 \\ T23 & 4 WF 13 & d & 0.96 & 0.114 & 0.114 & 0.93 & 0.93 & 1.00 \\ T25 * & 8 WF 31 & e & 0.88 & 0.696 & 0.631 & - & - & 0.91 \\ T26 & 4 WF 13 & c & 0.96 & 0.122 & 0.122 & 0.72 & 0.76 & 1.05 \\ T27\({}^{\text{z}}\) & 8 WF 31 & c & 0.825 & 0.503 & 0.503 & 0.21 & 0.20 & 0.94 \\ T32 & 4 WF 13 & c & 1.29 & 0.122 & 0.122 & 0.64 & 0.63 & 0.98 \\ \hline \hline \end{tabular}  
* Nomenclature: \(\overline{\lambda}\)—s slenderness ratio of the appropriate buckling axis, N\({}_{\text{pl}}\), M\({}_{\text{pl}}\)—the cross-section resistance in compression/bending, N\({}_{\text{ed,exp}}\), M\({}_{\text{ul,exp}}\)—the ultimate resistances in compression/bending acquired by the experiment, N\({}_{\text{ed,FEM}}\), M\({}_{\text{ul,FEM}}\)—the ultimate resistances in compression/bending acquired by the numerical analysis, \(\xi\)—the ratio of the experimentally acquired ultimate compression/bending resistances to the resistance obtained by numerical analysis.  
\end{table} 
Table 1: Comparison of ultimate resistance of beam-columns determined by numerical and experimental analysis. \[\frac{N_{Ed}}{\frac{\chi_{x}N_{Rg}}{\gamma_{M1}}}+k_{zy}\frac{M_{y,Ed}+\Delta M_{ y,Ed}}{\chi_{LT}\frac{M_{y,Rd}}{\gamma_{M1}}}+k_{zz}\frac{M_{z,Ed}+\Delta M_{z,Ed}}{ \frac{M_{z,Rd}}{\gamma_{M1}}}\leq 1, \tag{2}\]  
where: \(N_{Ed}\), \(M_{y,Ed}\), and \(M_{z,Ed}\) are the design values of the compression force and the maximum bending moments about the \(y\)-\(y\) and \(z\)-\(z\) axis along the member, \(\Delta M_{y,Ed}\) and \(\Delta M_{z,Ed}\) are the bending moments due to the shift of the centroid axis for class 4 of cross-sections, \(\chi_{y}\) and \(\chi_{z}\) are the reduction factors due to flexural buckling, \(\chi_{LT}\) is the reduction factor due to lateral torsional buckling, \(k_{yy}\), \(k_{yz}\), \(k_{zz}\), are the interaction factors that are relative to either method Annex A or Annex B of standard EN 1993-1-1, \(N_{Rk}\), \(M_{y,Rk}\), and \(M_{z,Rk}\) are the characteristic values of resistances to normal force and bending moments \(y\)-\(y\) and \(z\)-\(z\) axis, and \(\gamma_{M1}\) is the partial factor for resistance of members to instability assessed by member checks. A cross-sectional check is included in the buckling resistance formulas according to Eurocode 9 Section 6.3.3, so that only the lowest resistance from the buckling resistance formulas is considered in the verification. This is especially true in the case of a laterally unrestrained beam-column subjected to a combination of axial compressive force and bending moment about the major axis, as the failure can occur due to the large bending moment in the plane of bending and to the large axial force in the direction perpendicular to the plane of bending.