# Consistency of Imperfections in Steel Eurocodes
## 4 Amplitudes of the Equivalent Geometrical Imperfections according to Eurocodes [7,17]  
The values of the initial local bow imperfection \(e_{0}/L\) in the draft ForrEN 1993-1-1:2021-11-26 [17] were changed compared to the current Eurocode EN 1993-1-1 [7]. According to 7.3.3.1 in [17], the equivalent bow imperfection, \(e_{0}\), of the members for flexural buckling may be determined as follows:  
\[e_{0}=\alpha\beta\sqrt{\frac{f_{y}}{235\;\text{MPa}}}L, \tag{74}\]  
where the imperfection factor \(\alpha\) is in Table 6 and \(\beta\) is the relative bow imperfection according to Table 7. \begin{table}
\begin{tabular}{c c c} \hline
**Buckling Curve,** & **Elastic Analysis** & **Plastic Analysis** \\
**Imperfection Factor \(\alpha\)** & \(e_{0}/L\) & \(e_{0}/L\) \\ \hline \(a_{0}:\;\alpha=0.13\) & 1/350 & 1/300 \\ \hline \(a:\;\alpha=0.21\) & 1/300 & 1/250 \\ \hline \(b:\;\alpha=0.34\) & 1/250 & 1/200 \\ \hline \(c:\;\alpha=0.49\) & 1/200 & 1/150 \\ \hline \(d:\;\alpha=0.76\) & 1/150 & 1/100 \\ \hline \end{tabular}
\end{table} 
Table 6: Design value of initial local bow imperfection \(e_{0}/L\) according to 5.3.2b) in [7]. It is evident in the UGLI imperfection method that safety factor \(\gamma_{\text{M1}}\) cannot be removed from \(e_{0,\text{d},}\) and value \(e_{0,\text{k}}\) cannot be used in this method. See comparisons in Tables 8 and 9. From the comparisons in the above diagrams in Figures 19-22 for the values of bow imperfections \(e_{0}\) used in the second-order theory, the results read as so: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Numerical Example 1, Table 3**} & **FprEN 1993-1-1:2021 [17]** & **EN 1993-1-1:2005 [7]** \\ \hline \(e_{0,\text{k}}\) & \(e_{0,\text{d}}\) & \(e_{0,\text{el,y}}\left(e_{0,\text{pl,y}}\right)\) & \(e_{0,\text{el,y}}\left(e_{0,\text{pl,y}}\right)\) \\ \hline \(14.92\) mm & \(15.63\) mm & \(22.9,\ (33.6)\) mm & \(40,\ (48)\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Buckling about axis \(y-y\). Comparison of bow imperfections \(e_{0}\) for the second-order theory. Figure 21: Comparison of horizontal lines [7] to the curves [17] defining bow imperfection \(e_{0,\text{pl,y}}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Numerical Example 1, Table 4**} & **FprEN 1993-1-1:2021 [17]** & **EN 1993-1-1:2005 [7]** \\ \hline \(e_{0,\text{k}}\) & \(e_{0,\text{d}}\) & \(e_{0,\text{el,z}}\left(e_{0,\text{pl,z}}\right)\) & \(e_{0,\text{el,z}}\left(e_{0,\text{pl,z}}\right)\) \\ \hline \(8.09\) mm & \(10.51\) mm & \(18.55,\ (27.2)\) mm & \(24,\ (30)\) mm \\ \hline \(M_{\text{Ed,c}}^{\text{II}}\) calculated by FE-STAB [1] & for \(N_{\text{Ed}}=859.584\) kN, for the above \(e_{0}\) values with \(\gamma_{\text{M1}}=1.1\) \\ \hline \(22.96\) kNm & \(29.83\) kNm & \(52.65\) kNm & \(68.12\) kNm \\ \hline \multicolumn{2}{c}{\(U^{\text{II}}\); \((U_{\text{MN}}),\)\(U_{\text{N}}=0.348\)} \\ \hline \(0.85;\ (0.502)\) & \(1.0;\ (0.652)\) & \(1.499;\ (1.151)\) & \(1.837;\ (1.489)\) \\ \hline \multicolumn{2}{c}{\(N_{\text{Ed}}\) to obtain \(U^{\text{II}}=1.1\)} \\ \hline \(916\) kN & \(\begin{array}{c}N_{\text{Ed}}=N_{\text{b,Rd}}\\ =859.584\) kN \\ \end{array}\) & \(723\) kN & \(656\) kN \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Buckling about axis \(z-z\). Comparison of values \(e_{0},\ M_{\text{Ed,c}}^{\text{II}},\ U^{\text{II}}\) and \((U_{\text{MN}})\) of the second-order analysis. The \(e_{0}\) values for the plastic cross-section verification were higher than those for the elastic ones, \(e_{0,\mathrm{p}\mathrm{,p}\mathrm{l}}>e_{0,\mathrm{e}\mathrm{l}}\);  
The \(e_{0}\) values for the buckling about the axis \(y-y\) were higher than those for the buckling about the axis \(z-z\), \(e_{0,y}>e_{0,z}\);  
If \(e_{0,\mathrm{d}}\) was used in the second-order theory, identical results were obtained to the EM method in the ultimate state when \(N_{\mathrm{E}\mathrm{d}}=N_{\mathrm{b,Rd}}\). The UGLI imperfection method is based on this fact.