# Consistency of Imperfections in Steel Eurocodes
## 2 Equivalent Member Method  
The derivation of the reduction factor \(\chi\) for the relevant buckling curve: The basic formula for the flexural buckling curve to determine the characteristic resistance of a member under compression reads as so:  
\[\sigma=\frac{N_{\rm Ed}}{A}+\frac{M_{\rm Ed}^{\rm II}}{W}=f_{y}. \tag{1}\]  
After using the characteristic values of cross-section resistances:  
\[N_{\rm c,RK}=Af_{\rm y},\ M_{\rm c,RK}=Wf_{\rm y}, \tag{2}\]  
then Equation (1) may be rewritten as follows:  
\[\frac{N_{\rm Ed}}{N_{\rm c,RK}}+\frac{M_{\rm Ed}^{\rm II}}{M_{\rm c,RK}}=1.0. \tag{3}\]  
The factor by which the design value of the loading would have to be increased to cause elastic instability is as follows:  
\[\alpha_{\rm cr}=\frac{N_{\rm cr}}{N_{\rm Ed}}. \tag{4}\]  
The amplification factor after taking into account second-order effects is as follows:  
\[k=\frac{1}{1-\frac{N_{\rm Ed}}{N_{\rm cr}}}=\frac{1}{1-\frac{1}{\alpha_{\rm cr }}}=\frac{\alpha_{\rm cr}}{\alpha_{\rm cr}-1}. \tag{5}\]The bending moment at the member midspan according to the first-order theory is as follows:  
\[M_{\text{Ed}}^{\text{I}}=N_{\text{Ed}}e_{0,\text{k}}=N_{\text{Ed}}\alpha\big{(} \overline{\lambda}-\overline{\lambda}_{0}\big{)}\ \frac{W}{A}. \tag{6}\]  
The bending moment at the member midspan according to the second-order theory is as follows:  
\[M_{\text{Ed}}^{\text{II}}=kM_{\text{Ed}}^{\text{I}}=\frac{1}{1-\frac{N_{\text{ Ed}}}{N_{\alpha}}}N_{\text{Ed}}e_{0,\text{k}}=\frac{1}{1-\frac{N_{\text{ Ed}}}{N_{\alpha}}}N_{\text{Ed}}\alpha\big{(}\overline{\lambda}- \overline{\lambda}_{0}\big{)}\ \frac{W}{A}. \tag{7}\]  
The characteristic imperfection amplitude value is as follows:  
\[e_{0,\text{k}}=\alpha\big{(}\overline{\lambda}-\overline{\lambda}_{0}\big{)} \ \frac{M_{\text{c,Rk}}}{N_{\text{c,Rk}}}=\alpha\big{(}\overline{\lambda}- \overline{\lambda}_{0}\big{)}\ \frac{W}{A}. \tag{8}\]  
The shape of the equivalent geometrical initial imperfection \(\eta_{\text{init}}(x)\) is that of the structure's elastic critical buckling mode \(\eta_{\text{cr}}(x)\). For the investigated member, it comes in the form of a sinus half-wave and includes both structural and geometrical imperfections, as shown by the following:  
\[\eta_{\text{init}}(x)=e_{0,\text{k}}\eta_{\text{cr}}(x)=e_{0,\text{k}}\sin \Bigl{(}\frac{\pi x}{L}\Bigr{)}. \tag{9}\]  
The imperfection factors \(\alpha\) corresponding to the appropriate buckling curves are shown in Table 1, together with \(\alpha=0\), which is valid for the ideal member. The relative slenderness \(\overline{\lambda}\) and plateau length \(\overline{\lambda}_{0}\) of buckling curves are shown in Equation (10). In the steel Eurocode, \(\overline{\lambda}_{0}=0.2\). Different \(\overline{\lambda}_{0}\) values are used in the aluminium Eurocode. \[\overline{\lambda}=\sqrt{\frac{N_{\text{c,Rk}}}{N_{\text{cr}}}}\,\ \overline{ \lambda}_{0}=0.2. \tag{10}\]  
After inserting the reduction factor \(\chi\) (11) and fraction (12) from Equation (3), the following was obtained:  
\[N_{\rm c,RK}=Af_{\rm y},\ M_{\rm c,RK}=Wf_{\rm y}, \tag{2}\]0  
\[N_{\rm c,RK}=Af_{\rm y},\ M_{\rm c,RK}=Wf_{\rm y}, \tag{2}\]1  
The basic quadratic Equation (13) was obtained for reduction factor \(\chi\):  
\[N_{\rm c,RK}=Af_{\rm y},\ M_{\rm c,RK}=Wf_{\rm y}, \tag{2}\]2  
which may be rewritten in the following way:  
\[N_{\rm c,RK}=Af_{\rm y},\ M_{\rm c,RK}=Wf_{\rm y}, \tag{2}\]3  
The solution of quadratic Equation (14) leads to \"European column buckling curves\" (Figure 1). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Imperfection Factor \(\alpha\)**} & \multirow{2}{*}{
\begin{tabular}{c} **For Ideal** \\ **Member** \\ \end{tabular} } & \multicolumn{4}{c}{**For Buckling Curves (b.c.s)**} \\ \cline{2-6}  & & \(a_{0}\) & \(a\) & \(b\) & \(c\) & \(d\) \\ \cline{2-6}  & 0 & 0.13 & 0.21 & 0.34 & 0.49 & 0.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of imperfection factors \(\alpha\) for ideal member and buckling curves.