## 2 Method for Obtaining UGLI Imperfection Amplitude
#### Steps for Obtaining UGLI Imperfection  
The first step is obtaining the buckling load \(\alpha_{cr}\) and buckling shape \(\{\eta_{cr}\}\). They can be computed using finite element method, for instance, according to [PERSON] [27].  
The second step is the calculation of bending moments and stresses related to the buckling mode as follows:  
\[M_{y,\eta}\ =\ -EI_{y}\frac{d^{2}\eta_{cr,w}}{dx^{2}} \tag{10}\]  
\[\sigma_{M}\ =\ \frac{E_{y}\frac{d^{2}\eta_{cr,w}}{dx^{2}}}{W_{y}} \tag{11}\]  
The third step is the first iteration.  
Initial guess:  
\[\alpha_{ult,1}\ =\ \min(\frac{A\cdot f_{y}}{N_{Ed}})\rightarrow\tilde{\lambda_{1 }}\ =\ \sqrt{\frac{\alpha_{ult,1}}{\alpha_{cr}}}\rightarrow\chi_{1}\rightarrow\alpha _{b,1}\ =\ \frac{\alpha_{ult,1}\cdot\chi_{1}}{\gamma_{M1}} \tag{12}\]  
Another initial guess may be considered in the location where the stress calculated according Equation (11) has a maximum value:  
The imperfection (scale factor \(\Omega_{1}(x)\)) needed at each section in order to reach the cross-section resistance when the buckling load level is reached \(\alpha_{b,1}\) is calculated from formula:  
\[\frac{N_{Ed}\cdot\alpha_{b,1}}{A}+\frac{\Omega_{1}(x)}{\left(\frac{\alpha_{ cr}}{\alpha_{b,1}}-1\right)}\frac{E_{y}\frac{d^{2}\eta_{cr,w}}{dx^{2}}}{W_{y}}\ =\ \frac{f_{y}}{\gamma_{M0}} \tag{13}\]  
Figure 1: Flow chart describing the procedure on how to obtain the location of the critical section \(x_{\sigma}\) and the unique global and local initial (UGLI) imperfection amplitude \(\eta_{0}\). Illustrative example.  
\[\Omega_{1}(x)\ =\ \left(\frac{f_{y}}{\gamma_{\rm M0}}-\frac{N_{Ed}\cdot\alpha_{b,1}}{A }\right)\!\!\left(\!\frac{\alpha_{cr}}{a_{b,1}}-1\!\right)\!\frac{W_{y}}{EI_{y} \frac{\bar{a}^{2}\eta_{\rm Mcr}}{dx^{2}}} \tag{14}\]  
The minimum of those scale factors is the objective in the first iteration, i.e., \(\eta_{0,1}\) takes place at the critical section \(x_{cr,1}\):  
\[\eta_{0,1}\ =\ \min(\Omega_{1}(x))\ =\ \Omega_{1}(x_{cr,1}) \tag{15}\]  
The fourth step is the second iteration:  
\[\alpha_{ult,2}\ =\ \frac{A(x_{cr,1})\cdot f_{y}(x_{cr,1})}{N_{Ed}(x_{cr,1})} \rightarrow\bar{\lambda_{2}}\ =\ \sqrt{\frac{\alpha_{ult,2}}{\alpha_{cr}}}\rightarrow\chi_{2} \rightarrow\alpha_{b,2}\ =\ \frac{\alpha_{ult,2}\cdot\chi_{2}}{\gamma_{\rm M1}} \tag{16}\]  
The imperfection (scale factor \(\Omega_{2}(x)\)) needed at each section in order to reach the cross-section resistance when the buckling load level is reached \(\alpha_{b,2}\) is calculated as follows:  
\[\Omega_{2}(x)\ =\ \left(\frac{f_{y}}{\gamma_{\rm M0}}-\frac{N_{Ed}\cdot\alpha_ {b,2}}{A}\right)\!\!\left(\frac{\alpha_{cr}}{a_{b,2}}-1\!\right)\!\frac{W_{y}} {EI_{y}\frac{\bar{a}^{2}\eta_{\rm Mcr}}{dx^{2}}} \tag{17}\]  
\(\eta_{0,2}\) is the amplitude at the second iteration. It takes place at the critical section \(x_{cr,2}\):  
\[\eta_{0,2}\ =\ \min(\Omega_{2}(x))\ =\ \Omega_{2}(x_{cr,2}) \tag{18}\]  
If the critical section is the same as in the previous iteration, the critical section is found and consequently the value of UGLI imperfection amplitude may be calculated. If the critical section is different, then the other iteration is needed starting from the fourth step. Finally, the distribution of UGLI imperfection \(\eta_{init}(x)\) is calculated:  
\[\{\eta_{init}\}\ =\ \eta_{0}[\eta_{cr}] \tag{19}\]