## 3 Validation of Presented Procedure
The scale factor is plotted in Figure 4. The minimum value at the last iteration gives the amplitude of the imperfection \(\eta_{0}\) = 39 mm. [PERSON]'s values in the brackets confirm the very good agreements of the results (Figure 3). According to the direct method [23] the critical section takes place where the second order effects have a maximum. This method gives a value of amplitude \(\eta_{0}\) = 29 mm which differs from 39 mm a lot. In all the other examples, the direct method gave similar results as Eurocode's method. Figure 2: Example 1: description taken from [7], Section 5.4. _The value \(\eta_{0}=39\) mm given in Figure 4 was obtained after seven necessary iterations. Table 1 shows all the details of the iterations._ \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Iteration** & \(\alpha_{ult}\) & \(\widetilde{\lambda}\) & \(\chi\) & \(\alpha_{b}\) & \(x_{cr,i}\) & \(\frac{\eta_{0,i}}{\eta_{0}}\) \\ \hline
1 & 1.2228 & 0.4557 & 0.8106 & 0.9011 & 1.9200 & 1.6435 \\ \hline
2 & 1.5532 & 0.5136 & 0.7698 & 1.0870 & 1.0200 & 0.7413 \\ \hline
3 & 1.3762 & 0.4835 & 0.7910 & 0.9896 & 1.6800 & 1.2012 \\ \hline
4 & 1.5023 & 0.5051 & 0.7757 & 1.0594 & 1.3200 & 0.8779 \\ \hline
5 & 1.4311 & 0.4930 & 0.7842 & 1.0203 & 1.5000 & 1.0586 \\ \hline
6 & 1.4660 & 0.4990 & 0.7800 & 1.0396 & 1.4400 & 0.9702 \\ \hline
7 & 1.4542 & 0.4970 & 0.7814 & 1.0331 & 1.4400 & 1.0000 \\ \hline \end{tabular}
\end{table} 
Table 1: Example 1: key values for each iteration. Figure 4: Example 1: the scale factor \(\Omega(x)\) for utilization factor \(U_{\text{N+M}}=1.0\) in bay No.1. In other bays, the scale factor is greater than 10 times the minimum value. Figure 3: Example 1: UGLI imperfection \(\eta_{init}\), bending moment \(M\), shear force \(V\), utilization factor \(U\). Values in brackets are from [7]. **Example 2**. : _The Example 2 is described in Figure 5. It is a schema of the upper chord of the steel railway half-through truss bridge. More details about the bridge may be found in Section 5.5[7]. Steel grade S235 and \(\gamma_{M1}=1.1\) are taken into account._  
_The truss chord under compression that is subjected to the lateral buckling may be analyzed by modeling the elements as a column subjected to the compression force \(N_{Ed}\) and supported by discrete elastic restraints modeled as springs. The springs stiffnesses are \(C_{0}=11\) MN/m and \(C=5.068\) MN/m. Input values for the iterative procedure [7] are given in Figure 5. In the two edge fields on both sides of the truss chord: \(A=0.01909\)\(m^{2}\), \(I=96.4\times 10^{-6}\)\(m^{4}\), \(W=459\) cm\({}^{3}\), in the four middle fields of the truss chord: \(A=0.02898\)\(m^{2}\), \(I=177.525\times 10^{-6}\)\(m^{4}\), \(W=845\) cm\({}^{3}\). The loading is defined in Figure 5._  
_The distributions of UGLI imperfection, bending moment M(x) and shear force V(x) due to the imperfection and the utilization factors U(x) are plotted in Figure 6. The steps in the shear force distribution are due to steps in the axial forces and in spring supports. The critical section takes place at \(x_{cr}=13.15\) and \(18.25\) m simultaneously due to the symmetry of the problem. The scale factor is plotted in Figure 7. The scale factor minimum led to UGLI imperfection amplitude \(\eta_{0}=9\) mm. The significant values are shown in Table 2. Only one iteration was needed. The comparison with [PERSON]\({}^{\prime}\)s values given in brackets shows very good agreement._  
Figure 5: Example 2: description taken from [7] Section 5.5. Figure 4:0 Example 2: UGLI imperfection \(\eta_{init}\), bending moment \(M\), shear force \(V\), and utilization factor \(U\). Values in brackets are from [7]. _[PERSON] used the method also for more complicated cases, where the simplified method is not applicable, e.g., for the Maria Valeria Bridge across the Danube, Stirovo-Esztergom. See Note 5 in [7] referring to [29]._  
_According to the direct method, the same results are obtained._  
**Example 3**.