## 3 Validation of Presented Procedure  
[PERSON]'s complex examples [3; 7; 8] are very interesting in order to validate the presented method. The fourth example shows that differences are negligible between \(\alpha_{b}\) values calculated by the new procedure and \(\alpha_{b}\) values obtained by [PERSON] et al. [28] with the help of Geometrically and Materially Nonlinear Analysis with Imperfections (GMNIA). Plots of the following parameters are implemented to have a full understanding of the examples:  
* The relative UGLI imperfection is plotted along the beam. The values are divided by the maximum value of imperfection in order to obtain the relative UGLI imperfection with maximum value 1.0. * The distribution of relative bending moments due to UGLI imperfection is plotted along the beam, the values are divided by the maximum bending moment value in order to have a maximum value 1.0: \[\frac{\eta_{0}}{(\alpha_{cr}-1)}EI_{y}\frac{d^{2}\eta_{cr,w}}{dx^{2}}\] (20)
* The distribution of the relative shear forces due to UGLI imperfection is plotted along the beam. The values are divided by the maximum shear force in order to have a maximum value 1.0: \[\frac{\eta_{0}}{(\alpha_{cr}-1)}EI_{y}\frac{d^{3}\eta_{cr,w}}{dx^{3}}\] (21)
* The plot of utilization factors \(U\) distribution: due to axial forces \(U_{N}\), due to bending moments \(U_{M}\) and the global utilization factor \(U_{N+M}\). This factor shows how close each section is to its maximum strength and the influence of axial forces and bending moments:\[U_{N+M}\;=\;(U_{N})+[U_{M}]\;=\;\left(\frac{N_{Ed}}{A\left(\frac{f_{y}}{\gamma_{M0}} \right)}\right)+\left[\frac{\eta_{0}}{(\alpha_{cr}-1)}\frac{EI_{y}\frac{d^{2} \eta_{cr}}{dx^{2}}}{W_{y}\left(\frac{f_{y}}{\gamma_{M0}}\right)}\right] \tag{22}\]  
The value of the scale factor distribution along the member is plotted showing the first, the second and the last iteration. The minimum value of each plot is \(\eta_{0,i}\). The maximum value shown is 10 times \(\eta_{0}\):  
\[\Omega_{i}(x)\;=\;\left(\frac{f_{y}}{\gamma_{M0}}-\frac{N_{Ed}\cdot\alpha_{bj} }{A}\right)\cdot\left(\frac{\alpha_{cr}}{\alpha_{bj,i}}-1\right)\frac{W_{y}}{ EI_{y}\frac{d^{2}\eta_{cr}}{dx^{2}}} \tag{23}\]  
\[\eta_{0,j}\;=\;\min\{\Omega_{i}(x)\}\;=\;\Omega_{i}(x_{cri}) \tag{24}\]  
The values of UGLI imperfection amplitude \(\eta_{0}\), the location of critical section \(x_{cr}\) and other values have been compared with [PERSON]'s values which are given in brackets. Very good agreements were achieved. In the Examples 3 and 4, the influence of the local buckling is neglected. It is beyond the scope of this paper. The authors have solutions of these examples which also take into account local buckling. They will be published with all the necessary details in another paper. The Example 1 is described in Figure 2. It is the scheme of the bottom flange of the continuous composite steel and concrete road bridge. More details about the bridge may be found in Section 5.4 [7]. Steel grade S355 and \(\gamma_{M1}\) = 1.1 are taken into account. The bottom flange of the steel cross-section in compression is subjected to lateral buckling. It may be analyzed by modeling the elements as a column subjected to the compression force N\({}_{Ed}\) and supported by continuous and discrete elastic restraints modeled as springs (Figure 2). The springs stiffnesses are C1 = 10 kN\(\cdot\)mm\({}^{-1}\), C2 = C3 = C4 = 4 kN\(\cdot\)mm\({}^{-1}\). The elastic restraint due to the cross-section web is c = 5.863 kN\(\cdot\)m\({}^{-2}\). The axial force distribution is given by formula N\({}_{Ed}\) (x) = \(-\)9000 1062 x \(-\)17.49 x\({}^{2}\) (kN). The distributions of UGLI imperfection, bending moment M(x) and shear force V(x) due to UGLI imperfection and utilization factors U(x) are plotted in Figure 3. The critical section takes place after several iterations at \(x_{cr}\) = 1.44 \(m\). The significant step in shear force distribution V is located where the column is supported by the spring with the greatest stiffness.