## 3 Validation of Presented Procedure
: _The Example 3 is described in Figure 8, with steel grade S355 and \(\gamma_{M1}=1.0\)_  
_The member is fabricated from an IPE 400 section. It was cut in two pieces and consequently welded together in such a way that the depths of its cross-section at the ends are 560 and 240 mm._  
_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 4:1 The critical section is found at \(x_{cr}=12\ m\). The scale factor minimum led to UGLI imperfection amplitude \(\eta_{0}=36.31\ mm\) (Figure 10). The significant values for all iterations are shown in Figure 4:2 Only two iterations were needed._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Iteration** & \(\mathbf{\alpha_{ult}}\) & \(\widetilde{A}\) & \(X\) & \(\mathbf{\alpha_{b}}\) & \(\mathbf{x_{cr1}}\) & \(\frac{\eta_{0}}{\eta_{0}}\) \\ \hline \(1\) & \(1.9376\) & \(0.6777\) & \(0.7961\) & \(1.4023\) & \(13.2273\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example 2: key values for each iteration. Figure 8: Example 3: description taken from [9; 10] and [7] Section 5.2. Figure 7: Example 2: the scale factor \(\Omega(x)\). Utilization factor \(U_{N+M}=1.0\). Figure 10: Example 3: the scale factor \(\Omega(x)\). Utilization factor \(U_{N+M}=1.0\). Figure 9: Example 3: UGLI imperfection \(\eta_{init}\), bending moment \(M\), shear force \(V\), and utilization factor \(U\). Values in brackets are from [7,10]. [PERSON]'s values are in brackets and they confirm the very good agreement of the results (Figure 9). The direct method gives the value of imperfection amplitude \(\eta_{0}\) = 34.56 mm which approximately equals value 36 mm (Figure 10)._  
**Example 4**. : _(a) The Example 4 was published by [PERSON] et al. [28], with steel grade 235, and \(\gamma_{M1}\) = 1.0. It is a tapered column similar to that in Figure 8 but made from IPE 200, with different boundary conditions. The linearly varying height is defined by the taper ratio of \(h_{max}\)/\(h_{min}\) = 3. It is simply supported at both ends, \(L\) = 12.9 m. The member is restrained against out-of-plane buckling. The compression force N\({}_{Ed}\) = 500 kN was applied. The computed load factor \(\alpha_{b}\) = 1.004 with GMNIA. The method proposed by the authors leads to \(\alpha_{b}\) = 0.991. The difference is \(-\)1.29%._  
_In Example 4(a), GMNIA was performed by [PERSON] et al. [28] using ABAQUS software taking into account a geometrical imperfection with the shape of the buckling mode and an amplitude of L/1000, also considering the residual stresses of a welded cross-section. 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 11. The critical section is found at \(x_{cr}\) = 10.2 m. The scale factor minimum led to UGLI imperfection amplitude \(\eta_{0}\) = 28.04 mm (Figure 12). The significant values for all iterations are shown in Table 4. Three iterations were needed._In the Example 4(a), the maximum moment \(M_{max}(x=7.654~{}m)\) can be obtained from Equation (25), which serves to verify the value \(M_{max}(x=7.654~{}m)=30.441~{}\mathrm{kNm}\) given in Figure 11:  
\[M_{\max}~{}=~{}\frac{F_{Ed}\cdot\eta_{0}}{\left(1-\frac{1}{\alpha_{cr}}\right)}~ {}=~{}\frac{500\cdot 28/1000}{\left(1-\frac{1}{1.852}\right)}\approx 30.441~{} \mathrm{kN}\cdot\mathrm{m} \tag{25}\]  
According to the direct method, the imperfection amplitude is \(\eta_{0}=\)25.86 mm. It is a similar result. (b) In Figure 13, the \(\alpha_{b}\) values calculated for different slenderness at the critical section are compared with the GMNIA results of [PERSON] et al. [28]. The maximum difference is \(-8\%\) for relative slenderness \(\overline{\lambda}~{}=0.773\) (buckling curve b). This analysis is performed for a hot-rolled tapered column \(b=100~{}mm\), \(t_{f}=t_{w}=\)10 mm, \(h_{min}=100~{}mm\) and \(h_{max}=400~{}mm\), steel S235 with a linearly varying height, and a simply supported member.