## 1 Introduction
### Research Significance, Contributions and Basic Assumptions of Presented New Method  
The innovation of the presented research is the numerical method for obtaining the equivalent UGLI imperfection amplitude for metal members susceptible to buckling due to axial forces. The method is consistent with the Eurocode procedure and allows to obtain the location of critical section \(x_{cr}\) and consequently UGLI imperfection amplitude for any case.  
Several assumptions are considered in this work:  
1. The first assumption is that the first buckling mode of the examined structural member is dominant and therefore the effect of the higher buckling modes can be neglected. However, in some cases, the higher buckling modes have to be considered in the design. See suggestions by [PERSON] et al. (2017).
2. The second assumption is that [PERSON]'s method given in [3] remains valid. Quoting [PERSON] [3], where experimentally established values are unavailable, the amplitude of the equivalent imperfection in the shape of the elastic buckling mode may be determined assuming that the buckling resistance of a structure with axially loaded members shall be equal to the buckling resistance of the equivalent member. The buckling resistance of axially loaded columns is defined in clauses 6.3.1.1-6.3.1.4 [1]. The relative slenderness \(\overline{\lambda}\) relates to the critical section. The equivalent member has pinned ends, its cross-section and axial force are the same as in the critical cross-section \(m\) of the frame and its length is such that its critical force equals the axial force in the critical cross-section \(m\) at the critical loading of the structure. The position of the critical cross-section \(m\) is determined by the condition that the utilization \(U_{m}\), with allowance for the effect of the axial force and bending moments due to imperfections in the critical cross-section \(m\), is greater than the utilization \(U(x)\) at all other cross-sections of the verified member or frame structure.
3. The third assumption is that the linear interaction Formula (22) is used to verify the member according to Eurocodes and [25; 26].
4. The fourth assumption is that if plastic cross-section resistance is taken into account, a maximum plastic shape factor of 1.25 may be taken into account [4; 26]. This means that if the criterion is applied with reference to the plastic resistance, the design value of the moment resistance \(\mathrm{M_{c,Rd}}\) should be limited to 1.25 \(\mathrm{M_{el,Rd}}\) for both a strong axis and weak axis.
5. The fifth assumption accepts the common method for finding the equivalent local bow imperfections given in [1] 5.3.2 (3). For the new generation of Eurocodes, intensive investigations were done to enhance these values, now in draft [4] see 7.3.3.1 (1) [25; 26]. In practice, we often deal with members under compression and bending. If the external bending moments are present, the results may be more unfavorable, because a larger part of the cross-section in longitudinal direction may be plastified, thus leading to a stiffness reduction and therefore higher necessary bow imperfections for simplified calculations. If this is neglected, the load carrying capacity may be underestimated. In our cases of centrally loaded members, there is no need to use the fourth and fifth assumptions.