## 1 Introduction  
### Overview and Analysis of Current State  
According to Eurocode EN 1993-1-1 [1] and EN 1999-1-1 [2], second order effects and initial imperfection depending on the type of frame structure may be taken into account in global analysis. The design of steel structures with compression elements must consider the existence of residual stresses and geometric imperfections such as initial out-of-plumbness, flatness and fit tolerances. The negative effects of all these imperfections are covered in the global analysis by equivalent geometrical imperfection with a relevant shape and value of amplitude. There are the following types of imperfections in [1]: (i) global initial sway imperfection for the global analysis of frames; (ii) local initial bow imperfection for global and member analysis; (iii) imperfection for the analysis of bracing systems; and (iv) unique global and local initial (UGLI) imperfection based on elastic critical buckling modes. In this paper, the global analysis with UGLI imperfection is utilized. Section 5.3.2(11) of [1] provides formulae which are the base of a procedure for obtaining UGLI imperfection. Unfortunately, the procedure in [1] is limited to the structures with a uniform cross-section under uniform axial force distribution. No iterations are needed in such cases. The critical section will take place where the curvature has maximum value. The detailed description of the development of this method is given by [PERSON] in [3]. The procedure described in [3] is without the above limitations and may be used for the most practical cases. Generally, the critical section must be found by iterations. The more general procedure without above limitations is given in EN 1999-1-1 [2]. The new generation of Eurocodes is in preparation and it should be available to the technical public in 2023. In the working drafts prEN 1993-1-1 [4] and prEN 1999-1-1 [5] the more general procedure was improved. The formulation in draft [5] is more useful and clear compared to its former edition [2]. The working draft prEN 1993-1-1 [4] differs from its former edition [1] and from [2] and [5]. In Eurocode [4], the partial material safety factor \(\gamma_{\text{M1}}\) was removed from the formula for the imperfection amplitude \(e_{0}\) of the equivalent member. The consequence is that the value of imperfection amplitude \(e_{0}\) is in [1,2,5] greater than in [4]. The working draft Eurocode prEN 1999-1-1:2020 [5] contains the following formulae for:  
* The characteristic value of the initial imperfection amplitude of the equivalent member related to the critical section \(m\): \[e_{0,k,m}\ =\ \alpha\Big{(}\overline{\lambda}_{m}-\overline{\lambda}_{0} \Big{)}\frac{M_{Rk,m}}{N_{Rk,m}}\] (1)
* Design value of the initial imperfection amplitude of the equivalent member related to the critical section \(m\): \[e_{0,d,m}\ =\ e_{0,k,m}\frac{1-\frac{\chi_{m}\overline{\lambda}_{m}^{2}}{ \gamma_{\text{M1}}}}{1-\chi_{m}\overline{\lambda}_{m}^{2}}\ =\ \alpha\Big{(}\overline{\lambda}_{m}-\overline{ \lambda}_{0}\Big{)}\frac{M_{Rk,m}}{N_{Rk,m}}\frac{1-\frac{\chi_{m}\overline{ \lambda}_{m}^{2}}{\gamma_{\text{M1}}}}{1-\chi_{m}\overline{\lambda}_{m}^{2}}\] (2)  
Note that the symbol \(e_{0,d,m}\) used in Equation (2) must not be confused with the amplitude of the initial local bow imperfection \(e_{0}\) given in [1] or [2] in Table 5.  
3. Design value of UGLI imperfection amplitude \(\eta_{init,\max}\): \[\eta_{init,\max}\ =\ e_{0,d,m}\frac{N_{cr,m}}{EI_{m}\big{|}\eta^{\prime\prime} \ _{\sigma,m}\big{|}}\] (3) For the simply supported members with a uniform cross-section and uniform axial force distribution \(\eta_{init,\max}\ =\ e_{0,d,m}\). The symbols \(\epsilon_{0,k,m}\) and \(\eta_{init,\max}\) are used here for the better understanding of the procedure. They are not explicitly used in Eurocodes [1; 2; 4; 5]. The differences of Eurocodes [1; 2; 4] compared with the working draft Eurocode prEN 1999-1-1:2019 [5] are as follows:  
1.