## 1 Introduction
### Overview and Analysis of Current State
Current Eurocode EN 1999-1-1:2005 [1] contains the above Equations (2) and (3) but in Corrigendum AC (2009) the index \"d\" was removed and some print errors were corrected. 2. Current Eurocode EN 1999-1-1:2007 [2] contains the above Equations (2) and (3) but in Amendment A2 (2013) [2] there are several changes and index \"d\" was removed. 3. Working draft Eurocode prEN 1993-1-1:2020 [4] removed from the above Equations (2) and (3) the partial material safety factor \(\gamma_{\rm M1}\) and index \"d\". Removing \(\gamma_{\rm M1}\) from Equation (2) destroyed the main assumption of [PERSON]'s method defined below, but the code committee followed this idea because it leads to an ease of use in practical applications [6]. The authors intend to publish the results of the study showing the consequences of this decision leading to an important difference between the values of UGLI imperfection amplitude calculated according to [4] or [5]. Eurocodes use the symbol \(\eta_{init,\max}\) for the design value of UGLI imperfection amplitude. In the paper, the symbol \(\eta_{0}\ =\ \eta_{init,\max}\) is also used, which is not in Eurocodes. The symbol \(\eta_{0}\) is used in the text, Equations and Tables because it is shorter than \(\eta_{init,\max}\). The critical section is indicated in this paper by symbol \(x_{cr}\) and not by index \"\(m\)\" as it is done in Eurocodes. Equation (4) can be rearranged in the form of Equation (5):  
\[\eta_{init,m}(x)\ =\ e_{0,d,m}\frac{N_{cr,m}}{EI_{m}\big{|}\eta^{\prime\prime} \ _{\sigma,m}\big{|}}\eta_{cr}(x)\ =\ \eta_{init,\max}\eta_{cr}(x) \tag{4}\]  
where \(\chi\) is the reduction factor for the relevant buckling curve depending on the relevant cross-section, see 6.3.1. Equation (4) is used in Eurocodes in the clauses: 5.3.2(11) [1], 5.3.2(11) [2], 7.3.6(1) [4] and 7.3.2(11) [5]. Equation (5) is just another form of (4). It is more convenient for programming. When flexural buckling occurs about the strong axis, UGLI imperfection may be written in the form:  
\[\eta_{init,w}(x)\ =\ \left(\frac{a\big{(}\overline{\lambda}_{y}-\overline{ \lambda}_{0}\big{)}}{\overline{\lambda}_{y}^{\ 2}}\frac{1-\overline{\lambda}_{y}^{\ 2}\chi_{y}}{1-\overline{\lambda}_{y}^{\ 2}\chi_{y}}\frac{f_{y}}{E\left(\frac{h_{y}}{W_{y}} \frac{d_{\ \  
method (see pages 693-694 [7]), derived this method in a different way and published the details of a step-by-step procedure for non-uniform members in [11]. [PERSON] and [PERSON] published detailed illustrative numerical examples in [12; 13]. They investigated in [14] the in-plane stability of the large two-hinged arch bridge (Zd'akov bridge in Czech Republic). [PERSON] showed that for the uniform members, UGLI imperfection amplitude has important geometrical meaning, which enables doing calculation and results verification by \"hand calculation\". The importance of so called \"hand calculation\" was illustrated in several examples in [15]. In [3; 7; 8], [PERSON] solved in detail a member with a non-uniform cross-section and three-story building as in [8] and three new examples relating to the stability of parts of bridge structures: (i) the bottom flange of the continuous composite steel and concrete road bridge; (ii) the upper chord of the steel railway half-through truss bridge; and (iii) the basket handle type of the arch bridge. Numerical examples in [3; 7; 8] provide guidance of the proper application of [PERSON]'s method. [PERSON] showed there the good agreement with other methods of Eurocode EN 1993-1-1:2005 described in clauses 5.2.2(3) and 5.3.2(3). The guidance for the design of members of the steel basket handle arch bridges is given in the clause NB.3 of Slovak National Annex to Eurocode STN EN 1993-2 Design of steel structures--Part 2 Steel bridges. [PERSON] pointed out there that it is not enough for a such type of bridges to take into account only the first buckling mode. The members of such bridges are in biaxial bending. [PERSON] in [3; 7; 8] generalized his method for members when flexural buckling occurs about both axes.