Decision tree for local + global imperfection combinations in double-symmetric prismatic members – Practical recommendations in the framework of advanced analysis

© 2023 The Authors. Published by Ernst & Sohn GmbH, Berlin. Steel Construction 16 (2023) 1 DOI: 10.1002/stco.202200041 ARTICLE ARTICLE Decision tree for local + global imperfection combinations in double-symmetric prismatic members Practical recommendations in the framework of advanced analysis Andreas Müller, Martin Vild, Andreas Taras Better and simpler possibilities of structural optimization due toincreasing computational power but also for reasons of environ mental sustainability, the use of materials and their reusability lead to greater acceptance towards more advanced numerically intensive, so-called ‘design by analysis’ methods like geometrically and materially non-linear analyses with imperfections (GMNIA). The general choice of imperfections and their combination in such models, especially for slender cross sections of intermediate length prone to an interaction between a global and local plate buckling, is crucial in terms of the reached load-bearing capacity. Annex C of EN 1993-1-5:2010 makes use of the ‘70 %-rule’ for the combination of imperfection modes and amplitudes. This rule postulates that two GMNIA calculations should be conducted when local and global interactive buckling may be dominant; one with 100 % + 70 % of the maximum specified amplitude in either case. In addition, extended information is provided on the choice and combination of imperfections in the newly introduced and currently available draft of the prEN 1993-1-14:2020 (design assisted by finite element analysis). Although information is provided on how the local and global imperfections should be combined, it is not stated when it is relevant to consider those. Based on conducted GMNIA simulations on SHS/RHS (square and rectangular hollow sections) and I-shaped sections, this article presents general decision support on the choice of equivalent imperfections. On the basis of numerical analysis, the developed flow chart and design routine allow for the decision whether the consideration of the interaction of local and global imperfections is required or not. Keywords LBA simulations; GMNIA simulations; advanced non-linear analysis; equivalent imperfections; combination of local and global imperfections 1 Introduction The main motivation for this article is to provide practical recommendations for design of steel members by geometrically and materially non-linear analysis with imperfections (GMNIA). The necessity was recognized within parts of an industrial project between software company IDEA StatiCa as the client and the ETH Zurich, Department of Steel and Composite Structures as the contractor. Within the scope of numerical investigations, comparative calculations were performed for different verification purposes between ‘IDEA StatiCa Member’ application (software versions 20.1 and 21.0) and Abaqus CAE 2019 [1] simulations. Within the scope of the project, reports [2, 3] with recommendations for hot-rolled [4] and cold-formed [5] square and rectangular hollow sections as well as rolled I-shaped sections were developed. These recommendations are summarized as follows: 1. General model verification and discussion of selected model approaches and limitations. 2. Choice of local imperfections for I-shaped profiles based on EN 1993-1-5:2010 [6] and for RHS/SHS profiles based on the results of the RFCS (Research Fund for Coal and Steel) funded project HOLLOSSTAB [7–9]. 3. Selection of global imperfections for the case of flexural buckling following the provisions of DIN EN 19931-1:2010 [10] and prEN 1993-1-1:2020 [11]. 4. Derivation of practical decision criteria for the combination of local and global imperfections based on slenderness limits. In general, ‘IDEA StatiCa Member’ application offers the user, according to the concept of a component-based finite element method (CBFEM), the possibility to consider whole members with realistic end connections, without the usual modelling effort, using FE models based on shell elements. Thus, linear buckling analyses (LBA) and more advanced GMNIA are easier and more practical to perform compared to software packages, which are mainly used in multidisciplinary scientific fields. This software development in civil engineering applications towards more advanced analysis methods (e.g., GMNIA simulations) is one of the several reactions to the digitalization of the whole construction sector and current design trends in architecture and more specifically in nowadays modern steel applications. These trends are mostly driven by the wider acceptance and willingness towards simulation-based solutions, higher computational efficiency and the improved possibilities of geometric parameterization of entire structures in CAD software packages. One major problem within the application of such advanced numerical methods is the choice of appropriate imperfections, their amplitudes and possible combinations of local and global buckling cases. For this purpose, the currently valid EN 1993-1-1:2010 [10] provides values for bow imperfections to be applied on the basis of two formulations, i.e., length or slenderness affine equivalent bow imperfections; see Eq. (2) and Eq. (5). However, in the This is an open access article under the terms of the Creative Commons AttributionNonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2 Steel Construction 16 (2023) A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members delled according to the code provisions of EN 10210-2 [4] and EN 10219-2 [5] for hot-rolled and cold-formed steel, respectively. 2. To have compatibility with the IDEA StatiCa Member model, a bi-linear material model was used, with an yield plateau assumed at s von Mises = fy. Strain hardening is considered with a reduced elasticity modulus of E/1000 . 3. Imperfection amplitudes based on eigenmodes and with various amplitudes (see the following sections) were considered. 4. The load introduction and boundary conditions make use of a Multi-Point Constraint (MPC) type of constraints at the member ends, which imply a rigid connection between the nodes at the extremity and a reference node at the centroid of the respective sections. Loads were applied through reference points (RFPoints) at the top and the bottom, each connected through an MPC-Tie formulation to associated node sets on the outer extremities. 2.1 Model choice and validation of RHS/SHS profiles The Abaqus model used in this report was validated by an extensive analytical, numerical and experimental campaign over the course of the RFCS project HOLLOSSTAB. The reader is referred to the references of the project for the details [7–9]. 2.2 Model choice and validation of I-shaped profiles Three commonly used models for Iand H-shaped profiles are summarized in Fig. 1 and described regarding their advantages and disadvantages. Solid models (Fig. 1a) can lead to a realistic geometry approximation, including the influence of the fillets between the web and the flanges. Nevertheless, requiring the implementation of the whole cross-section geometry, the calculation process can become computationally time-consuming, leading to necessary simplifications within the models. A more simplified model is shown in Fig. 1b, where the flanges and the web are modelled with shell elements, without a surface interception but with additional beam elements as square hollow sections of variable depth and wall thickness at the top and the bottom of the web. The beam elements are designed in such a way that they have the same area A and torsional moment of inertia It as the missing fillets between the web and flanges. This modelling approach was also successfully used and verified in [20]. One further approach is the use of a shell model with three plates representing the web and the flanges, which are intercepting in the centerline (see Fig. 1c). The fillets are not modelled explicitly but are approximated by the overlap between the web and the flanges. Following this model assumption, not all cross-section values can be considered precisely for hot-rolled I-shaped sections; welded profiles are mostly excluded from this. In some cases, especially the torsional moment of inertia It can deviate, depending course of the revision of all Eurocodes, the currently valid EN 1993-1-1:2010 will be replaced by the newly introduced prEN 1993-1-1:2020 following the approach of new formulations; see Eq. (3) for the validation of the initial equivalent length affine bow imperfections. These changes go back mainly to the work of Lindner et al. [12–15] and are justified by the fact that the previously derived values for equivalent bow imperfections in [10] can lead to results on the unsafe side. With the introduction of an entirely new part, the prEN 1993-1-14:2020 [16], for the design assisted by FE analysis, an additional equation (see Eq. (4)) is introduced to calculate equivalent geometric imperfections for the use in GMNIA for flexural buckling [17]. Local imperfection amplitudes are in general chosen according to EN 1993-1-5:2010, Annex C [6]. The new introduced prEN 1993-1-14:2020 [16] adopts those explanations for equivalent geometric imperfections for cross sections of plated structures in FE-based simulations. All following investigations and their results on the effect of different imperfection amplitude formulations according to EN 1993-1-1:2010 [10], prEN 1993-1-1:2020 [11], prEN 1993-1-14 [16] and EN 1993-1-5:2010 [6] on the load-bearing capacity for the case of local and flexural buckling (see Section 3) as well as the choice of eigenmode shapes for interactive cases of global and local buckling (see Section 4) are based exclusively on FE models and approaches presented in Section 2 and are therefore only valid in this context. 2 FE modelling All calculations presented in this article were performed by using the commercial software Abaqus [1]. Throughout all simulations, the following two steps were applied. First, a linear buckling analysis (LBA) was performed to obtain the critical buckling load and the mode-dependent eigenshape. The critical buckling load is used to determine relative slenderness. This is done either on the global or local level depending on the instability case of flexural buckling or local instability. The mode-dependent eigenshape is used as an initial imperfection amplitude within the second simulation step, where GMNIA were used to obtain the maximum load-bearing capacity of the members in compression. Throughout all simulations, the steel grade S355 was set constant. The member length was varied to account for local and flexural buckling exclusively and in combination through the slenderness range. A short description of the model is given here as follows: 1. Isoparametric shell elements with reduced integration of type S4R are used. The web and the flanges of I-sections were discretized into 30 and 20 elements, respectively. Mesh density of 60 elements in circumferential is used for RHS and SHS profiles. In longitudinal direction, member is divided into 100 elements per meter. The roundings in the corner regions are mo18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Steel Construction 16 (2023) 3 A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members ARTICLE tional efficiency and better model homogeneity between hot-rolled and welded I-shaped profiles. Additional investigations on the eigenmodes from LBA simulations and GMNIA results between the considered models from Fig. 1 are presented within reports in [2] and [3]. A general validation against literature was conducted for model b (shell–beam model) as a preliminary step to verify an overall modelling correctness. Therefore, different N–M interactions were considered to calculate a range of critical bifurcation loads (combined lateral-torsional-bucklingand flexural buckling loads), normalized by the plastic cross-section resistance. The comparison with analytical solutions from Trahair [18] showed compliant results (see Fig. 2a). In addition, GMNIA calculations for the flexural buckling case about the weak axis were conducted for different I and H sections. A comparison with the originally developed ECCS (European Convention for Constructional Steelwork) curves for flexural buckling [19] showed a good agreement between the performed GMNIA calculations with published and accepted solutions (see Fig. 2b). Further details on an equivalent model validation can be taken from [20]. on the selected profile series, by around 30 % [21]. This can lead, in accordance to the observed problem, to lower capacity values, e.g., in the case of lateral torsional buckling (LTB). However, for local instability problems and flexural buckling cases, which were investigated throughout this article and the associated report on I-shaped profiles [3], LTB effects and therefore the abovementioned modelling shortfall are in general negligible. Tab. 1 shows exemplary the deviations from GMNIA calculations, using the modelling approaches from Fig. 1 on short HEA300 profiles with the length of 800 mm. The deviations between the highest (model b) and lowest (model c) load-bearing capacity are around 5 %. For this reason, model c was chosen in terms of higher computaFig. 1 Common model approaches for Iand H-shaped profiles: a) solid model; b) shell–beam model; c) shell model Fig. 2 FE model validation using a) LBA results in comparison with [18] and b) GMNIA results in comparison with [19] Tab. 1 Different model approaches and their results from and GMNIA evaluations GMNIA Reached bearing capacity [kN] Model a Model b Model c 4017.00 4019.00 3814.40 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4 Steel Construction 16 (2023) A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members 3.1.1. Equivalent local imperfections EN 1993-1-5:2010, Annex C [6], and prEN 1993-1-14:2020 [16] provide magnitudes for local imperfections through an assumed value of e0 = B/200. B is the smaller of the two corresponding dimensions, the height or the width. Nevertheless, referring to the findings of Rusch and Lindner [24] and Toffolon and Taras [7], a determined amplitude of e0 = B/400 was found to be more suitable to represent the design curve for local buckling (Winter curve) [6] in numerical calculations for SHS/ RHS profiles. Herein, several GMNIA calculations were performed for a centrically loaded cold-formed and hot-rolled SHS200 profile with a varying thickness and a constant length of 800 mm to ensure local buckling exclusively. The associated imperfection range varies between the upper bonds of B/200 and B/400 and an additional imperfection of B/300 in between. Fig. 3 shows the performed GMNIA calculations for hot-rolled [4] and cold-formed [5] SHS200 profiles. The calculations with imperfection amplitudes of B/200 are comparatively conservative lying below the local buckling curve of EN 1993-1-5:2010 [6]. However, results closer to the local buckling curve are obtained with imperfection amplitudes of B/400, showing a better agreement and confirm the observations by [7] and [24] for RHS/SHS profiles. Further considerations for I-shaped profiles were made in [3] with the conclusion that an imperfection amplitude of B/200 is more suitable. 3.1.2 Equivalent bow imperfections According to EN 1993-1-1:2010 and prEN 1993-1-1:2020, the equivalent bow imperfection amplitude e0 for flexural Generally, it should be pointed out that within advanced non-linear calculations, not only the maximum reached load-bearing capacity is the driving design criterion but also the deformations (or strains) that are reached throughout the calculation during the loading path before reaching the limit load. Those two criteria are described in prEN 1993-1-14:2020 and denominated as C1 and C2 within the evaluation method of material non-linear analysis. The latter criteria refer to the largest tolerable deformations or strains and must therefore be linked to material and structural boundary conditions referring, e.g., to [6, 22, 23]. 3 Initial bow imperfections for flexural buckling according to EC3 3.1 Introduction and background information The choice of the initial imperfection magnitude depends on different factors summarized as follows: i. The type of analysis according to the considered crosssection failure linked to the cross-section class. ii. The type of imperfection considered for further calculations, i.e., geometric and material imperfections (tolerances and residual stresses) or equivalent, buckling curve-dependent imperfections, specifying a bow imperfection amplitude (case of flexural buckling) to account for tolerances and residual stresses. iii. The benchmark resistance in terms of a plastic or elastic calculation, which specifies the subsequent choice of an imperfection amplitude dependent on a cross-section dependent imperfection factor α . This corresponds to the global buckling concept of EN 1993-1-1:2010 [10] and prEN 1993-1-1:2020 [11]. Fig. 3 GMNIA calculations for a SHS200 profile and comparison with code provisions of EN 1993-1-5:2010: a) cold-formed; b) hot-rolled 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Steel Construction 16 (2023) 5 A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members ARTICLE As the values in Tab. 2 were exclusively derived on the basis of buckling curves used for the equivalent member method, meaning that the members are loaded by axial forces only without the combined influence of external bending moments My or Mz. However, as shown in [12] and [13], a combination can often lead to unfavourable values of initial equivalent imperfections. As a result of these and further investigations [14, 15], a new formulation was developed and is introduced within the new code generation in the current draft of prEN 1993-1-1:2020 [11], where the current Tab. 2 (Eq. (5.6) with Tab. 5.1 in [10]) will be replaced by Eq. (3) in connection with Tab. 3 (Eq. (7.8) with Tab. 7.1 in [11]). αβ ε == ⋅ je L 0,d (3) where α is the imperfection factor, depending on the relevant buckling curve; b is the reference bow imperfection according to Tab. 3; ε =f235/ y is the material parameter considering the steel grade; and L is the member length. An additional, modified length affine formulation is implemented in the current draft of prEN 1993-1-14:2020: α =⋅≥e LL 150 1000 0,d (4) It is based on the basic formulation from Eq. (3) with a difference that the required equivalent bow imperfection is no longer a function of ε because the influence of material yielding was captured directly in the analysis during the derivation of this expression. The factor b from Eq. (3) was additionally calibrated to a value of 1⁄150 based on 646 beam FE simulations. The use of this imperfection formulation requires a modified Young’s modulus of E=200000 N/mm2. For detailed information on the derivation of Eq. (4), the reader is referred to the work of Walport [17]. The back-calculation of slenderness-based equivalent bow imperfections, in both EN 1993-1-1:2010 and prEN 19931-1:2020, is provided by Eq. (5). αλ () =⋅ −⋅eM N 0.2 0 Rk Rk (5) buckling can be determined using two approaches, considering either a tabulated length proportional value or a slenderness-based formulation based on the elastic critical buckling load from analytical considerations or numerical analysis. In both cases, those values represent equivalent imperfections based on the determination from a centrically loaded strut with an assumed initial bow imperfection e0 of a half-sine wave. This bow imperfection is determined by introducing a moment M, calculated by second-order theory using an appropriate interaction formula; EN 1993-1-1, Section 6.2.9.1, contains therefore various interaction formulae for class 1 and 2 cross-sections. This consideration leads to the general representation in Eq. (1): ψχλ χ () = ⋅− ⋅eM N 1 0 2 Rk Rk (1) where λ = NN / pl cr is the normalized relative slenderness; c is the reduction factor for critical buckling; y is the interaction factor for combined bending and axial force, e.g., EN 1993-1-1, Section 6.2.9.1; MRk is the characteristic moment resistance of the critical cross section; and NRk is the characteristic axial resistance of the cross-section. The equivalent imperfection representation is normalized by the member length L in terms of a non-dimensional representation (see Eq. (2)). =je L 0,d (2) Finally, Eq. (1), combined with Eq. (2), was evaluated for different interaction approaches and profiles and was classified within fixed, non-dimensional limits depending on the buckling curves (a0, a, b, c, d) and the type of cross-section verification, i.e., elastic or plastic cross-section verification. These values were implemented within the currently valid EN 1993-1-1:2010 (Tab. 5.1 [10] and Tab. 2 in this article). Tab. 2 Values of the initial bow imperfection amplitude e0,d/L for members according to DIN EN 1993-1-1:2010 [10] Buckling curve Elastic cross-section verification Plastic cross-section verification e0,d/L e0,d/L a01/350 1/300 a1/300 1/250 b1/250 1/200 c1/200 1/150 d1/150 1/100 Tab. 3 Reference relative bow imperfection b according to prEN 1993-1-1:2020, Tab. 7.1 [11]] Buckling about axis Elastic cross-section verification Plastic cross-section verification y–y 1/110 1/75 z–z 1/200 1/68 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6 Steel Construction 16 (2023) A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members Fig. 4 Length affine imperfection formulation; GMNIA calculations and comparison with code provisions for hot-rolled (a, c, e) and cold-formed (b, d, f) SHS200 × 12.5 profiles: a) imperfection amplitude according to [10], elastic cross-section verification (see Tab. 2); b) imperfection amplitude accordingto [10], elastic cross-sectional verification (see Tab. 2); c) imperfection amplitude according to [11], elastic cross-section verification (see Tab. 3); d)imperfection amplitude according to [11], elastic cross-section verification (see Tab. 3); e) imperfection amplitude according to [16]; f) imperfection amplitude according to [16] 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Steel Construction 16 (2023) 7 A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members ARTICLE profile was varied, while a constant thickness of 12.5 mm was defined to exclude local buckling effects. The diagrams in Fig. 4a,b show the results of GMNIA simulations, where length affine equivalent bow imperfections were considered using Eq. (2) in combination with Tab. 2, according to the definition in [10]. In addition, Fig. 4c,d represent the results based on the definition of equivalent bow imperfections from [11], using Eq. (3) in combination with Tab. 3 for the elastic cross-section verification. As Tab. 3 requires the choice of a buckling direction, here the values for buckling about y–y axis were used in the case of SHS profiles. Subsequently, Fig. 4e,f show the results for the definition of the imperfection amplitude according to Eq. (4). The results according to EN 1993-1-1:2010 and prEN 1993-1-1:2020 are similar (Fig. 4a,c and Fig. 4b,d), although the results based on Eq. (3) are slightly more fa3.2 Results and discussion of bow equivalent imperfections A summary of selected GMNIA calculation is presented in Fig. 4 to Fig. 7. The basic layout is in all cases the same, where the y-axis is represented by the global buckling reduction factor c , defined by the estimated global resistance and divided through the nominal plastic cross-section resistance Npl. The determined reduction factors are plotted over a normalized global slenderness l along the x-axis. Therefore, the global slenderness was calculated from the square root of the plastic cross-section resistance divided by the critical buckling load. Fig. 4 shows the simulated results for a hot-rolled and coldformed SHS200 profile using the geometric predefinitions according to EN 1993-1-1:2010 [10], prEN 1993-1-1:2020 [11], and prEN 1993-1-14:2020 [16]. The length of the Fig. 5 Slenderness affine [10, 11] imperfection formulation; GMNIA calculations and comparison with code provision of SHS200 × 12.5 profile: a) hot-rolled, elastic design; b) cold-formed, elastic design; c) hot-rolled, plastic design; d) cold-formed, plastic design 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. 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Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members Fig. 6 Length affine imperfection formulation; GMNIA calculations and comparison with code provisions for an I-shaped HEA300 profile for buckling around z–z axis (a, c, e) and y–y axis (b, d, f): a) imperfection amplitude according to [10], elastic cross-section verification (see Tab. 2); b) imperfection amplitude according to [10], elastic cross-section verification (see Tab. 2); c) imperfection amplitude according to [11], elastic cross-section verification (see Tab. 3); d) imperfection amplitude according to [11], elastic cross-section verification (see Tab. 3); e) imperfection amplitude according to [16], buckling curve c; f) imperfection amplitude according to [16], buckling curve b 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Steel Construction 16 (2023) 9 A. Müller, M. Vild, A. Taras: Decision tree for local + global imperfection combinations in double-symmetric prismatic members ARTICLE results that are slightly on the safe side below the European global buckling curves 4 Choice of eigenmode shapes for interactive cases of global and local buckling The strength predictions based on GMNIA calculations can be strongly dependent on the choice of imperfection shapes, which are determined in advance by LBA, and the respective associated imperfection amplitudes. Acommon approach is to use the shape of the first eigenmode as the applied initial imperfection for subsequent GMNIA calculations. However, in important cases, this approach could neglect a possible interaction between local and global buckling, with significant consequences for the estimated load-bearing capacity. For this reason, it is important to assure that both local and global imperfecvourable and closer (safe-sided) to the governing buckling curves a and c for the hot-rolled and cold-formed SHS200 profile, respectively. The modified imperfection amplitude formulation from prEN 1993-1-14:2020 and Eq. (4) leads to results closer to the relevant buckling curves a and c, although being slightly optimistic in both cases (Fig. 4e,f) in areas of intermediate global slenderness. The same consideration was made in the case of the HEA300 profile, presented in Fig. 6, for buckling around both axes using the elastic cross-section verification. The non-linear imperfection amplitudes calculated with Eq. (5) lead in further analysis to resistances which are closer to the European buckling curves for flexural buckling. Fig. 5 and 7 give an overview of GMNIA calculations for an SHS200 × 12.5 and an HEB300 profile for the elastic and plastic design consideration. For both profiles, the choice of plastic design, i.e., Wpl/A, leads to Fig. 7 Slenderness affine [10, 11] imperfection formulation; GMNIA calculations and comparison with code provision of HEA300 profile: a) buckling around y–y axis, elastic design; b) buckling around z–z axis, elastic design; c) buckling around y–y axis, plastic design; d) buckling around z–z axis, plastic design 18670539, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/stco.202200041 by Technical University In Brno, Wiley Online Library on [31/01/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License