Equivalent geometric imperfections for the LTB‐design of members with I‐sections

For the stability assessment of members and structures according to EN 1993‐1‐1, the equivalent member method, the geometrical non‐linear calculation with equivalent geometrical imperfections or the GMNIA analysis with geometrical imperfections and residual stresses can be used alternatively. The second possibility requires a corresponding model for the cross‐section resistance. For the verification of lateral torsional buckling, bow imperfections e0 out‐of‐plane are defined, which lead in combination with the given loading in‐plane and the geometrical non‐linear analysis to bending moments Mz and torsion of the members. The amplitudes of the imperfections are highly dependent on the nature of the approach (e.g., scaling of the buckling shape, assumption of bow imperfections) and the verification method for the members. Within the framework of the scientific work supervised by the TU Dresden and TU Darmstadt, extensive parametric studies were conducted to calibrate imperfections for lateral torsional buckling based on the GMNIA. After determining the nature of imperfections and the design models for section resistance, this article presents results of these parametric studies and shows the calibration of imperfections for a standardisation proposal based on EN1993‐1‐1. The evaluation of the data in combination with the necessary simplifications for the design practice leads to appropriate definitions of imperfection values e0,LT and the necessary differentiations.


Imperfection assumptions according to EN 1993-1-1 and prEN1993-1-1
The design process by a geometrically non-linear analysis using equivalent geometrical imperfections (GNIA) will be conducted in three steps by applying imperfections, determination of internal forces and moments using geometrically non-linear analysis and verification of the cross-section resistance at the most unfavourable point of the member. The lateral torsional buckling (LTB) resistance according to this design method depends on the size and shape of the geometrical imperfection, the moment distribution and possible additional normal forces, the cross-section shape, the verification method and the steel grade. In EN 1993-1-1 [2], imperfections for the LTB-de-sign of members are defined in Chapter 5.3.4, where the shape is specified as an initial bow imperfection out-ofplane. The amplitude is given as k · e 0 , where the recommended value for k is 0.5 and the basic values e 0 are defined in Tabs. 5.1 and 6.2 of EN1993-1-1. For hot-rolled I-sections, the relevant buckling curve for flexural buckling depends on the h/b-ratio, the plate thickness and steel grade. For slender cross-sections with h/b > 1.2, this results in smaller imperfection amplitudes than for stocky cross-sections with h/b ≤ 1.2. Among others, Kindmann and Beier-Tertel [3] have shown that this assumption is not suitable for the LTB-design. Sections with h/b ≤ 2.0 are more favourable than those with h/b > 2.0. This is also considered in the selection of the LTB-curves according to Tab. 6.5 of EN 1993-1-1. In the German National Annex of DIN EN1993-1-1 [4], partly deviating imperfection amplitudes are given, which result from the assignment to the LTB-curves. The k-factor is given as 1.0 in the medium range of slenderness (0.7 ≤ l LT ≤ 1.3). These rules are adopted and modified in the latest version of prEN1993-1-1 [5] by considering the material para meter e as an amplifier. The shape of the equivalent geometrical imperfection is still specified as a bow imperfection outof-plane (Tab. 1). Fig. 1 shows a comparison of the calculated load capacities for hot-rolled I-and H-sections made of S235 with fork bearings under constant bending moment according to prEN1993-1-1 [5]. The load capacities calculated with the GNIA using the elastic cross-section resistance were compared with the load capacities according to the equivalent member method (Eqs. (1) to (3)). Eq. (3) includes the bending moments M z,Ed and the warping moment B Ed (bi-moment), which result from the bow imperfection and the influence of the geometric non-linearity. (1) with M y,ult moment resistance M y from the GNIA analysis using the elastic interaction, Eq.

Imperfection assumptions from the literature
In [6], a method was published which, using equivalent geometric imperfections and a calculation according to second-order theory, checked the stability against flexural and lateral torsional buckling of components subjected to compression and bending. Both constant and variable cross-sections over the length of the member can be considered. By calculating separately the in-plane and out-ofplane stability, two associated utilisation ratios have to be determined. For out-of-plane design, equivalent geometrical imperfections affine to the eigenmode must be applied and the referred slenderness is l op . In general, the determination of the imperfection amplitude and the identification of the design-relevant cross-section require an iterative procedure.
Chladný and Štujberová [7,8] developed a calculation method using equivalent geometric imperfections for the flexural buckling analysis of components subjected to compression and bending that are part of an overall structure and can have a constant or variable cross-sectional shape. The imperfection is assumed affine to the flexural buckling mode. The determination of the amplitude requires, among other things, knowledge of the slenderness derived for the overall structure at the design point.
In [9], a method using equivalent geometrical imperfections for the stability analysis of lateral torsional buckling was presented. The imperfection must be applied affine to the LTB-eigenmode. The key idea in the calculation of the imperfection amplitude is the objective that the effect of a bending moment M y according to the equivalent member method M b,Rd = χ LT · M y,Rd leads to a 100 % utilisation of the cross-section at the most unfavourable position. The elastic and plastic cross-sectional resistance is checked using a linear interaction function.
Investigations by Ebel [10] on various structural systems and loads have shown that the structural behaviour may be better reflected by considering the shape of the first LTB-buckling mode for the equivalent geometrical imperfection. Snijder et al. [11] also considered shapes of imperfections based on LTB-modes and developed a proposal for geometrical and material non-linear analysis. In Hajdú et al. [12], a proposal is presented, which considers the combination of compression and bending. The imperfection shape is also based on the relevant LTB-mode.
M y,Ed , M z,Ed , B Ed internal moments from the GNIA calculation, M el,y,Rd , M el,z,Rd , B el,Rd elastic moment resistances for bending about the y-and z-axis as well as bi-moments, h ratio value of the moment resistances M y from the GNIA analysis and the equivalent member method.
The ratio values h in Fig. 1 show that the design using GNIA leads to more or less conservative results compared to the equivalent member method. This also applies to the linear interaction relationship, where the cross-sections are utilised plastically. Parametric studies in [1] also show that the influences of the cross-section shape, the moment distribution and the steel strength are not or not correctly covered by the approaches in prEN1993-1-1. This also applies to the EN1993-1-1, which is currently in use.

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type and course of the imperfections. Normative design rules must refer to this and be practicable.
For the calibration of imperfections, in addition to the influences described above, the reference values for the member resistance are also significant. Due to the high effort, test results for lateral torsional buckling are only available to a limited extent. In many cases, the documentation of the tests is not sufficient to conduct a calibration of imperfections on them. Another possibility is the calibration of equivalent geometrical imperfections to the buckling curves χ LT determined by tests and GMNIA calculations. In the χ LT buckling curves of prEN1993-1-1, the influences from the cross-sectional shape, the fabrication method (hot-rolled, welded), the ratio W el,y /W el,z and the moment distribution are considered. The stronger differentiation compared to EN1993-1-1 already shows the necessity of explicitly recording the influences mentioned. In contrast to the verification of flexural buckling, the influence of the yield strength is not considered in the buckling curves χ LT and their assignment.
Finally, there is the possibility of calibrating equivalent geometric imperfections with the results of GMNIA calculations. In addition to the careful development and validation of structural and material models, imperfection assumptions are again required. The latter are usually assumed in form of residual stresses and geometric imperfections. The prEN1993-1-14 [15] contains recommendations for this. An advantage of the procedure compared to the calibration of the equivalent imperfections based on the buckling curves χ LT is that the inaccuracies that have arisen with the necessary simplifications in the derivation and assignment of these functions are not adopted. Further advantages are the independent determination of the ultimate loads and the extension of the parameter ranges. Of course, when the data from the parametric studies are available, simplifications are again necessary when developing approaches for equivalent geometric imperfections to make the design models applicable.

General
The verification of the stability of beams susceptible to lateral torsional buckling using GNIA is conducted in three steps: 1) development of a structural and load model by considering equivalent geometrical imperfections, 2) geometric non-linear elastic analysis and 3) verification of the cross-sectional resistance at the most unfavourable point.
Each individual step influences the result and should be clearly regulated. To limit the scope, practical specifica-In Quan et al. [13], two procedures for the LTB-design with GMNIA are presented, which are applicable to members made of both mild steel and stainless steel. The equivalent imperfection is to be applied either affine to the LTB-eigenmode or as a bow imperfection about the weak axis. The latter imperfection form is composed of a combination of a sine half-wave and a sine wave. The amplitude of the equivalent imperfection is determined as a function of the associated buckling curve, linked by the imperfection coefficient a z according to EN1993-1-1 for structural steel and a according to EN1993-1-4 [14] for stainless steels. Different steel grades are considered analogously to the new regulations according to prEN1993-1-1 by the material factor e, which leads to larger imperfection amplitudes for higher steel grades.
Compared to the above approaches, the method described here does not specify the equivalent geometrical imperfection as a function of the member slenderness. The influencing factors from the moment distribution, the cross-section and the material properties are considered by individual factors and the imperfection size is specified in relation to the member length, analogous to the rules in prEN 1993-1-1. The advantage of this approach is a more flexible calibration of the relevant factors, considering the mutual interactions. The application is simpler because no iterative procedure is required for the determination of the imperfection quantity to be applied. The transfer to complex steel structures consisting of several components is also associated with less calculation effort because no identification of the significant slenderness ratio and the design-relevant location is necessary.

Calibration of imperfections
Various influences play a role in the calibration of imperfections. For example, when determining the internal forces or stresses, a distinction must be made between elastic and plastic calculations. In the latter, for example, plastic zones can be considered or plastic hinges can be applied. The cross-sections can be utilised elastically, partially plastically or fully plastically using linear or nonlinear interaction relationships. Furthermore, not only the maximum amplitudes are important for the imperfection assumptions, but also the type of imperfections and their gradients. Equivalent geometrical imperfections also consider residual stresses and plastic zone propagations in an elastic calculation. If these influences are already considered in the structural analysis, usually only pure geometrical imperfections are applied. The approach of equivalent geometrical imperfections affine to the first eigenmode usually requires smaller amplitudes than the assumption of bow imperfections out-of-plane. With the buckling modes, the influences from the structural system, the bearing conditions and the actions are recorded. It is clear from the above that the imperfection assumptions must always be made in conjunction with the calculation of the structure, the verification method and the

Proposal P-1: Eigenmode affine imperfection IMP-1
The calculation with the eigenmode affinity imperfection requires a preceding modal analysis and the possibility of scaling the eigenform and applying it as an imperfection. With this procedure, the specific boundary conditions and stress ratios are covered. Systems with specific bearing conditions, such as a bonded axis of rotation or asymmetrical moment distributions, can be calculated in this way.
The structure of the design proposal and the procedure for determining the imperfection size were chosen based on the regulations for imperfection assumptions given in prEN1993-1-1. The detailed consideration of different influences is done by additional factors. The amplitude of the equivalent geometrical imperfection is determined with Eq. (6).
with e 0 basic value of the equivalent imperfection (see Tab. 2), b s factor for the cross-section shape (see Tab. 2), b M factor to consider the bending moment diagram (see Tab.  The minimum size e 0,min of the equivalent geometrical imperfection is equated with the permissible manufacturing tolerance for bow imperfections according to EN 1090-2 with L/1000, were L is the member length.

Proposal P-2: Bow imperfection IMP-2
In previous design practice, the approach of a bow imperfection from the system plane in the form of a sinusoidal pre-curvature is common. In many standard cases, sufficiently accurate results can be achieved by applying suitable imperfection values. Therefore, within the framework of the thesis [1], a proposal for imperfection assumptions was also developed for this, which takes into account the essential influencing factors. The amplitude of the equivalent geometrical imperfection is determined with Eq. (7).
tions are made for the verification of the structural safety. The structural analysis is conducted geometrically nonlinear, usually according to second-order theory, considering bending about the weak axis and warping torsion. The verification of the cross-section resistance is conducted elastically or plastically depending on the crosssection classification using the linear interaction of the internal forces. Partially plastic cross-sectional resistance can also be used if the imperfections for the plastic design are applied.
For the elastic verification of the cross-section resistance, the limit criterion is achieved when the von Mises stress σ v,Ed reaches the yield strength f yd at the most critical point of the structural member.
For linear interaction, the cross-section resistance is reached when the sum of the utilisation ratios from the internal forces involved reaches the value 1.0. Where N is the normal force, M y and M z are the bending moments about the cross-sectional axes y and z, and B is the bimoment from warping torsion.
Both types of verification are already part of prEN 1993-1-1 and do not need to be newly introduced. In the plastic cross-section check, possible reductions of the load-bearing capacities in Eq. (5) must be considered if shear stress generating internal forces (V y,Ed , V z,Ed , T t,Ed , T w,Ed ) act simultaneously. The application of non-linear plastic crosssection interactions was not considered in the calibration of the imperfection approaches.
The shape of the equivalent imperfection significantly influences the load-bearing capacity of beams susceptible to lateral torsional buckling. Two different shapes of imperfections are considered, the eigenmode affine imperfection (IMP-1) and the sinusoidal bow imperfection out-ofplane (IMP-2), see Parameter studies using GMNIA calculations

Structural model and imperfection assumptions
For reasons of practicability and to achieve higher accuracy, extensive parametric studies were conducted on hot-rolled I-sections using GMNIA. The non-linear simulations were performed using the finite element analysis software ANSYS [16,17]. For this purpose, a hybrid shellbeam model was created modelling the I-section as a combination of shell and beam elements. The shell elements were located in the middle plane of the flanges and the web and idealise the flat parts of the sections. The beam elements were arranged to consider the stiffening effect of the rolled radii at the intersection of the flange and the web. The cross-section of the beam elements was defined as rectangular hollow section (RHS). The dimensional definition of the RHS is based on two criteria. The height of the RHS is at most twice the flange thickness, and the torsional stiffness of the whole cross-section consisting of three plates (shell elements) and two bars (beam elements) corresponds to the tabulated values given by the production standards for hot-rolled I-sections. In addition, the torsional stiffness is influenced by modification of the shear modulus. The nodes of the beam elements (type Beam188) and the shell elements (type Shell181) each have six degrees of freedom. The flange plates were discretised with 12 elements over the widths and the web plates with nine elements over the height.
with e 0 basic value of the equivalent imperfection (see Tab. 3), b s factor for the cross-section shape, for hot-rolled I-sections: for welded I-sections: factor to consider the bending moment diagram (see Tab.  Hot-rolled I-section

Overview of the parametric studies
The aim of the present investigations was to derive imperfection assumptions for the LTB design by GNIA calculations. The main influences of the cross-sectional shape, the yield strength, the moment distribution and the model for the cross-sectional resistance should be considered. For the comparative calculations, single-span beams of hot-rolled I-sections with fork end conditions were used (Fig. 5). Different moment distributions for bending about the major axis have been analysed. The investigations included 52 hot-rolled I-sections listed in Tab. 5. The table provides additional information on the different aspect ratios, which are explained in more detail in Section 3.3.
The model for determining the cross-sectional resistance has a significant influence on the size of the required equivalent imperfection. In addition to the elastic verification method, several plastic interaction formulas were investigated. As a result of this study, which is not presented in this article for reasons of scope, the linear plastic interaction has proven to be particularly suitable for members with cross-section classes 1 and 2, which are susceptible to lateral torsional buckling. On the one hand, the linear plastic interaction is easy to handle. On The boundary conditions are idealised using pilot nodes (reference nodes), whose degrees of freedom can be constrained arbitrarily. Concentrated loads on the supports are also applied to the pilot nodes. By linking the degrees of freedom of the pilot node with the other nodes within a cross-section, idealised boundary conditions such as the fork end conditions can be modelled.
Initial geometric imperfections were implemented into the FE-models using the first buckling mode from an elastic buckling analysis. The amplitude was scaled to L/1000.

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The influence of shear stresses can lead to a reduction of the full plastic section resistances (see EN1993-1-1).
The shape of the equivalent imperfection strongly influences the spatial structural behaviour of beams loaded by in-plane bending moments. Two common shapes of imperfection, the first elastic buckling mode for LTB (mode imperfection, IMP-1) and the sinusoidal bow imperfection out-of-plane (IMP-2), were investigated. For both imperfection shapes, the amplitude e 0 corresponds to the maximum deflection of a flange (Fig. 2).
In the following, the influences from the cross-section geometry, the moment distribution and the load application point as well as the influence from the yield strength will be discussed. With regard to validation of the design proposals for the assumption of equivalent geometric imperfections, reference is made to the thesis [1] to limit the scope.

Influence from the cross-section geometry
The behaviour of members susceptible to lateral torsional buckling is significantly influenced by the section properties. The amplitude of the required equivalent geometric imperfection is measured with the factor j, which describes the ratio between the member length L and the imperfection size e 0 . The factor j is determined by the condition that, considering the geometric imperfection, a calculation of the member according to the second-order the other hand, plastic cross-section reserves can usually only be partially exploited until the load-bearing capacity is reached. Due to the propagation of plastic zones during LTB, there is a considerable loss of stiffness, which excludes fully plastic cross-section utilisation in the medium and high slenderness range. As the size e 0 of the equivalent geometric imperfection is adapted to the model of the cross-section resistance, the linear plastic interaction is not to be evaluated as particularly conservative either. Only in the compact slenderness range, in which lateral torsional buckling is no longer important, underestimations of the load-bearing capacity can occur.
The linear elastic interaction formula (Eq. (8)) can be used for double-symmetric cross-sections, if the shear stresses are negligible.
For linear plastic interaction, the cross-section resistance is reached when the sum of the utilisation ratios from the internal forces involved reaches the value 1.0 (Eq. (9)). The two diagrams in Fig. 6 show a pronounced scattering of the j-curves associated with the different hot-rolled sections. This illustrates the dependence of the imperfections on the section. The influence of the elastic and plastic interaction applied in the cross-section verification is evident with the different heights of the j-curves in Fig. 6a,b. Furthermore, there is a mutual influence of section dependence and the applied interaction formula in the cross-section verification. Various investigations were conducted to identify the cross-section parameters that are significant for the size of the j-values. According EN 1993-1-1, the size of the geometric equivalent imperfection is given as a function of the h/b-ratio, and a cut-off theory results in the same load resistance as the calculation by GMNIA. Fig. 6 shows the course of the j-values for the sections made of a steel grade S235, according to Tab. 5, with a constant bending moment, the approach of a geometric imperfection affine to the first eigenmode and a variable member slenderness. For all j-curves, a uniform curve with the minimum value j min is found in the medium slenderness range. This typical j-curve course is independent of the considered section, the manufacturing process (hot-rolled or welded), the applied shape of geometric imperfection (affine to the buckling shape or bow imperfection), the steel grade or the load. For the development of the verification method and especially for the calibration of the imperfection size e 0 , the knowledge of the  To find a suitable parameter that accurately reflects the influence of the cross-section geometry on the required geometric imperfection, the dependencies of the j-values on different cross-section parameters were analysed. Within the framework of extensive parametric studies with different ratio values (among others I y /I z , I y /I T , I z /I T , I w /I T , W el , y /W el , z , W pl , y /W pl , z ), the cross-section parameter x (see Eq. (11)) derived from the equivalent member method turned out to be most suitable (Fig. 8).
The colour differentiation of the results between sections with h/b ≤ 1.2 and h/b > 1.2 as well as the elastic and plastic cross-section resistance illustrates the necessity to differentiate between these cases. The comparatively low scatter and the increase of the j-values underline the suitability of the cross-section parameter x to capture the section dependence. The equations j(x) for the respective cases can be described in a good approximation with linear functions, starting from the origin of the coordinate system. The inclination of the straight lines is defined by the basic values of the equivalent imperfection e 0 . As the j-values for the elastic and plastic calculation for h/b ≤ 1. In an analogous way as described before, the cross-section coefficient b s for welded sections was also determined. Furthermore, equivalent imperfections in the form of bow imperfections out-of-plane were derived in this  To derive the moment factor b M for linear varying moment diagrams, a differentiation between the imperfection influences on the respective cross-section shapes was first considered, which results from the h/b-ratio and the cross-section parameter x (Fig. 8).  Fig. 11).
In principle, the load application at the top chord is the most unfavourable case. Due to the distance to the shear centre, there are driven effects from the transverse load, which lead to an increase in the cross-sectional rotation in a geometric non-linear calculation (GNIA). Lower buckling moments M cr result. The eigenmode has larger components from the rotations around the longitudinal axis. The load application on the bottom chord, on the contrary, leads to torsional loads against the direction of rotation and thus to a stabilising effect. This leads to an way. The ratio I z /I T turned out to be suitable as a crosssection parameter (see chapter 2.3).

Influence of the moment diagram M y and the load application point
The influence of different moment diagrams on the size of the equivalent imperfection is captured by the moment factor b M . An overview of the investigated load cases is shown in Fig. 9. The ratios of hogging and sagging moments are considered as variable quantities (cf. Tab. 4). amplitudes e 0 or larger values j (Fig. 13). The influence of different load application points increases with increasing sensitivity of the sections to imperfections. The largest differences in the j-values occur for sections with x ≤ 0.8. For transverse loads acting on the top chord, a lower bound of the j-value to 400 is introduced for this. The load application at the centre of gravity and at the bottom chord, on the other contrary, requires larger imperfections e 0 or smaller values j in this limitation range (see Fig. 13).
In the case of a load application on the top chord, the influence can be considered by limiting the cross-section factor to the value b s = 1.0. For deviating load application increase of the buckling moment and a reduction in the cross-sectional rotation in the eigenmode (Fig. 12a).
The ultimate load according GMNIA is also influenced by the application point of the transverse loads (Fig. 12b). By using the slenderness ratio l LT in the figure, the influence is captured to a high degree and the differences are only small. Nevertheless, the buckling curve for transverse loads acting on the bottom chord (z p = +h/2) is above the other two curves. This indicates positive loadbearing effects, such as the later onset of yield zone formation.
When calibrating the imperfections for the verification according to the GNIA, there are interactions between the cross-section shape, the moment diagram and the load application point. Depending on the imperfection shape, affine to the buckling mode or as a bow imperfection, different interactions result between the load application point and the size of the imperfection.
In the approach of imperfections affine to the eigenmode, transverse loads acting on the top require smaller If the imperfection is applied as a bow imperfection (IMP-2), an opposite trend results with regard to the size of e 0 . The point of load application at the top chord requires the largest value, while the other two cases require smaller imperfections (Fig. 14). This influence is considered by the factor for the load application point of transverse loads b zp as a linearly variable function according to Eq. (12).
As can be seen from the structure of the equation, the value b zp = 1.0 results for a load application on the top chord, for z p = +h/2 the value drops to 0.8. Therefore, the consideration of b zp is optional and serves economic efficiency (see Eq. (7)).

Influence of the yield strength
The evaluation of the influence of the yield strength on the size of the equivalent geometric imperfections to be applied is comparatively complex. The reasons for this are manifold; there are dependencies on, among other things, the degree of slenderness, the approach of the residual stresses and the verification method as well as on the shape of the cross-section.
For members susceptible to buckling, the influence of the strength on the load-bearing capacity is slenderness-dependent. With short and stocky members, effective increases in the resistance of the member can be achieved by selecting higher steel grades. With increasing member length, however, the influence of the yield strength decreases. For slender members, the load-bearing capacities are mainly influenced by the stiffnesses.

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of residual stress to yield strength decreases. This enables higher additional stresses from external loads to be absorbed before the onset of yielding and yield zone propagation occur. The resulting loss of stiffness of the beam only begins at a higher load level.
In addition to the residual stresses, the geometric imperfections also lead to a material-and slenderness-dependent difference in the resistance. This effect occurs both in calculations according to GMNIA and GNIA. Fig. 16a shows the calculated buckling curves for the imperfection dimensions e 0 of L/1000, L/400 and L/200 for a simply supported beam IPE500 under constant bending moment made of steel grades S235, S355 and S460. The calculation was done according to the second-order theory including warping torsion using geometric imperfections with an affinity to the eigenmode (IMP-1). The cross-sectional check was performed using linear plastic interaction (see Eq. (9)). and S460 as a function of the beam length and the slenderness. The calculation was done with the GMNIA and the imperfection approaches and model assumptions according to Section 3.1. By representing the related moment resistance M y /M pl,y via the slenderness l LT , the previously mentioned effects are largely adjusted. The resistance curves belonging to the different steel grades show an almost identical course in the lower (l LT < 0.6) and higher (l LT > 1.6) slenderness range. In the medium slenderness range, there are differences of up to 8 % in favour of the S460 over the S235.
The failure of members of medium slenderness is caused by the stiffness-reducing yield zone formation and manifests itself through buckling failure of the partially plasticised system. The onset of yielding and the extent of yield zone propagation are significantly influenced by the residual stresses. In the calculation according to GMNIA, the magnitude of the residual stresses for hot-rolled sections is assumed to be independent of the yield strength (see Section 3.1). As the strength class increases, the ratio   Fig. 18 shows the data basis on which the derivation of the material factor b a was conducted. It can be seen that a larger scatter band results with increasing yield strength and that the approach using Eq. (13) provides predominantly conservative results.
As the approach to determine the equivalent geometric imperfections e 0,LT according to Eqs. (6) and (7) shows, further differentiations are required with respect to the factor b a . When applying eigenform affine imperfections and elastic cross-section resistance, an increase of e 0,LT can be omitted and the factor b a = 1.0 can be conservatively assumed. For welded I-sections, the amount of residual stresses is currently still assumed to be proportion- As expected, the related load capacity χ = M y /M pl,y decreases with increasing imperfection dimension e 0 . However, the difference in the χ-values of the strength classes increases for the same slenderness ratio. Fig. 16b shows the ratio values χ of the steel grades S460 to S235 for calculations according to the GNIA and GMNIA. The maximum ratio values χ for calculations according GNIA are between 104.1 % and 109.5 %. For comparison, the curve from calculations according GMNIA using L/1000 and residual stresses is compared (Fig. 16b). In general, it can be stated that the related load capacities χ according to GNIA and GMNIA are higher at higher yield strengths. However, the maximum relationships χ occur at different degrees of slenderness (Fig. 16b). The comparison for the base value e 0 = L/400 (see Tab. 2) with the course of the GMNIA leads to the slenderness range marked in red, in which the GNIA provides higher load resistances than the GMNIA. To obtain reliable results over the entire range l LT , the calculation according to GNIA for the considered profile IPE500 requires slightly larger imperfection measures e 0 at higher steel grades than for the grade S235.
When assessing the influence of the yield strength for hotrolled I-sections, it has to be noted that for h/b ≤ 1.2 the residual stress should be assumed to be 50 %, for h/b > 1.2 only 30 % of the yield strength of an S235. Compact HEMprofiles with thick flanges are less sensitive to geometric imperfections, so the effects are opposite. Fig. 17 shows the j-values for hot-rolled I-sections using the plastic interaction relationship for the cross-section check and the buckling shape affine imperfection approach (IMP-1). The slightly larger imperfections for higher strength classes can be approximately captured by the factor b a according to Eq. (13). interaction relations that capture the simultaneous effect of the internal forces M y , M z , B and N are not regulated in prEN 1993-1-1. The given imperfection approaches have also not been validated for this. In addition, the suitability of a sinusoidal bow imperfection out-of-plane as an imperfection assumption must be questioned, as it does not adequately address many system boundary and loading conditions.
Based on the findings from extensive parametric studies, design proposals were developed for double-symmetric I-shaped welded and hot-rolled sections, considering the essential dependencies. This includes the cross-section shape, the moment distribution, the load application point of transverse loads, the yield strength, the utilisation of the cross-sections (elastic or plastic) and the interaction relationship used as well as the type of imperfection assumption. As the latter has a significant influence on the calculated utilisation of the members, values e 0,LT were derived for the assumption of imperfections with an affinity to the eigenform (IMP-1) and for the approach of sinusoidal bow imperfection out-of-plane (IMP-2).
With regard to the practical application, the most striking differences between the two imperfection approaches arise particularly in conjunction with different bending moment diagrams. The equivalent geometric imperfection with an affinity to the buckling mode for the constant moment diagram requires the largest imperfection amplitude related to the member length and can thus be transferred to any moment diagram in a simplified and safe manner. On the contrary, the sinusoidal bow imperfection out-of-plane must always be determined considering the moment diagram to achieve safe results.
The design proposals were developed for single-span beams with fork bearings on both ends. The validation for cantilever systems, beams with bound axis of rotation and asymmetric bearing conditions is still pending. For the combination of bending and normal force loading, [1] contains further design proposals.
al to the yield strength. In this case, Eq. (14) provides a suitable approximation for increasing the imperfection value e 0,LT The prEN1993-1-1 provides for a similar increase of the equivalent geometric imperfection for lateral torsional buckling. Recent investigations of welded I-sections show that the residual stresses of higher steel grades do not increase proportionally to the yield strength, so that more favourable assignments to buckling curves are possible (see [18][19][20]).
When applying lateral bow imperfections (IMP-2), lower values e 0,LT result for hot-rolled sections regardless of the type of cross-section check for higher steel grades. These can be described with Eq. (15) for the elastic and plastic cross-section resistance.
For welded double-symmetric I-sections, Eq. (14) can be used again. Extensive evaluations and further background information on the approaches described above can be found in [1].

Conclusion and outlook
The present article deals with equivalent geometrical imperfections for the LTB-design using GNIA. Although this calculation method has been the subject of design standards for several decades, the rules in the current EN 1993-1-1 as well as the future version of this standard prEN 1993-1-1 are not satisfactory. Important influencing variables on the imperfection values are not or only insufficiently covered. In most applications, uneconomical results are obtained if the cross-sectional resistance is determined with the linear interaction. Non-linear plastic