Interactive Buckling of Steel LC-Beams Under Bending

The present paper deals with the interactive buckling of thin-walled lipped channel (LC) beams under the bending moment in the web plane when the shear lag phenomenon and distortional deformations are taken into account. A plate model (2D) was adopted for LC beams. The structures were assumed to be simply supported at the ends. A modal method of solution to the interactive buckling problem within Koiter’s asymptotic theory, using the semi-analytical method (SAM) and the transition matrix method, was applied. LC-beams, from short through medium-long via long to very long beams, were considered. The paper focuses on the influence of the secondary global buckling mode on the load carrying capacity for the steel LC-beams under bending.


Introduction
Thin-walled cold-formed steel (CFS) members (columns, beams, and beam-columns) are widely used in the construction industry. C-section and LC-section (i.e., lipped channel) beams are basic structural elements that are primarily subject to bending. A capacity for resistant loads in thin-walled beams is limited not just by their strength, but first of all due to their stability.
The numerical methods often applied in nonlinear analysis of stability and load carrying capacity are as follows: finite strip method (FSM), finite element method (FEM), and generalized beam theory (GBT).
Mode decomposition of thin-walled columns and beams with different cross-sections subjected to various loadings based on GBT has been analyzed in Reference [1]. References [2,3] deal with the constrained finite element method (cFEM) employed for the buckling analysis of columns with opened cross-sections. A new method with a modal decomposition feature, the so-called constrained finite element method (cFEM) is presented in Reference [4]. The method can be applied to analyze a wide range of thin-walled members, including members with holes and varying cross-sections and stiffened plates.
FSM is commonly used for nonlinear analysis of elastic stability due to its very high numerical capability, low computational costs and easy implementation in thin-walled elements. The method is limited mainly to simple geometries and boundary conditions. It has resulted in an introduction of numerous new FSM variants or expansions.
In FEM, two kinds of analyses are usually conducted, namely for (i) linear elastic buckling to enable determination of critical loads and the corresponding buckling modes, i.e., an eigenproblem, (ii) nonlinear post-buckling analysis to determine the performance curve in the full range of loading and/or load carrying capacity. In addition, the FEM model is affected by epistemic uncertainties.
The influence of distortional, global, and local buckling modes, and their combination, on post-buckling behavior is widely investigated by Martins, Camotim, Gonçalves, and Dinis [5,6]. They and used their modification (GBTUL-2.0) [5] and employed existing tools (GBT and DSM) [6]. A semi-analytic method based on Koiter's theory of the compressed thin-walled structure with cross-section deformation modes is analyzed in Reference [7]. Composite C-beams subjected to the influence of the secondary global distortional-lateral mode on the interactive buckling and assessment of the load carrying capacity of LC-beams of various lengths. The aim of this investigation is to determine what values of length of the beam influences on the load carrying capacity.
In the present study, Lagrange's description, full Green's strain tensor for thin-walled plates and second Piola-Kirchhoff's stress tensor, and the exact transition matrix method and the numerical method of the transition matrix using Godunov's orthogonalization are used. The shear lag phenomenon, an effect of cross-sectional distortions, as well as coupled conditions between all the walls of structures are included. The most important advantage of this method is that a complete range of behavior of thin-walled structures can be described [14].
The coupled buckling of thin-walled steel LC-beams under bending in the web plane from short ones through medium-long to long beams is analyzed here.

Formulation of the Problem
Prismatic thin-walled steel (i.e., isotropic) beams built of plates connected along longitudinal edges and under the uniform major-axis bending moment were considered. The beams were simply supported at their ends [13,14]. In order to account for all modes of global, local, and coupled buckling, a plate model (i.e., 2D) of thin-walled structures was applied. Moreover, it was assumed that the material of the structure is obeyed Hooke's law. Details can be found in References [13,14] or see Appendix A.

Analysis of the Calculations Results
Detailed numerical calculations for interactive buckling were conducted for three steel LC-section beams of the cross-section dimensions identical to those in Reference [19]. The beam geometrical dimensions under consideration together with the assumed notations are presented in Figure 1 and Table 1. The ratios of the main central moments of inertia I max /I min , which range from 3.6 to 6.0, are included there as well. In the present study, Lagrange's description, full Green's strain tensor for thin-walled plates and second Piola-Kirchhoff's stress tensor, and the exact transition matrix method and the numerical method of the transition matrix using Godunov's orthogonalization are used. The shear lag phenomenon, an effect of cross-sectional distortions, as well as coupled conditions between all the walls of structures are included. The most important advantage of this method is that a complete range of behavior of thin-walled structures can be described [14].
The coupled buckling of thin-walled steel LC-beams under bending in the web plane from short ones through medium-long to long beams is analyzed here.

Formulation of the Problem
Prismatic thin-walled steel (i.e., isotropic) beams built of plates connected along longitudinal edges and under the uniform major-axis bending moment were considered. The beams were simply supported at their ends [13,14]. In order to account for all modes of global, local, and coupled buckling, a plate model (i.e., 2D) of thin-walled structures was applied. Moreover, it was assumed that the material of the structure is obeyed Hooke's law. Details can be found in References [13,14] or see Appendix A.

Analysis of the Calculations Results
Detailed numerical calculations for interactive buckling were conducted for three steel LC-section beams of the cross-section dimensions identical to those in Reference [19]. The beam geometrical dimensions under consideration together with the assumed notations are presented in Figure 1 and Table 1. The ratios of the main central moments of inertia Imax/Imin, which range from 3.6 to 6.0, are included there as well.    The following material constants: E = 210 GPa, ν = 0.3 were assumed for the steel lip channels [19]. In the pre-buckling state, the beams were subjected to linearly variable loading ( Figure 1) resulting in bending in the web plane (i.e., the upper flange was tensioned, the lower one was compressed).

Example of LC-1 Beams
In this example, lip channel LC-1 beams of the dimensions listed in Table 1 were analyzed. Alternations in values of the critical moment M r as a function of the buckling half-wavelength L b in a wide variability range 100 ≤ L b ≤ 10,000 mm are presented in Figure 2. The following material constants: E = 210 GPa, ν = 0.3 were assumed for the steel lip channels [19]. In the pre-buckling state, the beams were subjected to linearly variable loading ( Figure 1) resulting in bending in the web plane (i.e., the upper flange was tensioned, the lower one was compressed).

Example of LC-1 Beams
In this example, lip channel LC-1 beams of the dimensions listed in Table 1 were analyzed. Alternations in values of the critical moment Mr as a function of the buckling half-wavelength Lb in a wide variability range 100 ≤ Lb ≤ 10000 mm are presented in Figure 2. The lower curve (denoted as curve 1) corresponds to the lowest values of buckling loads, often referred to as primary buckling loads. The upper curve (marked as curve 2) refers to higher critical values, which can be called secondary buckling loads.
The value of the critical moment Mr (curve 1) for the variability range under consideration attains its maximum at Lb = 100 mm and the minimum at Lb ≈ 350 mm, and then it grows monotonously up to Lb ≈ 1000 mm, where it reaches the local maximum. Within the range 1000 ≤ Lb ≤ 10000 mm, values of the moment decrease monotonously. The critical values corresponding to curve 2 grow in the range 100 ≤ Lb ≤ 350 mm and attain the maximal value at Lb ≈ 350 mm. Next, in the range 350 ≤ Lb ≤ 1000 mm, they decrease drastically to attain the minimal value for Lb ≈ 1000 mm. Within the range 1000 ≤ Lb ≤4000 mm, curve 2 grows slowly, and at Lb > 4000 mm, it is constant in practice. While comparing curves 1 and 2, one can state that for Lb ≈ 350 mm the lower curve attains its minimum, whereas the upper one attains its maximum, and then at Lb ≈ 1000 mm, an opposite relation takes place. At Lb > 4000 mm, the drop gradient of curve 1 is significantly lower than for 1000 ≤ Lb ≤ 4000 mm, whereas the values corresponding to curve 2 are actually constant.
In Table 2, critical values of the moments Mr for the LC-1 subject to bending for selected four values of the total length L are presented. The following index notations are introduced: 1 -the lowest value of the buckling moment corresponding to the local buckling mode for m ≠ 1, 2 -the value of the primary global buckling mode for m = 1 (curve 1 in Figure 2), 3 -the value of the secondary global buckling mode for m = 1 (curve 2 in Figure 2). For the local moment M1, a number of half-waves m along the longitudinal direction is quoted additionally in the brackets. The lower curve (denoted as curve 1) corresponds to the lowest values of buckling loads, often referred to as primary buckling loads. The upper curve (marked as curve 2) refers to higher critical values, which can be called secondary buckling loads.
The value of the critical moment M r (curve 1) for the variability range under consideration attains its maximum at L b = 100 mm and the minimum at L b ≈ 350 mm, and then it grows monotonously up to L b ≈ 1000 mm, where it reaches the local maximum. Within the range 1000 ≤ L b ≤ 10,000 mm, values of the moment decrease monotonously. The critical values corresponding to curve 2 grow in the range 100 ≤ L b ≤ 350 mm and attain the maximal value at L b ≈ 350 mm. Next, in the range 350 ≤ L b ≤ 1000 mm, they decrease drastically to attain the minimal value for L b ≈ 1000 mm. Within the range 1000 ≤ L b ≤ 4000 mm, curve 2 grows slowly, and at L b > 4000 mm, it is constant in practice. While comparing curves 1 and 2, one can state that for L b ≈ 350 mm the lower curve attains its minimum, whereas the upper one attains its maximum, and then at L b ≈ 1000 mm, an opposite relation takes place. At L b > 4000 mm, the drop gradient of curve 1 is significantly lower than for 1000 ≤ L b ≤ 4000 mm, whereas the values corresponding to curve 2 are actually constant.
In Table 2, critical values of the moments M r for the LC-1 subject to bending for selected four values of the total length L are presented. The following index notations are introduced: 1-the lowest value of the buckling moment corresponding to the local buckling mode for m 1, 2-the value of the primary global buckling mode for m = 1 (curve 1 in Figure 2), 3-the value of the secondary global buckling mode for m = 1 (curve 2 in Figure 2). For the local moment M 1 , a number of half-waves m along the longitudinal direction is quoted additionally in the brackets. For the lengths of LC-1 under consideration, the values of the critical moment M 1 do not alter significantly (less than 10%). The values of M 3 are at least sixfold higher than M 1 . At the length L ≈ 2050 mm, the value M 1 ≈ M 2 , whereas, at L = 1500 mm, we have M 2 /M 1 = 1.6, and for L = 500 mm it is M 2 /M 1 = 1.16. At the length L = 250 mm, the lowest critical value was attained for the local mode M 1 and for one buckling half-wave (m = 1). Therefore, the critical value M 2 corresponding to the global mode was not given. The value M 3 is almost 14 times higher than M 1 and also occurs for m = 1.
In Figure 3a-d, for the lengths of LC-1 beams considered in Table 2, the buckling modes corresponding to the three modes under analysis, except L = 250 mm, for which only two modes (i.e., mode 1 and mode 3) are considered, are shown.  For the lengths of LC-1 under consideration, the values of the critical moment M1 do not alter significantly (less than 10%). The values of M3 are at least sixfold higher than M1. At the length L ≈ 2050 mm, the value M1 ≈ M2, whereas, at L = 1500 mm, we have M2/M1 = 1.6, and for L = 500 mm it is M2/M1 = 1.16. At the length L = 250 mm, the lowest critical value was attained for the local mode M1 and for one buckling half-wave (m = 1). Therefore, the critical value M2 corresponding to the global mode was not given. The value M3 is almost 14 times higher than M1 and also occurs for m = 1.
In Figure 3a-d, for the lengths of LC-1 beams considered in Table 2, the buckling modes corresponding to the three modes under analysis, except L = 250 mm, for which only two modes (i.e., mode 1 and mode 3) are considered, are shown. In the nonlinear analysis of interactive buckling, the signs of complex absolute values of imperfections of each mode were selected in the safest way, i.e., to attain the lowest value of the limit load carrying capacity M s [11][12][13][14] in (A4). For actual LC-section beams, post-buckling equilibrium paths were determined on the assumption in (A4) that ζ * Table 2 also lists values of the limit load carrying capacity referring to the lowest value of the critical moment M min = M 1 , and accounts for M s1 /M min for a three-mode approach (i.e., J = 3 in (A4)) and M s2 /M min for a two-mode approach (i.e., J = 2). At L = 250 mm, due to the fact that both modes occur for m = 1, it was assumed on the contrary that ζ * 3 = |0.1|. For this length, interaction between buckling modes (denoted by indices r = 1 and r = 3) does not take place within the loading range M/M min under analysis. In Figure 4, on the basis of Equation (A6), a plot of M/M min versus the angle α/α min is presented. Curve 1 corresponds to a one-mode analysis, that is to say, when only the mode J = r = 1 is considered, whereas curve 2 corresponds to a two-mode analysis for J = 2 (for r = 1 and r = 3).   Table  2 and Figure 5), when the interaction of the three modes is taken into account, we have the limit value of Ms1/Mmin, whereas, for an interaction of two modes (i.e., J = 2 for r = 1 and r = 2), the theoretical limit load carrying capacity was not obtained.  (Table 2 and Figure 5), when the interaction of the three modes is taken into account, we have the limit value of M s1 /M min , whereas, for an interaction of two modes (i.e., J = 2 for r = 1 and r = 2), the theoretical limit load carrying capacity was not obtained.  (Table  2 and Figure 5), when the interaction of the three modes is taken into account, we have the limit value of Ms1/Mmin, whereas, for an interaction of two modes (i.e., J = 2 for r = 1 and r = 2), the theoretical limit load carrying capacity was not obtained.  An interaction of buckling modes [11,12] takes place via the coefficients of cubic terms a pqr ζ p ζ q ζ r in the expression for total potential energy (A3). Thus, values of the coefficients a pqr for all lengths L under study were analyzed. For a short beam of L = 250 mm, the terms including the coefficients ζ 2 1 ζ 3 in (A3) are very low and buckling can be treated as uncoupled (i.e., one-mode) for the loads M/M min under consideration. It is also due to very considerable differences in values of critical loads, because M 3 /M 1 ≈ 14. At L = 500 mm, the terms ζ 2 1 ζ 3 , ζ 2 2 ζ 3 decide the interaction, whereas, for L = 1500 mm and L = 2050 mm, these are the terms ζ 2 1 ζ 2 , ζ 2 1 ζ 3 . In Reference [19], local imperfections were taken as in the present work, whereas global imperfections were assumed for selected buckling modes, and their level was close to that assumed here. In Reference [19], for various global imperfections and at L = 2050 mm, M s /M min = 0.864 was attained, and, in this work, M s1 /M min = 0.675 was attained.

Example of LC-2 Beams
The geometrical dimensions of the LC-2 beam are listed in Table 2. Figure 6 shows a change in the critical bending moment M r [MNcm] as a function of the buckling half-wavelength L b in the range 100 ≤ L b ≤ 10,000 mm.

Example of LC-2 Beams
The geometrical dimensions of the LC-2 beam are listed in Table 2. Figure 6 shows a change in the critical bending moment Mr [MNcm] as a function of the buckling half-wavelength Lb in the range 100 ≤ Lb ≤ 10000 mm. Curve 1 corresponds to the lowest critical values of the bending moment, i.e., the primary buckling moments, whereas curve 2 corresponds to the secondary buckling moments. Curve 1 decreases in the range 100 ≤ Lb≤ 400 mm, and then it increases up to the maximal value at Lb = 1500 mm. For higher lengths Lb, the critical moment decreases monotonously. On the other hand, curve 2 grows monotonously for 100 ≤ Lb ≤ 500 mm to attain its maximal value at Lb = 500 mm. At 500 ≤ Lb ≤ 1500 mm, it decreases sharply to grow next in the range 1500 ≤ Lb ≤ 4000 mm, and then the critical values remain constant for Lb ≥ 4000 mm in point of fact.
To sum up, curve 1 attains its local minimum at Lb ≈ 400 mm, curve 2 has its maximum at Lb ≈ 500 mm, curve 1 attains its maximum and curve 2 its minimum at Lb ≈ 1500 mm.
In Table 3 the results for the assumed four total lengths of LC-2 beams are collected. The index notations were the same as in  Curve 1 corresponds to the lowest critical values of the bending moment, i.e., the primary buckling moments, whereas curve 2 corresponds to the secondary buckling moments. Curve 1 decreases in the range 100 ≤ L b ≤ 400 mm, and then it increases up to the maximal value at L b = 1500 mm. For higher lengths L b , the critical moment decreases monotonously. On the other hand, curve 2 grows monotonously for 100 ≤ L b ≤ 500 mm to attain its maximal value at L b = 500 mm. At 500 ≤ L b ≤ 1500 mm, it decreases sharply to grow next in the range 1500 ≤ L b ≤ 4000 mm, and then the critical values remain constant for L b ≥ 4000 mm in point of fact.
To sum up, curve 1 attains its local minimum at L b ≈ 400 mm, curve 2 has its maximum at L b ≈ 500 mm, curve 1 attains its maximum and curve 2 its minimum at L b ≈ 1500 mm.
In Table 3 the results for the assumed four total lengths of LC-2 beams are collected. The index notations were the same as in  In Figure 7a-d, buckling modes for LC-2 are shown. The local buckling mode (mode 1) for the four assumed lengths is the same.  Figure 7a-d, buckling modes for LC-2 are shown. The local buckling mode (mode 1) for the four assumed lengths is the same.  At L = 700 mm and mode 3, the maximal deflection of the web is slightly higher than the displacement of lower corners. For this length, the global mode (curve 2) is identical to the local mode (curve 1), while for L = 2000 mm, mode 2 corresponds to the distortional-lateral buckling mode, as there are no right angles in lower corners. Thus, all buckling modes (curves 1, 2, 3) are distortional modes.
In Table 3, the values of the ratio of the limit load carrying capacity to the minimal critical value for two-(J = 2) and three-(J = 3) mode approaches, M s2 /M min and M s1 /M min , respectively, are given. Like in example 3.1, the same values of imperfections were assumed.
For the lengths L = 250 mm and L = 400 mm, limit values were not attained. For these lengths as for LC-1, it was assumed that ζ * 3 = |0.1|, as m = 1. For the remaining two lengths, the limit load carrying capacity is lower for the three-mode approach than for the two-mode approach, identically as for LC-1.
In the two next figures (Figures 8 and 9), a relationship of M/M min versus the angle α/α min is presented according to formula (A6) for the length L = 250 mm and L = 400 mm.
carrying capacity is lower for the three-mode approach than for the two-mode approach, identically as for LC-1.
In the two next figures (Figures 8, 9), a relationship of M/Mmin versus the angle α/αmin is presented according to formula (A6) for the length L = 250 mm and L = 400 mm.   as for LC-1.
In the two next figures (Figures 8, 9), a relationship of M/Mmin versus the angle α/αmin is presented according to formula (A6) for the length L = 250 mm and L = 400 mm.   Curve 1 corresponds to the case of one-mode buckling (r = J = 1), while curve 2 corresponds to two-mode buckling (J = 2, r = 1, r = 3). Both the curves overlap, which proves a lack of an interaction between the modes in the range of loading under consideration. The dependence of α/α min on M/M min at L = 700 mm, for the two-(J = 2) and three-mode (J = 3) approach, correspondingly, is presented in Figure 10. In the case of J = 3, the limit load carrying capacity is M s1 /M min = 0.867, whereas, for J = 2, M s2 /M min cannot be determined.
In this case, a significant effect of the secondary global mode (r = 3) on the load carrying capacity can be seen. At L = 2000 mm, the quantities M s1 /M min and M s2 /M min differ slightly, i.e., by less than 2%.
For L = 250 mm and L = 400 mm, the nonlinear coefficients (A3) ζ 2 1 ζ 3 , responsible for the interaction of modes, are very low and, moreover, M 3 /M 1 > 6; thus, we encounter one-mode buckling for the loads M/M min under analysis. At L = 700 mm, the terms including the coefficients ζ 2 1 ζ 3 , ζ 2 2 ζ 3 , ζ 2 3 ζ 2 play an important role, whereas, at L = 2000 mm, the terms are ζ 2 1 ζ 2 , ζ 2 1 ζ 3 . In Reference [19], for the length L = 2000 mm, the dimensional load carrying capacity is equal to M s /M min = 0.919, and, in this work, it is M s1 /M min = 0.803. One should remember that the values of global imperfections were assumed differently. At L = 400 mm in Reference [19], the load carrying capacity was not determined either. two-mode buckling (J = 2, r = 1, r = 3). Both the curves overlap, which proves a lack of an interaction between the modes in the range of loading under consideration. The dependence of α/αmin on M/Mmin at L = 700 mm, for the two-(J = 2) and three-mode (J = 3) approach, correspondingly, is presented in Figure 10. In the case of J = 3, the limit load carrying capacity is Ms1/Mmin = 0.867, whereas, for J = 2, Ms2/Mmin cannot be determined. In Reference [19], for the length L = 2000 mm, the dimensional load carrying capacity is equal to Ms/Mmin = 0.919, and, in this work, it is Ms1/Mmin = 0.803. One should remember that the values of global imperfections were assumed differently. At L = 400 mm in Reference [19], the load carrying capacity was not determined either.

Example of LC-3 Beams
Like in earlier examples, detailed geometrical dimensions are listed in Table 2. In Figure 11, alternations in critical bending moments Mr as a function of the buckling half-wavelength Lb are presented.

Example of LC-3 Beams
Like in earlier examples, detailed geometrical dimensions are listed in Table 2. In Figure 11, alternations in critical bending moments M r as a function of the buckling half-wavelength L b are presented. Curve 1 corresponds to the lowest values of the critical moment, while curve 2 corresponds to higher values for 100 ≤ Lb ≤ 10000 mm. The plots of both curves are similar to the plots in Figure 2 (LC-1) and Figure 6 (LC-2). The minimal local value of the moment for curve 1 was attained at Lb ≈ 450 mm, the local maximum was attained at Lb ≈ 1500 mm, whereas curve 2 attains the maximal value of the moment for Lb ≈ 480 mm and the minimal value for Lb ≈ 1500 mm, respectively.
As in former examples, Table 4 lists values of critical loads for 4 selected lengths of beams L. At L = 300 mm, the lowest local mode M1 occurs for m = 1. Thus, mode 2 was not considered. The value M3 (for m = 1) is almost 10-times higher than M1. At L = 800 mm, we have M2/M1 = 1.4, and for L = 2500 mm it is M2/M1 = 1.3, whereas, at L = 4500 mm, the global value M2 is lower than M1, as M2/M1 = 0.5. The secondary value of M3 for the values of L under analysis is at least 7-times higher than M1.
In Figure 12a-d, buckling modes for selected lengths L are presented. Local buckling modes (mode 1) are practically the same for all lengths. Curve 1 corresponds to the lowest values of the critical moment, while curve 2 corresponds to higher values for 100 ≤ L b ≤ 10,000 mm. The plots of both curves are similar to the plots in Figure 2 (LC-1) and Figure 6 (LC-2). The minimal local value of the moment for curve 1 was attained at L b ≈ 450 mm, the local maximum was attained at L b ≈ 1500 mm, whereas curve 2 attains the maximal value of the moment for L b ≈ 480 mm and the minimal value for L b ≈ 1500 mm, respectively.
As in former examples, Table 4 lists values of critical loads for 4 selected lengths of beams L. At L = 300 mm, the lowest local mode M 1 occurs for m = 1. Thus, mode 2 was not considered.  In Figure 12a-d, buckling modes for selected lengths L are presented. Local buckling modes (mode 1) are practically the same for all lengths.
At L = 300 mm and mode 3 (m = 1), maximal deflections occur in the web. At L = 800 mm, also the global mode (mode 2) is identical to mode 1 (Figure 12b). The secondary global mode (mode 3) has the maximal deflection for lower compressed corners of LC-3. Mode 2 (curve 2) for the length L = 2500 mm and L = 4500 mm is a "pure" lateral buckling mode in principle. At L = 2500 mm and mode 3, a slight displacement of the corner connecting the web with the compressed lower flange takes place, whereas, for L = 4500 mm, displacements of both web corners occur.
Moreover, Table 4 also shows the dimensionless limit load carrying capacity for two-and three-mode approaches, M s2 /M min and M s1 /M min , respectively. At L = 2500 mm and L = 4500 mm, differences between both the approaches are inconsiderable.
For the range of loadings under consideration, there is no interaction between the modes. For L = 800 mm, in the case of the three-mode, we have M s1 /M min = 0.833, whereas, for the two-mode approach, there is no limit load carrying capacity ( Figure 14).
For L = 300 mm, the value of the coefficient at the term ζ 2 1 ζ 3 is inconsiderable, but at L = 800 mm, the terms ζ 2 1 ζ 3 , ζ 2 2 ζ 3 play a significant role. At L = 2500 mm and at L = 4500 mm, the coefficients at the terms ζ 2 1 ζ 2 , ζ 2 1 ζ 3 are important. For the length L = 4500 mm in Reference [19], the value of the load carrying capacity was M s /M min = 0.806, whereas, in the present analysis, it was M s1 /M min = 0.77.
For all the examples under analysis in Reference [19], higher values were attained than in the present study. One should note once more that the values of local imperfections in Reference [19] were assumed in a different way than here. Curve 1 corresponds to the lowest values of the critical moment, while curve 2 corresponds to higher values for 100 ≤ Lb ≤ 10000 mm. The plots of both curves are similar to the plots in Figure 2 (LC-1) and Figure 6 (LC-2). The minimal local value of the moment for curve 1 was attained at Lb ≈ 450 mm, the local maximum was attained at Lb ≈ 1500 mm, whereas curve 2 attains the maximal value of the moment for Lb ≈ 480 mm and the minimal value for Lb ≈ 1500 mm, respectively.
As in former examples, Table 4 lists values of critical loads for 4 selected lengths of beams L. At L = 300 mm, the lowest local mode M1 occurs for m = 1. Thus, mode 2 was not considered. The value M3 (for m = 1) is almost 10-times higher than M1. At L = 800 mm, we have M2/M1 = 1.4, and for L = 2500 mm it is M2/M1 = 1.3, whereas, at L = 4500 mm, the global value M2 is lower than M1, as M2/M1 = 0.5. The secondary value of M3 for the values of L under analysis is at least 7-times higher than M1.
In Figure 12a-d, buckling modes for selected lengths L are presented. Local buckling modes (mode 1) are practically the same for all lengths. At L = 300 mm and mode 3 (m = 1), maximal deflections occur in the web. At L = 800 mm, also the global mode (mode 2) is identical to mode 1 (Figure 12b). The secondary global mode (mode 3) has the maximal deflection for lower compressed corners of LC-3. Mode 2 (curve 2) for the length L = 2500 mm and L = 4500 mm is a "pure" lateral buckling mode in principle. At L = 2500 mm and mode 3, a slight displacement of the corner connecting the web with the compressed lower flange takes place, whereas, for L = 4500 mm, displacements of both web corners occur.
Moreover, Table 4 also shows the dimensionless limit load carrying capacity for two-and three-mode approaches, Ms2/Mmin and Ms1/Mmin, respectively. At L = 2500 mm and L = 4500 mm, differences between both the approaches are inconsiderable.  Figure 13, at the length L = 300 mm, curve 1 for the one-mode approach (J = 1) overlaps curve 2 for the two-mode approach (J = 2, r = 1, r = 3). For the range of loadings under consideration, there is no interaction between the modes. For L = 800 mm, in the case of the three-mode, we have Ms1/Mmin = 0.833, whereas, for the two-mode approach, there is no limit load carrying capacity (Figure14). For L = 300 mm, the value of the coefficient at the term 2 ζ ζ is inconsiderable, but at L = 800 For the range of loadings under consideration, there is no interaction between the modes. For L = 800 mm, in the case of the three-mode, we have Ms1/Mmin = 0.833, whereas, for the two-mode approach, there is no limit load carrying capacity (Figure14). For L = 300 mm, the value of the coefficient at the term For the length L = 4500 mm in Reference [19], the value of the load carrying capacity was Ms/Mmin = 0.806, whereas, in the present analysis, it was Ms1/Mmin = 0.77. The plots of variability in critical moments (curves 1 and 2) as a function of the half-wavelength L b shown in Figures 2, 6 and 11 as well as an analysis of buckling modes, the load carrying capacity and the effect of nonlinear coefficients at the first-order approximation terms allow one to classify, according to the conclusions expressed in Reference [13], the following lengths of LC-beams, namely: Compared to Reference [13], the term of very long beams, for which the secondary global mode M 3 is actually constant and the primary global mode M 2 has a low gradient of the value drop in comparison to long beams, is introduced additionally in the above-mentioned classification.
Particular attention was paid to the influence of secondary global distortional-lateral buckling mode on the load carrying capacity for the LC-beams under bending. As demonstrated in the paper, the most significant influence is for medium-long beams. In this case, disregarding the interaction of three modes, including two global (i.e., primary and secondary) and local ones, may lead to an incorrect assessment of the load carrying capacity of the two-mode approach for medium-long beams.

Conclusions
The stability and interactive buckling of steel LC-beams under bending, for three different cross-sections in a wide range of beam length variability, were investigated. Attention was paid in particular to an effect of the secondary global buckling mode on an interaction between modes, including distortional buckling modes. A classification of lengths of beams subjected to bending in the web plane, starting from short to medium-long up to long, or even very long ones, is proposed. In the cases under consideration, the influence of the secondary global mode on the load carrying capacity is most evident for medium-long beams. It is advisable to extend this analysis onto an effect of the length of edge reinforcements and the wall thickness of LC-beams on the interactive buckling and load carrying capacity. It is important to conduct validation of the described and analyzed phenomenon, using the semi-analytical method SAM through the use of more comprehensive modeling using FEM.

Conflicts of Interest:
The authors declare no conflict of interest.