Lateral Torsional Buckling of Steel Beams Elastically Restrained at the Support Nodes

The study shows the results of theoretical investigations into lateral torsional buckling of bisymmetric I-beams elastically restrained against warping and against rotation in the plane of lateral torsional buckling (i.e., against lateral rotation) at the support nodes. The analysis accounted for the whole variation range of node stiffnesses, from complete warping freedom to full restraint, and from complete lateral rotation freedom to full restraint. It was assumed the beams are simply supported against bending about the major axis of the section. To determine the critical moment, the energy method was used. Both the twist angle function and the lateral deflection function of the beam were described using power polynomials with simple physical interpretation. Computer programmes were developed to make numerical and symbolic “computations”. General approximation formulas for the critical moment for lateral torsional buckling were derived. The formulas covered the basic and most frequently found loading diagrams. Detailed computations were performed for different values of the index of fixity against warping and against rotation in the plane of lateral torsional buckling. The critical moments determined using the programmes devised and approximation formulas were compared with the values obtained with LTBeam software (FEM). A very good congruence of results was found.


Introduction
Due to high strength of steel, beams of this material used in structures are characterized by small thicknesses of section walls. Therefore, they are susceptible to various forms of stability loss. One of the basic forms of general stability loss of beams in bending is the lateral torsional buckling. Consequently, in the design of steel beams, lateral torsional buckling should be taken into account, as it can significantly reduce load-bearing capacity and affect the safety of the entire structure.
In such cases, bisymmetrical I-sections are the most commonly used. This happens because, compared with other profiles of similar heights (e.g., C or Z sections), I-sections show considerable stiffness in warping torsion [1], which increases the beam resistance to the lateral torsional buckling. In

Elastic Restraint Against Warping and Against Lateral Rotation at the Support Nodes
The static diagram of the beam elastically restrained against warping and against lateral rotation at the support nodes is shown in Figure 1. The springs in colour symbolically represent the elastic restraint in the support sections, i.e., in red-warping restraint (α ω ), and in blue-lateral rotation restraint (α u ).
Appl. Sci. 2019Sci. , 9, 1944 should be noted that earlier versions of LTBeam, i.e., 1.0.11, contained an error in the inputting of the data on the degree of elastic restraint against warping. The error was eliminated from the latest version of the LTBeamN, namely 1.0.3. With regard to the analysis of lateral torsional buckling with the Abaqus software, it is much more complicated to correctly define spring stiffeners at the supports. It is obvious the best solution would be to model a larger part of structure with the representation of relevant details of connection to the beam of concern. Such models, however, take much longer time to build and make it necessary for the designer to have wide experience in FEM spatial modelling. A design engineer needs a faster tool for estimating the critical moment, even at the expense of lower approximation accuracy. Therefore, LTBeam software and approximation formulas need to be relied on. The simplest solution is obviously to assume the fork support regardless of the node structure, yet such an approach will gradually become outdated. Modern computational models aim at more precise rendering of the actual conditions of the structure operation. The goal is to adopt a more informed approach to the structural reliability of members, not relying on unknown bearing capacity reserves but on objective criteria.
In this study, the authors were concerned with lateral torsional buckling of single-span beams with bisymmetric I-sections. They are elastically restrained against warping and against rotation in the plane of lateral torsional buckling (i.e., against lateral rotation) at the support nodes. In bending with respect to the major axis of section stiffness, simple support conditions are found at the supports. In the analysis of lateral torsional buckling, the energy method [10] was used. The twist angle function and the function of the beam lateral deflection were approximated with appropriately selected power polynomials [34]. Programmes for numerical and symbolic "computations" were developed and approximation formulas were derived to estimate the elastic critical moment for lateral torsional buckling for most frequently found loading diagrams. Detailed computations were made for beams with different values of the index of fixity (against warping κω and against lateral rotation κu) with the assumption of the symmetry of boundary conditions with respect to the beam midspan. The results received were compared with FEM results (LTBeam [40]).

Elastic Restraint Against Warping and Against Lateral Rotation at the Support Nodes
The static diagram of the beam elastically restrained against warping and against lateral rotation at the support nodes is shown in Figure 1. The springs in colour symbolically represent the elastic restraint in the support sections, i.e., in red-warping restraint (αω), and in blue-lateral rotation restraint (αu). The degree of elastic restraint against warping εω [33][34][35] can be determined from formula: The degree of elastic restraint against warping ε ω [33][34][35] can be determined from formula: where: α ω -stiffness of the elastic restraint against warping [33][34][35] acc. formula: where: B-bimoment at the site of beam support, φ-twist angle, dφ dx -warping of the section. The degree of elastic restraint ε ω acc. formula (1), ranges from ε ω = 0 for complete warping freedom to ε ω = ∞ for full prevention of warping.
In this study, the degree of elastic restraint against rotation in the plane of lateral torsional buckling ε u , was derived. It was written in the following form: where: α u -stiffness of the elastic restraint against lateral rotation acc. formula: where: M z -bending moment with respect to the minor axis of the section at the support, u-lateral deflection, du dx -rotation about axis z. The degree of elastic restraint ε u , acc. formula (3), ranges from ε u = 0 for complete rotational freedom to ε u = ∞ for full prevention of rotation in the plane of lateral torsional buckling.
In order to account, in an explicit manner, for the beam elastic restraint, in the twist angle function φ(x) and the lateral deflection function u(x), the dimensionless indexes of fixity against warping κ ω [34] and against lateral rotation κ u were introduced acc. formulas: Indexes of fixity range from κ ω = 0 (κ u = 0) for complete freedom of warping or lateral rotation, respectively, to κ ω = 1 (κ u = 1) for complete blockage of warping or lateral rotation.

Twist Angle Function and Lateral Deflection Function
In study [42], in which lateral torsional buckling of fork-supported beams was analysed, the function of the twist angle of the section was approximated using power polynomials that described "the deflection function" of the hinged beam (Table 1, polynomials W Pi ). In Table 1, formulas for the polynomials of deflection were written in the dimensionless coordinates ρ = x/L. To account for the elastic restraint against warping in the beam support section, in study [34], the twist angle function (φ) was extended. That was done by introducing additional polynomials that describe "the deflection function" of the restrained beam (Table 1, polynomials W Ui ). "Hinged" polynomials (W Pi ) were coupled with "restrained" polynomials (W Ui ) by means of the index of fixity against warping κ ω (5a) [34] acc. formula: where: a i -free parameters of the twist angle function, W Pi , W Ui -polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].
In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: Appl. Sci. 2019, 9, 1944 6 of 18 deflection function" of the restrained beam (Table 1, polynomials WUi). "Hinged" polynomials (WPi) were coupled with "restrained" polynomials (WUi) by means of the index of fixity against warping κω (5a) [34] acc. formula: where: ai-free parameters of the twist angle function, WPi, WUi-polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].
In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: Appl. Sci. 2019, 9, 1944 6 of 18 deflection function" of the restrained beam (Table 1, polynomials WUi). "Hinged" polynomials (WPi) were coupled with "restrained" polynomials (WUi) by means of the index of fixity against warping κω (5a) [34] acc. formula: where: ai-free parameters of the twist angle function, WPi, WUi-polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].

Item
Polynomials Physical Interpretation In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: Appl. Sci. 2019, 9, 1944 6 of 18 deflection function" of the restrained beam (Table 1, polynomials WUi). "Hinged" polynomials (WPi) were coupled with "restrained" polynomials (WUi) by means of the index of fixity against warping κω (5a) [34] acc. formula: where: ai-free parameters of the twist angle function, WPi, WUi-polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].

Item
Polynomials Physical Interpretation In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: Appl. Sci. 2019, 9, 1944 6 of 18 deflection function" of the restrained beam (Table 1, polynomials WUi). "Hinged" polynomials (WPi) were coupled with "restrained" polynomials (WUi) by means of the index of fixity against warping κω (5a) [34] acc. formula: where: ai-free parameters of the twist angle function, WPi, WUi-polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].

Item
Polynomials Physical Interpretation In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: Appl. Sci. 2019, 9, 1944 6 of 18 deflection function" of the restrained beam (Table 1, polynomials WUi). "Hinged" polynomials (WPi) were coupled with "restrained" polynomials (WUi) by means of the index of fixity against warping κω (5a) [34] acc. formula: where: ai-free parameters of the twist angle function, WPi, WUi-polynomials acc. Table 1. Table 1. Polynomials used and their physical interpretation (where ρ = x/L) [34,43].

Item
Polynomials Physical Interpretation The polynomials used (Table 1) satisfy the boundary conditions of the twist angle function for the fork support WPi (ϕ = 0, ϕ" = 0 for x = 0 and x = L), and for full restraint WUi (ϕ = 0, ϕ' = 0 for x = 0 and x = L), respectively.
In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (WPi) and "restrained" polynomials (WUi) were "coupled" by means of the index of fixity κu (5b) acc. formula: where: bi-free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support WPi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint WUi (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κω acc. (5a) and κu acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (Mcr) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κω, κu) at the support nodes. The total potential energy of the beam-load system was determined from formula: The polynomials used (Table 1) satisfy the boundary conditions of the twist angle function for the fork support W Pi (φ = 0, φ" = 0 for x = 0 and x = L), and for full restraint W Ui (φ = 0, φ' = 0 for x = 0 and x = L), respectively.
In this study, to account for the elastic restraint of the beam against lateral rotation at the support nodes, the lateral deflection function (u) was written in a way analogous to (8). Thus "hinged" polynomials (W Pi ) and "restrained" polynomials (W Ui ) were "coupled" by means of the index of fixity κ u (5b) acc. formula: where: b i -free parameters of the lateral deflection function. The polynomials used (Table 1) satisfy the boundary conditions of the lateral deflection function for the fork support W Pi (u = 0, u" = 0 for x = 0 and x = L) and for full restraint W Ui (u = 0, u' = 0 for x = 0 and x = L).
Functions (8) and (9) make it possible to model boundary conditions at the nodes for the elastic restraint against warping and against lateral rotation for arbitrary (i.e., from 0 to 1 interval) values of the indexes of fixity κ ω acc. (5a) and κ u acc. (5b).

The Critical Moment for Lateral Torsional Buckling
The energy method was used to determine the critical moment (M cr ) of a single-span beam with bisymmetric I-section while taking into account elastic restraints (κ ω , κ u ) at the support nodes. The total potential energy of the beam-load system was determined from formula: where: U s,1 -elastic strain energy of beam bending and torsion; U s,2 -energy of the elastic restraint against warping; U s,3 -energy of the elastic restraint against lateral rotation; T-work done by external forces. The elastic strain energy of the beam bending and torsion [10] was expressed with the equation: The energy of the elastic restraint against warping (U s,2 ) [34] and against lateral rotation (U s,3 ) was determined from formulas: The work done by external forces is a function of the loading diagram and the co-ordinate of the point of load application. For instance, for a simply supported beam ( Figure 2), loaded with linearly varied distribution of transverse load, for an arbitrary co-ordinate (z g ) of the point of load application over the section height, the work done by external forces can be written as follows: Appl. Sci. 2019, 9, 1944 7 of 18 where: Us,1-elastic strain energy of beam bending and torsion; Us,2-energy of the elastic restraint against warping; Us,3-energy of the elastic restraint against lateral rotation; T-work done by external forces. The elastic strain energy of the beam bending and torsion [10] was expressed with the equation: The energy of the elastic restraint against warping (Us,2) [34] and against lateral rotation (Us,3) was determined from formulas: The work done by external forces is a function of the loading diagram and the co-ordinate of the point of load application. For instance, for a simply supported beam (Figure 2), loaded with linearly varied distribution of transverse load, for an arbitrary co-ordinate (zg) of the point of load application over the section height, the work done by external forces can be written as follows: To determine Mcr, a programme for numerical computations, namely McrLT_elastic_warping_rotation_2.nb (MLTB,EL,2 for short) was developed in the environment of the Mathematica® package. The programme allows the determination of the critical load as a function of the indexes of fixity (κω, κu), for arbitrary geometric parameters of the bisymmetric I-section, an arbitrary value of the co-ordinate (zg) of the load application point (see Figure 2), for beam loading diagrams that are most commonly found in practice ( Table 2; Table 3). In the programme, the first three terms (a1,2,3) of the twist angle function acc. (8) and the first three terms (b1,2,3) of the lateral deflection function acc. (9) were employed.
In study [34], McrLT_elastic_fix.on.warp._sym.cal.nb programme was developed to make symbolic "computations" for those cases, in which elastic restraint against warping (κω) occurs. In order to receive possibly simple approximation formulas, only the first term {a1((1 − κω)WP1 + κωWU1)} of the beam twist angle function (8) was employed. Still, a very good congruence between the results thus obtained and those produced by FEM (LTBeam, Abaqus) was noted.
In this study, McrLT_elastic_warping_rotation_sym.cal.nb programme was formulated, in analogous terms, in the environment of the Mathematica® package. The programme is geared towards symbolic "computations", and it accounts for both elastic restraint against warping (κω) and against rotation in the plane of lateral torsional buckling (κu). In this case, for the loading diagrams shown in Tables 2 and 3, the first term of the twist angle function (8) and the first or the second term {bi((1 − κu)WPi + κuWUi)}i = 1 or 2 of the lateral deflection function (9) were employed. To determine M cr , a programme for numerical computations, namely McrLT_elastic_warping_rotation_2.nb (M LTB,EL,2 for short) was developed in the environment of the Mathematica ® package. The programme allows the determination of the critical load as a function of the indexes of fixity (κ ω , κ u ), for arbitrary geometric parameters of the bisymmetric I-section, an arbitrary value of the co-ordinate (z g ) of the load application point (see Figure 2), for beam loading diagrams that are most commonly found in practice ( Table 2; Table 3). In the programme, the first three terms (a 1,2,3 ) of the twist angle function acc. (8) and the first three terms (b 1,2,3 ) of the lateral deflection function acc. (9) were employed.
In study [34], McrLT_elastic_fix.on.warp._sym.cal.nb programme was developed to make symbolic "computations" for those cases, in which elastic restraint against warping (κ ω ) occurs. In order to receive possibly simple approximation formulas, only the first term {a 1 ((1 − κ ω )W P1 + κ ω W U1 )} of the beam twist angle function (8) was employed. Still, a very good congruence between the results thus obtained and those produced by FEM (LTBeam, Abaqus) was noted.
In this study, McrLT_elastic_warping_rotation_sym.cal.nb programme was formulated, in analogous terms, in the environment of the Mathematica ® package. The programme is geared towards symbolic "computations", and it accounts for both elastic restraint against warping (κ ω ) and against rotation in the plane of lateral torsional buckling (κ u ). In this case, for the loading diagrams shown in Tables 2  and 3, the first term of the twist angle function (8) and the first or the second term {b i ((1 − κ u )W Pi + κ u W Ui )} i = 1 or 2 of the lateral deflection function (9) were employed. to centre of the section sheer (see Figure 2), has the following form: where: B1, B2, B3, B4 and D1-coefficients acc. Table 2.
For loads applied at the section shear centre (zg = 0), the formula for the critical moment is reduced to the following form: where: B1, B2, B3, B4 and D1-coefficients acc. Table 2.   For loads applied at the section shear centre (zg = 0), the formula for the critical moment is reduced to the following form:   For loads applied at the section shear centre (zg = 0), the formula for the critical moment is reduced to the following form: Table 3. Coefficients C 1 , C 2 , C 3 and D 1 for the beam loaded with moments concentrated at supports.

Item Load Diagram Coefficients
As regards the beam loaded with moments concentrated at the ends (for −1 ≤ ψ ≤ 1, Table 3), the formula for the critical moment for lateral torsional buckling, which accounts for arbitrary (0 ÷ 1) values of the indexes of fixity (κω, κu), has the following form: where: C1, C2, C3 and D1-coefficients acc. Table 3. Table 3. Coefficients C1, C2, C3 and D1 for the beam loaded with moments concentrated at supports.    (16), derived for concentrated moments from the interval −0.5 < ψ ≤ 1 (see Table 3), in which symmetric or slightly asymmetric (with respect to the beam midspan) lateral torsional

Item Load Diagram Coefficients
κ ω + β 6 κ 6 ω ·e 3ψ auxiliary coefficients: As regards the beam loaded with moments concentrated at the ends (for −1 ≤ ψ ≤ 1, Table 3), the formula for the critical moment for lateral torsional buckling, which accounts for arbitrary (0 ÷ 1) values of the indexes of fixity (κω, κu), has the following form: where: C1, C2, C3 and D1-coefficients acc. Table 3.    (16), derived for concentrated moments from the interval −0.5 < ψ ≤ 1 (see Table 3), in which symmetric or slightly asymmetric (with respect to the beam midspan) lateral torsional buckling mode occurs, was approximated with the first term of the twist angle function (8) and the first term of the lateral deflection function (9). However, for the interval −1 ≤ ψ ≤ −0.5, where much more asymmetric mode of lateral torsional buckling is found, the best results were obtained for the ω ·e −2,5ψ auxiliary coefficients: The formula for the critical moment for lateral torsional buckling, which addresses the indexes of fixity (κ ω , κ u ) and an arbitrary ordinate (z g ) of the point of transverse load application with respect to centre of the section sheer (see Figure 2), has the following form: where: B 1 , B 2 , B 3 , B 4 and D 1 -coefficients acc. Table 2.
For loads applied at the section shear centre (z g = 0), the formula for the critical moment is reduced to the following form: As regards the beam loaded with moments concentrated at the ends (for −1 ≤ ψ ≤ 1, Table 3), the formula for the critical moment for lateral torsional buckling, which accounts for arbitrary (0 ÷ 1) values of the indexes of fixity (κ ω , κ u ), has the following form: where: C 1 , C 2 , C 3 and D 1 -coefficients acc. Table 3.
Formula (16), derived for concentrated moments from the interval −0.5 < ψ ≤ 1 (see Table 3), in which symmetric or slightly asymmetric (with respect to the beam midspan) lateral torsional buckling mode occurs, was approximated with the first term of the twist angle function (8) and the first term of the lateral deflection function (9). However, for the interval −1 ≤ ψ ≤ −0.5, where much more asymmetric mode of lateral torsional buckling is found, the best results were obtained for the first term of the twist angle function (8) and the second term of the lateral deflection function (9).
The design of approximation formulas makes it possible to develop relatively simple spreadsheets.

FEM Verification
To verify the results of numerical calculations performed acc. M LTB,EL,2 programme and the results of analytical calculations made with approximation Formulas (14), (15) and (16), LTBeam software (FEM) [40] was used. The software allows the adoption of the classic boundary conditions i.e., fork support or complete fixity. Also, it accounts for the beam elastic restraint against the section warping and against rotation in the plane of lateral torsional buckling. As mentioned already, the LTBeam software version 1.0.11 contains an error in the units of the coefficient of the elastic restraint against warping (α ω ). The error results in the lowering of the actual value of α ω when it is given in the commonly used unit [kNcm 3 /rad]. The drawback was eliminated in the LTBeamN latest version, i.e., 1.0.3.
For the sake of comparison, in checking computations, predetermined values of the indexes of fixity κ ω and κ u were assumed. In this case, the stiffness of the elastic restraint against warping (α ω ) and the stiffness of the elastic restraint against lateral rotation (α u ), which are necessary to make LTBeam computations were determined from Formula (6ab). Figure 3 shows the form of lateral torsional buckling of the beam (Figure 3b) for the critical moment determined with the LTBeam programme.
An exemplary IPE300 beam, with span L = 5 m, was loaded with a concentrated force applied at the midspan to the upper flange of the beam (z g = +h/2). The elastic restraint against warping (κ ω = 0.75) and lateral rotation (κ u = 0.5) of the beam at support nodes were taken into account. The critical moment of the lateral torsional buckling of beam was obtained, having the value of M cr = 163.71 kNm (Figure 3a). The critical moments for beams analysed in this paper, for different types of sections, spans and load diagrams, and for different degrees of elastic restraint against warping and against lateral rotation, were determined in the way presented above. The results are discussed in Section 6.
For the sake of comparison, in checking computations, predetermined values of the indexes of fixity κω and κu were assumed. In this case, the stiffness of the elastic restraint against warping (αω) and the stiffness of the elastic restraint against lateral rotation (αu), which are necessary to make LTBeam computations were determined from Formula (6ab). Figure 3 shows the form of lateral torsional buckling of the beam (Figure 3b) for the critical moment determined with the LTBeam programme. An exemplary IPE300 beam, with span L = 5m, was loaded with a concentrated force applied at the midspan to the upper flange of the beam (zg = +h/2). The elastic restraint against warping (κω = 0.75) and lateral rotation (κu = 0.5) of the beam at support nodes were taken into account. The critical moment of the lateral torsional buckling of beam was obtained, having the value of Mcr = 163.71 kNm (Figure 3a). The critical moments for beams analysed in this paper, for different types of sections, spans and load diagrams, and for different degrees of elastic restraint against warping and

Examples
To make a comparative analysis, steel beams (E = 210GPa, G = 81GPa) made from IPE300, HEA300, HEB300 sections with a span of L = 5 and 7 m, and beams fabricated from IPE500, HEA500, HEB500 sections with a span of L = 8 and 10 m were assumed. In computations, the loads were as those in the diagrams shown in Tables 2 and 3. For the diagrams in Table 2, transverse loads were applied to the top flange (z g = +h/2), to the section weight axis (z g = 0) and to the bottom flange (z g = −h/2). When the loads were moments concentrated at supports (Table 3), the whole range of variation of the ratio of the moments (−1 ≤ ψ ≤ 1) was taken into account for the following parameter values ψ = {−1; −0.75; −0.5; −0.25; 0; 0.25; 0.5; 0.75; 1}. Analyses were carried out for the full range of variation of the index of fixity against warping κ ω (from 0 to 1) and against lateral rotation κ u (from 0 to 1) for the following values of κ i = {0; 0.25; 0.5; 0.75; 0.9; 1}. Computations were run for various combinations of the values of indexes κ ω and κ u . For each of the beams analysed, the critical moment for lateral torsional buckling was determined acc. M LTB,EL,2 programme using 3 terms of both series (8) and (9). The critical moment was estimated with Formulas (14), (15) and (16), and then compared with the values obtained from FEM (LTBeam). As the number of received values of the critical moment of beams was large, the paper reports only selected results of detailed cases (Tables 4 and 5, Figure 4; Figure 5). Cumulative analyses of results (for all cases included in the paper) are presented in Table 6. for lateral torsional buckling was determined acc. MLTB,EL,2 programme using 3 terms of both series (8) and (9). The critical moment was estimated with Formulas (14), (15) and (16), and then compared with the values obtained from FEM (LTBeam). As the number of received values of the critical moment of beams was large, the paper reports only selected results of detailed cases (Tables 4 and 5, Figure 4; Figure 5). Cumulative analyses of results (for all cases included in the paper) are presented in Table 6. Table 4 lists exemplary results of calculations obtained for IPE300 beam, with a span of L = 5 m, loaded with a concentrated force applied to the upper flange (zg = +h/2) at the midspan. Percentage differences in the results obtained with MLTB,EL,2 programme (Column 6) relative to the LTBeam programme (Column 5) are shown in Column 7. Analogous comparison of the results obtained with Formula (14) (Column 8) and the LTBeam programme (Column 5) can be seen in Column 9. When compared with LTBeam, the critical moments determined using MLTB,EL,2 programme showed the differences of +0.6 to +1.9% (Table 4). The application of the approximation Formula (14) produced the values that differed from −3.4 to −0.5% in comparison with FEM. Table 5 lists exemplary results of computations for IPE300 beam, L = 5 m, and selected values of indexes κω and κu, and loading diagrams acc. Tables 2 and 3. The percentage differences in the results obtained with the MLTB,EL,2 programme (Column 7) relative to the LTBeam programme (Column 6) are shown in Column 8. Analogous comparison of the results obtained from the Formulas (14) and (16) (Column 9) and the LTBeam programme (Column 6) is given in Column 10. When compared with LTBeam, the critical moments determined using MLTB,EL,2 programme showed the differences of +0.6 to +1.9% (Table 4). The application of the approximation Formula (14) produced the values that differed from −3.4 to −0.5% in comparison with FEM. Table 5 lists exemplary results of computations for IPE300 beam, L = 5 m, and selected values of indexes κω and κu, and loading diagrams acc. Tables 2 and 3. The percentage differences in the results obtained with the MLTB,EL,2 programme (Column 7) relative to the LTBeam programme (Column 6) are shown in Column 8. Analogous comparison of the results obtained from the Formulas (14) and (16) (Column 9) and the LTBeam programme (Column 6) is given in Column 10.   is strongly non-linear throughout the whole range of the κω restraint index (from 0 to 1). Figure 5 shows the courses of variation of critical moments of lateral torsional buckling of the beam, for geometric parameters according to Figure 4, depending on the value of the index of fixity  In the case ( Figure 4; Figure 5) warping and lateral rotation are fully restrained at supports (i.e., κω = κu = 1) of the IPE300 beam with the span L = 5 m, nearly +120% (Figures 4a and 5a) and +159% (Figures 4b and 5b) increase in the critical moment Mcr was found compared with the conditions of full freedom of warping and lateral rotation (i.e., κω = κu = 0), which correspond to fork support.

Conclusions
A natural trend in the development of modern design methods is to account for the factors that influence the structure bearing capacity and reliability.   Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using LTBeam for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500. In addition to the data shown in Table 6, it should be noted that for Scheme 1, the values received with MLTB,EL,2 programme differed from +0.5 to +2.6% (HEB300, L = 5 m), and the results obtained acc. Formula (14) showed differences of from −3.8 (IPE500, L = 10 m) to +4.1% (HEA300, L = 5 m) compared with LTBeam. As regards Scheme 2, MLTB,EL,2 program gave critical moments that   Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using LTBeam for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500. In addition to the data shown in Table 6, it should be noted that for Scheme 1, the values received with MLTB,EL,2 programme differed from +0.5 to +2.6% (HEB300, L = 5 m), and the results obtained acc. Formula (14) showed differences of from −3.8 (IPE500, L = 10 m) to +4.1% (HEA300, L = 5 m) compared with LTBeam. As regards Scheme 2, MLTB,EL,2 program gave critical moments that   Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using LTBeam for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500. In addition to the data shown in Table 6, it should be noted that for Scheme 1, the values received with MLTB,EL,2 programme differed from +0.5 to +2.6% (HEB300, L = 5 m), and the results obtained acc. Formula (14) showed differences of from −3.8 (IPE500, L = 10 m) to +4.1% (HEA300, L = 5 m) compared with LTBeam. As regards Scheme 2, MLTB,EL,2 program gave critical moments that   Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using LTBeam for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500. In addition to the data shown in Table 6, it should be noted that for Scheme 1, the values received with MLTB,EL,2 programme differed from +0.5 to +2.6% (HEB300, L = 5 m), and the results obtained acc. Formula (14) showed differences of from −3.8 (IPE500, L = 10 m) to +4.1% (HEA300, L = 5 m) compared with LTBeam. As regards Scheme 2, MLTB,EL,2 program gave critical moments that  Table 4 lists exemplary results of calculations obtained for IPE300 beam, with a span of L = 5 m, loaded with a concentrated force applied to the upper flange (z g = +h/2) at the midspan. Percentage differences in the results obtained with M LTB,EL,2 programme (Column 6) relative to the LTBeam programme (Column 5) are shown in Column 7. Analogous comparison of the results obtained with Formula (14) (Column 8) and the LTBeam programme (Column 5) can be seen in Column 9.
When compared with LTBeam, the critical moments determined using M LTB,EL,2 programme showed the differences of +0.6 to +1.9% (Table 4). The application of the approximation Formula (14) produced the values that differed from −3.4 to −0.5% in comparison with FEM. Table 5 lists exemplary results of computations for IPE300 beam, L = 5 m, and selected values of indexes κ ω and κ u , and loading diagrams acc. Tables 2 and 3. The percentage differences in the results obtained with the M LTB,EL,2 programme (Column 7) relative to the LTBeam programme (Column 6) are shown in Column 8. Analogous comparison of the results obtained from the Formulas (14) and (16) (Column 9) and the LTBeam programme (Column 6) is given in Column 10.
In addition to the comparison of the values of M cr , the results compiled in Tables 4 and 5 can be employed in the tests on the correctness of the design of Formulas (14) and (16) in spreadsheets. Table 6 lists the maximum percentage differences between the results obtained by the authors and those produced using LTBeam for beams IPE300, HEA300, HEB300, IPE500, HEA500 and HEB500.
The comparison of the critical moments of lateral torsional buckling (Figure 4), obtained for the full warping restraint (κ ω = 1) in relation to its full freedom (κ ω = 0), shows +71% ( Figure 4a) and +81% (Figure 4b) increase in M cr , basically regardless of the value of the κ u index. The dependence M cr (κ ω ) is strongly non-linear throughout the whole range of the κ ω restraint index (from 0 to 1). Figure 5 shows the courses of variation of critical moments of lateral torsional buckling of the beam, for geometric parameters according to Figure 4, depending on the value of the index of fixity against lateral rotation κ u (from 0 to 1) for selected values of the index of fixity against warping κ ω = {0; 0.25; 0.5; 0.75; 0.9; 1}.
The comparison of the critical moments of lateral torsional buckling ( Figure 5), obtained for the full lateral rotation restraint (κ u = 1) in relation to its full freedom (κ u = 0), shows +29% ( Figure 5a) and +43% (Figure 5b) increase in M cr , basically regardless of the value of the κ ω index. In this case, the dependence M cr (κ u ) is mildly non-linear throughout the entire range of the κ u restraint index (from 0 to 1).
In the case (Figure 4; Figure 5) warping and lateral rotation are fully restrained at supports (i.e., κ ω = κ u = 1) of the IPE300 beam with the span L = 5 m, nearly +120% (Figures 4a and 5a) and +159% (Figures 4b and 5b) increase in the critical moment M cr was found compared with the conditions of full freedom of warping and lateral rotation (i.e., κ ω = κ u = 0), which correspond to fork support.

Conclusions
A natural trend in the development of modern design methods is to account for the factors that influence the structure bearing capacity and reliability.
When actual conditions of beam support at the nodes are well represented, critical moments can be computed more accurately. Consequently, the coefficient of lateral torsional buckling and the design resistance of the beam can also be calculated more precisely. Such an approach allows taking more informed decision regarding the structural reliability of members. Intuitive estimation of the bearing capacity reserves is substituted with objective criteria.
The comparison of the critical moments (Tables 4-6), which were determined using M LTB,EL,2 programme and estimated from Formulas (14), (15) and (16), with the values obtained from LTBeam revealed a very good congruence of the results. The critical loads were computed for: (1) different variants in the selection of the indexes of fixity (κ ω , κ u ) which changed in the interval from 0 to 1; (2) various (characteristic) points at which transverse loads were applied (top flange, weight axis of the section and bottom flange); and (3) full range of variation in the ratio of the moments concentrated at the supports (−1 ≤ ψ ≤ 1).
The results obtained by the authors indicate that the estimations of the critical moments produced with the Formulas (14), (15) and (16) derived in the study, give approximations that are sufficient from the engineering standpoint. If the formulas mentioned above are written in the spreadsheet, it is necessary to compare the results obtained with Tables 4 and 5.
With an increase in the indexes of fixity (κ ω , κ u ), the value of the critical load of the beams grows. The critical moment for lateral torsional buckling is affected, to a greater extent, by the restraint of the support sections of the beam against warping.
Finally, in order to ensure the recommended level of structural reliability already at the design stage, it should be recommended to check computer calculations with the use of an available analytical method.
Author Contributions: Introduction was prepared by R.P. and A.S. Mathematical description was written by R.P. and A.S. Results were obtained by R.P. The analysis of the results and conclusions were written by R.P. and A.S.

Funding:
Project financed under the programme of the Minister of Science and Higher Education under the name "Regional Initiative of Excellence" in the years 2019-2022 project number 025/RID/2018/19 amount of financing 12 000 000 PLN.

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