Lateral-torsional Buckling Moment of Simply Supported Unrestrained Monosymmetric Beams

The stability equations for simply supported beams were solved approximately using the Bubnov-Galerkin method. We omit the influence of displacements in the plane of bending. The analytical solutions were used for checking the stability of laterally unrestrained mono-symmetric beams. The lateral-torsional buckling (LTB) moment depends on bending distribution and on the load height effect. Each of applied concentrated and distributed loads, may have arbitrary direction and optional coordinate for the applied force along the cross section’s height. Derived equations allow for simple, yet fast control of lateral buckling moment. Numerical examples show good consistency of analytic results with those obtained from Finite Element Method (FEM) software.


Introduction
The lateral-buckling resistance assessment is usually based on buckling curves and requires the computation of the elastic critical moment. To determine LTB moment of the monosymmetric I-shaped beams the LTBeam [1] programme is commonly used. In this paper, a contribution to the LTB of unrestrained thin-walled elements with monosymmetric open sections is investigated. According to the classical paper [2] an analytical solution is used for checking the lateral stability of beams. It depends on bending distribution, on load height effect and on degree of the monosymmetry of the section called Wagner's parameter. Coefficients C 1 , C 2 and C 3 are respectively affected to these parameters. In presented work simply supported beams is analysed -the most common case in design practice. Non-linear distributions of the bending moment along the element are considered in the C 1 coefficient. It is determined on the basis of the absolute values of the bending moment in 1/4, 1/2, 3/4 of the beam span and its maximum absolute value. There are several C 1 approximation methods based on these parameters in the literature [3,4,5].
Coefficient C 2 take into account of effect of position of load application with respect to height of the element cross-section for arbitrary loads [5,6,7,8]. It was found, in available literature, only two general approaches of solving this problem [5,9]. Formula presented in work [5] is correct only in some cases (Table 4). The second method [9] is the modification of the C 1 coefficient on the basis of numerical simulations. There, the case was considered, when the load was applied at the same height of the cross section. In this publication arbitrary load may be applied at different heights of the cross section. The method presented here is the significant extension existing methods. This is a development of the author's previous work [10].
The elastic LTB moment of mono-symmetric thin-walled beams may be computed using the expression [2,6]:  (1) where: N cr,z =π 2 EI z /L 2 , D = I w /I z + GI T /N cr , z (E -Young's modulus, I z -second moment of area about the z-z axis, L -beam span, I w -warping moment of inertia, G -shear modulus, I T -St. Venant torsion constant).
Coefficient B we calculate from the formula: where: e g -ordinate of the load application position relative to the shear centre M (Figure 2), β z -parameter of asymmetry.
The LTB moment M 0 b is designated from the stability equation as: (3) where: M 0 -max. absolute value of bending moment (M 0 = max│M y (x)│ for 0≤x≤L ), μ b -the smallest absolute value of critical load multiplier. Table 1 shows the coefficients C 1 , C 2 and C 3 for the two basic load schemes. If we use this table, the z axis must coincide with the load direction ( Figure 2) and pay attention to the correct sign of the asymmetry parameter β z (Figure 3). In order to take into account position of the load with respect to height of the cross-section [5] proposed the following relationship: C 2 =0.4 C 1 . As the example in Table 4 shows, this is incorrect for some static schemes. The error in calculating the LTB moment in the case of load applied to the upper flange is important (Table 4). Using the same as [5] approach to determining the coefficient C 1 , based on [10], correct formulas for the coefficient C 2 can be obtained. The results of these analyses are presented in Tables 2 and 3. Table 2. Coefficients C 1 , C 2 for special case In columns 5, 6 Table 3 the LTB moment is presented for beams with the span L = 8 and 12 m together with the relative error of approximation in relation to the results of FEM [1]. The z axis and load direction they are pointing down. The load is applied to the upper flange I-section IPE 500.  Table 3. Coefficients C 1 , C 2 and LTB moment for different values of the coefficient ψ obtained based on the basis of Table 2 (1)  Table 4.
Comparison of values derived from [5,10] [10] [5] A major advantage of some codes, such as the American AISC LRFD [4] is that they provide closed-form expressions to compute the C 1 coefficient for any moment distribution. In this code the C 1 coefficient can be estimated based on the absolute values of bending moments: Equation (4) is some modification of formula shown by [3]. Analogous dependencies can be found in the works [5,10].  (4) and (5), respectively However, in the case of a linear distribution of the bending moment coefficients C 1 , C 3 approximated by the following relations: where: ψ is the ratio of the moments at the ends of the beam (-1 ≤ ψ ≤1, Figure 1b ).

Analytical solution for checking the stability of laterally unrestrained monosymmetric beams 2.1. Differential equation of lateral-torsional buckling of monosymmetric beams
Let's consider the simply supported beam. It is loaded by the moments, which are concentrated at its ends and the lateral load at span (Figure 1). Any distributed and concentrated loads may have an arbitrary coordinate for the force applied at the height of the cross section. The centre of gravity of the cross section is denoted as G and the centre of shear as M. Figure 2 shows the positive support moments for the adopted coordinate system. Considering the boundary conditions for the simply supported beam, the differential equation of the flexural-torsional loss of stability, as the function of the torsion angle φ(x) can be written as [7]: The upper index b refers to the critical state. In this differential equation, φ(x) describes buckling mode angle of the cross-section. The remaining parameters in the critical state are determined from the dependence: where: M y (x) -distribution of the bending moment along the beam, q z (x) -distributed load in the z direction, N -the number of concentrated forces, Q zk -k th concentrated load in the z direction, x k -coordinate of applied k th concentrated load, Δ -Dirac's function. The place where the load is applied at the cross-section height in the case of distributed e A and concentrated e k load is determined in relation to the shear centre M (Figure 2).
The only difference in the differential equation (6) in relation to the work [7] is shown in formula (7c) and refers to the place where the load is applied at the cross-sectional height.
The distribution of the bending moment along the x-axis can be written using formula: where: M L , M R -concentrated moments applied to the left and right end of the beam at the supports respectively ( Figure 2).

The Bubnov-Galerkin's orthogonalization method
The stability equations (6) were solved approximately using the Bubnov-Galerkin method. In case of simply supported beams with free warping, sinusoidal mode is assumed for the torsion angle. The angle is approximated using only the first term of series. After integration, the lateral buckling moment is given by the roots of the quadratic equation, with regard to the absolute of the maximum bending moment M 0 (Figure 2), where: where: a 1 -coefficient describes the influence of distribution of bending moment along the length of an element, a 2 -the influence load height effect while a 3 -degree of the monosymmetry of the crosssection: where: ξ k =x k /L.
Equation (9) can be further converted into: where: Derived from the solution of quadratic equation (13) the value of critical moment is expressed as:  Next, for both types of loads, we'll calculate the coefficient C 2z from equation (15b) and C 3 from equation (15c), respectively. The comparison of LTB moments obtained with equation (16) and FEM [1] program is presented at the Table 4. In both cases the estimation error is less 1 %.

Conclusions
Proposed method allows for determination of critical moment in case, where each load is applied in different point, with respect to height of the cross-section and has any (upward or downward) direction. Shown examples prove that it allows for estimation of critical moment with enough accuracy from practical point of view.
In order to take into account position of the load with respect to height of the cross-section, Trahair proposed the estimation of coefficient C 2 =0.4C 1. It has been shown that this estimation is incorrect.
It should be emphasized that the obtained formulas give a very good LTB moment estimate for bisymmetrical beams. In the case of monosymmetric sections, the errors are also small, but only if the