TORSIONAL BUCKLING OF THIN-WALLED BEAMS IN PRESENCE OF BIMOMENT INDUCED BY AXIAL LOADS

Original scientific paper In this paper the influence of the bimoment induced by external axial loads on the elastic torsional buckling of thin-walled beams with open cross-section is studied. The governing differential equations of a deformed thin-walled beam consistent with Vlasov’s classical assumptions are derived applying the principle of virtual displacements. It is shown that in the cross sections with lack of symmetry the bimoment can considerably affect the torsional buckling load. The obtained results are verified using ANSYS finite element software.


Introduction
Thin-walled beams of open cross-section are widely used in structural engineering due to high bearing strength when compared to self weight.The assessment of elastic stability of thin-walled beams is one of the most important issues in the analysis of thin-walled structures.For hot rolled sections which in the most cases, according to Eurocode 3 [1] belong to classes 1 or 2, the webthickness and flange-thickness rations satisfy the requirements for preventing local buckling and distortion of the cross-section.The global stability of centrically or eccentrically compressed members is in the most cases the result of flexural buckling around minor axis.The analysis of pure torsional buckling may be of interest in sections with low torsional rigidity and/or in the cases when external supports decrease flexural but not torsional buckling lengths.The analysis of these elements is mainly covered by the assumptions of the classical Vlasov's theory [2].In plated girders the longitudinal stiffeners are added to prevent local buckling while distortion is restrained by cross-frames and diaphragms.The cold form steel elements have very small slenderness and they belong to the cross-section class 4, so the consideration of distortional and local buckling effect is of great importance.The finite element method [3] or the finite strip method [4] as well as the models based on one dimensional generalized beam theory (GBT) [5] can be applied for the analysis of these elements.
In this paper, starting from the assumptions of the classical Vlasov's theory, the stability of axially compressed members is studied in the specific case of asymmetric cross-sections when the loading point of the external axial force has a non zero value of warping function.According to our knowledge there is a lack of detailed investigation in this area.Focussing on the pure torsional buckling of a beam with Z cross-section, it is demonstrated that the bimoment produced by axial forces has the effect which is not negligible, so the critical force is increased or decreased with respect to the traditional approach of stability analysis with the force resultant and without bimoment.

Kinematics of deformation
The assumptions of classical (Vlasov's) theory for thin-walled beams with open cross-sections are used: cross-section is assumed to be perfectly rigid in its own plane; shear deformation in the middle surface of each thin-walled plate is neglected; there is no shear deformation of thin-walled plates in the plane (z-n) perpendicular to the middle surface.According to the adopted assumptions in a straight thin-walled beam with longitudinal axis z, the displacement components (u, v, w) of an arbitrary point of a thin-walled beam (Fig. 1), in the directions x, y and z, are given with .
Note that x, y coincide with the principal axes in the plane of the cross-section before deformation.In Eq. (1) u D (z), v D (z) are displacements of the shear centre D, 0 ( ) w z is the translation of the cross-section along z, ϕ(z) is the infinitesimal angle of twist around shear centre, while ω(x,y) is the normalized warping function.Primes denote the total derivative with respect to z.
The components of finite strain tensor referring to the axis z are given with ( ) where r is the position vector of an arbitrary point after deformation δ zα is Kronecker delta symbol, while s and e are the coordinates along the middle line of the cross-section and perpendicular to the middle surface.The virtual strains are obtained from the variations of strain components The virtual displacement along axes x and y are .) ( Note that δϕ denotes the virtual rotation around the initial longitudinal axes.Virtual displacements along the beam z axis are derived taking into account the assumption that there is no shear deformation in the middle surface s-z and in the plane z-e, i.e. for e=0, δe zs =0 and δe zn =0, which yields Finally, from Eq. ( 4) the virtual shear strain δe zs and the virtual strain δe zz along the beam axis are , , , , , where in the second equation the higher order term z z w w , , δ is neglected.The remaining virtual strains components are zero.

Equations of equilibrium
The equilibrium conditions of the deformed element could be derived using the principle of virtual work, applying variation of displacements of infinitesimal element between the cross-sections z and z+dz.The virtual work δW E of external forces per unit length is ( ) where u T ={u, v, w} denotes the displacement vector, The work of internal forces per unit length is In Eq. ( 10) the normal and shear stress are found from deformation, Eq. ( 2), retaining only linear terms (11) where E = Young's modulus and G = shear modulus.
Using the total virtual work δW E +δW I =0 and by neglecting the second powers of the displacement gradients, the following equations of equilibrium according to the second order theory are obtained (12) (12) In Eqs. ( 12) N, M x , M y , M ω denote the axial force, bending moments and bimoment; Q x , Q y , T D are shear forces and torsional moment around shear centre; p x , p y , p z are distributed forces; m x , m y , m D denote distributed moments while m ω is the external distributed bimoment.Neglecting the contribution of shear stresses to the equilibrium in z direction, the underlined term in Eq. (12.1) can be omitted.In addition, it is assumed that in products of static and deformation variables the values are found from the first order theory.The result of this assumption is that the underlined terms in Eqs.(12.4-6) are equal to zero.Finally, considering Eqs.(11) and the definition of sectional forces (see Appendix), the following differential equations are obtained (13) The Eqs. (13.2) to (13.4) define the coupled bending and torsion according to the linearized second order theory.The underlined term in Eq. (13.4) represents the influence of bimoment due to second order effect which is important only in the cross sections with lack of symmetry, i.e. β ω ≠0.In many classical books referring to stability of thin-walled beams (e.g.[6 ÷ 9]) the influence of M ω is ignored.It is recognized by Vlasov [2] and by some authors (e.g.[10 ÷ 12]) and in [13] where the influence of the bimoment on lateral buckling, caused by the transverse force, is considered.In this paper, as we underlined in the introduction, the influence of the bimoment on torisonal stability of axially loaded thinwalled beams is studied.

Stability analysis -axial loading
The stability of thin walled-beam is defined with the homogenous part of Eqs.(13).In the case of a concentrated compressive axial force P with eccentricities e x , e y the thin-walled beam is stressed with N=−P, constant bending moments M x =−Pe x , M y =−Pe y and the variable bimoment M ω (z)=λ(z)M ω0 where M ω0 =−Pω (P) is the bimoment at the end cross-sections, ω (P) is the warping function of the loading point, while λ(z) is the pre-buckling bimoment distribution function.The Eqs. (13.2-4) reduce to (14) Note that in Eqs. ( 14) u D , v D and ϕ denote displacements due to buckling while the pre-buckling deformations are neglected.
In order to obtain closed form solution of the buckling load from Eqs. ( 14), the variable distribution of bimoment is approximated with its average value M ω (z)=λ m M ω0 , where λ m is the mean value of λ(z) in the range 0≤z≤L.For a simply supported beam the end conditions are , and 0 for 0 The boundary conditions (i.e.Eqs. ( 15)), are satisfied with the following buckling displacement field .π sin Following the well-known procedure [6] and substituting Eqs.(17) into Eqs.( 14) yields (for each n =1, 2...) the buckling equation , 0 In the general case of flexural-torsional buckling, the Eq. ( 18) is in the form polynomial of the third order.
If the axial load is applied at the shear centre the pure torsional critical load is * * The second term in the denominator of Eq. ( 20) represents the influence of the bimoment.Depending on the sign of the warping function in the loading point the buckling force is increased or decreased with respect to the critical force obtained by the classical approach [6] when the presence of the bimoment is ignored.

Numerical examples
A straight simply supported thin-walled beam with Z cross-section (Fig. 2a) is analyzed in order to investigate the effect of the bimoment on the elastic torsional buckling.The following elastic constants are used: Young's modulus E=210 GPa and Poisson's ratio ν=0,3.
The geometrical properties of the cross section of the beams, in the succeeding examples, were calculated using the computer program given in the [14].The two load cases are considered: the uniformly distributed axial load p=P/h along the web (load case 1 -LC1) and load case with two concentrated axial forces P/2 (load case 2 -LC2) acting at the flange tips -points A and B (Fig. 2a).In both cases, besides the axial force N=−P the bimoment exist, where at the beam ends M ω0 =−ΣP i ω (P)i .For the LC1 M ω0 =−Pω (D) while for the LC2 M ω0 =−P/2(ω (A) + ω (B) ) =−Pω (A) -see Fig. 2b for the warping function.
The results for LC1 and LC2 are given in Table 1 and in Fig. 3.In Table 1 the span L is varied from 2,0 m to 6,0 m, the web height and flange width are kept constant h=0,30 m and b=0,12 m while in Fig. 3 the following parameters are varied: h=0,12 m; 0,18 m; 0,24 m the span is in the range L=2 m to 6 m, while b=0,12 m.In both cases the thickness is t=0,01 m.The buckling loads for the first torsional modes are calculated and compared with the results found by ANSYS finite element model using thin shell 4-node SHELL181 elements.The distortion of the cross section in the finite element model is suppressed by providing the transverse diaphragms infinitely rigid in the plane of the cross-section and zero out of plane stiffness.As an illustration, the first torsional buckling mode for h=0,30 m, b=0,12m t=0,01m, L=5,0m obtained by ANSYS is presented in Fig. 4.
By the analysis of the results in Tab. 1 and Fig. 3 it can be seen the significant influence of the bimoment on the buckling load.If λ=0 in LC1 the underestimated values of the buckling force are obtained.On the other hand the buckling load for LC2 calculated with the presence of bimoment is considerably decreased from the critical force obtained with λ=0.The comparison with ANSYS values justifies the approach with mean value of bimoment for the global torsional stability analysis.The analysis with the exact distribution of λ is of interest for longer spans and higher values of parameter k.In the next example the thin-walled beam with Z section is subjected to axial load uniformly distributed at the end cross-sections (load case LC0).In this case no bimoment is produced (λ=0).In Fig. 5 this load case is compared with the load case LC1 in order to demonstrate the influence of flange width and span length on the buckling load.Note that influence of the flange width is more pronounced in the presence of bimoment.

Conclusions
The influence of the bimoment on the torsional bifurcation stability of thin-walled beams with open cross-section subjected to axial loads is presented.The governing equations are based on the classical (Vlasov's) theory.The torsional stability of simply supported beam with Z cross-section due to different axial load distributions at beam ends is studied and the analytically calculated critical loads are compared with the results obtained by the finite element model.Obtained results clearly demonstrate the significant influence of the bimoment induced by axial loads on the global elastic torsional critical load.For other cross-sections with lack of symmetry and the compressed beams with other boundary conditions, the contribution of the bimoment to the stability analysis has to be investigated.

Figure 1
Figure 1 Cross-section geometry

σ
T ={t xz , t yz, t zz } is the stress vector with the components resolved along the initial unit basis and the surface forces applied along the middle surface.The components t zx , t zy and t zz are obtained from the true stress components σ zx , σ zy and σ zz by the transformation due to infinitesimal rotations around axes x, y and z