Elastic buckling of thin-walled beam-columns based on a refined energy formulation

Modern Trends in Research on Steel, Aluminium and Composite Structures – Gizejowski et al (Eds) © 2021 Copyright the Author(s), ISBN 978-0-367-67637-7 ▪ Elastic buckling of thin-walled beam-columns based on a refined energy formulation M.A. Giżejowski & A.M. Barszcz Department of Concrete and Metal Structures, Warsaw University of Technology, Warsaw, Poland J. Uziak Faculty of Engineering and Technology, University of Botswana, Gaborone, Botswana ABSTRACT: The paper discusses the effects of both in-plane displacements and second order P–δ bending on the elastic flexural-torsional buckling of beam-columns. An energy based solution of the elastic flexural-torsional buckling limit curves under arbitrary proportion between the major axis bending moment and the axial force is presented. The novelty of the approach is related to the development of an improved closed-form solution, in which the equivalent uniform moment modification factor should vary not only with the minor axis buckling force utilization ratio N/Nz but also with that of major axis buckling N/Ny represented by the factored ratio N/Nz(1-k1). Investigations include the effect of in-plane displacements resulting from an arbitrary moment gradient on the elastic flexural–torsional buckling of thin-walled narrow flange and wide flange double-tee section members. The obtained solution is illustrated by elastic flexural-torsional buckling curves for different values of the factor k1 of a beam-column subjected to unequal end moments. 1 INTRODUCTION The elastic flexural-torsional instability of bisymmetric open section members belongs to the classical buckling problem investigated with the use of the Vlasov theory of thin-walled members. Many analytical closed-form and approximate solutions have been obtained for unrestrained beam-columns under major axis bending and presented in scientific papers published worldwide and summarized in the textbooks, e.g. Trahair (1993). The investigations expose a strong interaction between the column buckling and the beam lateral buckling, in which the buckling mode is the combination of flexural or torsional ones in relation to compression and lateral-torsional mode with respect to the major axis bending. The classical stability theory of thin-walled members based on the Vlasov theory uses the assumption of small, in-plane, bending displacements allowing to ignore the effect of prebuckling deflections on the flexural-torsional critical state. It has to be noted that the Linear Eigenvalue Analysis (LEA) equations are quite conservative for wide flange sections, leading to uneconomical design of beams and beam-columns made of wide flange rolled steel or equivalent welded H-sections. Conservatism is more pronounced for double-tee sections, in which case the minor axis moment of inertia is closer to that of the major axis. For rather imaginary situation, when the moments of inertia are equal, the flexural-torsional mode of buckling is proved to not affect the ultimate state of beam and beam-column members. This fact has been well accepted in the literature for a couple of decades, when the closed-form solution, for the case of double-tee bisymmetric beams under uniform moment of My, by including the second order prebuckling effects on the lateral-torsional buckling was presented. The buckling state of fork-supported beam-columns of such a section, for uniform bending, was presented by Trahair et al. (2008): DOI: 10.1201/9781003132134-19 171 where, the correction factors to be applied for the classical LEA solution are as follows: – correction factor representing the second order effect of minor axis flexure: – correction factor representing the second order effect of torsion: where, Mcr,0 = critical moment in uniform bending My,max; Ny, Nya = major axis flexural buckling lowest and the second lowest bifurcation forces; Nz, Nza = minor axis flexural buckling lowest and the second lowest bifurcation forces; NT = torsional buckling lowest bifurcation force; E = Young modulus; G = Kirchhoff modulus; Iy = major axis moment of inertia; Iz = minor axis moment of inertia; Iw = section warping constant; IT = section torsion constant; i0 = polar radius of gyration. Investigations for more accurate solutions have been carried out in the last decades for the critical state predictions of wide flange double-tee section thin-walled members. The mode of flexural-torsional buckling was extensively studied by Mohri et al. (2008) in relation to the bisymmetric double-tee sections. Because of existing coupling between the minor axis displacement and twist rotation, as well as their derivatives, the derivation of strain components for stability analysis needs to be thoroughly investigated in order to account for all the important factors affecting the buckling state formulation based on the thin-walled member theory. The classical energy equation is no longer valid when the effect of prebuckling displacements has an important effect on the buckling state. To facilitate the formulation presented in this paper, a certain level of approximation, aligned with the development of energy equation, is maintained, so that the refined classical energy formulation from this study yields a required level of accuracy in combination with the simplicity of energy equation. This means that in the proposed refined energy equation only the effect described by k1 factor in Eq. (1) is maintained while that concerned with k2 factor is neglected. In this paper, a derivation of the non-classical energy equation in relation to bisymmetric I-sections is discussed first, then its application for the development of a general closed-form solution for the flexural-torsional buckling of fork supported beam-columns under an arbitrary loading condition is presented. The formulation developed proved that the energy based solution of present study allows for sufficiently accurate predictions of the critical state of wide flange double-tee section beam-columns. It is shown that the moment modification factor is a function of not only N/Nz, like in LEA formulation, but also of N/Ny. To the authors’ best knowledge, this is the first study that shows, in a general form, the effect of combined flexural in-plane and out-of-plane critical force utilization ratios on the equivalent uniform moment modification factor of the flexural-torsional buckling state. 2 FIELD DISPLACEMENTS The accurate rotation matrix R was studied by Pi & Trahair (1994) and then by Pi & Bradford 2001). Two parallel Cartesian coordinate systems (x, y, z) are chosen, one fixed in space and the other attached to the deformed elemental length Δx of the member. The basis vectors in 172 the initial state of the axis system are (px, py, pz) while these vectors become (qx, qy, qz) in the deformed state. The rotation θ is a compound rotation from the basis vectors (px, py, pz) to the basis vectors (qx, qy, qz), and represented by a rotation matrix R. In Pi & Bradford investigations, the vector algebra was used to obtain the rotation matrix R. The expressions for the components of the rotation matrix R are related to the choice of the technique for representing the sequence of rotations, so that the resultant rotation matrix components may not be easy interpreted in view of the explicit form of the displacement field components in the deformed state in reference to the initial state. In the present study, the starting point is to formulate the rotation matrix for the deflected configuration, assuming small rotations in that configuration. This allows to write a general matrix relationship for the displacement field in the deflected configuration as: where, ω = sectional warping coordinate; κx ðxÞ = twist along the axis indicated by the subscript symbol; u; v; w = displacements along the axes x, y, z; f ð. . .Þ = variable f being a function of selected arguments listed in the round bracket that indicate the coordinates of adopted Cartesian system (in the following, the arguments are dropped for the convenience of notation), and: df where, the symbol f 0 ¼ dx and the coordinate system is the same as used in Barszcz et al. (2021). The rotation matrix R defined in the member deflected configuration with regard to the fixed Cartesian coordinate system of the initial configuration takes the form: where, ¢i (i=x, y, z) = angles of rotation in the deflected configuration with reference to the initial configuration; is the fiber length measured along the x-axis in the deflected configuration; is the normal strain measure of the section fiber including the bowing effect of ; u0 = displacement along the member axis. The vector of rotation angles in the deflected configuration may be related to those in the initial configuration through the cosines direction matrix T Rθ as follows: where, the direction cosines matrix: The angle θx is the twist rotation and the flexural rotations θy , θz , are given by: 173 where, v0 ; w0 = displacements of the member axis along y and z coordinates, respectively. After decomposing the square rotation matrix in Eq. (6) into two components, namely into the vector corresponding to dx and the rectangular rotation matrix component Ryz corresponding to the section coordinates (y, z), the displacement field may be expressed as follows: in which, the framed variables are the terms of direction cosines matrix. The rotation submatrix R takes the form: yz 3 VARIATION OF THE TOTAL POTENTIAL ENERGY AT THE BUCKLING STATE The strain components depend upon the gradient of the displacement field vector given by: in which: and the curvatures in the deflected configuration: y y Let us acknowledge that the second order in-plane moment My is the sum of the following symmetric and antisymmetric components of the transverse loading system: 174 ∂u By defining the nonzero linear normal strain component εxx ¼ ∂x and shear strain compo∂u ∂v ∂w nents εxy ¼ ∂y þ ∂x and εxz ¼ ∂u þ in the deflected configuration, the strain energy may be ∂z ∂x evaluated. Considering the strain energy terms belonging to the out-of-plane bifurcation problem, neglecting the terms of higher order than two and including the potential energy of the loading system measured from the untwist and laterally unbent configuration, the variation of the total potential energy at the buckling state becomes: where, k1 = according to Eq. (2); ¢ = angle of twist rotation ¢x (subscript x is dropped hereafter for notation convenience); M II is the x-coordinate dependent prebuckling major axis second order moment; is the x-coordinate dependent buckling minor axis second order moment; qzi = uniformly distributed load component (UDL); Qzj = point load component (PL), zq = section coordinate of applied UDL; zQ = section coordinate of applied PL. 4 BUCKLING SOLUTION In the following, beam-columns being simply supported and free warped at the end sections are taken into consideration. Refining the linear initial stress term My δðv00 ¢Þ in Eq. (16), v00 is calculated from the following second order differential equilibrium equation: and the minor axis displacements are approximated by: where, vs ; va – minor axis displacements corresponding to symmetric and antisymmetric comI I ponents; Mys ; Mya – first order moment equations corresponding to symmetric and antisymmetric components. Substituting the results to Eq. (16), the energy equation may be approximated by: 175 Figure 1. Comparison of the moment conversion factor C1bc for unequal end moments. The last integral in Eq. (19) is concerned with bifurcation modes under the axial force without any bending action effect about the major axis. By equating the relationship given by Eq. (19) to zero, then approximating the buckling modes by conventional trigonometric functions (combination of sinus half wave and wave functions for the minor axis displacement and sinus half wave for the angle of twist rotation), the following solution governs: where, Cbc = equivalent uniform moment factor (Mcr=Cbc Mcr,0 where Mcr,0 is the critical moment in uniform bending): The moment distribution dependent parameters are given by: where, mIys ; mIya = dimensionless bending moment functions corresponding to symmetric and antisymmetric load effect components, obtained by dividing the moment functions by Mys;max and Mya;max , respectively. 176 One has to notice that for the narrow flange I-section beam-columns that are laterally and torsionally unrestrained between supports, both in-plane effects may be ignored, i.e. assuming in Eqs. (20) and (21) the following: k1 ¼ 1, NNz ð1 k1 Þ ¼ 0 and NNza ð1 k1 Þ ¼ 0. In Figure 1, the inversion of the equivalent uniform moment conversion factor Cbc of present study is compared for beam-columns subjected to unequal end moments. This asymmetric loading system is described by the moment gradient parameter ψM ¼ My;min =My;max so that the following relationships hold: and the factors given by Eqs. (22) are the same as those obtained for bending without compression, cf. Barszcz et al. (2021). 5 CONCLUDING REMARKS The paper discusses issues related to the consistent formulation of the energy equation for beam-column flexural-torsional buckling. Equation (19) obtained in the paper differs from the classical one for which the nonlinear moment term in square bracket under the second integral of Eq. (16) must be replaced by its linear counterpart It has been shown that the equivalent uniform moment modification factor should vary not only with the minor axis buckling force utilization ratio N/Nz but also with that of major axis buckling N/Ny represented by the factored ratio N/Nz(1-k1). The obtained solution, for the uniform bending, appears to be the same as that of the closed form solution, cf. Eq. (1) for k2=1 and Mohri et al. (2008). When k1=1, the obtained solution appears to be that obtained with the assumption of disregarding the effect of prebuckling displacements on the critical state. Contrary, when k1=0, the LHS of Eq. (20) becomes zero (the critical moment becomes infinity) and the buckling state is entirely controlled by the compressive axial force, with no effect of the bending moment on the buckling state. Moreover, when either N/Nz=1 or N/NT=1, the RHS of Eq. (20) becomes zero (the critical moment becomes zero). REFERENCES Trahair, N.S. 1993. Flexural-Torsional Buckling of Structures. Boca Raton: CRC Press. Trahair, N.S., Bradford, M.A., Nethercot, D.A. & Gardner, L. 2008. The behaviour and design of steel structures to EC3 (4th Edition). London-New York: Taylor and Francis. Mohri, F., Bouzerira, Ch. & Potier-Ferry, M. 2008. Lateral buckling of thin-walled beam-column elements under combined axial and bending loads. Thin-Walled Structures 46(3): 290–302. Pi Y-L. & Trahair N.S. 1994. Nonlinear inelastic analysis of steel beam-columns. Journal of Structural Engineering, Part I: Theory 120(7):2041–2061, Part II: Applications. 120(7):2967–2985. Pi, Y.L., Bradford, M.A. 2001. Effects of approximations in analyses of beams of open thin-walled cross-section. International Journal for Numerical Methods in Engineering, Part I: Flexural–torsional stability 51: 757–72,Part II: 3D non-linear behaviour 51: 773–790. Barszcz, A.M., Giżejowski, M.A. & Stachura, Z. 2021. On elastic lateral-torsional buckling analysis of simply supported I-shape beams using Timoshenko’s energy method. Proc. 14th International Conference on Metal Structures [submitted]. 177