Keywords: flexural-torsional buckling, Timoshenko beam, nonuniform torsion, warping, bar, twist, boundary element method, shear deformation.

Elastic stability of beams is one of the most important criteria in the design of structures subjected to compressive loads. This beam buckling analysis becomes much more complicated in the case the cross section's centroid does not coincide with its shear center (asymmetric beams), leading to the formulation of the flexural-torsional buckling problem. Moreover, unless the beam is very "thin" the error incurred from the ignorance of the effect of shear deformation is substantial, and an accurate analysis requires its inclusion.

In this investigation, an integral equation technique is developed for the solution of the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing the concept of the analog equation. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows:

1. The proposed method can be applied to beams having an arbitrary simply or multiply connected constant cross section and not to a necessarily thin-walled one.
2. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not necessarily coincide with the principal bending one.
3. For the first time in the literature the shear deformation effect is taken into account in the flexural-torsional buckling analysis of a beam of a non-symmetric constant cross section.
4. Torsional warping arising from nonuniform torsion is taken into account.
5. The beam is supported by the most general linear boundary conditions including elastic support or restraint.
6. The shear deformation coefficients are evaluated using an energy approach, instead of Timoshenko's and Cowper's definitions, for which several authors have pointed out that one obtains unsatisfactory results or definitions given by other researchers, for which these factors take negative values.
7. The effect of the material's Poisson ratio nu is taken into account.
8. The proposed method employs a pure BEM approach (requiring only boundary discretization) resulting in line or parabolic elements instead of area elements of the FEM solutions (requiring the whole cross section to be discretized into triangular or quadrilateral area elements), while a small number of line elements are required to achieve high accuracy.

Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-tube theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

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