Keywords: lateral torsional buckling, beam, geometrical non-linear.

The buckling moment of spatial beams under different types of end bending moment and compressive axial force is investigated using finite element method. A beam under the action of bending moment may exhibit a buckling behavior. The buckling analysis of spatial beams under end moment was extensively studied. However, most researches were linear buckling analysis and the axial force was not considered. A limitation of the linear buckling analysis has been the omission of any consideration of the effect of prebuckling deflections of the beam. In many cases, however, the effect of the prebuckling deflections must be taken into account if the buckling load is to be determined with accuracy.

The nature of a moment depends on the mechanism that generates the moment. Different ways for generating configuration dependent moments were proposed in the literature[1,2]. When the beams are clamped at one end and free at another end, the buckling behavior of beams under different types of end moment may be different. However, little information in the literature is available, especially when the axial compressive load is also considered.

The objective of the present paper is to investigate the buckling moment of beams under different types of end moment and the axial force using finite element method. Here, the ways for generating conservative moment proposed in[2] are employed here. In [3], Hsiao and Lin presented a co-rotational total Lagrangian finite element formulation for the geometrical nonlinear analysis of doubly symmetric thin-walled beam. The element has two nodes with seven degrees of freedom per node. The element nodes are chosen to be located at the shear centers of the end cross sections of the beam element and the shear center axis is chosen to be the reference axis. The deformations of the beam element are described in the current element coordinate system, which is constructed at the current configuration of the beam element. In element nodal forces, all coupling among bending, twisting, and stretching deformations of the beam element is considered by consistent second-order linearization of the fully geometrically nonlinear beam theory. The element is proven very effective for geometrical nonlinear buckling and postbuckling analysis by numerical examples studied by [3]. Thus, the beam element proposed in [3] is employed here.

An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the effect of compressive force on the buckling moment of spatial beams under different types of bending moment.

1

H. Ziegler, "Principles of Structural Stability", Birkhauser Verlag Basel, 1977.

2

K.M. Hsiao, R.T. Yang and W.Y. Lin, "A consistent finite element formulation for linear buckling analysis of spatial beams", Comput. Meth. Appl. Mech. Engng, 156, 259-276 1998. doi:10.1016/S0045-7825(97)00210-7

3

K.M. Hsiao and W.Y. Lin, "Co-rotational Formulation for Geometric Nonlinear Analysis of Doubly Symmetric Thin-Walled Beams", Comput. Methods Appl. Mech. Engrg. (accepted), 2001.

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