Keywords: beam, finite element, non-linear, open section, post-buckling, stability, thin-walled.

Thin-walled elements with isotropic or anisotropic composite materials are extensively used as beams and columns in engineering applications, ranging from buildings to aerospace and other many industry fields where the requirements of weight saving are the most important. As a result of their particular shapes resulting from the fabrication process, these structures always have arbitrary open cross sections which make them highly sensitive to torsion and instabilities.

It has been shown that beam elements available in the commercial codes are not able to compute accurately the buckling loads of beams with arbitrary open cross section shapes. Therefore recourse to shell elements is recommended. But in this case, the model preparation and computation process needs a long time, especially when the analysis of nonlinear behaviour is undertaken with the presence singular points on the equilibrium curves. Hence a finite element model for thin walled beams with arbitrary sections undergoing large displacements and large torsion required. The authors have developed a finite element model for thin walled beams with arbitrary cross sections for large torsion. The equilibrium equations and the material behaviour are established without any assumption concerning the twist angle amplitude. Trigonometric functions of the twist angle were included as additional variables in the model. Pre-buckling deflections and flexural-torsional coupling are naturally included. This model was extended to a finite element formulation in the same way. In the meshing process, three-dimensional beams with seven degrees of freedom including warping are considered.

Some other finite element models have been investigated for the stability of thin-walled beams where the trigonometric functions were approximated using polynomial functions such as linear, quadratic and cubic functions. These approximations were made for the discretisation of the equilibrium equations with finite element approaches or in semi-analytical analyses. In the present paper, a finite element formulation is derived according to cubic, quadratic and linear approxiamtions. For each element, the flexural-torsional coupling is taken into account in kinematics (displacement and Green's tensor components) and in the equilibrium. The tangent stiffness matrix of each approximation is carried out and is incorporated in a finite element code. Many comparison examples are considered. They concern the non-linear behaviour of beams of arbitrary cross section under a twist moment and the post-buckling behaviour of columns and beams under axial loads or bending loads. For each element, the flexural-torsional equilibrium paths are obtained under the same conditions. The efficiency of the large torsion element is confirmed.

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