Co-rotational and Lagrangian formulations of elastic three-dimensional beam finite elements
It has been pointed out in a previous paper by the authors that conservative internal moments of a spatial beam are of the so-called fourth kind, and that the rotation variables which are energy-conjugate with these moments are vectorial rotations. Vectorial rotations of a spatial Euler–Bernoulli beam have nonlinear relationships with its transverse displacement derivatives. This implies that strictly speaking, the first partial derivative of the strain energy with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation), and that modifications should be introduced to the conventional Hermitian shape functions employed in the Rayleigh–Ritz method of finite element analysis. On the other hand, the neglect of the rotational behaviour of nodal moments has led to an incorrect stability matrix in the literature, and it is shown through numerical examples that this incorrect stability matrix cannot detect the flexural-torsional buckling load of spatial structures in which the members are not connected collinearly. The validity of the Wagner hypothesis on the coupling of axial and torsional deformations for a spatial beam-column is illustrated through a numerical example. Finally, one issue related to the Updated Lagrangian formulation addressed in this paper is the oft-used assumption of a straight configuration at the last known state.
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