Co-rotational and Lagrangian formulations for elastic spatial beam elements

It is explained in the companion paper that conservative internal moments of a beam are of the so-called fourth kind. It is also noted that the rotation variables that are energy-conjugate with these moments are vectorial rotations. Meanwhile, Donald & Kleeman (1971) have shown that vectorial rotations have second-order relationships with transverse displacement derivatives. This implies that the first partial derivative of the strain energy of a beam with respect to a transverse displacement derivative is not a bending moment (even if we ignore the axial deformation). Furthermore, the shape functions employed in the Rayleigh-Ritz method of finite element analysis should be applied directly to the vectorial rotations rather than the transverse displacements of arbitrary points along the beam element. 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 beam structures.