Spatial Rotation Kinematics and Flexural–Torsional Buckling

This paper aims to clarify the intricacies of spatial rotation kinematics as applied to three-dimensional (3D) stability analysis of metal framed structures with minimal mathematical abstraction. In particular, it discusses the ability of the kinematic relationships traditionally used for a spatial Euler–Bernoulli beam element, which are expressed in terms of transverse displacement derivatives, to detect the flexural–torsional instability of a cantilever and of an L-shaped frame. The distinction between transverse displacement derivatives and vectorial rotations is illustrated graphically. The paper also discusses the symmetry and asymmetry of tangent stiffness matrices derived for 3D beam elements, and the concepts of semitangential moments and semitangential rotations. Finally, the fact that the so-called vectorial rotations are independent mathematical variables are pointed out.