On torsion and shear of Saint-Venant beams

Torsion and shear stress fields of a Saint-Venant beam and the relative location of shear and twist centres are investigated for sections of any degree of connectedness. The sliding-torsional compliance tensor of a Timoshenko beam is evaluated by an energy equivalence with Saint-Venant theory. Accordingly, the mutual sliding-torsional term is shown to depend linearly on the relative position of shear and twist centres and the standard definition of shear centre in a Timoshenko beam is found to be coincident with Saint-Venant twist centre. Coincidence of shear and twist centres is assessed for sections with vanishing Poisson ratio and for open, closed and multi-cell thin-walled cross sections. The eigenvalues of the shear factors tensor and the torsion factor are shown to be greater than unity, with the principal directions of shearing and bending compliances non necessarily coincident for non-symmetric cross sections. Numerical examples are developed to provide evidences of the location of the centres and of the principal shearing directions, for non-symmetric L-shaped cross sections with various thickness ratios.