The Role of Saint Venant’s Solutions in Rod and Beam Theories

Within small displacement linear elasticity, Saint Venant derived semi-inverse solutions for bending and torsion of rods and beams. The resulting linear expressions for bending moments and torque in terms of curvatures and torsion are often taken to be those required by elastica theory in describing general configurations of rods and beams. It is here shown, by using a perturbation procedure based on an appropriate scaling of axial variations, that Saint Venant’s solutions combine naturally with elastica theory to describe configurations involuing large rotations and displacements. Also it is shown that the classical solution for beam flexure is essentially a bending solution with varying curvature. The contribution involving the bending function is, for general beam loading, only part of the first-order correction.