On stress function in Saint-Venant beams

A general discussion concerning existence of stress function in Saint-Venant beams with simply or multiply connected cross-sections under shear and torsion is performed. It is shown that the tangential stress fields can be expressed in terms of a stress function only when the centroids of the cross-section and of the holes are located on a line orthogonal to the shearing vector. The analysis performed by Muskhelishvili under the special assumption that the shear force acts along a principal direction is recovered as a special case.