Membrane Analogy for Saint-Venant Torsion: New Results

The Saint-Venant torsion problem for a twisted shaft may be formulated in terms of the Prandtl stress function (PSF) or the conjugate warping function (CWF), each of which may be interpreted as the transverse deflection of a thin membrane via the well-known membrane (soap-film) analogies. The deflection of the PSF membrane is associated with a uniform transverse pressure loading, whereas the CWF membrane is nonpressurized—its deformation is due solely to imposed deflections along its edge. The utility of the PSF analogy stems from the simple relationships that exist between the geometry of the deformed membrane (slope, enclosed volume) and the shaft quantities (shear stress, rigidity). However, despite the fact that the CWF analogy involves a simpler loading than its PSF counterpart, similar relationships relating the deformed CWF membrane to the shaft quantities do not appear in the literature. The purpose of the present note is to complete the CWF analogy by deriving such expressions. The use of the new relationships is illustrated in a purely geometric derivation of the solution for a circular cylindrical shaft. The results may also furnish a new means of visualizing the torsional behavior of shafts of other cross-sectional shapes.