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 warpin g function (CWF), each of which may be interpreted as the transverse d eflection of a thin membrane via the well-known membrane (soap-film) a nalogies, The deflection of the PSF membrane is associated with a unif orm transverse pressure loading, whereas the CWF membrane is nonpressu rized-its deformation is due solely to imposed deflections along its e dge. The utility of the PSF analogy stems from the simple relationship s that exist between the geometry of the deformed membrane (slope, enc losed volume) and the shaft quantities (shear stress, rigidity). Howev er, despite the fact that the CWF analogy involves a simpler loading t han its PSF counterpart, similar relationships relating the deformed C WF membrane to the shaft quantities do not appear in the literature. T he purpose of the present note is to complete the CWF TF analogy by de riving such expressions. The use of the new relationships is illustrat ed in a purely geometric derivation of the solution for a circular cyl indrical shaft. The results may also furnish a new means of visualizin g the torsional behavior of shafts of other cross-sectional shapes.