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.