On the modeling of restrained torsional warping: an analysis of two formulations

This work considers the modeling of torsion in elastic shafts accounting for the non-uniform warping of the cross sections along them. In this paper, we present an analysis of (1) the original formulation of Timoshenko–Wagner–Kappus–Vlasov, consisting of the underlying Saint-Venant torsion but with a non-constant rate of twist defining the warping magnitude, and (2) the alternative formulation first considered by Reissner–Benscoter–Vlasov involving an independent field for the warping amplitude. The theoretical results presented here characterize the kinematically constrained character of the first of these formulations, noting in the process the anomalies resulting from the full restrainment of the warping at a cross section in this setting. New explicit expressions for the warping shear stress and other features are obtained in this context. The second formulation relaxes the constraint between warping and twisting, with the analyses presented here identifying explicitly for the first time how its enforcement can be achieved in a limit process controlled by a parameter depending on the cross-sectional geometry. Hence, it avoids the anomalies of the first formulation, but at the price to involve local stresses not in equilibrium, a situation that does not seem to be much present in the literature.