On the restrained torsional warping of thin‐walled elastic shafts

Abstract

AbstractThis paper studies the torsion of thin‐walled shafts when warping is restrained. It focuses on the treatment of the warping–twist constraint (whereby the twisting of the shaft is constrained to control the warping of its cross sections) by different formulations of warping torsion. This constraint underlies the kinematics of the classical TWKV formulation of Timoshenko‐Wagner‐Kappus‐Vlasov, whereas it is relaxed in the widely‐used RBV formulation of Reissner‐Benscoter‐Vlasov and the new mixed formulation recently proposed by the author. These two unconstrained formulations are still able to enforce the constraint in a penalty‐type limit process depending on specific combinations of the section's torsional constants, some new to the literature. Here we obtain rigorous estimates of these torsional constants and penalty parameters in the thin‐wall limit, thus providing an alternative understanding of the physical response of these sections. In particular, we prove that the constrained limit is attained in the thin‐wall limit for open sections but not for closed (multiply‐connected) cross sections. These analyses also provide an understanding of how those three formulations model the torsion of elastic shafts. To this purpose, we also undertake a detailed numerical evaluation of these considerations. The results presented here show the improved performance gained by the new mixed formulation of warping torsion, avoiding the anomalies observed for the TWKV formulation and the incorrect (not in equilibrium) RBV stresses.