A hybrid formulation for determining torsion and warping constants

Abstract

Theoretically, the torsion constant of a bar subjected to a torque can be calculated irrespective of its cross-sectional shape, either with a potential energy (displacement) or with a complementary energy (stress function) formulation. The potential energy formulation, however, also provides the so-called warping constant, which may be important in the case of thin-walled sections, but the complementary energy formulation does not. On the other hand, the complementary energy formulation leads to simple analytical expressions for the torsion constant of thin-walled sections, while a finite element calculation based on potential energy requires a relatively large number of elements to obtain comparable results. Since hybrid formulations combine some potential energy features with complementary ones, a hybrid formulation was derived for the torsion problem, with the purpose of combining the advantages of both—i.e. an ultimate displacement formulation and possibly greater accuracy. Both the torsion and warping constants can be calculated consistently with this formulation. The hybrid variational formulation that was derived, together with the general discretization procedure, can be used to formulate finite elements for calculating the torsional constants for all kinds of cross-section, whether solid or thin-walled, open or closed, symmetrical or unsymmetrical. The initial application, to rectangular solid cross-sections, resulted in an improved warping constant.