An approximate procedure for a general analytical expression of the Prandtl membrane analogy in linear elastic pure torsion

Abstract

For pure torsion in linear elasticity, if the shape of cross-section is defined by a closed-form analytical expression satisfying certain conditions, the shear stress in each of the principal Cartesian directions can be obtained by the derivation of the stress function, whose analytical expression can be obtained from the Prandtl analogy. In this paper, a general methodology to obtain an approximate analytical expression of the Prandtl membrane is presented. The proposed methodology uses two variable quadratic piecewise functions to define the Prandtl membrane. This approximate procedure can be applied to any cross-section shape, and it has been proved be especially suitable for steel shapes, for which the proposed method leads to values close to the ones obtained with more precise methods. Several examples are presented.