Shear Properties of Isotropic and Homogeneous Beam-Like Solids Having Arbitrary Cross Sections

Abstract The paper addresses the theoretical framework and the numerical procedure developed to carry out the sectional analysis preliminary to the use of a recently formulated isotropic and homogeneous beam model consistently derived from the Saint Venant solid one. Specifically, in order to ensure the equivalence of the new beam model with the 3D solid one, both in terms of elastic energy and displacements of the beam axis, one requires the evaluation of a symmetric second-order tensor, accounting for the shear deformation, besides the center of twist and the torsional stiffness factor. Common property to such geometrical quantities is to be expressed as integrals of harmonic functions solution of Neumann problems that are defined over the section domain. Hence, with a view towards numerical applications, they are reformulated as integrals defined over the section boundary, assumed to be of arbitrary polygonal shape. The numerical evaluation of the resulting line integrals is obtained by exploiting a recently formulated boundary element approach in which the harmonic potential functions are expressed as polynomials defined on suitably defined subsets of each edge of the section boundary. As an outcome of the extensive numerical tests that have been carried out on compact and thin-walled sections a general criterion, established elsewhere for properly selecting the best combination of polynomial degree and edge discretization, is shown to be successful also for carrying out the shear analysis of beam sections having arbitrary shape.