The line element-less method analysis of orthotropic beam for the De Saint Venant torsion problem

Abstract

This paper deals with the extension of a novel numerical technique, labelled line element-less method (LEM), in order to provide approximate solutions of the De Saint Venant torsion problem for orthotropic beams having simply and multiply connected cross-section. A suitable transformation of coordinates allows to take full advantage of the theory of analytic complex functions as in the isotropic case. A complex potential function analytic in all the transformed domain whose real and imaginary parts are related to the shear stress components and to the orthotropic ratio is introduced and expanded in the double-ended Laurent series involving harmonic polynomials. An element-free weak-form procedure has been proposed imposing that the square of the net flux of the shear stress across the border is minimized with respect to the series coefficients. Numerical implementation of the LEM results in system of linear algebraic equations involving symmetric and positive-definite matrices. All the integrals are transferred into the boundary without requiring any discretization neither in the domain nor in the contour. The technique provides the complete shear stress field as shown by some numerical applications in order to assess the efficiency and the accuracy of the method to handle shear stress problems in presence of orthotropic material.