Solution of De Saint Venant flexure-torsion problem for orthotropic beam via LEM (Line Element-less Method)

Abstract

Abstract In this paper the numerical technique, labelled Line Element-less Method (LEM), is employed to provide approximate solutions of the coupled flexure-torsion De Saint Venant problem for orthotropic beams having simply and multiply-connected cross-section. The analysis is accomplished with a suitable transformation of coordinates which allows to take full advantage of the theory of analytic complex functions as in the isotropic case. A boundary value problem is formulated with respect to a novel complex potential function whose real and imaginary parts are related to the shear stress components, the orthotropic ratio and the Poisson coefficients. This potential function is analytic in all the transformed domain and then expanded in the double-ended Laurent series involving harmonic polynomials. The solution is provided employing an element-free weak form procedure imposing that the squared net flux of the shear stress across the border is minimum 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 evaluation of the shear stress field at any interior point as shown by some numerical applications worked out to illustrate the efficiency and the accuracy of the developed method to handle shear stress problems in presence of orthotropic material.