Analysis of Thin-Walled Beams via a One-Dimensional Unified Formulation Through a Navier-Type Solution

Abstract

A unifying approach to formulate several axiomatic theories for beam structures is addressed in this paper. A [Formula: see text]-order polynomials approximation is assumed on the beam cross-section for the displacement unknown variables, being [Formula: see text] a free parameter of the formulation. Classical beam theories, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. According to the proposed unified formulation, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The linear static analysis of thin-walled beams is carried out through a closed form, Navier-type solution. Simply supported beams are, therefore, presented. Box, C- and I-shaped cross-sections are accounted for. Slender and deep beams are investigated. Bending and torsional loadings are considered. Results are assessed toward three-dimensional finite element solutions. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and the loading conditions.