Analysis of thin-walled beams via a one-dimensional unified formulation through a navier-type solution

A unifying approach to formulate several axiomatic theories for beam structures is addressed in this paper. A N-order polynomials approximation is assumed on the beam cross-section for the displacement unknown variables, N being 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.