A Refined Beam Theory with Only Displacement Variables and Deformable Cross-Section

Abstract

This paper proposes several refined theories for the linear static analysis of beams made of orthotropic materials. A hierarchical scheme is obtained by extending Carrera’s Unified Formulation (CUF), which has previously been proposed for plates and shells, to beam structures. An N-order approximation via Mac Laurin’s polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli’s and Timoshenko’s, are obtained as particular cases. According to CUF, 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 governing differential equations are solved via the Navier type, closed form solution. Poisson’s locking correction is assumed for the first-order and classical models. Rectangular and I-shaped cross-sections are considered. Beams are considered to undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons to three-dimensional exact solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional stress field 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 loading conditions.