Levinson Beam Theory: An Analytical Solution to the First-Order Analysis of Uniform Beams with Simply Symmetrical Cross-sections

This paper presents an analytical solution to the Levinson beam theory (LBT) for the first-order analysis of uniform beams with simple symmetrical cross-sections, contrarily to most papers on LBT that analyzed beams with rectangular or double symmetrical cross-sections. LBT is a higher-order shear deformation theory characterized by a displacement field which includes warping of the cross-section and satisfies the shear free conditions on the lower and upper surfaces of the beam. In this study, the shear stresses were assumed to have their maximal values at the centroidal axis. This led to a displacement field that was fourth-order for beams with simple symmetrical cross-sections and third-order for beams with double symmetrical cross-sections. The equilibrium equations set on a vectorial basis were composed of two coupled differential equations combining the transverse deflection and the rotation of the cross-section at the centroidal axis: after some manipulations the governing equation (a fourth-order differential equation), the efforts and deformations were expressed in terms of the transverse deflection. Finally, closed-form solutions for efforts and deformations were presented for various loading and support conditions, as well as element stiffness matrices. The results were in agreement with those in the literature for beams with rectangular cross-sections.