Stresses in Half-Elliptic Curved Beams Subjected to Transverse Tip Forces

In-plane bending of curved beams produces substantial through-thickness normal and shear stresses that can result in structural failures. A half-elliptic curved beam, having a known prescribed variable radius of curvature, is studied as an extension of the previously published circular arc beam. The equations for the normal, tangential, and shear stresses are developed for a curved beam outlined by two confocal half ellipses loaded by a pair of concentrated perpendicular forces on its ends. Closed-form analytical solutions for the stresses are found using an elasticity approach, where the solution is found by using selected terms of the biharmonic equation in elliptic coordinates. For the case of an elliptic beam with an aspect ratio of very close to unity, the solution closely agrees with published circular beam solutions. For other elliptic beam aspect ratios, the calculated stresses display good correlation to detailed finite element model solutions for thickness to semi-axis ratios < 0.1. A parametric study revealed that the maximum normal stress is located at the midplane for high-aspect ratio (a/b ≥ 1) half-elliptic beams, but shifts toward the load tip for low aspect ratio (a/b < 1) beams due to local curvature effects. Moreover, the peak shear stress location moves toward the midplane and the magnitude greatly increases as the aspect ratio is increased. Thus, there are large normal and shear stress interactions occurring near the midplane for high-aspect ratio half-elliptic beams, which is not observed for circular beams. These stress interactions can produce unique failures in materials having low shear strength and through-thickness strength. The current closed-form solution is an improvement on previously published approximate solutions.