Elasticity solution for vibration of 2-D curved beams with variable curvatures using a spectral-sampling surface method

Abstract

Summary An accurate spectral-sampling surface method for the vibration analysis of 2-D curved beams with variable curvatures and general boundary conditions is presented. The method combines the advantages of the sampling surface method and spectral method. The formulation is based on the 2-D elasticity theory, which provides complete accuracy and efficiency for curved beams with arbitrary thicknesses and variable curvatures because no other assumptions on the deformations and stresses along the thickness direction are introduced. Specifically, a set of non-equally spaced sampling surfaces parallel to the beam's middle surface are primarily collocated along the thickness direction, and the displacements of these surfaces are chosen as fundamental beam unknowns. This fact provides an opportunity to derive elasticity solutions for thick beams with a prescribed accuracy by selecting sufficient sampling surfaces. Each of the fundamental beam unknowns is then invariantly expanded as Chebyshev polynomials of the first kind, and the problems are stated in variational form with the aid of the penalty technique and Lagrange multipliers, which provide complete flexibility to describe any arbitrary boundary conditions. Finally, the desired solutions are obtained by the variational operation. Copyright © 2016 John Wiley & Sons, Ltd.