A spectral-sampling surface method for the vibration of 2-D laminated curved beams with variable curvatures and general restraints

Study of the vibration characteristics of thick composite laminated and sandwich curved beams with variable curvatures as well as general restraints is a challenging work since it requires the use of layerwise laminate theory that contains full 2-D kinematics and constitutive relations. In the present paper, an accurate spectral-sampling surface method is developed to fulfill the task. The method combines the advantages of the sampling surface method with those of the spectral method. The formulation is based on the two dimensional theory of elasticity and no other assumption on deformations and stresses along the thickness direction is introduced. In this method, a certain number of non-equally-spaced sampling surfaces which parallel to the beam middle surface are primarily collocated in each beam layer and, the displacements of these surfaces are introduced as the fundamental beam unknowns. This fact gives the opportunity to derive the elasticity solutions for thick laminated curved beams with a prescribed accuracy by utilizing a sufficiently large number of sampling surfaces. Each of the fundamental beam unknowns is then invariantly expanded as Chebyshev polynomials of first kind and, the problems are stated in a variational form by the aid of penalty parameters and Lagrange multipliers which provides complete flexibility to describe any arbitrary boundary conditions. Finally, the desired solutions are obtained by the variational operation.