A series solution for the vibrations of composite laminated deep curved beams with general boundaries

In this paper, a series solution is derived for the vibration analysis of composite laminated deep curved beams with general boundary conditions. The effects of shear deformation, inertia rotary and deepness term are considered in the formulation. Under the current framework, the governing equations and the related boundary equations are obtained via the Hamilton’s principle. And each of beam displacements, regardless of boundary conditions, is expanded as a modified Fourier series composed of a standard cosine Fourier series and certain supplementary terms introduced to remove the potential discontinuities at the ends, thus ensure and accelerate the convergence of the series representation. The characteristic equations are then derived directly in an exact sense by solving the equations of motion in matrix form by combining the associated boundary equations and the modified Fourier series representation. The convergence and accuracy of the solution are tested and validated by several numerical cases against the results available in the literature, with excellent agreements obtained. A systematic parametric study is also performed regarding the effects of shear deformation and inertia rotary, deepness term, boundary conditions, lamination schemes, material and geometrical parameters. Finally, several numerical results of composite laminated deep and shallow curved beams with different geometry dimensions are presented for various boundary conditions and lamination schemes, which may serve as benchmark solutions for the future researches in this field.