Free vibration analysis of composite beams via refined theories

Abstract

This paper presents a free-vibration analysis of simply supported, cross-ply beams via several higher-order as well as classical theories. The three-dimensional displacement field is approximated along the beam cross-section in a compact form as a generic N-order polynomial expansion. Several higher-order displacements-based theories accounting for non-classical effects can be, therefore, formulated straightforwardly. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the principle of virtual displacements. Thanks to the compact form of the displacement field approximation, governing equations are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations. Besides the fundamental natural frequency, natural frequencies associated to higher modes (such as torsional, axial, shear and mixed ones) are investigated. A half waves number equal to one is considered. The effect of the length-to-thickness ratio, lamination, aspect ratio and material properties on: (1) the accuracy of the proposed theories and (2) the natural frequencies and modes is presented and discussed. For the latter case, the modes change in order of appearance (modes swapping) and in shape (modes mutation) is investigated. Results are assessed towards three-dimensional FEM solutions. Numerical results show that, upon the choice of the appropriate approximation order, very accurate results can be obtained for all the considered modes.