Free vibration analysis of laminated beam by polynomial, trigonometric, exponential and zig-zag theories

A number of refined beam theories are discussed in this paper to trace the free vibration response of laminated beams, including thin-walled boxes. By expanding the unknown displacement variables over the beam section axes using Taylor type expansions, trigonometric series, exponential, hyperbolic and zig-zag functions, many new displacement fields were obtained and, for the first time, evaluated for the dynamic analyses of composite structures. The finite element method is used to derive governing equations in weak form. These equations are written using the unified formulation introduced by the first author, in terms of fundamental nuclei, whose forms do not depend on the expansions used. The natural frequencies are compared with results available in the literature or with those obtained by the finite element models related to commercial software. A number of analyses were conducted to compare various theories, including Euler-Bernoulli and Timoshenko models. The advantages/disadvantages of using the different theories are discussed for significant problems related to laminated beams as well as thin-walled boxes. It is shown that refined kinematic theories are able to yield a very accurate evaluation of fundamental as well as higher mode frequencies in a way comparable to three-dimensional analysis, but it is obtained with a strong reduction of computational costs.