Three higher order refined displacement models are proposed for the free vibration analysis of sandwich and composite beam fabrications. These theories model the warping of the cross-section by taking the cubic variation of axial strain and they eliminate the need for a shear correction coefficient by assuming a quadratic shear strain variation across the depth of the cross-section. Numerical experiments for various lamination schemes, boundary conditions and aspect ratios are carried out to compare these models with the first order shear deformation theory, earlier investigations and also among themselves to ascertain the most efficient one. Numerical results for deep sandwich and composite beams are also presented for future references.

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Vibrations of angle ply laminated beams are studied using the higher order theory and isoparametric 1d finite element formulations through proper constitution of elasticity matrix. Subsequent to the validation of the formulation, deep sandwich and composite beams are critically analyzed for various boundary conditions. Frequencies classified based on their spectrum are presented along with those of first order theories for comparison.

When finite element formulations are used to study the non-linear vibration problems, some simplifications, which are not consistent with the governing variational principles, are commonly employed. Three such simplifications are critically reviewed here, through beam finite element models. The first one, 'equivalent/ quasi-linearisation technique' is shown to have a reduced non-linear stiffness. The second, where in 'neglect of in plane displacements' takes place, is seen to register an excessive non-linear stiffness. Thirdly, when both these simplifications are introduced together, they produce results closer to those of variationally correct ones, rather fortuitously. The objective of this paper is to highlight the necessity of formulating this class of problems in a variationally correct and consistent manner. Numerical computations are performed systematically, using two different beam finite element models for various commonly studied boundary conditions and suitable conclusions are drawn.Keywords Finite elements Á Non-linear beam vibrations AE Quasi-linearisation Á Neglect of in plane displacements 1 Introduction

Abstract-Higher-order shear-deformable refined theories, based on isoparametric elements, are adopted for transient dynamic analysis of symmetric and unsymmetric sandwich and composite beam constructions. These shear-correction coefficient free theories model cross sectional warping using nonlinear variation of inplane displacements across the depth. They also incorporate transverse shear stress in the formulation. A special lumping scheme is employed for the evaluation of diagonal mass matrix, and a central difference scheme is used for carrying out the integration of the equation of motion, to obtain the response history. Through numerical experiments, the efficacy of higher-order models in predicting displacements and stress, resultants over from the first-order theory, with respect to time, is clearly brought out in this paper. 0 1997 Elsevier Science Ltd.