TWO GENERALIZED HIGHER ORDER THEORIES IN FREE VIBRATION STUDIES OF MULTILAYERED PLATES

Abstract

This paper presents an extension of two-dimensional models for the analysis of freely vibrating laminated plates. The extension concerns the enlargement of higher order theories, recently introduced by different authors in several forms, to encompass higher order terms over the cubic one usually taken into consideration. Higher order effects such as rotatory inertia and transverse shear stress are naturally included without any shear correction factors. Namely, two different models are introduced by expanding, on different functional bases, displacements (D2D) and transverse shear stresses in conjunction with displacements (M2D). The expansion is considered to be consistent with the traction-type boundary condition on the external surfaces of the plate. The governing equations and associated boundary conditions are consistently obtained by the classical Hamilton's variational principle and Reissner's mixed variational theorem. Both models are equivalent single layer type and, therefore, differ according to the layer-wise descriptions, preserve the independence of the number of unknown variables on the number of layers. However, this feature is presented together with intrinsic physical violations for both models. Model D2D violates the interlaminar stress continuity requirement and model M2D violates in a weaker from the same requirement (derivatives are not piecewise continuous), besides neglecting the transverse normal stress. The importance of completely fulfilling the mentioned continuity is then discussed once the relevant governing equations are tailored for the cylindrical bending condition. The effectiveness of the models is indicated by making numerical comparisons with the exact three-dimensional theory of the elasticity for several lamination schemes, angle/cross-ply lay-ups, and characteristic geometric ratios for low and higher frequencies.