Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic) in-plane forces are analyzed in the frequency domain. In¯uences of shear, rotatory inertia, transverse normal stress and of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner±Mindlin type are considered. These theories are written in a unifying manner using tracers to account for the various in¯uencing parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin de¯ections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and by imposed ®ctitious``thermal'' curvatures. These de¯ections are further split into de¯ections of linear elastic prestressed membranes with effective stiffness, mass and load. This analogy for the de¯ections is con®rmed by utilizing D'Alembert's dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and polygonal Mindlin plates are yielded and veri®ed by analogy in a quick and simple manner.Key words thick plate, vibrations, frequency domain, Kirchhoff plate, membrane
IntroductionIn order to account for the in¯uence of shear, rotatory inertia and transverse normal stress, various celebrated engineering approximations of the three-dimensional equations of linear elasticity have been developed in the literature, in re®nement of the classical LagrangeKirchhoff theory for¯exural deformations of thin plates. For some important references on re®ned plate theories, see e.g. the re¯ections presented in [1] and the monograph [2]. Since the complexity of the resulting equations of the re®ned theories is quite high, there remains a need for benchmark solutions, capable for a calibration of numerical routines. Furthermore, the question whether results of the thin-plate theory may be linked directly to those of the re®ned theories remains a challenging task.Subsequently, this problem is studied by means of various analogies between the re®ned problem and the classical theory. Free and forced vibrations of moderately thick,
