International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 12 Issue: 12 | Dec 2025
p-ISSN: 2395-0072
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Analytical solution for free vibration analysis of Laminated Sandwich plate R.N.Mokal1, S.A.Ahire2 1,Department of Mechanical Engineering Mit coe engineering Dhanore yeola-423401
2Department of Mechatronicss Engineering SND coe R&C Bhabulgaon yeola—423401
Maharashtra India -------------------------------------------------------------------------***------------------------------------------------------------------------
ABSTRACT- In this paper a variationally consistent polynomial shear deformation theory is presented for the free
vibration of thick isotropic square and rectangular plate. In this displacement based theory, the in-plane displacement field use parabolic function in terms of thickness coordinate to include the shear deformation effect. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results of frequency are obtained from free vibration of simply supported isotropic square and rectangular plates and compared with those of other refined theories and frequencies from exact theory. Keywords: Shear deformation; laminated plate; Shear correction factor; free vibration.
1 INTRODUCTION Plates are the basic structural components that are widely used in various engineering disciplines such as aerospace, civil, marine and mechanical engineering. The transverse shear and transverse normal deformation effects are more pronounced in shear flexible plates which may be made up of isotropic, orthotropic, anisotropic or laminated composite materials. In order to address the correct structural behavior of structural elements made up of these materials; development of refined theories, which take into account refined effects in static and dynamic analysis of structural elements, becomes necessary. The study of plate vibration dates back to the early eighteen century, with the German physicist, who observed the nodal patterns for a flat square plate. Since then there has been a tremendous research interest in the subject of plate vibrations. Several thin plate vibration solutions based on Kirchhoff’s plate theory are available in the literature. The classical plate theory based on Kirchhoff's hypothesis [1] is not adequate for the analysis of shear flexible plates due to the neglect of transverse shear deformation and the rotary inertia in the theory; as a consequence, it under predicts deflections and over predicts all the vibration frequencies for thick plates, and the higher frequencies for the thin plates. The most suitable starting point for the analysis of both thin and thick plates seems to be a theory in which the classical hypothesis of zero transverse shear strains is relaxed. At first, Reissner proposed that the rotations of the normal to the plate mid-surface in the transverse plane could be introduced as independent variables in the plate theory. Raiser has developed a stress based theory which incorporates the effect of shear. Mindlin [2] simplified Reissner’s assumption that normal to the plate mid-surface before deformation remains straight but not necessarily normal to the plate mid-surface after deformation and the stress normal to the mid-surface is disregarded as in the case of classical plate theory of Kirchhoff. Mindlin employed displacement based approach. In Mindlin’s theory, transverse shear stress is assumed to be constant through the thickness of the plate, but this assumption violates the shear stress free surface conditions. The theory includes both the shear deformation and rotary inertia effects. Both effects decrease the frequencies. There are still other effects not accounted for by the Mindlin are stretching in the thickness direction and the warping of the normal to the mid-plane, which are more important in case of thick plates. Mindlin’s theory satisfies constitutive relations for transverse shear stresses and shear strains by using shear correction factor. The value of this factor is not unique but depends on the material, geometry, loading and boundary condition parameters. Wang discussed these theories in detail and developed the relationships between bending solutions of Reissner and Minding plate theory. Usually, in two dimensional plate theories, displacement components are considered power series expressions in thickness coordinate (z). Depending on the number of terms retained in the power series expressions, various higher order theories for homogeneous and laminated plates can be developed Reddy [3,4] utilize some simplification of the generalized displacement function.The simplified higher order theories, generally third order shear deformation theories give parabolic variation of transverse shear stress through the thickness of the plate satisfying the shear stress free boundary conditions on the top and bottom surfaces of the plate. Thus, these theories do not require shear correction factors. Levinson formulated a theory based on displacement approach which does not require shear correction factor.
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