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Comprehensive study of free vibration of rectangular mindlin’s plates with rotationally constrained edges using dynamic timoshenko trial functions

Verma Y.

Engineering Transactions

2018

Vol. 66, iss. 2

129--160

A comprehensive theoretical study of the free vibration of rotationally restrained rectangular uniform isotropic Mindlin’s plate is presented. The plate mode shape is assumed to be a weighted combination of the product of the Timoshenko beam functions in the either direction, which are previously generated for rotationally constrained boundary conditions. The effect of the uniformly distributed rotational spring constant (modelling the edge) participates in the potential energy of the plate. The Rayleigh-Ritz method has been used to generate the natural frequencies and plate mode shapes for various intermediate boundary conditions, asymptoting to those of the plates with all possible (six) classical boundary conditions. Plates with various thickness ratios have been studied to converge the results to the corresponding Kirchhoff’s frequencies. The eigenvectors from the eigenvalue problem have been scrutinized to establish the beam-wise modal participation from either direction into the final plate mode shape. The square Mindlin’s plate mode shapes have been generated to establish the various types of frequencies; which have been innovatively named and categorized as the (i) single frequencies, (ii) repeated frequencies (identical twins) and (iii) non-repeated frequencies(fraternal twins). Plates with different rectangular aspect ratios have been also analysed to show the deviation in the frequencies and mode shapes from the square plate. Also, their asymptotic behaviour to the corresponding Timoshenko beam at extreme aspect ratios has been established.

Effects of elastically restrained edges on FG sandwich annular plates by using a novel solution procedure based on layerwise formulation

Alipour M. M.

Archives of Civil and Mechanical Engineering

2016

Vol. 16, no. 4

678--694

In practical applications, sandwich plates are often connected to other members, supported by damaged clamped/simply supported boundary conditions or supported by elastic restraints. Therefore, the mentioned structures may not always be simulated by the classical boundary conditions, i.e., ideal simply supported, clamped and free edges. Also, these structures may be subjected to various loads. In this study, for the first time, a novel economical analytical solution procedure is presented for axisymmetric static analysis of sandwich annular plates, by using the layerwise and 3D elasticity theories. Based on the proposed approach, functionally graded sandwich annular plates with various elastically restrained edges under arbitrary distributed loads may be analyzed and all of the displacements and stresses components may be exactly achieved. Also, imposed loads at the boundaries may be evaluated. Transverse shear and normal stresses boundary conditions on the top and bottom of the sandwich plate and the interlaminar continuity conditions of the in-plane displacement, transverse shear and normal stresses are exactly satisfied. Accuracy and efficiency of the presented solution procedure are demonstrated by comparing the obtained results for sandwich plates with the classical edge conditions as some special cases of the elastic supports with results of the three-dimensional theory of elasticity.