Benchmark Solution for Free Vibration of Moderately Thick Functionally Graded Sandwich Sector Plates on Two-Parameter Elastic Foundation with General Boundary Conditions

Abstract

The free vibration analysis of moderately thick functionally graded (FG) sector plates resting on two-parameter elastic foundation with general boundary conditions is presented via Fourier-Ritz method, which is composed of the modified Fourier series approach and the Ritz procedure. The material properties are assumed to vary continuously along the thickness according to the powerlaw distribution. The bilayered and single-layered functionally graded sector plates are obtained as the special cases of sandwich plates. The first-order shear deformation theory (FSDT) is adopted to construct the theoretical model. Under current framework, regardless of boundary conditions, each displacement and each rotation of plates is represented by the modified Fourier series consisting of a standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. Then, the accurate solutions are obtained by using the Ritz procedure based on the energy function of sector plates. The present method shows good convergence, reliability, and accuracy by comprehensive investigation with some selected classical boundary conditions. Numerous new vibration results for moderately thick FG sandwich sector plates are provided. The effects of the elastic restraint parameters and so forth on free vibration characteristic of sector plates are presented.

A Series Solution for the Vibration of Mindlin Rectangular Plates with Elastic Point Supports around the Edges

“…According to the Mindlin plate theory, the transverse displacement of the plate middle surface and the rotations of the cross section, respectively, along the direction and the direction are utilized. Based on the traditional solution method, the admissible functions usually express a Fourier series expansion, because the Fourier functions constitute a complete set and exhibit an excellent numerical stability in the previous study [37][38][39][40][41][42][43]. We found the conventional Fourier series expression to have some defects which contain the convergence problem along the boundary edges except for a few simple boundary conditions, and the derivatives of a Fourier series cannot be obtained simply through term-byterm differentiation.…”

An Accurate Solution Method for the Static and Vibration Analysis of Functionally Graded Reissner‐Mindlin Rectangular Plate with General Boundary Conditions

“…However, the conventional Fourier series just adapt to a few of simple boundary conditions due to the convergence problem along the boundary conditions. Recently, a modified Fourier series technique proposed by Li [50] has been widely applied in the vibration problems of plates and shells subject to different boundary conditions by the Ritz method, e.g., [42,[51][52][53][54][55][56][57][58][59][60]. In this technique, each displacement of the structure under study is written in the form of a conventional cosine Fourier series and several supplementary terms.…”

Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

“…The functionally graded (FG) materials are widely used in aerospace, automobile, and civil engineering due to continuous variation of material properties along the thickness direction [1][2][3][4][5][6][7]. As is known to us, the FG shallow shellsstructures, i.e., FG plate, FG circular cylindrical shallow shell, FG spherical shallow shell, and FG hyperbolic paraboloidal shallow shell, have attracted considerable attention for its high strength and stiffness [8][9][10][11][12].…”