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

Xiancheng Wang,Wei Li,Tang Sheng,Yu Fu,Jianjun Yao,Zhenshuai Wan

doi:10.1155/2019/7527148

Abstract

In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.

Highlights

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]
Extensive research efforts focused on vibration of FG shallow shells have been made by various shell theories, such as classical shallow shell theory (CSST) [16], first-order shear deformation theory (FSDT) [17], and higher-order shear deformation theory (HSDT) [18]
This paper presents a modified Fourier method for free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations based on first-order shear deformation shallow shell theory (FSDSST)

Summary

Introduction

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]. It is noticed that the FG shallow shells are unavoidably suffered from dynamic loads, which can lead to fatigue wear and structural damage [13,14,15]. It is essential to study the free vibration characteristics of FG shallow shell structures. Extensive research efforts focused on vibration of FG shallow shells have been made by various shell theories, such as classical shallow shell theory (CSST) [16], first-order shear deformation theory (FSDT) [17], and higher-order shear deformation theory (HSDT) [18]. Numerous calculation methods, i.e., Ritz method, finite element method (FEM), generalized differential quadrature method, and wave propagation approach, have been developed [19,20,21,22,23,24,25,26,27]

Methods

Results

Conclusion