Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

Abstract

<jats:p> The purpose of the paper is to study stability and free vibrations of laminated plates and shallow shells composed of functionally graded materials. The approach proposed incorporates the Ritz method and the R-functions theory. It is assumed that the shell consists of three layers and is loaded in the middle plane. The both cases of uniform as well as non-uniform load are possible. The power-law distribution in terms of volume fractions is applied to get effective material properties for the layers. These properties are calculated for different arrangements and thicknesses of the layers by the analytical formulae obtained in the paper. The mathematical formulation is carried out in framework of the first-order shear deformation theory. The proposed approach consists of two steps. The first step is to define the pre-buckling state by solving the respective elasticity problem. The critical buckling load and frequencies of functionally graded material shallow shells are determined in the second step. The highlight of the method proposed is that it can be used for vibration and buckling analysis of plates and shallow shells of complex shape. The numerical results for frequencies and buckling load of plates and shallow shells of complex shape and different curvatures are presented to demonstrate the potential of the method developed. Different functionally graded material plates and shallow shells composed of a mixture of metal and ceramics are studied. The effects of the power law index, boundary conditions, thickness of the core, and face sheet layers on the fundamental frequencies and critical loads are discussed in this paper. The main advantage of the method is that it provides an analytical representation of the unknown solution, which is important when solving nonlinear problems. </jats:p>

“Nonlinear Dynamics of Mechanical Systems” dedicated to 75th birthday and 52 years scientific contribution of Prof. Katica R. (Stevanovic) Hedrih

“…Jiang et al based on a nonlinearity quantification of the complexity of the trajectories with nonlinearity measures a method for nonlinear system identification method is shown, and in rotor bearing system is applied. Buckling and free of force dynamic analyses of functionally graded sandwich plates and shallow shells using the R-functions in Kurpa and Shmatko 6 with good agreement with the literature results is shown. Mohajed et al 1 with nonlinear dynamic analysis showed the existence of a particular type of modal interactions with a self-adeptness of an attachment to narrow or broadband targeted energy transfers in low or high frequencies.…”

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

“…Thus, by putting the value of M ly from equations (7) to (17), one can obtain equation (18), which is the final governing equation of partially cracked orthotropic micro-plate with non-uniform thickness…”

“…An analytical investigation of the dynamic behaviour and natural frequencies of laminated nanocomposite plates have done by Khoa and Nguyen 16 Parida and Mohanty 17 used a higher-order shear deformation theory to develop a finite element model for the free vibration analysis of a rotating functionally graded material plate in the thermal environment. Kurpa and Shmatko 18 have proposed the Ritz method and the R-functions theory to study the stability and free vibrations of functionally graded sandwich plates and shallow shells. Civalek 19 has used a discrete singular convolution method to estimate an isotropic and orthotropic plate's natural frequencies with linearly varying thickness.…”

Vibration and buckling analyses of functionally graded plates based on refined plate theory using airy stress function

“…27,28 It should be noted that wide usage of FG materials were studied more in detail for not only rectangular plates but also for other various mechanical models/components, for example, nano-plates, shells, cylinders, disks, beams, vessels, and spheres. [29][30][31][32][33][34][35][36][37] In this paper, a new method based on the RPT is proposed to investigate the vibration and buckling of rectangular FG plates. This approach does not require a shear correction factor while satisfying shear stress free surface conditions.…”