This paper deals with the free vibration response of rectangular functionally graded material sandwich nanoplates with simply supported boundary conditions. The material properties of the FGM layers are temperature-dependent and supposed to be graded continuously along the thickness direction. A simple power-law distribution in terms of the volume fractions of the material constituents is employed to obtained the effective material properties. Eringen’s nonlocal elasticity model is incorporated in order to take into account the small size effects. Two types of functionally graded material sandwich nanoplates are considered: a sandwich with functionally graded material face layers and homogeneous core, and a sandwich with homogeneous face layers and functionally graded material core. The equations of motion of the functionally graded material sandwich nanoplates are derived by using the higher shear deformation theory and the Hamilton’s variational principle, and solved using the Navier’s solutions. Several numerical results indicate the influence of the power–law index, the nonlocal parameter, the geometrical parameters of the nanoplate, and the temperature variation on the free vibration response are presented.

The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.

In this article, dynamic analysis of viscoelastic beam, which is made of porous functionally graded materials (FGM), is investigated by using the generalised differential quadrature method (GDQM). The material properties for FG beam are supposed to be graded in the thickness direction by two types of porosities and estimated using a modified power-law form. The surrounding visco-Pasternak foundation is made of viscous layer symbolised by dashpots, parallel springs and a shear layer. The governing equations achieved by Hamilton’s principle are solved implementing GDQM. Afterwards, the effects of types of porosity distribution and its pattern, foundation type, power-law index, slenderness ratio, structural damping coefficient and boundary conditions on the damping frequency of the present viscoelastic FG beams are explored.

This article aims to investigate the buckling and free vibration behaviors of nanocomposite beams reinforced with functionally graded (FG) agglomerated carbon nanotubes (CNTs) in polymeric beams matrix. In order to estimate the material properties of the FG CNTs reinforced beam, the Eshelby–Mori–Tanaka rule on the basis of equivalent fiber is employed. Moreover, the Lagrange’s principle is used to achieve the governing equations under supposition of the refined higher order beam theory without need of any shear correction factor. The differential quadrature finite element method (DQFEM) as a robust and precise numerical procedure is utilized to achieve the natural frequencies and buckling loads with various boundary supports. Then, the assessment of the effects of different factors such as of CNTs agglomeration parameters, different patterns of CNTs distribution, length to thickness quotients, and mixed boundary conditions on the vibrational and buckling responses of agglomerated CNTs reinforced beam are explored. It is concluded that the current numerical solution based DQFEM is efficient and accurate for predicting the mechanical behavior of the nanocomposite structures. Moreover, the stability and vibration responses of the NC beams can be significantly affected by tailoring the agglomeration parameters.

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

The contribution provided in this article is to examine free vibration and stability response of bidirectional graded material beams (BDFB) considering different boundary conditions. The stretch effect by means of a normal deformation in conjunction with the shear deformation influence are integrated for accurate results. The beam materials features are expected to depend on gradation pattern over the length and transverse directions via power law form. Lagrange’s principle is employed in order to extract the governing equations of motion. A novel approach namely differential quadrature-based Galerkin method (DQGM) is used as an effective and accurate semi-analytical tool to compute the critical stability loads and natural frequencies of (BD-FG) beam for the first time. The approach takes features from simplicity and reduced computational cost of the DQ in combination with the efficacy of the Galerkin method for its analytical aspect and easy dealing with mixed boundary conditions resulting in more accurate results. The correctness of the results obtained from the present approach for two-directional functionally graded materials (FGMs) beams have been verified. Finally, the impacts of design parameters such as material grading exponent distributions, length to thickness ratios, porosity amount, and end support types on the natural frequencies and maximum of stability loads related to the present 2D-FGM beams are examined.