• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.161-169
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• 2017

In this paper, using consistent couple stress theory and Hamilton's principle, the free vibration analysis of Euler-Bernoulli nano-beams made of bi-directional functionally graded materials (BDFGMs) with small scale effects are investigated. To the best of the researchers' knowledge, in the literature, there is no study carried out into consistent couple-stress theory for free vibration analysis of BDFGM nanostructures with arbitrary functions. In addition, in order to obtain small scale effects, the consistent couple-stress theory is also applied. These models can degenerate into the classical models if the material length scale parameter is taken to be zero. In this theory, the couple-tensor is skew-symmetric by adopting the skew-symmetric part of the rotation gradients as the curvature tensor. The material properties except Poisson's ratio are assumed to be graded in both axial and thickness directions, which it can vary according to an arbitrary function. The governing equations are obtained using the concept of Hamilton principle. Generalized differential quadrature method (GDQM) is used to solve the governing equations for various boundary conditions to obtain the natural frequencies of BDFG nano-beam. At the end, some numerical results are presented to study the effects of material length scale parameter, and inhomogeneity constant on natural frequency.

• Steel and Composite Structures
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• v.24 no.6
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• pp.711-726
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• 2017

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded (FG) nanocomposite sandwich plates resting on Pasternak foundation are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. To determine the effect of CNT agglomeration on the elastic properties of CNT-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight CNTs. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of two-parameter elastic foundation modulus, geometrical and material parameters together with the boundary conditions on the frequency parameters of the laminated FG nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of CNTs agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated plates.

• Advances in materials Research
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• v.6 no.3
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• pp.279-301
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• 2017

Thermo-mechanical vibration characteristics of in homogeneousporous functionally graded (FG) micro/nanobeam subjected to various types of thermal loadings are investigated in the present paper based on modified couple stress theory with consideration of the exact position of neutral axis. The FG micro/nanobeam is modeled via a refined hyperbolic beam theory in which shear deformation effect is verified needless of shear correction factor. A modified power-law distribution which contains porosity volume fraction is used to describe the graded material properties of FG micro/nanobeam. Temperature field has uniform, linear and nonlinear distributions across the thickness. The governing equations and the related boundary conditions are derived by Extended Hamilton's principle and they are solved applying an analytical solution which satisfies various boundary conditions. A comparison study is performed to verify the present formulation with the known data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as thermal loadings, porosity volume fraction, power-law exponents, slenderness ratio, scale parameter and various boundary conditions on natural frequencies of porous FG micro/nanobeams in detail.

• Steel and Composite Structures
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• v.29 no.3
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• pp.363-377
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• 2018

An efficient and free of shear locking finite element model is developed and employed to study free vibration of tapered bidirectional functionally graded material (BFGM) beams. The beam material is assumed to be formed from four distinct constituent materials whose volume fraction continuously varies along the longitudinal and thickness directions by power-law functions. The finite element formulation based on the first-order shear deformation theory is derived by using hierarchical functions to interpolate the displacement field. In order to improve efficiency and accuracy of the formulation, the shear strain is constrained to constant and the exact variation of the cross-sectional profile is employed to compute the element stiffness and mass matrices. A comprehensive parametric study is carried out to highlight the influence of the material distribution, the taper and aspect ratios as well as the boundary conditions on the vibration characteristics. Numerical investigation reveals that the proposed model is efficient, and it is capable to evaluate the natural frequencies of BFGM beams by using a small number of the elements. It is also shown that the effect of the taper ratio on the fundamental frequency of the BFGM beams is significantly influenced by the boundary conditions. The present results are of benefit to optimum design of tapered FGM beam structures.

• Steel and Composite Structures
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• v.39 no.5
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• pp.493-509
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• 2021

There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature. The main purpose of this study is to develop a reliable and efficient computational modeling using an efficient functional in MFEM for free vibration and static analysis of FGM composite beams subject to high order shear deformation effects. The modeling of material properties was performed using mixture rule and Mori-Tanaka scheme which are more realistic determination techniques. This method based on the assumption that a two phase composite material consisting of matrix reinforced by spherical particles, randomly distributed in the beam. To explain the displacement components of the shear deformation effects, it was accepted that the shear deformation effects change sinusoidal. Partial differential field equations were obtained with the help of variational methods and then these equations were transformed into a novel functional for FGM beams with the help of Gateaux differential derivative operator. Thanks to the Gateaux differential method, the compatibility of the field equations was checked, and the field equations and boundary conditions were reflected to the function. A MFEM model was developed with a total of 10 degrees of freedom to apply the obtained functional. In the numerical applications section, free vibration and flexure problems solutions of FGM composite beams were compared with those predicted by other theories to show the effects of shear deformation, thickness changing and boundary conditions.

• Wind and Structures
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• v.23 no.6
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• pp.543-558
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• 2016

The response of functionally graded ceramic-metal plates is investigated using theoretical formulation, Navier's solutions, and a new displacement based on the high-order shear deformation theory are presented for static analysis of functionally graded plates. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. Numerical results of the new refined plate theory are presented to show the effect of the material distribution on the deflections, stresses and fundamental frequencies. It can be concluded that the proposed theory is accurate and simple in solving the static and free vibration behavior of functionally graded plates.

• Advances in nano research
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• v.6 no.3
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• pp.257-278
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• 2018

Vibration of axially functionally graded nano-rods and beams is investigated. It is assumed that the material properties change along the rod and beam length. The Ritz method with algebraic polynomials is used in the formulation of the problems. Stress gradient elasticity theory is utilized in order to include the nonlocal effects. Frequencies are obtained for different boundary conditions, geometrical and material properties. Nonlocal parameter is assumed as changing linearly or quadratically along the length of the nanostructure. Frequencies are compared to constant nonlocal parameter cases and considerable differences are observed between constant and variable nonlocal parameter cases. Mode shapes in various cases are depicted in order to explain the effects of axial grading.

• Smart Structures and Systems
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• v.17 no.5
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• pp.837-857
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• 2016

In the present study, for first time the size dependent vibration behavior of a rotating functionally graded (FG) Timoshenko nanobeam based on Eringen's nonlocal theory is investigated. It is assumed that the physical and mechanical properties of the FG nanobeam are varying along the thickness based on a power law equation. The governing equations are determined using Hamilton's principle and the generalized differential quadrature method (GDQM) is used to obtain the results for cantilever boundary conditions. The accuracy and validity of the results are shown through several numerical examples. In order to display the influence of size effect on first three natural frequencies due to change of some important nanobeam parameters such as material length scale, angular velocity and gradient index of FG material, several diagrams and tables are presented. The results of this article can be used in designing and optimizing elastic and rotary type nano-electro-mechanical systems (NEMS) like nano-motors and nano-robots including rotating parts.

• Structural Engineering and Mechanics
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• v.60 no.2
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• pp.313-335
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• 2016

In this article, a four-variable refined plate theory is presented for buckling analysis of functionally graded plates subjected to uniform, linear and non-linear temperature rises across the thickness direction. The theory accounts for parabolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. Young's modulus and Poisson ratio of the FGM plates are assumed to remain constant throughout the entire plate. However, the coefficient of thermal expansion of the FGM plate varies according to a power law form through the thickness coordinate. Equilibrium and stability equations are derived based on the present theory. The influences of many plate parameters on buckling temperature difference such ratio of thermal expansion, aspect ratio, side-to-thickness ratio and gradient index will be investigated.

• Smart Structures and Systems
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• v.26 no.3
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• pp.361-371
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• 2020

Exact solution for nonlinear behavior of clamped-clamped functionally graded (FG) buckled beams is presented. The effective material properties are considered to vary along the thickness direction according to exponential-law form. The in-plane inertia and damping are neglected, and hence the governing equations are reduced to a single nonlinear fourth-order partial-integral-differential equation. The von Kármán geometric nonlinearity has been considered in the formulation. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. Based on the mode of the corresponding linear problem, which readily satisfy the boundary conditions, the frequencies for the nonlinear problem are obtained using the Jacobi elliptic functions. The effects of various parameters such as the Young's modulus ratio, the beam slenderness ratio, the vibration amplitude and the magnitude of axial load on the nonlinear behavior are examined.