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Nonlinear Free Vibration of Functionally Graded Nanobeams on Nonlinear Elastic Foundation

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Iranian Journal of Science and Technology, Transactions of Civil Engineering Aims and scope Submit manuscript

Abstract

Nonlinear free vibration of functionally graded (FG) nanobeams on nonlinear elastic foundation is studied. The elastic foundation with cubic nonlinearity and a shearing layer is considered. The governing equations are derived based on the first-order shear deformation theory in conjunction with the von Kármán’s assumption and the Eringen’s nonlocal elasticity theory. Differential quadrature method as an efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. The discretized equations are transformed from time domain to frequency domain. The direct displacement control iterative method is used to solve the nonlinear system of algebraic equations, and the nonlinear frequency is obtained. Some examples are solved to show the applicability, rapid rate of convergence and accuracy of the method by comparing the results with those available in the literature. The effects of nonlocal parameter, boundary conditions, length-to-thickness ratios and the foundation parameters on the nonlinear free vibration of the FG nanobeams are studied.

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Abbreviations

A ij :

Extensional stiffness of the beam

\( A_{ij}^{x(r)} \) :

Weighting coefficient of the rth order

\( A_{ij}^{x} \) :

Weighting coefficients of the first-order derivative

\( B_{ij}^{x} \) :

Weighting coefficients of the second-order derivative

B :

Width of the beam

D ij :

Bending stiffness of the beam

\( D_{ij}^{x} \) :

Weighting coefficients of the fourth-order derivative

E :

Young’s modulus

G :

Shear modulus

h :

Total thickness of the beam

L :

Length of the beam

\( I_{\text{oo}} ,I_{22} \) :

Inertia of the beam

k 1, k 2 :

Linear and nonlinear coefficients of elastic foundation

k g :

Coefficient of shearing layer of the elastic foundation

\( K_{ 1} , \, K_{ 2} \) :

Nondimensional linear and nonlinear elastic foundation coefficients [\( {=}(k_{ 1} ,\frac{{k_{ 2} }}{{h^{2} }})\frac{{D_{11} }}{{L^{4} }} \)]

K g :

Nondimensional coefficients of shearing layer elastic foundation k g D 11/L 2

M xx :

Bending moment

\( M_{xx}^{nl} \) :

Nonlocal bending moment

N x :

Number of grid points in x-directions

N xx :

In-plane normal force resultant in x-directions

(N xx ) ij :

Discretized in-plane normal force resultant in x-directions

n :

Volume fraction coefficient

P :

FG beams material property

Q :

Transformed reduced stiffness

V :

Volume fraction

{u}:

The axial displacement degrees of freedom vector

w c :

Center deflection of transverse displacement

{w}:

The transverse displacement degrees of freedom vector

u, w :

Displacement component in the x and transverse direction of a point on midline of beam, respectively

W :

Amplitude of transverse displacement

x, z :

The Cartesian coordinate variables

ρ :

Mass density

ε 0 :

Convergence tolerance

ϕ x :

Bending rotation about y-axes

{ϕ x}:

The bending rotation degrees of freedom vector

κ :

Shear correction factor

μ :

Small-scale parameter

υ :

Poisson’s ratio

\( \omega_{l}^{l} \) :

Linear frequency obtained from local elasticity theory

\( \omega_{nl}^{l} \) :

Nonlinear frequency obtained from local elasticity theory

\( \omega_{nl}^{l} \) :

Linear frequency obtained from nonlocal elasticity theory

\( \omega_{nl}^{nl} \) :

Nonlinear frequency obtained from nonlocal elasticity theory

ω r :

Frequency at rth iteration

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, School of Engineering Shiraz University, Shiraz, Islamic Republic of Iran

    A. R. Vosoughi

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Vosoughi, A.R. Nonlinear Free Vibration of Functionally Graded Nanobeams on Nonlinear Elastic Foundation. Iran. J. Sci. Technol.Trans. Civ. Eng. 40, 23–32 (2016). https://doi.org/10.1007/s40996-016-0012-5

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