Abstract A modified functionally graded beam theory based on the neutral axis is exploited to investigate natural frequencies of macro/nanobeams. The location of neutral axis is determined, where material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The FG nanobeam is investigated on the basis of the nonlocal Eringen continuum model. The kinematic assumption of nanobeam is proposed according to Euler–Bernoulli beam theory. The finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion. The verification of the model is obtained by comparing the current results with previously published work. Numerical results are presented to show the significance of neutral axis position, the material distribution profile, and nonlocal effect on the fundamental frequencies of nanobeams.

Determination of neutral axis position and its effect on natural frequenciesof functionally graded macro/nanobeams

In the paper, a novel model of Euler–Bernoulli beams made of bi-directional (2D) functionally graded materials (FGMs) is presented to study the nonlinear free vibration. We found that the 2D FGMs may induce asymmetric modes in free vibration, which is distinctly different from previous research. The Hamilton's principle is applied to derive the nonlinear governing equation of the beam and associated boundary conditions based on the geometric nonlinearity. The generalized differential quadrature method (GDQM) is used to predict the vibration modes. The closed-form solutions of the nonlinear free vibration of the beam are determined by the homotopy analysis method. The effects of the material distributions, length-thickness ratio and initial amplitude on the nonlinear free vibration are discussed in details. It is notable that the nonlinear dynamic properties are highly dependent on materials properties, which suggests that the vibration behaviors of the beam may be tailored/tuned by multi-direction FGMs.

Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration

State of the art in functionally graded materials

Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads

In this paper, vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory. The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs. The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams. The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton’s principle, which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions. Based on the numerical experiments, it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.

Chebyshev collocation approach for vibration analysis of functionally graded porous beams based on third-order shear deformation theory

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Free flexural vibration of tapered beams

Coupled axial-bending dynamic stiffness matrix for beam elements

The dynamic stiffness matrix of a coupled axial-bending Timoshenko beam is developed to investigate the free vibration behaviour of such beams and their assemblies. Applying Hamilton's principle, the governing differential equations of motion of a Timoshenko beam in free vibration is derived by considering the axial-bending coupling effect arising from the mass axis eccentricity with the elastic axis of the beam cross-section. The differential equations are then solved in an exact sense, giving expressions for the axial and bending displacements as well as the bending rotation. The expressions for axial force, shear force and bending moment are formed using the natural boundary conditions which resulted from the Hamiltonian formulation. Next, the frequency-dependent dynamic stiffness matrix of the coupled axial-bending Timoshenko beam is derived by relating the amplitudes of the axial force, shear force and bending moment to the corresponding amplitudes of axial displacement, bending displacement and bending rotation. The resulting dynamic stiffness matrix is effectively applied to investigate the free vibration behaviour of axial-bending coupled Timoshenko beams by making use of the Wittrick-Williams algorithm as solution technique. The results with emphasis on the axial-bending coupling effects and the importance of the shear deformation and rotatory inertia in free vibration behaviour of coupled axial-bending Timoshenko beams and frameworks are discussed with significant conclusions drawn.

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A. Ananthapuvirajah

Papers

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

An exact dynamic stiffness matrix for a beam incorporating Rayleigh–Love and Timoshenko theories

Free flexural vibration of tapered beams

Coupled axial-bending dynamic stiffness matrix for beam elements

Coupled axial-bending dynamic stiffness matrix and its applications for a Timoshenko beam with mass and elastic axes eccentricity