• Structural Engineering and Mechanics
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• v.75 no.6
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• pp.713-722
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• 2020

This article aims to illustrate the damped dynamic responses of layered functionally graded (FG) thick 2D beam under dynamic pulse sinusoidal load by using finite element method, for the first time. To investigate the response of thick beam accurately, two-dimensional plane stress problem is assumed to describe the constitutive behavior of thick beam structure. The material is distributed gradually through the thickness of each layer by generalized power law function. The Kelvin-Voigt viscoelastic constitutive model is exploited to include the material internal damping effect. The governing equations are obtained by using Lagrange's equations and solved by using finite element method with twelve -node 2D plane element. The dynamic equation of motion is solved numerically by Newmark implicit time integration procedure. Numerical studies are presented to illustrate stacking sequence and material gradation index on the displacement-time response of cantilever beam structure. It is found that, the number of waves increases by increasing the graduation distribution parameter. The presented mathematical model is useful in analysis and design of nuclear, marine, vehicle and aerospace structures those manufactured from functionally graded materials (FGM).

• Structural Engineering and Mechanics
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• v.69 no.2
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• pp.205-219
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• 2019

This paper analyzes nonlinear free vibration of the circular nano-tubes made of functionally graded materials in the framework of nonlocal strain gradient theory in conjunction with a refined higher order shear deformation beam model. The effective material properties of the tube related to the change of temperature are assumed to vary along the radius of tube based on the power law. The refined beam model is introduced which not only contains transverse shear deformation but also satisfies the stress boundary conditions where shear stress cancels each other out on the inner and outer surfaces. Moreover, it can degenerate the Euler beam model, the Timoshenko beam model and the Reddy beam model. By incorporating this model with Hamilton's principle, the nonlinear vibration equations are established. The equations, including a material length scale parameter as well as a nonlocal parameter, can describe the size-dependent in linear and nonlinear vibration of FGM nanotubes. Analytical solution is obtained by using a two-steps perturbation method. Several comparisons are performed to validate the present analysis. Eventually, the effects of various physical parameters on nonlinear and linear natural frequencies of FGM nanotubes are analyzed, such as inner radius, temperature, nonlocal parameter, strain gradient parameter, scale parameter ratio, slenderness ratio, volume indexes, different beam models.

• Advances in nano research
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• v.5 no.2
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• pp.141-169
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• 2017

In this paper, a nanobeam connected to a rotating molecular hub is considered. The vibration behavior of rotating functionally graded nanobeam based on Eringen's nonlocal theory and Euler-Bernoulli beam model is investigated. Furthermore, axial preload and porosity effect is studied. It is supposed that the material attributes of the functionally graded porous nanobeam, varies continuously in the thickness direction according to the power law model considering the even distribution of porosities. Porosity at the nanoscopic length scale can affect on the rotating functionally graded nanobeams dynamics. The equations of motion and the associated boundary conditions are derived through the Hamilton's principle and generalized differential quadrature method (GDQM) is utilized to solve the equations. In this paper, the influences of some parameters such as functionally graded power (FG-index), porosity parameter, axial preload, nonlocal parameter and angular velocity on natural frequencies of rotating nanobeams with pure ceramic, pure metal and functionally graded materials are examined and some comparisons about the influence of various parameters on the natural frequencies corresponding to the simply-simply, simplyclamped, clamped-clamped boundary conditions are carried out.

• Advances in nano research
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• v.16 no.3
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• pp.313-324
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• 2024

The main aims of this study are to develop a new nonlocal quasi-3D theory for the free vibration behaviors of the functionally graded sandwich nanobeams. The sandwich beams consist of a ceramic core and two functionally graded material layers resting on elastic foundations. The two layers, linear spring stiffness and shear layer, are used to model the effects of the elastic foundations. The size-effect is considered using nonlocal elasticity theory. The governing equations of the motion of the functionally graded sandwich nanobeams are obtained via Hamilton's principle in combination with nonlocal elasticity theory. Then the Navier's solution technique is used to solve the governing equations of the motion to achieve the nonlocal free vibration behaviors of the nanobeams. A deep parametric study is also provided to demonstrate the effects of some parameters, such as length-to-height ratio, power-law index, nonlocal parameter, and two parameters of the elastic foundation, on the free vibration behaviors of the functionally graded sandwich nanobeams.

• Structural Engineering and Mechanics
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• v.54 no.4
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• pp.693-710
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• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• Steel and Composite Structures
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• v.18 no.2
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• pp.409-423
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• 2015

In the present work, a simple and refined trigonometric higher-order beam theory is developed for bending and vibration of functionally graded beams. The beauty of this theory is that, in addition to modeling the displacement field with only 3 unknowns as in Timoshenko beam theory, the thickness stretching effect (${\varepsilon}_Z{\neq}0$) is also included in the present theory. Thus, the present refined beam theory has fewer number of unknowns and equations of motion than the other shear and normal deformations theories, and it considers also the transverse shear deformation effects without requiring shear correction factors. The neutral surface position for such beams in which the material properties vary in the thickness direction is determined. Based on the present refined trigonometric higher-order beam theory and the neutral surface concept, the equations of motion are derived from Hamilton's principle. Numerical results of the present theory are compared with other theories to show the effect of the inclusion of transverse normal strain on the deflections and stresses.

• Steel and Composite Structures
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• v.43 no.1
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• pp.31-54
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• 2022

This paper is concerned with the buckling behavior of functionally graded graphene reinforced porous nanocomposite beams based on the finite element method (FEM) using two variables trigonometric shear deformation theory. Both Young's modulus and material density of the FGP beam element are simultaneously considered as grading through the thickness of the beam. The finite element approach is developed using a nonlocal strain gradient theory. The governing equations derived here are solved introducing a 3-nodes beam element, and then the critical buckling load is calculated with different porosity distributions and GPL dispersion patterns. After a convergence and validation study to verify the accuracy of the present model, a comprehensive parametric study is carried out, with a particular focus on the effects of weight fraction, distribution pattern of GPL reinforcements on the Buckling behavior of the nanocomposite beam. The effects of various structural parameters such as the dispersion patterns for the graphene and porosity, thickness ratio, boundary conditions, and nonlocal and strain gradient parameters are brought out. The results indicate that porosity distribution and GPL pattern have significant effects on the response of the nanocomposite beams, and the results allows to identify the most effective way to achieve improved buckling behavior of the porous nanocomposite beam.

• Steel and Composite Structures
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• v.43 no.6
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• pp.797-808
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• 2022

This paper presents an investigation on the free vibration characteristics of functionally graded nanocomposite double-beams reinforced by single-walled carbon nanotubes (SWCNTs). The double-beams coupled by an interlayer spring, resting on the elastic foundation with a linear layer and shear layer, and is simply supported in thermal environments. The SWCNTs gradient distributed in the thickness direction of the beam forms different reinforcement patterns. The materials properties of the functionally graded carbon nanotube-reinforced composites (FG-CNTRC) are estimated by rule of mixture. The first order shear deformation theory and Euler-Lagrange variational principle are employed to derive the motion equations incorporating the thermal effects. The vibration characteristics under several patterns of reinforcement are presented and discussed. We conducted a series of studies aimed at revealing the effects of the spring stiffness, environment temperature, thickness ratios and carbon nanotube volume fraction on the nature frequency.

• Steel and Composite Structures
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• v.49 no.3
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• pp.293-306
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• 2023

This article focuses on the study of the buckling behavior of two-dimensional functionally graded (2D-FG) nanosize tubes, including porosity, based on the first shear deformation and higher-order theory of the tube. The nano-scale tube is simulated using the nonlocal gradient strain theory, and the general equations and boundary conditions are derived using Hamilton's principle for the Zhang-Fu's tube model (as a higher-order theory) and Timoshenko beam theory. Finally, the derived equations are solved using a numerical method for both simply-supported and clamped boundary conditions. A parametric study is performed to investigate the effects of different parameters, such as axial and radial FG power indices, porosity parameter, and nonlocal gradient strain parameters, on the buckling behavior of the bi-dimensional functionally graded porous tube. Keywords: Nonlocal strain gradient theory; buckling; Zhang-Fu's tube model; Timoshenko theory; Two-dimensional functionally graded materials; Nanotubes; Higher-order theory.

• Steel and Composite Structures
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• v.20 no.5
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• pp.963-981
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• 2016

A nonlocal trigonometric shear deformation beam theory based on neutral surface position is developed for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The present model is capable of capturing both small scale effect and transverse shear deformation effects of FG nanobeams, and does not require shear correction factors. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived by employing Hamilton's principle, and the physical neutral surface concept. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.