Dynamic response of axially functionally graded beam with longitudinal–transverse coupling effect

Xie, Ke; Wang, Yuewu; Fu, Tairan

doi:10.1016/j.ast.2018.12.004

Cited by 26 publications

(9 citation statements)

References 38 publications

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“…The vibration problems of a homogenous beam subjected to moving loads were extensively studied by [71][72][73][74][75]. The dynamic performance of one-directional FGM beams undergoing a moving load was studied based on the material gradation in the thickness direction by [76][77][78][79][80][81][82] or the material gradation in the length direction by [83][84][85]. Taking the features of BDFG into account, Şimşek [86] investigated the free and forced vibration responses of BDFG Timoshenko beams acted upon by moving loads using Ritz and the implicit time integration methods.…”

Section: Introductionmentioning

confidence: 99%

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Attia

Shanab

2022

Acta Mech

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This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

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Section: Introductionmentioning

confidence: 99%

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Attia

Shanab

2022

Acta Mech

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“…Gauss quadrature is used herein to evaluate the integrals in Eqs. ( 28)- (31). The highest order of the polynomials in ( 28) -( 31) is six, and thus the integrals can be evaluated exactly by 4 Gauss points.…”

Section: Solution Methodsmentioning

confidence: 99%

“…The shift of the physical neutral surface from the mid-plane was taken into account in deriving governing equation of the beams. Nonlinear free and forced vibration of of axially FGM beams under moving loads was considered by Xie et al [31] with the aid of Newton-Raphson and Newmark methods. Both Euler-Bernoulli and Timoshenko beam theories were adopted by the authors in deriving the governing equations of the beams.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Free Vibration of Functionally Graded Sandwich Beams Based on a Sinusoidal Shear Deformation Theory

Nguyen²

2022

Preprint

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Nonlinear free vibration of functionally graded sandwich beams is studied in this paper on the basis of a sinusoidal shear deformation theory and von Kármán geometric nonlinearity. The sandwich beams are composed of a pure ceramic core and two metal/ceramic functionally graded face sheets. The material properties of the face sheets are considered to be graded in the thickness by a power-law distribution, and they are evaluated by the Mori-Tanaka scheme. Nonlinear differential equations of motion are derived from Hamilton’s principle, and they are discretized by a two-node beam element. The direct iterative method is adopted to evaluate nonlinear frequencies of the sandwich beams with various boundary conditions. The numerical result reveals that the increase of the nonlinear frequency ratio by increasing the vibration amplitude is more significant for the beam associated with a higher power-law index. A parametric study is carried out to highlight the effect of the material and geometrical parameters on nonlinear vibration of the sandwich beams.

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“…Ghayesh [13] performed nonlinear forced vibration analysis on non-uniform AFGM beams utilising 3 rd order shear deformation theory and Galerkin's method where Hamilton's principle was used to derive the governing equations. Xie et al [14] studied the dynamic response of an AFGM beam subjected to moving transverse and longitudinal harmonic forces using Lagrange's equations and Newmark method. Zhang et al [15] used Jacobi polynomial theory to carry out free vibration analysis on AFGM beams where the beam was modelled using Euler-Bernoulli, Timoshenko and nonlocal strain gradient beam theories.…”

Section: Introductionmentioning

confidence: 99%

Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation

Kumar

2022

Trends Sci

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Free vibration analysis is carried out on axially inhomogeneous beams resting on Winkler-Pasternak elastic foundation. The material properties of the beam like Young’s modulus, modulus of rigidity and material density are considered to be varying along the length direction following constant, linear and exponential material models. The beam is subjected to different combinations of clamped and simply supported boundary conditions. The formulation is based on Timoshenko beam theory and energy method along with Hamilton’s principle is used to derive the governing equations. The effect of material gradation and the 2 parameters of elastic foundation on the natural frequencies are studied in detail. The present results are validated by comparing them with established ones and satisfactory matching is observed. HIGHLIGHTS Free vibration behavior of axially functionally graded beams in investigated Three different material models and three boundary conditions are considered Effect of two parameter elastic foundation is considered Timoshenko beam theory and Rayleigh Ritz method are employed It is found that the stiffness of elastic foundation significantly affects the natural frequency of the beam GRAPHICAL ABSTRACT

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Dynamic response of axially functionally graded beam with longitudinal–transverse coupling effect

Cited by 26 publications

References 38 publications

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Nonlinear Free Vibration of Functionally Graded Sandwich Beams Based on a Sinusoidal Shear Deformation Theory

Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation

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