Dynamic Response of Shear Deformable Functionally Graded Porous Beams

2016 ◽  
Vol 846 ◽  
pp. 434-439 ◽  
Author(s):  
Da Chen ◽  
Sritawat Kitipornchai ◽  
Jie Yang
Keyword(s):  
Dynamic Response ◽  
Metal Foam ◽  
Mass Density ◽  
Slenderness Ratio ◽  
Ritz Method ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Shear Strain ◽  
Harmonic Load ◽  
Shear Deformable

This paper analyzes the dynamic response of a shear deformable functionally graded porous beam under a point harmonic load. Timoshenko beam theory is employed to include the effect of transverse shear strain. The elasticity moduli and mass density of the porous beam vary continuously in the thickness direction based on two different porosity distributions. The relationship between porosity and density coefficients are determined according to the mechanical property of an open-cell metal foam. The equations of motion are derived and solved by applying Ritz method in the space domain and Newmark-β method in the time domain. The dynamic deflections are obtained for porous beams with different boundary conditions. A detailed numerical analysis is presented to show the effects of porosity coefficient and slenderness ratio on the dynamic response of porous beams. The influence of porosity distribution pattern is highlighted to shed a useful insight into the design of functionally graded porous structures.

Free and Forced Vibration of Beams Reinforced by 3D Graphene Foam

2020 ◽  
Vol 12 (05) ◽  
pp. 2050056 ◽  
Author(s):  
Feng Liu Yang ◽  
Yan Qing Wang
Keyword(s):  
Forced Vibration ◽  
Lagrange Equation ◽  
Mass Density ◽  
Slenderness Ratio ◽  
Ritz Method ◽  
Beam Theory ◽  
Dynamic Responses ◽  
Graphene Foam ◽  
3D Graphene ◽  
Reinforced Beams

In this paper, free and forced vibrations of nanocomposite beams reinforced by 3D graphene foam (3D-GrF) are studied. Different distributions of 3D-GrF in the beam thickness direction are considered. In accordance with the rule of mixture, the effective Young’s modulus, Poisson’s ratio and mass density of the 3D-GrF reinforced beams are predicted. Based on the Timoshenko beam theory, the governing equation of the 3D-GrF reinforced beam is derived by using the Lagrange equation. The natural frequencies and dynamic responses of the 3D-GrF reinforced beam are solved by the Ritz method and the Newmark-[Formula: see text] method, respectively. Results show that the foam coefficient, the 3D-GrF distribution, the slenderness ratio and the 3D-GrF mass fraction play important roles on free and forced vibration characteristics of the 3D-GrF reinforced beams.

Three-Dimensional Dynamics Analysis of Rotating Functionally Gradient Beams Based on Timoshenko Beam Theory

2019 ◽  
Vol 11 (04) ◽  
pp. 1950040 ◽  
Author(s):  
Ding Zhou ◽  
Jianshi Fang ◽  
Hongwei Wang ◽  
Xiaopeng Zhang
Keyword(s):  
Timoshenko Beam ◽  
Chebyshev Polynomials ◽  
Slenderness Ratio ◽  
Ritz Method ◽  
Three Dimensional ◽  
Beam Theory ◽  
Functionally Graded ◽  
Timoshenko Beam Theory ◽  
Strengthening Effect ◽  
Graded Materials

Through the Timoshenko beam theory (TBT), the 3D dynamics of a rotary functional gradient (FG) cantilever beam are investigated. Material capabilities alter continuously throughout the thickness obeying the power law. It is assumed that the Poisson’s ratio does not change. Based on the von Kármán nonlinearity, the governing equation is determined through the Hamilton principle, which includes the Coriolis effects. The couplings among the axial, flapwise and chordwise deformations caused by the usage of the functionally graded materials (FGMs) are revealed. Chebyshev polynomials are utilized to construct trial functions of deformations in the Rayleigh–Ritz method. The centrifugal strengthening effect caused by the rotational motion is described through the nonlinear axial shortening deformations derived from transverse deformations. The influences of the dimensionless angular velocity, FG index and slenderness ratio on vibration characteristics are studied. It is proved that the FG index significantly affects the dynamic response of deformation. For high-frequency external excitation cases, selection of Chebyshev polynomials as trial functions is more stable and effective than other polynomials.

scholarly journals Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

2021 ◽  
Vol 18 (9) ◽  
Author(s):  
Wachirawit SONGSUWAN ◽  
Monsak PIMSARN ◽  
Nuttawit WATTANASAKULPONG
Keyword(s):  
Dynamic Response ◽  
Elastic Foundation ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Sandwich Beams ◽  
Moving Loads ◽  
Layer Thickness Ratio ◽  
Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

scholarly journals Finite Element Analysis of Functionally Graded Beams using Different Beam Theories

2020 ◽  
Vol 6 (11) ◽  
pp. 2086-2102
Author(s):  
Farshad Rahmani ◽  
Reza Kamgar ◽  
Reza Rahgozar
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Power Law ◽  
Volume Fraction ◽  
Slenderness Ratio ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Transverse Shear Strain ◽  
Material Distribution ◽  
Beam Theories

The present study deals with buckling, free vibration, and bending analysis of Functionally Graded (FG) and porous FG beams based on various beam theories. Equation of motion and boundary conditions are derived from Hamilton’s principle, and the finite element method is adopted to solve problems numerically. The FG beams are graded through the thickness direction, and the material distribution is controlled by power-law volume fraction. The effects of the different values of the power-law index, porosity exponent, and different boundary conditions on bending, natural frequencies and buckling characteristics are also studied. A new function is introduced to approximate the transverse shear strain in higher-order shear deformation theory. Furthermore, shifting the position of the neutral axis is taken into account. The results obtained numerically are validated with results obtained from ANSYS and those available in the previous work. The results of this study specify the crucial role of slenderness ratio, material distribution, and porosity condition on the characteristic of FG beams. The deflection results obtained by the proposed function have a maximum of six percent difference when the results are compared with ANSYS. It also has better results in comparison with the Reddy formulae, especially when the beam becomes slender. Doi: 10.28991/cej-2020-03091604 Full Text: PDF

Static Analysis of a Functionally Graded Piezoelectric Beam Using Finite Element Method

2010 ◽  
Author(s):  
Iman Eshraghi ◽  
Aghil Yousefi-Koma
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shear Deformation ◽  
Static Analysis ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Functionally Graded ◽  
Bending Behavior ◽  
Functionally Graded Piezoelectric ◽  
Element Method

In this study static analysis of functionally graded piezoelectric material (FGPM) beams is performed using finite element modeling. First order shear deformation beam theory (Timoshenko beam theory) with the assumption of linear strain-displacement relations is used for modeling of displacement and strain fields in the beam. Theoretical formulations are derived employing Hamilton’s principle using linear constitutive relations of piezoelectric materials and including the effect of transverse shear deformation. Finite element method with one dimensional linear continuum isoparametric element, three displacement mechanical degrees of freedom, and one electric potential degree of freedom assigned to each node is then used to investigate the bending behavior of FGPM beam actuator under thermo-electro-mechanical loads. Consequently, a parametric study of the bending behavior of an FGPM beam is performed. The effects of slenderness ratio and fraction of volume of constituent materials, on the thermo-electro-mechanical characteristics are studied. It is shown that under electrical loading the beam represents the so-called non-intermediate behavior.

Dynamic Responses of Functionally Graded Sandwich Beams Resting on Elastic Foundation Under Harmonic Moving Loads

2018 ◽  
Vol 18 (09) ◽  
pp. 1850112 ◽  
Author(s):  
Wachirawit Songsuwan ◽  
Monsak Pimsarn ◽  
Nuttawit Wattanasakulpong
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Moving Load ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Harmonic Load ◽  
Sandwich Beams

This paper investigates the free vibration and dynamic response of functionally graded sandwich beams resting on an elastic foundation under the action of a moving harmonic load. The governing equation of motion of the beam, which includes the effects of shear deformation and rotary inertia based on the Timoshenko beam theory, is derived from Lagrange’s equations. The Ritz and Newmark methods are employed to solve the equation of motion for the free and forced vibration responses of the beam with different boundary conditions. The results are presented in both tabular and graphical forms to show the effects of layer thickness ratios, boundary conditions, length to height ratios, spring constants, etc. on natural frequencies and dynamic deflections of the beam. It was found that increasing the spring constant of the elastic foundation leads to considerable increase in natural frequencies of the beam; while the same is not true for the dynamic deflection. Additionally, very large dynamic deflection occurs for the beam in resonance under the harmonic moving load.

Dynamic Analysis of a Rotating Double-Tapered FGM Beam with Various Shear Models Using a Constrained Variational Method

2021 ◽  
Author(s):  
Zixuan Zhou ◽  
Xiuchang Huang ◽  
Hongxing Hua
Keyword(s):  
Variational Method ◽  
Shear Deformation ◽  
Power Law ◽  
Slenderness Ratio ◽  
Mathematical Formulation ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Functionally Graded ◽  
Geometrically Exact Beam ◽  
Beam Theories

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively.

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