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.

scholarly journals Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zakaria Ibnorachid ◽  
Lhoucine Boutahar ◽  
Khalid EL Bikri ◽  
Rhali Benamar
Keyword(s):  
Shear Deformation ◽  
Elastic Foundation ◽  
Power Law ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Slenderness Ratio ◽  
Higher Order ◽  
Functionally Graded ◽  
Transverse Displacement

In this paper, free vibrations of Porous Functionally Graded Beams (P-FGBs), resting on two-parameter elastic foundations, and exposed to three forms of thermal field, uniform, linear, and sinusoidal, are studied using a Refined Higher-order shear Deformation Theory. The present theory accounts for shear deformation by considering a constant transverse displacement and a higher-order variation of the axial displacement through the thickness of the beam. The stress-free boundary conditions are satisfied on the upper and lower surfaces of the beam without using any shear correction factor. The material properties are temperature-dependent and vary continuously through the depth direction of the beam, based on a modified power-law rule, in which two kinds of porosity distributions, uniform, and nonuniform, through the cross-section area of the beam, are considered. Hamilton’s principle is applied to obtain governing equations of motion, which are solved using a Navier-type analytical solution for simply supported P-FGB. Numerical examples are proposed and discussed in detail, to prove the effect of the thermal environment, the porosity distribution, and the influence of several parameters such as the power-law index, porosity volume fraction, slenderness ratio, and elastic foundation parameters on the critical buckling temperatures and the natural frequencies of the P-FGB.

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.

Buckling capacity of functionally graded beams: A comparative study of different shear deformation beam theories

2019 ◽  
Vol 08 (02) ◽  
pp. 1950003
Author(s):  
Abdellatif Selmi
Keyword(s):  
Shear Deformation ◽  
Young’S Modulus ◽  
Phase Contrast ◽  
Young's Modulus ◽  
Beam Theory ◽  
Functionally Graded ◽  
Thickness Ratio ◽  
Reinforcement Distribution ◽  
Beam Edge ◽  
Beam Theories

This paper investigates the postbuckling response of simply supported functionally graded beams under axial loading using several beam theories; classical beam theory, CBT, Timoshenko beam theory, TBT and parabolic shear deformation beam theory, PSDBT. Hamilton’s principle is used to derive the governing equations which are solved by closed-form method. It is assumed that the Young’s modulus is varying continuously in the thickness direction according to power-law form while all other material properties are taken to be constant. The effects of the reinforcement distribution, the beam edge-to-thickness ratio and the phase contrast (ratio of the reinforcement Young’s modulus to the matrix Young’s modulus) on the postbuckling behavior and critical buckling load of functionally graded beam are studied. Results demonstrate the important contribution of the shear effect to both the buckling and postbuckling behaviors. Finally, the combinations of reinforcement distribution, beam edge-to-thickness ratio and phase contrast that correspond to the highest and the lowest buckling capacities are identified.

Free vibration analysis of imperfect functionally graded sandwich plates: analytical and experimental investigation

2021 ◽  
Vol 111 (2) ◽  
pp. 49-65
Author(s):  
E.K. Njim ◽  
S.H. Bakhy ◽  
M. Al-Waily
Keyword(s):  
Free Vibration ◽  
Power Law ◽  
Vibration Analysis ◽  
Sandwich Plate ◽  
Plate Theory ◽  
Slenderness Ratio ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Sandwich Plates ◽  
Classical Plate Theory

Purpose: This paper develops a new analytical solution to conduct the free vibration analysis of porous functionally graded (FG) sandwich plates based on classical plate theory (CPT). The sandwich plate made of the FGM core consists of one porous metal that had not previously been taken into account in vibration analysis and two homogenous skins. Design/methodology/approach: The analytical formulations were generated based on the classical plate theory (CPT). According to the power law, the material properties of FG plates are expected to vary along the thickness direction of the constituents. Findings: The results show that the porosity parameter and the power gradient parameter significantly influence vibration characteristics. It is found that there is an acceptable error between the analytical and numerical solutions with a maximum discrepancy of 0.576 % at a slenderness ratio (a/h =100), while the maximum error percentage between the analytical and experimental results was found not exceeding 15%. Research limitations/implications: The accuracy of analytical solutions is verified by the adaptive finite elements method (FEM) with commercial ANSYS 2020 R2 software. Practical implications: Free vibration experiments on 3D-printed FGM plates bonded with two thin solid face sheets at the top and bottom surfaces were conducted. Originality/value: The novel sandwich plate consists of one porous polymer core and two homogenous skins which can be widely applied in various fields of aircraft structures, biomedical engineering, and defense technology. This paper presents an analytical and experimental study to investigate the free vibration problem of a functionally graded simply supported rectangular sandwich plate with porosities. The objective of the current work is to examine the effects of some key parameters, such as porous ratio, power-law index, and slenderness ratio, on the natural frequencies and damping characteristics.

Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams

2017 ◽  
Vol 28 (15) ◽  
pp. 2007-2022 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohammad Reza Barati
Keyword(s):  
Elastic Foundation ◽  
Power Law ◽  
Slenderness Ratio ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Electric Voltage ◽  
Electro Magnetic ◽  
Functionally Graded Nanobeam

This article investigates vibration behavior of magneto-electro-elastic functionally graded nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of magneto-electro-elastic functionally graded nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen’s nonlocal elasticity theory which captures the small size effects and using Hamilton’s principle, the nonlocal governing equations of motions are derived and then solved analytically. Then, the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index, and slenderness ratio on the frequencies of the embedded magneto-electro-elastic functionally graded nanobeams are studied.

Natural frequency analysis of a functionally graded rotor-bearing system with a slant crack subjected to thermal gradients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arnab Bose ◽  
Prabhakar Sathujoda ◽  
Giacomo Canale
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Power Law ◽  
Beam Theory ◽  
Functionally Graded ◽  
Thermal Gradients ◽  
Element Analysis ◽  
Rotor Bearing ◽  
Natural Frequency Analysis ◽  
Bearing System

Abstract The present work aims to analyze the natural and whirl frequencies of a slant-cracked functionally graded rotor-bearing system using finite element analysis for the flexural vibrations. The functionally graded shaft is modelled using two nodded beam elements formulated using the Timoshenko beam theory. The flexibility matrix of a slant-cracked functionally graded shaft element has been derived using fracture mechanics concepts, which is further used to develop the stiffness matrix of a cracked element. Material properties are temperature and position-dependent and graded in a radial direction following power-law gradation. A Python code has been developed to carry out the complete finite element analysis to determine the Eigenvalues and Eigenvectors of a slant-cracked rotor subjected to different thermal gradients. The analysis investigates and further reveals significant effect of the power-law index and thermal gradients on the local flexibility coefficients of slant-cracked element and whirl natural frequencies of the cracked functionally graded rotor system.

scholarly journals A Refined Theory for Bending Vibratory Analysis of Thick Functionally Graded Beams

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1422
Author(s):  
Youssef Boutahar ◽  
Nadhir Lebaal ◽  
David Bassir
Keyword(s):  
Beam Theory ◽  
Functionally Graded ◽  
Effective Characteristics ◽  
Hyperbolic Function ◽  
Transverse Displacement ◽  
Refined Theory ◽  
Distribution Law ◽  
Fg Beam ◽  
The Impact ◽  
Beam Theories

A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams.

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