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.

Analytical and numerical investigation of the free vibration of functionally graded materials sandwich beams

2021 ◽  
Vol 2 (110) ◽  
pp. 72-85
Author(s):  
S.H. Bakhy ◽  
M. Al-Waily ◽  
M.A. Al-Shammari
Keyword(s):  
Analytical Solution ◽  
Free Vibration ◽  
Functionally Graded Materials ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Functionally Graded ◽  
Gradient Index ◽  
Sandwich Beams ◽  
Graded Materials ◽  
The Impact

Purpose: In this study, the free vibration analysis of functionally graded materials (FGMs) sandwich beams having different core metals and thicknesses is considered. The variation of material through the thickness of functionally graded beams follows the power-law distribution. The displacement field is based on the classical beam theory. The wide applications of functionally graded materials (FGMs) sandwich structures in automotive, marine construction, transportation, and aerospace industries have attracted much attention, because of its excellent bending rigidity, low specific weight, and distinguished vibration characteristics. Design/methodology/approach: A mathematical formulation for a sandwich beam comprised of FG core with two layers of ceramic and metal, while the face sheets are made of homogenous material has been derived based on the Euler–Bernoulli beam theory. Findings: The main objective of this work is to obtain the natural frequencies of the FG sandwich beam considering different parameters. Research limitations/implications: The important parameters are the gradient index, slenderness ratio, core metal type, and end support conditions. The finite element analysis (FEA), combined with commercial Ansys software 2021 R1, is used to verify the accuracy of the obtained analytical solution results. Practical implications: It was found that the natural frequency parameters, the mode shapes, and the dynamic response are considerably affected by the index of volume fraction, the ratio as well as face FGM core constituents. Finally, the beam thickness was dividing into frequent numbers of layers to examine the impact of many layers' effect on the obtained results. Originality/value: It is concluded, that the increase in the number of layers prompts an increment within the frequency parameter results' accuracy for the selected models. Numerical results are compared to those obtained from the analytical solution. It is found that the dimensionless fundamental frequency decreases as the material gradient index increases, and there is a good agreement between two solutions with a maximum error percentage of no more than 5%.

scholarly journals Bending of Symmetric Sandwich FGM Beams with Shear Connectors

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Dung Nguyen Thai ◽  
Phung Van Minh ◽  
Cuong Phan Hoang ◽  
Tam Ta Duc ◽  
Nhung Nguyen Thi Cam ◽  
...  
Keyword(s):  
Finite Element ◽  
Mechanical Response ◽  
Volume Fraction ◽  
Beam Theory ◽  
Numerical Data ◽  
Functionally Graded ◽  
Engineering Practice ◽  
Graded Materials ◽  
Shear Connectors ◽  
The Impact

This paper carries out the static bending analysis of symmetric three-layer functionally graded sandwich beams, in which each layer is made from different functionally graded materials, and they are connected by shear connectors due to sliding movement. The finite element formulations are based on Timoshenko’s first-order shear deformation beam theory (FSDT) and the finite element method to establish the equilibrium equation of beams. The calculation program is coded in the MATLAB environment, and then verification examples are given out to compare the numerical data of present work with those of exact open sources. The impact of several geometrical and material parameters on the mechanical response of the structure, such as the height-to-length ratio, boundary conditions, volume fraction index, and especially the shear coefficient of connectors, is being explored. When designing and using these types of structures in engineering practice, the computed results can be utilized as a valid reference.

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 Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

2020 ◽  
Vol 41 (9) ◽  
pp. 1303-1320
Author(s):  
Zhu Su ◽  
Lifeng Wang ◽  
Kaipeng Sun ◽  
Jie Sun
Keyword(s):  
Transverse Shear ◽  
Three Dimensional ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Penalty Function Method ◽  
Transverse Displacement ◽  
Three Dimensional Model ◽  
Special Cases

Abstract A quasi-three dimensional model is proposed for the vibration analysis of functionally graded (FG) micro-beams with general boundary conditions based on the modified strain gradient theory. To consider the effects of transverse shear and normal deformations, a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness. The conventional beam theories including the classical beam theory, the first-order beam theory, and the higherorder beam theory are regarded as the special cases of this model. The material properties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme. The energy-based formulation is derived by a variational method integrated with the penalty function method, where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables. The formulation is validated by some comparative examples, and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

Surface Effect on Static Bending of Functionally Graded Porous Nanobeams Based on Reddy’s Beam Theory

2019 ◽  
Vol 19 (06) ◽  
pp. 1950062 ◽  
Author(s):  
Jie Su ◽  
Yang Xiang ◽  
Liao-Liang Ke ◽  
Yue-Sheng Wang
Keyword(s):  
Surface Stress ◽  
Surface Effect ◽  
Beam Theory ◽  
Functionally Graded ◽  
Transverse Load ◽  
Transverse Displacement ◽  
Porosity Distribution ◽  
Residual Surface Stress ◽  
Porosity Coefficient ◽  
Distribution Type

In this paper, the surface effect on the static bending behavior of functionally graded porous (FGP) nanobeams subjected to a concentrated transverse load is studied by using Reddy’s higher-order beam theory. Three types of porosity distributions are considered for the nanobeam, i.e. uniform porosity distribution, symmetric and asymmetric non-uniform porosity distributions. With the consideration of the surface effect, the nanobeams can be abstracted as a composite beam composed of a surface layer and a bulk volume. According to the generalized Young–Laplace equation, the normal stress discontinuity across a surface due to the effect of surface stress is taken into consideration. The analytical solutions of the static bending problem of FGP nanobeams are obtained for the beams with hinged-hinged, clamped-clamped and clamped-free boundary conditions. The effects of the residual surface stress, porosity distribution type, porosity coefficient and length-to-thickness ratio on the transverse displacement of the FGP beams are discussed.

scholarly journals Nonlinear Analysis of a Functionally Graded Beam Resting on the Elastic Nonlinear Foundation

2014 ◽  
Vol 44 (2) ◽  
pp. 71-82 ◽  
Author(s):  
M. Arefi
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Beam Theory ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Nonlinear Foundation ◽  
Simply Supported ◽  
Nonlinear Responses ◽  
Fg Beam ◽  
Linear And Nonlinear

Abstract This paper evaluates the nonlinear responses of a function- ally graded (FG) beam resting on a nonlinear foundation. After derivation of fundamental nonlinear differential equation using the Euler-Bernouli beam theory, a semi analytical method has been used to study the response of the problem. The responses can be evaluated for both linear and nonlinear isotropic and FG beams individually. Adomians Decomposition and successive approximation methods have been used for solution of nonlinear differential equation. As numerical investigation, the beams with simply supported ends and linear and nonlinear foundations have been analyzed using this method.

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.

scholarly journals Fundamental frequencies of bidirectional functionally graded sandwich beams partially supported by foundation using different beam theories

2021 ◽  
Vol 72 (4) ◽  
pp. 452-467
Author(s):  
Ninh Vu Thi An
Keyword(s):  
Sandwich Beam ◽  
Beam Theory ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Height Ratio ◽  
Fundamental Frequencies ◽  
The Difference ◽  
Foundation Parameters ◽  
Material Gradation ◽  
Beam Theories

Investigation on the influence of beam theory and partial foundation support on natural frequencies play an important role in design of structures. In this paper, fundamental frequencies of a bidirectional functionally graded sandwich (BFGSW) beam partially supported by an elastic foundation are evaluated using various beam theories. The core of the sandwich beam is homogeneous while its two face sheets are made from three distinct materials with material properties varying in both the length and thickness directions by power gradation laws. The finite element method is employed to derive equation of motion and to compute the frequencies of the beam. The effects of the material gradation, the foundation parameters and the span to height ratio on the frequencies are studied in detail and highlighted. The difference of the frequencies obtained by different beam theories is also examined and discussed. The numerical results of the paper are useful in designing BFGSW beams with desired fundamantal frequencies.

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.

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