Free vibration and buckling analysis of elastically supported transversely inhomogeneous functionally graded nanoplate in thermal environment using Rayleigh–Ritz method

2020 ◽  
pp. 107754632096693
Author(s):  
Piyush P Singh ◽  
Mohammad S Azam
Keyword(s):  
Free Vibration ◽  
Thermal Environment ◽  
Ritz Method ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Small Scale ◽  
Pasternak Foundation ◽  
Differential Model ◽  
Aspect Ratios ◽  
Elastically Supported

In this study, free vibration and buckling behaviors of a functionally graded nanoplate supported by the Winkler–Pasternak foundation using a nonlocal classical plate theory are investigated. Eringen’s nonlocal differential model has been used for considering the small-scale effect. The properties of the functionally graded nanoplate are considered to vary transversely following the power law. The governing vibration and buckling equations of an elastically supported functionally graded nanoplate have been derived using the principle of virtual work, and the solution is obtained using the Rayleigh–Ritz method and characteristic polynomials. The advantage of this method is that it disposes of all the drawbacks regarding edge constraints. The objective of the article is to see the effect of edge constraints, aspect ratios, material property exponent, nonlocal parameter, and foundation parameters on the nondimensionalized frequency and the buckling load of an embedded functionally graded nanoplate in a thermal environment. The study highlights that the nonlocal effect is pronounced for higher modes and/or higher aspect ratios and need to be considered for the analysis of the nanoplate. Further, it is observed that the effect of the Pasternak foundation is prominent on nondimensionalized frequencies and buckling of the functionally graded nanoplate.

Free vibration analysis of functionally graded double-tapered beam rotating in thermal environment considering geometric nonlinearity, shear deformability, and Coriolis effect

2017 ◽  
Vol 232 (12) ◽  
pp. 2244-2262 ◽  
Author(s):  
Suman Pal ◽  
Debabrata Das
Keyword(s):  
Mathematical Model ◽  
Free Vibration ◽  
Material Properties ◽  
Volume Fraction ◽  
Thermal Environment ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Coriolis Effect ◽  
Governing Equations

The present work investigates the free vibration behavior of double-tapered functionally graded beams rotating in thermal environment, using an improved mathematical model. The functional gradation for ceramic–metal compositions, following power-law, is considered to be symmetric with respect to the mid-plane, leading to metal-rich core and ceramic-rich outer surfaces of the beam. The temperature dependence of the material properties are considered using Touloukian model. The nonlinearity in strain–displacement relationships for both the axial and transverse shear strains are considered. Firstly, the governing equations for deformed beam configuration under time-independent centrifugal loading are obtained using minimum total potential energy principle, and the solution is obtained following Ritz method. Then the free vibration problem of the centrifugally deformed beam is formulated employing Lagrange’s principle and considering tangent stiffness of the deformed beam configuration. Coriolis effect is considered in the mathematical model, and the governing equations are transformed to the state-space for obtaining an eigenvalue problem. The results for the first two modes of both chord-wise and flap-wise vibrations are presented in nondimensional plane to show the effects of taperness parameter, root-offset parameter, volume fraction exponent, operating temperature, and functionally graded material composition. The results in comparative form are presented for both temperature-dependent and temperature-independent material properties.

Thermoelastic Vibration and Flexural Behavior of FG-CNT Reinforced Composite Curved Panel

2017 ◽  
Vol 09 (04) ◽  
pp. 1750046 ◽  
Author(s):  
Kulmani Mehar ◽  
Subrata Kumar Panda ◽  
Bhumesh Kumar Patle
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Thermal Environment ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Flexural Behavior ◽  
Composite Panel ◽  
Aspect Ratios ◽  
Reinforced Composite ◽  
Curved Panel

The free vibration and flexural behavior of functionally graded carbon nanotube reinforced composite curved panel is investigated under uniform and linear thermal environment. The carbon nanotube reinforced composite curved panel has been modeled mathematically based on the higher-order shear deformation theory. The nanotube properties are assumed to be depended on the temperature and graded in the thickness direction using different grading rules. The governing equations for the static and vibration analysis of the functionally graded carbon nanotube reinforced composite panel are obtained using the variational method. Further, isoparametric finite element steps are implemented for the discretization of the governing equation and solved numerically via a specialized computer code developed in MATLAB environment. The rate of convergence and the validity of the presently developed numerical model have been checked. Finally, the effect of different geometrical and material parameters (thickness ratios, support conditions, volume fractions, thermal load, aspect ratios, and type of grading) on the free vibration and flexural behavior of functionally graded carbon nanotube reinforced composite are examined and discussed detail under thermal environment.

FREE VIBRATION OF FUNCTIONALLY GRADED THIN RECTANGULAR PLATES RESTING ON WINKLER ELASTIC FOUNDATION WITH GENERAL BOUNDARY CONDITIONS USING RAYLEIGH–RITZ METHOD

2014 ◽  
Vol 06 (04) ◽  
pp. 1450043 ◽  
Author(s):  
S. CHAKRAVERTY ◽  
K. K. PRADHAN
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Power Law ◽  
Ritz Method ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Aspect Ratios ◽  
Winkler Elastic Foundation ◽  
Special Cases

In this paper, free vibration of functionally graded (FG) rectangular plates subject to different sets of boundary conditions within the framework of classical plate theory is investigated. Rayleigh–Ritz method is used to obtain the generalized eigenvalue problem. Trial functions denoting the displacement components are expressed in simple algebraic polynomial forms which can handle any sets of boundary conditions. Material properties of the FG plate are assumed to vary continuously in the thickness direction of the constituents according to power-law form. The objective is to study the effects of constituent volume fractions, aspect ratios and power-law indices on the natural frequencies. New results for frequency parameters are incorporated after performing a test of convergence. Comparison with the results from the existing literature are provided for validation in special cases. Three-dimensional mode shapes are presented for FG square plates having various boundary conditions at the edges for different power-law indices. The present investigation also involves the rectangular FG plate to lay on a uniform Winkler elastic foundation. New results for the eigenfrequencies associated with foundation parameters are also reported here with the validation in special cases after checking a convergence pattern.

scholarly journals Free vibration of piezoelectric FGM sandwich beam with porous core embedded in thermal environment and elastic foundation

2021 ◽  
Vol 15 (4) ◽  
pp. 15-28
Author(s):  
Tran Quang Hung ◽  
Tran Minh Tu ◽  
Do Minh Duc
Keyword(s):  
Temperature Distribution ◽  
Free Vibration ◽  
Elastic Foundation ◽  
Thermal Environment ◽  
Ritz Method ◽  
Sandwich Beam ◽  
Beam Theory ◽  
Functionally Graded ◽  
Lagrange Equations ◽  
Linear Temperature

This paper aims to present thermo-electrical free vibration characteristics of functionally graded material (FGM) sandwich beam placed on the two-parameter elastic foundation. The beam is constructed of a foam core, two middle FGM layers, and two outer piezoelectric layers. It is assumed that the beam is subjected to a constant voltage and a uniform/linear temperature distribution. Physical properties of the core and two middle layers vary smoothly through the thickness according to the cosine and power-law forms, respectively. Lagrange equations in conjunction with the Reddy third-order beam theory is employed to derive the governing equations of motion. A simple polynomial trial function-based Ritz method is adopted for the approximation of the displacement field to obtain the vibration response. The correctness of the study is verified by comparisons with other authors’ published results. Influences of geometry parameters, material property distribution, applied voltage, elastic foundation, temperature distribution, temperature change, porosity coefficient, span-to-height ratio, and boundary conditions are investigated through parametric studies.

scholarly journals Free Vibration of FGSW Plates Partially Supported by Pasternak Foundation Based on Refined Shear Deformation Theories

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Cong Ich Le ◽  
Vu Nam Pham ◽  
Dinh Kien Nguyen
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Pasternak Foundation ◽  
Aspect Ratios ◽  
Hermitian Polynomials ◽  
Quadrilateral Finite Element

A refined third-order shear deformation theory (RTSDT), in which the transverse displacement is split into bending and shear parts, is employed to formulate a four-node quadrilateral finite element for free vibration analysis of functionally graded sandwich (FGSW) plates partially supported by a Pasternak foundation. An element based on the refined first-order shear deformation theory (RFSDT) which requires a shear correction factor is also derived for comparison purpose. The plates consist of a fully ceramic core and two functionally graded skin layers with material properties varying in the thickness direction by a power gradation law. The Mori–Tanaka scheme is employed to evaluate the effective moduli. The elements are derived using Lagrangian and Hermitian polynomials to interpolate the in-plane and transverse displacements, respectively. The numerical result reveals that the frequencies obtained by the RTSDT element are slightly higher than the ones using the RFSDT element. It is also shown that the foundation supporting area plays an important role on the vibration of the plates, and the effect of the material distribution on the frequencies is dependent on this parameter. A parametric study is carried out to highlight the effects of the material inhomogeneity, the foundation stiffness parameters, and the foundation supporting area on the frequencies and vibration modes. The influence of the layer thickness and aspect ratios on the frequencies is also examined and highlighted.

scholarly journals Free vibration analysis on axially graded beam resting on variable Pasternak foundation

2021 ◽  
Vol 1206 (1) ◽  
pp. 012016
Author(s):  
Saurabh Kumar
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Axial Direction ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Material Gradation ◽  
Euler Bernoulli Beam

Abstract Free vibration analysis is conducted on axially functionally graded Euler-Bernoulli beam resting on variable Pasternak foundation. The material properties of the beam and the stiffness of the foundation are considered to be varying linearly along the axial direction. Two types of boundary conditions namely; clamped and simply supported are used in the analysis. The problem is formulated using Rayleigh-Ritz method and governing equations are derived with the help of Hamilton’s principle. The numerical results are generated for different material gradation parameter, foundation parameter and boundary conditions and the effect of these parameters on the free vibration behaviour of the beam is discussed.

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