scholarly journals Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

2019 ◽  
Vol 2019 ◽  
pp. 1-19
Author(s):  
Xiancheng Wang ◽  
Wei Li ◽  
Jianjun Yao ◽  
Zhenshuai Wan ◽  
Yu Fu ◽  
...  
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Ritz Method ◽  
Functionally Graded ◽  
General Boundary ◽  
Future Research ◽  
Geometrical Parameters ◽  
Shallow Shells ◽  
Fourier Cosine ◽  
Benchmark Solutions

In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.

scholarly journals The Modified Fourier-Ritz Approach for the Free Vibration of Functionally Graded Cylindrical, Conical, Spherical Panels and Shells of Revolution with General Boundary Condition

2017 ◽  
Vol 2017 ◽  
pp. 1-32 ◽  
Author(s):  
Lijie Li ◽  
Haichao Li ◽  
Fuzhen Pang ◽  
Xueren Wang ◽  
Yuan Du ◽  
...  
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
General Boundary ◽  
General Boundary Condition ◽  
Shells Of Revolution ◽  
Fourier Cosine ◽  
Cosine Series

The aim of this paper is to extend the modified Fourier-Ritz approach to evaluate the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.

scholarly journals A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First-Order Shear Deformation Theory

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Fuzhen Pang ◽  
Cong Gao ◽  
Jie Cui ◽  
Yi Ren ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Numerical Verification ◽  
First Order

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

A two-dimensional free vibration analysis of functionally graded sandwich beams under thermal environment

2012 ◽  
Vol 226 (12) ◽  
pp. 2860-2873 ◽  
Author(s):  
AR Setoodeh ◽  
M Ghorbanzadeh ◽  
P Malekzadeh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Vibration Analysis ◽  
Thermal Environment ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Geometrical Parameters ◽  
Sandwich Beams ◽  
Two Dimensional

In this article, free vibration analysis of elastically supported sandwich beams with functionally graded face sheets subjected to thermal environment is presented. In order to accurately include the transverse shear deformation and the inertia effects, two-dimensional elasticity theory is used to formulate the problem. The layerwise theory in conjunction with the differential quadrature method is employed to discretize the governing equations in the thickness and axial directions, respectively. The material properties of functionally graded face sheets are assumed to be temperature-dependent and graded in the thickness direction according to a power-law distribution. For the purpose of comparison, the problem under consideration is also solved using two-dimensional finite element method and the first-order shear deformation theory. The accuracy, convergence, and versatility of the method are demonstrated by comparing the results with those of the two aforementioned approaches and also with the existing solutions in literature. Eventually, some new numerical results are presented which depict the effects of different material and geometrical parameters on natural frequencies and mode shapes of the beam.

scholarly journals Free Vibration Of Functionally Graded Carbon Nanotube Reinforced Composite Annular Sector Plate With General Boundary Supports

2018 ◽  
Vol 5 (1) ◽  
pp. 49-67 ◽  
Author(s):  
Fuzhen Pang ◽  
Haichao Li ◽  
Yuan Du ◽  
Yanhe Shan ◽  
Fang Ji
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Ritz Method ◽  
Admissible Function ◽  
Functionally Graded ◽  
General Boundary ◽  
Reinforced Composite ◽  
Annular Sector ◽  
Sector Plate ◽  
Annular Sector Plate

Abstract In this paper, an efficient and unified approach for free vibration analysis of the moderately thick functionally graded carbon nanotube reinforced composite annular sector plate with general boundary supports is presented by using the Ritz method and the first-order shear deformation theory. For the distribution of the carbon nanotubes in thickness direction, it may be uniform or functionally graded. Properties of the composite media are based on a refined rule of the mixture approach which contains the efficiency parameters. A modified Fourier series is chosen as the basic function of the admissible function to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges. To establish the general boundary supports of the annular sector plate, the artificial spring boundary technique is implemented at all edges. The desired solutions are obtained through the Ritz-variational energy method. Some numerical examples are considered to check the accuracy, convergence and reliability of the present method. In addition, the parameter studies of the functionally graded carbon nanotube reinforced composite annular sector plate are carried out as well.

scholarly journals Static analysis of four-parameter functionally graded plates with general boundary conditions

2019 ◽  
Vol 13 (2) ◽  
pp. 12-23
Author(s):  
Hoang Thu Phuong ◽  
Tran Huu Quoc ◽  
Ho Thi Hien
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Material Properties ◽  
Load Distribution ◽  
Plate Theory ◽  
Ritz Method ◽  
Functionally Graded ◽  
General Boundary ◽  
Geometrical Parameters ◽  
General Boundary Conditions

In this study, the Ritz variational method is used to analyze and solve the bending problem of rectangular functionally graded material plate with general boundary conditions and subject to some types of load distribution over the entire plate domain. Based on the Kirchoff plate theory, the equilibrium equations are obtained by minimizing the total potential energy. The material properties are assumed to be graded through the thickness of the plates according to a power law with four parameters. The accuracy of the solution has been checked and validated through different comparisons to that available literature. A wide variety of examples have been carried out to reveal the influences of different geometrical parameters, FGM power law index, type of load distribution and boundary conditions on the bending responses of the plates. The results show that the gradients in material properties play an important role in determining the response of the FGM plates.Keywords: FGM; Kirchhoff plate; Ritz method; boundary conditions.

Free vibration of moderately thick functionally graded parabolic and circular panels and shells of revolution with general boundary conditions

2017 ◽  
Vol 34 (5) ◽  
pp. 1598-1641 ◽  
Author(s):  
Qingshan Wang ◽  
Dongyan Shi ◽  
Qian Liang ◽  
Fuzhen Pang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Power Law ◽  
Ritz Method ◽  
Functionally Graded ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Shells Of Revolution ◽  
Content Type ◽  
Vibration Behavior

Purpose The purpose of this work is to apply the Fourier–Ritz method to study the vibration behavior of the moderately thick functionally graded (FG) parabolic and circular panels and shells of revolution with general boundary conditions. Design/methodology/approach The modified Fourier series is chosen as the basis function of the admissible functions of the structure to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges, and the vibration behavior is solved by means of the Ritz method. The complete shells of revolution can be achieved by using the coupling spring technique to imitate the kinematic compatibility and physical compatibility conditions of FG parabolic and circular panels at the common meridian of θ = 0 and 2π. The convergence and accuracy of the present method are verified by other contributors. Findings Some new results of FG panels and shells with elastic restraints, as well as different geometric and material parameters, are presented and the effects of the elastic restraint parameters, power-law exponent, circumference angle and power-law distributions on the free vibration characteristic of the panels are also presented, which can be served as benchmark data for the designers and engineers to avoid the unpleasant, inefficient and structurally damaging resonant. Originality/value The paper could provide the reference for the research about the moderately thick FG parabolic and circular panels and shells of revolution with general boundary conditions. In addition, the change of the boundary conditions can be easily achieved by just varying the stiffness of the boundary restraining springs along all the edges of panels without making any changes in the solution procedure.

OUT-OF-PLANE FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR CURVED BEAMS SUPPORTED ON ELASTIC FOUNDATION

2010 ◽  
Vol 02 (03) ◽  
pp. 635-652 ◽  
Author(s):  
P. MALEKZADEH ◽  
M. R. GOLBAHAR HAGHIGHI ◽  
M. M. ATASHI
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Curved Beams ◽  
Out Of Plane ◽  
Benchmark Solutions

As a first endeavor, the out-of-plane free vibration analysis of thin-to-moderately thick functionally graded (FG) circular curved beams supported on two-parameter elastic foundation is presented. The formulation is derived based on the first-order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be graded in the direction normal to the plane of the beam curvature. The differential quadrature method (DQM), as an efficient and accurate method, is employed to discretize the equations of motion and the related boundary conditions. In order to assure the accuracy of the formulation and the method of solution, convergence behavior of the nondimensional natural frequencies is examined for FG circular curved beams and comparison studies with those of isotropic curved beams, available in the literature, are performed. The effects of the elastic foundation coefficients, boundary conditions, the material graded index and different geometrical parameters on the natural frequency parameters of the FG circular curved beams are investigated. The new results can be used as benchmark solutions for future research works.

On buckling and free vibration studies of sandwich plates and cylindrical shells: A review

2018 ◽  
Vol 33 (5) ◽  
pp. 673-724 ◽  
Author(s):  
Pavan Kumar ◽  
CV Srinivasa
Keyword(s):  
Free Vibration ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Future Research ◽  
Sandwich Plates ◽  
Composite Sandwich ◽  
Sandwich Plates And Shells ◽  
Graded Material ◽  
Plates And Shells

Many review articles were published on free vibration and buckling of laminated composites, sandwich plates, and shells. The present article reviews the literature on the buckling and free vibration analysis of shear deformable isotropic and laminated composite sandwich plates and shells using various methods available for plates in the past few decades. Various theories, finite element modeling, and experimentations have been reported for the analysis of sandwich plates and shells. Few papers on functionally graded material plates, plates with smart skin (electrorheological, magnetorheological, and piezoelectric), and also viscoelastic materials were also reviewed. The scope for future research on sandwich plates and shells was also accessed.

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