Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method
TL;DR: In this paper, the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions is performed based on the classical and first order shear deformation beam theories.
Abstract: Present investigation is concerned with the free vibration analysis of functionally graded material (FGM) beams subjected to different sets of boundary conditions. The analysis is based on the classical and first order shear deformation beam theories. Material properties of the beam vary continuously in the thickness direction according to the power-law exponent form. Trial functions denoting the displacement components of the cross-sections of the beam are expressed in simple algebraic polynomial forms. The governing equations are obtained by means of Rayleigh–Ritz method. The objective is to study the effects of constituent volume fractions, slenderness ratios and the beam theories on the natural frequencies. To validate the present analysis, comparison studies are also carried out with the available results from the existing literature.
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TL;DR: In this article, the effects of material property distribution, spring constants and porosity volume fraction on linear and nonlinear frequencies of functionally graded materials (FGMs) beams with porosity phases were investigated.
339 citations
TL;DR: In this article, free and forced vibration of a bi-directional functionally graded (BDFG) Timoshenko beam under the action of a moving load was investigated by means of Lagrange equations based on TBT and Euler-Bernoulli beam theory.
196 citations
TL;DR: In this article, a finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory is presented, where the core of sandwich beam is fully metal or ceramic and skins are composed of a functionally graded material across the depth.
177 citations
Cites background from "Free vibration of Euler and Timoshe..."
...Based on the different shear deformation theories, though many works on these problems for FG beams are available ([1]-[13]), research on vibration and buckling of FG sandwich beams is a few in number....
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TL;DR: In this paper, the authors used the modified rule of mixture to approximate material properties of the FGM beams including the porosity volume fraction and the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams.
Abstract: Flexural vibration analysis of beams made of functionally graded materials (FGMs) with various boundary conditions is considered in this paper. Due to technical problems during FGM fabrication, porosities and micro-voids can be created inside FGM samples which may lead to the reduction in density and strength of materials. In this investigation, the FGM beams are assumed to have even and uneven distributions of porosities over the beam cross-section. The modified rule of mixture is used to approximate material properties of the FGM beams including the porosity volume fraction. In order to cover the effects of shear deformation, axial and rotary inertia, the Timoshenko beam theory is used to form the coupled equations of motion for describing dynamic behavior of the beams. To solve such a problem, Chebyshev collocation method is employed to find natural frequencies of the beams supported by different end conditions. Based on numerical results, it is revealed that FGM beams with even distribution of porosities have more significant impact on natural frequencies than FGM beams with uneven porosity distribution.
167 citations
TL;DR: In this paper, a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams is proposed.
Abstract: This paper proposes a new higher-order shear deformation theory for buckling and free vibration analysis of isotropic and functionally graded (FG) sandwich beams. The present theory accounts a new hyperbolic distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion are derived from Lagrange's equations. Analytical solutions are presented for the isotropic and FG sandwich beams with various boundary conditions. Numerical results for natural frequencies and critical buckling loads obtained using the present theory are compared with those obtained using the higher and first-order shear deformation beam theories. Effects of the boundary conditions, power-law index, span-to-depth ratio and skin-core-skin thickness ratios on the critical buckling loads and natural frequencies of the FG beams are discussed.
147 citations
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TL;DR: In this paper, a higher-order shear deformation theory of laminated composite plates is developed, which accounts for parabolic distribution of the transverse shear strains through the thickness of the plate.
Abstract: A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano (1970), but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.
3,504 citations
Book•
29 Mar 1991
TL;DR: In this article, the authors present a general solution based on the principle of virtual work for two-dimensional linear elasticity problems and their convergence rates in one-dimensional dimensions. But they do not consider the case of three-dimensional LEL problems.
Abstract: Mathematical Models and Engineering Decisions. Generalized Solutions Based on the Principle of Virtual Work. Finite Element Discretizations in One Dimension. Extensions and Their Convergence Rates in One Dimension. Two-Dimensional Linear Elastostatic Problems. Element-Level Basis Functions in Two Dimensions. Computation of Stiffness Matrices and Load Vectors for Two Dimensional Elastostatic Problems. Potential Flow Problems. Assembly, Constraint Enforcement, and Solution. Extensions and Their Convergence Rates in Two Dimensions. Computation of Displacements, Stresses and Stress Resultants. Computation of the Coefficients of Asymptotic Expansions. Three-Dimensional Linear Elastostatic Problems. Models for Plates and Shells. Miscellaneous Topics. Estimation and Control of Errors of Discretization. Mathematical Models. Appendices. Index.
2,748 citations
Book•
01 Jan 1982
TL;DR: In this paper, the Finite Element Method is used to derive a system equation from a set of finite element vectors and matrices and then to solve the problem of finding the solution.
Abstract: 1. Overview of the Finite Element Method, 2. Discretization of the Domain, 3. Interpolation Models, 4. Higher Order and Isoparametric Elements, 5. Derivation of Element Matrices and Vectors, 6. Assembly of Element Matrices and Vectors and Derivation of System Equations, 7. Numerical Solution of Finite Element Equations, 8. Basic Equations and Solution Procedure, 9. Analysis of Trusses, Beams and Frames, 10. Analysis of Plates, 11. Analysis of Three-Dimensional Problems, 12. Dynamic Analysis, 13. Formulation and Solution Procedure, 14. One-Dimensional Problems, 15. Two-Dimensional Problems, 16. Three-Dimensional Problems, 17. Basic Equations of Fluid Mechanics, 18. Inviscid and Incompressible Flows, 19. Viscous and Non-Newtonian Flows, 20. Solution of Quasi-Harmonic Equations, 21. Solution of Helmhotz Equation, 22. Solution of Reynolds Equation, Appendix-A Green Greass Theorem.
1,247 citations
TL;DR: In this article, a study on the vibration of cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented, the objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies.
726 citations
TL;DR: In this article, an elasticity solution for a functionally graded beam subjected to transverse loads is obtained, where Young's modulus of the beam is assumed to vary exponentially through the thickness, and the Poisson ratio is held constant.
603 citations