Free Vibration of Non-prismatic Sandwich Beams Using the Chebyshev Series

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%.

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An efficient semi-analytical static and free vibration analysis of laminated and sandwich beams based on linear elasticity theory

The Journal of Strain Analysis for Engineering Design ◽

10.1177/03093247211062688 ◽

2021 ◽

pp. 030932472110626

Author(s):

Zhao Yin ◽

Hangduo Gao ◽

Gao Lin

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Free Vibration ◽

Mechanical Load ◽

Geometric Model ◽

Free Vibration Analysis ◽

Sandwich Beams ◽

Beam Structure ◽

Precise Integration ◽

Element Method

Based on the two-dimensional (2D) elastic theory without enforcing any beam assumption, an efficient semi-analytical scaled boundary finite element method (SBFEM) is proposed to solve the bending and free vibration responses of composite laminated and sandwich beams under the mechanical load. The scaled center is placed at infinity, which produces the accurate result by discretizing only the longitudinal direction of the beam structure treated as a one-dimensional (1D) discretization problem. A new kind of 1D high-order spectral element shape functions with the advantages of high accuracy and superior convergence is introduced in SBFEM coordinate system to approximate the geometric model and corresponding variables. The principle of weighted residual in conjunction with the Green’s theorem are applied to obtain the SBFEM governing equation of each layer with respect to radial displacement fields. The solution of equation is indicated analytically by a matrix exponential function, which can be accurately solved by using the precise integration technique (PIT). Finally, an effective and simple stiffness matrix is obtained. By comparing two examples with the solutions based on the finite element method (FEM), the results show that the proposed method has good accuracy and rapid convergence with only a few meshes. The numerical examples are given to investigate the parametric effects of the stacking sequence, thickness ratio, boundary condition, and load form on the variation of the displacement, stress and natural frequency. The results validate that the present technique is also applicable to the complex beam structure with softcore layer inside.

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