Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory
Introduction
In recent years, the application of functionally graded (FG) sandwich structures in aerospace, marine, civil construction is growing rapidly due to their high strength-to-weight ratio. There exist two common types: sandwich structures with FG core and sandwich structures with FG skins. With the wide application of FG sandwich structures, understanding vibration and buckling of FG sandwich structures becomes an important task. Based on the different shear deformation theories, though many works on these problems for FG beams are available [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], research on vibration and buckling of FG sandwich beams is a few in number. Di Sciuva and Gherlone [14] developed finite element formulations of the Hermitian Zig-Zag model to investigate the static and dynamic analyses of sandwich beams. Bhangale and Ganesan [15] derived finite element model to study thermal buckling and vibration analysis of a FG sandwich beam having constrained viscoelastic layer in thermal environment. Amirani et al. [16] used the element free Galerkin method for free vibration analysis of sandwich beam with FG core. Bui et al. [17] investigated transient responses and natural frequencies of sandwich beams with inhomogeneous FG core using a truly meshfree radial point interpolation method.
In this paper, which is extended from the previous work [18], finite element model for vibration and buckling of FG sandwich beams is presented. The developed theory accounts for parabolical variation of the transverse shear strain and stress through the beam depth, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. The core of sandwich beam is fully metal or ceramic and skins are composed of a FG material across the beam depth. Governing equations of motion and boundary conditions are derived from the Hamilton’s principle. Effects of power-law index, span-to-height ratio, core thickness and boundary conditions on the natural frequencies, critical buckling loads and load–frequency curves of sandwich beams are discussed. Numerical results show that the above-mentioned effects play very important role on the vibration and buckling analysis of FG sandwich beams.
Section snippets
Kinematics
Consider a FG sandwich beam, composed of ”Layer 1”, ”Layer 2”, and ”Layer 3”, as shown in Fig. 1. The x-, y-, and z-axes are taken along the length (L), width (b), and height (h) of the beam, respectively. The core of sandwich beam is fully metal or ceramic and skins are composed of a FG material across the beam depth. The vertical positions of the bottom and top, and of the two interfaces between the layers are denoted by , respectively. The effective material properties for
Variational formulation
In order to derive the equations of motion, Hamilton’s principle is used:where and denote the virtual variation of the strain energy, kinetic energy and potential energy, respectively.
The variation of the strain energy can be stated as:where and are the stress resultants, defined as:
Constitutive equations
The linear constitutive relations of a FG sandwich beam can be written as:
By using Eqs. (4), (9), and (14), the constitutive equations for stress resultants and strains are obtained:where are the stiffnesses of FG sandwich beams and given by:
Governing equations of motion
The equilibrium equations of the present study can be obtained by integrating the derivatives of the varied quantities by parts and collecting the coefficients of and :
The natural boundary conditions are of the form:
By substituting Eqs. (6), and (15) into
Finite element formulation
The present theory for FG sandwich beams described in the previous section was implemented via a displacement based finite element method. The variational statement in Eq. (13) requires that the bending and shear components of transverse displacement and be twice differentiable and -continuous, whereas the axial displacement u must be only once differentiable and -continuous. The generalized displacements are expressed over each element as a combination of the linear interpolation
Numerical examples
For verification purpose, the fundamental natural frequencies and critical buckling loads of FG beams with different values of span-to-height ratio for three boundary conditions, which are clamped-clamped (C-C), clamped-free (C-F) and simply-supported (S-S) are given in Table 1, Table 2, Table 3, Table 4. FG material properties are assumed to be [6]: Aluminum (Al: ) and Alumina (Al2O3: ). For buckling analysis, Li and Batra [11] used
Conclusions
Based on refined shear deformation theory, vibration and buckling of FG sandwich beams is presented. Governing equations of motion and boundary conditions are derived from the Hamilton’s principle. Finite element model is developed to determine the natural frequencies, critical buckling loads and load–frequency curves as well as corresponding mode shapes of FG sandwich beam with homogeneous hardcore and softcore. Effects of power-law index, span-to-height ratio, core thickness and boundary
Acknowledgements
The first author gratefully acknowledges research support fund for UoA16 from Northumbria University. The third author gratefully acknowledges financial support from Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2012.07. The fifth author gratefully acknowledges financial support by the Basic Research Laboratory Program of the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology
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