Elsevier

Engineering Structures

Volume 64, 1 April 2014, Pages 12-22
Engineering Structures

Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory

https://doi.org/10.1016/j.engstruct.2014.01.029Get rights and content

Abstract

Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory is presented. The core of sandwich beam is fully metal or ceramic and skins are composed of a functionally graded material across the 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 functionally graded sandwich beams.

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 h0=-h2,h1,h2,h3=h2, respectively. The effective material properties for

Variational formulation

In order to derive the equations of motion, Hamilton’s principle is used:t1t2(δK-δU-δV)dt=0where δU,δK and δV denote the virtual variation of the strain energy, kinetic energy and potential energy, respectively.

The variation of the strain energy can be stated as:δU=0l0bn=13hn-1hn(σx(n)δx+σxz(n)δγxz)dzdydx=0l(Nxδz°+Mxbδκxb+Mxsδκxs+Qxzδγxz°)dxwhere Nx,Mxb,Mxs and Qxz are the stress resultants, defined as:Nx=n=13hn-1hnσx(n)bdzMxb=n=13hn-1hnσx(n)zbdzMxs=n=13hn-1hnσx(n)fbdzQxz=n=13hn-

Constitutive equations

The linear constitutive relations of a FG sandwich beam can be written as:σx(n)=E(n)xσxz(n)=E(n)21+ν(n)γxz=G(n)γxz

By using Eqs. (4), (9), and (14), the constitutive equations for stress resultants and strains are obtained:NxMxbMxsQxz=R11R12R130R22R230R330sym.R44x°κxbκxsγxz°where Rij are the stiffnesses of FG sandwich beams and given by:R11=n=13hn-1hnE(n)bdzR12=n=13hn-1hnzE(n)bdzR13=n=13hn-1hnfE(n)bdzR22=n=13hn-1hnz2E(n)bdzR23=n=13hn-1hnzfE(n)bdzR33=n=13hn-1hnf2E(n)bdzR44=n=13hn-1

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 δu,δwb and δws:Nx=m0u¨-m1w¨b-mfw¨sMxb-P0(wb+ws)=m0(w¨b+w¨s)+m1u¨-m2w¨b-mfzw¨sMxs+Qxz-P0(wb+ws)=m0(w¨b+w¨s)+mfu¨-mfzw¨b-mf2w¨s

The natural boundary conditions are of the form:δu:Nxδwb:Mxb-P0(wb+ws)-m1u¨+m2w¨b+mfzw¨sδwb:Mxbδws:Mxs+Qxz-P0(wb+ws)-mfu¨+mfzw¨b+mf2ws¨δws:Mxs

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 wb and ws be twice differentiable and C1-continuous, whereas the axial displacement u must be only once differentiable and C0-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: Em=70GPa,νm=0.3,ρm=2702kg/m3) and Alumina (Al2O3: Ec=380GPa,νc=0.3,ρc=3960kg/m3). For buckling analysis, Li and Batra [11] used νm=νc

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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