Ich Cong Le


In this paper, free vibration of a bidirectional functionally graded sandwich (BFGSW) beams partially resting on a Pasternak foundation is studied. The beams with three layers, an axially functionally graded core and two bidirectional functionally graded face sheets, are made from a mixture of metal and ceramic. The material properties of the face sheets are considered to vary continuously in both the thickness and length directions by the power-law distributions, and they are estimated by Mori-Tanaka scheme. A sinusoidal shear deformation theory, in which the transverse displacement is split into bending and shear parts, is employed to derive energy expressions of the beam. A finite element formulation is formulated and employed to compute vibration characteristics. Numerical result reveals that the ratio of foundation support to the beam length plays an important role on the vibration behaviour, and the dependence of the frequencies upon the material grading indexes is governed by this ratio. Numerical investigation is carried out to highlight the effects of the material distribution, the layer thickness ratio, the foundation stiffness on the vibration characteristics of the beams. The influence of the aspect ratio on the frequencies of the beams and is also examined and discussed.


BFSW beam, Pasternak foundation, Trigonometric beam theory, Free vibration, Finite element formulation

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Fukui Y. - Fundamental investigation of functionally graded materials manufacturing system using centrifugal force, Japan Society of Mechanical Engineering International Journal, Series III, 34, (1991), pp. 144-148.

Koizumi M. - FGM activities in Japan, Composites Part B, 28, (1997), pp. 1-4.

Vo T.P., Thai H.T., Nguyen T.K., Inam F., Lee J. - A quasi-3D theory for vibration and buckling of functionally graded sandwich beams, Composite Structures, 119, (2015), pp. 1-12.

Thai H.T., Vo T.P. - A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates, Applied Mathematical Modelling, 37(5), (2013), pp.3269-3281.

Nguyen D.K., Nguyen Q.H., Tran T.T., Bui V.T. - Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, 228(1), pp. 141-155.

Vo T.P., Thai H.T., Nguyen T.K., Inam F., Lee J. - Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory, Engineering Structures, 64, pp. 12-22.

Nguyen T.K., Vo T.P., Nguyen B.D., Lee J. - An analytical solution for buckling and vibration analysis of functionally graded sandwich beams using a quasi-3D shear deformation theory, Compos Struct, 156, pp. 238-252.

Apetre N.A., Sankar B.V., Ambur D.R. - Analytical modelling of sandwich beams with functionally graded core, J. Sandw. Struct.Mater., 10, (2008), pp. 53–74.

Amirani M.C., Khalili S.M.R., Nemati N. - Free vibration analysis of sandwich beam with FG core using the element free Galerkin method, Composite Structures, 90, (2009), pp. 373–379.

Sakiyama T., Matsuda H., Morita C. - Free vibration analysis of sandwich beam with elastic or viscoelastic core by applying the discrete Green function, J. Sound Vib, 191, (1996), pp. 189–206.

Rahmani O., Khalili S.M.R., Malekzadeh K., Hadavinia H. - Free vibration analysis of sandwich structures with a flexible functionally graded syntactic core, Compos Struct, 91, (2009), 229-235.

Trinh L.C., Vo T.P., Osofero A.I., Lee J. - Fundamental frequency analysis of functionally graded sandwich beams based on the state space approach, Compos. Struct., 156, (2016), pp. 263–275.

Karamanli A. - Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory. Composite Structures, 174, (2017), pp.70-86.

Nguyen V.C., Le C. I., Le T.N.A., Nguyen D.K. - Elastostatic bending of 2D-FGSW beams under nouniform distributed loads. Vietnam Journal of Science and Technology, 57, no. 3, (2019), pp. 381-400.

Su Z., Jin G., Wang Y., Ye X. - A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations, Acta Mech., 227, (2016), pp. 1493–1514.

Tossapanon P., Wattanasakulpong N. - Stability and free vibration of functionally graded sandwich beams resting on two parameter elastic foundation, Compos. Struct., 142, (2016), pp. 215–225.

Zenkour A.M., Allam M.N.M., Sobhy M. - Bending analysis of FG viscoelastic sandwich beams with elastic cores resting on Pasternak’s elastic foundations. Acta Mechanicca, 212, (2010), pp. 233-252.

Songsuwan W., Pimsarn M., Wattanasakulpong N. - Dynamic responses of functionally graded sandwich beams resting on elastic Foundation under harmonic moving loads. International Journal of Structural Stability and Dynamics, 18, no. 9, (2018) , DOI: 10.1142/S0219455418501122.

Eisenberger M., Yankelevsky D. Z., Adin M. A. - Vibration of beams fully or partially supported on elastic foundation. Earthquake Engineering and Structural Dynamics, 13, (1985), pp. 651-660.

Le C.I., Pham V.N., Nguyen D.K. - Free vibration of FGSW plates partially supported by Pasternak foundation based on refined shear deformation theories, Mathematical Problems in Engineering, 2020}, (2020), pp. 1–13.

Mori T., Tanaka K. - Average stress in the matrix and average elastic energy of materials with misfitting inclusions, Acta Metallurgica, 21, (1973), pp. 571–574.

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