Abstract
Free vibration analysis of functionally graded sandwich beams with general boundary conditions and resting on a Pasternak elastic foundation is presented by using strong form formulation based on modified Fourier series. Two types of common sandwich beams, namely beams with functionally graded face sheets and isotropic core and beams with isotropic face sheets and functionally graded core, are considered. The bilayered and single-layered functionally graded beams are obtained as special cases of sandwich beams. The effective material properties of functionally graded materials are assumed to vary continuously in the thickness direction according to power-law distributions in terms of volume fraction of constituents and are estimated by Voigt model and Mori–Tanaka scheme. Based on the first-order shear deformation theory, the governing equations and boundary conditions can be obtained by Hamilton’s principle and can be solved using the modified Fourier series method which consists of the standard Fourier cosine series and several supplemented functions. A variety of numerical examples are presented to demonstrate the convergence, reliability and accuracy of the present method. Numerous new vibration results for functionally graded sandwich beams with general boundary conditions and resting on elastic foundations are given. The influence of the power-law indices and foundation parameters on the frequencies of the sandwich beams is also investigated.
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Iyengar K.T.S.R., Raman P.V.: Free vibration of rectangular beams of arbitrary depth. Acta Mech. 32, 249–259 (1979)
Thambiratnam D., Zhuge Y.: Free vibration analysis of beams on elastic foundation. Comput. Struct. 60, 971–980 (1996)
Lai H.Y., Hsu J.C.: An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method. Comput. Math. Appl. 56, 3204–3220 (2008)
Shafiei M., Khaji N.: Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load. Acta Mech. 221, 79–97 (2011)
Wang Z., Hong M., Xu J.C., Cui H.Y.: Analytical and experimental study of free vibration of beams carrying multiple masses and springs. J. Mar. Sci. Appl. 13, 32–40 (2014)
Chen W.Q., Lü C.F., Bian Z.G.: A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Appl. Math. Model. 28, 877–890 (2004)
Alshorbagy A.E., Eltaher M.A., Mahmoud F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35, 412–425 (2011)
Atmane, H.A., Tounsi, A., Meftah, S.A., Belhadj, H.A.: Free vibration behavior of exponential functionally graded beams with varying cross-section. J. Vib. Control 17, 311–318 (2010). doi:10.1177/1077546310370691
Li X.F.: A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J. Sound Vib. 318, 1210–1229 (2008)
Sina S.A., Navazi H.M., Haddadpour H.: An analytical method for free vibration analysis of functionally graded beams. Mater. Des. 30, 741–747 (2009)
Shahba A., Attarnejad R., Marvi M.T., Hajilar S.: Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos. Part B Eng. 42, 801–808 (2011)
Pradhan K.K., Chakraverty S.: Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos. Part B Eng. 51, 175–184 (2013)
Thai H.T., Vo T.P.: Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories. Int. J. Mech. Sci. 62, 57–66 (2012)
Şimşek M.: Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories. Nucl. Eng. Des. 240, 697–705 (2010)
Giunta G., Crisafulli D., Belouettar S., Carrera E.: Hierarchical theories for the free vibration analysis of functionally graded beams. Compos. Struct. 94, 68–74 (2011)
Rajabi K., Kargarnovin M.H., Gharini M.: Dynamic analysis of a functionally graded simply supported Euler–Bernoulli beam subjected to a moving oscillator. Acta Mech. 224, 425–446 (2013)
Wattanasakulpong N., Prusty B.G, Kelly D.W.: Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams. Int. J. Mech. Sci. 53, 734–743 (2011)
Lü C.F., Chen W.Q., Xu R.Q., Lim C.W.: Semi-analytical elasticity solutions for bi-directional functionally graded beams. Int. J. Solids Struct. 45, 258–275 (2008)
Chakraborty A., Gopalakrishnan S.: A higher-order spectral element for wave propagation analysis in functionally graded materials. Acta Mech. 172, 17–43 (2004)
Backström D., Nilsson A.C.: Modelling the vibration of sandwich beams using frequency-dependent parameters. J. Sound Vib. 300, 589–611 (2007)
Banerjee J.R., Sobey A.J.: Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam. Int. J Solids Struct. 42, 2181–2197 (2005)
Banerjee J.R., Cheung C.W., Morishima R., Perera M., Njuguna J.: Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment. Int. J. Solids Struct. 44, 7543–7563 (2007)
Banerjee J.R.: Free vibration of sandwich beams using the dynamic stiffness method. Comput. Struct. 81, 1915–1922 (2003)
Chen W.Q., Lv C.F., Bian Z.G.: Free vibration analysis of generally laminated beams via state-space-based differential quadrature. Compos. Struct. 63, 417–425 (2004)
Khalili S.M.R., Nemati N., Malekzadeh K., Damanpack A.R.: Free vibration analysis of sandwich beams using improved dynamic stiffness method. Compos. Struct. 92, 387–394 (2010)
Sokolinsky V.S., Von Bremen H.F., Lavoie J.A., Nutt S.R.: Analytical and experimental study of free vibration response of soft-core sandwich beams. J Sandw. Struct. Mater. 6, 239–261 (2004)
Amirani M.C., Khalili S.M.R., Nemati N.: Free vibration analysis of sandwich beam with FG core using the element free Galerkin method. Compos. Struct. 90, 373–379 (2009)
Apetre N.A., Sankar B.V., Ambur D.R.: Analytical modeling of sandwich beams with functionally graded core. J. Sandw. Struct. Mater. 10, 53–74 (2008)
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, 229–235 (2009)
Chakraborty A., Gopalakrishnan S., Reddy J.N.: A new beam finite element for the analysis of functionally graded materials. Int. J. Mech. Sci. 45, 519–539 (2003)
Pradhan S.C., Murmu T.: Thermo-mechanical vibration of an FGM sandwich beam under variable elastic foundations using differential quadrature method. J. Sound Vib. 321, 342–362 (2009)
Chakraborty A., Gopalakrishnan S.: A spectrally formulated finite element for wave propagation analysis in functionally graded beams. Int. J. Solids Struct. 40, 2421–2448 (2003)
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 Mech. 212, 233–252 (2010)
Vo T.P., Thai H.T., Nguyen T.K., Maheri A., Lee J.: Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng. Struct. 64, 12–22 (2014)
Bui T.Q., Khosravifard A., Zhang C. et al.: Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method. Eng. Struct. 47, 90–104 (2013)
Li W.L: Vibration analysis of rectangular plates with general elastic boundary supports. J. Sound Vib. 273, 619–635 (2004)
Beslin O., Nicolas J.: A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions. J. Sound Vib. 202, 633–655 (1997)
Ye T.G., Jin G.Y., Ye X.M., Wang X.R.: A series solution for the vibrations of composite laminated deep curved beams with general boundaries. Compos. Struct. 127, 450–465 (2015)
Su Z., Jin G.Y., Shi S.X., Ye T.G., Jia X.Z.: A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions. Int. J. Mech. Sci. 80, 62–80 (2014)
Su Z., Jin G.Y., Wang X.R.: Free vibration analysis of laminated composite and functionally graded sector plates with general boundary conditions. Compos. Struct. 132, 720–736 (2015)
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Su, Z., Jin, G., Wang, Y. et al. A general Fourier formulation for vibration analysis of functionally graded sandwich beams with arbitrary boundary condition and resting on elastic foundations. Acta Mech 227, 1493–1514 (2016). https://doi.org/10.1007/s00707-016-1575-8
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DOI: https://doi.org/10.1007/s00707-016-1575-8