Analysis of Bi-directional FG Porous Sandwich Beams in Hygrothermal Environment Resting on Winkler/Pasternak Foundation, Based on the Layerwise Theory and Chebyshev Tau Method

Document Type: Original Article

Authors

1 Department of Mathematics, University of Mazandaran, Babolsar, Iran.

2 Department of Mechanical Engineering, University of Mazandaran, Babolsar, Iran.

10.22084/jrstan.2019.18781.1090

Abstract

In this paper, for the first time, displacement and stress analysis of bidirectional functionally graded (BDFG) porous sandwich beams are developed using the Chebyshev tau method. Based on the presented approach, sandwich beams under non-uniform load rested on Winkler/Pasternak foundation are analyzed. The material properties of core and each face sheet can be varied continuously in the axial and thickness directions, also the material properties are affected by the variation of temperature and moisture. To overcome some of the shortcomings of the traditional equivalent single layer theories for analysis of sandwich structures, governing equations are extracted based on the layerwise theory and five coupled differential equations are obtained. The resulting differential equations are solved using the Chebyshev tau method (CTM). The effectiveness of the CTM is demonstrated by comparing the obtained results with those extracted from the ABAQUS software. The comparisons indicate that the applied method to solve the systems of ordinary differential equations is efficient and very good accurate.

Keywords


[1] H. Niknam, A. Fallah, M.M. Aghdam, Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading, Int. J. Non-Linear Mech., 65 (2014) 141-147.
[2] S.R. Li, R.C. Batra, Relations between buckling loads of functionally graded Timoshenko and homogeneous euler-bernoulli beams, Compos. Struct., 95 (2013) 5-9.
[3] Y. Huang, X.F. Li, A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, J. Sound Vib., 329(11) (2010) 2291-2303.
[4] Y. Huang, L.E. Yang, Q.Z. Luo, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Compos. Part B: Eng., 45(1) (2013) 1493-1498.
[5] C.F. Lu, W.Q. Chen, R.Q. Xu, C.W. Lim, Semianalytical elasticity solutions for bi-directional functionally graded beams, Int. J. Solids Struct., 45(1) (2008) 258-275.
[6] H. Deng, W. Cheng, Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams, Compos. Struct., 141 (2016) 253-263.
[7] Z.H. Wang, X.H. Wang, G.D. Xu, S. Cheng, T. Zeng, Free vibration of two-directional functionally graded beams, Compos. Struct., 135 (2016) 191-198.
[8] X. Chen, Y. Lu, Y. Li, Free vibration, Buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium, Appl. Math. Modell., 67 (2019) 430-448.
[9] X. Chen, X. Zhang, Y. Lu, Y. Li, Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams, Int. J. Mech. Sci., 151 (2019) 424-443.
[10] M. Simsek, Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions, Compos. Struct., 149 (2016) 304-314.
[11] M. Simsek, Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of  Timoshenko beams with various boundary conditions, Compos. Struct., 133 (2015) 968-978.
[12] J. Li, Y. Guan, G. Wang, G. Zhao, J. Lin, H. Naceur, D. Coutellier, Meshless modeling of bending behavior of bi-directional functionally graded beam structures, Compos. Part B: Eng., 155 (2018) 104-111.
[13] S. Rajasekaran, H. Bakhshi Khaniki, Sizedependent forced vibration of non-uniform bidirectional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass, Appl. Math. Modell., 72 (2019) 129-154.
[14] T.T. Nguyen, J. Lee, Interactive geometric interpretation and static analysis of thin-walled bidirectional functionally graded beams, Compos. Struct., 191 (2018) 1-11.
[15] J. Fariborz, R.C. Batra, Free vibration of bidirectional functionally graded material circular beams using shear deformation theory employing logarithmic function of radius, Compos. Struct., 210 (2019) 217-230.
[16] A. Karamanli, Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory, Compos. Struct., 189 (2018) 127-136.
[17] A. Pydah, A. Sabale, Static analysis of bidirectional functionally graded curved beams, Compos. Struct., 160 (2017) 867-876.
[18] J. Lei, Y. He, ZH. Li, S. Guo, D. Liu, Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory, Compos. Struct., 209 (2019) 811-829.
[19] M.M. Alipour, M. Shariyat, Stress analysis of twodirectional FGM moderately thick constrained circular plates with non-uniform load and substrate stiffness distributions, J. Solid Mech., 2(4) (2010) 316-331.
[20] M.M. Alipour, M. Shariyat, A power series solution for free vibration of variable thickness Mindlin circular plates with two-directional material heterogeneity and elastic foundations, J. Solid Mech., 3(2) (2011) 183-197.
[21] C.I. Gheorghiu, Spectral Methods for Differential Problems, Tiberiu Popoviciu, Institute of Numerical Analysis, Cluj-Napoca Publisher, Romania, (2007).
[22] M. Shariyat, A.A. Jafari, M.M. Alipour, Investigation of the thickness variability and material heterogeneity effects on free vibration of the viscoelastic circular plates, Acta Mech. Solida Sin., 26(1) (2013) 83-98.
[23] C. Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, (1956) 464-517. 
[24] D. Johnson, Chebyshev Polynomials in the Spectral Tau Method and Applications to Eigenvalue Problems, University of Florida Gainesville Publisher, Florida, (1996).
[25] S. Etehadi, M. Botshekanan Dehkordi, Effect of axial stresses of the core on the free vibration response of a sandwich beam with FG carbon nanotube faces and stiff and flexible cores, J. Stress Anal., 3(2) (2019) 1-14.
[26] M. Shaban, Elasticity solution for static analysis of sandwich structures with sinusoidal corrugated cores, J. Stress Anal., 1(1) (2016) 23-31.
[27] H.I. Siyyam, M.I. Syam, An accurate solution of the Poisson equation by the Chebyshev-Tau method, J. Comput. Appl. Math., 85(1) (1997)1-10.
[28] M.R. Ahmadi, H. Adibi, The Chebyshev Tau technique for the solution of Laplace’s equation, Appl. Math. Comput., 184(2) (2007) 895-900.
[29] A. Saadatmandi, M. Dehghan, Numerical solution of hyperbolic telegraph equation using the Chebyshev Tau method, Numer. Methods Partial Differ., 26(1) (2010) 239-252.
[30] H. Wang, An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger Equations, Comput. Phys. Commun., 181(2) (2010) 325-340.
[31] M. Saravi, On the clenshaw method for solving linear ordinary differential equations, Am. J. Comput. Appl. Math., 1(2) (2011) 74-77.
[32] E. Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, Arch. Comput. Meth. Engng., 9(2) (2002) 87-140.
[33] E. Carrera, S. Brischetto, A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates, Appl. Mech. Rev., 62(1) (2009) 010803.
[34] M.M. Alipour, M Shariyat, Analytical zigzagelasticity transient and forced dynamic stress and displacement response prediction of the annular FGM sandwich plates, Compos. Struct., 106 (2013) 426-445.
[35] M.M. Alipour, A novel economical analytical method for bending and stress analysis of functionally graded sandwich circular plates with general elastic edge conditions, subjected to various, Compos. Part B: Eng., 95 (2016) 48-63.
[36] M.M. Alipour, M. Shariyat, Analytical layerwise stress and deformation analysis of laminated composite plates with arbitrary shapes of interfacial imperfections and discontinuous lateral deflections, Compos. Struct., 200 (2018) 88-102.
[37] M.M. Alipour, Effects of elastically restrained edges on FG sandwich annular plates by using a novel solution procedure based on layerwise formulation, Arch. Civ. Mech. Eng., 16(4) (2016) 678-694.
[38] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC Publisher, (2003).
[39] D.R. Gardner, S.A. Trogdon, R.W. Douglass, A modified tau spectral method that eliminates spurious eigenvalues, J. Comput. Phys., 80(1) (1989) 137-167.
[40] L. Fox, Chebyshev methods for ordinary differential equations, Comput. J., 4(4) (1962) 318-331.
[41] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods, SIAM Publisher, Philadelphia,(1977).
[42] W. Rudin, Real and Complex Analysis, McGrawHill, (1987).
[43] C. Canuto, M.Y. Hussaini, A.M. Quarteroni, T.H.A.J. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, Springer-Verlag Publisher, (1988).