American Journal of Mechanical Engineering
ISSN (Print): 2328-4102 ISSN (Online): 2328-4110 Website: Editor-in-chief: Kambiz Ebrahimi, Dr. SRINIVASA VENKATESHAPPA CHIKKOL
Open Access
Journal Browser
American Journal of Mechanical Engineering. 2016, 4(1), 11-20
DOI: 10.12691/ajme-4-1-3
Open AccessArticle

A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution

Fatemeh Farhatnia1,

1Mechanical Engineering department, Islamic Azad University Khomeinishahr Branch, Isfahan, Iran

Pub. Date: January 11, 2016

Cite this paper:
Fatemeh Farhatnia. A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution. American Journal of Mechanical Engineering. 2016; 4(1):11-20. doi: 10.12691/ajme-4-1-3


In this paper, we proposed a simple mathematical procedure to solve the differential equations governing the buckling and bending analysis of FG thick rectangular plates resting on two-parametric foundation based on Mindlin assumption. All edges are set on the simply supported conditions. Young modulus of the FG plate was assumed to vary according to a simple four-parameter power law across the thickness direction. For bending analysis, the plate was subjected to two kinds of loading: sinusoidal and uniform. For bucking analysis, two kinds of in-plane loading were applied to the plate: uniaxial and biaxial. Variations of FG material variation profile, thickness ratio, and foundation parameters on buckling critical load and out-plane displacement were examined. The distribution of axial and shear stress across the thickness, when the plate is exposed to uniform transverse loading, was further studied.

Mindlin rectangular plates power law FG distribution two parametric elastic foundations

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


Figure of 7


[1]  Shenshen H., Functionally Graded Materials Nonlinear Analysis of Plates and Shells, CRC Press, Taylor and Francis Group, USA , 2009.
[2]  Tounsi, A., Houari, M.S.A., Benyoucef, S., Adda Bedia, E.A., “A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates”, Aerosp Sci Technol, 24.209-220. 2013.
[3]  Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., Anwar Bég, O., “An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates”, Compos Part- B, 60. 274-283. 2014.
[4]  Meziane, A. A., Abdelaziz, H.H., Tounsi, A., “An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions“, J. Sandwich Struct. Mater. 16(3). 293-318. 2014.
[5]  Hebali, H., Tounsi, A., Hourai, M.S.A., Bessian, A., “A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates”, ASCE J. Eng. Mech., 140.374-383. 2014.
[6]  Mahi, A., Adda Bedia, E.A., Tounsi, A. (2015), "A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates”, Appl. Math. Modell, 39. 2489-2508.
[7]  Ait Yahia, S., Ait Atmane, H., Houari, M.S.A., Tounsi, A., “Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories”, Struct. Eng. Mech., 53(6). 1143-1165. 2015.
[8]  Bourada, M., Kaci, A., Houari, M.S.A., Tounsi, A., “A new simple shear and normal deformations theory for functionally graded beams”, Steel Compos. Struct. 18(2), 409-423. 2015.
[9]  Szilard, R., Theories and Applications of Plate Analysis, John Wiley and Sons, New Jersy, USA, 2004.
[10]  Lévy, M., “Mémoire sur la théorie des plaques élastiques “, J. Math. Pure Appl., 30. 219-306. 1877.
[11]  Reissner, E., “On the theory of bending of elastic plates”, J. Math Phys. 23, 184-191, 1944.
[12]  Mindlin, R.D., “Influence of rotary inertia on flexural motions of isotropic elastic plates”, J. Appl. Mech., 18.31-38. 1951.
[13]  Levinson, M.,”An accurate, simple theory of the statics and dynamics of elastic plates”, Mech. Res. Commun., 7(6). 343-350.1980.
[14]  Lanhe, W., “Thermal buckling of a simply supported moderately thick rectangular FGM plate”, Compos. Struct., 64. 211-218.2004.
[15]  Liew, K.M.,Chen X.L., “Buckling of rectangular Mindlin plates subjected to partial in-plane edge loads using the radial point interpolation method”, Int. J. Solids Struct., 41(5-6). 1677-1695.2004.
[16]  Shimpi, R. P. ,Patel, H. G. , Arya, H. , New first-order shear deformation plate theories, J. Appl. Mech., 74. 523-533. 2007.
[17]  Morimoto T., Tanigawa, Y.,”Linear buckling analysis of orthotropic inhomogeneous rectangular plates under uniform in-plane compression”, Acta Mech., 187(1), 219-229. 2006.
[18]  Abrate, S., “Functionally graded plates behave like homogeneous plates”, Compos. Part B: Eng. 39 (1). 151-158. 2008.
[19]  Bousahla, A.A., Houari, M.S.A., Tounsi, A., Adda Bedia, E.A., “A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates”, Int. J. Comput. Methods, 11(6), 1350082, (2014).
[20]  Abbasi, S., Farhatnia, F., Jazi, S. R., “A semi-analytical solution on static analysis of circular plate exposed to non-uniform axisymmetric transverse loading resting on Winkler elastic foundation”, Arch. Civil Mech. Eng. (ACME), 14. 476-488. 2014.
[21]  Birman, V., Plates Structures, Springer, New York, NY, USA, 2011.
[22]  Gupta, U.S., Ansari, A.H., Sharma, S. “Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation”, J. Sound Vib., 297. 457-476. 2006.
[23]  Rashed, Y.F., “A boundary integral transformation for ending analysis of thick plates resting on Bi-parameter foundation, Adv. Struct. Eng., 5(1). 13-22.2009.
[24]  Wen, P.H., “The fundamental solution of Mindlin plates resting on an elastic foundation in the Laplace domain and its applications”, Int. J. Solids Struct., 45. 1032-1050. 2008.
[25]  Civalek, O., “Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method”, Int. J. Mech. Sci., 49. 752-765. 2008.
[26]  Akhavan, H., Hosseini-Hashemi, S., Damavandi-Taher, HR, Alibeigloo A., Vahabi, S., “Exact solutions for rectangular Mindlin plates under in-plane loads resting on Pasternak elastic foundation, Part I: buckling analysis”, Comput. Mater. Sci., 44(3). 968-978. 2009.
[27]  Hosseini-Hashemi, Sh., Karimi, M., Hossein Rokni, D.T., “Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads”, Soil Dyn. Earthq. Eng., 30, 1487-1499. 2010.
[28]  Bouderba, B., Tounsi, A., Hourai, M.S.A., “Thermomechanical bending response of FGM thick plates resting on Winkler–Pasternak elastic foundations”, Steel Compos. Struct, 14(1). 85-104. 2013.
[29]  Zidi, M., Tounsi, A., Houari, M.S.A., Adda Bedia, E.A., Anwar Bég, O. (2014), “Bending analysis of FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory”, Aerosp. Sci. Techno., 34. 24-34. 2014.
[30]  Hamidi, A., Houari, M.S.A., Mahmoud, S.R., Tounsi, A.,“A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates”, Steel Compos. Struct.,, 18(1), 235-253. 2015.
[31]  Bennoun, M., Houari, M.S.A., Tounsi, A., “A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates”, Mech. Adv. Mat. Struct., 23(4). 423-431. 2016.
[32]  Tornabene F.,“Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution”, Comput. Methods Appl. Mech. Eng., 198, 2911-2935. 2009.
[33]  Latifi, M., Farhatnia F. and Kadkhodaei, M. “Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion”, Eur. J. Mech.-A/Solids, 41. 16-27. 2013.
[34]  Thai, H., Kim, S., “Closed-form solution for buckling analysis of thick functionally graded plates on elastic foundation”, Int. J. Mech. Sci., 5. 34-44. 2015.