Frequency Analysis of FG Sandwich Rectangular Plates with a Four-Parameter Power-Law Distribution

Document Type: Research Paper


1 Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University

2 Department of Mechanical Engineering, Razi University, Kermanshah

3 Young Researchers and Elite Club, Central Tehran Branch, Islamic Azad University


An accurate solution procedure based on the three-dimensional elasticity theory for the free vibration analysis of Functionally Graded Sandwich (FGS) plates is presented. Since no assumptions on stresses and displacements have been employed, it can be applied to the free vibration analysis of plates with arbitrary thickness. The two-constituent FGS plate consists of ceramic and metal graded through the thickness, from one surface of the each sheet to the other according to a generalized power-law distribution with four parameters. The benefit of using generalized power-law distribution is to illustrate and present useful results arising from symmetric, asymmetric and classic profiles. Using the Generalized Differential Quadrature (GDQ) method through the thickness of the plate, further allows one to deal with FG plates with an arbitrary thickness distribution of material properties. The fast rate of convergence and accuracy of the method are investigated through the different solved examples. The effects of different geometrical parameters such as the thickness-to-length ratio, different profiles of materials volume fraction and four parameters of power-law distribution on the vibration characteristics of the FGS plates are investigated. Interesting result shows that by utilizing a suitable four-parameter model for materials volume fraction, frequency parameter can be obtained more than the frequency parameter of the similar FGS plate with sheets made of 100% ceramic and at the same time lighter. Also, results show that frequencies of symmetric and classic profiles are smaller and larger than that of other types of FGS plates respectively. The solution can be used as benchmark for other numerical methods and also the refined plate theories.


[1] Tornabene F., Viola E., 2009, Free vibration analysis of four-parameter functionally graded parabolic panels and shells of revolution, European Journal of Mechanics - A/Solids 28:991-1013.

[2] Sobhani B., Yas M.H., 2010, Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels, International Journal of Mechanical Sciences 52:1047-1063.

[3] Sobhani B., Yas M.H., 2010, Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution, Acta Mechanica 215:155-173.

[4] Pourasghar A., Yas M.H., Kamarian S., 2013, Local aggregation effect of CNT on the vibrational behavior of four-parameter continuous grading nanotube reinforced cylindrical panels, Polymer Composites 34(5):707-721.

[5] Malekzadeh P., 2008, Three-dimensional free vibrations analysis of thick functionally graded plates on elastic foundations, Composite Structures 89(3):367-373.

[6] Yas M.H., Sobhani B., 2010, Free vibration analysis of continuous grading fiber reinforced plates on elastic foundation, International Journal of Engineering Science 48:1881-1895.

[7] Matsunaga H., 2008, Free vibration and stability of functionally graded plates according to a 2D higher-order deformation theory, Composite Structures 82:499-512.

[8] Li Q, Iu VP, Kou KP, 2008, Three-dimensional vibration analysis of functionally graded material sandwich plates, The Journal of Sound and Vibration 311(1–2):498–515.

[9] Zenkour AM., 2005, A comprehensive analysis of functionally graded sandwich plates: Part1- deflection and stresses. International Journal of Solids and Structures 42:5224–5242.

[10] Zenkour AM., 2005, A comprehensive analysis of functionally graded sandwich plates: Part2- buckling and free vibration deflection and stresses, International Journal of Solids and Structures 42:5243–5258.

[11] Khalili S.M.R., Mohammadi Y., 2012, Free vibration analysis of sandwich plates with functionally graded face sheets and temperature dependent material properties: A new approach, European Journal of Mechanics - A/Solids 35:61–74.

[12] Natarajan S., Manickam G., 2012, Bending and vibration of functionally graded material sandwich plates using an accurate theory, Finite Elements in Analysis and Design 57:32–42.

[13] Neves A.M.A., Ferreira A.J.M., Carrera E., Cinefra M., Roque C.M.C., Jorge R.M.N., Soares C.M.M., 2013, Static free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique, Composites Part B: Engineering 44(1):657–674.

[14] Sobhy M., 2012, Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions, Composite Structures 99:76-87.

[15] Song X., Gui-wen K., Ming-sui Y., Yan Z., 2013, Natural frequencies of sandwich plate with functionally graded face and homogeneous core, Composite Structures 96:226–231.

[16] Bellman R, Kashef B.G., Casti J., 1972, Differential Quadrature: a technique for a rapid solution of non linear partial differential equations, Journal of Computational Physics 10:40–52.

[17] Shu C., 2000, Differential Quadrature and Its Application in Engineering, Berlin, Springer.

[18] Kamarian S., Yas M.H., Pourasghar A., 2012, Free Vibrations of Continuous Grading Fiber Orientation Beams on Variable Elastic Foundations, Journal of Solid Mechanic 4(1): 75-83.

[19] Bert CW., Malik M., 1996, Differential quadrature method in computational mechanics, a review, Applied Mechanics Reviews 49:1-28.

[20] Yas M. H., Kamarian S., Eskandari J., Pourasghar A., 2011, Optimization of functionally graded beams resting on elastic foundations, Journal of Solid Mechanic 3(4):365-378.

[21] Yas M.H., Kamarian S., Pourasghar A.,2012, Application of imperialist competitive algorithm a and neural networks to optimize the volume fraction of three-parameter functionally graded beams, Journal of Experimental & Theoretical Artificial Intelligence , doi:10.1080/0952813X.2013.782346.