Free Vibrations of Three-Parameter Functionally Graded Plates Resting on Pasternak Foundations

Document Type : Research Paper


1 Composite Materials and Technology Center, MUT, Tehran

2 Department of Mechanical Engineering, Ilam Branch, Islamic Azad University

3 Young Researchers Club, Islamic Azad University, Tehran Markaz- Branch


In this research work, first, based on the three-dimensional elasticity theory and by means of the Generalized Differential Quadrature Method (GDQM), free vibration characteristics of functionally graded (FG) rectangular plates resting on Pasternak foundation are focused. The two-constituent functionally graded plate consists of ceramic and metal grading through the thickness. A three-parameter power-law distribution is considered for the ceramic volume fraction. The benefit of using a three-parameter power-law distribution is to illustrate and present useful results arising from symmetric, asymmetric and classic profiles. A detailed parametric study is carried out to highlight the influences of different profiles of fiber volume fraction, three parameters of power-law distribution and two-parameter elastic foundation modulus on the vibration characteristics of the FG plates. The main goal of the structural optimization is to minimize the weight of structures while satisfying all design requirements imposed. Thus, for the second aim of this paper, volume fraction optimization of FG plates with objective of minimizing the density to achieve a specified fundamental frequency is presented. The primary optimization variables are the three parameters of the volume fraction of ceramic. Since the optimization processes is complicated and too much time consuming, a novel meta–heuristic called Imperialist Competitive Algorithm (ICA) which is a socio-politically motivated global search strategy and Artificial Neural Networks (ANNs) are applied to obtain the best material profile through the thickness. The performance of ICA is evaluated in comparison with other nature inspired technique Genetic Algorithm (GA). Comparison shows the success of combination of ANN and ICA for design of material profile of FG plates. Finally the optimized material profile for the considered optimization problem is presented.


[1] Hosseini-Hashemi Sh., Akhavan H., Rokni Damavandi Taher H., Daemi N., Alibeigloo A., 2010, Differential quadrature analysis of functionally graded circularand annular sector plates on elastic foundation, Materials and Design 31: 1871–1880.
[2] Naderi A., Saidi A.R., 2011, Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation, Composite Structures 93: 629–638.
[3] Matsunaga H., 2008, Free vibration and stability of functionally graded plates according to a 2D higher-order deformation theory, Composite Structures 82: 499–512.
[4] Malekzadeh P., 2009, Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations, Composite Structures 89: 367–373.
[5] Yas M.H., Sobhani B., 2010, Free vibration analysis of continuous grading fibre reinforced plates on elastic foundation, International Journal of Engineering Science 48: 1881-1895.
[6] Viola E., Tornabene F., 2010, Free vibrations of three-parameter functionally graded parabolic panels of revolution, Mech Res Commun 163: 51-59.
[7] Yas M. H., Kamarian S., Eskandari J., Pourasghar A., 2012, Optimization of functionally graded beams resting on elastic foundations, Journal of Solid Mechanic.
[8] 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.
[9] Shu C., 2000, Differential quadrature and its application in engineering, Springer, Berlin.
[10] Shu C., Richards BE., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier Stockes equations, International Journal Numer Meth Fluid 15: 791-798.
[11] Abouhamze M., Shakeri M., 2007, Multi-objective stacking sequence optimization of laminated cylindrical panels using a genetic algorithm and neural networks, Composite Structures 81: 253–263.
[12] Walker M., Smith R., 2003, A technique for the multi objective optimization of laminated composite structures using genetic algorithms and finite element analysis, Composite Structures 62: 123–128.
[13] Jacob L. Pelletier, Senthil S. Vel., 2006, Multi-objective optimization of fiber reinforced composite laminates for strength, stiffness and minimal mass, Computers and Structures 84: 2065–2080.
[14] Atashpaz-Gargari E., Hashemzadeh F., Rajabioun R., Lucas C., 2008, Colonial competitive algorithm, a novel approach for PID controller design in MIMO distillation column process, International Journal of Intelligent Computing and Cybernetics 1: 337–355.
[15] Biabangard-Oskouyi A., Atashpaz-Gargari E., Soltani N., Lucas C., 2009, Application of imperialist competitive algorithm for materials property characterization from sharp indentation test, International Journal of Engineering Simulation 11–12.
[16] Khabbazi A., Atashpaz E., Lucas C., 2009, Imperialist competitive algorithm for minimum bit error rate beam forming, International Journal Bio-Inspired 1: 125 - 133.
[17] A. M. Jasour, E. Atashpaz, C. Lucas, 2008, Vehicle fuzzy controller design using imperialist competitive algorithm, Second Iranian Joint Congress on Fuzzy and Intelligent Systems, Tehran, Iran.
[18] Jodaei A., Jalal M., Yas M.H., 2012, Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN, Composites: Part B 43: 340-353.
[19] Ootao Y., Tanigawa Y., Nakamura T., 1999, Optimization of material composition of FGM hollow circular cylinder under thermal loading, a neural network approach Composites Part B 30: 415–422.
[20] Han X., Xu D., Liu G.R., 2003, A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network, Neuro computing 51: 341 – 360.
[21] W.Q. Chen, Z.G. Chen, 2003, Elasticity solution for free vibration of laminated beam, Composite Structures 62: 75–82.
[22] Atashpaz-Gargari E., Lucas C., 2007, imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition, IEEE congress on evolutionary computation 4661-4667.
[23] D. Zhou, YK. Cheung, SH. Lo, FTK. Au, 2004,Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundation, International Journal for Numerical Methods in Engineering 59 : 1313–1334.
[24] Shakeri M., Akhlaghi M., Hoseini S.M., 2006, Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder 76: 174–181.