Applying Differential Transform Method on the Effect of the Elastic Foundation on the out - Plane Displacement of the Functionally Graded Circular Plates

Subject Areas : Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering

سمیه عباسی ¹ , فاطمه فرهت نیا ² , سعید رسولی جزی ³

1 - کارشناس ارشد، دانشکده مکانیک، دانشگاه آزاد اسلامی واحد خمینی شهر
2 - استادیار، دانشکده مکانیک، دانشگاه آزاد اسلامی واحد خمینی شهر
3 - استادیار، دانشکده مکانیک، دانشگاه آزاد اسلامی واحد خمینی شهر

Received: 2015-09-28 Accepted : 2015-09-28 Published : 2012-06-21

Keywords: Circular plates, Differential transform method (DTM), Winkler elastic foundation, Functionally graded materials (FGM), Out of plane displacement,

Abstract :

In this paper, the effect of elastic foundation on the out of plane displacement of functionally graded circular plates using differential transform method is presented. Differential transform method is a semi-analytical-numerical solution technique that is capable to solve various types of differential equations. Using this method, governing differential equations are transformed into recursive relations and boundary conditions are changed into algebraic equations. Since the problem of plates on elastic foundation have a great practical importance in modern engineering structures and Winkler foundation model is widely used, plate is assumed on Winkler elastic foundation. In this article functionally graded plate is considered in which material properties vary through the thickness direction by power-law distribution. Analysis results of out of plane displacement of plate on elastic foundation under uniform transverse loads are obtained in different terms of foundation stiffness, material properties and boundary conditions. In order to validate the solution technique, results obtained are compared with the results of the finite element method (FEM).

References:

1] Yamanouchi M., Koizumi M., Shiota I., Proceedings of the first international symposium on functionally gradient materials, Sendai, Japan, 1990.

[2] Reddy J.N., Wang C.M., Kitipornchai S., Axisymmetric bending of functionally graded circular and annular plate, European Journal of Mechanics A/Solids, vol. 18, 1999, pp.185–199.

[3] Najafzadeh M.M., Eslami M.R., Buckling analysis of circular plates of functionally graded materials under uniform radial compression, International Journal of Mechanical Sciences, vol. 44, 2002, pp.  2479–2493

[4] Abrate S., Free vibration, buckling, and static deflections of functionally graded plates, Composites Science and Technology, vol. 66, 2006, pp. 2383–2394.

[5] Bayat M., Sahari B.B., Saleem M., Ali A., Wong S.V., Bending analysis of a functionally graded rotating disk based on the first order shear deformation theory, Applied Mathematical Modeling, vol. 33, 2009, pp. 4215-4230.

[61] Bodaghi M., Saidi A.R.,Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation, Archive of Applied Mechanic, vol. 81, 2011, pp. 765–780.

[7] Zenkour A.M., Allam M.N.M., Shaker M.O, Radwan A.F, On the simple and mixed first-order theories for plates resting on elastic foundations, Acta Mechanical, vol. 220, 2011, pp. 33–46.

[8] Gupta U.S., Ansari A.H., Sharma S, Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation, Journal of Sound and Vibration, vol. 297,  2006, pp. 457–476.

[9] Matsunaga H., Vibration and buckling of deep beam-columns on two-parameter elastic foundations, Journal of Sound and Vibration, vol. 228, 1999, pp. 359-376.

[10] El-Mously M., Fundamental frequencies of Timoshenko beams mounted on Pasternak foundatıon ,Journal of Sound and Vibration, vol. 228, 1999, pp. 452-457.

[11] Chen C.N., Vibration of prismatic beam on an elastic foundation by the differential quadrature element method, Computers and Structures, vol. 77, 2000, pp. 1–9.

[12] Coşkun İ, The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load, European Journal of Mechanics A/Solids, vol. 22, 2003, pp. 151–161.

[13] 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, Applied Mathematical Modeling, vol. 28, 2004, pp. 877–890.

[14] Luo H., Pozrikidis C., Buckling of a circular plate resting over an elastic foundation in simple shear flow, Journal of Applied Mechanics, vol. 75, 2008, pp. 1-6.

[15] Benyoucef S., Mechab I., Tounsi A., Fekrar A., Atmane H.A., Bedia E.A., Bending of thick functionally graded plates resting on Winkler- Pasternak elastic foundations, Mechanics of Composite Materials, vol. 4, 2010, pp. 425-434.

[16] Zhou J.K., Differential transformation and its applications for electrical circuits, Huarjung University Press, Wuuhahn, China, 1986.

[17] Arikoglu A., Ozkol I., Solution of boundary value problems for integro-differential equations by using differential transform method, Applied Mathematics and Computation, vol. 168, 2005, pp. 1145–1158.

[18] Momani Sh., Noor M.A., Numerical comparison of methods for solving a special fourth-order boundary value problem, Applied Mathematics and Computation, vol. 191, 2007, pp. 218–224.

[19] Yalcin H.S., Arikoglu A., Ozkol I., Free vibration analysis of circular plates by differential transformation method, Applied Mathematics and Computation, vol. 212, 2009, pp. 377–386.

[20] Chen C.K., Ho S.H., Solving partial differential equations by two dimensional differential transform, Applied Mathematics and Computation, vol. 106, 1999, pp. 171–179.

[21] Yeh Y., Jang M.J., Wang Ch., Analyzing the free vibrations of a plate using finite difference and differential transformation method, Applied Mathematics and Computation, vol. 178, 2006, pp. 493–501.

 [22] Özdemir Ö, Kaya MO, Flap wise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method, Journal of Sound and Vibration, vol. 289, 2006, pp. 413–420.

[23] Odibat Z., Momani Sh., A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters, vol. 21, 2008, pp. 194–199.

[24] Zhang D.G., Zhou Y.H., A theoretical analysis of FGM thin plates based on physical neutral surface, Computational Materials Science, vol. 44, 2008, pp. 716-720.

[25] Timoshenko S., Woinowsky- Krieger S.,Theory of Plates and Shells, second ed., McGraw-Hill Inc., New York, 1959, pp. 259-281.

[26] Li X.Y., Ding H.J., Chen W.Q., Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qr^k, International Journal of Solids Structures, vol. 45, 2008, pp. 191-210.