Free vibration analysis of two dimensional functionally graded materials Timoshenko beam

Subject Areas : Information Technology in Engineering Design (ITED) Journal

Mohammad Rahim Torshizian ¹ , Abdalhosein Okhovatpour ²

1 - Mechanical Engineering Department, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
2 - MS.c., Mechanical Engineering Department, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Received: 2017-03-13 Accepted : 2017-03-13 Published : 2015-08-23

Keywords:

Abstract :

In this research is investigated the free vibration analysis of two dimensional functionally graded material beams with considering Timoshenko's theory. Material properties of the beam are assumed to vary continuously in the length and thickness direction and follow from exponentially function. The governing equations of motions are derived by using Hamilton's principle. Both axial and rotary inertia of the beam are considered in present analysis. The governing differential equations of motions are converted to a set of first order differential equations using the state space technique and natural frequencies are determined. The effects of length to thickness ratio, gradient indices and boundary conditions on natural frequencies are investigated.

References:

[1]     B.V. Sankar,"An elasticity solution for functionally graded beams", Composites Science and Technology, Vol. 61, pp. 689-696, 2001.

[2]     S.H. Chi, Y.L. Chung, "Mechanical behavior of functionally graded material plates under transverse load-part I: Analysis", International Journal of Solids and Structures, Vol. 43, pp. 3657-3674,  2006.

[3]     M. Aydogdu, V. Taskin, "Free vibration analysis of functionally graded beams with simply supported edges", Materials and Design, Vol. 28, pp. 1651-1656, 2007.

[4]     H.J Xiang, J. Yang, "Free and forced vibration of a laminated FGM Timoshenko beam of variable  thickness under heat condition", Composites: Part B, Vol. 39, pp. 292-303, 2008.

[5]     M. Nemat-Alla, "Reduction of thermal stresses by developing two-dimensional functionally graded materials", International Journal of Solids and Structures, Vol. 40, pp. 7339-7356, 2003.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[6]     C.F. Lu, W.Q. Chen, R.Q. Xu, C.W. Lim, "Semi-analytical elasticity solutions for bi-directional functionally graded beams", International Journal of Solids and Structures, Vol. 45, pp. 258-275,  2008.

[7]     M.R. Torshizian, M.H. Kargarnovin, C. Nasirai, "Mode Ш fracture of an arbitrary oriented crack in two dimensional functionally graded materials", Mechanics Research Communications, Vol. 38, pp. 164-169,  2011.

[8]     S.A. Sina, H.M. Navazi, H. Haddadpour, "An analytical method for free vibration analysis of functionally graded beams", Materials and Design, Vol. 30, pp. 741-747,  2009.

[9]     C.M. Wang, J.N. Reddy, K.H. Lee, "Shear deformable beams and plates", First Ed. Elsevier, 2000, pp. 11-37.

[10] J.N Reddy, "Energy principles and variational methods in appllied mechanics", First Ed. John Eily & Sons, 1984, pp. 177-204.