Static Analysis of a Functionally Graded Beam under a Uniformly Distributed Load by Ritz Method

Abstract

Static analysis of a functionally graded (FG) simply-supported beam subjected to a uniformly distributed load has been investigated by using Ritz method within the framework of Timoshenko and the higher order shear deformation beam theories. The material properties of the beam vary continuously in the thickness direction according to the power-law form. Trial functions denoting the transverse, the axial deflections and the rotation of the cross-sections of the beam are expressed in trigonometric functions. In this study, the effect of various material distributions on the displacements and the stresses of the beam are examined. Numerical results indicate that stress distributions in FG beams are very different from those in isotropic beams

Keywords

• Beams, functionally graded materials, Timoshenko beam theory, the higher order shear deformation theory, Ritz method

References

1. [1] Sankar B.V., An elasticity solution for functionally graded beams. Composites Sciences and Technology, 61(5), 689-696, 2001.
2. [2] Chakraborty A., Gopalakrishnan S, Reddy J.N., A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45(3), 519-539, 2003.
3. [3] Chakraborty A., Gopalakrishnan S., A spectrally formulated finite element for wave propagation analysis in functionally graded beams. International Journal of Solids and Structures, 40(10), 2421-2448, 2003.
4. [4] Aydogdu M., Taskin V., Free vibration analysis of functionally graded beams with simply supported edges. Materials & Design, 28(5), 1651-1656, 2007.
5. [5] Zhong Z., Yu T., Analytical solution of a cantilever functionally graded beam. Composites Sciences and Technology, 67(3-4), 481-488, 2007.
6. [6] Ying J., Lü C.F., Chen W.Q., Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structures, 84(3), 209-219, 2008.
7. [7] Kapuria S., Bhattacharyya M., Kumar A.N., Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation. Composite Structures, 82(3), 390-402, 2008.
8. [8] Yang J., Chen Y., Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures, 83(1), 48-60, 2008.

9. [9] Li X.F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. Journal of Sound and Vibration, 318(4-5), 1210-1229, 2008.
10. [10] Yang J., Chen Y., Xiang Y., Jia X.L., Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. Journal of Sound Vibration, 312(1-2), 166-181, 2008.
11. [11] Kadoli R., Akhtar K., Ganesan N., Static analysis of functionally graded beams using higher order shear deformation theory. Applied Mathematical Modelling, 32(12), 2509-2525, 2008.
12. [12] Benatta M.A., Mechab I., Tounsi A., Adda Bedia E.A., Static analysis of functionally graded short beams including warping and shear deformation effects. Computational Materials Science, 44(2), 765-773, 2008.
13. [13] Sallai B.O., Tounsi A., Mechab I., Bachir B.M., Meradjah M., Adda B.E.A., A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Computational Materials Science, 44(4), 1344-1350, 2009.
14. [14] Sina S.A., Navazi H.M., Haddadpour H., An analytical method for free vibration analysis of functionally graded beams. Materials and Design, 30(3), 741-747, 2009.
15. [15] Şimşek M., Kocatürk T., Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Composite Structures, doi:10.1016/j.compstruct.2009.04.24, 2009.
16. [16] Şimşek M., Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures, submitted.
17. [17] Reddy J.N, Energy and Variational Methods in Applied Mechanics, John Wiley New York, 1984.
18. [18] Wakashima K., Hirano T., Niino M., Space applications of advanced structural materials, SP, 303-397, ESA 1990.