Large Deflections of a Non-linear Cantilever Functionally Graded Beam

Abstract

The analysis of the large deformation of a non-linear cantilever functionally graded material (FGM) beam is made. When subjected to an end moment, explicit expressions for deflection and rotation are derived for a functionally graded beam with work hardening of power law. The effects of the gradient distribution of Young’s modulus and the material non-linearity parameter on the deflections of the FGM beam are analyzed. Our results show that depth-dependent Young’s modulus and material non-linearity have a significant influence on the deflections of the beam, and a FGM beam can bear larger applied load than a homogeneous beam. Moreover, to determine an optimal gradient distribution, an optimum design of a beam of a lighter weight and larger stiffness is given. The influence of the geometric non-linearity of the beam is also studied. Large and small deformation theories predict nearly the same deflections with 5% error when rotation is less than 45°, and the predictions based on the small deformation theory are overestimated to exceed 10% when rotation is greater than 60°.

Keywords

functionally graded beam non-linear material large deflection.

Get full access to this article

View all access options for this article.

References

Suresh, S. ( 1997). Modeling and Design of Multi-layered and Graded Materials, Prog. Mater. Sci., 42: 243-251.

Miyamoto, Y. , Kaysser, W.A. , Rabin, B.H. , Kawasaki, A. and Ford, R.G. ( 1999). Functionally Graded Materials: Design, Processing and Applications, Kluwer Academic Publishers, Dordrecht.

Sankar, B.V. (2001). An Elasticity Solution for Functionally Graded Beams, Comp. Sci. Tech., 61: 689-696.

Zhong, Z. and Yu, T. ( 2007). Analytical Solution of a Cantilever Functionally Graded Beam, Comp. Sci. Tech., 67: 481-488.

Nirmala, K. , Upadhyay, P.C. , Prucz, J. and Lyons, D. ( 2006). Thermo-elastic Stresses in Composite Beams with Functionally Graded Layer, J. Reinf. Plast. Comp., 25: 1241-1254.

Babaei, M.H. , Abbasi, M. and Eslami, M.R. (2008). Coupled Thermoelasticity of Functionally Graded Beams, J. Therm. Stress, 31: 680-697.

Li, X.-F. (2008). A Unified Approach for Analyzing Static and Dynamic Behaviors of Functionally Graded Timoshenko and Euler-Bernoulli Beams, J. Sound Vib., 318: 1210-1229.

Pitakthapanaphong, S. and Busso, E.P. (2002). Self-consistent Elastoplastic Stress Solutions for Functionally Graded Material Systems Subjected to Thermal Transients, J. Mech. Phys. Solids, 50: 695-716.

Frisch-Fay, R. ( 1962). Flexible Bars, Butter Worths, London.

10.

Fertis, D.G. ( 1999). Nonlinear Mechanics, CRC Press , New York.

11.

Lo, C.C. and Gupta, S.D. ( 1978). Bending of a Non-linear Rectangular Beam in Large Deflection , J. Appl. Mech., 45: 213-215.

12.

Lau, J.H. (1982). Large Deflections of Beams with Combined Loads, ASCE J. Eng. Mech. Div., 108: 180-185.

13.

Chucheepsakul, S. , Buncharoen, S. and Wang, C.M. (1994). Large Deflection of Beams Under Moment Gradient, ASCE J. Eng. Mech., 120: 1848-1960.

14.

Wang, C.M. , Lam, K.Y. , He, X.Q. and Chucheepsakul, S. ( 1997). Large Deformation of an End Supported Beam Subjected to a Point Load, Int. J. Mech. Non-Linear Mech., 32: 63-72.

15.

Lee, K. ( 2002). Large Deflections of Cantilever Beams of Non-linear Elastic Material Under a Combined Loading, Int. J. Mech. Non-Linear Mech., 37: 439-443.

16.

Lewis, G. and Monasa, F. ( 1981). Large Deflections of Cantilever Beams of Nonlinear Materials, Comp. Struct., 14: 357-360.

17.

Lewis, G. and Monasa, F. ( 1982). Large Deflections of Cantilever Beams of Non-linear Materials of the Ludwick Type Subjected to an End Moment, Int. J. Non-Linear Mech., 17: 1-6.

18.

Baykara, C. , Guven, U. and Bayer, I. (2005). Large Deflections of a Cantilever Beam of Nonlinear Bimodulus Material Subjected to an End Moment , J. Reinf. Plast. Comp., 24: 1321-1326.

19.

Shatnawi, A.S. and Al-Sadder, S. (2007). Exact Large Deflection Analysis of Nonprismatic Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment, J. Reinf. Plast. Comp., 26: 1253-1268.

20.

Brojan, M. , Videnic, T. and Kosel, F. ( 2007). Non-prismatic Non-linearly Elastic Cantilever Beams Subjected to an End Moment, J. Reinf. Plast. Comp., 26: 1071-1082.

21.

Denton, A.A. (1966). Plane Strain Bending with Work Hardening, J. Strain Anal., 1: 196-203.

22.

Bassett, M.B. and Johnson, W. (1966). The Bending of Plate Using a Three-roll Pyramid Type Plate Bending Machine, J. Strain Anal., 1: 398-414.

23.

Gere, J.M. and Timoshenko, S.P. ( 1991). Mechanics of Materials, Chapman-Hall , London.