Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach

- Journal title : Structural Engineering and Mechanics
- Volume 59, Issue 1, 2016, pp.101-122
- Publisher : Techno-Press
- DOI : 10.12989/sem.2016.59.1.101

Title & Authors

Exact analysis of bi-directional functionally graded beams with arbitrary boundary conditions via the symplectic approach

Zhao, Li; Zhu, Jun; Wen, Xiao D.;

Zhao, Li; Zhu, Jun; Wen, Xiao D.;

Abstract

Elasticity solutions for bi-directional functionally graded beams subjected to arbitrary lateral loads are conducted, with emphasis on the end effects. The material is considered macroscopically isotropic, with Young`s modulus varying exponentially in both axial and thickness directions, while Poisson`s ratio remaining constant. In order to obtain an exact analysis of stress and displacement fields, the symplectic analysis based on Hamiltonian state space approach is employed. The capability of the symplectic framework for exact analysis of bi-directional functionally graded beams has been validated by comparing numerical results with corresponding ones in open literature. Numerical results are provided to demonstrate the influences of the material gradations on localized stress distributions. Thus, the material properties of the bi-directional functionally graded beam can be tailored for the potential practical purpose by choosing suitable graded indices.

Keywords

bi-directional functionally graded materials;analytical elasticity solutions;symplectic approach;state space;eigenfunction;

Language

English

Cited by

References

1.

Chu, P., Li, X.F., Wu, J.X. and Lee, K.Y. (2015), "Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending", Acta Mech., 226(7), 2235-2253.

2.

Ding, H.J., Huang, D.J. and Chen, W.Q. (2007), "Elasticity solutions for plane anisotropic functionally graded beams", Int. J. Solid. Struct., 44(1), 176-196

3.

Ebrahimi, M.J. and Najafizadeh, M.M. (2014), "Free vibration analysis of two-dimensional functionally graded cylindrical shells", Appl. Math. Model., 38, 308-324.

4.

Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.K. and Jorge, R.M.N. (2006), "Natural frequencies of functionally graded plates by a meshless method", Compos. Struct., 75, 593-600.

5.

Hedia, H.S. (2005), "Comparison of one-dimensional and two-dimensional functionally graded materials for the backing shell of the cemented acetabular cup", J. Biomed. Mater. Res.: Part B. Appl. Biomater., 74B(2), 732-739.

6.

Huang, D.J., Ding, H.J. and Chen, W.Q. (2009), "Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading", Sci. China Ser. G: Phys. Mech Astron, 52(8), 1244-1256.

7.

Huang, Y. and Li, X.F. (2011), "Buckling analysis of nonuniform and axially graded columns with varying flexural rigidity", J. Eng. Mech., 137(1), 73-81.

8.

Khalili, S.M.R., Jafari, A.A. and Eftekhari, S.A. (2010), "A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads", Compos. Struct., 92, 2497-2511.

9.

Koizumi, M. (1993), "The concept of FGM", Trans. Am. Ceram. Soc., 34, 3-10.

10.

Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B: Eng., 28(1-2), 1-4.

11.

Kuo, H.Y. and Chen, T.Y. (2005), "Steady and transient Green's functions for anisotropic conduction in an exponentially graded solid", Int. J. Solid. Struct., 42(3-4), 1111-1128.

12.

Leung, A.Y.T. and Zheng. J.J. (2007), "Closed form stress distribution in 2D elasticity for all boundary conditions", Appl. Math. Mech. Eng. Ed., 28(12), 1629-1642.

13.

Lezgy-Nazargah, M. (2015), "Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach", Aero. Sci. Tech., 45,154-164.

14.

Lu, C.F., Chen, W.Q., Xu, R.Q. and Lim, C.W. (2008), "Semi-analytical elasticity solutions for bi-directional functionally graded beams", Int. J. Solid. Struct., 45(1), 258-275.

15.

Lu, C.F., Lim, C.W. and Chen, W.Q. (2009), "Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions", Int. J. Numer. Meth. Eng., 79(1), 25-44.

16.

Nemat-Alla, M. (2003), "Reduction of thermal stresses by developing two-dimensional functionally graded materials", Int. J. Solid. Struct., 40(26), 7339-7356.

17.

Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Roque, C.M.C., Cinefra, M. and Jorge R.M.N. (2011), "Bending of FGM plates by a sinusoidal plate formulation and collocation with radial basis functions", Mech. Res. Commun., 38(5), 368-371.

18.

Nie, G. and Zhong, Z. (2010), "Dynamic analysis of multi-directional functionally graded annular plates", Appl. Math. Model., 34, 608-616.

19.

Qian, L.F. and Batra, R.C. (2005), "Design of bidirectional functionally graded plate for optimal natural frequencies", J. Sound Vib., 280(1-2), 415-424.

20.

Reddy, J.N. (2000), "Analysis of functionally graded plates", Int. J. Numer. Meth. Eng., 47, 663-684.

21.

Sallai, B.O., Tounsi, A., Mechab, I., Bachir, B.M., Meradjah, M. and Adda Bedia, E.A. (2009), "A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams", Comput. Mater. Sci., 44, 1344-1350.

22.

Sankar, B.V. (2001), "An elastic solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696.

23.

Shahba, A. and Rajasekaran, S. (2012), "Free vibration and stability of tapered Euler-Bernoulli beams made of axially functionally graded materials", Appl. Math. Model., 36(7), 3094-3111.

24.

Shahba, A., Attarnejad, R. and Hajilar, S. (2012), "A mechanical-based solution for axially functionally graded tapered Euler-Bernoulli beams", Mech. Adv. Mater. Struct., 20(8), 696-707.

25.

Shahba, A., Attarnejad, R. and Hajilar, S. (2011), "Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams", Shock & Vibration, 18(5), 683-696.

26.

Shahba, A., Attarnejad. R., Marvi, M.T. and Hajilar, S. (2011), "Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions", Compos. Part B: Eng., 42, 801-808.

27.

Shariyat, M. and Alipour, M.M. (2013), "A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations", Appl. Math. Model., 37, 3063-3076.

28.

Simsek, M. and Cansiz, S. (2012), "Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load", Compos. Struct., 94, 2861-2878.

29.

Simsek, M. (2009), "Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method", Int. J. Eng. Appl. Sci., 1, 1-11.

30.

Simsek, M. (2015), "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions", Compos. Struct., 133, 968-978.

31.

Simsek, M. and Reddy, J.N. (2013), "A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory", Compos. Struct., 101, 47-58.

32.

Simsek, M., Kocaturk, T. and Akbas, S.D. (2013), "Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory", Compos. Struct., 95,740-747.

33.

Sobhani Aragh, B., Hedayati, H., Borzabadi Farahani, E. and Hedayati, M. (2011), "A novel 2-D six-parameter power-law distribution for free vibration and vibrational displacements of two-dimensional functionally graded fiber-reinforced curved panels", Eur. J. Mech. A/Solid., 30, 865-883.

34.

Sutradhar, A. and Paulino, G.H. (2004), "The simple boundary element method for transient heat conduction in functionally graded material", Comp. Meth. Appl. Mech. Eng., 193(42-44), 4511-4539.

35.

Timoshenko, S.P. and Goodier, J.N. (1970), Theory of Elasticity, McGraw-Hill, New York, NY, USA.

36.

Yao, W.A., Zhong, W.X. and Lim, C.W. (2009), Symplectic Elasticity, World Scientific Publishing Company, New Jersey, USA.

37.

Zhao, L. and Wei, Z.G. (2015), "Analytical solutions for functionally graded beams under arbitrary distribution loads via the symplectic approach", Advan. Mech. Eng., Article ID, 321263.

38.

Zhao, L., Chen, W.Q. and Lu, C.F. (2012a), "New assessment on the Saint-Venant solutions for functionally graded materials beams", Mech. Res. Commun., 43, 1-6.

39.

Zhao, L., Chen, W.Q. and Lu, C.F. (2012b), "Two-dimensional complete rational analysis of functionally graded beams within symplectic framework", Appl. Math. Mech. Eng. Ed., 33(10), 1225-1238.

40.

Zhao, L., Chen, W.Q. and Lu, C.F. (2012c), "Symplectic elasticity for bi-directional functionally graded materials", Mech. Mater., 54, 32-42.

41.

Zhong, W.X. (1995), A New Systematic Methodology for Theory of Elasticity, Dalian University of Technology Press, Dalian, DL, China. (in Chinese)