Non-linear free vibrations and post-buckling analysis of shear flexible functionally graded beams

다국어 초록 (Multilingual Abstract)

Large amplitude free vibration and thermal post-buckling of shear flexible Functionally Graded Material (FGM) beams is studied using finite element formulation based on first order Timoshenko beam theory. Classical boundary conditions are considered. The ends are assumed to be axially immovable. The von-Karman type strain-displacement relations are used to account for geometric non-linearity. For all the boundary conditions considered, hardening type of non-linearity is observed. For large amplitude vibration of FGM beams, a comprehensive study has been carried out with various lengths to height ratios, maximum lateral amplitude to radius of gyration ratios, volume fraction exponents and boundary conditions. It is observed that, for FGM beams, the non-linear frequencies are dependent on the sign of the vibration amplitudes. For thermal post-buckling of FGM beams, the effect of shear flexibility on the structural response is discussed in detail for different volume fraction exponents, length to height ratios and boundary conditions. The effect of shear flexibility is observed to be predominant for clamped beam as compared to simply supported beam.

참고문헌 (Reference)

1 Birman, V., "Vibrations of damaged cantilever beams manufactured from functionally graded materials" 45 (45): 2747-2757, 2007

2 Prakash, T., "Vibrations and thermal stability of functionally graded spherical caps" 24 (24): 447-462, 2006

3 Rao, S. S., "Vibration of Continuous System" John Wiley & Sons, Inc 2007

4 Raju, K.K., "Towards improved evaluation of large amplitude free vibration behavior of uniform beams using multi-term admissible functions" 282 : 1238-1246, 2005

5 Sang-Lae, L., "Thermal stability boundary of FG panel under aerodynamic load" 1 (1): 105-110, 2007

6 K. Sanjay Anandrao, "Thermal post-buckling analysis of uniform slender functionally graded material beams" 국제구조공학회 36 (36): 545-560, 2010

7 Rao, G.V., "Thermal post buckling of uniform Columns : A simple intuitive method" 40 (40): 2138-2140, 2002

8 Rao, G.V., "Thermal post buckling of columns" 22 (22): 850-851, 1984

9 Thivend, J., "Thermal post buckling analysis of FGM beams" 2008

10 Hesham, H. I., "Thermal buckling and nonlinear flutter behavior of functionally graded material panels" 44 (44): 1610-1618, 2007

11 Timoshenko, S.P., "Theory of Elastic Stability" McGraw-Hill 1970

12 Woinowsky - Krieger, S., "The effect of an axial force on the vibrations of hinged bars" 17 : 35-36, 1950

13 Koizumi, M., "The concept of FGM" 34 (34): 3-10, 1993

14 Gupta, R. K., "Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams" 45 : 624-631, 2009

15 Singh, G., "Re-investigation of large amplitude free vibrations of beams using finite elements" 143 : 351-355, 1990

16 Rao, G.V., "Post-buckling of uniform cantilever columns-Galerkin finite element formulation" 9 : 1-4, 1977

17 Sathyamoorthy, M., "Nonlinear Analysis of Structures, CRC Mechanical Engineering Series" CRC Press 26-38, 1998

18 Kitipornchai, S., "Non-linear vibration of edge cracked functionally graded Timoshenko beams" 324 : 962-982, 2009

19 Deschilder, M., "Non-linear static analysis of a beam made of functionally graded material" 2006

20 Batra, R.C., "Natural frequencies of a functionally graded anisotropic rectangular plate" 282 : 509-516, 2005

21 Reddy, J. N., "Mechanics of Laminated Composite Plates and Shells: Theory and Analysis" CRC Press 2003

22 Sung-Cheon, H., "Mechanical vibration and buckling analysis of FGM plates and shells using a four node quasi conforming shell element" 8 (8): 203-229, 2008

23 Rao, G.V., "Large amplitude vibrations of beams with elastically restrained ends" 57 (57): 302-304, 1978

24 Singh, G., "Large amplitude free vibrations of beams-a discussion on various formulations and assumptions" 142 (142): 77-85, 1990

25 Srinivasan, A. V., "Large amplitude free oscillations of beams and plates" 3 (3): 1951-1953, 1965

26 Aboudi, J., "Higher order theory for functionally graded materials" 30 : 777-832, 1999

27 Alshorbagy, A.E., "Free vibration characteristics of a functionally graded beam by finite element method" 35 : 412-425, 2011

28 Yang, J., "Free vibration and buckling analyses of functionally graded beams with edge cracks" 83 : 48-60, 2008

29 Mei, C., "Finite element analysis of non-linear vibrations of beam columns" 11 : 115-117, 1973

30 M. Jabbari, "Exact solution for asymmetric transient thermal and mechanical stresses in FGM hollow cylinders with heat source" 국제구조공학회 29 (29): 551-565, 2008

31 Prakash, T., "Asymmetric flexural vibration and thermoelastic stability of FGM plates using finite element method" 37 : 642-649, 2006

32 Ke, L. L., "An analytical study on the nonlinear vibration of functionally graded beams" 45 : 743-752, 2010

33 Sina, S. A., "An analytical method for free vibration analysis of functionally graded beams" 30 : 741-747, 2009

34 Reddy, J. N., "An Introduction to Non-linear Finite Element Analysis" Oxford University Press 2004

35 ANSYS Inc., "ANSYS package version 10.0" Canons Burgh

36 Li, X. F., "A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams" 318 : 1210-1229, 2008

37 Rao, G.V., "A simple method to predict the thermal post-buckling behavior of columns on Pasternak foundation" 10 : 177-182, 2003

38 Rao, G. V., "A simple energy method to predict the thermal post buckling behavior of columns" 55 (55): 141-143, 2003

39 Raju, K.K., "A note on large amplitude vibrations" 18 (18): 1189-1191, 1984

40 Chakraborty, A., "A new beam finite element for the analysis of functionally graded materials" 45 : 519-539, 2003

41 Rao, G.V., "A direct numerical integration method to study the large amplitude vibration of slender beams with immovable ends" 83 : 42-44, 2002