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Mathematical solution for free vibration of sigmoid functionally graded beams with varying cross-section

  • Atmane, Hassen Ait (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Tounsi, Abdelouahed (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Ziane, Noureddine (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Mechab, Ismail (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes)
  • Received : 2011.03.14
  • Accepted : 2011.08.18
  • Published : 2011.11.25
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Abstract

This paper presents a theoretical investigation in free vibration of sigmoid functionally graded beams with variable cross-section by using Bernoulli-Euler beam theory. The mechanical properties are assumed to vary continuously through the thickness of the beam, and obey a two power law of the volume fraction of the constituents. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the S-FGM beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Results show that, all other parameters remaining the same, the natural frequencies of S-FGM beams are always proportional to those of homogeneous isotropic beams. Therefore, one can predict the behaviour of S-FGM beams knowing that of similar homogeneous beams.

Keywords

References

  1. Abrate, S. (2006), "Free vibration, buckling, and static deflections of functionally graded plates", Compos. Sci. Technol., 66(14), 2383-2394. https://doi.org/10.1016/j.compscitech.2006.02.032
  2. Bao, G. and Wang, L. (1995), "Multiple cracking in functionally graded ceramic/metal coatings", Int. J. Solids Struct., 32(19), 2853-2871. https://doi.org/10.1016/0020-7683(94)00267-Z
  3. Bathe, K. J. (1982), Finite Element Procedures in Engineering Analysis, Englewood Cliffs, New Jersey: Prentice-Hall.
  4. Benatta, M.A., Mechab, I., Tounsi, A. and Adda Bedia, E.A. (2008), "Static analysis of functionally graded short beams including warping and shear deformation effects", Comput. Mater. Sci., 44(2), 465-773.
  5. Benatta, M.A, Tounsi A., Mechab I. and Bachir Bouiadjra M. (2009), "Mathematical solution for bending of short hybrid composite beams with variable fibers spacing", Appl. Math. Comput., 212(2), 337-348. https://doi.org/10.1016/j.amc.2009.02.030
  6. Beskos, D. E. (1987), "Boundary element methods in dynamic analysis", Appl. Mech Rev., 40(1), 1-23. https://doi.org/10.1115/1.3149529
  7. Beskos, D. E. (1979), "Dynamics and stability of plane trusses with gusset plates", Comput. & Struct., 10(5), 785-795. https://doi.org/10.1016/0045-7949(79)90042-7
  8. Beskos, D. E. and Narayanan, G. V. (1983), "Dynamic response of frameworks by numerical Laplace transforms",Comput. Meth. Appl. Mech. Eng., 37(3), 289-307. https://doi.org/10.1016/0045-7825(83)90080-4
  9. Caruntu, D. (2000), "On nonlinear vibration of non-uniform beam with rectangular cross-section and parabolic thickness variation", Solid Mechanics and its Applications, 73.Kluwer Academic Publishers, Dordrecht, Boston,London, 109-118.
  10. Chung, Y.L. and Chi, S.H. (2001), "The residual stress of functionally graded materials", Journal of the Chinese Institute of Civil and Hydraulic Engineering., 13, 1-9.
  11. Chu, C. H. and Pilkey, D. W. (1979), "Transient analysis of structural members by the CSDT Riccati transfer matrix method", Comput. & Struct., 10(4), 599-611. https://doi.org/10.1016/0045-7949(79)90004-X
  12. Cranch, E.T. and Adler, A.A. (1956), "Bending vibration of variable section beams", ASME J. Appl. Mech., 23, 103-108.
  13. Datta, A.K. and Sil, S.N. (1996), "An analysis of free undamped vibration of beams of varying cross-section", Comput. & Struct., 59(3), 479-483. https://doi.org/10.1016/0045-7949(95)60270-4
  14. Delale, F. and Erdogan, F. (1983), "The crack problem for a nonhomogeneous plane", ASME J. Appl. Mech., 50(3), 609-614. https://doi.org/10.1115/1.3167098
  15. Elishako, I. and Johnson, V. (2005), "Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass", J. Sound Vib., 286(4-5), 1057-1066. https://doi.org/10.1016/j.jsv.2005.01.050
  16. Elishako, I. (2005), Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton.
  17. Jang, S.K. and Bert, C.W. (1989b), "Free vibration of stepped beams: Higher mode frequencies and effects of steps on frequencies", J. Sound Vib., 32, 164-168.
  18. Gorman, D.J. (1975), Free Vibration Analysis of Beams and Shafts, Wiley, New York.
  19. Jabbari, M., Vaghari, A.R., Bahtui, A. and Eslami, M.R. (2008), "Exact solution for asymmetric transient thermal and mechanical stresses in FGM hollow cylinders with heat source", Struct. Eng. Mech., 29, 244-254.
  20. Jang, S.K. and Bert, C.W. (1989a), "Free vibration of stepped beams: Exact and numerical solutions", J. Sound Vib., 30, 342-346.
  21. Guven, U., Celik, A., Baykara, C. (2004), "On transverse vibrations of functionally graded polar orthotropic rotating solid disk with variable thickness and constant radial stress", J. Reinf. Plast. Comp., 23(12), 1279-1284. https://doi.org/10.1177/0731684404035600
  22. Huang, X.L. and Shen, H.S. (2004), "Nonlinear vibration and dynamic response of functionally graded plates in thermal environment", Int. J. Solids Struct., 41(9-10), 2403-27. https://doi.org/10.1016/j.ijsolstr.2003.11.012
  23. Just, D. J. (1977), "Plane frameworks of tapered box and I-section", ASCE J. Struct. Eng., 103, 71-86.
  24. Laura, P.A.A., Gutierrez, R.H. and Rossi, R.E. (1996), "Free vibration of beams of bi-linearly varying thickness", Ocean Engineering., 23, 1-6. https://doi.org/10.1016/0029-8018(95)00029-K
  25. Lee, S. Y., Ke, H. Y. and Kuo, Y. H. (1990), "Analysis of non-uniform beam vibration", J. Sound Vib., 142, 15-29. https://doi.org/10.1016/0022-460X(90)90580-S
  26. Lee, Y.D. and Erdogan, F. (1995), "Residual/thermal stress in FGM and laminated thermal barrier coatings", Int.J. Fract., 69(2), 145-165. https://doi.org/10.1007/BF00035027
  27. 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(4-5), 1210-1229. https://doi.org/10.1016/j.jsv.2008.04.056
  28. Mushelishvili, N. I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen.
  29. Ovunk, B. A. (1974), "Dynamics of frameworks by continuous mass methods", Comput. & Struct., 4(5), 1061-1089. https://doi.org/10.1016/0045-7949(74)90024-8
  30. Pradhan, S.C., Sarkar, A. (2009), "Analyses of tapered fgm beams with nonlocal theory", Struct. Eng. Mech., 32, 811-833. https://doi.org/10.12989/sem.2009.32.6.811
  31. Pradhan, S.C. and Phadikar, J.K. (2009), "Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory", Struct. Eng. Mech., 33, 193-213. https://doi.org/10.12989/sem.2009.33.2.193
  32. Sallai, B.O., Tounsi, A., Mechab, I., Bachir Bouiadjra, M., Meradjah, M. and Adda Bedia, E.A. (2009), "A theoretical analysis of flexional bending of $Al/Al_{2}O_{3}$ S-FGM thick beams", Comput. Mater. Sci., 44, 1344-1350. https://doi.org/10.1016/j.commatsci.2008.09.001
  33. Sanjay Anandrao, K., Gupta, R.K., Ramchandran, P. and Venkateswara Rao, G. (2010), "Thermal post-buckling analysis of uniform slender functionally graded material beams", Struct. Eng. Mech., 36, 545-560. https://doi.org/10.12989/sem.2010.36.5.545
  34. Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696. https://doi.org/10.1016/S0266-3538(01)00007-0
  35. Simsek, M. and Kocaturk, T. (2009), "Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load", Compos. Struct., 90(4), 465-473. https://doi.org/10.1016/j.compstruct.2009.04.024
  36. Simsek, M. (2010), "Vibration analysis of a functionally graded beam under a moving mass by using different beam theories", Compos. Struct., 92(4), 904-917. https://doi.org/10.1016/j.compstruct.2009.09.030
  37. Simsek, M. (2010), "Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories", Nuclear Engineering and Design, 240(4), 697-705. https://doi.org/10.1016/j.nucengdes.2009.12.013
  38. Simsek, M. (2010), "Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load", Compos. Struct., 92(10), 2532-2546. https://doi.org/10.1016/j.compstruct.2010.02.008
  39. Sina, S.A., Navazi, H.M. and Haddadpour, H. (2009), "An analytical method for free vibration analysis of functionally graded beams", Materials and Design., 30(3), 741-747. https://doi.org/10.1016/j.matdes.2008.05.015
  40. Tong, X., Tabarrok, B. and Yeh, K.Y. (1995), "Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section", J. Sound Vib., 186(5), 821-835. https://doi.org/10.1006/jsvi.1995.0490
  41. Toso, M. and Baz, A. (2004), "Wave propagation in periodic shells with tapered wall thickness and changing material properties", Shock Vib., 11(3-4), 411-432. https://doi.org/10.1155/2004/456982
  42. Yang, B., Ding, H.J. and Chen, W.Q. (2008), "Elasticity solutions for a uniformly loaded annular plate of functionally graded materials", Struct. Eng. Mech., 30, 501-512. https://doi.org/10.12989/sem.2008.30.4.501
  43. Yas, M.H., Sobhani Aragh, B. and Heshmati, M. (2011), "Three-dimensional free vibration analysis of functionally graded fiber reinforced cylindrical panels using differential quadrature method", Struct. Eng. Mech.,37, 301-313.
  44. Yeh, K. Y. (1979), "General solutions on certain problems of elasticity with nonhomogeneity and variable thickness, part IV: bending, buckling, and free vibration of nonhomogeneous variable thickness beams", Journal of Lanzhou University., 1, 133-157.
  45. Yeh, K. Y., Tong, X., Ji and Z. Y. (1992), "General analytic solution of dynamic response of beams with nonhomogeneity and variable cross section", Appl. Math. Mech., 13(9), 779-791. https://doi.org/10.1007/BF02481798
  46. Ying, J., Lu, C.F. and Chen and W.Q. (2008), "Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations", Compos. Struct., 84(3), 209-219. https://doi.org/10.1016/j.compstruct.2007.07.004
  47. Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol., 67(3-4), 481-488. https://doi.org/10.1016/j.compscitech.2006.08.023

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