Browse > Article
http://dx.doi.org/10.12989/scs.2011.11.6.489

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)
Publication Information
Steel and Composite Structures / v.11, no.6, 2011 , pp. 489-504 More about this Journal
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
functionally graded materials; beams; variable cross-section; free vibration;
Citations & Related Records
Times Cited By KSCI : 5  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
연도 인용수 순위
1 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.   DOI
2 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.   DOI   ScienceOn
3 Just, D. J. (1977), "Plane frameworks of tapered box and I-section", ASCE J. Struct. Eng., 103, 71-86.
4 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.   DOI   ScienceOn
5 Lee, S. Y., Ke, H. Y. and Kuo, Y. H. (1990), "Analysis of non-uniform beam vibration", J. Sound Vib., 142, 15-29.   DOI   ScienceOn
6 Lee, Y.D. and Erdogan, F. (1995), "Residual/thermal stress in FGM and laminated thermal barrier coatings", Int.J. Fract., 69(2), 145-165.   DOI   ScienceOn
7 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.   DOI   ScienceOn
8 Mushelishvili, N. I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen.
9 Ovunk, B. A. (1974), "Dynamics of frameworks by continuous mass methods", Comput. & Struct., 4(5), 1061-1089.   DOI   ScienceOn
10 Pradhan, S.C., Sarkar, A. (2009), "Analyses of tapered fgm beams with nonlocal theory", Struct. Eng. Mech., 32, 811-833.   DOI
11 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.   DOI
12 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.   DOI   ScienceOn
13 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.   DOI
14 Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696.   DOI   ScienceOn
15 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.   DOI   ScienceOn
16 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.   DOI   ScienceOn
17 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.   DOI
18 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.   DOI
19 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.   DOI
20 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.   DOI   ScienceOn
21 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.   DOI
22 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.   DOI
23 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.   DOI   ScienceOn
24 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.
25 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.
26 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.   DOI   ScienceOn
27 Zhong, Z. and Yu, T. (2007), "Analytical solution of a cantilever functionally graded beam", Compos. Sci. Technol., 67(3-4), 481-488.   DOI   ScienceOn
28 Bathe, K. J. (1982), Finite Element Procedures in Engineering Analysis, Englewood Cliffs, New Jersey: Prentice-Hall.
29 Abrate, S. (2006), "Free vibration, buckling, and static deflections of functionally graded plates", Compos. Sci. Technol., 66(14), 2383-2394.   DOI   ScienceOn
30 Bao, G. and Wang, L. (1995), "Multiple cracking in functionally graded ceramic/metal coatings", Int. J. Solids Struct., 32(19), 2853-2871.   DOI   ScienceOn
31 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.
32 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.   DOI   ScienceOn
33 Beskos, D. E. (1987), "Boundary element methods in dynamic analysis", Appl. Mech Rev., 40(1), 1-23.   DOI
34 Beskos, D. E. (1979), "Dynamics and stability of plane trusses with gusset plates", Comput. & Struct., 10(5), 785-795.   DOI   ScienceOn
35 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.   DOI   ScienceOn
36 Cranch, E.T. and Adler, A.A. (1956), "Bending vibration of variable section beams", ASME J. Appl. Mech., 23, 103-108.
37 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.
38 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.
39 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.   DOI   ScienceOn
40 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.   DOI   ScienceOn
41 Delale, F. and Erdogan, F. (1983), "The crack problem for a nonhomogeneous plane", ASME J. Appl. Mech., 50(3), 609-614.   DOI
42 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.   DOI   ScienceOn
43 Elishako, I. (2005), Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions, CRC Press, Boca Raton.
44 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.
45 Gorman, D.J. (1975), Free Vibration Analysis of Beams and Shafts, Wiley, New York.
46 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.
47 Jang, S.K. and Bert, C.W. (1989a), "Free vibration of stepped beams: Exact and numerical solutions", J. Sound Vib., 30, 342-346.