, Volume 44, Issue 3, pp 255–281 | Cite as

Free vibration analysis of functionally graded panels and shells of revolution



The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one.

The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning eight types of shell structures illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behaviour of parabolic and circular shell structures.


Functionally graded materials Doubly curved shells FSD theory Free vibrations Generalized differential quadrature 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Reddy JN (2003) Mechanics of laminated composites plates and shells. CRC, New York Google Scholar
  2. 2.
    Viola E, Artioli E (2004) The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural elements. Struct Eng Mech 17:789–817 Google Scholar
  3. 3.
    Artioli E, Gould P, Viola E (2005) A differential quadrature method solution for shear-deformable shells of revolution. Eng Struct 27:1879–1892 CrossRefGoogle Scholar
  4. 4.
    Artioli E, Viola E (2005) Static analysis of shear-deformable shells of revolution via G.D.Q. method. Struct Eng Mech 19:459–475 Google Scholar
  5. 5.
    Artioli E, Viola E (2006) Free vibration analysis of spherical caps using a G.D.Q. numerical solution. J Press Vessel Technol 128:370–378 CrossRefGoogle Scholar
  6. 6.
    Abrate S (2006) Free vibration, buckling, and static deflection of functionally graded plates. Compos Sci Technol 66:2383–2394 CrossRefGoogle Scholar
  7. 7.
    Arciniega RA, Reddy JN (2007) Large deformation analysis of functionally graded shells. Int J Solids Struct 44:2036–2052 MATHCrossRefGoogle Scholar
  8. 8.
    Elishakoff I, Gentilini C, Viola E (2005) Forced vibrations of functionally graded plates in the three-dimensional setting. AIAA J 43:2000–2007 CrossRefADSGoogle Scholar
  9. 9.
    Elishakoff I, Gentilini C, Viola E (2005) Three-dimensional analysis of an all-around clamped plate made of functionally graded materials. Acta Mech 180:21–36 MATHCrossRefGoogle Scholar
  10. 10.
    Matsunaga H (2008) Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Compos Struct 82:499–512 CrossRefGoogle Scholar
  11. 11.
    Najafizadeh MM, Isvandzibaei MR (2007) Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support. Acta Mech 191:75–91 MATHCrossRefGoogle Scholar
  12. 12.
    Nguyen T-K, Sab K, Bonnet G (2008) First-order shear deformation plate models for functionally graded materials. Compos Struct 83:25–36 CrossRefGoogle Scholar
  13. 13.
    Patel BP, Gupta SS, Loknath MS, Kadu CP (2005) Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory. Compos Struct 69:259–270 CrossRefGoogle Scholar
  14. 14.
    Pelletier JL, Vel SS (2006) An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int J Solids Struct 43:1131–1158 MATHCrossRefGoogle Scholar
  15. 15.
    Roque CMC, Ferreira AJM, Jorge RMN (2007) A radial basis function for the free vibration analysis of functionally graded plates using refined theory. J Sound Vib 300:1048–1070 CrossRefADSGoogle Scholar
  16. 16.
    Singh BM, Rokne J, Dhaliwal RS (2006) Torsional vibration of functionally graded finite cylinders. Meccanica 41:459–470 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Singh BM, Rokne J, Dhaliwal RS (2008) Vibrations of a solid sphere or shell of functionally graded materials. Eur J Mech–A/Solids 27:460–468 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sofiyev AH (2003) Dynamic buckling of functionally graded cylindrical thin shells under non-periodic impulsive loading. Acta Mech 165:151–163 MATHCrossRefGoogle Scholar
  19. 19.
    Sofiyev AH, Deniz A, Akçay IH, Yusufoğlu E (2006) The vibration and stability of a three-layered conical shell containing an FGM layer subjected to axial compressive load. Acta Mech 183:129–144 MATHCrossRefGoogle Scholar
  20. 20.
    Yang J, Shen HS (2003) Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J Sound Vib 261:871–893 CrossRefADSGoogle Scholar
  21. 21.
    Vena P (2005) Thermal residual stresses in graded ceramic composites: a microscopic computational model versus homogenized models. Meccanica 40:163–179 MATHCrossRefGoogle Scholar
  22. 22.
    Wu CP, Tsai YH (2004) Asymptotic DQ solutions of functionally graded annular spherical shells. Eur J Mech–A/Solids 23:283–299 MATHCrossRefGoogle Scholar
  23. 23.
    Zenkour AM (2006) Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Model 30:67–84 MATHCrossRefGoogle Scholar
  24. 24.
    Zhou Z-G, Wang B (2006) An interface crack for functionally graded strip sandwiched between two homogeneous layers of finite thickness. Meccanica 41:79–99 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Shu C (2000) Differential quadrature and its application in engineering. Springer, Berlin MATHGoogle Scholar
  26. 26.
    Tornabene F (2007) Modellazione e soluzione di strutture a guscio in materiale anisotropo. PhD Thesis. University of Bologna, DISTART Department Google Scholar
  27. 27.
    Tornabene F, Viola E (2007). Free vibration analysis of functionally graded doubly curved shell structures using GDQ method. In: Proceedings of XVIII° National Conference of Italian Association of Theoretical and Applied Mechanics (AIMETA 2007) - Brescia, Italy, 11–14 September 2007 Google Scholar
  28. 28.
    Tornabene F, Viola E (2007) Vibration analysis of spherical structural elements using the GDQ method. Comput Math Appl 53:1538–1560 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Tornabene F, Viola E (2008). 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method. Eur J Mech–A/Solids. Available online 4 March 2008 Google Scholar
  30. 30.
    Viola E, Tornabene F (2005) Vibration analysis of damaged circular arches with varying cross-section. Struct Integr Durab (SID-SDHM) 1:155–169 Google Scholar
  31. 31.
    Viola E, Tornabene F (2006) Vibration analysis of conical shell structures using GDQ method. Far East J Appl Math 25:23–39 MATHMathSciNetGoogle Scholar
  32. 32.
    Viola E, Dilena M, Tornabene. F (2007) Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches. J Sound Vib 299:143–163 CrossRefADSGoogle Scholar
  33. 33.
    Marzani A, Tornabene F, Viola E (2008) Nonconservative stability problems via generalized differential quadrature method. J Sound Vib 315:176–196 CrossRefADSGoogle Scholar
  34. 34.
    Alfano G, Auricchio F, Rosati L, Sacco E (2001) MITC finite elements for laminated composite plates. Int J Numer Methods Eng 50:707–738 MATHCrossRefGoogle Scholar
  35. 35.
    Auricchio F, Sacco E (1999) A mixed-enhanced finite-element for the analysis of laminated composite plates. Int J Numer Methods Eng 44:1481–1504 MATHCrossRefGoogle Scholar
  36. 36.
    Auricchio F, Sacco E (2003) Refined first-order shear deformation theory models for composite laminates. J Appl Mech 70:381–390 MATHCrossRefGoogle Scholar
  37. 37.
    Toorani MH, Lakis AA (2000) General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. J Sound Vib 237:561–615 CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.DISTART - Department, Faculty of EngineeringUniversity of BolognaBolognaItaly

Personalised recommendations