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Archive of Applied Mechanics

, Volume 79, Issue 12, pp 1083–1098 | Cite as

Free vibration analysis of third-order shear deformable composite beams using dynamic stiffness method

  • Li Jun
  • Li Xiaobin
  • Hua Hongxing
Original

Abstract

The dynamic stiffness method is introduced to investigate the free vibration of laminated composite beams based on a third-order shear deformation theory which accounts for parabolic distribution of the transverse shear strain through the thickness of the beam. The exact dynamic stiffness matrix is found directly from the analytical solutions of the basic governing differential equations of motion. The Poisson effect, shear deformation, rotary inertia, in-plane deformation are considered in the analysis. Application of the derived dynamic stiffness matrix to several particular laminated beams is discussed. The influences of Poisson effect, material anisotropy, slenderness and end condition on the natural frequencies of the beams are investigated. The numerical results are compared with the existing solutions in literature whenever possible to demonstrate and validate the present method.

Keywords

Generally laminated beams Third-order shear deformation theory Free vibration Dynamic stiffness matrix Poisson effect 

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References

  1. 1.
    Krishna Murty A.V.: On the shear deformation theory for dynamic analysis of beams. J. Sound Vib. 101, 1–12 (1985)CrossRefGoogle Scholar
  2. 2.
    Bhimaraddi A., Chandrashekhara K.: Some observations on the modeling of laminated composite beams with general lay-ups. Compos. Struct. 19, 371–380 (1991)CrossRefGoogle Scholar
  3. 3.
    Soldatos K.P., Elishakoff I.: A transverse shear and normal deformable orthotropic beam theory. J. Sound Vib. 154, 528–533 (1992)CrossRefGoogle Scholar
  4. 4.
    Chandrashekhara K., Bangera K.M.: Free vibration of composite beams using a refined shear flexible beam element. Comput. Struct. 43, 719–727 (1992)zbMATHCrossRefGoogle Scholar
  5. 5.
    Singh M.P., Abdelnaser A.S.: Random response of symmetric cross-ply composite beams with arbitrary boundary conditions. Am. Inst. Aeronaut. Astronaut. J. 30, 1081–1088 (1992)zbMATHGoogle Scholar
  6. 6.
    Khdeir A.A.: Dynamic response of antisymmetric cross-ply laminated composite beams with arbitrary boundary conditions. Int. J. Eng. Sci. 34, 9–19 (1996)zbMATHCrossRefGoogle Scholar
  7. 7.
    Marur S.R., Kant T.: On the performance of higher order theories for transient dynamic analysis of sandwich and composite beams. Comput. Struct. 65, 741–759 (1997)zbMATHCrossRefGoogle Scholar
  8. 8.
    Kant T., Marur S.R., Rao G.S.: Analytical solution to the dynamic analysis of laminated beams using higher order refined theory. Compos. Struct. 40, 1–9 (1997)CrossRefGoogle Scholar
  9. 9.
    Karama M., Abou Harb B., Mistou S., Caperaa S.: Bending, buckling and free vibration of laminated composite with a transverse shear stress continuity model. Compos. B 29B, 223–234 (1998)CrossRefGoogle Scholar
  10. 10.
    Shimpi R.P., Ainapure A.V.: Free vibration analysis of two layered cross-ply laminated beams using layer-wise trigonometric shear deformation theory. Commun. Numer. Methods Eng. 15, 651–660 (1999)CrossRefGoogle Scholar
  11. 11.
    Shi G., Lam K.Y.: Finite element vibration analysis of composite beams based on higher-order beam theory. J. Sound Vib. 219, 707–721 (1999)CrossRefGoogle Scholar
  12. 12.
    Shimpi R.P., Ainapure A.V.: A beam finite element based on layerwise trigonometric shear deformation theory. Compos. Struct. 53, 153–162 (2001)CrossRefGoogle Scholar
  13. 13.
    Ghugal Y.M., Shimpi R.P.: A review of refined shear deformation theories for isotropic and anisotropic laminated beams. J. Reinf. Plast. Compos. 20, 255–272 (2001)CrossRefGoogle Scholar
  14. 14.
    Rao M.K., Desai Y.M., Chitnis M.R.: Free vibrations of laminated beams using mixed theory. Compos. Struct. 52, 149–160 (2001)CrossRefGoogle Scholar
  15. 15.
    Arya H., Shimpi R.P., Naik N.K.: A zigzag model for laminated composite beams. Compos. Struct. 56, 21–24 (2002)CrossRefGoogle Scholar
  16. 16.
    Karama M., Afaq K.S., Mistou S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003)zbMATHCrossRefGoogle Scholar
  17. 17.
    Kapuria S., Dumir P.C., Jain N.K.: Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Compos. Struct. 64, 317–327 (2004)CrossRefGoogle Scholar
  18. 18.
    Murthy M.V.V.S., Mahapatra D.R., Badarinarayana K., Gopalakrishnan S.: A refined higher order finite element for asymmetric composite beams. Compos. Struct. 67, 27–35 (2005)CrossRefGoogle Scholar
  19. 19.
    Aydogdu M.: Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method. Int. J. Mech. Sci. 47, 1740–1755 (2005)CrossRefGoogle Scholar
  20. 20.
    Aydogdu M.: Free vibration analysis of angle-ply laminated beams with general boundary conditions. J. Reinf. Plast. Compos. 25, 1571–1583 (2006)CrossRefGoogle Scholar
  21. 21.
    Subramanian P.: Dynamic analysis of laminated composite beams using higher order theories and finite elements. Compos. Struct. 73, 342–353 (2006)CrossRefGoogle Scholar
  22. 22.
    Jones R.M.: Mechanics of Composite Materials. McGraw-Hill, New York (1976)Google Scholar
  23. 23.
    Wolfram S.: Mathematica: A System for Doing Mathematics by Computer, 2nd edn. Addison-Wesley, MA (1991)Google Scholar
  24. 24.
    Wittrick W.H., Williams F.W.: A general algorithm for computing natural frequencies of elastic structures. Q. J. Mech. Appl. Math. 24, 263–284 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Teboub Y., Hajela P.: Free vibration of generally layered composite beams using symbolic computations. Compos. Struct. 33, 123–134 (1995)CrossRefGoogle Scholar
  26. 26.
    Ritchie I.G., Rosinger H.E., Shillinglaw A.J., Fleury W.H.: The dynamic elastic behaviour of a fiber reinforced composite sheet: Part I. The precise experimental determination of the principal elastic moduli. J. Phys. D Appl. Phys. 8, 1733–1749 (1975)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Vibration, Shock and Noise InstituteShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.School of Transportation, Wuhan University of TechnologyWuhanPeople’s Republic of China

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