Mechanics of Composite Materials

, Volume 49, Issue 2, pp 171–180 | Cite as

Axisymmetric Vibrations of Stepped Cylindrical Shells Made of Composite Materials. Part 21


Axisymmetric vibrations of stepped cylindrical shells are studied. The shells are made of elastic unidirectionally reinforced composite materials and have a piecewise constant (stepped) thickness. It is assumed that, at the reentrant corners of steps, stable circumferential cracks of constant depth are located. The method of a “massless rotational spring” developed in earlier papers for investigation of homogeneous shells made of isotropic materials is extended to the case of anisotropic composite materials. Analytical expressions are derived for shells with an arbitrary number of steps, and numerical results are presented for shells with one and two steps.


anisotropy shell vibration crack 


Acknowledgements. A partial support from the Grant 9110 of the ESF “Optimization of structural elements” and from the target financed project SF0180081s08 “Models of applied mathematics and mechanics” is acknowledged.


  1. 1.
    G. R. Irwin, “Fracture Mechanics,” in: Eds. By Goodier J. N., Hoff N. J., Structural Mechanics, Pergamon Press, Oxford, 1960.Google Scholar
  2. 2.
    H. Okamura, H. W. Liu, Chu Chorng-Shin, and H. Liebowitz, “A cracked column under compression,” Engineering Fracture Mechanics, No. 1, 547–564 (1969).Google Scholar
  3. 3.
    H. Anifantis and A. Dimarogonas, “Stability of columns with a single crack subjected to follower and vertical loads,” Int. J. of Solids and Structures, 19, 281–291 (1983).CrossRefGoogle Scholar
  4. 4.
    H. Anifantis and A. Dimarogonas, “Post buckling behavior of transversely cracked columns,” Composite Structures, 18, 351–356 (1984).CrossRefGoogle Scholar
  5. 5.
    Q. S. Li, “Buckling of multi-step non-uniform beams with elastically restrained boundary conditions,” J. of Constructional Steel Research, 57, 753–777 (2001).CrossRefGoogle Scholar
  6. 6.
    J. Lellep and E. Sakkov, “Buckling of stepped composite columns,” Mech. Compos. Mater., 42, No. 1, 63–72 (2006).CrossRefGoogle Scholar
  7. 7.
    S. Caddemis and I. Calio, “Exact solution of the multi-cracked Euler-Bernoulli column,“ Int. J. of Solids and Structures, 45, 1332–1351 (2008).CrossRefGoogle Scholar
  8. 8.
    L. Zhou and Y. Huang, “Crack effect the elastic buckling behavior of axially and eccentrically loaded columns,” Structural Engng and Mech., 22, 169–184 (2006).Google Scholar
  9. 9.
    Q. Wang and J. G. Chase, “Buckling analysis of cracked column structures and piezoelectric-based repair, and enhancement of axial load capacity,” Int. J. of Structural Stability and Dynamics, 3, 17–112 (2003).CrossRefGoogle Scholar
  10. 10.
    G. Bamnios and A. Trochides, “Dynamic behavior of a cracked cantilever beam,” Appl. Acoustics, 45, 97–61 (1995).CrossRefGoogle Scholar
  11. 11.
    S. Orhan, “Analysis of free and forced vibration of a cracked cantilever beam,” N D. T E International, 40, 443–450 (2007).CrossRefGoogle Scholar
  12. 12.
    M. Krawczuk and W. Ostachowicz, “Damage indicators for diagnostic of fatigue cracks in structures by vibration measurements – a Survey,” J. of Theoretical and Appl. Mech., 34, No. 2, 307–326 (1996).Google Scholar
  13. 13.
    J. Lee, “Identification of multiple cracks in a beam using vibration amplitudes,” J. of Sound and Vibration, 326, 205–212 (2009).CrossRefGoogle Scholar
  14. 14.
    J. Yang and Y. Chen, “Free vibration and buckling analyses of functionally graded beam with edge cracks,” Compos. Struct., 83, 48–60 (2008).CrossRefGoogle Scholar
  15. 15.
    K. Nikpour, “Diagnosis of axisymmetric cracks in orthotropic cylindrical shells by vibration measurement,” Compos. Sci. Technol., 39, 45–61 (1990).CrossRefGoogle Scholar
  16. 16.
    J. Lellep and L. Roots, “Vibrations of cylindrical shells with circumferential cracks,” WSEAS Transactions on Mathematics, 9, 689–699 (2010).Google Scholar
  17. 17.
    J. Lellep and L. Roots, “Axisymmetric vibrations of orthotropic circular cylindrical shells with cracks. Part 1,” Mech. Compos. Mater., 49, No. 1, 59–68 (2013).CrossRefGoogle Scholar
  18. 18.
    J. R. Rice and N. Levy, “The part-through surface crack in an elastic plate,” J. of Appl. Mech., 3, 185–194 (1972).CrossRefGoogle Scholar
  19. 19.
    A. D. Dimarogonas, “Vibration of cracked structures: a state of the art review,” Eng. Fracture Mech., 55, 831–857 (1996).CrossRefGoogle Scholar
  20. 20.
    T. G. Chondros, A. D. Dimarogonas, and J. Yao, “A continuous cracked beam vibration theory,” J. of Sound and Vibration, 215, No. 1, 17–34 (1998).CrossRefGoogle Scholar
  21. 21.
    T. G. Chondros and A. D. Dimarogonas, “Vibration of a cracked cantilever beam,” Trans. ASME, J. of Vibration and Acoustics, 120, 742–746 (1998).CrossRefGoogle Scholar
  22. 22.
    G. Bao, S. Ho, Z. Suo, and B. Fan, “The role of material orthotropy in fracture specimens for composites,” Int. J. of Solids and Struct., 29, No. 9, 1105–1116 (1992).CrossRefGoogle Scholar
  23. 23.
    H. Tada, P. C. Paris, and G. R. Irwin, Stress Analysis of Cracks, Handbook, ASME, N. Y., 2000.CrossRefGoogle Scholar
  24. 24.
    K. Nikpour and A. Dimarogonas, “Local compliance of composite cracked bodies,” Compos. Sci. Technol.,. 38, 209–223 (1988).CrossRefGoogle Scholar
  25. 25.
    M. D. Isaac, I. Ori, Engineering Mechanics of Composite Materials. Oxford University Press, New York, 1994.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Tartu, Institute of MathematicsTartuEstonia

Personalised recommendations