Skip to main content
SpringerLink
Log in

The influence of the axial force on the vibration of the Euler–Bernoulli beam with an arbitrary number of cracks

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The exact closed-form solution for the vibration modes and the eigen-value equation of the Euler–Bernoulli beam-column in the presence of an arbitrary number of concentrated open cracks is proposed. The solution is provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads the exact evaluation of the vibration frequencies as well as the buckling load of the beam-column and the corresponding eigen-modes. Furthermore, the presented solution allows a comprehensive evaluation of the influence of the axial load on the modal parameters of the beam. The cracks, which are not subjected to the closing phenomenon, are modelled as a sequence of Dirac’s delta generalised functions in the flexural stiffness. The eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any further continuity condition. The influence of the axial load on the vibration modes of beam-columns with different number and position of cracks, under different boundary conditions, has been analysed by means of the proposed closed-form expressions. The presented parametric analysis highlights some abrupt changes of the eigen-modes and the corresponding frequencies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Caddemi S., Caliò I.: Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J. Sound Vib. 327, 473–489 (2009)

    Article  Google Scholar 

  2. Kasper D.G., Swanson D.C., Reichard K.M.: Higher-frequency wavenumber shift and frequency shift in a cracked, vibrating beam. J. Sound Vib. 312, 1–18 (2008)

    Article  Google Scholar 

  3. Loya J.A., Rubio L., Fernandez-Saez J.: Natural frequencies for bending vibrations of Timoshenko cracked beams. J. Sound Vib. 290, 640–653 (2006)

    Article  Google Scholar 

  4. Kisa M., Gurel A.: Modal analysis of multi-cracked beams with circular cross section. Eng. Fract. Mech. 73, 963–977 (2006)

    Article  Google Scholar 

  5. Ranjbaran A.: Analysis of cracked members: the governing equations and exact solutions. Iran. J. Sci. Technol. Trans. B Eng. 34(B4), 407–417 (2010)

    MATH  Google Scholar 

  6. Viola E., Ricci P., Aliabadi M.H.: Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method. J. Sound Vib. 304, 124–153 (2007)

    Article  Google Scholar 

  7. Binici B.: Vibration of beams with multiple open cracks subjected to axial force. J. Sound Vib. 287, 277–295 (2005)

    Article  Google Scholar 

  8. Li Q.S.: Free vibration analysis of non-uniform beams with an arbitrary number of cracks and concentrated masses. J. Sound Vib. 252(3), 509–525 (2002)

    Article  Google Scholar 

  9. Caddemi S., Caliò I., Marletta M.: The non-linear dynamic response of the Euler–Bernoulli beam with an arbitrary number of switching cracks. Int. J. Non-Linear Mech. 45, 714–726 (2010)

    Article  Google Scholar 

  10. Caddemi S., Caliò I.: Exact solution of the multi-cracked Euler–Bernoulli column. Int. J. Solids Struct 45(16), 1332–1351 (2008)

    Article  MATH  Google Scholar 

  11. Ostachowicz W.M., Krawczuk C.: Analysis of the effect of cracks on the natural frequencies of a cantilever beam. J. Sound Vib. 150(2), 191–201 (1991)

    Article  Google Scholar 

  12. Dimarogonas A.D.: Vibration of cracked structures: a state of the art review. Eng. Fract. Mech. 55, 831–857 (1996)

    Article  Google Scholar 

  13. Rizos P.F., Aspragathos N., Dimarogonas A.D.: Identification of crack location and magnitude in a cantilever beam from the vibration modes. J. Sound Vib. 138(3), 381–388 (1990)

    Article  Google Scholar 

  14. Chondros T.J., Dimarogonas A.D., Yao J.: A continuous cracked beam vibration theory. J. Sound Vib. 215(1), 17–34 (1998)

    Article  MATH  Google Scholar 

  15. Liebowitz H., Claus W.D.S. Jr.: Failure of notched columns. Eng. Fract. Mech. 1, 379–383 (1968)

    Article  Google Scholar 

  16. Liebowitz H., Vanderveldt H., Harris D.W.: Carrying capacity of notched column. Int. J. Solids Struct. 3, 489–500 (1967)

    Article  Google Scholar 

  17. Okamura H., Liu H.W., Chorng-Shin Chu, Liebowitz H.: A cracked column under compression. Eng. Fract. Mech. 1, 547–564 (1969)

    Article  Google Scholar 

  18. Bilello, C.: Theoretical and experimental investigation on damaged beams under moving systems. Ph.D. Thesis, Università degli Studi di Palermo, Palermo, Italy (2001)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Civile e Ambientale, Università di Catania, viale A. Doria 6, Catania, Italy

    S. Caddemi & I. Caliò

Authors
  1. S. Caddemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. Caliò
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Caddemi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caddemi, S., Caliò, I. The influence of the axial force on the vibration of the Euler–Bernoulli beam with an arbitrary number of cracks. Arch Appl Mech 82, 827–839 (2012). https://doi.org/10.1007/s00419-011-0595-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-011-0595-z

Keywords

Search

Navigation