Effects of damage on the response of Euler-Bernoulli beams traversed by a moving mass

  • Cristiano Bilello
Part of the International Centre for Mechanical Sciences book series (CISM, volume 471)


The perturbation induced by damage in the dynamic response of Euler-Bernoulli beams traversed by a moving mass is investigated. The structure is discretized into segments of constant bending stiffness, connected together by elastic hinges representing damaged sections. The beam-moving mass interaction force is modelled in the most accurate way by taking into account the effective structural mass distribution and the convective acceleration terms, often omitted in similar studies. The analytical response is obtained through a series expansion of the unknown deflection in a basis of the beam eigenfunctions. The results of experimental tests, performed on a small-scale model of a prototype bridge structure, are presented for a validation of the proposed solution procedure, and to provide data of practical engineering relevance. A good agreement with theoretical predictions is observed, moreover, the nature of the applied load seems to increase the structural damage sensitivity.


Structural Health Monitoring Damage Beam Beam Response Entrance Speed Damage Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. J. E. Akin and M. Mofid. Numerical solution for response of beam with moving mass. Journal of Structural Engineering, 115: 120–131, 1997.CrossRefGoogle Scholar
  2. N. Anifantis and A. Dimarogonas. Stability of columns with a single crack subjected to follower and vertical loads. International Journal of Solids Structures, 19: 281–291, 1983.CrossRefMATHGoogle Scholar
  3. C. Bilello. Theoretical and experimental Investigation on Damaged Beams under Moving Systems. PhD. Thesis, Dip. Ing. Strut. e Geotec., Università degli Studi di Palermo, 2001.Google Scholar
  4. C. Bilello and L. A. Bergman. Vibration of damaged beams under a moving mass: theory and experimental validation. Journal of Sound and Vibration, under review, 2002.Google Scholar
  5. C. Bilello, L. A. Bergman, and D. Kuchma. Experimental investigation of a small scale bridge model under a moving mass. Journal of Structural Engineering, under review, 2002.Google Scholar
  6. P.K. Chatterjee, T. K. Datta, and C. S. Surana. Vibration of continuous bridges under moving vehicles. Journal of Sound and Vibration, 169: 619–632, 1994.CrossRefMATHGoogle Scholar
  7. T. G. Chondros and A. D. Dimarogonas. Vibration of a cracked cantilever beam. Journal of Vibration and Acoustics, 120: 742–746, 1998.CrossRefGoogle Scholar
  8. T. G. Chondros, A. D. Dimarogonas, and J. A. Yao. A continuous cracked beam vibration theory. Journal of Vibration and Acoustics, 217: 17–34, 1997.Google Scholar
  9. S. Christides and A. D. S. Barr. One-dimensional theory of cracked Bernoulli-Euler beams. International Journal of Mechanical Science, 26: 639–648, 1984.CrossRefGoogle Scholar
  10. A. D. Dimarogonas. Vibration Engineering. West Publisher, 1976.Google Scholar
  11. A. D. Dimarogonas. Vibration of cracked structures: a state of the art. Engineering Fracture Mechanics, 55: 831–857, 1996.CrossRefGoogle Scholar
  12. L. Fryba. Vibration of Solids and Structures under Moving Mass. Noordhoff International Publishing, 1972.Google Scholar
  13. T. Hayashikawa and N. Watanabe. Dynamic behavior of continuous beams with moving loads. Journal of the Engineering Mechanics Division, 107: 229–246, 1981.Google Scholar
  14. J.L. Humar and A.M. Kashif. Dynamic response of bridges under traveling loads. Canadian Journal of Civil Engineers, 20: 287–298, 1993.CrossRefGoogle Scholar
  15. G. R. Irwin. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 24: 361–364, 1957.Google Scholar
  16. J. Lee and L. A. Bergman. The vibration of stepped beams and rectangular plates by an elemental dynamic flexibility method. Journal of Sound and Vibration, 171: 617–640, 1994.CrossRefMATHGoogle Scholar
  17. M. A. Mahamoud. Stress intensity factors for single and double edge cracks in a simple beam subjected to a moving load. International Journal of Fracture, 111: 151–161, 2001.CrossRefGoogle Scholar
  18. L. Meirovitch. Analytical Methods in Vibrations. MacMillan, 1967.Google Scholar
  19. M. Mofid and M. Shadam. On the response of beams with internal hinges under moving mass. Advances in Engineering Software, 31: 323–328, 2000.CrossRefGoogle Scholar
  20. C. A. Papadopoulus and A. D. Dimarogonas. Coupled longitudinal and bending vibrations of a rotating shaft with an open crack. Journal of Sound and Vibration.Google Scholar
  21. D. R. Parhi and A. K. Behera. Dynamic deflection of a cracked shaft subjected to moving mass. Transactions of the CSME, 21: 295–316, 1997.Google Scholar
  22. E. C. Pestel and F. A. Leckie. Matrix Methods in ElastoMechanics. McGraw and Hill, 1983.Google Scholar
  23. A. V. Pesterev and L. A. Bergman. Vibration of elastic continuum carrying moving linear oscillator. Journal of Engineering Mechanics, 123: 878–884, 1997a.CrossRefGoogle Scholar
  24. A. V. Pesterev and L. A. Bergman. Vibration of elastic continuum carrying accelerating oscillator. Journal of Engineering Mechanics, 123: 886–889, 1997b.CrossRefGoogle Scholar
  25. A. V. Pesterev and L. A. Bergman. A contribution to the moving mass problem. Journal of Vibration Acoustics, 120: 824–826, 1998.CrossRefGoogle Scholar
  26. A. V. Pesterev, L. A. Bergman, and C. A. Tan. Response and stress calculation of an elastic continuum carrying multiple oscillators. In Proceedings of Advance in Structural Dynamics Vol. I, pages 545–552, 2000.Google Scholar
  27. A. V. Pesterev, L. A. Bergman, and C. A. Tan. Revisiting the moving force problem. Journal of Sound and Vibration,in press, 2002.Google Scholar
  28. J. R. Rice and N. Levy. The part-through a surface crack in an elastic plate. Journal of Applied Mechanics, 3: 185–195, 1972.CrossRefGoogle Scholar
  29. S. Sadiku and H.H.E. Leipholz. On the dynamics of elastic systems with moving concentrated mass. Ingenieur Archive, 57: 223–242, 1987.CrossRefMATHGoogle Scholar
  30. M.-H. H. Shen and C. Pierre. Natural modes of Bernoulli-Euler beams with symmetric cracks. Journal of Sound and Vibration, 138: 115–134, 1990.CrossRefGoogle Scholar
  31. M. M. Stanisic, J. A. Euler, and S. T. Montgomery. On a theory concerning the dynamic behavior of structures carrying moving mass. Ingenieur Archives, 43: 295–305, 1974.CrossRefMATHGoogle Scholar
  32. H. Tada, P. C. Paris, and G. R. Irwin. The Stress Analysis and Cracks Handbook. Del Research Corporation, 1985.Google Scholar
  33. B. Yang, C. A. Tan, and L. A. Bergman. Direct numerical procedure for solution of moving oscillator problems. Journal of Engineering Mechanics, 126: 62–69, 2000.Google Scholar
  34. Y. B. Yang and J. D. Yau. Vehicle-bridge interaction element for dynamic analysis. Journal of Structural Engineering, 123: 1512–1518, 1997.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Cristiano Bilello
    • 1
  1. 1.Dipartimento di Ingegneria Strutturale e GeotecnicaUniversitá degli Studi di PalermoPalermoItaly

Personalised recommendations