International Journal of Steel Structures

, Volume 19, Issue 2, pp 635–649 | Cite as

Vibrations of a Box-Sectional Cantilever Timoshenko Beam with Multiple Cracks

  • Ahmet Can AltunışıkEmail author
  • Fatih Yesevi Okur
  • Volkan Kahya


This paper considers a Timoshenko cantilever beam with box cross-section including multiple cracks. Under six damage scenarios, the problem is solved analytically by the transfer matrix method, and numerically by the finite element method. Results are validated by the experimental measurements with the aid of ambient vibration tests, that use Enhanced Frequency Domain Decomposition and Stochastic Subspace Identification methods. Measured and calculated natural frequencies and mode shapes for undamaged and damaged beams are compared with each other. Modal assurance criterion and coordinated modal assurance criterion values are obtained from two set of measurements to establish the correlation between the measured and calculated mode shapes for the damage location identification.


Damage Vibration Transfer matrix method Finite element analysis Operational modal analysis Timoshenko beam 


  1. Allemang, R. J., & Brown, D. L. (1982). A correlation coefficient for modal vector analysis. In Proceedings of the 1st international modal analysis conference (pp. 110–116).Google Scholar
  2. Altunışık, A. C., Okur, F. Y., & Kahya, V. (2017a). Automated model updating of multiple cracked cantilever beams for damage detection. Journal of Constructional Steel Research, 138, 499–512.CrossRefGoogle Scholar
  3. Altunışık, A. C., Okur, F. Y., & Kahya, V. (2017b). Structural identification of a cantilever beam with multiple cracks: modelling and validation. International Journal of Mechanical Sciences, 130, 74–89.CrossRefGoogle Scholar
  4. ANSYS Engineering Analysis System. (2015). Swanson Analysis Systems, USA.Google Scholar
  5. Attar, M. (2012). A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. International Journal of Mechanical Sciences, 57, 19–33.CrossRefGoogle Scholar
  6. Baghiee, N., Esfahani, R. M., & Moslem, K. (2009). Studies on damage and FRP strengthening of reinforced concrete beams by vibration monitoring. Engineering Structures, 31, 875–893.CrossRefGoogle Scholar
  7. Behera, R. K., Pandey, A., & Parhi, D. R. (2014). Numerical and experimental verification of a method for prognosis of inclined edge crack in cantilever beam based on synthesis of mode shapes. Procedia Technology, 14, 67–74.CrossRefGoogle Scholar
  8. Behzad, M., Meghdari, A., & Ebrahimi, A. (2005). A new approach for vibration analysis of a cracked beam. International Journal of Engineering, 18(4), 319–330.zbMATHGoogle Scholar
  9. Castel, A., Vidal, T., & François, R. (2012). Finite-element modeling to calculate the overall stiffness of cracked reinforced concrete beams. Journal of Structural Engineering, ASCE, 138(7), 889–898.CrossRefGoogle Scholar
  10. Chen, H., Kurt, M., Lee, Y. S., McFarland, D. M., Bergman, L. A., & Vakakis, A. F. (2014). Experimental system identification of the dynamics of a vibro-impact beam with a view towards structural health monitoring and damage detection. Mechanical Systems and Signal Processing, 46(1), 91–113.CrossRefGoogle Scholar
  11. Chondros, T. G., Dimarogonas, A. D., & Yao, J. (1998). A continuous cracked beam vibration theory. Journal of Sound and Vibration, 215(1), 17–34.CrossRefzbMATHGoogle Scholar
  12. Christides, S., & Barr, A. D. S. (1984). One-dimensional theory of cracked Euler-Bernoulli beams. International Journal of Mechanical Science, 26, 639–648.CrossRefGoogle Scholar
  13. Dimarogonas, A. D. (1996). Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 55, 831–857.CrossRefGoogle Scholar
  14. Doebling, S.W., Farrar, C. R., Prime, M. B., & Shevitz, D. W. (1996). Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review. Report no. LA-13070-MS, Los Alamos National Laboratory.Google Scholar
  15. Dona, M., Palmeri, A., Lombardo, M., & Cicirello, A. (2015). An efficient two-node finite element formulation of multi-damaged beams including shear deformation and rotatory inertia. Computers and Concrete, 147, 96–106.Google Scholar
  16. FEMtools. (2016). Model updating theoretical manual, dynamic design solutions, version 3.8. Leuven, Belgium.Google Scholar
  17. Gounaris, G., & Dimarogonas, A. D. (1988). A finite-element of a cracked prismatic beam for a structural analysis. Computer and Structures, 28, 309–313.CrossRefzbMATHGoogle Scholar
  18. Gudmunson, P. (1983). The dynamic behaviour of slender structures with cross-sectional cracks. Journal of Mechanics and Physics of Solids, 31, 329–345.CrossRefGoogle Scholar
  19. Joshi, A., & Madhusudhan, B. S. (1991). A unified approach to free vibration of locally damaged beams having various homogeneous boundary conditions. Journal of Sound and Vibration, 147, 475–488.CrossRefGoogle Scholar
  20. Kim, J. H., Jeon, H. S., & Lee, C. W. (1992). Application of the modal assurance criteria for detecting and locating structural faults. In Proceedings of the 10th international modal analysis conference, San Diego, CA, USA, 3–7 February (pp. 536–540).Google Scholar
  21. Labib, A., Kennedy, D., & Featherson, C. (2014). Free vibration analysis of beams and frames with multiple cracks for damage detection. Journal of Sound and Vibration, 333, 4991–5003.CrossRefGoogle Scholar
  22. Lin, H. P. (2004). Direct and inverse methods on free vibration analysis of simply supported beams with a crack. Engineering Structures, 26, 427–436.CrossRefGoogle Scholar
  23. Lin, H. P., Chen, S. C., & Wu, J. D. (2002). Beam vibrations with an arbitrary number of cracks. Journal of Sound and Vibration, 258(5), 987–999.CrossRefGoogle Scholar
  24. Loya, J. A., Rubio, L., & Fernández-Sáez, J. (2006). Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration, 290, 640–653.CrossRefGoogle Scholar
  25. Morassi, A. (1993). Crack-induced changes in eigenfrequencies of beam structures. Journal of Engineering Mechanics, ASCE, 119, 1768–1803.CrossRefGoogle Scholar
  26. OMA. (2006). Software: Operational modal analysis, Release 4.0. Aalborg: Structural Vibration Solution A/S.Google Scholar
  27. PULSE. (2006). Analyzers and solutions, release 11.2. Aalborg: Bruel and Kjaer, Sound and Vibration Measurement A/S.Google Scholar
  28. Rizos, P. F., Aspragatos, N., & Dimarogonas, A. D. (1990). Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 138, 381–388.CrossRefGoogle Scholar
  29. Ruotolo, R., & Surace, C. (2004). Natural frequencies of a bar with multiple cracks. Journal of Sound and Vibration, 272, 301–316.CrossRefGoogle Scholar
  30. Shafiei, M., & Khaji, N. (2011). Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load. Acta Mechanica, 221, 79–97.CrossRefzbMATHGoogle Scholar
  31. Sinha, J. K., Friswell, M. I., & Edwards, S. (2002). Simplified models for the location of cracks in beam structure using measured vibration data. Journal of Sound and Vibration, 251, 13–38.CrossRefGoogle Scholar
  32. Tondreau, G., & Deraemaeker, A. (2015). Automated data-based damage localization under ambient vibration using local modal filters and dynamic strain measurements: Experimental applications. Journal of Sound and Vibration, 333(26), 7364–7385.CrossRefGoogle Scholar
  33. Yuen, M. M. (1985). A numerical study of the eigenparameters of a damaged cantilever. Journal of Sound and Vibration, 103, 301–310.CrossRefGoogle Scholar
  34. Zheng, D. Y., & Fan, S. C. (2003). Vibration and stability of cracked hollow-sectional beams. Journal of Sound and Vibration, 267, 933–954.CrossRefGoogle Scholar

Copyright information

© Korean Society of Steel Construction 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringKaradeniz Technical UniversityTrabzonTurkey

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