A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

Nguyen Tien Khiem, Duong The Hung
Author affiliations


  • Nguyen Tien Khiem Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam
  • Duong The Hung Thai Nguyen University of Technology, Vietnam




Timoshenko beams, multiple cracked beams, natural frequencies, sensitivity analysis


A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.


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How to Cite

N. T. Khiem and D. T. Hung, A closed-form solution for free vibration of multiple cracked Timoshenko beam and application, Vietnam J. Mech. 39 (2017) 315–328. DOI: https://doi.org/10.15625/0866-7136/9641.



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