A New Form of Frequency Equation for Functionally Graded Timoshenko Beams with Arbitrary Number of Open Transverse Cracks

  • Tran Van LienEmail author
  • Ngo Trong Duc
  • Nguyen Tien Khiem
Research Paper


The present paper addresses the problem of free vibration in multiple cracked beams made of functionally graded material (FGM). Governing equations of the beam vibration are established using the Timoshenko beam theory, power law of material grading along the beam thickness and rotational spring model of crack. The actual position of neutral plane differed from the mid-plane due to the material properties variation is also taken into account. The frequency equation obtained in a simple form provides an efficient tool for natural frequency analysis and may also be used for solving the inverse problems such as identification of material properties or cracks in the beams from natural frequencies. Numerical examples have been carried out to validate the proposed theory and to investigate the effect of cracks, material and geometry parameters on natural frequencies of the beams.


Timoshenko beam FGM Transverse crack Natural frequency 



This research is funded by National University of Civil Engineering (NUCE) under grant number 215-2018/KHXD-TĐ.


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Copyright information

© Shiraz University 2018

Authors and Affiliations

  • Tran Van Lien
    • 1
    Email author
  • Ngo Trong Duc
    • 2
  • Nguyen Tien Khiem
    • 3
  1. 1.National University of Civil EngineeringHanoiVietnam
  2. 2.Design Consultant and Investment of Construction, Ministry of DefenceHanoiVietnam
  3. 3.Institute of MechanicsVietnam Academy of Science and TechnologyHanoiVietnam

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