Advertisement

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
  • 47 Downloads

Abstract

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.

Keywords

Timoshenko beam FGM Transverse crack Natural frequency 

Notes

Acknowledgements

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

References

  1. Akbas SD (2014) Wave propagation in edge cracked functionally graded beams under impact force. J Vib Control.  https://doi.org/10.1177/1077546314547531 MathSciNetGoogle Scholar
  2. Aydin K (2013) Free vibration of functional graded beams with arbitrary number of cracks. Eur J Mech A Solid 42:112–124CrossRefzbMATHGoogle Scholar
  3. Banerjee A, Panigrahi B, Pohit G (2015) Crack modelling and detection in Timoshenko FGM beam under transverse vibration using frequency contour and response surface model with GA. Nondestruct Test Eval.  https://doi.org/10.1080/10589759.2015.1071812 Google Scholar
  4. Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201CrossRefGoogle Scholar
  5. Erdogan F, Wu BH (1997) The surface crack problem for a plate with functionally graded properties. J Appl Mech 64:448–456CrossRefzbMATHGoogle Scholar
  6. Huyen NN, Khiem NT (2016) Uncoupled vibrations in functionally graded Timoshenko beam. J Sci Technol VAST 54(6):785–796.  https://doi.org/10.15625/0866-708X/54/6/7719 Google Scholar
  7. Ke LL, Yang J, Kitipornchai S, Xiang Y (2009) Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech Adv Mater Struct 16:488–502CrossRefGoogle Scholar
  8. Khiem NT, Huyen NN (2017) A method for multiple crack identification in functionally graded Timoshenko beam. Nondestruct Test Eval 32(3):319–341CrossRefGoogle Scholar
  9. Khiem NT, Lien TV (2001) A simplified method for natural frequency analysis of multiple cracked beam. J Sound Vib 245(4):737–751CrossRefGoogle Scholar
  10. Khiem NT, Lien TV (2002) The dynamic stiffness matrix method in forced vibration analysis of multiple cracked beam. J Sound Vib 254(3):541–555CrossRefGoogle Scholar
  11. Khiem NT, Kien ND, Huyen NN (2014) Vibration theory of FGM beam in the frequency domain. In: Proceedings of national conference on engineering mechanics celebrating 35th anniversary of the institute of mechanics, VAST, vol 1, pp 93–98, Apr 9 (in Vietnamese)Google Scholar
  12. Kitipornchai S, Ke LL, Yang J, Xiang Y (2009) Nonlinear vibration of edge cracked functionally graded Timoshenko beams. J of Sound Vib 324:962–982CrossRefGoogle Scholar
  13. Li XF (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib 318:1210–1229CrossRefGoogle Scholar
  14. Lien TV, Duc NT, Khiem NT (2016) Free vibration analysis of functionally graded Timoshenko beam using dynamic stiffness method. J Sci Technol Civ Eng Natl Univ Civ Eng 31:19–28Google Scholar
  15. Matbuly MS, Ragh O, Nassar M (2009) Natural frequencies of a functionally graded cracked beam using differential quadrature method. Appl Math Comput 215:2307–2316MathSciNetzbMATHGoogle Scholar
  16. Sherafatnia K, Farrahi GH, Faghidian SA (2014) Analytic approach to free vibration and bucking analysis of functionally graded beams with edge cracks using four engineering beam theories. Int J Eng 27(6):979–990Google Scholar
  17. Su H, Banerjee JR (2015) Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beam. Comput Struct 147:107–116CrossRefGoogle Scholar
  18. Swamidas ASJ, Yang X, Seshadri R (2004) Identification of cracking in beam structures using Timoshenko and Euler formulations. J Eng Mech 130(11):1297–1308CrossRefGoogle Scholar
  19. Yan T, Kitipornchai S, Yang J, He XQ (2011) Dynamic behaviour of edge-cracked shear deformable functionally graded beams on an elastic foundation under a moving load. Compos Struct 93:2992–3001CrossRefGoogle Scholar
  20. Yang J, Chen Y (2008) Free vibration and buckling analyses of functionally graded beams with edge cracks. Compos Struct 83:48–60CrossRefGoogle Scholar
  21. Yang J, Chen Y, Xiang Y, Jia XL (2008) Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. J Sound Vib 312:166–181CrossRefGoogle Scholar
  22. Yu ZG, Chu FL (2009) Identification of crack in functionally graded material beams using the p-version of finite element method. J Sound Vib 325(1–2):69–84CrossRefGoogle Scholar
  23. Zhao X, Zhao YR, Gao XZ, Li XY, Li YH (2016) Green’s functions for the forced vibrations of cracked Euler–Bernoulli beams. Mech Syst Signal Process 68:155–175CrossRefGoogle Scholar
  24. Ziou H (2016) Numerical modelling of a Timoshenko FGM beam using the finite element method. Int J Struct Eng 7(3):239–261CrossRefGoogle Scholar

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

Personalised recommendations