Skip to main content
SpringerLink
Log in

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dimarogonas, A.D.: Vibration of cracked structures: a state of the art review. Eng. Fract. Mech. 55(5), 831–857 (1996)

    Article  Google Scholar 

  2. Gounaris, G.D., Papadopoulos, C.A., Dimarogonas, A.D.: Crack identification in beams by coupled response measurements. Comput. Struct. 58(2), 299–305 (1996)

    Article  MATH  Google Scholar 

  3. Lin, H.P., Chang, S.C., Wu, J.D.: Beam vibrations with an arbitrary number of cracks. J. Sound Vib. 258(5), 987–999 (2002)

    Article  Google Scholar 

  4. Khiem, N.T., Lien, T.V.: The dynamic stiffness matrix method in forced vibration analysis of multiple-cracked beam. J. Sound Vib. 254(3), 541–555 (2002)

    Article  Google Scholar 

  5. Wang, C.Y., Wang, C.M., Aung, T.M.: Buckling of a weakened column. ASCE J. Eng. Mech. 130(11), 1373–1376 (2004)

    Article  Google Scholar 

  6. Binici, B.: Vibration of beams with multiple open cracks subjected to axial force. J. Sound Vib. 287(1–2), 277–295 (2005)

    Article  Google Scholar 

  7. Caddemi, S., Calio, I.: Exact solution of the multi-cracked Euler–Bernoulli column. Int. J. Solids Struct. 45(5), 1332–1351 (2008)

    Article  MATH  Google Scholar 

  8. El Bikri, K., Benamar, R., Bennouna, M.M.: Geometrically non-linear free vibrations of clamped–clamped beams with an edge crack. Comput. Struct. 84(7), 485–502 (2006)

    Article  Google Scholar 

  9. Kisa, M., Brandon, J., Topcu, M.: Free vibration analysis of cracked beams by a combination of finite elements and component mode synthesis methods. Comput. Struct. 67(4), 215–223 (1998)

    Article  MATH  Google Scholar 

  10. Zheng, D.Y., Fan, S.C.: Natural frequency changes of a cracked Timoshenko beam by modified Fourier series. J. Sound Vib. 246(2), 297–317 (2001)

    Article  Google Scholar 

  11. Loya, J.A., Rubio, L., Fernández-Sáez, J.: Natural frequencies for bending vibrations of Timoshenko cracked beams. J. Sound Vib. 290(3–5), 640–653 (2006)

    Article  Google Scholar 

  12. Nayfeh, A.H., Pal, P.F.: Nonlinear non-planar parametric responses of an inextensional beam. Int. J. Non-Linear Mech. 24(2), 138–158 (1989)

    Article  Google Scholar 

  13. Zavodney, L.D., Nayfeh, A.H.: The nonlinear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment. Int. J. Non-Linear Mech. 24(2), 105–125 (1989)

    Article  MATH  Google Scholar 

  14. Abou-Rayan, A.M., Nayfeh, A.H., Mook, D.T., Nayfeh, M.A.: Nonlinear response of a parametrically excited buckled beam. Nonlinear Dyn. 4, 499–525 (1993)

    Article  Google Scholar 

  15. Kar, R.C., Dwivedy, S.K.: Nonlinear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances. Int. J. Non-Linear Mech. 34(3), 515–529 (1999)

    Article  MATH  Google Scholar 

  16. Anderson, T.J., Nayfeh, A.H., Balachandran, B.: Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam. ASME J. Vib. Acous. 118(1), 21–27 (1996)

    Article  Google Scholar 

  17. Delale, F., Erdogan, F.: The crack problem for a nonhomogeneous elastic medium. ASME J. Appl. Mech. 50(3), 609–614 (1983)

    Article  MATH  Google Scholar 

  18. Noda, N., Jin, Z.H.: A crack in functionally gradient materials under thermal shock. Arch. Appl. Mech. 64(2), 99–110 (1994)

    MATH  Google Scholar 

  19. Jin, Z.H., Batra, R.C.: Some basic fracture mechanics concepts in functionally graded materials. J. Mech. Phys. Solids 44(8), 1221–1235 (1996)

    Article  Google Scholar 

  20. Erdogan, F., Wu, B.H.: The surface crack problem for a plate with functionally graded properties. J. Appl. Mech. ASME 64(3), 448–456 (1997)

    Article  Google Scholar 

  21. Praveen, G.N., Reddy, J.N.: Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)

    Article  MATH  Google Scholar 

  22. Yang, J., Shen, H.S.: Dynamic response of initially stressed functionally graded rectangular thin plates resting on elastic foundations. Compos. Struct. 54(4), 497–508 (2001)

    Article  Google Scholar 

  23. Yang, J., Shen, H.S.: Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J. Sound Vib. 261(5), 871–893 (2003)

    Article  Google Scholar 

  24. Yang, J., Liew, K.M., Kitipornchai, S.: Dynamic stability of laminated FGM plates based on higher-order shear deformation theory. Comput. Mech. 33(4), 305–315 (2004)

    Article  MATH  Google Scholar 

  25. Huang, X.L., Shen, H.S.: Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int. J. Solids Struct. 41(9–10), 2403–2427 (2004)

    Article  MATH  Google Scholar 

  26. Sundaraja, N., Prakash, T., Ganapathi, M.: Nonlinear free flexural vibration of functionally graded rectangular and skew plates under thermal environments. Finite Elem. Anal. Des. 42(2), 152–168 (2005)

    Article  Google Scholar 

  27. Yang, J., Liew, K.M., Wu, Y.-F., Kitipornchai, S.: Thermo-mechanical post-buckling of FGM cylindrical panels with temperature-dependent properties. Int. J. Solids Struct. 43(2), 323–340 (2006)

    Article  Google Scholar 

  28. Yang, J., Liew, K.M., Kitipornchai, S.: Second-order statistics of the elastic buckling of functionally graded rectangular plates. Compos. Sci. Technol. 65(7–8), 1165–1175 (2005)

    Article  Google Scholar 

  29. Hao, Y.X., Chen, L.H., Zhang, W., Lei, J.G.: Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J. Sound Vib. 312, 862–892 (2008)

    Article  Google Scholar 

  30. Zhang, W., Yang, J., Hao, Y.X.: Chaotic vibration of orthotropic FGM rectangular plates based on the third-order shear deformation theory. Nonlinear Dyn. 59(4), 619–660 (2010)

    Article  MATH  Google Scholar 

  31. Sridhar, R., Chakraborty, A., Gopalakrishnan, S.: Wave propagation analysis in anisotropic and inhomogeneous uncracked and cracked structures using pseudospectral finite element method. Int. J. Solids Struct. 43(16), 4997–5031 (2006)

    Article  MATH  Google Scholar 

  32. Yang, J., Chen, Y., Xiang, Y., Jia, X.L.: Free and forced vibration of cracked inhomogeneous beams under an axial force and a moving load. J. Sound Vib. 312(1–2), 166–181 (2008)

    Article  Google Scholar 

  33. Yang, J., Chen, Y.: Free vibration and buckling analyses of functionally graded beams with edge cracks. Compos. Struct. 83(1), 48–60 (2008)

    Article  Google Scholar 

  34. Ke, L.L., Yang, J., Kitipornchai, S., Xiang, Y.: Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mech. Adv. Mat. Struct. 16(6), 488–502 (2009)

    Article  Google Scholar 

  35. Yang, J., Hao, Y.X., Zhang, W., Kitipornchai, S.: Nonlinear dynamic response of a functionally graded plate with a through-width surface crack. Nonlinear Dyn. 59(1–2), 207–219 (2010)

    Article  MATH  Google Scholar 

  36. Broek, D.: Elementary Engineering Fracture Mechanics. Martinus Nijhoff, Dordrecht (1986)

    Book  Google Scholar 

  37. Lu, C.F., Chen, W.Q.: Free vibration of orthotropic functionally graded beams with various end conditions. Struct. Eng. Mech. 20(4), 465–476 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, China

    T. Yan & S. Kitipornchai

  2. School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, VIC, 3083, Australia

    J. Yang

Authors
  1. T. Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Kitipornchai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Yang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yan, T., Yang, J. & Kitipornchai, S. Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation. Nonlinear Dyn 67, 527–540 (2012). https://doi.org/10.1007/s11071-011-0003-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-0003-9

Keywords

Search

Navigation