Skip to main content
SpringerLink
Log in

The inverse medium problem for Timoshenko beams and frames: damage detection and profile reconstruction in the time-domain

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

We discuss a systematic methodology that leads to the reconstruction of the material profile of either single, or assemblies of one-dimensional flexural components endowed with Timoshenko-theory assumptions. The probed structures are subjected to user-specified transient excitations: we use the complete waveforms, recorded directly in the time-domain at only a few measurement stations, to drive the profile reconstruction using a partial-differential- equation-constrained optimization approach. We discuss the solution of the ensuing state, adjoint, and control problems, and the alleviation of profile multiplicity by means of either Tikhonov or total variation regularization. We report on numerical experiments using synthetic data that show satisfactory reconstruction of a variety of profiles, including smoothly and sharply varying profiles, as well as profiles exhibiting localized discontinuities. The method is well suited for imaging structures for condition assessment purposes, and can handle either diffusive or localized damage without need for a reference undamaged state.

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. Adams RD, Cawley P, Pye CJ, Stone BJ (1978) A vibration technique for non-destructively assessing the integrity of structures. J Mech Eng Sci 20(2): 93–100

    Article  Google Scholar 

  2. Binici B, Kallivokas LF (2004) Crack identification in beams using multiple spectra. In: Third European conference on structural control, Vienna, Austria

  3. Cao TT, Zimmerman DC (1997) Application of load dependent ritz vectors for structural damage detection. In: Proceedings of the 15th international modal analysis conference, Orlando, FL, USA, pp 1319–1324

  4. Cao TT, Zimmerman DC (1997) A procedure to extract ritz vectors from dynamic testing data. In: Proceedings of the 15th international modal analysis conference, Orlando, FL, USA, pp 1036–1042

  5. Cerri MN, Vestroni F (2000) Detection of damage in beams subjected to diffused cracking. J Sound Vib 234(2): 259–276

    Article  Google Scholar 

  6. Graff KF (1971) Wave motion in elastic solids. Springer, New York

    Google Scholar 

  7. Kang JW, Kallivokas LF (2010) The inverse medium problem in 1D PML-truncated heterogeneous semi-infinite domains. Inverse Probl Sci Eng 18(6): 759–786

    Article  MATH  MathSciNet  Google Scholar 

  8. Kapur KK (1966) Vibrations of a Timoshenko beam, using finite-element approach. J Acoust Soc Am 40: 1058–1063

    Article  MATH  Google Scholar 

  9. Liang RY, Choy FK, Hu J (1991) Detection of cracks in beam structures using measurements of natural frequencies. J Franklin Inst 328(4): 505–518

    Article  MATH  Google Scholar 

  10. Lions J (1971) Optimal control of systems governed by partial differential equations. Springer, New York

    MATH  Google Scholar 

  11. Na S-W, Kallivokas LF (2008) Continuation schemes for shape detection in inverse acoustic scattering problems. Comput Model Eng Sci 35(1): 73–90

    MATH  MathSciNet  Google Scholar 

  12. Nocedal J, Wright SJ (2006) Numerical optimization. Springer, New York

    MATH  Google Scholar 

  13. Pandey AK, Biswas M, Samman KK (1991) Damage detection from changes in curvature mode shapes. J Sound Vib 145(2): 321–332

    Article  Google Scholar 

  14. Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Methods Appl Mech Eng 149: 113–132

    Article  MATH  Google Scholar 

  15. Rizos PF, Aspragathos N, Dimarogonas AD (1990) Identification of crack location and magnitude in a cantelever beam from the vibration modes. J Sound Vib 138(3): 381–388

    Article  Google Scholar 

  16. Ruotolo R, Surace C (1997) Damage assessment of multiple cracked beam : numerical results and experimental validation. J Sound Vib 206(4): 567–588

    Article  Google Scholar 

  17. Sansalone M, Carino NJ (1986) Impact echo: a method for flaw detection in concrete using transient stress waves. J Res Natl Bur Stand 86(3452): 222

    Google Scholar 

  18. Sansalone M, Streett WB (1997) Impact echo: nondestructive testing of concrete and masonry. Bullbrier Press, New Jersey Shore

    Google Scholar 

  19. Thomas J, Abbas AH (1975) Finite element model for dynamic analysis of Timoshenko beam. J Sound Vib 41(3): 291–299

    Article  Google Scholar 

  20. Tikhonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Sov Math Doklady 4: 1035–1038

    Google Scholar 

  21. Timoshenko SP (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag Lett 41: 744–746

    Google Scholar 

  22. Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross section. Philos Mag Lett 43: 125–131

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, 1 University Station, C1748, Austin, TX, 78712, USA

    Pranav M. Karve, Jun Won Kang & Loukas F. Kallivokas

  2. VLFS Development TF, Samsung Heavy Industries, Co., Ltd., 24th Fl., Samsung Life Insurance Seocho Tower, 1321-15, Seocho-dong, Seocho-Gu, Seoul, 137-955, Korea

    Seong-Won Na

Authors
  1. Pranav M. Karve
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seong-Won Na
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Won Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Loukas F. Kallivokas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Loukas F. Kallivokas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karve, P.M., Na, SW., Kang, J.W. et al. The inverse medium problem for Timoshenko beams and frames: damage detection and profile reconstruction in the time-domain. Comput Mech 47, 117–136 (2011). https://doi.org/10.1007/s00466-010-0533-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-010-0533-x

Keywords

Search

Navigation