Analysis and Damage Identification of a Moderately Thick Cracked Beam Using an Interdependent Locking-Free Element


The Timoshenko interdependent interpolation element, based on the assumption of cubic interpolation for the transverse displacement and quadratic interpolation for the rotation, is developed for both the static and the dynamic problems. Next, the different behavior of a beam due to the presence of a damaged zone is investigated and the problem of identifying diffused crack affecting a portion of the beam using natural frequencies is studied. The damaged zone can be completely taken into account by introducing only three parameters, and for the inverse problem, numerical optimization is applied to define their values.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20


  1. 1.

    Casciati, S.: Stiffness identification and damage localization via differential evolution algorithms. Struct. Control Health Monit. 15(3), 436–449 (2008).

    Article  Google Scholar 

  2. 2.

    Casciati, S., Elia, L.: The potential of the firefly algorithm for damage localization and stiffness identification. Stud. Comput. Intell. 585, 163–178 (2015).

    Article  Google Scholar 

  3. 3.

    Casciati, F., Faravelli, L.: Sensor placement driven by a model order reduction (mor) reasoning. Smart Struct. Syst. 13(3), 343–352 (2014).

    Article  Google Scholar 

  4. 4.

    Friedman, Z., Kosmatka, J.: An improved two-node Timoshenko beam finite element. Comput. Struct. 47(3), 473–481 (1993).

    Article  MATH  Google Scholar 

  5. 5.

    Reddy, J.: On locking-free shear deformable beam finite elements. Comput. Methods Appl. Mech. Eng. 149(1–4), 113–132 (1997).

    Article  MATH  Google Scholar 

  6. 6.

    Reddy, J.: On the dynamic behaviour of the Timoshenko beam finite elements. Sadhana Acad. Proc. Eng. Sci. 24(3), 175–198 (1999).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Mukherjee, S., Reddy, J., Krishnamoorthy, C.: Convergence properties and derivative extraction of the superconvergent Timoshenko beam finite element. Comput. Methods Appl. Mech. Eng. 190(26–27), 3475–3500 (2001).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Hearn, G., Testa, R.: Modal analysis for damage detection in structures. J. Struct. Eng. (U.S.) 117(10), 3042–3063 (1991).

    Article  Google Scholar 

  9. 9.

    Davini, C., Gatti, F., Morassi, A.: A damage analysis of steel beams. Meccanica 28(1), 27–37 (1993).

    Article  Google Scholar 

  10. 10.

    Bicanic, N., Chen, H.P.: Damage identification in framed structures using natural frequencies. Int. J. Numer. Methods Eng. 40(23), 4451–4468 (1997).<4451::AID-NME269>3.0.CO;2-L

    Article  MATH  Google Scholar 

  11. 11.

    Lew, J.S.: Using transfer function parameter changes for damage detection of structures. AIAA J. 33(11), 2189–2193 (1995).

    Article  MATH  Google Scholar 

  12. 12.

    Lin, C.: Location of modeling errors using modal test data. AIAA J. 28(9), 1650–1654 (1990).

    Article  Google Scholar 

  13. 13.

    Cornwell, P., Doebling, S., Farrar, C.: Application of the strain energy damage detection method to plate-like structures. J. Sound Vib. 224(2), 359–374 (1999).

    Article  Google Scholar 

  14. 14.

    Wang, Z., Lin, R., Lim, M.: Structural damage detection using measured frf data. Comput. Methods Appl. Mech. Eng. 147(1–2), 187–197 (1997).

    Article  MATH  Google Scholar 

  15. 15.

    Thyagarajan, S., Schulz, M., Pai, P., Chung, J.: Detecting structural damage using frequency response functions. J. Sound Vib. 210(1), 162–170 (1998).

    Article  Google Scholar 

  16. 16.

    Lofrano, E., Paolone, A., Vasta, M.: A perturbation approach for the identification of uncertain structures. Int. J. Dyn. Control 4(2), 204–212 (2016).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lofrano, E., Paolone, A., Vasta, M.: Identification of uncertain vibrating beams through a perturbation approach. ASCE ASME J. Risk Uncertainty Eng. Syst. Part A Civ. Eng. (2016).

    Article  Google Scholar 

  18. 18.

    Worden, K., Farrar, C., Manson, G., Park, G.: The fundamental axioms of structural health monitoring. Proc. R. Soc. A 463, 1639–1664 (2007).

    Article  Google Scholar 

  19. 19.

    Sayyad, F., Kumar, B.: Identification of crack location and crack size in a simply supported beam by measurement of natural frequencies. JVC J. Vib. Control 18(2), 183–190 (2012).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Sayyad, F., Kumar, B., Khan, S.: Approximate analytical method for damage detection in free–free beam by measurement of axial vibrations. Int. J. Damage Mech. 22(1), 133–142 (2013).

    Article  Google Scholar 

  21. 21.

    Vestroni, F., Capecchi, D.: Damage evaluation in cracked vibrating beams using experimental frequencies and finite element models. JVC J. Vib. Control 2(1), 69–86 (1996).

    Article  Google Scholar 

  22. 22.

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

    Article  Google Scholar 

  23. 23.

    Vestroni, F., Capecchi, D.: Damage detection in beam structures based on frequency measurements. J. Eng. Mech. 126(7), 761–768 (2000).

    Article  Google Scholar 

  24. 24.

    Capecchi, D., Vestroni, F.: Monitoring of structural systems by using frequency data. Earthq. Eng. Struct. Dyn. 28(5), 447–461 (1999).<447::AID-EQE812>3.0.CO;2-2

    Article  Google Scholar 

  25. 25.

    Sinha, J., Friswell, M., Edwards, S.: Simplified models for the location of cracks in beam structures using measured vibration data. J. Sound Vib. 251(1), 13–38 (2002).

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Marco Pingaro.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pingaro, M., Maurelli, G. & Venini, P. Analysis and Damage Identification of a Moderately Thick Cracked Beam Using an Interdependent Locking-Free Element. J Optim Theory Appl 187, 800–821 (2020).

Download citation


  • Interdependent interpolation element
  • Damage parameters
  • Direct problem
  • Inverse problem
  • Numerical optimization

Mathematics Subject Classification

  • 49J53
  • 49K99
  • 49K35